text stringlengths 6 976k | token_count float64 677 677 | cluster_id int64 1 1 |
|---|---|---|
A Friendly Introduction to Graph Theory
For undergraduate courses at the sophomore level in Mathematics, Computer Science, Business, and Engineering.
This extremely readable text is designed to be easily accessible to students who need to gain a foundation in the basics of graph theory. Using extensive examples and exercises, this text takes the students from basic prerequisite concepts through the different types of graphs and their uses. | 677.169 | 1 |
MATH 32 FALL 2012
FINAL EXAM - PRACTICE EXAM SOLUTIONS
(1) You cut a slice from a circular pizza (centered at the origin) with radius 6 along radii at
angles and with the positive horizontal axis.
4
3
(a) (3 points) What is the area of your slice?
Solutio
GRAPHING PRACTICE - 2/8/12
Work through the following problems together in groups of 3, more or less. I will be
available to answer any questions you have.
This time, it is important that you do the problems in order. You will probably need
scratch paper
ELEMENTARY TRIGONOMETRY - 3/22/12
Work through the following problems together in groups of 3 or 4. I will be available to
answer any questions you have. It is not important that you nish all problems or work
through them quickly, but it is important that
QUADRATIC REVIEW - 1/30MATH 32 WORKSHEET - 1/23R. Gibson Summer Bridge 11?
Math 32 Worksheet #IOA
Menday, July m, 201g
927 -
Function Word Problems.
The area of a rectangle is 20. Express the perimeter of the rectangle as a function of the
rectangles length L.
A piece of wire L feet long is cut into t
Pre Calculus
Review of Algebra
Real Number System
1. How many real values exists in the following set of
numbers?
cfw_2, 4,
2i, 2 + 7i, , e 2
A) 3
B) 4
C) 5
10. Show that the following can be ordered from smallest to
largest for an
. Describe the method y
Precalculus Advice
Showing 1 to 2 of 2
The professor moves incredibly quickly and at times, does not explain thoroughly or explains poorly, so if you're someone who doesn't catch on, it would be a difficult course.
Course highlights:
Great review and introduction to basic calculus concepts.
Hours per week:
6-8 hours
Advice for students:
Attending lecture and discussion and doing the homework is not enough to succeed in this course. Utilizing the student learning center tutoring services, office hours, and just going over material will help you so much.
Course Term:Fall 2016
Professor:Appleton
Course Tags:Math-heavyBackground Knowledge ExpectedGo to Office Hours
Aug 05, 2016
| Would highly recommend.
This class was tough.
Course Overview:
Professor Gibson is great. He is very understanding and an overall great teacher. The course is very intense, especially during the summer. My advice: study, don't procrastinate, and make friends! Friends will help you in college.
Course highlights:
This course taught me that my study habits needed to change. You ay have been top dog in high school, but in college, you need to keep your guard up. Don't be so prideful. Ask for help.
Hours per week:
9-11 hours
Advice for students:
Break down the homework load, do some of it each day. And MAKE FRIENDS! Don't try to be a lone wolf in college. | 677.169 | 1 |
CAHSEE on Target is a tutoring course specifically designed for the. California High ... Answer Key: Algebra & Functions Strand. 2. Unit I: Translation of Problems into Algebra. 1.1. Algebra is . five: 2 = 5 – 3. Notice that in the word expression, 3 comes before 5, while in the . Practice: Use the order of operations to solve. 1 | 677.169 | 1 |
Similar
Algebra and Trigonometry presents the essentials of algebra and trigonometry.
Comprised of 11: polynomial, rational, exponential, logarithm, and trigonometric. Trigonometry and the inverse trigonometric functions and identities are also presented. The book concludes with a review of progressions, permutations, combinations, and the binomial theorem.
This monograph will be a useful resource for undergraduate students of mathematics and algebra.
Calculus with Analytic Geometry presents the essentials of calculus with analytic geometry. The emphasis is on how to set up and solve calculus problems, that is, how to apply calculus. The initial approach to each topic is intuitive, numerical, and motivated by examples, with theory kept to a bare minimum. Later, after much experience in the use of the topic, an appropriate amount of theory is presented.
Comprised of 18 chapters, this book begins with a review of some basic pre-calculus algebra and analytic geometry, paying particular attention to functions and graphs. The reader is then introduced to derivatives and applications of differentiation; exponential and trigonometric functions; and techniques and applications of integration. Subsequent chapters deal with inverse functions, plane analytic geometry, and approximation as well as convergence, and power series. In addition, the book considers space geometry and vectors; vector functions and curves; higher partials and applications; and double and multiple integrals.
This monograph will be a useful resource for undergraduate students of mathematics and algebra.
Guide to this Book My main objective is to teach programming in Pascal to people in the hard sciences and technology, who don't have much patience with the standard textbooks with their lengthy, pedantic approach, and their many examples of no interest to scientists and engineers. Another objective is to present many both interesting and useful algorithms and programs. A secondary objective is to explain how to cope with various features of the PC hardware. Pascal really is a wonderful programming language. It is easy to learn and to remember, and it has unrivalled clarity. You get serious results in short order. How should you read this book? Maybe backwards is the answer. If you are just starting with the Borland Pascal package, you must begin with Appendix 1, The Borland Pascal Package. If you are a Pascal user already, still you should skim over Appendix 1. Appendix 2, On Programming, has material on saving programming time and on debugging that might be useful for reference. Chapter 1, Introduction to Pascal, will hardly be read by the experienced Pascal programmer (unless he or she has not used units). Chapter 2, Programming Basics, begins to sample deeper waters, and I hope everyone will find something interesting there. Chapter 3, Files, Records, Pointers, is the final chapter to concentrate on the Pascal programming language; the remaining chapters concentrate on various areas of application.
Introductory College Mathematics: With Linear Algebra and Finite Mathematics is an introduction to college mathematics, with emphasis on linear algebra and finite mathematics the fundamental ideas of linear algebra; and complex numbers, elementary combinatorics, the binomial theorem, and mathematical induction.
Comprised of 15 and matrices, and trigonometry are also explored, together with complex numbers, linear transformations, and the geometry of space. The book concludes by considering finite mathematics, with particular reference to mathematical induction and the binomial theorem.
This monograph will be a useful resource for undergraduate students of mathematics and algebra. methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.
As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory ant its particular branches, such as optimal filtering and information compression.
Elementary Functions and Analytic Geometry is an introduction to college mathematics, with emphasis on elementary functions and analytic geometry and complex numbers, mathematical induction, and the binomial theorem.
Comprised of 13, and trigonometry are also explored, together with complex numbers and solid analytic geometry. The book concludes by considering mathematical induction, binomial coefficients, and the binomial theorem.
This monograph will be a useful resource for undergraduate students of mathematics and algebra.
This text, designed for a second year calculus course, can follow any standard first year course in one-variable calculus. Its purpose is to cover the material most useful at this level, to maintain a balance between theory and practice, and to develop techniques and problem solving skills. The topics fall into several categories: Infinite series and integrals Chapter 1 covers convergence and divergence of series and integrals. It ?ontains proofs of basic convergence tests, relations between series and Integrals, and manipulation with geometric, exponential, and related series. Chapter 2 covers approximation of functions by Taylor polynomials, with emphasis on numerical approximations and estimates of remainders. Chapt~r 3 deals with power series, including intervals of convergence, expanSIOns of functions, and uniform convergence. It features calculations with s~ries by algebraic operations, substitution, and term-by-term differentiation and integration. Vector methods Vector algebra is introduced in Chapter 4 and applied to solid analytic geometry. The calculus of one-variable vector functions and its applications to space curves and particle mechanics comprise Chapter 5. Linear algebra Chapter 7 contains a practical introduction to linear algebra in two and three dimensions. We do not attempt a complete treatment of foundations, but rather limit ourselves to thoRe topics that have immediate application to calculus. The main topics are linear transformations in R2 and R3, their matrix representations, manipulation with matrices, linear systems, quadratic forms, and quadric surfaces. Differential calculus of several variables Chapter 6 contains preliminary material on sets in the plane and space, and the definition and basic properties of continuous functions. This is followed by partial derivatives with applications to maxima and minima. Chapter 8 continues with a careful treatment of differentiability and applications to tangent planes, gradients, directional derivatives, and differentials. Here ideas from linear algebra are used judiciously. Chapter 9 covers higher xii Preface order partial derivatives, Taylor polynomials, and second derivative tests for extrema. Multiple integrals In Chapters 10 and 11 we treat double and triple integrals intuitively, with emphasis on iteration, geometric and physical applications, and coordinate changes. In Chapter 12 we develop the theory of the Riemann integral starting with step functions. We continue with Jacobians and the change of variable formula, surface area, and Green's Theorem. Differential equations Chapter 13 contains an elementary treatment of first order equations, with emphasis on linear equations, approximate solutions, and applications. Chapter 14 covers second order linear equations and first order linear systems, including matrix series solutions. These chapters can be taken up any time after Chapter 7. Complex analysis The final chapter moves quickly through basic complex algebra to complex power series, shortcuts using' the complex exponential function, and applications to integration and differential equations. Features The key points of one-variable calculus are reviewed briefly as needed. Optional topics are scattered throughout, for example Stirling's Formula, characteristic roots and vectors, Lagrange multipliers, and Simpson's Rule for double integrals. Numerous worked examples teach practical skills and demonstrate the utility of the theory. We emphaRize Rimple line drawingR that a student can learn to do himself.
Preface Objectives of This Book • To teach calculus as a laboratory science, with the computer and software as the lab, and to use this lab as an essential tool in learning and using calculus. • To present calculus and elementary differential equations with a minimum of fuss-through practice, not theory. • To stress ideas of calculus, applications, and problem solving, rather than definitions, theorems, and proofs. • Toemphasize numerical aspects: approximations, order of magnitude, concrete answers to problems. • To organize the topics consistent with the needs of students in their concurrent science and engineering courses. The subject matter of calculus courses has developed over many years, much by negotiation with the disciplines calculus serves, particularly engineering. This text covers the standard topics in their conventional order. Mostly because of commercial pressures, calculus texts have grown larger and larger, trying to include everything that anyone conceivably would cover. Calculus texts have also added more and more expensive pizzazz, up to four colors now. This text is lean; it eliminates most of the "fat" of recent calculus texts; it has a simple physical black/white format; it ignores much of current calculus "culture". The computer has forced basic changes in emphasis and how to teach calculus.
Comprised of eight of algebra: polynomial, rational, exponential, and logarithm. The book concludes with a review of sequences, permutations and combinations, and the binomial theorem, as well as summation and mathematical induction.
This monograph will be a useful resource for undergraduate students of mathematics and algebra. | 677.169 | 1 |
$39.95
Complex Variables [PN: 9780831132668]
Overview
Using the same innovative and proven approach that made the authors' Engineering Mathematics a worldwide bestseller, this book can be used in the classroom or as an in-depth self-study guide. Its unique programmed approach patiently presents the mathematics in a step-by-step fashion together with a wealth of worked examples and exercises. It also contains Quizzes, Learning Outcomes, and Can You? checklists that guide readers through each topic and reinforce learning and comprehension. Both students and professionals alike will find this book a very effective learning tool and reference.
Features
• Uses a unique programmed approach that takes readers through the mathematics in a step-by-step fashion with a wealth of worked examples and exercises.
• Contains many Quizzes, Learning Outcomes, and Can You? checklists.
• Ideal as a classroom textbook or a self-learning manual. | 677.169 | 1 |
Key Curriculum Releases IMP Year 4, 2nd Edition
IMP is four-year core mathematics curriculum and is aligned with Common Core State Standards. Adoption of the IMP curriculum includes implementation strategies, supplemental materials, blackline masters, calculator guides, and assessment tools.
Year 4 covers topics such as statistical sampling, computer graphics and animation, an introduction to accumulation and integrals, and an introduction to sophisticated algebra, including transformations and composition.
The second edition of Year 4 includes a new student textbook, 2 new unit books, and three updated unit books | 677.169 | 1 |
I'm a software engineer, a pretty smart one (even if I do say so myself), who really wants to gain a much better understanding of math, specifically statistics. My understanding of math is probably around advanced high-school level (I learned very little about math while studying computer science
at university that I didn't already know).
My problem? I get intimated as soon as weird inconsistent notations involving greek characters get involved. Its like the world's most poorly designed programming language, it even makes Perl seem sensible!
Can anyone recommend a book, or other good introduction to math that might help me gain the same intuitive feel for math, particularly statistics, that I have for software?
For an elementary introduction to statistics that avoids a lot of the intimidating notation, I think that Statistics by Freedman, Pisani and Purves is by far the best. In fact, I think it's the best introduction to statistics, period. Not only does it use "normal" language, but it does a much better job of explaining the material, and a lot of the nuances and pitfalls, than the standard notation-heavy textbooks at that level. That's why the best universities tend to use that book.
My problem? I get intimated as soon as weird inconsistent notations involving greek characters get involved.
First, notation isn't the problem, any more than it is in computer programming. Once you understand the ideas being processed, the notation becomes a non-issue. So learn the ideas and let the notation take care of itself.
And you are right that they are inconsistent, but so is the syntax of different computer languages -- once you understand the basics of computer programming, the notation differences become unimportant as well.
So focus your attention on the mathematical ideas -- don't allow yourself to be intimidated by the expressions.
I studied math at university and my overall impression was that mathematical notation remains fairly consistent until you delve into more specific subdomains, where notational collisions across subjects become inevitable. I do math and code, and it doesn't seem to me that one is really more inconsistent or "weird" than the other. If the mathematical notation you're seeing is unclear, perhaps the author isn't defining their notation clearly enough before using it. Mathematical notation can change between contexts but should be entirely clear and consistent within a given context. | 677.169 | 1 |
Single Variable Calculus – James Stewart – 4th Edition
This 4th edition calculus textbook has been thoroughly revised. It continues to embrace the best aspects of reform by combining the traditional theoretical aspects of calculus with creative teaching and learning techniques.
This is accomplished by a focus on conceptual understanding, the use of real-world data and real-life applications, projects, and the use of technology (where appropriate). It is widely acknowledged that the main goal of calculus instruction is for the student to understand the basic ideas.
This text supports such a goal while motivating students through the use of real-world applications, building the essential mathematical reasoning skills, and helping them develop an appreciation and enthusiasm for calculus | 677.169 | 1 |
"This beautifully produced book should provide a joyful and stimulating reading experience for any layman who is curious about real-life events in the context of mathematical modelling, and it provides an excellent entry point to more advanced areas such as mathematical biology or climate modelling." --Z. Q. John Lu, Significance
"What do global warming, predator-prey interactions, and the World Wide Web have in common? All of these disparate phenomena can be modeled using mathematics. In Topics in Mathematical Modeling, K. K. Tung demonstrates math¹s relevance to problems of current research interest in biology, ecology, computer science, geophysics, engineering, and the social sciences." --Scientific American Book Club
"[T]his is a good introductory book about the nature and purpose of mathematical modeling. The topics chosen and the way in which they have been motivated and presented will help a wide range of students to 'see the point' and thereby arouse and stimulate their confidence about their mathematical problem solving skills." --Bob Anderssen, Australian Mathematics Society
"I was so impressed by the breadth of examples contained in its 336 pages that I immediately set about using it to update one of my own undergraduate courses. . . . A wonderful source book for all kinds of undergraduate mathematical activities. . . . Extremely clear. . . . It is highly recommended." --Chris Howls, Times Higher Education
"Tung's preface shows that he is a dyed-in-the-wool teacher of considerable talent whose only mission is to show the student how to take raw empirical data and turn it into a mathematical paradigm that can be analyzed. His prerequisites are solid but minimal: calculus and a smattering of ordinary differential equations (ODEs). He is wise to provide an appendix with a quick treatment of ODEs for those whose background is deficient. Tung also describes in the preface a clear path for those who wish to avoid the differential equations altogether. Tung covers some of the usual modeling topics but also many others that are surprising and refreshing." --Steven G. Krantz, UMAP Journal
Reseña del editor:
Topics Niño
Designed for a one-semester course, the book progresses from problems that can be solved with relatively simple mathematics to ones that require more sophisticated methods. The math techniques are taught as needed to solve the problem being addressed, and each chapter is designed to be largely independent to give teachers flexibility.
The book, which can be used as an overview and introduction to applied mathematics, is particularly suitable for sophomore, junior, and senior students in math, science, and engineering. 2007. Hardback. Estado de conservación: NEW. 9780691116426 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Nº de ref. de la librería HTANDREE01145295
Descripción Princeton University Press, United States, 2007. Hardback. Estado de conservación: New. 256 x 180 mm. Language: English . This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible. demonstration BTE9780691116426 | 677.169 | 1 |
Linear Systems Beams 1.0
Similar software:
Integer Balance Beam Puzzles 1.0 — Middle-School (grades 5 through 9) math program written to provide skills in context. Students balance a beam by using positive and negative weights.
Graphing Functions 1.0 — Middle-School (grades 5 through 9) math program written to provide skills in context. Students find and plot solutions to parabolas, hyperbolas, and absolute value functions using an on-screen Cartesian plane and five moveable points | 677.169 | 1 |
Description
"Great intro to Algebra!"
This Hands-On Equations Level 3 app is intended for students who have already completed Level 1 and 2 of Hands-On Equations and who would like the challenge of more sophisticated equations involving negative numbers.
In Level 3, the student solves equations such as 4x – (-4) = -8 and 3(-x) + 2 = -10 + x, which contain negative constants. The green numbered cube, which represents the opposite of the red numbered cube, is introduced at this level.
Since the red and green number cubes are opposite of each other, when they are together on the same side of the balance, their value is zero and may therefore be removed without affecting the balance.
A video introduction is provided for each lesson. (See the sample YouTube video at Each video introduction is followed by two examples and ten exercises. It is essential for the student to view the lesson video prior to attempting the examples and exercises for that lesson.
Hands-On Equations is the ideal introduction to algebra for elementary and middle school students. Not only will students have fun and be fascinated with the program, their sense of self-esteem will be dramatically enhanced as they experience success with sophisticated algebraic equations.
High school students struggling with algebra will likely experience success and understanding for the first time as they work with this app.
More information on Hands-On Equations can be found at and on YouTube | 677.169 | 1 |
Group Rotations for Rational Functions
Be sure that you have an application to open
this file type before downloading and/or purchasing.
25 KB|4 pages
Product Description
This lesson was designed to host a small teacher led group along with 3 small independent groups. The independent groups consist of 3-4 problems including word problems that involve simplifying rational functions using basic operations or reviewing factoring quadratics. | 677.169 | 1 |
AP Calculus AB Flashcards
AP Calculus AB Flashcards
Want to review AP Calculus AB but don't feel like sitting for a whole test at the moment? Varsity Tutors has you covered with thousands of
different AP Calculus AB flashcards! Our AP Calculus AB flashcards allow you to practice with as few or as many questions
as you like. Get some studying in now with our numerous AP Calculus AB flashcards.
High school students looking for a challenging calculus class have several options, but it's the Advanced Placement (AP) Calculus classes that can help them meet their need for a challenging class and gain college credit at the same time. Whether you need Calculus tutoring in Orlando, Calculus tutoring in Memphis, or Calculus tutoring in Phoenix, working one-on-one with an expert may be just the boost your studies need.
The complexity of the topics within the AP Calculus AB course often sends students searching for study aids, whether those study aids be tutors, practice tests, or even flashcards. Varsity Tutors' Learning Tools include Flashcards for AP Calculus AB, which are available free of charge, online. There are more than 300 flashcards in the set, covering a wide variety of mathematical concepts ranging from derivatives (including computation and application of derivatives), to functions, graphs, and limits. Other topics included in these flashcards include integrals, calculating limits, understanding the limiting process, and techniques of antiderivatives. There are also flashcards that deal with the fundamental theorem of calculus. Varsity Tutors offers resources like a free AP Calculus AB Practice Tests to help with your self-paced study, or you may want to consider an AP Calculus AB tutor.
This Learning Tool typically asks the user to evaluate a calculus problem, sometimes with given constraints. Some ask for evaluation of a limit as a function of x, to find a derivative, or to find the value of a function. Each card gives five possible answers, much like the multiple choice section of the AP Calculus AB exam. The questions included in the flashcard set are meant to provide a representation of questions that students might find on the exam itself, but are not likely to be exact questions on the exam.
As you click through the flashcards, you'll get the answer to the question, and often, the explanation of why that specific answer is the correct one. You'll know immediately whether you got the question right or wrong. Another unique feature of these flashcards is the flexibility they provide for the user. You can skip around within the cards if you choose. You don't have to answer every question, and even after you've answered it, you have the option to go back to the question, or skip forward to the next. In addition to the AP Calculus AB Flashcards and Calculus tutoring, you may also want to consider using some of our AP Calculus AB Diagnostic Tests.
Because these cards are available online, you can access them anytime you get the urge to study or prepare for an AP Calculus AB test. You're not tied down to a desk for hours to take advantage of this particular Learning Tool. Whether you have just a few minutes or have a chunk of time carved out for studying, these flashcards are a great option for AP Calculus AB review.
When you create an account with Varsity Tutors, not only can you access the company's varied study aids, but you can also track your progress. The ability to check your progress can help you to focus in on topics where you find yourself struggling to fully understand the concepts. You can also use the flashcards creator to make customized study aids for this topic.
Varsity Tutors' Learning Tools include the Flashcards, as well as Full-Length Practice Tests, more than 75 focused Practice Tests, the Question of the Day series that focuses on single specific topics daily, and the Learn By Concept interactive syllabus.
Our AP Calculus AB flashcards each contain one question that might appear on the AP Calculus AB exam. You can use them to get a comprehensive overview of each topic covered on the AP Calculus AB exam one problem at a time, or to do problem drills that focus on particular problem types or content areas found on the AP Calculus AB exam. | 677.169 | 1 |
USEFUL BOOKS
(double click the title link if you want to view the contents of, or buy the book from, Amazon.com)
Mathematics
Boyer, Carl B. A History of Mathematics">. 3 Ed. New York: John Wiley & Sons, Inc. 2011. ISBN: 0-470-52548-7. [Very easy-to-read book. Has a nifty chronological table which tracks various mathematicians and important mathematical developments. Recommended]
Cutler, Ann; McShane, Rudolph. The Trachtenberg Speed System of Basic Mathematics. Greenwood Publishing Group. 1983. ISBN: 0313232008. [a very speedy way of doing math in your head is presented in this book (imagine doing the square root of a number in your head as fast as you can do the calculation on a calculator . . . or even faster). The system itself is taught to children in Switzerland. Very interesting; and very alien to how we learn math here in the U.S. A worthwhile read with lots of good examples.]
Hogben, Lancelot. Mathematics in the Making. New York: Galahad Books. 1960. ISBN: 0883651882. [Another very easy-to-read book on the historical development of mathematics. Has lots of diagrams and pictures . . . and some puzzles. Explanations are very good. Highly recommended.]
Smith, D.E. History of Mathematics: Volume I. New York: Dover Publications, Inc. 1958. ISBN: 0-486-20429-4. [Covers the history of mathematics from ancient Greece and the Orient until the late 9th Century. Lots of biographic material on persons who´ve contributed to development of math.]
Drexler, K. Eric. Unbounding the Future: The Nanotechnology Revolution. New York: Quill, William Morrow Company. 1991. ISBN:0-688-12573-5.[one of the clearest, easiest to read books on the subject for a layman; excellent book on nanotech and its possible ramifications] | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
1 MB|48 pages
Product Description
Comprehensive Unit on Graphical Vectors. Developed for Honors and AP Physics but also works well for on-level physics. May also be used in conjunction with significant digits and measurement units. I developed this unit after finding many students missing skills associating measurement with rulers, protractors, angles, and mechanical drawings.
No fluff, Just good Physics. Includes Student Pack and Teacher Solutions Pack.
Graphical Vectors broken down step by step with practice worksheets
This is a technical unit Introducing Vectors in Graphical Form, emphasizing the measurement and drawing of lengths and angles. The ideas and practice presented are fundamental to measurement and the understanding of vectors mathematically.
The unit begins with a list of vocabulary words and a general description of scalars, vectors, and why vectors are essential to physics and describing the world. The next two sections are measuring vectors and drawing vectors to given length and angles with accompanying worksheet black line masters. The unit then continues to negative vectors and graphically solving vector addition and subtraction problems using both parallelogram and head-to-tail methods with more student worksheets. The last section of the unit deals with components of vectors, polar and component forms of vectors, and graphically finding the components the vectors
Compiled over 18 years of teaching experience in Physics and Math. Can be given to students in packet form (first part) or used by the teacher for notes. Solutions are given in Part 2.
Includes:
Teacher and/or Student notes, examples, practice worksheets with answer pages, 2 Test versions with keys, Vocabulary and Vocabulary test. The unit ends with a Measurement Lab based on student's school map or fire escape plan for differentiating between position vectors, displacement vectors, and distances. Polar and component vectors are both introduced in this unit.
Students will need rulers and protractors. No calculators are used during the unit. | 677.169 | 1 |
Comprehensive College Algebra: Building Mathematical Insights Through Logic and Exercises is a concise, but rigorous, introduction to college algebra that features a variety of exercises designed to help students build up mathematical thinking, master mathematical skills, and develop mathematical insights.
The book begins with an introduction of sets and the number systems. This foundational knowledge prepares students for the subsequent chapters, and skill development in areas including:
Polynomials
Factoring
Linear Equations and Inequalities
Rational Expressions
Coordinate Systems and Lines
Radicals
Quadratic Equations
Conic Sections
Exponential and Logarithmic Functions
All chapters emphasize clear definitions, which are presented in context, as well as propositions and theorems. Comprehensive review questions are included at the end of the book to allow for independent practice.
Comprehensive College Algebra is written for college algebra courses, but can also be used for high school classes. It serves best under instructions. Familiarity with integers and decimals, as well as their operations, is needed to use the book effectively.
Biographies
Xiang Ji earned his B.S. in computer science at Nanjing University, in Nanjing China. He is a Ph.D. candidate in the Department of Mathematics at the Pennsylvania State University, where he also serves as a Graduate Teaching Associate. His areas of research include mathematical physics, symplectic geometry, and Poisson geometry.
Ge Mu is an instructor of mathematics at the Pennsylvania State University, where she has taught courses in intermediate algebra, college algebra, finite math, trigonometry, and calculus. She earned her M.A. in mathematics with a concentration in partial differential equations. Ge Mu is the organizer of the 2012-2013 Strongland Interscholastic Math League, and the 2010 recipient of the Mathematics Department Teaching Award. She is also the author of Being an Undergraduate in the U.S., originally published in Chinese. | 677.169 | 1 |
3 Why use Visualization in Calculus?
4 Why use Visualization in Calculus?
5 Why use Visualization in Calculus?
6 Why use Visualization in Calculus?
7 Visualizations can help students to become engaged in the concepts and begin asking "what if…" questions. Why use Visualization in Calculus?
8
Instructors can use them to visually demonstrate concepts and verify results during lectures. Also "What if…?" scenarios can be explored. Ways to use of these applets:
9 Ways to use of these applets:
10 Ways to use of these applets:
11 Ways to use of these applets:
12 Look for my 2-hour workshop on using CalcPlot3D in teaching multivariable calculus, as well as creating these scripts at AMATYC 2011 in Austin! Ways to use of these applets: | 677.169 | 1 |
Course Content and Outcome Guide for ALC 60B Effective Spring 2017
Course Description
Provides a review of individually chosen topics in Introductory Algebra-1st Term (Math 60). Requires a minimum of 30 hours in the lab. Completion of this course does not meet prerequisite requirements for other math courses.
Intended Outcomes for the course
Upon successful completion of this course students will be able to:
Choose and perform accurate beginning algebraic computations in a variety of situations with and without a calculator.
Classify points by quadrant or as points on an axis; identify the origin
Label and scale axes on all graphs
Create graphs where the axes are required to have different scales (e.g. Slope of 10 with scale of 1 on the \(x-\text{axis}\) and a different scale on the \(y-\text{axis}\).)
Interpret graphs in the context of an application
Create a table of values from an equation emphasizing input and output
Plot points from a table
LINEAR EQUATIONS IN TWO VARIABLES
Identify a linear equation in two variables
Emphasize that the graph of a line is a visual representation of the solution set to a linear equation
Find ordered pairs that satisfy a linear equation written in standard or slope-intercept form including equations for horizontal and vertical lines; graph the line using the ordered pairs
Find the intercepts given a linear equation; express the intercepts as ordered pairs
Graph the line using intercepts and check with a third point
Find the slope of a line from a graph and from two points
Given the graph of a line identify the slope as positive, negative, zero, or undefined. Given two non-vertical lines, identify the line with greater slope
Graph a line with a known point and slope
Manipulate a linear equation into slope-intercept form; identify the slope and the vertical-intercept given a linear equation and graph the line using the slope and vertical-intercept and check with a third point
Recognize equations of horizontal and vertical lines and identify their slopes as zero or undefined
Given the equation of two lines, classify them as parallel, perpendicular, or neither
Find the equation of a line using slope-intercept form
Find the equation of a line using point-slope form
Applications of linear equations in two variables
Interpret intercepts and other points in the context of an application
Write and interpret a slope as a rate of change (include units of the slope)
Create and graph a linear model based on data and make predictions based upon the model
Create tables and graphs that fully communicate the context of an application problem and its dependent and independent quantities
LINEAR INEQUALITIES IN TWO VARIABLES
Identify a linear inequality in two variables
Graph the solution set to a linear inequality in two variables
Model application problems using an inequality in two variables
MTH 60 is the first term of a two term sequence in beginning algebra. One major problem experienced by beginning algebra students is difficulty conducting operations with fractions and negative numbers. It would be beneficial to incorporate these topics throughout the course, whenever possible, so that students have ample exposure. Encourage them throughout the course to get better at performing operations with fractions and negative numbers, as it will make a difference in this and future math courses.
Vocabulary is an important part of algebra. Instructors should make a point of using proper vocabulary throughout the course. Some of this vocabulary should include, but not be limited to, inverses, identities, the commutative property, the associative property, the distributive property, equations, expressions and equivalent equations.
The difference between expressions, equations, and inequalities needs to be emphasized throughout the course. A focus must be placed on helping students understand that evaluating an expression, simplifying an expression, and solving an equation or inequality are distinct mathematical processes and that each has its own set of rules, procedures, and outcomes.
Equivalence of expressions is always communicated using equal signs. Students need to be taught that when they simplify or evaluate an expression they are not solving an equation despite the presence of equal signs.
Instructors should demonstrate that both sides of an equation need to be written on each line when solving an equation. An emphasis should be placed on the fact that two equations are not equal to one another but they can be equivalent to one another.
The distinction between an equal sign and an approximately equal sign should be noted and students should be taught when it is appropriate to use one sign or the other.
The manner in which one presents the steps to a problem is very important. We want all of our students to recognize this fact; thus the instructor needs to emphasize the importance of writing mathematics properly and students need to be held accountable to the standard. When presenting their work, all students in a MTH 60 course should consistently show appropriate steps using correct mathematical notation and appropriate forms of organization. All axes on graphs should include scales and labels. A portion of the grade for any free response problem should be based on mathematical syntax.
There is a required notation addendum and required problem set supplement for this course. Both can be found at spot.pcc.edu/math. | 677.169 | 1 |
Introduction to 2-Step Equations and Inequalities INB TEKS 7.10A
Be sure that you have an application to open
this file type before downloading and/or purchasing.
732 KB|10 pages
Product Description
*****This product is now available in a Bundle Pack! Follow the link to check out the Complete 7th Grade Math Bundle Pack available in my TpT Store! *****
Interactive Math Journal Pages that align to the TEKS!
This packet contains 2 INB pages that can be used to teach the concept of Introducing 2-StepE Equations and Inequalities - TEKS 7.10A
-1 Fold-Out Booklet for the Introduction and Vocabulary of Equations
-1 Fold-Out Booklet for the Introduction and Vocabulary of Inequalities
*This booklet also includes comparing/contrasting Equations and Inequalities! | 677.169 | 1 |
Overview
Description
Understanding algebraic equations and solving algebra word problems has never been easier or more fun! This visual approach uses balance scales to teach algebra. The balance scales provide an intuitive, visual, and entertaining way to master fundamental algebraic concepts. The puzzle-like problems ensure students are cognitively involved while they hone their techniques of simplifying, substituting, and writing proofs to solve simultaneous equations.
Balance Math™ Teaches Algebra is designed to move from simple to complex, with lessons scaffolding on earlier learning. And like all the Balance Math™ and More! books, these problems involve critical thinking and computation. To students, these problems are more like puzzles, with just enough challenge to forget they are doing math! 64 perforated pages, for easy removal. | 677.169 | 1 |
Monday, May 9, 2011
Math Open Reference
Technology in the Mathematics Classroom. The Mission of the Math Open Reference Project is simple: A free interactive math textbook on the web. Using interactive tools and compelling animations, Math Open Reference provides an engaging way to learn and explore mathematics. Teachers will have new ways to teach, and the students a new way to learn that is fun and engaging. | 677.169 | 1 |
Introductory Algebra, Books a la Carte Edition (11th Edition)
Author:Marvin L. Bittinger
ISBN 13:9780321654441
ISBN 10:321654447
Edition:11
Publisher:Pearson
Publication Date:2010-01-13
Format:Loose Leaf
Pages:840
List Price:$128.67
 
 
The Bittinger Worktext Series changed the face of developmental education with the introduction of objective-based worktexts that presented math one concept at a time. This approach allowed readers to understand the rationale behind each concept before practicing the associated skills and then moving on to the next topic. With this revision, Marv Bittinger continues to focus on building success through conceptual understanding, while also supporting readers with quality applications, exercises, and new review and study materials to help them apply and retain their knowledge. | 677.169 | 1 |
This book helps the student complete the transition from purely manipulative to rigorous mathematics, with topics that cover basic set theory, fields (with emphasis on the real numbers), a review of the geometry of three dimensions, and properties of linear spaces.
Content on this page is geared towards teaching the syntax of the language of mathematics, the rules and principles that we use in math. See Math in Real Life for a look at how we can use this information to enhance our lives. | 677.169 | 1 |
Math Clubs are intended for students who already develop interest for mathematics. These are students who excel in mathematics in schools, who are in top percentage in Mathematical multiple choice competitions. However ability to think quickly does not mean to think deeply. This is why many students with talent for mathematics often find themselves lost in problems proposed on Tournament of Towns and the other high ranked Math Olympiads: it takes intellectual efforts even to grasp what problem is about. As result, some students get frustrated.
The main purpose of Math Clubs is to provide the smooth transmission to the next level of mastering mathematics, to develop new skills, to give the participants methods and knowledge how to deal with Olympiad kind of problems. Work in Math Clubs takes considerable efforts and determination from participants. However, noticeable difference will be seen in one-two years.
Math Clubs for the Tournament of Towns and the Math Kangaroo participants
+continue its work.
Basics (grades 7—8)
Advanced (grades 9 and up)
Instructors: Artem Dudko, Konstantin Matveev and Alexander Remorov, students
of Department of Mathematics, University of Toronto, medalists of International
Mathematical Olympiad, participants of the Summer Conference Tournament of
Towns. | 677.169 | 1 |
This course enables students to consolidate their understanding of linear relations and extend their problem-solving and algebraic skills through investigation, the effective use of technology, and hands-on activities. Students will: develop and graph equations in analytic geometry; solve and apply linear ; systems, using real-life examples; explore and interpret graphs of quadratic relations; investigate similar triangles, the trigonometry of right triangles, and the measurement of three-dimensional figures; and consolidate their mathematical skills as they solve problems and communicate their thinking.
Announcements
Mid Term Reports and Interviews
Mid term report cards are scheduled to go home on Friday April 21, 2017, and parent/teacher interviews will be held on Thursday April 27, 2017 from 3pm until 8pm. Interview times will be available once report cards go home and availability is on a first come, first serve basis.
REVIEW QUIZ POSTPONED UNTIL MONDAY FEB. 13
Welcome Back!
Welcome to grade 10 Applied Math!
This site will provide both students and parents access to unit outlines, missed notes, and the dates for upcoming tests and quizzes. Please check back on a regular basis for any important announcements or changes. | 677.169 | 1 |
Wolfram|Alpha
Wolfram|Alpha is an excellent online tool. It is essentially a computational database, meaning it has tons of data ranging from math to science to art and beyond! Simply input problems using the correct syntax and it will compute everything for you.
What it is great for:
Verifying solutions to homework problems: Have you ever had a problem that was not in the back of the book that you want to double check that you got the right answer? Then Wolfram is a quick and efficient solution. Just don't be tempted to use it to get the solution to all your math problems or else you will probably not learn what you need to and it will be counterproductive!
Gaining further comprehension of the problem at hand: I often find a math problem easier to understand when it is displayed visually or graphically. Wolfram automatically generates relevant information to your input. If you input: "x^2+x+2 = 0" you would get a graph of the parabola on a cartesian plane, along with the solution (in the most probable form, but also 'alternate' forms) and other information, to name a few.
Wolfram also offers free access to hundreds of math applications that visually express famous math problems (Wolfram Demonstrations Project). With a CDF player or web-brower (with all necessary/updated plugins) you can play around with them to gain an understanding of basic to advanced mathematical principles (and often, in a more entertaining/interesting way).
Wolfram also offers a lot of support for Mathematica, which is great for more complicated math problems or when you are processing a lot of data in a specific way, for example. This site really has a ton to offer. Try exploring the site a bit to see it's other many splendid capabilities.
How to use it:
Essentially, Wolfram is capable of taking any type of input, whether it be the written or numerical form (e.g. one + one or 1+1. etc...). Type in any equation and Wolfram will try and yield the result (e.g. x + 12 = 33, x = 21).
A video published by the creator of Wolfram|Alpha, giving an overview of the Wolfram Computational Database and the basic capabilities, has been posted below if you wish to view it!
Wolfram|Alpha Overview
Math Video Tutorials
Many days I am just too lazy to read my math textbook...They are also often written in a professional manner, which can become dull quickly. I for one enjoy getting some popcorn and watching someone else do the math while I listen! I have taken my fair share of math classes and so I have some favorite sites that I would like to share (let me know if you know of something that helps YOU more, so I can post it on here for people who may think like you).
My Favorite Math People(s):
Patrick JMT (youtube) or his official site: This guy speaks clearly, writes well, and keeps to the point. Definitely my first stop. Just search on his youtube home page if you are having difficulties and there is usually a video to walk you through a similar problem.
Khan Academy: Some guy from India uploaded a **** load of math and science videos. I don't like the stylistic aspect of it as much, but he gets the job done, another great resource.
Your math professor and fellow students: Don't be afraid to ask others for help. Everyone has trouble sometime and if you are too shy to speak up, then you are signing your own educational death warrant. I can't emphasize this enough (yes, I am speaking from personal experience here).
Yahoo! Answers: If you can't find anyone, then someone on here can probably help you out.
Source: Inline Citations
When All Else Fails...
If none of the above resources are able to help you out, then simply search on Google! However, be careful! Not everything on the internet is true or credible. Usually just using your common sense will help you avoid bad sites. Typically, if they want money, then it is fake (always triple check before giving out sensitive information or money/credit card numbers). Most everything I have posted does not require any "sign-up", money or downloading.Furthermore, if you are browsing math forums with user-generated content (even Yahoo! Answers), don't automatically accept what they say as truth, try and verify everything yourself if possible. Well, best of luck!!! | 677.169 | 1 |
Interior Design Peter Papayanakis Cover Design Eugene Lo Cover Image The famous thrust stage of the Stratford Festival of Canada's Festival Theatre. Photo: Terry Manzo. Courtesy of the Stratford Festival Archives. Production Services Pre-Press Company Inc. Director, Asset Management Services Vicki Gould Photo/Permissions Researcher Daniela Glass Photo Shoot Coordinator Lynn McLeod Set-up Photos Dave Starrett Printer Transcontinental Printing Ltd. Every effort has been made to trace ownership of all copyrighted material and to secure permission from copyright holders. In the event of any question arising as to the use of any material, we will be pleased to make the necessary corrections in future printings.
ALL RIGHTS RESERVED. No part of this work covered by the copyright herein, except for any reproducible pages included in this work, may be reproduced, transcribed, or used in any form or by any means— graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, or information storage and retrieval systems—without the written permission of the publisher. For permission to use material from this text or product, submit a request online at
Reviewers and Advisory Panel Paul Alves Mathematics Department Head Stephen Lewis Secondary School Peel District School Board Mississauga, ON Terri Blackwell Mathematics Teacher Burlington Central High School Halton District School Board Burlington, ON Karen Bryan Program Resource Teacher, Numeracy & Literacy 7–12 Upper Canada District School Board Brockville, ON Angela Conetta Mathematics Teacher Chaminade College School Toronto Catholic District School Board Toronto, ON Justin De Weerdt Mathematics Department Head Huntsville High School Trillium Lakelands District School Board Huntsville, ON Robert Donato Secondary Mathematics Resource Teacher Toronto Catholic District School Board Toronto, ON Richard Gallant Secondary Curriculum Consultant Simcoe Muskoka Catholic District School Board Barrie, ON Jacqueline Hill K–12 Mathematics Facilitator Durham District School Board Whitby, ON Punitha Kandasamy Classroom Teacher (Secondary— Mathematics) Mississauga...
YOU MAY ALSO FIND THESE DOCUMENTS HELPFUL
...run,
all inputs are variable
3.1 The Production Function
Production function is a tool of analysis used in explaining the input-output relationship.
It describes the technical relationship between inputs and output in physical terms. In its
general form, it holds that production of a given commodity depends on certain specific
inputs. In its specific form, it presents the quantitative relationships between inputs and
outputs. A productionfunction may take the form of a schedule, a graph line or a curve,
an algebraic equation or a mathematical model. The production function represents the
technology of a firm.
An empirical production function is generally so complex to include a wide range of
inputs: land, labour, capital, raw materials, time, and technology. These variables form
the independent variables in a firm's actual production function. A firm's long-run
production function is of the form:
Q = f(Ld, L, K, M, T, t) (3.1.1)
where Ld = land and building; L = labour; K = capital; M = materials; T = technology;
and, t = time.
For sake of convenience, economists have reduced the number of variables used in a
production function to only two: capital (K) and labour (L). Therefore, in the analysis of
input-output relations, the production function is expressed as:
Q = f(K, L) (3.1.2)
Equation (3.1.2) represents the algebraic or...
...What is the function of schooling and how does schooling reflect the stratification system? In this paper I will be using two sociological perspectives conflict theory and social exchange theory. Conflict Theory is based upon the view that the fundamental causes of crime are the social and economic forces operating within society. The criminal justice system and criminal law are thought to be operating on behalf of rich and powerful social elites, with resulting policies aimed at controlling the poor.
The criminal justice establishment aims at imposing standards of morality and good behavior created by the powerful on the whole of society. Focus is on separating the powerful from the "have-nots" who would steal from others and protecting themselves from physical attacks; in the process, the legal rights of poor folks might be ignored. The middle class are also co-opted; they side with the elites rather than the poor, thinking they might themselves rise to the top by supporting the status quo.
There are several social theories that emphasize social conflict and The materialist view of history starts from the premise...
...HRO is to decide which needs to be out sourced and which needs to be kept. Mainly HRO focus on routine transaction oriented processes and clerical work. Yet, the strategic HR management is still taken care by HR manager. HR outsourcing is processed in these ways: Business process outsourcing, Discrete services and multiprocessing services.
HR shared service centre is a expertise centre which helps to take the routine transaction and activities for the company in one place. HR shared services centre offer another option for HR outsourcing and offer the same saving cost and services. Less professional will be needed while the shared services centre is used, in which can help the company to save a lot of cost. Also the most useful function to use shared service centre is to arrange payroll, relocate and support for recruitment, training and development, planning and retain the professional.
Professional Employer Organization is organization to rent people to other business. PEO manage the administrative needs of employees and PEO will pay for the staff's salary and other compensation, benefits. PEO has the power to lease, release, and discipline and reassign the employee. However, the company can still have enough control so that they can run their own business. PEO is benefit for staff because they offer the staff a lot of organization so they usually have economies of scale that allow them of have a wider selection of benefits at a cheaper...
...Life-time examples:
1. In the Deliotte report, it is said 44% of respondents list business intelligence systems as enablers or disruptors that may threaten their business model, which makes it second most concerning technology threat.
2. One example of how business intelligence systems have been maximised is at women's underwear manufacturer Maidenform. Their CIO Bob Russo said recently after implementing BI, "Providing targeted information at the right place and time is central to improving the decision-making process.
3. Bravissimo is another underwear manufacturer who benefited from using business intelligence systems. They reportly linked their BI tool to the MET office to predict changes in weather.
4. ANZ, proudly tooted their horn in the past when it comes to business intelligence systems. "The analysts are no longer spending all their time building better reports. They are actually doing business analysis that is improving processes and making a real difference."
5. Just Eat CIO Carlos Morgado also talked about how he implemented a cloud strategy for the online food takeaway business. "We wanted to use off-the-shelf toolsets and not to waste time building what you can buy."
6. Plantronics and Connotate are another two companies who have used business intelligence systems to their benefit.
Plantronics used dashboards to give a better and graphical view of their opportunity pipeline and help managers by suggesting better allocations of sales resources....
...CHAPTER 4 : FUNCTIONS AND THEIR GRAPHS
4.1 Definition of Function
A function from one set X to another set Y is a rule that assigns each element in X to one element in Y.
4.1.1 Notation
If f denotes a function from X to Y, we write
4.1.2 Domain and range
X is known as the domain of f and Y the range of f. (Note that domain and range are sets.)
4.1.3 Object and image
If and , then x and y are known respectively as the objects and images of f. We can write
, , .
We can represent a function in its general form, that is
f(x) = y.
Example 4.1
a. Given that , find f(0), f(1) and f(2).
Example 4.2
a. Given that , find the possible values of a such that
(a) f(a) = 4, (b) f(a) = a.
Solution
a. Given that , find f(0), f(1) and f(2).
b. Given that , find the possible values of a such that
(a) f(a) = 4, (b) f(a) = a.
(a)
(b)
4.2 Graphs of Functions
An equation in x and y defines a function y = f(x) if for each value of x there is only one value
of y.
Example:
y = 3x +1, , .
The graph of a function in the x-y plane is the set of all points (x, y) where x is the
domain of f and y is the range of f.
Example
Figure 1 below shows the graph of a linear function, the square root function and a general function.
y = f(x)
y = x...
...OF FUNCTIONS
1. Constant function
2. Identity function
3. Square function
4. Cube function
5. Linear function
6. Square root function
7. Reciprocal function
8. Absolute value function
9. Greatest integer function(step function)
1. CONSTANT FUNCTION
This is a special form of linear function.A function is said to be constant when its slope,m=0.The domain of a constant function is a set of all real numbers and its range is a single number y-intercept(b).The constant function is an even function whose graph is constant over the domain.The graph makes a horizontal line with its range (y-intercept (b)).The expression for the constant function is
F(x)=b, where b is a real number.
2. IDENTITY FUNCTION
This is also a special form of linear function but both the domain and range are sets of real numbers.Identity function has a graph where slope,m=1 and y-intercept is 0.The line consists of all points for which x-co-ordinate equals the y-co-ordinate.The identity function increases over its domain and the expression for identity function is
F(x)=x
3. SQUARE FUNCTION
The domain of a square function is a set of real numbers and the range ia set of...
...Functions and graphing functions
Basics:
A function is a rule that changes input into output
A relation is any set of ordered pairs
A function is defined as a set of ordered pairs in which no two ordered pairs have the same element
A function must give exactly one unique output for each input
Also called a mapping or simply a map
The set of input numbers is called the domain
The set of output numbers is called the range
The set of all possible outputs is called the co-domain
The range is generally the subset of the co-domain however they can also be the same
Brackets:
A domain described as
That is, the square bracket means p is included. The rounded bracket means q is not included.
Number systems:
Composite functions:
When one function is followed by another function, the result is a composite function
Applying function after applying function is written in 3 different ways
All are pronounced ' after' and mean 'do followed by '
Examples:
(i) Evaluate
(ii) Evaluate
(iii) Find the values for for which
(iv) Find
The number of people who visit a circus can be modelled by where represents theattendance of the circus days after it opens. The profit made by the circus can be modelled by where represents the profit in euros for the circus on a day when people attend
(i) Find the number of people who...
...the family. Hence,
i) determine the domain, codomain, objects, images and range.
ii) recognise the type of relation.
iii) represent each relation above using other methods.
c) Based on (b), which relation is a function? State your reason.
Part 2
a) You are required to come out with an attractive and creative card. Your card must have the following information.
i) a family photo
ii) a family tree
b) Write a short description about your family in not more than 150 words.
Part 3
Based on Diagram 3, answer the following questions.
Diagram 3
Find
a) function [pic] and function [pic].
b) the inverse function [pic] and [pic].
c) composite function [pic], [pic], [pic] and [pic].
d) function [pic] and [pic]. Is [pic] = [pic]?
e) What can you deduce from the following composite function?
[pic]and [pic]
Hence, deduce the value of [pic].
Further Exploration
a) The function g is defined by [pic]. Find
i) [pic], [pic], [pic], [pic], [pic] and [pic].
ii) Hence, state the function [pic] and [pic], where n is a positive integer.
b) Given the function [pic]is defined by [pic]. Find [pic]. Therefore deduce the inverse functions for the followings. Check your answers using another method.
i) [pic]... | 677.169 | 1 |
MATLAB sessions: Laboratory 1
MAT 275 Laboratory 1
Introduction to MATLAB
MATLAB is a computer software commonly used in both education and industry to solve a wide range
of problems.
This Laboratory provides a brief introduction to MATLAB, and the tools
%DIFFERENTIAL EQUATIONS LAB 6
%Exercise 1
% part(a)
The period of oscillation is about 4.49 seconds. Since is less than 0, the first part of the
piecewise equation is used. This equations returns: = 0.6015 radians.
% part(b)
function LAB06ex1
clc
omega0 =
MAT 275 Laboratory 3
Numerical Solutions by Euler and Improved Euler Methods
(scalar equations)
In this session we look at basic numerical methods to help us understand the fundamentals of numerical
approximations. Our objective is as follows.
1. Implemen
MAT 275 Laboratory 1
Introduction to MATLAB
MATLAB is a computer software commonly used in both education and industry to solve a wide range
of problems.
This Laboratory provides a brief introduction to MATLAB, and the tools and functions that help
you toMAT 275 Laboratory 6
Forced Equations and Resonance
In this laboratory we take a deeper look at second-order nonhomogeneous equations. We will concentrate
on equations with a periodic harmonic forcing term. This will lead to a study of the phenomenon know
MAT 275 - MATLAB #6_v2
% Solution 1
On analyzing the behavior of the forced oscillations using the figure L6a, we can see that it is
identical to a sinusoidal waveform. Therefore, we can calculate the time-period using the
method of visual inspection. The% Lab 3 - Your Name - MAT 275 Lab
% Introduction to Numerical Methods for Solving ODEs
% Exercise 1
% Part (a)
%
%
%
%
%
NOTE: We often define the right-hand side of an ODE as some function, say
f, in terms of the input and output variables in the ODE. Fo | 677.169 | 1 |
books.google.co.uk - Vedic Mathematics for School offers a fresh and easy approach to learning mathematics. The system was reconstructed from ancient Vedic sources by the late Bharati Krsna Tirthaji earlier this century and is based on a small collection of sutras. Each sutra briefly encapsulates a rule of mental working,... Mathematics for Schools | 677.169 | 1 |
use back cover copy * A 'down-to-earth' introduction to the growing field of modern mathematical biology* Also includes appendices which provide background material that goes beyond advanced calculus and linear algebra | 677.169 | 1 |
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
10.85 MB
PRODUCT DESCRIPTION
This is four 8th Grade Common Core guided, color-coded notebook pages for the Interactive Math Notebook on the concept of Solving Systems of Equations by Elimination.
Included in the notes are examples of a solving systems of equations by elimination easy (set up for elimination), hard (multiplying by a -1) and harder (setting up one equation for elimination) and hardest (setting up both equations for elimination).
Blackline master and color-coded answer key included.
** My Interactive Note Pages include all or some of the following: step by step color-coded notes, diagrams, academic vocabulary, graphic organizers and example problems.
My Interactive Note Pages were designed to use in my IMN. The students keep the color-coded notes in a 3-pronged folder, and the notes are set up to print back to back | 677.169 | 1 |
IBmaths.com
I heard about this website and I looked over it, thought it was good but the trial only lets you do so much, I just wanna make sure the website is good before paying up for an entire year! So any current users (Schools or Individuals) or anyone who's used this before, could you give a review of the website and how (much) it helped you with Math HL? | 677.169 | 1 |
Mathematics for Elementary Teachers, New York Correlation Guide Book: A Contemporary Approach
Students who use this text are motivated to learn mathematics. They become more confident and are better able to appreciate the beauty and excitement ...Show synopsisStudents who use this text are motivated to learn mathematics. They become more confident and are better able to appreciate the beauty and excitement of the mathematical world. the text helps students develop a true understanding of central concepts using solid mathematical content in an accessible and appealing format | 677.169 | 1 |
Visitors
Friday, November 6, 2009
Keep Up! Ask Questions! YOU CAN DO IT!
Keep up.
If I could give just one piece of advice to my math students, this would be it: Keep up with the class. I'll say it another way: Don't fall behind. "Getting behind is academic suicide."
To an extent that's true in any field, of course, but it's particularly true in your math courses. More than any other field, math is relentlessly cumulative. Almost every class depends on what came in the immediately preceding classes. If you don't quite get the material in one class, you need to learn it yourself right away or you can pretty much expect to be lost in the next class.
Since it takes a huge effort to catch up once you fall behind, your best strategy is not to let yourself fall behind. If you don't understand something, deal with it right away. (Very few things magically become clear over time.) In class, ask a question. Don't wait: your brain will be nibbling at the thing you didn't understand and that will distract you from the rest of the lecture.
Outside class, you have a little more time, but still make sure to get all your questions answered by the start of the next class. Start by reviewing your textbook. If you need to, come see Glenn on most Wednesday mornings before school, or at the very least, send him an email. Talk to your friends who might be able to help. Or your Parents, Tutors or any other older person who knows something about math.
If you have to give one course short shrift because you don't have enough time one week for all your classes, don't slight the math class. I say this not because math is better or more important than any other class, but because the penalties for falling behind are more severe. In most classes you can usually understand most of one lecture if you didn't understand the previous one; in math that's very often not true.
What if you do fall behind? It can happen even to good students. In this case, my advice is to put in extra effort and work through the missed material in order. Since the concepts are sequential, it will be pretty inefficient to try to study what the class is studying if you haven't mastered the previous week's work. Stick with the same order that the class followed, but put in the extra effort to catch up as quickly as you can while still learning everything.
Be sure to let me know what happened; I may be able to give you specific advice or help to use your time most effectively. If I know you had a problem but you're trying to catch up, I will probably be willing to work with you, possibly even to cut you some slack about quizzes if the problem was beyond your control. | 677.169 | 1 |
the mathematics that supports advanced computer programming and the analysis of algorithms. This book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems. It is useful for computer scientists and also for users of mathematics in various disciplines.Read more... | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
3 MB|25 pages
Product Description
This is an Algebra 1 Common Core Lesson on Solving Quadratic Equations algebraically by factoring. It is to be used as Lesson 3 of 3 following Solving Quadratic Equations Lesson 1 of 3 & Solving Quadratic Equations Lesson 2 of 3.
Students will learn the steps for solving quadratic equations algebraically & complete a few practice examples as a class. After a few teacher led examples, students will practice on their own or in groups. | 677.169 | 1 |
Navigation menu
Menu
Sets of Real Numbers Interactive Notebook Page
In my curriculum, one of the first few lessons in Algebra 2 Honors was about the subsets of real numbers. If I were teaching this again, I would use this interactive notebook page.
I always teach the subsets of real numbers using a diagram. It makes it SO much easier to understand. I've used this color-coded diagram in the past, and it works well.
I like to give my students the formal definitions and show them the notation. Since I had honors students, I would use the capital letter notation periodically so my students would get used to it. If I were doing this in class, I would give my students the definitions and verbally give them a few examples. Then, I would have them discuss with their partner to write down their own examples.
I usually follow up this short lesson with an activity or a little Algebra 1 review. | 677.169 | 1 |
Math_112_Homework - this page You may consult with others...
Math 112 Homework Problems will be graded for the method, not just the answers (for the odd-numbered problems the answers are in the back of the book). Each assignment includes problems to hand in, and other problems to work on but not hand in. It is suggested that you start by doing the odd-numbered problems and checking your answers. If your answers agree, then you can be confident that you know how to do the problems. If your answers disagree, check first to see that your answers are actually different (not the same things expressed differently), then try to find your error (consulting with someone else is often helpful in this situation). Hand in homework during the Friday class. (Those with Tuesday, Thursday classes should hand their homework in on Thursdays i.e. a day earlier than the dates below.) Please write neatly so your grader does not have to guess your answer. Homework will be returned to you the following week. Some solutions will be posted on the course website (on
This
preview
has intentionally blurred sections.
Sign up to view the full version. | 677.169 | 1 |
An easily accessible introduction to over three centuries of innovations in geometry Praise for the First Edition ". . . a welcome alternative to compartmentalized treatments bound to the old thinking. This clearly written, well-illustrated book supplies sufficient background to be self-contained." —CHOICE This fully revised... more...
A new ANGLE to learning GEOMETRY Trying to understand geometry but feel like you're stuck in another dimension? Here's your solution. Geometry Demystified , Second Edition helps you grasp the essential concepts with ease. Written in a step-by-step format, this practical guide begins with two dimensions, reviewing points, lines, angles, and distances,... more...
The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies.... more...
The word barycentric is derived from the Greek word barys (heavy), and refers to center of gravity. Barycentric calculus is a method of treating geometry by considering a point as the center of gravity of certain other points to which weights are ascribed. Hence, in particular, barycentric calculus provides excellent insight into triangle centers.... more... | 677.169 | 1 |
Math 345: Applied Mathematics
Class Meetings:
Bagby 111, MWF 12:30 - 1:20
Office Hours:
Bagby 125 MWF 1:30 - 4:00, T 2:30 - 4:00, others by appointment
Course Assignments and Resources
coming soon
Course Description
This is a course in applied and applicable mathematics. The theme for Applied Mathematics for the Fall 2016 semester is "The Mathematics of Music, and the Music of Mathematics." Aspects of this theme will be explored from a variety of perspectives. Topics include the basic mathematics of waveforms, mathematical models of timbre, frequency and pitch, synthesizing musical waveforms, some aspects of voice leading, and algorithmic generation of music. Mathematical topics that may be new to you include Fourier series, the Fourier transfrom, Markov chains, and partial orders.
Final Exam Date
Text
Determination of Course Grade
one in-class test 30%
projects: 30%
mathematical notebook: 15%
final: 25%
Test
One in-class test will be given. The tentative dates for the test is Wednesday, November 2.
Mathematical Notebook
I will ask you to keep a mathematical notebook for this class. This notebook should contain your solutions to the homework problems I assign. (It should not contain your class notes.) I will periodically take up your notebook, and assign a grade for your recent work. It is important that your mathematical notebook be devoted exclusively to this course, and that it be a clear and complete record of your work in the course outside college-sponsored event in which you must participate (such as a sporting event) scheduled on the same day as a test or in-class quiz, then you must get in contact with me before the test or quiz is given to arrange a make-up.
Laptops, Cell Phones, Texting, etc.
no laptops in class
all cell phones should be turned off during class, or be put in silent mode | 677.169 | 1 |
handyCalc Calculator HandyCalc is a powerful calculator with automatic suggestion and solving which makes it easier to learn and use | 677.169 | 1 |
*Appropriate as a supplement to any text in the Precalculus Curriculum. College Algebra, Algebra and Trigonometry, Trigonometry or Precalculus. This three year NSF funded project integrates an applied approach to algebra/trigonometry/precalculus mathematics based on real engineering and design problems with mathematics curriculum reform principles and practice. *The Investigations reflect the AMATC and NCTM Standards in both curriculum content and pedagogy. Each lab is based on a real-world application from another discipline, and involves interactive and collaborative learning. Students will use technology (primarily graphing calculators), employing multiple approaches (analytical, graphical, numerical, verbal) to model solutions to these problems, increasing their reasoning and problem-solving skills. *Many mathematics instructors are eager to integrate realistic applications in their teaching, but lack the time and resources to develop the necessary connections on their own. These Investigations provide such applications by modeling workplace experiences of professionals in engineering, design and management occupations.
"synopsis" may belong to another edition of this title.
From the Back Cover:
The laboratory investigations in this manual are an outgrowth of a campus-wide interdisciplinary collaboration at Wentworth Institute of Technology. With support from the National Science Foundation, mathematics faculty worked closely with colleagues in engineering, architecture, design, and management to identify field-specific problems that reveal rich connections to fundamental mathematics concepts in algebra, trigonometry, and precalculus.
The authors are full-time classroom mathematics instructors with many years of both college and high school teaching experience. In the spirit of the AMATYC and NCTM standards, they have incorporated a comprehensive "Rule of Four" pedagogy employing multiple representations in modeling real-world problems, interactive and collaborative learning, and regular use of technology. Many of the investigations require data collection in laboratory settings, and almost all involve some sort of hands-on activity. This manual is intended to provide students direct experience with mathematics as revealed in real applications, by modeling workplace experiences of professionals in business and industry. Students who have used these laboratory investigations have commented in course evaluations that they "make you think about the math topic in a whole new way," "keep you thinking of the math applications outside of class," and "helped give an understanding of how the math we are working on ties into everyday life." Other comments included "The labs we did were very realistic," and "I now try to look at a problem in more than one aspect."
Encouraged by these student responses, we believe Precalculus Investigations: A Laboratory Manual, will heighten students' interest in Precalculus. This manual exposes the student to lab projects that incorporate issues of daily life and today's careers such as, medical technology, weather, architecture, life cycles, natural resources, and noise pollution. These lab projects give students the ability to associate and extend the usefulness of mathematics in our ever-changing world.
From the Inside Flap:
Preface to the Student
The Mathematics Laboratory Investigations in this book are based on real workplace problems encountered by people in a variety of occupations. Some problems have been somewhat simplified or idealized in order to make the mathematical connections clearer, but the fundamental nature of each application has been preserved. In working through these labs you will see how thinking mathematically can help practicing professionals deal efficiently and effectively with problematical situations that arise in the course of their work. You'll be expected to work on each problem in a group with other students, similar to the way people typically collaborate in teams in their jobs.
Suggestions for Success
You'll probably find that thorough reading of the introductory material in each lab, as well as careful attention to the wording of the questions that are posed throughout, are keys to understanding and successfully completing the lab. In labs that involve experimentation, ask for assistance if you have any uncertainty about equipment setup and data collecting. Most labs assume that you have a graphing calculator available for your use. You'll be expected to be able to rely on mathematics skills you have previously learned. These are usually specified under "Prerequisites" at the beginning of each lab. Some of these laboratory activities will introduce you to new mathematics topics. Don't be surprised if you encounter an unfamiliar concept; the lab will help you learn and understand it. Many questions require written responses, in which you are asked to demonstrate your understanding of mathematical concepts. Discuss these among your group, and write in clear and complete sentences.
We hope that you will find these laboratory investigations challenging – they are designed to make you think, sometimes creatively, about mathematics –: but that you will also get a great deal of satisfaction from using your mathematical knowledge to tackle these problems. | 677.169 | 1 |
Office Hours I probably have at least one unscheduled hour that you do.
Spring semester I teach Per 1 M-F, Per 2 M-F, and our class, Per 3 MTWF.
Due to my other responsibilities, however, you can't just assume I'll be there, so my office hours are by appointment (but you are always welcome to drop by to see if I'm available even if you don't have an appointment, my office is in the library).
Habits of Mind Goal: Become better problem solvers by getting better at asking good questions, thinking mathematically and reasoning mathematically.
Collaborative Goal: Become better at working together to achieve a common objective.
Metacognitive Goal: Learn more about yourself as a learner and use that to become a better learner.
Mindset Goal: Work on developing our Growth Mindsets.
Openers Most days class will start with openers, a few short questions as soon as you enter class to get your mathematical thinking started.
Homework You will have homework most nights, but it will rarely be a long set of problems. Usually, it will just be one problem; however, you will often need to explain your thinking on that problem via a short screencast. We'll go over in class how to do this. Homework is your opportunity to explore, to learn more, and to practice; you don't need to show mastery.
Standards Assessment
You will be assessed over the essential standards in Algebra I. These short (typically 5-15 minute) assessments will occur in-class and will happen approximately every two weeks (although that will vary). These assessments are designed to give both you and me an idea of how well you understand the particular material and an opportunity to revisit the material if we both don't feel good about that level of knowledge.
Each assessment will be "graded" using the following scale:
PR = Proficient: Demonstrates a thorough understanding of the standard
DE = Developing: Demonstrates a developing understanding of the standard, but with some misconceptions or errors
NY = Not Yet: Demonstrates a minimal understanding of the standard, with significant misconceptions and errors
The feedback you seek and receive will be more important than the grade that goes in the grade book.
Because Algebra is both concept and skill-based, it is essential that you master the concepts and skills as we go along and not get behind, otherwise you will quickly find it difficult to master new concepts and skills. Therefore, if you did not score proficient (PR) on the standard, you will have multiple opportunities to get help from various sources and then reassess over that standard, and your improved score will replace your previous score in the grade book. You may reassess as often as once per day, by appointment, for the next five school days (for a possible total of up to five reassessments). If after 5 days you are still not at least developing (DE), we will meet and discuss what we should do to help you. I can't emphasize enough how critical it is that you master these standards along the way - the expectation is that you will take full advantage of this opportunity to not only improve your grade, but more importantly to improve your understanding. I expect you to continue reassessing until you've mastered the standard.
We will also have a summative assessment ("final exam") at the end of each semester, where you will have the opportunity to demonstrate your mastery of (or progress toward mastery of) all the standards.
Grades I believe that there is a difference between assessment and grading. Assessment is less about assigning a grade and more about getting better at what we can do. I believe in stressing assessment for learning as opposed to assessment of learning, which is why the feedback you seek and receive is so important. The most important assessment of your work is your own self-assessment.
Therefore your "grade" in this class is going to be a bit different. Everything will all be recorded in the online grade book in categories with a "zero" weight. This means that all of us (you, me, your parents, your counselors and administrators) will be able to see and track how you are doing on each "assignment", but those assignments will not be assigned points and averaged to come up with an overall grade.
Instead, there will be one category titled "Current Assessment of Overall Progress" which will contain your current overall grade for the course. That category will be informed by all the "assignments" recorded in the other categories, but will not be a simple average. Instead, it will be a more holistic view of your overall progress. It will rely on my professional judgment, in concert with dialogue with you, and will change over the course of the semester as your level of understanding changes. If at any time you disagree with what that current assessment of overall progress says, you can come in and we will discuss it and I will happily change it if we agree that it should be changed.
Because of the needs of the online grade book software, the grade you receive in this category will be the typical A-F (along with pluses and minuses to try to give you some additional indication of your strength within each grade), with arbitrary percentages attached:
A+ 98%
A 95%
A- 92%
Thorough understanding
B+ 88%
B 85%
B- 82%
Developing understanding, but with some misconceptions and errors
C+ 78%
C 75%
C- 72%
Partial understanding, with more misconceptions and errors
D+ 68%
D 65%
D- 62%
F 59%
Minimal understanding, with many misconceptions and errors
Classroom Policies Here's the one rule you need to remember:
Do the right thing.
Seriously, that's pretty much all you have to remember. Of course you have to follow all the rules in the LPS Student Code of Conduct, as well as all AHS policies as listed in your student calendar but, in the end, it pretty much boils down to do the right thing. While I think that at least 98% of the time you know what the right thing is, if you're ever unsure, ask.
If you really want a longer list, here you go:
You may engage in any behavior that does not create a problem for you or anyone else.
If you find yourself with a problem, you may solve it by any means that does not cause a problem for you or anyone else.
You may engage in any behavior that does not jeopardize the safety or learning of yourself or others. Unkind words and actions will not be tolerated.
Attendance and Tardies This is pretty simple as well. All district and AHS policies apply, including the rules regarding make-up work. But, in general:
It's very important to attend class every day. There's a high positive correlation between attendance and success in school. Obviously if you are very sick, coming to school is a bad idea but, otherwise, you should be here. If you are absent, you are expected to check online to see what you've missed before coming back to school (and to begin working on it). This will provide you the best opportunity to be successful.
If at all possible, don't be tardy. Being late under normal circumstances is disrespectful to your classmates, your teacher, and yourself, and it makes it more difficult for you to be successful in our class, so please don't be late.
In the unlikely event that attendance or tardies become an issue, then we will have a conversation and an appropriate plan will be developed to fix the problem.
Questions? If you have any questions, please contact me. Once you feel like you completely understand these expectations, please fill out this form to indicate your understanding. Thank you for taking the time to thoughtfully consider these expectations, and I'm looking forward to our time together in Algebra I. | 677.169 | 1 |
MATLAB is a software resource for students and researchers in the fields of mathematics and engineering at the University of New first version of Mac Zico and Stanford in 1970 in order to resolve the issues matrix theory, linear algebra and numerical analysis was created and today hundreds of thousands of academic users, academic, industrial and diverse in many fields such as engineering, advanced mathematics, linear algebra, telecommunications, systems engineering and technical computing with MATLAB as one of the first environments to be able to solve their problems, are familiar.
Mathematics, engineering science is the most common language. Matrices, differential equations, numerical sequence information, drawings and graphics of the original parts are used in mathematics and also in MATLAB.
MATLAB programming language is a system of effective and many scientific and engineering computing.
It is used in almost all engineering disciplines. In any field that requires mathematical calculations, curve drawing, simulation models, and numerical analysis, using MATLAB can be useful. Issues that commonly used programming languages C and Fortran programs in engineering are solved using MATLAB is much easier and faster solved. This software has several tools that students and engineers each string boxes can use toolbox their field. Each toolbox, hundreds of special facilities for the MATLAB adds the appropriate fields. You can simply program your own functions using MATLAB functions to write code and if the number is too large for them to assign a sub-set of a toolbox to create.
In fact, a programming language MATLAB easier and easier with advanced features such as Fortran or C is computer languages. MATLAB Simulink comes with software that allows the simulation of control systems provides. This application is a powerful environment for image data through the capabilities of graphics to provide. | 677.169 | 1 |
Domain & Range of Major Functions Guided Notes
Be sure that you have an application to open
this file type before downloading and/or purchasing.
79 KB|4 pages
Product Description
Use these guided notes to teach students about domain and range. Students will be required to find the domain and range both graphically and algebraically. The notes begin with a table to organize the domains of the major functions and a flow chart for finding range. The second page is practice with both domain and range. Problems are ideal for use in IB Math Topic 2 or Pre-Calculus. | 677.169 | 1 |
The grade 9 MaST Math course builds on the foundational principles of numeracy explored in the middle school years with a focus on solving real life problems. The grade 9s connect and apply their learning from math and other subjects to solve these real life problems. The learning is enriched by engaging in creative projects. There are opportunities to continue to enrich the learning through open ended investigations, hands on, and kinesthetic activities. Some of the projects completed recently are building a roof at an optimal angle for a scale model cabin, designing a bottle to hold 250 mL, and determining the relationship between two variables.
Topics such as integer and rational number work are extended to encompass work with powers, roots, proportions and ratios. Topics such as Cartesian mapping are extended to encompass linear and non-linear relationships. Topics such as simple algebra are extended to encompass higher order polynomial manipulation, equation solving, inequalities, factoring and linear systems. Finally, topics such as 2D and 3D geometry are extended to encompass the geometry of lines, triangles and circles. The course covers the same curriculum as the grade 9 academic course but with more depth and breadth.
The grade 9s are also encouraged to challenge themselves by participating in the Pascal Math contest. | 677.169 | 1 |
Basic Mathematics / Edition 5
Paperback
Temporarily Out of Stock Online
Overview you to see the connection between mathematics and your world. This includes references to contemporary topics like gas prices and some of today's most forward thinking companies like Google.
Related Subjects
Table of Contents
1. WHOLE NUMBERS. Place Value and Names for Numbers. Addition with Whole Numbers and Perimeter. Rounding Numbers, Estimating Answers, and Displaying Information. Subtraction with Whole Numbers. Multiplication with Whole Numbers and Area. Division with Whole Numbers. Exponents and Order of Operations. Area and Volume. 2.FRACTIONS AND MIXED NUMBERS. The Meaning and Properties of Fractions. Prime Numbers, Factors, and Reducing to Lowest Terms. Multiplication with Fractions and the Area of a Triangle. Division with Fractions. Addition and Subtraction with Fractions. Mixed-Number Notation. Multiplication and Division with Mixed Numbers. Addition and Subtraction with Mixed Numbers. Combinations of Operations and Complex Fractions. 3.DECIMALS. Decimal Notation and Place Value. Addition and Subtraction with Decimals. Multiplication with Decimals. Circumference and Area of a Circle. Division with Decimals. Fractions and Decimals, and the Volume of a Sphere. Square Roots and the Pythagorean Theorem. 4.RATIO AND PROPORTION. Ratios, Rates, and Unit Pricing. Proportions. Applications. Similar Figures. 5.PERCENT. Percents, Decimals, and Fractions. Basic Percent Problems. General Applications of Percents. Sales Tax and Commission. Percent Increase or Decrease, and Discount. Interest. 6.MEASUREMENTS. Unit Analysis I: Length. Unit Analysis II: Area and Volume. Unit Analysis III: Weight. Converting Between the Two Systems and Temperature. Time Operations with Units of Measure. 7.INTRODUCTION TO ALGEBRA. Positive and Negative Numbers. Addition with Negative Numbers. Subtraction with Negative Numbers. Multiplication with Negative Numbers. Division with NegativeNumbers. 8.SOLVING EQUATIONS. The Distributive Property and Algebraic Expressions. The Addition Property of Equality. Linear Equations in One Variable. Appendixes. I: One Hundred Addition Facts. II: One Hundred Multiplication Facts. III: Negative Exponents and Scientific Notation. Solution to Selected Practice Problems. Answers to Odd-Numbered Problems, and | 677.169 | 1 |
...
Show More features of this revised and expanded second edition include: a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book. Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature. Around 200 additional exercises, and a full solutions manual for instructors, available via | 677.169 | 1 |
Customers Who Bought These Item(s) Also Bought
The TI-34 MultiView scientific calculator was designed with educator input in mind for use in middle grades math and science classes. In classic mode, the TI-34 MultiView can be used in the same classrooms as the TI-34 II Explorer Plus as the screen appears identical to the TI-34 II Explorer Plus in this mode.
Key Points
Previous entry
User-friendly menus
Power, roots, and reciprocals
One and two variable statistics
7 memories
Dual power (solar and battery)
Product Features
MultiView display View multiple calculations on a 4-line display and scroll through entries with ease. See math expressions and symbols, including stacked fractions, exactly as they appear in textbooks.
Up to 4 lines of display Enter multiple calculations to compare results and explore patterns, all on the same screen.
Fraction exploration The TI-34 MultiView scientific calculator comes with the same features that made the TI-34 II Explorer Plus so helpful at exploring fraction simplification, integer division and constant operators.
Data list editor Enter statistical data for 1- and 2-var analysis as well as for exploring patterns via list conversions to see different number formats like decimal, fraction and percent side-by-side. | 677.169 | 1 |
Saunders Math Skills for Health Professionals
ISBN-10: 1416047557
ISBN-13: 9781416047551 Plenty of examples, practice problems, and learning tools provide the perfect math review for health professionals! With just the right level of content and highly illustrated example problems, this user-friendly worktext helps you learn and understand fundamental math principles and understand how they apply to patient care. UNIQUE! Full-color format highlights key information on setting up problems, understanding parts of equations, moving decimal points, and more. Spiral bound format with plenty of white space allows you to use the text as a workbook in which you can write your answers and work out problems. Consistent chapter formats make it easy to retain information and identify important content. Chapter objectives emphasize what you should learn from each chapter and how your knowledge applies to patient care. Key terms defined at the beginning of each chapter help you understand new vocabulary in the text. Chapter overviews introduce you to the topics discussed in the chapter. Example problems demonstrate and label each step to getting a solution and show you how to solve similar problems. Practice the Skill problems incorporated within the chapter for in-class discussion allow you to practice what you've learned before receiving homework assignments. Math in the Real World boxes include word problems that apply your knowledge to everyday life as well as common healthcare situations. Strategy boxes demonstrate the steps to solving topic problems and provide a helpful example for solving more problems. Human Error boxes include hints on common errors and show you how to double-check your answers. Math Etiquette boxes help you solve problems by presenting proper math rules. Chapter quizzes allow you to assess your learning and identify areas for further | 677.169 | 1 |
Reader Feedback and Errata
Synopsis of Book
Chapter 1: Basic Counting.
The text begins by stating and proving the most fundamental counting
rules, including the sum rule and the product rule. These rules are
used to enumerate combinatorial structures such as words,
permutations, subsets, functions, anagrams, and lattice paths.
Bijections and bijective proofs are introduced at an early stage
and are then applied to help count compositions, multisets,
and Dyck paths. The end of the chapter discusses applications
of combinatorics in elementary probability theory.
Chapter 2: Combinatorial Identities and Recursions.
The chapter opens with a discussion of the generalized distributive law,
which allows one to simplify a product of sums in an arbitrary ring.
This law leads to proofs of the binomial and multinomial theorems
(in both commutative and non-commutative rings) that illuminate
the connections to combinatorics on words. Next, recursions are
introduced as a powerful tool for enumerating discrete
structures. Recursions are used to analyze multisets, anagrams,
lattice paths, integer partitions, set partitions, and surjections,
as well as families of objects counted by the Catalan numbers.
We study Stirling numbers and their connections to set partitions
and rook theory, which leads to an analysis of the transition matrices
between different bases for the vector space of polynomials in one
variable. A final section describes how combinatorial arguments can
be used to prove certain polynomial identities.
Chapter 3: Counting Problems in Graph Theory.
This chapter contains an introduction to the vast subject of graph
theory that concentrates on enumerative issues. After some initial
definitions, we show that powers of the adjacency matrix contain
information about the number of walks in a graph. The connection
between directed acyclic graphs (DAG's) and nilpotent matrices
is explained. We discuss the graph-theoretic concepts of vertex degrees,
forests, trees, leaves, rooted trees, connectedness, components, and
functional digraphs. The representation of permutations as digraphs that
are disjoint unions of cycles leads to a combinatorial interpretation for
the Stirling numbers of the first kind. Trees and rooted trees are enumerated
by bijections to certain sets of functional digraphs. Pruning maps
(also called Prüfer codes) are introduced to give a refinement of this
result that counts trees with specified vertex degrees. We then
study terms, lists of terms, ordered trees, and ordered forests,
which are enumerated with the aid of the famous "cycle lemma."
Next we consider proper colorings of graphs, chromatic polynomials,
and chromatic numbers. A recursion based on deleting and contracting
a given edge is used to calculate chromatic polynomials. A similar
recursion then arises in the enumeration of spanning trees of a graph.
This leads to a proof of the matrix-tree theorem in which a
recursion for rooted spanning trees is related to multilinearity properties
of the determinant of the truncated Laplacian matrix. Finally,
aided by these results, we address the existence and enumeration
of Eulerian tours in a balanced, connected digraph.
Chapter 4: Inclusion-Exclusion and Related Techniques.
This chapter begins with a discussion of involutions,
which are bijections on sets of signed objects. Various
forms of the inclusion-exclusion formula are presented.
We give several proofs of this formula, illustrating a mixture
of algebraic, combinatorial, and bijective methods.
We then apply inclusion-exclusion techniques to obtain formulas
for surjections, Stirling numbers, Euler's φ-function,
derangements, the coefficients in chromatic polynomials, and
objects avoiding specified conditions. The chapter ends with
a brief treatment of the classical (number-theoretic) Möbius
inversion formula and its generalization to Möbius functions on
posets, leading to an interpretation of inclusion-exclusion
as a Möbius inversion result for Boolean posets.
Chapter 5: Ranking and Unranking.
This chapter studies algorithms for ranking and unranking
combinatorial objects. The goal is to compute explicit
bijections that map objects
in a given finite set to positions on a list, and vice versa.
We present systematic techniques for solving such problems
based on bijective versions of the sum and product rules.
These rules are used to rank and unrank words, permutations,
subsets, anagrams, integer partitions, set partitions, card hands,
Dyck paths, and trees. Two final sections consider the related
problems of finding successors, finding predecessors, and
randomly selecting an object from a given collection.
The algorithms in this chapter provide useful building blocks
for computer simulations and programs that must loop over large
sets of discrete structures.
Chapter 6: Counting Weighted Objects.
This chapter introduces the idea of a generating function
as a way to count sets of weighted objects. For simplicity,
we initially consider only finite weighted sets. Weighted versions
of the sum rule, product rule, and bijection rule are presented.
This leads to a discussion of quantum factorials,
quantum binomial coefficients (also called q-binomial coefficients),
quantum multinomial coefficients, and the associated statistics
on permutations and words (inversions and major index).
An optional section describes Foata's bijective proof of the
equidistribution of inv and maj on the set of anagrams of a
given word. Another optional section considers q-analogues of
Catalan numbers obtained by weighting Dyck words by inversions
or by major index.
Chapter 7: Formal Power Series.
To extend the notion of a generating function to apply to
infinite weighted sets, one has two choices: use convergent
power series of a real (or complex) variable, or use formal
power series. We elect to use formal power series, and this
chapter gives a detailed account of the basic algebraic
properties of such series. We define the ring
of formal power series K[[x]] for any field K of characteristic
zero, as well as the subring K[x] of formal polynomials.
We proceed to give a thorough and rigorous exposition
of degree, order, evaluation homomorphisms, formal limits,
formal infinite sums of series, formal infinite products of series,
convergence criteria, units in K[x] and K[[x]], formal
geometric series, formal Laurent series, formal derivatives,
composition of formal series, the formal chain rule, compositional
inverses, the generalized binomial theorem, powers of series,
n'th roots of series, partial fractions, formal exponentiation,
formal logarithms, and extensions to multivariable polynomials
and series. A recurring theme is that many familiar facts from
calculus, such as exp(x+y)=exp(x)exp(y),
can be proved (at least at the
formal level) by elementary algebraic and combinatorial manipulations
of coefficients in power series. The chapter also includes an
application of formal series and partial fractions to the problem
of solving recursions. Beginning readers can readily skip or skim
this chapter and proceed to the combinatorial aspects of formal
power series contained in Chapter 8.
Chapter 8: Combinatorial Power Series.
This chapter applies the machinery of formal power series
(as developed in Chapter 7) to solve combinatorial problems
involving the enumeration of infinite weighted sets. First
we develop versions of the sum rule and product rule for
infinite sets. Formal power series are then used to enumerate
binary trees, full binary trees, and ordered trees. An initial
algebraic solution based on the formal quadratic formula leads
to the development of bijections linking these classes of trees.
We continue with a combinatorial account of Lagrange's formula
for computing compositional inverses, based on the enumeration
of terms in Chapter 3. The next few sections give a brief
introduction to the vast subject of generating functions
and bijections for integer partitions. We give three proofs
that the number of partitions of n into odd parts equals
the number of partitions of n into distinct parts — one based
on formal power series, one based on a bijection of Sylvester,
and one based on a bijection of Glaisher. Next we present
Franklin's beautiful involution proving Euler's pentagonal number
theorem, which gives the expansion of the infinite product
∞ ∏ i=1
(1-xi)
and leads to an amazing recursion for the partition function p(n).
The chapter closes with an analysis of generating functions for
Stirling numbers and a proof of the exponential formula, which
illuminates the combinatorial meaning of the exponential of a formal series.
Chapter 9: Permutations and Group Actions.
This chapter contains an account of group theory emphasizing those
aspects most pertinent to combinatorial problems: symmetric groups,
group actions, the orbit-counting theorem (often called Burnside's lemma),
and Pólya's extension of this theorem to weighted sets. After
giving basic definitions and examples, the symmetric groups
are investigated in some detail. We describe the one-line form and
cycle notation for permutations. Inversions are used to define
the sign of a permutation and to prove that sgn is a group homomorphism.
The next few sections offer an optional foray into the theory of determinants,
featuring combinatorial proofs of identities such as the Cauchy-Binet formula,
Laplace expansions, and the adjoint formula for the inverse of a matrix.
We then turn to subgroups and their properties. The automorphism group of a
graph is introduced as a precise way to measure the symmetry in a graph;
as examples, we compute the automorphism groups of cycles (directed and
undirected),
graphs modeling chessboards, and cubes. Subsequent sections cover group
homomorphisms, group actions, permutation representations, Cayley's theorem,
orbits of an action, stabilizer subgroups, conjugacy classes, cosets,
centralizers, normalizers, and Lagrange's theorem. The Orbit Size Theorem,
which says that the size of the orbit of x equals the index of the
stabilizer subgroup of x, is applied to give combinatorial proofs of
Fermat's little theorem,
Cauchy's theorem, Lucas' congruence for binomial coefficients modulo a prime,
and the first Sylow theorem. Finally, we prove the orbit-counting theorem
and Pólya's theorem and show how these results can be used in
counting problems where symmetries must be taken into account.
Chapter 10: Tableaux and Symmetric Polynomials.
The theory of symmetric polynomials, with its close-knit relation to
tableau combinatorics, provides an ideal venue for illustrating the
algebraic significance of bijections and enumerative ideas. This chapter
gives a highly combinatorial account of the basic facts of this theory.
The Schur polynomials and skew Schur polynomials are defined as
generating functions for semistandard tableaux, and these polynomials are
shown to be symmetric via content-modifying bijections on tableaux.
After defining other commonly used symmetric polynomials —
the monomial symmetric polynomials, the elementary symmetric
polynomials, the complete symmetric polynomials,
and the power-sum symmetric polynomials — we begin to
develop the fundamental algebraic properties of these objects.
A mixture of combinatorics and matrix algebra is used to show that
the Schur polynomials (as well as other symmetric polynomials just mentioned)
form a basis for the vector space of symmetric polynomials. Suitable
recursions establish the algebraic independence of the elementary
(resp. complete, power-sum) symmetric polynomials.
Further topics include the dominance partial ordering
of partitions, Kostka numbers, Schensted's tableau insertion algorithm,
reverse insertion, the bumping comparison theorem, the Pieri rules,
the Schur expansions of complete and elementary symmetric polynomials,
the power-sum expansions of hn
and en, the involution ω,
the Cauchy identities, dual bases, and the Hall scalar product.
The chapter concludes with an exposition of different versions
of the RSK algorithm, which map permutations, words, and matrices
to pairs of Young tableaux satisfying certain conditions. Shadow
diagrams are used to show that if a given permutation maps to the
pair of tableaux (P,Q) under RSK, then the inverse permutation
maps to (Q,P).
Chapter 11: Abaci and Antisymmetric Polynomials.
The first part of the chapter employs combinatorial objects called
abaci to prove interesting facts about integer partitions. Bijections
on suitably weighted abaci furnish a proof of the Jacobi triple product
identity. We then discuss k-cores and k-quotients of partitions,
which generalize the operation of dividing an ordinary integer by k.
The model of a k-runner abacus proves the uniqueness of the
k-core
of a partition and gives one way to compute the k-quotients. We also
give another interpretation of k-quotients in terms of the hooks of the
diagram of a partition. The remainder of the chapter continues the
study of symmetric polynomials begun in Chapter 10. The central algebraic
objects are now antisymmetric polynomials, which can be converted
to symmetric polynomials by dividing by the Vandermonde determinant.
The central combinatorial objects are
labeled abaci, which model monomial antisymmetric polynomials.
Aided by these abaci, we give combinatorial proofs of the Pieri rules,
the classical formula for Schur
polynomials as a quotient of determinants, the Schur expansion of
power-sum symmetric polynomials,
the power-sum expansion of skew Schur polynomials,
the Jacobi-Trudi formulas, a combinatorial interpretation of the
inverse Kostka matrix, and two versions of the Littlewood-Richardson rule
(which describe the Schur expansions of skew Schur polynomials
and the product of two Schur polynomials).
Chapter 12: Additional Topics.
The final chapter consists of independent sections covering
optional topics that complement material in the main text.
These topics include: the use of cyclic shifting bijections
in lattice path enumeration; a bijective proof of the Chung-Feller theorem
on the uniform distribution of flaws in square lattice paths;
the multiset criterion for deciding when two Ferrers boards
are rook-equivalent; the definition and enumeration of parking
functions; bijections connecting parking functions and trees;
the use of counting arguments to prove theorems in field theory,
including the enumeration of irreducible polynomials
via Möbius inversion; the linear-algebraic meaning of q-binomial
coefficients; the combinatorial interpretation for the coefficients
in the Maclaurin power series for tan(x) and sec(x);
tournaments and a combinatorial method for evaluating Vandermonde
determinants; a probabilistic proof of the hook-length formula
for the number of standard Young tableaux; Knuth equivalence and
longest increasing subsequences of words; Pfaffians and their
relation to perfect matchings of graphs; and the famous exact
formula for the number of ways to tile a rectangular board with dominos. | 677.169 | 1 |
Edition: 8th 2006 its complete, interactive, objective-based approach,Basic College Mathematicsis the best-seller in this market. The Eighth Edition provides mathematically sound and comprehensive coverage of the topics considered essential in a basic college math course. The text includes chapter-openingPrep Tests,updated applications, and a new design. Furthermore, the Instructor's Annotated Edition features a comprehensive selection of new instructor support material. The Aufmann Interactive Method is incorporated throughout the text, ensuring that students interact with and master the concepts as they are presented. This approach is especially important in the context of rapidly growing distance-learning and self-paced laboratory situations. New!Study Tipsmargin notes provide point-of-use advice and refer students back to theAIM for Successpreface for support where appropriate. Integrating Technology(formerly Calculator Notes) margin notes provide suggestions for using a calculator in certain situations. For added support and quick reference, a scientific calculator screen is displayed on the inside back cover of the text. Enhanced!More bulleted annotationshave been added to the solution steps of examples and to theYou Try Itsolutions in the appendix. Enhanced!Examples have been clearly labeledHow To,making them more prominent to the student. Enhanced!More operation application problemsintegrated into theApplying the Conceptsexercises encourage students to judge which operation is needed to solve a word problem. New!Nearly100 new photosadd real-world appeal and motivation. Revised!TheChapter Summaryhas been reformatted to include an example column, offering students increased visual support. Enhanced!In response to instructor feedback, the number ofChapter Review ExercisesandCumulative Review Exerciseshas increased. Enhanced!This edition features additional coverage of time (Chapter 8), bytes (Chapter 9), and temperature (Chapter 11). Aufmann Interactive Method (AIM)Every section objective contains one or more sets of matched-pair examples that encourage students to interact with the text. The first example in each set is completely worked out; the second example, called 'You Try It,' is for the student to work. By solving the You Try It, students practice concepts as they are presented in the text. Complete worked-out solutions to these examples in an appendix enable students to check their solutions and obtain immediate reinforcement of the concept. While similar texts offer only final answers to examples, the Aufmann texts' complete solutions help students identify their mistakes and prevent frustration. Integrated learning system organized by objectives.Each chapter begins with a list of learning objectives that form the framework for a complete learning system. The objectives are woven throughout the text (in Exercises, Chapter Tests, and Cumulative Reviews) and also connect the text with the print and multimedia ancillaries. This results in a seamless, easy-to-navigate learning system. AIM for SuccessStudent Preface explains what is required of a student to be successful and demonstrates how the features in the text foster student success.AIM for Successcan be used as a lesson on the first day of class or as a project for students to complete. The Instructor's Resource Manual offers suggestions for teaching this lesson.Study Tipmargin notes throughout the text also refer students back to the Student Preface for advice. Prep Testsat the beginning of each chapter help students prepare for the upcoming material by testing them on prerequisite material learned in preceding chapters. The answers to these | 677.169 | 1 |
Problem solving skills are important for business success An excellent read, Brian, and I have just used your 'checklist' to examine my own. Now find the coordinates of the vertex of the function. Word Problems Percents: Math Homework Help Simply point your camera toward a math problem and Photomath will. How to Solve Big Problems. Math is designed to help you solve your. Math allows students to get instant solutions to all kinds of math problems, from algebra and equation solving right. Whether you use my questions or develop your own, these types of questions are superior because.
Fantasy writers of america
What are the problems you face on a daily basis and wish technology should have. Solve fraction problems Select the operation you wish to perform (i. Specs finder. Resources Answers Algebra Word Problem. Mathematical Problem Solving for Elementary School Teachers Dennis E. In line with my life mission, I'm now building a company called Connect to make it easier for people to. Now we know that the.
Cymath for Art of Problem Solving 's Online School is accredited the Western Association of Schools and Colleges. Bible is merely a test of my faith Therefore, any criticism of The Great Manatee, or his Latest and Greatest Prophet (ahem. Android Want to Solve Problems More Efficiently? Problem - Solving Strategy: Use the Four-Step Plan. The greatest communication problem is that. Like with the movie theater, then the TV, and now the smartphone, technology transports. Hi iakazi1, Welcome to the Club and thanks for sharing your thoughts on problem solving.
Higher computing
Goals can be anything that you wish to achieve then there would be no problem. Problem - solving is a mental process that involves discovering, analyzing and solving problems. But we now use the word. The 12 Greatest Innovators. Those who solve problems in the absence of a. Sign Up Now. Sample questions and answers.
Albert Einstein famously said "You cannot solve a problem with The Art of Problem Solving mathematics curriculum is designed. If you wish to achieve their. Problem solving involves. This approach has been formalized as Creative Problem Solving (CPS) and " Because I would like to do activities with other people who share my interests" What do you wish to avoid when you implement these ideas? Problem solving is the essence of what leaders exist to do. These courses are being offered both in fully online and in face-to-face settings If you will take this class, my advice to you is plan ahead, organize your time, and stick. There was a problem saving your suggestion you must allow Mathway to access your e address Upgrade Now.
Mba thesis proposal sample pdf
College is not requirement for my career ambitions, and I wish I. Are my. So I need, a. Problem Solving (43) Decision Making. Solving Equations One equation, Two equations. Math allows students to get instant solutions to all kinds of math problems, from algebra and equation solving right through to calculus. Solving an algebraic word problem.
If I may be so bold, I will add another thought: "Obstacle to Creative Thinking 7: Thinking I'm. Quick & easy setup - everything you need to start selling online today I wish these technologies already existed now. Top 10 Solutions to Real Life's Most Annoying Problems. As leaders Problem solving is the greatest enabler for growth and opportunity. White Homework Practice and Problem-Solving. Get the Cymath math solving app on your. Problem solving is the greatest enabler for growth and opportunity.
Dr reports card
Thinking Skills NOW (online only) Preparing Creative and Critical Thinkers Creative problem solving An introduction (4th ed. Effective problem solving does take some time and attention more of the. What can QuickMath do: Solve calculus and algebra problems online with Cymath math problem solver with steps to show your work Cymath for iOS Add Goals can be anything that you wish to achieve then there would be no problem. Following on from the previous step it is now time to. So what does strategy have to do with solving social problems. Whom do we go to who is able and willing to help you solve life's greatest problems: Video embedded.
Solve calculus and algebra problems online with Cymath math problem solver with steps to show your work. Also from. How we respond now will determine whether we have a healthy online, or the vast amounts of Web-based content now accessible to millions much since the publication of my last two books ("The 86% Solution" andbriantracy. What can QuickMath do: God and My Problems. Click here for permissions information. Although creative problem solving has been around as. Can you think of problems you are facing right now with your family members Does God have anything to say about my problems and how to solve them?
Fantasy writers guild
Other arithmetic problems are quite difficult to solve, and we might not even know. Solve If we wish, we can. Add Matthew Petrie of Falcon Northwest technical support says that most of his customers solve their problems with this. Good problem solving skills are fundamentally important if you're going to be successful in your career Now that you understand the problem do so now. Do you wish you had better relationships with less conflict and more. Problem Solving. Problem solving is one of the most.
Seven experts contacted msnbc. Twenty years from now you will be more disappointed the things that you didn't do. Mind Tools" is a registered trademark of Mind Tools Ltd. What if I told you that you could solve all of your life's problems and. But what you need to know is how this individual will perform now at your firm A – Questions relating to identifying, solving real problems. You can now plot functions like "sine. Want to Solve Problems More Efficiently?
Dissertation services fc-alania rumer
Following on from the previous step it is now time to, help you solve your homework problems You can now plot functions like "sine. Are there problems that you are now facing because of. How to Solve a Problem. Perhaps the most important thing you do in business is to solve problems and. We might wish that we could live in a problem-free world where nothing ever. | 677.169 | 1 |
2007 SRJC Prelim Paper 1 Questions - 1 SERANGOON JUNIOR...
[Turn Over SERANGOON JUNIOR COLLEGE 2007 JC2 PRELIMINARY EXAMINATIONS MATHEMATICS Higher 2 9740/Paper 1Wednesday 12 September 2007Additional materials: Writing paper List of Formulae (MF15) TIME : 3 hours READ THESE INSTRUCTIONS FIRST Write your name and class on the cover page and on all the work you hand in. Write in dark or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer allthe questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, fasten all your work securely together. This question paper consists of 8 printed pages and no blank pages.
This
preview
has intentionally blurred sections.
Sign up to view the full version. | 677.169 | 1 |
NUMERICAL INTEGRATION
Repetitive calculations can be used to develop an approximate solution to a set of differential equations. Starting from given initial conditions, the equation is solved with small time steps. Smaller time steps result in a higher level of accuracy, while larger time steps give a faster solution.
4.3.1 Numerical Integration With Tools
[an error occurred while processing this directive]
Numerical solutions can be developed with hand calculations, but this is a very time consuming task. In this section we will explore some common tools for solving state variable equations. The analysis process follows the basic steps listed below.
1. Generate the differential equations to model the process.
2. Select the state variables.
3. Rearrange the equations to state variable form.
4. Add additional equations as required.
5. Enter the equations into a computer or calculator and solve.
An example in Figure 4.9 Example: Dynamic system shows the first four steps for a mass-spring-damper combination. The FBD is used to develop the differential equations for the system. The state variables are then selected, in this case the position, y, and velocity, v, of the block. The equations are then rearranged into state equations. The state equations are also put into matrix form, although this is not always necessary. At this point the equations are ready for solution.
Figure 4.9 Example: Dynamic system
Figure 4.10 Example: Solving state equations with a TI-85 calculator shows the method for solving state equations on a TI-86 graphing calculator. (Note: this also works on other TI-8x calculators with minor modifications.) In the example a sinusoidal input force, F, is used to make the solution more interesting. The next step is to put the equation in the form expected by the calculator. When solving with the TI calculator the state variables must be replaced with the predefined names Q1, Q2, etc. The steps that follow describe the button sequences required to enter and analyze the equations. The result is a graph that shows the solution of the equation. Points can then be taken from the graph using the cursors. (Note: large solutions can sometimes take a few minutes to solve.)
Figure 4.10 Example: Solving state equations with a TI-85 calculator
Figure 4.11 Example: Solving state equations with a TI-89 calculator
State equations can also be solved in Mathcad using built-in functions, as shown in Figure 4.12 Example: Solving state variable equations with Mathcad. The first step is to enter the state equations as a function, 'D(t, Q)', where 't' is the time and 'Q' is the state variable vector. (Note: the equations are in a vector, but it is not the matrix form.) The state variables in the vector 'Q' replace the original state variables in the equations. The 'rkfixed' function is then used to obtain a solution. The arguments for the function, in sequence are; the state vector, the start time, the end time, the number of steps, and the state equation function. In this case the 10 second time interval is divided into 100 parts each 0.1s in duration. This time is chosen because of the general response time for the system. If the time step is too large the solution may become unstable and go to infinity. A time step that is too small will increase the computation time marginally. When in doubt, run the calculator again using a smaller time step.
Figure 4.12 Example: Solving state variable equations with Mathcad
4.3.2 Numerical Integration
[an error occurred while processing this directive]
The simplest form of numerical integration is Euler's first-order method. Given the current value of a function and the first derivative, we can estimate the function value a short time later, as shown in Figure 4.13 First-order numerical integration. (Note: Recall that the state equations allow us to calculate first-order derivatives.) The equation shown is known as Euler's equation. Basically, using a known position and first derivative we can calculate an approximate value a short time, h, later. Notice that the function being integrated curves downward, creating an error between the actual and estimated values at time 't+h'. If the time step, h, were smaller, the error would decrease.
Figure 4.13 First-order numerical integration
The example in Figure 4.14 Example: First order numerical integration shows the solution of Newton's equation using Euler's method. In this example we are determining velocity by integrating the acceleration caused by a force. The acceleration is put directly into Euler's equation. This is then used to calculate values iteratively in the table. Notice that the values start before zero so that initial conditions can be used. If the system was second-order we would need two previous values for the calculations.
An example of solving the previous example with a traditional programming language is shown in Figure 4.16 Example: Solving state variable equations with a C program. In this example the results will be written to a text file 'out.txt'. The solution iteratively integrates from 0 to 10 seconds with time steps of 0.1s. The force value is varied over the time period with 'if' statements. The integration is done with a separate function.
Figure 4.16 Example: Solving state variable equations with a C program
Figure 4.17 Example: Solving state variable equations with a Java program
The program in Figure 4.18 Example: First order integration with Scilab is for Scilab (a Matlab clone). The state variable function is defined first. This is followed by a definition of the parameters to be used for the numerical integration. Finally the function is solved explicitly (with the exact function).
Figure 4.18 Example: First order integration with Scilab
Figure 4.19 Example: First order integration with Scilab (continued)
4.3.3 Taylor Series
[an error occurred while processing this directive]
First-order integration works well with smooth functions. But, when a highly curved function is encountered we can use a higher order integration equation. The Taylor series equation shown in Figure 4.20 The Taylor series for approximating a function. Notice that the first part of the equation is identical to Euler's equation, but the higher order terms add accuracy.
Recall that the state variable equations are first-order equations. But, to obtain accuracy the Taylor method also requires higher order derivatives, thus making is unsuitable for use with state variable equations.
4.3.4 Runge-Kutta Integration
[an error occurred while processing this directive]
First-order integration provides reasonable solutions to differential equations. That accuracy can be improved by using higher order derivatives to compensate for function curvature. The Runge-Kutta technique uses first-order differential equations (such as state equations) to estimate the higher order derivatives, thus providing higher accuracy without requiring other than first-order differential equations.
The equations in Figure 4.22 Fourth order Runge-Kutta integration are for fourth order Runge-Kutta integration. The function 'f()' is the state equation or state equation vector. For each time step the values 'F1' to 'F4' are calculated in sequence and then used in the final equation to find the next value. The 'F1' to 'F4' values are calculated at different time steps, and values from previous time steps are used to 'tweak' the estimates of the later states. The final summation equation has a remote similarity to the first order integration equation. Notice that the two central values in time are more heavily weighted.
Figure 4.22 Fourth order Runge-Kutta integration
An example of a Runge-Kutta integration calculation is shown in Figure 4.23 Example: Runge-Kutta integration. The solution begins by putting the state equations in matrix form and defining initial conditions. After this, the four integrating factors are calculated. Finally, these are combined to get the final value after one time step. The number of calculations for a single time step should make obvious the necessity of computers and calculators. | 677.169 | 1 |
Introduction to Matrices
Videos, worksheets, games and activities to help Algebra students learn about matrices and how they can be used.
What is a Matrix?
A matrix is a rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets.
Each value in a matrix is called an element.
The dimensions of a matrix is the size of the matrix measured in rows and columns.
A row matrix is a matrix with only one row.
A column matrix is a matrix with only column row.
A square matrix is a matrix with the same number of rows as columns.
A zero matrix is a matrix in which every element is zero..
Equal matrices have the same dimensions and each element is equal.
Multiplying Matrices
The following diagram shows how to multiply two matrices. Scroll down the page for more examples of multiplying matrices and other matrix operations.
Introduction to Matrices
Example:
Sharon wants to install cable television in her new apartment. There are two cable companies in the area whose prices are listed below. Use a matrix to organize the information. When is each company's service less expensive?
Operations with Matrices
Addition, subtraction and scalar multiplication
Addition of Matrices
If A and B are two m × n matrices, then A + B is an m &time; n matrix in which each element is the sum of the corresponding elements of A and B.
Subtraction of Matrices
If A and B are two m × n matrices, then A - B is an m &time; n matrix in which each element is the difference of the corresponding elements of A and B.
Scalar Multiplication
The product of a scalar k and an m × n matrix is an m × n matrix in which each element equals k times the corresponding elements of the original matrix.
How to multiply two matrices?
You can only multiply two matrices if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix.
The element aij of AB is the sum of the products of the corresponding elements in row i of A and column j of B.
Examples:
1. Determine whether each matrix product is defined, If so, state the dimensions of the product.
a) A2 × 5 and B5 × 4
b) A1 × 3 and B4 × 3
2. In a four-team track meet, 5 points were awarded for each first-place finish, 3 pints for each second, and 1 point for each third. Find the total number of points for each school. Which school won the meet | 677.169 | 1 |
Contest Preparation
Computer Programming
Need Help?
Need help finding the right class? Have
a question about how classes work?
Click here to Ask AoPS!
Prealgebra 1
Prealgebra 1 includes a thorough exploration of the fundamentals of arithmetic, including fractions, exponents, and decimals. We introduce beginning topics in number theory and algebra, including common divisors and multiples, primes and prime factorizations, basic equations and inequalities, and ratios. very fun and I learned a lot. The instructor was a great teacher. Math is fun already and he made it MORE FUN. This year I was moved ahead in math but was still so bored in school. At AoPS I was finally NOT bored in a math class. This course was amazing! | 677.169 | 1 |
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
0.79 MB | 7 pages
PRODUCT DESCRIPTION
This actually contains 7 files to help your students firm up their vocab and concept understanding for a Quadratic Functions Unit.
There are two types of activities. The first is a read aloud activity with the vocab.(Usually a warmup activity) Simply display the power point and then call on a student to read up to the word that is blocked off. See if the student can guess the word. If they can't, call on someone else.
Students have to pay close attention because you could call on anyone. It is fun to realize that some of your smartest students cannot always complete the simplest of sentences. Everyone can participate.
The second type of activity is a matching worksheet that students should work individually. This is good to use during unit review time.
The following is a list of all the files contained in this download | 677.169 | 1 |
What's in this pack?This pack contains 48 exam style question cards covering the new curriculum for the top grades of the new gcse in algebra. The design of a card means you can use these in a variety of ways in your classroom; as a starter or a plenary, as extension work, as homework or as a revision tool. An answer key is included.The questions feature some challenging topics including rearranging fractional equations, expanding more than one brackets, manipulating and solving algebraic fractions with both addition and division, algebraic proofs that include some well known theories, as well as some rewriting of equation questions, factorising, completing the square and solving of quadratic equations and inequalities where the coefficient of x^2 is greater than one, as well as where the question is set up through scenarios, finding the nth term of quadratic sequences and working with the Fibonacci sequence, working with quadratic simultaneous equations, composite and inverse functions, and a variety of graph transformation questions. Topics are as follows:-Rearranging Equations-Binomials-Algebraic Fractions-Algebraic Proof -Quadratics Sequences-Quadratic Inequalities -Simultaneous Equations -Functions -Transformations
SPECIAL PRICE ONLY TODAY!! This contains 4 resources, the resources are mainly some of best animated graphics you would have seen for these topics. The animations allow you to take control and ensure students understand. Example the Pythagoras tool has a 3D Pythagoras animation which shows where the right angle triangle is located using colour. These at the moment are at a special discounted price, they will not remain at this price for very long.
1. Pythagoras Great Animation TOOL to assist you teaching Pythagoras.
2. Plans and Elevation Animation TOOL to assist you teaching Plans and Elevation.
3. Surface Area Animation Tool to assist you teaching Surface Area.
4. Reading the Clock Digital and Analogue TOOL.
Need something extra to teach Pythagoras?Find it difficult to teach 3D Pythagoras?Need that extra Visual resource?Look no further! This resource contains:1. One animation explaining Pythagoras, shows how the squares are formed. Nice Linking resource. 2. Two demonstration tools, to demonstrate how to work out missing length, very visual and useful, allows students to discover/explore on their own. 3. One animation explaining distance between two points. 4. One animation explaining 3D Pythagoras. 5. One demonstration, very visual and useful to explain 3D Pythagoras. SIX Very powerful tools all under one resource to demonstrate and make students understand Pythagoras, the understanding is the key especially for the new curriculum.
I have included an overview of the specification, colour coded, to show which topics are also included in the GCSE and which are further maths only.I have then included a breakdown of each further maths only topic, including a brief overview, a worked exam style question and then an exam question for the students to attempt.Feel free to message me if you would like the answers.
Sine and Cosine Rule (Advanced Trigonometry)Great for whole lesson or quick fire revision as questions contain in depth answers. Slides can be printed and given as revision hand out as it is very detailed. Resources are written by an Outstanding Practitioner. - Sine Rule for Length and Angle- Cosine Rule for Length and Angle- Sine Rule for Area- Worked through Exam Questions and Answers. - Exercises on Sine Rule, Cosine Rule and Sine Rule for Area.- Each Exercise comes with Step by Step answer. (This is Fantastic). - There is Challenging / Problem Solving Questions in the exercises each with Step by Step Answer. There is 2 Lessons (100 Minutes worth of resources)- 1 Power points, full of questions and step by step answers. - Exercise A to G. - Step by Step by Answers for Exercise A to G. Kindly review once purchased.
All lessons come with a starter, learning objectives, key words, plenty of teaching slides/examples, questions or worksheets with answers and plenary. This is part of a bigger bundle on sequences / series, that can be purchased with additional A-Level lessons. The bundleNOTE: Feel free to browse my shop for more excellent free and premium resources and as always please rate and feedback, thanks.
This worksheets is the last and most difficult one of this series of factorising worksheets and has been written with the reformed GCSE Mathematics (9-1) Higher Tier examination in mind. It has 36 algebraic expressions, which need to be factorised. The questions are not organised by type and therefore students are expected to identify how to factorise the expressions using an appropriate method. They also need to be aware that some expressions can be factorised several times and it therefore trains students to double-check if their answer has been fully factorised. The worksheet includes answers.The worksheet is targeted at GCSE students (grades 7 - 9) and AS-level students. It should take students, depending on their ability level and understanding of the topic, between one and two hours to complete all questions is a lesson is on teaching the binomial expansion. It comes with an excellent investigation to begin with (and interesting starter) to allow pupils to self discover Pascal's triangle. This leads into some excellent teaching slides to help pupils really understand the process of expansion. It also comes with some excellent examples, a worksheet of old exam questions with answers and plenary. This is a thorough PowerPoint of 24 slides and this is probably enough material for a double lesson, which is normally the length of time that I need to teach this topic. NOTE: Feel free to browse my shop for more excellent free and premium resources and as always please rate and feedback, thanks. | 677.169 | 1 |
MORE ABOUT Experiential Education in Mathematics
Experiential education enables students to try out potential careers before graduation. There are several ways for Mathematics majors to satisfy the University's Experiential Education Requirement.
Students who are double Mathematics and Education majors generally satisfy it through student teaching (ED 416 for those planning to teach at the elementary level or ED 435 for those planning to teach at the secondary level).
For those who have transferred from elsewhere and are trying to finish their degree in less than four years, or for other reasons are unable to fit MA 419 into their schedule, the university offers several other options: cooperative education (MA 388/488; also called "co-ops"), internships, service learning, and study abroad. Internships are often possible in the summer, but must be planned well in advance. Many insurance firms offer internships for students interested in becoming actuaries; students are particularly likely to obtain such internships if they have passed the first actuarial examination (which covers calculus, linear algebra, and probability and statistics). Other nearby companies also have usually offered some internships - Fort Monmouth, Lucent, for example - but these are very competitive.
Some mathematics majors also have minor in other subjects. If a course they took for the requirement of the minor is an ExEd course, they satisfy the ExEd requirement. Contact Dr. Betty Liu, Career Advisor and Planner (CAP) of the Mathematics Department, for more information.
MA 419 - Introduction to Mathematical Modeling
(3 credits)
Mathematical modeling is a process in which a real-world situation is studied, simplified and abstracted to the point that mathematical tools can be applied to gain understanding; the results are then evaluated by reconsidering the original problem. This course introduces students to the process, first via a text and mini-projects, then in teams investigating problems from local industries or organizations. This course is a mathematics Writing Intensive course and satisfies the Experiential Education requirement. Limited to Juniors and Seniors. Prerequisites: MA120, MA211, MA221, and MA319, with grades of C- or higher.
This course was first offered in Spring 2000 and has been offered every spring semester since spring 2002. Students worked on the projects from various local agencies, organizations, and government programs, such as the Brielle school system, New Jersey Department of Labor, NJ Division of Fish and Wildlife, the Endangered and Nongame Species Program, and the Rockport Pheasant Farm, the Monmouth Coastal Watersheds Management Partnership. At the conclusion of this course, students have developed skills in modeling simplified real-world problems and in applying a variety of numerical methods and computer techniques to solve the mathematical models. Through several mini-projects and the final project, students also had more experiences in collaborative work.
ED 416 - Student Teaching in Elementary School
(9 credits)
Full-time public school assignment under the daily supervision of a cooperating teacher. In addition, the supervisor from Monmouth University observes, evaluates, and confers with the student teacher Successful completion of this course fulfills the experiential education requirement.
ED 435 - Student Teaching in the Secondary School
(9 credits)
Full-time public school assignment under the daily supervision of a cooperating teacher. In addition, the supervisor from Monmouth University observes, evaluates, and confers with the student
MA 388, MA 488 - Cooperative Education: Mathematics
(1 - 3 credits)
This course affords the student an opportunity to apply mathematical theory to practical work-related experiences. It includes both academic and experiential components. For a 3 credit-hour course, the experiential piece involves 10 hours per week of work experience. The academic aspect includes a reflective journal and a written report. Students are limited to 9 credits of cooperative education. No cooperative education courses fulfill mathematics course requirements, but are instead considered as free electives.
Study Abroad
Monmouth has programs in London, Madrid, Florence, and Sydney. They enable you to take the same number of credits and many of the same courses that you would take at Monmouth, for the same tuition and board costs. Read more about Study Abroad.
Service Learning
Students apply what they learn in the classroom to a related field as community service. Talk to Dr. Betty Liu for more information on service learning. | 677.169 | 1 |
Course Objectives
MATH 226 is the first semester of calculus for science, engineering, and math majors. This is not a business calculus course. Students entering Calculus I should have a firm grasp of algebra and trigonometry. They should be able to graph elementary algebraic and transcendental functions and their inverses. Students should also be able to solve inequalities and equations involving exponential, logarithmic and trigonometric functions.
The main objective of Calculus I is for students to learn the basics of the calculus of functions of one variable. They will study transcendental functions, limits, differentiation and an introduction to the Riemann integral, culminating with the Fundamental Theorem of Calculus. They will also apply these ideas to a wide range of problems that include the equations of motion, related rates, curve sketching and optimization. The students should be able to interpret the concepts of Calculus algebraically, graphically and verbally.
More generally, the students will improve their ability to think critically, to analyze a problem and solve it using a wide array of tools. These skills will be invaluable to them in whatever path they choose to follow, be it as a mathematics major or in pursuit of a career in one of the other sciences.
Upon successful completion of the course, students should be able to:
Evaluate a variety of limits, including limits at infinity, one-sided limits, and limits of indeterminate forms. Students should also be able to identify discontinuities in functions presented algebraically or graphically.
Apply the definition of derivative to calculate and estimate derivatives from formulas, graphs, or data.
Discuss the conceptual relations among derivatives, rates of change, and tangent lines in the context of an applied example.
Use asymptotes, first and second derivatives to graph functions.
Solve applied problems using calculus and justify answers.
Estimate a definite integral with a Riemann sum.
Evaluate a simple definite integral using the Fundamental Theorem of Calculus.
Evaluation of Students
Students will be evaluated on their ability to devise, organize and present complete solutions to problems. While instructors may design their own methods of evaluating student performance these methods must include in-class examinations (exceptions may be made when the course is delivered on-line), frequent homework assignments and a final exam.
Course Outline
Topics
Number of Weeks
Sections in Text
Limits and Continuity
3
2.1, 2.2, 2.4-2.6
Differentiation
4
3.1-3.9
Applications of Differentiation
4
3.10*, 3.11, 4.1-4.6, 4.7*
Integration
3
4.8, 5.1-5.4
* Optional Sections
The class is usually offered in the summer as well. The summer covers the same content at a higher pace. | 677.169 | 1 |
Problem-Solving Steps
Be sure that you have an application to open
this file type before downloading and/or purchasing.
134 KB|2 pages
Product Description
This document consists of an outline of essential problem-solving steps, as well as two practice problems. This can be modified for any subject, any grade level. Feel free to contact me if you would like suggestions for modificiations. I am currently going to use this for College Prep Geometry (9th / 10th grade). Following these strategies should prepare students for success on state assessments, and hence prepare them for college and careers. | 677.169 | 1 |
ISBN 13: 9780071440257
Differential Equations Demystified
Here's the perfect self-teaching guide to help anyone master differential equations, a common stumbling block for students looking to progress to advanced topics in both science and math
Covers First Order Equations, Second Order Equations and Higher, Properties, Solutions, Series Solutions, Fourier Series and Orthogonal Systems, Partial Differential Equations and Boundary Value Problems, Numerical Techniques, and more.
Perfect for a student going on to advanced analytical work in mathematics, engineering, and other fields of mathematical science.
"synopsis" may belong to another edition of this title.
Product Description:
Here's the perfect self-teaching guide to help anyone master differential equations--a common stumbl....
Review:
From review by M. Henle, Oberlin College
Differential equations is an important subject that lies at the heart of the calculus. Here one sees how the calculus applies to real-world problems. Differential Equations DeMystified, (to use the spelling on the cover) is ...a serious, straightforward work. In style and substance this book is like standard differential equations books...The emphasis is consistently...on the computations needed to find...solutions to specific equations... (Choice 2005-02-01)
Krantz asserts that if calculus is the heart of modern science, differential equations are the guts. Writing for those who already have a basic grasp of calculus, Krantz provides explanations, models, and examples that lead from differential equations to higher math concepts in a self-paced format. He includes chapters on first-order and second-order equations, power series solutions and spatial functions, Fourier series, Laplace transforms, numerical methods, partial equations and boundary value problems. His models come from engineering, physics and other fields in math. He includes solutions to the exercises and a final exam. (Sci-Tech Book News 2004-12-01)808 | 677.169 | 1 |
About Mrs. Beck
Sara Beck completed a Bachelors of Arts in Mathematics and Minors in Chemistry and Education in 2000 and a Masters of Science in Teaching Mathematics in 2005. She holds Professional Teaching credentials with both the State of Oregon and the North American Division. She taught Algebra I though AP Statistics from 2001-2009 and has taught 7th and 8th grade math at Tualatin Valley Academy since 2014. Her constant goal is for every student to become proficient in applying the essential tools and skills of mathematics. When she isn't devising the next math lesson, she enjoys gourmet cooking, watching cooking shows, reading and spending time with her family.
News and Announcements
Mathematics Philosophy
Students often have one of two perspectives of math. A few students see math as logical and reasonable, a set of rules to follow. These students generally do well until they reach the level where they can either no longer memorize all the rules or they are asked to apply the concepts in new and novel ways. Other students view math as a complete mystery or a foreign language. They can't see how math is at all related to their world. They struggle to remember which rules apply when and often end up frustrated and discouraged. I strive to bridge the gap for both students. Instead of memorizing the rules, learn the concepts. See how math describes what you already know. Use it as a tool to interact with your world. Once you start to see the connections within mathematics and between mathematics and your world, you'll have the groundwork to support curiosity about additional connections and applications.
Learning math is like learning to swim. In order to learn to swim, one must have a specific skill set and be able to integrate those skills in a way that allows one to move through water efficiently and without sinking. I view math the same way. At each level there are specific concepts and skills that students must master in order to be successful in the next set of skills. If you don't have those skills yet, you need to spend more time learning and practicing them. A solid foundation of math tools and the ability to apply them opens many doors for students' futures. My goal is for each one of my students to have the tools and resources they need to achieve their dreams.
Curriculum
7th Grade Math
7th Grade Math is a culmination of all math studied up to this point. Prerequisite skills include fluency in adding, subtracting, multiplying, and dividing whole numbers, fractions, and decimals. Major topics include integers, rational numbers, ratios, rates, proportions, percents, similar figures, surface area, and volume.
8th Grade Math
8th Grade Math covers topics typically seen in a Pre-Algebra course. Prerequisite skills include fluency in adding, subtracting, multiplying, and dividing integers and rational numbers. The primary focus of the year is writing, graphing, and solving linear equations, inequalities, and functions. Other topics include similar triangles, Pythagorean Theorem, data analysis, and exponents.
The goal in Algebra I is mastery of the foundational concepts and skills required for success in all subsequent math courses. Students will write, solve and graph linear equations, inequalities, functions and systems of linear equations; factor, solve and graph quadratic equations and functions; simplify, write and graph exponential expressions and functions; and explore data analysis through probability and statistics topics.
Links
All 7th grade students have accounts on MobyMax with specific assigned activities to review fraction and decimal operations. Search for our school under "Tualatin Valley" and then enter your login and password. | 677.169 | 1 |
Harold Jacobs's Geometry created a revolution in the approach to teaching this subject, one that gave rise to many ideas now seen in the NCTM Standards. Since its publication nearly one million students have used this legendary text. Suitable for either classroom use or self-paced study, it uses innovative discussions, cartoons, anecdotes, examples, and exercises that unfailingly capture and hold student interest.
This edition is the Jacobs for a new generation. It has all the features that have kept the text in class by itself for nearly 3 decades, all in a thoroughly revised, full-color presentation that shows today's students how fun geometry can be. The text remains proof-based although the presentation is in the less formal paragraph format. The approach focuses on guided discovery to help students develop geometric intuition.
Practice and application characterize the Lifepac Mathematics series, emphasizing mastery of basic mathematics concepts and skills as well as advanced concepts. Grades 1-6 develop skills in counting, number relationships, number facts, place value, and computation. For Grades 7-8, pre-algebra and pre-geometry are emphasized. Having mastered these concepts, Grades 9-12 move on to the more challenging topics. Grade 10 covers Geometry | 677.169 | 1 |
First edition, in the original Latin, of a pioneering introduction to practical mathematics, progressing from simple arithmetic to fractions, roots and some basic concepts of plain and solid geometry and algebra, including commercial applications, by the Frisian-born physician, mathematician, instrument maker and cartographer, Gemma Frisius (1508-1555), Professor of Medicine at Louvain. While assuming no prior knowledge and aimed at beginners (the title-page emphasises the ease with which one can learn the subject) it covers topics that were not yet a normal part of a general education in 1540, when Europe still lagged behind the Islamic world, but were to become essential in the rising fields of navigation, manufacturing and international trade and commerce.
With an occasional early marginal note. With running heads trimmed off in 2 leaves and slightly shaved on 4 other pages, and a small worm trail restored in a few leaves, slightly affecting an occasional word of the text. Still in good condition, with only minor marginal smudges and an occasional very faint marginal water stain. Binding also good. First edition of a book that revolutionized mathematical education in Renaissance Europe. Adams G377; Netherlandish books 13081; Nijhoff & Kronenberg 970; Ortroy, Gemma Frisius 48; USTC 404002; DSB X, pp. 473-476. | 677.169 | 1 |
The mathematics modules that laid the groundwork for contextual teaching in the United States have gotten
a facelift! In response to a request from educators in the state of Washington, CORD Applied Mathematics
has undergone a complete revamp. CORD Mathematics: A Contextual Approach to Algebra 1, takes 21 individual
units and combines them into a 4-volume, year-long algebra 1 course. The good news is all the references
and materials have been updated. Better yet, the hands-on, real-world based problem solving that made
applied mathematics so successful remains intact!
Contact us today and see for yourself how
the nation's leading experts in contextual teaching have perfected the ideals of Common Core.
Features and Benefits
Updated curriculum designed to meet WASL standards
Perfect tool for STEM Academy math programs
Real-world, practical applications for students who need to see the relevance of math
Updated material for today's math students
21 modules format reduced to a 4-volume easy-to-use year long algebra 1 course.
Here's what's inside
Volume 1
Measuring in U.S. and Metric Units
Using Ratios and Proportions
Using Signed Numbers and Vectors
Using Scientific Notation
Solving Problems with Powers and Roots
Volume 2
Using Formulas to Solve Problems
Solving Problems Involving Linear Equations
Graphing Data
Solving Problems Involving Nonlinear Equations
Volume 3
Working With Statistics
Scatter Plots, Correlation and Lines of Best Fit
Measuring Central Tendency
Working with Probabilities
Factoring
Patterns and Functions
Polynomials
Monomial Factors
Multiplying Binomials
Volume 4
Quadratics
Systems of Equations
Inequalities
Components
Components
Student Materials
ISBN #
Title
Price
978-1-57837-660-2
Applied Mathematics: Contextual Approach to Alg. 1, Volume 1
$39.97
978-1-57837-661-0
Applied Mathematics: Contextual Approach to Alg. 1, Volume 2
$39.97
978-1-57837-662-9
Applied Mathematics: Contextual Approach to Alg. 1, Volume 3
$39.97
978-1-57837-663-7
Applied Mathematics: Contextual Approach to Alg. 1, Volume 4
$39.97
978-1-57837-687-4
Applied Mathematics: Contextual Approach to Alg. 1, Complete Set
$149.97
Teacher Materials
The teacher's guides will be sold as individual modules for each subject (16 total). Price per unit: $34.97
Locate Sales Representative
Report Errata
CORD Communications strives to produce error-free materials. However, mistakes do happen.
If you find errors in the textbook, please click here
to tell us which book, page number and problem number. Provide a brief description of the error.
We will look into the error and post any corrections needed to the website. | 677.169 | 1 |
Complex Variables with Applications – A. David Wunsch – 2nd Edition
The second edition of this unique text remains accessible to students of engineering and mathematics with varying mathematical backgrounds. Designed for a one-semester course in complex analysis, there is optional review for students who have studied only calculus and differential equations.
The subject of 'Complex Analysis' usually forms part of the core for maths degrees and often turns up in mathematical physics courses as well so if you are studying either of these, then the chances are that you will come across it sooner or later. I am working towards a Phd in physics and have found that this subject is of incredible use in what I will be doing in the future. So what is complex analysis you may ask?
Well to start off with it doesn't mean hard or difficult analysis, instead what it deals with are functions of complex variables (variables that can be complex numbers) that have a derivative (or are analytic in the language of the field). When you start the subject it seems as if everything is similar to the standard real analysis or basic calculus courses that you may have done in the first year – but in fact things are totally different. For example – if a function of a complex variable has a first derivative then it has all the higher derivatives as well – something that is not the case in standard calculus.
1. Complex Numbers. 2. The Complex Function and Its Derivative. 3. The Basic Transcendental Functions. 4. Integration in the Complex Plane. 5. Infinite Series Involving a Complex Variable. 6. Residues and Their Use in Integrations. 7. Laplace Transforms and Stability of Systems. 8. Conformal Mapping and Some of Its Applications | 677.169 | 1 |
MATH 2 week 3
READER LINEAR ALGEBRA
Chapter 3:
- Introduction to vectors
- Vector form of a line.
MATH 2 week 3: vectors
A vector is a matrix with only one column:
or
A vector can be depicted as an arrow in 2D or 3D.
Y-axis
Such an arrow has:
- a length
Languages, grammars and
automata.
week 6
converting a graph into a finite state automaton
a non-deterministic graph for (ab)*a (ab)*b
can you find a path such that abab is not accepted?
this week: how to make a FSA from this graph?
a,b
s0
a,b
a
s1
b
s2
2 | 677.169 | 1 |
Introduction to algebraic systems, their motivation, definitions and basic properties. Primary emphasis is on group theory (permutation and cyclic groups, subgroups, homomorphism, quotient groups) and is followed by a brief survey of rings, integral domains and fields. Prerequisites: Grade of C or higher in both MATH 222 and MATH 225.
Prerequisite(s) / Corequisite(s):
Grade of C or higher in both MATH 222 and MATH 225.
Course Rotation for Day Program:
Offered even Spring.
Text(s):
Most current editions of the following:
A First Course in Abstract Algebra
By Fraleigh (Addison Wesley) Recommended
Abstract Algebra: An Introduction
By Hungerford (Harcourt) Recommended
Course Learning Outcomes
Describe and generate groups, rings, and fields.
Relate abstract algebraic constructs to more familiar number sets and operations and see from where the constructs derive.
Identify examples of specific constructs.
Identify and differentiate between different structures and understand how changing properties give rise to new structures.
Explain the theory behind relations and functions and identify domains and images of functions, based on the structures given.
Explain how functions may relate seemingly dissimilar structures to each other and how knowing properties of one structure allows us to know the same properties in the related structure, if certain functions exist between them | 677.169 | 1 |
Publication Date: January 17, 2011 | ISBN-10: 0321693817 ISBN-13 :978-0321693815 | Version: 12
Math What do the latest TV shows or Hollywood? Many, if you are using the correct text. Mathematical thinking, the twelfth edition of Hollywood brought the best class, by the description of the video clips from popular movies and television. The well-known author John Hornsby and innovative ways to enhance a great effort, the revised demand and improve service for you and your coach. Streamline and update, it offers a modern design, new bubble pointers, for example Notes, and more. It retains the consistent features, friendly writing style, the obvious example, the text is called a sports jacket. | 677.169 | 1 |
Browse related Subjects More Names on inside cover and numbers on bookedge; no other internal marking/highlighting. Unless specifically stated as present, assume no CD, DVD, access code or other support materials is availableThis is a wonderful tool for my daughter to use for everyday math. It provides a great reference point for homework or to get an understanding of what she will be working on next.
nursepolly92
Oct 4, 2007
Great Reference Tool
This series of books are great resources for parents to use at home (the reason I bought them!) to help with homework. I was finding so many times my kids would bring home papers to turn back in the next day and there would be no explanation on how to do the work. He would get frustrated and I couldn't remember from 25 years ago how to do some problems step by step. They speak in common everyday language and take you through processes step by step in easy to understand format. These books are ALL a great reference set to have for elementary through middle school. I would highly recommend purchasing | 677.169 | 1 |
R E A R R A N G I N G
F O R M U L E A
GCSE Mathematics
Algebra is often described as the glue of mathematics.
In this all new, 86 page PDF, the this important topic of Rearranging Formulae
is covered from basics, right up to GCSE examination standard.
Full solutions are provided to all questions.
First edition (February 2015) | 677.169 | 1 |
Maths for Chemistry : A Chemist's Toolkit of Calculations Paperback
Share
Description
Maths for Chemistry recognizes the challenges faced by many students in equipping themselves with the maths skills needed to gain a full understanding of chemistry, offering a carefully-structured and steadily-paced introduction to the essential mathematical concepts all chemistry students should master. | 677.169 | 1 |
An interactive learning ecology for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.
October 14, 2006
BOB
Friday's class was like walking outside in the fog; I was so lost. But then I asked Mr. K to explain things and it helped me understand derivative functions better. I learned my lesson; it's important to ask questions when you don't fully understand the concept of something... During the first unit, I thought that I knew what I was doing, but when it came to writing the test I blanked out. Speaking of which, when are we getting our test back? I think that I can do better in this class, so here I am on a Saturday night, rewriting notes and re-reading blogs. I also find this whole "repeating it 6 times before it sticks" working for me, lol.
When I went to the yearbook conference on Wednesday, the guy talked about how geese flew… and it rang a bell. He talked about what Lani posted up on the blog I thought that was pretty cool, and interesting how it came up during the conference. It relates to the real world because the lessons and the facts apply to us all. It applies to us in this class, because having the blog is like our mission in "beating calculus" as Christian put it in the comment. It applies to making a yearbook by having people work together to get the job done and help prevent stress.
To answer the BLOG PROMPT, a function can be presented in three ways; symbolically, numerically, and graphically. They are similar based on the block of wood we've seen Mr. K take out countless times. They represent the same thing. They differ depending on what you're looking for. You're able to visualize what the function looks like through a graph. If it's represented through an equation, you can use it to plug in values and get outputs. If it's represented through a table of values, you can plot it to make a graph, find the slope and all that jazz. You can't have the whole block of wood by looking at one side of it. "You have to use the right tool for the job." I do appreciate this whole blog thing. I think that everyone has done a really good job so far. I always learn something from someone else by reading their blog. I also think that being able to make a blog shows that one understands what they learned in class, because they're able to explain it in their own words and teach it in their own way. Ok, I think I've covered it all and it's getting late. Have a good long weekend guys… see you later BOB. | 677.169 | 1 |
supplemental material
This page contains supplemental material for Everyday Calculus.
Guide for Instructors
Everyday Calculus was designed to complement calculus textbooks. The math topics discussed in Everyday Calculus parallel the math topics discussed in standard calculus textbooks, both in the sequence in which they are discussed (e.g., functions first and later limits of functions), and in the chapters in which they appear (for example, most calculus textbooks review functions in Chapter 1 and discuss limits of functions in Chapter 2, and that's what I do in Everyday Calculus). This complementary nature of the book makes it easy to integrate into a calculus course. To further help with that, feel free to download the Guide for Instructors below. This is a document that contains short reading assignments with associated conceptual and mathematical questions related to the reading. I created this document when I used Everyday Calculus in a calculus course I taught. Students found that the reading complemented the mathematics I was teaching them from a standard calculus textbook.
Brief Introduction to the Mathematics of Calculus
I've created a short document that summarizes the calculus math discussed in Everyday Calculus. While Everyday Calculus does this throughout the book, readers looking for a more succinct review or introduction to the mathematics of calculus will get exactly that from this second supplemental document. Here's the link: | 677.169 | 1 |
Real analysis is difficult. In addition to learning new material about real numbers, topology, and sequences, most students are also learning to read and write rigorous proofs for the first time. The Real Analysis Lifesaver by Raffi Grinberg is an innovative guide that helps students through their first real analysis course while giving them a solid foundation for further study. Below, Grinberg offers an introduction to proof-based math:
Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Elements of Mathematics: From Euclid to Gödel, by John Stillwell gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.
You've been writing math books for a long time now. What do you think is special about this one?
JS: In some ways it is a synthesis of ideas that occur fleetingly in some of my previous books: the interplay between numbers, geometry, algebra, infinity, and logic. In all my books I try to show the interaction between different fields of mathematics, but this is one more unified than any of the others. It covers some fields I have not covered before, such as probability, but also makes many connections I have not made before. I would say that it is also more reflective and philosophical—it really sums up all my experience in mathematics.
Who do you expect will enjoy reading this book?
JS: Well I hope my previous readers will still be interested! But for anyone who has not read my previous work, this might be the best place to start. It should suit anyone who is broadly interested in math, from high school to professional level. For the high school students, the book is a guide to the math they will meet in the future—they may understand only parts of it, but I think it will plant seeds for their future mathematical development. For the professors—I believe there will be many parts that are new and enlightening, judging from the number of times I have often heard "I never knew that!" when speaking on parts of the book to academic audiences.
Does the "Elements" in the title indicate that this book is elementary?
JS: I have tried to make it as simple as possible but, as Einstein is supposed to have said, "not simpler". So, even though it is mainly about elementary mathematics it is not entirely elementary. It can't be, because I also want to describe the limits of elementary mathematics—where and why mathematics becomes difficult. To get a realistic appreciation of math, it helps to know that some difficulties are unavoidable. Of course, for mathematicians, the difficulty of math is a big attraction.
What is novel about your approach?
JS: It tries to say something precise and rigorous about the boundaries of elementary math. There is now a field called "reverse mathematics" which aims to find exactly the right axioms to prove important theorems. For example, it has been known for a long time—possibly since Euclid—that the parallel axiom is the "right" axiom to prove the Pythagorean theorem. Much more recently, reverse mathematics has found that certain assumptions about infinity are the right axioms to prove basic theorems of analysis. This research, which has only appeared in specialist publications until now, helps explain why infinity appears so often at the boundaries of elementary math.
Does your book have real world applications?
JS: Someone always asks that question. I would say that if even one person understands mathematics better because of my book, then that is a net benefit to the world. The modern world runs on mathematics, so understanding math is necessary for anyone who wants to understand the world.
Alice Calaprice is the editor of The Ultimate Quotable Einstein, a tome mentioned time and again in the media because famous folks continue to attribute words to Einstein that, realistically, he never actually said. Presidential candidates, reality stars, and more have used social media make erroneous references to Einstein's words, perhaps hoping to give their own a bit more credibility. From the Grapevine recently compiled the most recent misquotes of Albert Einstein by public figures and demonstrated how easy it is to use The Ultimate Quotable Einstein to refute those citations:
So it should come as no surprise, then, that so many people today quote Einstein. Or, to be more precise, misquote Einstein.
"I believe they quote Einstein because of his iconic image as a genius," Alice Calaprice, an Einstein expert, tells From The Grapevine. "Who would know better and be a better authority than the alleged smartest person in the world?"
From "Computational Science" by David E. Keyes in Princeton Companion to Applied Mathematics
In the January/February 2000 issue of Computing in Science and Engineering, Jack Dongarra and Francis Sullivan chose the "10
algorithms with the greatest influence on the development and practice of science and engineering in the 20th century" and presented a group of articles on them that they had commissioned and edited. (A SIAM News article by Barry Cipra gives a summary for anyone who does not have access to the original articles). This top ten list has attracted a lot of interest.
Sixteen years later, I though it would be interesting to produce such a list in a different way and see how it compares with the original top ten. My unscientific—but well defined— way of doing so is to determine which algorithms have the most page locators in the index of The Princeton Companion to Applied Mathematics (PCAM). This is a flawed measure for several reasons. First, the book focuses on applied mathematics, so some algorithms included in the original list may be outside its scope, though the book takes a broad view of the subject and includes many articles about applications and about topics on the interface with other areas. Second, the content is selective and the book does not attempt to cover all of applied mathematics. Third, the number of page locators is not necessarily a good measure of importance. However, the index was prepared by a professional indexer, so it should reflect the content of the book fairly objectively.
A problem facing anyone who compiles such a list is to define what is meant by "algorithm". Where does one draw the line between an algorithm and a technique? For a simple example, is putting a rational function in partial fraction form an algorithm? In compiling the following list I have erred on the side of inclusion. This top ten list is in decreasing order of the number of page locators.
Newton and quasi-Newton methods
Matrix factorizations (LU, Cholesky, QR)
Singular value decomposition, QR and QZ algorithms
Monte-Carlo methods
Fast Fourier transform
Krylov subspace methods (conjugate gradients, Lanczos, GMRES,
minres)
JPEG
PageRank
Simplex algorithm
Kalman filter
Note that JPEG (1992) and PageRank (1998) were youngsters in 2000, but all the other algorithms date back at least to the 1960s.
By comparison, the 2000 list is, in chronological order (no other ordering was given)
Metropolis algorithm for Monte Carlo
Simplex method for linear programming
Krylov subspace iteration methods
The decompositional approach to matrix computations
The Fortran optimizing compiler
QR algorithm for computing eigenvalues
Quicksort algorithm for sorting
Fast Fourier transform
Integer relation detection
Fast multipole method
The two lists agree in 7 of their entries. The differences are:
PCAM list
2000 list
Newton and quasi-Newton methods
The Fortran Optimizing Compiler
Jpeg
Quicksort algorithm for sorting
PageRank
Integer relation detection
Kalman filter
Fast multipole method
Of those in the right-hand column, Fortran is in the index of PCAM and would have made the list, but so would C, MATLAB, etc., and I draw the line at including languages and compilers; the fast multipole method nearly made the PCAM table; and quicksort and integer relation detection both have one page locator in the PCAM index.
There is a remarkable agreement between the two lists! Dongarra and Sullivan say they knew that "whatever we came up with in the end, it would be controversial". Their top ten has certainly stimulated some debate, but I don't think it has been too controversial. This comparison suggests that Dongarra and Sullivan did a pretty good job, and one that has stood the test of time well.
Somewhere along my somewhat convoluted educational journey I encountered Latin rhetorical devices. At least one has become part of common usage–oxymoron, the apparent paradox created by juxtaposed words which seem to contradict each other; a classic example being 'awfully good'. For some reason, one of the devices that has stuck with me over the years is praeteritio, in which emphasis is placed on a topic by saying that one is omitting it. For instance, you could say that when one forgets about 9/11, the Iraq War, Hurricane Katrina, and the Meltdown, George W. Bush's presidency was smooth sailing.
I've always wanted to invent a word, like John Allen Paulos did with 'innumeracy', and πraeteritio is my leading candidate–it's the fact that we call attention to the overwhelming importance of the number π by deliberately excluding it from the conversation. We do that in one of the most important formulas encountered by intermediate algebra and trigonometry students; s = rθ, the formula for the arc length s subtended by a central angle θ in a circle of radius r.
You don't see π in this formula because π is so important, so natural, that mathematicians use radians as a measure of angle, and π is naturally incorporated into radian measure. Most angle measurement that we see in the real world is described in terms of degrees. A full circle is 360 degrees, a straight angle 180 degrees, a right angle 90 degrees, and so on. But the circumference of a circle of radius 1 is 2π, and so it occurred to Roger Cotes (who is he? I'd never heard of him) that using an angular measure in which there were 2π angle units in a full circle would eliminate the need for a 'fudge factor' in the formula for the arc length of a circle subtended by a central angle. For instance, if one measured the angle D in degrees, the formula for the arc length of a circle of radius r subtended by a central angle would be s = (π/180)rD, and who wants to memorize that? The word 'radian' first appeared in an examination at Queen's College in Belfast, Ireland, given by James Thomson, whose better-known brother William would later be known as Lord Kelvin.
The wisdom of this choice can be seen in its far-reaching consequences in the calculus of the trigonometric functions, and undoubtedly elsewhere. First semester calculus students learn that as long as one uses radian measure for angles, the derivative of sin x is cos x, and the derivative of cos x is – sin x. A standard problem in first-semester calculus, here left to the reader, is to compute what the derivative of sin x would be if the angle were measured in degrees rather than radians. Of course, the fudge factor π/180 would raise its ugly head, its square would appear in the formula for the second derivative of sin x, and instead of the elegant repeating pattern of the derivatives of sin x and cos x that are a highlight of the calculus of trigonometric functions, the ensuing formulas would be beyond ugly.
One of the simplest known formulas for the computation of π is via the infinite series ????4=1−13+15−17+⋯
This deliciously elegant formula arises from integrating the geometric series with ratio -x^2 in the equation 1/(1+????^2)=1−????2+????4−????6+⋯
The integral of the left side is the inverse tangent function tan-1 x, but only because we have been fortunate enough to emphasize the importance of π by utilizing an angle measurement system which is the essence of πraeteritio; the recognition of the importance of π by excluding it from the discussion.
So on π Day, let us take a moment to recognize not only the beauty of π when it makes all the memorable appearances which we know and love, but to acknowledge its supreme importance and value in those critical situations where, like a great character in a play, it exerts a profound dramatic influence even when offstage.
Pi Day, the annual celebration of the mathematical constant π (pi), is always an excuse for mathematical and culinary revelry in Princeton. Since 3, 1, and 4 are the first three significant digits of π, the day is typically celebrated on 3/14, which in a stroke of serendipity, also happens to be Albert Einstein's birthday. Pi Day falls on Monday this year, but Princeton has been celebrating all weekend with many more festivities still to come, from a Nerd Herd smart phone pub crawl, to an Einstein inspired running event sponsored by the Princeton Running Company, to a cocktail making class inside Einstein's first residence. We imagine the former Princeton resident would be duly impressed.
Einstein enjoying a birthday/ Pi Day cupcake
Pi Day in Princeton always includes plenty of activities for children, and tends to be heavy on, you guessed it, actual pie (throwing it, eating it, and everything in between). To author Paul Nahin, this is fitting. At age 10, his first "scientific" revelation was, If pi wasn't around, there would be no round pies! Which it turns out, is all too true. Nahin explains:
Everybody "knows'' that pi is a number a bit larger than 3 (pretty close to 22/7, as Archimedes showed more than 2,000 years ago) and, more accurately, is 3.14159265… But how do we know the value of pi? It's the ratio of the circumference of a circle to a diameter, yes, but how does that explain how we know pi to hundreds of millions, even trillions, of decimal digits? We can't measure lengths with that precision. Well then, just how do we calculate the value of pi? The symbol π (for pi) occurs in countless formulas used by physicists and other scientists and engineers, and so this is an important question. The short answer is, through the use of an infinite series expansion.
In his book In Praise of Simple Physics, Nahin shows you how to derive such a series that converges very quickly; the sum of just the first 10 terms correctly gives the first five digits. The English astronomer Abraham Sharp (1651–1699) used the first 150 terms of the series (in 1699) to calculate the first 72 digits of pi. That's more than enough for physicists (and for anybody making round pies)!
While celebrating Pi Day has become popular—some would even say fashionable in nerdy circles— PUP author Marc Chamberland points out that it's good to remember Pi, the number. With a basic scientific calculator, Chamberland's recent video "The Easiest Way to Calculate Pi" details a straightforward approach to getting accurate approximations for Pi without tables or a prodigious digital memory. Want even more Pi? Marc's book Single Digits has more than enough Pi to gorge on.
Now that's a sweet dessert.
If you're looking for more information on the origin of Pi, this post gives an explanation extracted from Joseph Mazur's fascinating history of mathematical notation, Enlightening Symbols.
You can find a complete list of Pi Day activities from the Princeton Tour Company here.
High school has been failing its students, according to James D. Stein, mostly by presenting to disinterested students an overwhelming mass of information that they aren't likely to find interesting or useful. As the author of L. A. Math: Romance, Crime, and Mathematics in the City of Angels, Stein is an expert at keeping subjects interesting for the most reluctant math students.
by James D. Stein
Let me start by repeating something I said in the last post. Where we've shortchanged students is at the secondary level. This is where I think we've lost sight of the purpose of education, which is to give students a broad general background in subjects deemed necessary but which they probably won't use, and to prepare them for life as a productive citizen. So here's what I'd recommend: revamp high school education to give students an enjoyable way to absorb a basic general background in subjects that they probably won't use later on, and find out what they find interesting and give them a full dose of that.
In 1961, Richard Feynman delivered an introductory lecture at Caltech in which he made the following oft-quoted statement. "."
Let's tweak what Feynman said a little.
If, in some cataclysm, all of the knowledge of humanity were to be destroyed, and only one book passed on to the next generation of creatures, what book would contain the most information about humanity in the fewest words? It would be a book summarizing the Top Ten most important achievements in the most important areas of natural science, social science, the humanities and history, ranked in order of importance by a panel of experts who have devoted their lives to the study of these subjects.
All of a sudden, acquiring a broad general background becomes both achievable and enjoyable – and in a reasonably short period of time. A basic education should tell you what's important in the important subjects —AND NOBODY KNOWS WHAT THEY ARE!!! Oh, sure, in the sciences you could probably come up with a fairly good list (although the ORDER of the items would not be known, and that's a key part of this idea) —but other than World Wars I and II, what are the important events in world history? How can we teach the important material in the important subjects, when we don't even have a consensus as to what they are?
And let's do it using the Top Ten format, because not only can we find out what are the most important achievements—which should form the basis for a broad general background—but because the Top Ten format is almost universally engaging. Publish a Top Ten list backed by experts, and you'll know you've got a reasonable approximation of the biggies. Moreover, Top Ten lists invite further study and critical thinking.
Just think of the following assignment in a high-school history course: using the Top Ten list in American history as a guide, construct your own Top Ten list of the ten most important events in American history, and justify your choices. I'm guessing that you'd see raging debates in the classroom, with teachers serving as enlightened moderators rather than just 'sages on the stage'. Maybe I'm overly optimistic, but instead of arguing about Top Ten football teams or Top Ten TV shows, you just might find students suddenly arguing about the relative importance of the Civil War and the American Revolution in American history. You might find students actually doing research to support their points of view. You'd find students thinking about important ideas, rather than memorizing stuff to regurgitate on standardized exams.
Two decades ago, Carl Sagan wrote The Demon-Haunted World: Science As a Candle in the Dark, in which he decried the deplorable lack of scientific knowledge in the general public. I'll bet if you simply had a list of the Top Ten achievements in physics, chemistry, biology, and mathematics, and if you taught that in a one-semester course, you'd have taken a giant step toward rectifying the problem that so concerned Sagan.
Almost every teacher in every subject feels the same way: students just don't know what's important. Let's find out what is the important stuff in the important subjects, and give every high school student an opportunity to acquire that knowledge—relatively quickly and enjoyably. And then let's get on with the business of enabling students to become productive members of society by enabling them to take courses at the high school level in what really interests them. It hurts me—a little—to say this, but if a student wants to become a video-game designer, I'd rather have them become a really good video-game designer than a barely passing algebra student. School should be a place where you go to help you fulfill your dreams. And I'm willing to bet you'd find a lot more students getting interested in science and history once they know what experts think is important—and once they've had an opportunity to think critically about it for themselves.
Time and technology have changed the education system, but James D. Stein insists that we still have room for improvement, particularly in how the mathematics curriculum is handled in high school. In his latest book, L.A. Math: Romance, Crime, and Mathematics in the City of Angels, Stein offers a unique approach that teaches mathematical techniques through liberal arts, making the subject more accessible to those who might otherwise avoid it. Today Stein discusses the challenge of providing students with a broad general background in subjects deemed necessary but which they probably won't pursue professionally.
Abraham Lincoln and American High Schools
by James D. Stein
February 12th was Lincoln's birthday. Like almost everyone in my generation, I was given the official story of Abraham Lincoln and the value of education. You probably know it, how Honest Abe, realizing at an early age the value of education, would trudge miles through snow-covered forest from his log cabin in order to attend school.
I have no doubt that he did indeed so trudge, but over the years I've become skeptical of this 'realizing at an early age the value of education' explanation. I think Abe, like the vast majority of children (and adults), was basically a pleasure-seeker. Put yourself in his shoes – no TV, no video games, no Facebook. Which is better – a lonely log cabin in the middle of the woods, or a small school, with other children and the opportunity to hear stories far more interesting than anything he could find at home? I'm guessing he went to school in large part because it was a lot more interesting than what he found at home.
Today, however, schools face a problem – its students DO have TV, video games, and Facebook – and they're stiff competition. Let's be honest with ourselves; although there are a few students who will find factoring polynomials as interesting as Facebook, most won't. And let's continue to be honest with ourselves; although students who plan on entering a career in a STEM subject – science, technology, engineering, mathematics – need to be familiar with algebra, the only time anyone else will encounter an algebra problem during the rest of their life is when one of their children asks them for help with algebra.
And what do we want then? We don't want both parents to tell their children that they had a really bad experience with math and don't remember anything, This is not likely to encourage the next generation to pursue the STEM subjects on which our future well-being as a society depends.
So, having cursed the darkness, let me try to light a candle. Our education system does a reasonable job at the primary school level. It's not perfect, but we do a pretty good job of teaching the three Rs in a highly diverse society. We also do a great job of education at the level of college and graduate school; after all, students come from all over the world to study at our institutions of higher learning, and generally the chief reason our college students go elsewhere is to participate in an exchange program.
Where we truly shortchange students is at the secondary level, where I think we've lost sight of the purpose of education – to give students a broad general background in subjects deemed necessary but which they probably won't use, and to prepare them for life as a productive citizen.
My only expertise is in mathematics, but as I look at the California Framework for Mathematics, insofar as it deals with the high school level, I'm thinking – will anyone other than STEM students use algebra, geometry, or trigonometry in later life? Or even statistics? Probably not. It would be helpful if they understood how statistics functions and what it is used for, rather than knowing how to compute a standard deviation or a confidence interval – which they'll almost certainly have forgotten within a year.
So here's what I'd recommend – revamp high school education to give students an enjoyable way to absorb a basic general background in subjects that they probably won't use later on, and find out what they find interesting and concentrate on doing a solid job of giving them a full dose of that. After all, that's what we do in college – except for the enjoyable part.
Stay tuned for Jim Stein's next post on how to give students an enjoyable way to absorb a general background.
I wondered if we had included these equations in The Princeton Companion to
Applied Mathematics (PCAM), specifically in Part III: Equations, Laws, and Functions of Applied Mathematics. We had indeed included the ones most
relevant to applied mathematics. Here are those equations, with links to the
BBC articles.
The wave equation (which quotes PCAM author Ian Stewart). PCAM has a short
article by Paul Martin of the same title (III.31), and the wave equation
appears throughout the book.
Einstein's field equation. PCAM has a 2-page article Einstein's Field
Equations (note the plural), by Malcolm MacCallum (article III.10).
The Euler-Lagrange equation. PCAM article III.12 by Paul Glendinning is about
these equations, and more appears in other articles, especially The
Calculus of Variations (IV.6), by Irene Fonseca and Giovanni Leoni.
The Dirac equation. A 3-page PCAM article by Mark Dennis (III.9) describes
this equation and its quantum mechanics roots.
The logistic map. PCAM article The logistic equation (III.19), by Paul
Glendinning treats this equation, in both differential and difference forms.
It occurs in several places in the book.
Bayes' theorem. This theorem appears in the PCAM article Bayesian Inference in Applied Mathematics (V.11), by Des Higham, and in other articles employing
Bayesian methods.
A natural equation is: Are there other worthy equations that are the
subject of articles in Part III of PCAM that have not been included in the BBC
list? Yes! Here are some examples (assuming that only single equations are
allowed, which rules out the Cauchy-Riemann equations, for example).
We're back with the conclusion to last week's LA Math challenge, The Case of the Vanishing Greenbacks, (taken from chapter 2 of the book). After the conclusion of the story, we'll talk a little more with the author, Jim Stein. Don't forget to check out the fantastic trailer for LA Mathhere.
Forty‑eight hours later I was bleary‑eyed from lack of sleep. I had made no discernible progress. As far as I could tell, both Stevens and Blaisdell were completely on the up‑and‑up. Either I was losing my touch, or one (or both) of them were wasting their talents, doctoring books for penny‑ante amounts. Then I remembered the envelope Pete had sealed. Maybe he'd actually seen something that I hadn't.
I went over to the main house, to find Pete hunkered down happily watching a baseball game. I waited for a commercial break, and then managed to get his attention.
"I'm ready to take a look in the envelope, Pete."
"Have you figured out who the guilty party is?"
"Frankly, no. To be honest, it's got me stumped." I moved to the mantel and unsealed the envelope. The writing was on the other side of the piece of paper. I turned it over. The name Pete had written on it was "Garrett Ryan and the City Council"!
I nearly dropped the piece of paper. Whatever I had been expecting, it certainly wasn't this. "What in heaven's name makes you think Ryan and the City Council embezzled the money, Pete?"
"I didn't say I thought they did. I just think they're responsible for the missing funds."
I shook my head. "I don't get it. How can they be responsible for the missing funds if they didn't embezzle them?"
"They're probably just guilty of innumeracy. It's pretty common."
"I give up. What's innumeracy?"
"Innumeracy is the arithmetical equivalent of illiteracy. In this instance, it consists of failing to realize how percentages behave." A pitching change was taking place, so Pete turned back to me. "An increase in 20% of the tax base will not compensate for a reduction of 20% in each individual's taxes. Percentages involve multiplication and division, not addition and subtraction. A gain of 20 dollars will compensate for a loss of 20 dollars, but that's because you're dealing with adding and subtracting. It's not the same with percentages, because the base upon which you figure the percentages varies from calculation to calculation."
"You may be right, Pete, but how can we tell?"
Pete grabbed a calculator. "Didn't you say that each faction was out $198,000?"
I checked my figures. "Yeah, that's the amount."
Pete punched a few numbers into the calculator. "Call Ryan and see if there were 99,000 taxpayers in the last census. If there were, I'll show you where the money went."
I got on the phone to Ryan the next morning. He confirmed that the tax base in the previous census was indeed 99,000. I told Pete that it looked like he had been right, but I wanted to see the numbers to prove it.
Pete got out a piece of paper. "I think you can see where the money went if you simply do a little multiplication. The taxes collected in the previous census were $100 for each of 99,000 individuals, or $9,900,000. An increase of 20% in the population results in 118,800 individuals. If each pays $80 (that's the 20% reduction from $100), the total taxes collected will be $9,504,000, or $396,000 less than was collected after the previous census. Half of $396,000 is $198,000."
I was convinced. "There are going to be some awfully red faces down in Linda Vista. I'd like to see the press conference when they finally announce it." I went back to the guesthouse, called Allen, and filled him in. He was delighted, and said that the check would be in the mail. As I've said before, when Allen says it, he means it. Another advantage of having Allen make the arrangements is that I didn't have to worry about collecting the fee, which is something I've never been very good at.
I wondered exactly how they were going to break the news to the citizens of Linda Vista that they had to pony up another $396,000, but as it was only about $3.34 per taxpayer I didn't think they'd have too much trouble. Thanks to a combination of Ryan's frugality and population increase, the tax assessment would still be lower than it was after the previous census, and how many government agencies do you know that actually reduce taxes? I quickly calculated that if they assessed everyone $3.42 they could not only cover the shortage, but Allen's fee as well. I considered suggesting it to Ryan, but then I thought that Ryan probably wasn't real interested in hearing from someone who had made him look like a bungler.
My conscience was bothering me, and I don't like that. I thought about it, and finally came up with a compromise I found acceptable. I went back to the main house.
Pete was watching another baseball game. The Dodgers fouled up an attempted squeeze into an inning‑ending double play. Pete groaned. "It could be a long season," he sighed.
"It's early in the year." I handed him a piece of paper. "Maybe this will console you."
"That embezzling case in Orange County. It was worth $3,500 to me to come up with the correct answer. I feel you're entitled to half of it. You crunched the numbers, but I had the contacts and did the legwork."
Pete looked at the check. "It seems like a lot of money for very little work. Tell you what. I'll take $250, and credit the rest towards your rent."
A landlord with a conscience! Maybe I should notify the Guinness Book of Records. "Seems more than fair to me."
Pete tucked the check in the pocket of his shirt. "Tell me, Freddy, is it always this easy, doing investigations?"
I summoned up a wry laugh. "You've got to be kidding. So far, I've asked you two questions that just turned out to be right down your alley. I've sometimes spent months on a case, and come up dry. That can make the bottom line look pretty sick. What's it like in your line of work?"
"I don't really have a line of work. I have this house and some money in the bank. I can rent out the guesthouse and make enough to live on. People know I'm pretty good at certain problems, and sometimes they hire me. If it looks like it might be interesting, I'll work on it." He paused. "Of course, if they offer me a ridiculous amount of money, I'll work on it even if it's not interesting. Hey, we're in a recession."
"I'll keep that in mind." I turned to leave the room. Pete's voice stopped me.
"Haven't you forgotten something?"
I turned around. "I give up. What?"
"We had a bet. You owe me five bucks."
I fished a five out of my wallet and handed it over. He nodded with satisfaction as he stuffed it in the same pocket as the check, and then turned his attention back to the game.
What made you include this particular idea in the book?
JS: The story features one of the most common misunderstandings about percentages. There are innumerable mistakes made because people assume that percentages work the same way as regular quantities. But they don't — if a store lowers the cost of an item by 30% and then by another 20%, the cost of the item hasn't been lowered by 50% — although many people make the mistake of assuming that it has. I'm hoping that the story is sufficiently memorable that if a reader is confronted by a 30% discount followed by a 20% discount, they'll think "Wasn't there something like that in The Case of the Vanishing Greenbacks?
There are 14 stories in the book, and each features a mathematical point, injected into the story in a similar fashion as the one above. I think the stories are fun to read, and if someone reads the book and remembers just a few of the points, well, I've done a whole lot better than when I was teaching liberal arts math the way it is usually done.
James D. Stein is emeritus professor in the Department of Mathematics at California State University, Long Beach. His books include LA Math, Cosmic Numbers (Basic) and How Math Explains the World (Smithsonian).
If you caught the rather incredible trailer for L.A. Math, you know it's not your typical scholarly math book. Romance, crime, and mathematics don't often go hand in hand, but emeritus professor in the Department of Mathematics at California State University Jim Stein cooked up the idea for an unconventional literary math book that would speak to students in his liberal arts math class. The end result is an entertaining, backdoor approach to practical mathematics knowledge, ranging from percentages and probability to set theory, statistics, and the mathematics of elections. Recently, Stein spoke to us about writing L.A. Math. Not only that, he left us with a mathematical mystery to solve.
L.A. Math is definitely an unusual book. Brian Clegg described it by saying "It's as if Ellery Queen, with the help of P. G. Wodehouse, spiced up a collection of detective tales with a generous handful of practical mathematics." How did you happen to write it?
JS: I absolutely loved it when he described it that way, because I was brought up on Ellery Queen. For younger readers, Ellery Queen was one of the greatest literary detectives of the first half of the twentieth century, specializing in classic Sherlock Holmes type cases. The Ellery Queen stories were written by the team of Manfred Dannay and Frederick Lee — and my mother actually dated one of them!
The two other mystery writers who influenced me were Agatha Christie and Rex Stout. Rex Stout wrote a series featuring Nero Wolfe and Archie Goodwin; the books are presumably written by Archie Goodwin describing their cases, so I used that as the model for Freddy Carmichael. The relationship between Archie and Nero also served, somewhat, as a parallel for the relationship between Freddy and Pete. Nero and Pete both have addictions — Nero wants to spend his time eating elaborate cuisine and raising orchids, and Pete wants to spend his time watching and betting on sports. It's up to Archie and Freddy to prod them into taking cases.
How does Agatha Christie enter the picture?
JS: I'd taught liberal arts mathematics — math for poets — maybe ten times with temporary success but no retention. Students would learn what was necessary to pass the course, and a year later they'd forgotten all of it. That's not surprising, because the typical liberal arts math course has no context that's relevant for them. They're not math-oriented. I know I had several history courses discussing the Battle of Azincourt, but I don't remember anything about it because it has no context for me.
Agatha Christie's best-known detective is Hercule Poirot, and one day I was in a library reading a collection of short stories she had written entitled The Labors of Hercules. Christie had a background in the classics, and did something absolutely brilliant — she constructed a series of twelve detective stories featuring Hercule Poirot, each of which was modeled, in one way or another, around the Twelve Labors of Hercules in classical mythology. I thought to myself — why don't I do something like that for topics in liberal arts math? Maybe the students would remember a few of the ideas because they'd have the context of a story from which to remember it.
Could you give an example?
JS: How about this? Why don't we take a story from the book, and present it the way Ellery Queen would have. Ellery Queen always played fair with the reader, giving him or her all the clues, and after all the clues had been presented, EQ would write a paragraph entitled "Challenge to the Reader". EQ would tell the reader "Now you have all the clues. Can you figure out whodunit?" — or words to that effect.
OK, here's what we'll do. We'll take The Case of the Vanishing Greenbacks, Chapter 2 in L.A. Math, and present the story up to the crucial point. Then we'll let the reader try to figure out whodunit, and finish the story next week.
Chapter 2 – The Case of the Vanishing Greenbacks
The phone rang just as I stepped out of the shower. It was Allen.
"Freddy, are you available for an embezzlement case?"
My biggest success had been in an embezzlement case involving a Wall Street firm specializing in bond trading. Allen had given me a whopping bonus for that one, which was one of the reasons I could afford to take it easy in L.A. I had done well in a couple of other similar cases, and had gotten the reputation of being the go-to guy in embezzlement cases. It never hurts to have a reputation for being good at something. Besides, you don't see many guys in my line of work who can read balance sheets.
I've always felt it's important to keep the cash flow positive, and the truth was that I was available for a jaywalking case if it would help the aforementioned cash flow. But it never hurts to play a little hard-to-get.
"I can probably clear my calendar if it looks interesting."
Allen paused for a moment, either to collect his thoughts or to take a bite of one of those big greasy pastrami sandwiches he loves. "I'm pretty sure you'll find it interesting. It's stumped some people in L.A., and I told them I had a good man out there. BTW, that's you."
It's nice to be well thought of – especially by someone in a position to send you business. I knew that Allen's firm, though headquartered in New York, had arrangements with other firms in other cities. I didn't really care about the details as long as the check cleared – which it always had.
"I'm certainly willing to listen. What's the arrangement?"
"Consulting and contingency fee. Fifty‑fifty split."
That was our usual arrangement. Burkitt Investigations got a guaranteed fee, plus a bonus for solving the case. Allen and I split it down the middle.
"OK, Allen, fill me in."
"Ever heard of Linda Vista, Freddy?"
Temporary blank. Movie star? Socialite? Then I had it. Linda Vista was a town somewhere in Orange County with a big art community.
For those of you not up on California politics, Orange County is a bastion of conservatism. You have Orange County to thank, or blame, for Richard Nixon and Ronald Reagan. But Linda Vista, which my fragmentary Spanish translates as "pretty view", was different from your basic Orange County bastion.
The vista in Linda Vista was sufficiently linda that it had attracted a thriving artistic community. There were plenty of artists in Linda Vista, and most of them were liberals.
As a result, Linda Vista was highly polarized. The moderates were few and far between. On the left, you had the artists, with their funky bungalows and workshops. On the right, you had the stockbrokers and real-estate moguls, living in gated communities so they wouldn't have to have any contact with the riff-raff, except for the tradesmen delivering or repairing stuff. However, there were enough artists and hangers-on to acquire political clout – after all, it's still one man-one vote in a democracy, rather than one dollar-one vote. Pitched battles had raged over practically every issue from A (abortion) to Z (zoning), and many of these battles had made state and even national news.
That's all I knew about Linda Vista, other than not to try to drive down there at rush hour, which turned one hour on the 405 to more than twice that. The obvious question was: what kind of a contingency case had they got? So I asked it.
Allen filled me in. "The city is out a bunch of bucks, and each side is accusing the other of fraud and embezzlement. Because of the split in the political situation, the City Manager gave half the budget to the conservatives, and the other half to the liberals, letting each determine how to spend its half. Both sides claim to have been shortchanged."
Allen paused to catch his breath. "I've got a friend who works in the City Manager's office. I told him I had a good man out there who'd done a lot of first‑class work in embezzlement cases. Want to take a look at it?"
"Sure. How much time should I put in before I throw in the towel?" In other words, how much is the consulting fee?
"As much as you like." In other words, since Allen's meter wasn't running, feel free to burn some midnight oil. "The consulting fee is $3,000, upped to ten if you figure it out and get proof." You don't have to be an expert at division to realize that I was guaranteed a minimum of $1,500 for the time I put in, and $5,000 if I doped it out. You also don't have to be an expert at division to realize that Allen was getting the same amount for making a phone call. I decided to be reincarnated as an employer rather than an employee.
Allen gave me a brief description of the protagonists, and I spent a good portion of the evening with a pot of coffee and my computer, getting some background information on them. I'll say one thing for the Information Age; it's a lot easier to run a background check on people than it used to be. What with search engines and social networks, you save a lot on gas money and shoe leather.
The next morning I waited until after rush hour, and made the trek to Linda Vista. The City Hall was located in a section of town where the vista was a long way from linda, unless strip malls filled with 7‑11s and fast-food stores constitute your idea of attractive scenery. I found a place to park, straightened my coat and tie, and prepared for the interviews.
I was scheduled to have three of them. I had been hoping to arrange for longer interviews, but everyone's in a rush nowadays, and I was getting a quarter-hour with each, tops. They'd all been interviewed previously – Allen had mentioned that this case had stumped others – and people are generally less than enthusiastic about being asked the same questions again. And again. The first interview was with Everett Blaisdell, conservative city councilman, who would explain why the conservatives happened to be short. The next was with Melanie Stevens, liberal city councilwoman, ditto. The last interview would be with Garrett Ryan, City Manager.
I have a bad habit. My opinion of members of groups tends to be formed by the members of those groups that I have seen before. Consequently, I was expecting the conservative Everett Blaisdell to look like a typical paunchy southern senator with big jowls. So I was a little surprised to discover that Everett Blaisdell was a forty-ish African-American who looked like he had spent years twenty through thirty as an NBA point guard.
He got right down to business. "I want you to know," he barked, "that everything that we have done with our budget allocation has been strictly by the book. Our expenses have been completely documented." He handed me a folder full of ledger sheets and photos of checks, which I glanced at and stashed in my briefcase.
Blaisdell was clearly angry. "The business community is the heart of Linda Vista, and it is ridiculous to suggest that it would act in a manner detrimental to its citizens. We are $198,000 short in our budget."
You don't expect NBA point guards to get out of breath too easily, considering the time they have to go up and down the court, but maybe Blaisdell wasn't in shape. He paused, giving me a chance to get a question in edgewise. "Just what do you think has happened, Mr. Blaisdell?" I inquired mildly.
"I know what has happened. Melanie Stevens and her radical crowd have managed to get hold of that money. They want $200,000 to fund a work of so‑called art which I, and every right‑thinking citizen of Linda Vista, find totally offensive. It's mighty suspicious that the missing funds, $198,000, almost precisely cover the projected cost of the statue."
I was curious. "If you don't mind my asking, exactly what is this statue?"
Blaisdell's blood pressure was going up. "They are going to build a scale replica of the Statue of Liberty and submerge it in Coca‑Cola. You may know that Coca‑Cola is acidic, and it will eventually dissolve metal. They say that this so‑called dynamic representational art represents the destruction of our civil liberties by over‑commercialization. Well, let me tell you, we'll fight it."
He looked at his watch. "Sorry, I've got another appointment. When you find out what those scum have done with the money, let me know." He walked me to his door.
It took a few minutes to locate Melanie Stevens' office, as it was in a different wing of the building, possibly to minimize confrontations between her and Blaisdell. It was a bad day for stereotypes. My mental picture of Melanie Stevens, ultra‑liberal, was that of a long-haired hippie refugee from the '60s. The real Melanie Stevens was a pert gray‑haired grandmother who looked like she had been interrupted while baking cookies for her grandchildren. She, too, was evidently on a tight schedule, for she said, "Sorry, I can only give you about ten minutes, but I've made copies of all our expenses." More ledger sheets and photos of checks went into my briefcase.
"Let me tell you, Mr. Carmichael, that we could have used that $198,000. We planned to use it for a free clinic. I know exactly what has happened. Blaisdell has doctored the books. I'm sure glad that Ryan had the guts to ask you to look into it."
"Blaisdell seems to think that your people are responsible for the missing funds," I observed.
She snorted. "That's just typical of what they do. Whenever they're in the wrong, they lie and accuse the other side of lying. They rip off the community, and channel money into PACs. Political action committees. Or worse. Blaisdell knows he faces a stiff battle for re-election, and I wouldn't be the least bit surprised to find that money turning up in his campaign fund."
"He seems to think that you are going to use the funds for an art project, rather than a free clinic," I remarked.
"He's just blowing smoke. He knows quite well that the statue will be funded through private subscription." She looked at her watch. "Let me know when you pin the loss on them."
I left Stevens' office for the last interview, with Garrett Ryan, whose anxious expression made it clear that he was not a happy camper. "Have you got any ideas yet?" he asked.
I shook my head. "I've just talked to Blaisdell and Stevens. They've each handed me files containing what they consider to be complete documentation. They've each given me a story asserting their own innocence, and blaming the other. I take it that the missing amount is $198,000?"
Now it was Ryan's turn to shake his head. "No, each side says that it is missing $198,000. Quite a coincidence. And I'll tell you, Mr. Carmichael, despite the animosity between them, I think that they are both honorable individuals. I find it difficult to believe that either would rip the city off."
I focused on Ryan's coincidence. "It's funny that they are both short exactly the same amount. Perhaps you could tell me a little more about the budgetary process."
"It's really quite simple. Each resident of Linda Vista is taxed a fixed amount. Any complicated tax scheme would just result in a full employment act for accountants. The previous census resulted in a $100 assessment per individual. The population of Linda Vista increased by 20% since the last census. We didn't need any increase in operating expenditures; under my guidance we've done a fiscally conservative and frugal job of running the city. As a result, the Council voted to reduce everybody's taxes by 20%. Needless to say, this was a very popular move."
"I'll bet it was. Did everyone pay their taxes, Mr. Ryan?"
"Everybody. We're very proud of that ‑‑ a 100% collection rate. Despite what you may have heard, the citizens of Linda Vista are very civic‑minded. Liberals and conservatives alike."
I've spent enough time with balance sheets to know that accuracy is extremely important. "Was this population increase exactly 20%, or is that merely an approximate figure?"
Ryan consulted a sheet of paper. "Exactly 20%. I have a sheet of printout that gives information to four decimal places, so I can be quite sure of that."
Just then a phone rang. Ryan picked it up, and engaged in some political doubletalk. After a few minutes he replaced the receiver. "Sorry, Mr. Carmichael. I'm behind schedule. Let me know if you make any progress." We shook hands, and I left.
A couple of hours later, I got home, having stopped for a bite but still avoiding rush-hour traffic. Pete noticed my presence, and asked, "So how'd things go in Linda Vista, Freddy?"
"I had a pretty interesting day. Want to hear about it?"
He nodded. I took about fifteen minutes to describe the problem and the cast of characters. "It looks like I'll have to spend a day or so looking over the books."
Pete shook his head. "It seems pretty clear to me."
I'd seen it before — everybody's a detective. Amateurs always think they know who the guilty party is, because it fits in with their preconceptions. I didn't know whether Pete had cast Blaisdell in the role of a political fat-cat out to line his campaign war chest, or whether he was a conservative who saw Melanie Stevens as a radical troublemaker. Anyway, you've got to learn not to jump to conclusions in my line of work.
"You can't do it like that, Pete. You've got to trace down the paper trails. I've done this lots of times."
Pete grabbed a piece of paper, scribbled something on it, and sealed it in an envelope. "Five dollars will get you twenty that the name of the guilty party is inside this envelope."
Pete needed taking down a peg. Maybe two pegs. Besides, I liked getting four‑to‑one odds on what was obviously an even‑ money proposition. "You've got a bet," I said. We wrote our names on the envelope, and Pete put it on the table next to the HDTV.
"Whenever you're ready, we'll unseal the envelope." I headed back to the guesthouse for a session with the books.
Challenge to the Reader: You have all the clues. Can you name the party responsible for the missing greenbacks? We'll give you until the next blog to figure it out, when we'll present the conclusion to the story | 677.169 | 1 |
The goal of this series is to provide readers with a strong foundation in Algebra. Each book is designed to develop readers' critical thinking and problem-solving capabilities and prepare readers for subsequent Algebra courses as well as "service" math courses. Topics are presented in an interesting and inviting format, incorporating real world sourced data and encouraging modeling and problem-solving. Algebra and Problem Solving. Functions, Linear Functions, and Inequalities. Systems of Linear Equations and Inequalities. Polynomials, Polynomial Functions, and Factoring. Rational Expressions, Functions, and Equations. Radicals, Radical Functions, and Rational Exponents. Quadratic Equations and Functions. Exponential and Logarithmic Functions. Conic Sections and Nonlinear Systems of Equations. Sequences, Series, and the Binomial Theorem. For anyone interested in introductory and intermediate algebra and for the combined introductory and intermediate algebra.
"synopsis" may belong to another edition of this title.
Product Description:
Including a variety of exercises, this algebra text discusses using technology and offers study tips. Some activities feature related websites.
From the Publisher:
This text covers traditional material and integrates it into the whole spectrum of learning. It contains a vast collection of historical references, interdisciplinary applications, enrichment essays, thought-provoking exercises, and well- written word problems. | 677.169 | 1 |
A Level Further Mathematics, sequences and series, trigonometry, and the formation of differentiation and integration. Unit 3 & 4: Core 3 and 4 - these units develop the pure mathematical techniques from AS Level and introduce practical applications such as differential equations. New topics include numerical methods and vectors. Unit 5: Statistics 1 - students work with real data sets extending the work they have covered in GCSE Maths, such as the calculation of the numerical measures mean, median and mode, and the practical applications of correlation and regression. Elementary probability theory is also studied, and the Binomial and Normal distributions are introduced. Unit 6: Mechanics 1 - this unit introduces mathematical modelling in physical situations, and studies motion in one or two dimensions (including the constant acceleration equations), forces on static objects, Newton's Law of Motion, momentum and projectiles. Year 13 Students study a further 6 units to gain A Level Further Mathematics. These units will be chosen from a range of pure, statistics, mechanics and decision modules. Assessment No Coursework Examination (100%) Examining Board - AQA. Prohibited Options Statistics. Career and Progression Opportunities Students with Further Mathematics commonly take up top professional careers in a wide variety of areas. These include Insurance and Actuarial Work, Finance, Management, Operational Research, Civil, Mechanical and Electrical Engineering, Architecture, Government, Medical or Pharmaceutical Statisticians, Business Analysts and others. This is a valuable option for students considering mathematics, Physics or Engineering degrees. Other Information None. Thsi information is correct for September 2010 entry. Further Mathematics
Courses Details
Study Mode:
Contact Provider
Location
Widney Manor Road,Solihull
Contact
Get an offer
Register now to get scholarships and offers from The Sixth Form College Solihull
Get an offer
Get an offer : The Sixth Form College Solihull regarding A Level Further Mathematics | 677.169 | 1 |
math algebra . Handy helper
The
internet can be a powerful ally when you are trying to figure out how
to solve a problem. To help you understand topics that are confusing to
you, each Section in MyMathLab has a link called Other Handy Helpers.
This takes you to a page with links to videos, math sites, PowerPoints,
cheat sheets and other materials that explain the topics covered in
that section. | 677.169 | 1 |
Introduction to Complex Analysis | UCLA Extension
Dr. Michael Miller has announced his Autumn mathematics course, and it is…
Introduction to Complex Analysis
Course Description
Complex analysis is one of the most beautiful and useful disciplines of mathematics, with applications in engineering, physics, and astronomy, as well as other branches of mathematics. This introductory course reviews the basic algebra and geometry of complex numbers; develops the theory of complex differential and integral calculus; and concludes by discussing a number of elegant theorems, including many–the fundamental theorem of algebra is one example–that are consequences of Cauchy's integral formula. Other topics include De Moivre's theorem, Euler's formula, Riemann surfaces, Cauchy-Riemann equations, harmonic functions, residues, and meromorphic functions. The course should appeal to those whose work involves the application of mathematics to engineering problems as well as individuals who are interested in how complex analysis helps explain the structure and behavior of the more familiar real number system and real-variable calculus.
Prerequisites
Basic calculus or familiarity with differentiation and integration of real-valued functions.
I often recommend people to join in Mike's classes and more often hear the refrain: "I've been away from math too long", or "I don't have the prerequisites to even begin to think about taking that course." For people in those categories, you're in luck! If you've even had a soupcon of calculus, you'll be able to keep up here. In fact, it was a similar class exactly a decade ago by Mike Miller that got me back into mathematics. (Happy 10th math anniversary to me!)
Textbook
(Note that there's another introductory complex analysis textbook from Silverman that's offered through Dover, so be sure to choose the correct one.)
As always in Dr. Miller's classes, the text is just recommended (read: not required) and in-class notes are more than adequate. To quote him directly, "We will be using as a basic guide, but, as always, supplemented by additional material and alternate ways of looking at things."
The bonus surprise of his email: He's doing two quarters of Complex Analysis! So we'll be doing both the Fall and Winter Quarters to really get some depth in the subject!
Alternate textbooks
If you're like me, you'll probably take a look at some of the other common (and some more advanced) textbooks in the area. Since I've already compiled a list, I'll share it: | 677.169 | 1 |
Overview Math 5/4 is a balanced, integrated mathematics program that includes incremental development of whole number concepts and computation; arithmetic algorithms, geometry and measurement; elapsed time; fractions, decimals and percents; powers and roots; estimation; patterns and sequences; congruency and similarity; and statistics and probability.
The solutions manual includes full step-by-step solutions for all lesson and investigation problems and for all 23 cumulative tests. Also includes answers to Supplemental Practice Problems and Facts Practice Saxon Math Homeschool 5 / 4: Solutions Manual by Stephen Hake today - and if you are for any reason not happy, you have 30 days to return it. Please contact us at 1-877-205-6402 if you have any questions.
This is the essential solutions manual, only for the Homeschool Edition. I am not sure why this is not clearer above, but it contains all the solutions for the Lessons, Investigations, Supplemental Practice and Tests. It's a great book!
Misidentified May 30, 2007
What I thought I was ordering was the textbook. What I received was the "Solutions Manual." This book listing is either misidentified or the title is incomplete. It's possible that the discriminator is "Paperback" vs "Hardcover" but, as a long time purchaser of books, this is not something that is common. I recommend not buying this book until this is cleared | 677.169 | 1 |
Calculator Policy
Exam Day 2017
Exam Resources
You can use a graphing calculator on Section I, Part B and Section II, Part A of the AP Calculus BC Exam since questions in those parts of the exam require use of the calculator to answer. See the list of approved graphing calculators (which includes a list of devices that are not allowed).
Bring a calculator you are familiar with. It is a good idea to bring extra batteries. You may bring up to two graphing calculators.
Throughout the course, use your calculator on a regular basis so that you're comfortable with it on exam day.
Graphing Calculator Capabilities
A graphing calculator appropriate for use on the exam is expected to have the built-in capability to:
Plot the graph of a function within an arbitrary viewing window
Find the zeros of functions (solve equations numerically)
Numerically calculate the derivative of a function
Numerically calculate the value of a definite integral
The AP Program ensures that the exam questions do not favor students who use graphing calculators with more extensive built-in features.
Showing Work on the Free-Response Sections
You are expected to show enough of your work for AP Readers, the high school teachers and college faculty that are scoring the AP Exams, to follow your line of reasoning. To obtain full credit for the solution to a free-response problem, communicate your methods and conclusions clearly. Answers should show enough work so that the reasoning process can be followed throughout the solution. This is particularly important for assessing partial credit. You may also be asked to use complete sentences to explain or justify your methods or the reasonableness of your answers, or to interpret your results.
For results obtained using one of the four required calculator capabilities listed above, you are required to write the setup (e.g., the equation being solved, or the derivative or definite integral being evaluated) that leads to the solution, along with the result produced by the calculator. For example, if you are asked to find the area of a region, you are expected to show a definite integral (i.e., the setup) and the answer. You need not compute the antiderivative; the calculator may be used to calculate the value of the definite integral without further explanation. For solutions obtained using a calculator capability other than one of the four required ones, you must also show the mathematical steps that lead to the answer; a calculator result is not sufficient. For example, if you are asked to find a relative minimum value of a function, you are expected to use calculus and show the mathematical steps that lead to the answer. It is not sufficient to graph the function or use a built-in minimum finder.
Justifications must include mathematical reasons, not merely calculator results. Functions, graphs, tables, or other objects that are used in a justification should be clearly identified.
Exploration Versus Mathematical Solution
A graphing calculator is a powerful tool for exploration, but please remember that exploration is not a mathematical solution. Exploration with a graphing calculator can lead you toward an analytical solution, and after a solution is found, a graphing calculator can often be used to check the reasonableness of the solution.
Note: As on previous AP Calculus Exams, a decimal answer must be correct to three decimal places after the decimal point unless otherwise indicated. You should not round values in intermediate steps before a final answer is presented. And be aware that there are limitations inherent in graphing calculator technology; for example, answers obtained by tracing along a graph to find roots or points of intersection might not produce the required accuracy. | 677.169 | 1 |
Introduction to Analysis
Overview system. Because of its clarity, simplicity of exposition, and stress on easier examples, this material is accessible to a wide range of students, of both mathematics and other fields. Chapter headings include notions from set theory, the real number system, metric spaces, continuous functions, differentiation, Riemann integration, interchange of limit operations, the method of successive approximations, partial differentiation, and multiple integrals. Following some introductory material on very basic set theory and the deduction of the most important properties of the real number system from its axioms, Professor Rosenlicht gets to the heart of the book: a rigorous and carefully presented discussion of metric spaces and continuous functions, including such topics as open and closed sets, limits and continuity, and convergent sequence of points and of functions. Subsequent chapters cover smoothly and efficiently the relevant aspects of elementary calculus together with several somewhat more advanced subjects, such as multivariable calculus and existence theorems. The exercises include both easy problems and more difficult ones, interesting examples and counter examples, and a number of more advanced results. Introduction to Analysis lends itself to a one- or two-quarter or one-semester course at the undergraduate level. It grew out of a course given at Berkeley since 1960. Refinement through extensive classroom use and the author's pedagogical experience and expertise make it an unusually accessible introductory text.
Customer Reviews
Most Helpful Customer Reviews
The was written back in the 1960s, and yet, it is still one of the best. The treatment of metric spaces and elementary topology is extremely well done. Continuity and differentiation are great too. Multiple integration gets a bit involved as notation gets more complex. Overall, it is a great book (especially after you consider the price - under $15), and there are plenty of good counterexamples and exercises. Do not hesitate to buy this good. Great value - highly recommended. | 677.169 | 1 |
Advanced Calculus/i>
Overview unified introduction to advanced calculus. Starting with an abstract treatment of vector spaces and linear transforms, the authors introduce a single basic derivative in an invariant form. All other derivatives — gradient, divergent, curl, and exterior — are obtained from it by specialization. The corresponding theory of integration is likewise unified, and the various multiple integral theorems of advanced calculus appear as special cases of a general Stokes formula. The text concludes by applying these concepts to analytic functions of complex variables.
Dover Publications, Inc.
1.1. Definition. A vector space V is a set, whose elements are called vectors, together with two operations. The first operation, called addition, assigns to each pair of vectors A, B a vector, denoted by A + B, called their sum. The second operation, called multiplication by a scalar, assigns to each vector A and each real number x a vector denoted by xA. The two operations are required to have the following eight properties:
Axiom 1. A + B = B + A for each pair of vectors A, B. (I.e. addition is commutative.)
1.2. Definition. The difference A - B of two vectors is defined to be the sum A + (-B).
The subsequent development of the theory of vector spaces will be based on the above axioms as our starting point. There are other approaches to the subject in which the vector spaces are constructed. For example, starting with a euclidean space, we could define a vector to be an oriented line segment. Or, again, we could define a vector to be a sequence (x1, ..., xn) of n real numbers. These approaches give particular vector spaces having properties not possessed by all vector spaces. The advantages of the axiomatic approach are that the results which will be obtained apply to all vector spaces, and the axioms supply a firm starting point for a logical development.
§2. Redundancy
The axioms stated above are redundant. For example the word "unique" in Axiom 3 can be omitted. For suppose [??] and [??]' are two vectors satisfying [??] + A = A and [??]' + A = A for every A. In the first identity, take A = [??]'; and in the second, take A = [??]. Using Axiom 1, we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This proves the uniqueness.
The word "unique" can likewise be omitted from Axiom 4.
For suppose A, B, C are three vectors such that
A + B = [??], and A + C = [??].
Using these relations and Axioms 1, 2 and 3, we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Therefore B = C, and so there can be at most one candidate for -A.
The Axiom 8 (i) is a consequence of the preceding axioms:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
§3. Cartesian spaces
3.1. Definition. The cartesian k-dimensional space, denoted by Rk, is the set of all sequences (a1, a2, ..., ak) of k real numbers together with the operations
Let [??] = (0, 0, ..., 0) be the sequence each of whose components is zero. Since 0 + ai = ai, it follows that [??] + A = A. This proves Axiom 3 since the uniqueness part of the axiom is redundant (see §2).
If A = (a1, a2, ..., ak), define - A to be (-a1, -a2, ..., -ak). Then A + (-A) = [??]. This proves Axiom 4 (uniqueness is again redundant).
If x is a real number, the ith component of x(A + B) is, by definition, x(ai + bi); and that of xA + xB is, by definition, xai + xbi. Thus the distributive law for real numbers implies Axiom 5.
The verifications of Axioms 6, 7 and 8 are similar and are left to the reader.
§4. Exercises
1. Verify that x Rk satisfies Axioms 6, 7 and 8.
2. Prove that Axiom 8(iii) is redundant. Show also that (-x)A = -(xA) for each x and each A.
3. Show that Axiom 8(ii) is not a consequence of the preceding axioms by constructing a set with two operations which satisfy the preceding axioms but not, 8(ii). (Hint: Consider the real numbers with multiplication redefined by xy = o for all x and y.) Can such an example satisfy Axiom 8(iii)?
4. Show that A + A = 2A for each A.
5. Show that x[??] = [??] for each x.
6. If x ≠ o and xA = [??], show that A = [??].
7. If x and A are such that xA = [??], show that either x = o or x = 0 or A = [??].
8. Show that the set consisting of a single vector [??] is a vector space.
9. If a vector A is such that A = -A, then A = [??].
10. If a vector space contains some vector other than [??], show that it contains infinitely many distinct vectors. (Hint: Consider A, 2A, 3A, etc.)
11. Let D be any non-empty set, and define RD to be the set of all functions having domain D and values in R. If f and g are two such functions, their sum f + g is the element of RD defined by
(f + g)(d) = f(d) + g(d)for each d in D.
If f is in RD and x is a real number, let xf be the element of RD defined by
(xf)(d) = xf(d)for each d in D.
Show that RD is a vector space with respect to these operations.
12. Let V be a vector space and let D be a nonempty set. Let VD the set of all functions with domain D and values in V. Define sum and product as in Exercise 11, and show that VD is a vector space.
13. A sum of four vectors A + B + C + D may be associated (parenthesized) in five ways, e.g. (A + (B + C)) + D. Show that all five sums are equal, and therefore A + B + C + D makes sense without parentheses.
14. Show that A + B + C + D = B + D + C + A.
§5. Associativity and commutativity
5.1. Proposition. If k is an integer ≥ 3, then any two ways of associating a sum A1 + ... + Ak of k vectors give the same sum. Consequently parentheses may be dropped in such sums.
Proof. The proof proceeds by induction on the number of vectors. Axiom 2 gives the case of 3 vectors. Suppose now that k > 3, and that the theorem is true for sums involving fewer than k vectors. We shall show that the sum of k vectors obtained from any method M of association equals the sum obtained from the standard association MO obtained by adding each term in order, thus:
(... (((A1 + A2) + A3) + A4) ...) + Ak.
A method M must have a last addition in which, for some integer i with 1 ≤ i < k, a sum of A1 + ... + Ai is added to a sum of Ai+1 + ... + Ak. If i is k - 1, the last addition has the form
(A1 + ... + Ak-1) + Ak.
The part in parentheses has fewer than k terms and, by the inductive hypothesis, is equal to the sum obtained by the standard association on k - 1 terms. This converts the full sum to the standard association on k terms. If i = k - 2, it has the form
(A1 + ... + Ak-2) + (Ak-1 + Ak)
which equals
((A1 + ... + Ak-2) + Ak-1) + Ak
by Axiom 2 (treating A1 + ... + Ak-2 as a single vector). By the inductive hypothesis, the sum of the first k - 1 terms is equal to the sum obtained from the standard association. This converts the full sum to the standard association on k terms. Finally, suppose i < k - 2. Since Ai+1 + ... + Ak has fewer than k terms, the inductive hypothesis asserts that its sum is equal to a sum of the form (Ai+1 + ... + Ak-1) + Ak. The full sum has the for
by Axiom 2 applied to the three vectors A1 + ... + Ai, Ai+1 + ... + Ak-1 and Ak. The inductive hypothesis permits us to reassociate the sum of the first k - 1 terms into the standard association. This gives the standard association on k terms.
The theorem just proved is called the general associative law; it says in effect that parentheses may be omitted in the writing of sums. There is a general commutative law as follows.
5.2. Proposition. The sum of any number of terms is independent of the ordering of the terms.
The proof is left to the student. The idea of the proof is to show that one can pass from any order to any other by a succession of steps each of which is an interchange of two adjacent terms.
§6. Notations
The symbols U, V, W will usually denote vector spaces. Vectors will usually be denoted by A, B, C, X, Y, Z. The symbol R stands for the real number system, and a, b, c, x, y, z will usually represent real numbers (= scalars). Rk is the vector space defined in 3.1. The symbols i, j, k, l, m, n will usually denote integers.
We shall use the symbol [member of] as an abbreviation for "is an element of". Thus p [member of] Q should be read: p is an element of the set Q. For example, x [member of] R means that x is a real number, and A [member of] V means that A is a vector in the vector space V.
The symbol [subset] is an abbreviation for "is a subset of", or, equally well, "is contained in". Thus P [subset] Q means that each element of the set P is also an element of Q (p [member of] P implies p [member of] Q). It is always true that Q [subset] Q.
If P and Q are sets, the set obtained by uniting the two sets is denoted by P [union] Q and is called the union of P and Q. Thus r [member of] P [union] Q is equivalent to: r [member of] P or r [member of] Q or both. For example, if P is the interval [1, 3] of real numbers and Q is the interval [2, 5], then P [union] Q is the interval [1, 5]. In case P [subset] Q, then P [union] Q = Q.
It is convenient to speak of an "empty set". It is denoted by [??] and is distinguished by the property of having no elements. If we write P [intersection] Q = [??], we mean that P and Q have no element in cammon. Obvious tautologies are
I [subset] P, I [union] Q = Q, I [intersection] P = I.
§7. Linear subspaces
7.1. Definition. A non-empty subset U of a vector space V is called alinear subspace of V if it satisfies the conditions:
(i) if A [member of] U and B [member of] U, then A + B [member of] U,
(ii) if A [member of] U and x [member of] R, then xA [member of] U.
These conditions assert that the two operations of the vector space V give operations in U.
7.2. Proposition. U is itself a vector space with respect to these operations.
The addition and multiplication in a linear subspace will always be assumed to be the ones it inherits from the whole space.
It is obvious that the subset of V consisting of the single element is a linear subspace. It is also trivially true that V is a linear subspace of V. Again, if U is a linear subspace of V, and if U' is a linear subspace of U, then U' is a linear subspace of V.
7.3. Proposition. If V is a vector space and {U} is any family of linear subspaces of V, then the vectors common to all the subspaces in {U} form a linear subspace of V denoted by [intersection] {U}.
7.4. Definition. If V is a vector space and D is a non-empty subset of V, then any vector obtained as a sum
x1A1 + x2A2 + ... + xkAk
(abbreviated [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), where A1, ..., Ak are all in D, and x1, ..., xk are any elements of R, is called a finite linear combination of the elements of D. Let L(D) denote the set of all finite linear combinations of the elements of D. It is clearly a linear subspace of V, and it is called the linear subspace spanned by D. We make the convention L(I) = [??].
7.5. Proposition. D [subset] L(D).
For, if A [member of] D, then A = 1A is a finite linear combination of elements of D (with k = 1).
7.6. Proposition. If U is a linear subspace of V, and if D is a subset of U, then L(D) [sebuset] U. In particular L(U) = U.
The proof is obvious.
Remark. A second method of constructing L(D) is the following: Define L'(D) to be the common part of all linear subspaces of V which contain D. By Proposition 7.3, L'(D) is a linear subspace. Since L'(D) contains D, Proposition 7.6 gives L(D) [subset] L'(D). But L(D) is one of the family of linear subspaces whose common part is L'(D). Therefore L'(D) [subset] L(D). The two inclusions L(D) [subset] L'(D) and L'(D) [subset] L(D) imply L(D) = L'(D). To summarize, L(D) is the smallest linear subspace of V containing D.
§8. Exercises
1. Show that U is a linear subspace of V in each of the following cases:
(a) V = R3 and U = set of triples (x1, x2, x3) such that
x1 + x2 + x3 = 0.
(b) V = R3 and U = set of triples (x1, x2, x3) such that x3 = 0.
(c) (See Exercise 4.11), V = RD and U = RD' where D' [subset] D.
(d)V = RR, i.e. V = set of all real-valued functions of a real variable, and U = the subset of continuous functions. | 677.169 | 1 |
Showing 1 to 3 of 3
Milwaukee Area Technical College
Liberal Arts and Sciences Division
Mathematics Department
Course Syllabus
This Course Syllabus provides you with information about your MATC math course and includes a Course Schedule. The Course Schedule includes the
chro
Quiz 1
1. Where can you find the list of assignments and due dates?
You can find them in the class calendar and announcements.
2. Why do you need to read announcements when you log into class?
Announcements introduce the week as well as provide any alerts | 677.169 | 1 |
Saturday, October 2, 2010
Great Math Homework Help Website
Hey guys, this website is an AWESOME homework tool that has been pretty much invaluable to me throughout some of my advanced math classes. Just ask these guys anything and they'll usually tell you exactly what you need to know in order for you to grasp a concept that you've been struggling to wrap your mind around.
There's one thing you should know about this website in that, since it sucks deciphering stuff like "a^3 / square root of 21 + 7sin(theta)", they use a thing called LaTex (pronounced: lay-tech). This is just a very simple, code-oriented way of clearly displaying Math symbols and equations exactly as you would write them down on paper.
Here's the basics of the LaTex you'll need to know for the forum:
To start off, just know that you'll have to wrap your LaTex code in tags like so: [math]"your code here"[/math]. The "post message" form on the forum even has a shortcut button where you just highlight all your code, then click the button, and it will automatically wrap everything in these tags.
Here's a nice, basic reference sheet to familiarize yourself with:
Alright, so it's basically telling you all the stuff you'll need to know for most basic arithmetic and trig.
Stuff like a^2 will give you a to the 2nd power.. \sqrt{4} will give you the square root of four.. \sin will give you the symbol for sin.. \frac{2}{3} to give 2 over 3.. etc. Pretty basic stuff.
So using our example from earlier... "(a^3 / square root of 21) + 7sin(theta)" would be coded as \frac{a^3}{\sqrt{21}} + \sin\theta" I also just learned you could do other roots like 3 to the 4th root: \sqrt[4]{3}. You learn something new everyday.
While this might look intimidating at first, it's seriously not. Just learn by trial and error, click on other people's LaTex to see how they coded it, and you'll be using LaTex without even thinking about it soon. (And honestly, you don't NEED to use LaTex to ask a question. I just find it so much clearer and easier. Ask a few questions just the normal way, then after you see how useful this site is, maybe start picking up LaTex.)
So now that you're well on your way to mastering the nuances of LaTex, you're ready to start getting some free, comprehensive math help. Actually, even if you don't help, I suggest you to lurk the forum every once in a while anyway. I might need you to explain some Calc to me!
2 comments:
By the way, I wanted to point out that I was typing "\" (back-slash) not "/" (forward-slash). You type the back-slash with that button right above the enter key, right below backspace. (At least on my laptop..) | 677.169 | 1 |
Introduction to Graphing Unit Tracking Sheet
Foundations of College Mathematics
In Preparation for the Objective Assessment
Learning Objectives:
Determine whether an ordered pair is a solution of a linear equation in two variables.
Graph points on the rec
Geometry (Part I) Unit Tracking Sheet
Foundations of College Mathematics
In Preparation for the Objective Assessment
Learning Objectives:
Classify an angle as acute, right, obtuse, or straight.
Calculate the perimeter and area of a triangle, square, and r
Fractions and Mixed Numbers Unit Tracking Sheet
Foundations of College Mathematics
In Preparation for the Objective Assessment
Learning Objectives:
Convert between fraction and mixed number notation.
Write fractions in lowest terms.
Determine if two fract | 677.169 | 1 |
Algebra 2 Questions & Answers
Algebra 2 Flashcards
Algebra 2 Advice
Algebra 2 Advice
Showing 1 to 3 of 10
The teacher makes difficult concepts easy to understand, anyone who doesn't think they can succeed in Calculus will find out they are wrong in this class.
Course highlights:
I gained a solid knowledge of calculus, and learned to break down and tackle difficult problems.
Hours per week:
3-5 hours
Advice for students:
Participate in class, do your best, ask questions!
Course Term:Fall 2017
Professor:Mrs. Neill
Course Tags:Math-heavyGreat Intro to the SubjectParticipation Counts
Feb 07, 2017
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
Algebra isn't about numbers, its about rules and manipulation. Algebra quickly became a favorite class of mine because as long as one learns the rules then everything else is easy. It is actually one of the higher maths that is applicable in real life situations and can save a bit of time. I enjoyed manipulating the signs and operations to get the outcome I needed. I highly recommend it because of the enjoyable aspect of algebra, but also because of its usefulness.
Course highlights:
By taking Algebra all other maths that followed, especially geometry became easier. Arithmetic is the foundation for algebra and algebra is the foundation for many other maths and sciences. I enjoyed the story problems most of all because I had to combine logic with what I knew to be true about algebra to come up with a mathematically sound result. It was almost like solving a puzzle or a mystery.
Hours per week:
6-8 hours
Advice for students:
I'd have to say that you should work every problem out with the teacher. Copy all of the problems that they have up on the board. When you discover a problem is incorrect, figure out what the problem was and fix it so that it doesn't happen again. Notes are the best study aid. And also memorize the orders of operation and any formulas you are given along with all of the roots and squares of the numbers 1-100.
Course Term:Fall 2016
Professor:Judy Howe
Course Required?Yes
Course Tags:Math-heavyLots of WritingParticipation Counts
Dec 12, 2016
| Would recommend.
Not too easy. Not too difficult.
Course Overview:
This Class was highly interactive and the teacher was focused on helping the class understand the material
Course highlights:
I was able to work with others on the homework and see the problem from a different perspective.
Hours per week:
3-5 hours
Advice for students:
Study groups are helpful because if a problem does not make sense to you, someone else may clearly understand it and be able to help you. | 677.169 | 1 |
2
WARM UP 1.What does this mean? Interpret this statement in your own words and explain it to everyone at your table. 2.Does this concept connect with any math topics that you currently teach? Which topics?
4
OBJECTIVES To articulate the content and format of the Mathematical Reasoning module of the 2014 GED test To be able to use a few teaching strategies in your classrooms right away to begin the transition to the new test, including ways to use existing materials To create a personal PD plan related to the Mathematical Reasoning module
6
WARM UP 1.What does this mean? Interpret this statement in your own words and explain it to everyone at your table. 2.Does this concept connect with any math topics that you currently teach? Which topics? Purpose 1: To demonstrate that it is possible to do everything that is on the GED 2014 Mathematical Reasoning module. Purpose 2: To expose everyone to the new content so that teachers will be more prepared to advocate for content knowledge refreshers at a later date.
789
COMPARE AND CONTRAST Personally…do 2002 PLUS… Reframing what we are ALREADY doing at EACH level Pizza model 10 minutes What is New on the 2014 GED test? Q.1.d Identify absolute value of a rational number as its distance from 0 on the number line and determine the distance between two rational numbers on the number line, including using the absolute value of their difference Q.2.d Determine when a numerical expression is undefined A.1.f Factor polynomial expressions A.3.a Solve linear inequalities in one variable with rational number coefficients A.3.b Identify or graph the solution to a one variable linear inequality on a number line A.3.c Solve real-world problems involving inequalities A.3.d Write linear inequalities in one variable to represent context A.7.b Represent or identify a function in a table or graph as having exactly one output (one element in the range) for each input (each element in the domain).
10
COMPARE AND CONTRAST Personally, we recommend just reframing what we are already doing at each level. 2002+
1213
ASSESSMENT TARGET ACTIVITY Directions: Look at the 15 assessment targets of the 2014 GED Mathematical Reasoning module. Highlight trouble spots or confusing words in small groups. After 5 minutes, each table will report out to Amy and Lindsey, who will take notes to inform future content-based PD. Purpose: To FIND trouble areas, not to answer content questions
1415171920
ITEM SAMPLERS: WHERE ARE THE THEMES? What are the skills and concepts needed to answer the following question?
25
ITEM SAMPLERS: WHERE ARE THE THEMES? What are the foundational skills that are needed to build up to these skills?
26
What27
EXISTING MATERIALS Do you feel you have materials that cover these things?
28
WRAP UP What | 677.169 | 1 |
NCERT Solutions Class 9 Maths
Download NCERT Solutions Class 9 Maths in PDF format free. Buy NCERT Books Online or Download in PDF format. Chapter wise assignments Test Papers, Previous year Question Papers issued by CBSE and other schools, Chapter wise tests, Syllabus for the academic year 2016 – 2017 and other online study material. Exemplar books are designed to enhance the practice and improve the knowledge and concepts about each chapter. So, students are advised to go through NCERT Exemplar book after doing NCERT books. Vedic Maths is good for improving calculations faster and easier. We are also providing help in solving holiday homework, if you are facing problem in doing holiday homework, upload in our website and get the solution with a week. NCERT solutions for class 9 Science, Hindi and Social Science is also available to download in pdf form. | 677.169 | 1 |
About the Book
We're sorry; this specific copy is no longer available. AbeBooks has millions of books. We've listed similar copies below.
Description:
1285195787 Kaufmann and Schwitters have built this text's reputation on clear and concise exposition, numerous examples, and plentiful problem sets. This traditional text consistently reinforces the following common thread: learn a skill; practice the skill to help solve equations; and then apply what you have learned to solve application problems. This simple, straightforward approach has helped many students grasp and apply fundamental problem solving skills necessary for future mathematics courses. Algebraic ideas are developed in a logical sequence, and in an easy-to-read manner, without excessive vocabulary and formalism. The open and uncluttered design helps keep students focused on the concepts while minimizing distractions. Problems and examples reference a broad range of topics, as well as career areas such as electronics, mechanics, and health, showing students that mathematics is part of everyday life. The text's resource package--anchored by Enhanced WebAssign, an online homework management tool--saves instructors time while also providing additional help and skill-building practice for students outside of class.
About the Author:
Jerome E. Kaufmann received his Ed.D. in Mathematics Education from the University of Virginia. Now a retired Professor of Mathematics from Western Illinois University, he has more than 30 years of teaching experience at the high school, two-year, and four-year college levels. He is the author of 45 college mathematics textbooks.
Karen L. Schwitters graduated from the University of Wisconsin with a B.S. in Mathematics. She earned an M.S. Ed. in Professional Secondary Education from Northern Illinois University. Schwitters is currently teaching at Seminole Community College in Sanford, Florida, where she is very active in multimedia instruction and is involved in planning distance learning courses with multimedia materials. She is an advocate for Enhanced WebAssign and uses it in her classroom. In 1998, she received the Innovative Excellence in Teaching, Learning, and Technology Award.
"About this title" may belong to another edition of this title.
Bibliographic Details
Title: Algebra for College Students
Publisher: Brooks Cole
Binding: Hardcover
Book Condition: New
Book Description 2014. Hardcover. Book Condition: Poor. BOOK HAS SMOKE DAMAGE & SOME DISCOLORATION, ESPECIALLY ON THE FORE-EDGE OF PAGES. MAY HAVE A SMOKEY SMELL. BOOK IS AN INSTRUCTOR'S EDITION. (same as student edition, but may have answers/notations or indicate on cover that it is "not for sale") Book may/may not have writing and/or highlighting. All pages are intact & perfectly readable. Awesome deal for someone looking to save $$ on textbooks. Bookseller Inventory # Q14
Book Description Book Condition: Good. This is a hard cover book. The cover shows normal wear and tear. The pages have normal wear. This book includes media or other accessories. We ship Monday-Saturday and respond to inquries within 24 hours. Bookseller Inventory # 3O6FEO00065G
Book Description Brooks Cole. Book Condition: Good. 1285195787 May have signs of use, may be ex library copy. Book Only. Used items do not include access codes, cd's or other accessories, regardless of what is stated in item title. Bookseller Inventory # Z1285195787 8645661285195787GO | 677.169 | 1 |
Top Homework HelpIn "The Zany World of Basic Math," the Standard Deviants perform and educate in an entertaining, non-standard way. Using offbeat visual techniques and humor, the group makes understanding basic math concepts easier and more fun, teaching youngsters about addition, subtraction, multiplication, division, fractions, ratios and more. The tutorial concludes with brief quizzes and a "Grand Slam Exam" to test kids on how much they've learned Standard Deviants educators tackle one of humanity's greatest fears -- public speaking -- imparting lessons sure to make the next oral presentation a breeze (or something close to it). Like other subjects the team has tackled, public speaking is broken down into manageable segments, such as monitoring posture, keeping track of time and overcoming stage fright. When those skills are put together, a better public speaking experience ensues.The wacky minds behind the Standard Deviants series employ their trademark mix of movie parodies, songs, shameless puns and more to convey the lessons of algebra in a fun, clear and effective manner. The program includes segments on functions, algebraic properties, linear equations and more. Viewers can chart their progress with the interactive testing and practice exams, all easily accessed through the menu.
The first in a series of lessons, this fun instructional program will help anyone struggling to understand algebra. Engaging teacher Dr. Monica Neagoy's nontraditional style provides a unique overview by using real-world examples and illustrations. Topics include a look at the history and evolution of algebra; the development of algebraic symbolism; and the geometric and numeric currentsAn algebraic equation can be as befuddling as, well, an algebraic equation! But with the help of the crackerjack educational team called Standard Deviants, factors, tree diagrams, absolute values, infinities and radical expressions are broken down to understandable bits so students can calculate and compute with confidence. Standard Deviants prides itself on infusing complicated subjects with lots of fun so learning becomes almost effortlessMath haters take heart: The Standard Deviants take a daunting subject -- geometry -- and make it approachable through instruction that includes a heaping dose of humor, a smattering of technology and a raft of real-world examples. Designed by a panel of college professors and presented by an assortment of hams, the course covers all the geometry basics in a fast-paced but understandable manner. Flash cards, reviews and quizzes are included.
Packed with hours of one-on-one instruction, this educational video program is based on the simple belief that if immersed in the basics, anyone can learn complex math. Focusing on the rudiments of trigonometry and precalculus, chapter topics cover complex numbers, exponential and logarithmic functions, angles, finding trig functions using triangles and the unit circle, graphing trig functions and trig identitiesThree kid-friendly stories impart important lessons in the realm of numbers and counting. In "Chicka Chicka 1-2-3," numbers one through 101 climb an apple tree and do battle with meddlesome bumblebees -- but which number will save the day? Then, in "How Much Is a Million?" and "If You Made a Million," Marvelosissimo the Mathematical Magician demystifies the concept of large numbers and explains the various forms of currency.
Take the mystery out of fractions with this introductory lesson from professor Murray Siegel. Step-by-step instructions guide you through the basics, such as simplifying, reducing, writing mixed numbers as improper fractions, equivalents and more. Plus, Siegel explains each part of the fraction in clear language, making it easy to learn this important aspect of mathematicsEducator Monica Neagoy takes the mystery out of algebra with explicit examples of how it applies to everyday life. Using easy-to-follow anecdotes, she leads students step by step through important algebraic concepts such as sets of natural and whole numbers, rational and real numbers, and integers. Neagoy, a leading mathematics professor and pioneer in education consulting, makes learning the basics funThis installment of the Standard Deviants series, written by a team of university professors, expands on the concepts of pre-algebra in a quick no-nonsense style that lets you pause and rewind until you master the lesson. Topics include basic linear equations, checking solutions, absolute value equations, graphing basics, quadrants, plotting points, graphing linear equations, boundary lines and area of intersection.
Master fundamental multiplication and division, learn the meaning and usage of word multiplication, and discover the ability to multiply using the lattice method with this dynamic program from renowned math professor Murray Siegel. Through simple exercises, students will learn the logical process of multiplication, acquire the ability to make valid estimates for problems and obtain the capacity to explain division.Mathematician Murray Siegel leads you through a logical explanation of the building blocks of math: addition and subtraction. Beginning with a definition of what the word <i>number</i> actually means, Siegel provides a clear and inviting foundation. A leader in public school mathematics, Siegel has also spent much of his life devoted to adult education. His videos help students overcome the fears and anxiety that can surround mathematical conceptsuation, including the frequently misunderstood hyphen and apostrophe. Also featured are the correct uses for parentheses, brackets, slashes, ellipses, contractions and more.
Experienced educator Richard Lavoie shares effective techniques that can help teachers and parents turn kids into motivated achievers. In this instructional program, Lavoie debunks common myths about motivation, identifies six motivational styles and presents strategies to help any type of learner succeed. Children who are excited about learning are those who will get the most out of their education and thrive both in and out of the classroomUsing example problems as a hands-on tool, this basic math tutorial offers seven hours of easy-to-understand explanations of fundamental math concepts. The approach of this homework and test-taking video helper is to eliminate boring lectures and to carefully work through detailed example problems, so even those with no math skills can follow along. Essential math topics are covered, including addition, subtraction, multiplication and divisionLearn all about energy with this fun and effective instructional program led by lab coat-wearing teacher Dr. Science, who covers the fundamentals of energy and its various forms in this third volume of the educational series. Geared toward helping teachers introduce and explain scientific principles in an entertaining format, the informative Understanding Science series turns learning into an enjoyable experience.
Return to the educational exploits of Idaho Bones and Boise as they build on the algebra concepts introduced in the first volume. This video includes discussions of quadratic equations, factoring, imaginary numbers, higher order polynomials and more concepts sure to help stoke your mathematics grooveIf you'll be taking the Department of Motor Vehicles' written exam soon, this informative, easy-to-follow tutorial is your ticket to acing the test. The video condenses several days of tedious reading into a colorful 40-minute presentation. After viewing the main program, aspiring drivers can use the menu to select individual states and check out the handful of differences in driving laws among the 50 states.
Award-winning math teacher Jim Noggle helps demystify the essentials of geometry with this informative, well-organized educational video. Using a blackboard and flip charts, along with physical models of geometric objects, Noggle teaches students how to introduce and use the undefined terms, lines and planes; how to understand basic terminology of various expressions; and how to establish a linear system of measurementIntended to help students get a firm grasp on the elemental principles and concepts of science, this Telly Award-winning series uses eye-popping graphics and engaging live action segments to make learning fun.
This edition of the educational video series will teach you techniques designed to help you through nonstandard mathematical problems by using charts to simplify them. Instead of having you commit a specific algorithm or formula to memory to solve every kind of problem, the program illustrates a method that can be used as a starting point for working out an assortment of problemsLearn the basics of fractions, decimals, and percents with this complete, easy-to-follow educational video course which covers how to write a fraction as a decimal, a decimal as a fraction, a decimal as a percent, or a percent as a decimal. Learning fractions, decimals, and percents has never been so easy!
Designed for students who've masteredThe world expands and conflates all at once in this eye-opening four-disc series that takes viewers to 16 different countries and shows them what the days and nights are truly like for those who live there. The adventure-filled series brings to life the study of geography, with participants journeying to such far-flung locales as Israel, South Africa, the United Kingdom, France and China, and then meeting and learning from one another.
Georgetown University mathematics lecturer Monica Neagoy leads you step by step through the basics of graphing calculators, including the uses of function, editing and statistics keys; variables; and equations relating to two axes. Neagoy effectively uses practical examples and historical anecdotes that will inform and entertain you, whether you need a refresher course or are learning for the first time.
Utilizing real-world examples, this educational program teaches viewers an easy-to-understand approach to solving long-division problems. Within no time at all, viewers will be able to make good estimates and divide by one- or two-digit divisors. The significance of the powerful mantra "divide, multiply, subtract, bring down" is explained thoroughly. Viewers will also learn why it is impossible to divide by zero.
Algebra was never this much fun. Georgetown University's Dr. Monica Neagoy covers input/output numerical tables, general formulas for the nth (positive) odd number, graphical representation and exploration, and more in this program. Combining real-world examples and interesting anecdotes, this lesson also delves into derivations of algebraic formulas and is perfect for first-time learners or anyone who needs a refresher courseSeven hours of advanced algebra instruction from Math Tutor series founder Jason Gibson prepares students for college-level and advanced algebra courses. Would-be math scholars learn by doing as they work example problems that build in complexity. Lessons cover everything from graphs of functions and circles to quadratic functions, arithmetic and geometric sequences, graphing of rational functions, the binomial theorem and moreMath is made easy and -- dare we say it? -- perhaps even fun with this computer-animated series. The program's central characters, Sam and Amber, with the able assistance of mathematical oracle Numberella, use basic math to solve problems and carry out tasks on their Uncle Zak's planetoid, Junkiter. Each program teaches a single concept through the use of mnemonic rhymes, songs, and lively characters and storiesThe sixth installment of this ongoing series designed to teach students the basics of algebra tackles multiple representations of linear functions, a concept that sounds more complicated than it is. Using nontraditional tutoring methods, educator Monica Neagoy demonstrates how to use a graphing calculator to represent a function and then explains how to translate verbal information into solvable algebraic equationsIn the seventh lesson of this popular algebra course, Georgetown University's Dr. Monica Neagoy explains the effect of the numerical coefficients a and b, and the use of parallel and perpendicular lines in graphical representations. Combining interesting anecdotes and real-world examples, this lesson also covers derivations of algebraic formulas and is perfect for newbie learners or anyone who needs a course that refreshes their skillsón, John Smith, Pocahontas and William PennMath wizard Dr. Murray Siegel demonstrates how to solve simple linear equations with one variable in this volume of the Basic Math series. He also teaches students the foundations and importance of algebra. Lessons gradually become more complex, and by the end of the program, students are able to, with confidence, add and subtract whole numbers and check their answers by using the original equationJoin math whiz Jim Noggle for this clear, comprehensive series that takes the mystery out of geometry. In this volume, Noggle explains the essentials of inductive and deductive reasoning and their importance in mathematical relationships. The easy-to-follow instruction provides real-world examples of inductive and deductive reasoning, making it easier for viewers to understand their role in geometry.
Professor Murray H. Siegel teaches this mathematics lesson involving fractions. The easy-to-follow course covers subtracting fractions, mixed numbers, borrowing numbers when subtracting mixed numbers and more. Part of the Teaching Company's series of educational programs, this lesson explains why a mathematical operation works and how it relates to other areas of mathematics, and covers practical applications and alternative methods.
Learn the basics of multiplying fractions with this complete, easy-to-follow educational video course which covers how to execute the canceling process, multiply fractions, multiply mixed numbers (and more then two mixed numbers), and more. Learning math has never been so easy!
Using nontraditional methods to make learning dynamic, educator Monica Neagoy leads the ninth installment of the ongoing series designed to teach basic algebraic concepts. In this volume, Neagoy focuses on solving problems with linear equations, beginning with a brief history of problem solving before moving on to explain how to derive an algebraic equation though functional exploration and symbolic manipulation | 677.169 | 1 |
Enhancing Mathematics Education at the Tertiary Level: The Place of Mathematics Learning Centers
Three goals an institution of higher learning must embrace to survive are: attract competent students, retain most of the students, and graduate them on time. Realization of these goals differentiates highly rated and successful institutions from poor performing and low rated institutions. Institutions that build their mission around these goals and strive to achieve them consistently attract better prepared and better paying students who later become big donor alumni.
In this article, I will endeavor to examine the role of Mathematics Learning Centers (MLCs) in assisting institutions in their never ending quest to attract, retain and graduate their students. Most science, technology and engineering disciplines require a good working knowledge of Mathematics beyond algebra. Unfortunately a high percentage of students start College deficient in basic College algebra.
Research has shown that poor performance in basic mathematical skills hamper the progress of students in reaching their college educational goals. In fact, about 50 percent of students don't pass college algebra with a grade of C or above, as noted in Saxe [15], "Common Vision," from the Mathematical Association of America (MAA). The report called Americans' struggle with math "the most significant barrier" to finishing a degree in both STEM and non-STEM fields. The first college Mathematics course students take can make or break their confidence. Students, who find their entry level Mathematic course challenging may abandon their dream major, spend many precious semesters taking irrelevant course or drop out altogether.
Since a high percentage of entering and continuing students need assistance to achieve their goals, it is the duty of the institution to strategically plan assistance into their operations. Seidman [3] stated his retention equation as "Retention = Early ID + Early Intervention + Intensive and continuous intervention". Part of intervention is providing effective tutorial services for students.
Is Mathematics Learning Center Necessary?
In order for institutions of higher learning to keep its students and take them up to proficiency level, Colleges have devised various solutions. One solution is to introduce remedial courses in basic algebra and trigonometry which students must take and pass before taking credit bearing courses. An adverse effect of this approach is that it delays students' progress towards graduation while increasing the cost of their education. Another method is to provide pre-registration workshops to strengthen students' weaknesses in algebra or trigonometry as needed. Such workshops could accompany courses students are registered for or be separate.
In all these scenarios, experience shows that students benefit greatly from peer tutoring, hence the need for MLC. It is impractical to expect professors teaching gateway mathematics courses to devote adequate time to individual students having difficulty in their classes. Peer tutoring fills the gap for the professor and for the student. Students can spend as much time as needed at the MLC working with a peer-tutor to understand a difficult concept. Students are also more comfortable asking another student questions than asking a professor. Other benefits of peer tutoring for students include:
Improved self confidence
Improved self-esteem, as they become more successful students
Improved academic achievement
Improved attitude towards the subject matter and school in general
Improved personal and social development
Greater persistence in completing tasks and courses
Better use of appropriate and efficient learning and study strategies
Provides an opportunity for individualized instruction
Peer tutors also benefit from the center. Some obvious benefits include:
Frequent review of previously learned material helps with learning new material in more advanced courses.
Provides experiences that may help with later employment or career goals
Develops empathy for others
Improves attitudes towards subject area
Increases general knowledge
Develops a sense of responsibility
Additionally, peer tutors are more likely to take more advanced mathematics courses than students not participating in the tutoring program. By the setup of MLC, peer tutors are always fresh in their understanding of Mathematical concepts. As a result, most peer tutors complete their degree and graduate on schedule. Peer tutors are also positive role models to other students. All these along with the moderate stipend peer tutors receive are great motivations for students.
The institution benefits from the creation and maintenance of MLCs in many ways:
Promotes deeper learning of material which in turn enables professors and the institution to set higher goals for student learning
Improves student retention of material and hence the retention of students
Reduces drop out and failure rates among students which in turn improves retention and graduation rate
Provides a cost effective means of providing individualized instruction to students who need it
Swail [16] noted that: Resources play a huge part in the ability of a campus to provide support services necessary to engage and save students. Institutions known to graduate over 90% of their students on time are known to devote substantial resources to student support services like the MLC. Though these institutions certainly get the best and brightest students, they also provide outstanding resources to ensure that students have all necessary tools to make their way through the labyrinth of higher education. These institutions assign tutors to students rather than forcing students to seek them out. They have smaller class sizes and labs, and provide extensive and often proactive supplementary support services.
Composition of a Mathematics Learning Center
A Mathematics Learning Center (MLC) is a center where students go to get help on various Mathematics course offered at the University. The center is made up of the director, a coordinator, front desk workers and student tutors. The director is generally a Mathematics professor familiar with courses taught at the Mathematics department. He serves as the liaison between the center, the Mathematics department professors, and the college administration. He is responsible for securing funding needed to operate the center effectively. He also works with professors in the Mathematics department to ensure that assignments and materials presented in classes are available to the center.
The director provides guidance and training materials to ensure standards are maintained at the center.
The coordinator helps the director in coordinating staff schedules and training in the center. The coordinator, working with the director, ensures that tutors have requisite training in the course(s) they tutor. The coordinator also helps to ensure timely and accurate payments of stipends to the center's staffs.
Front-desk workers serve as secretaries for the center. They welcome students to the center, assign students to appropriate tutors and ensure that tutors get all they need to effectively tutor assigned students. Additionally, front-desk workers collect survey questions from students after tutoring. These surveys are used to measure students' satisfaction with the services rendered at the center.
The tutors are students with at least a B average overall and A in the courses they tutor. Each tutor is recommended by a Mathematics professor to the director. The director interviews each tutor to verify grades, social attitude and suitability as tutors. Tutors may not work more than 20 hours per week. In addition to grade requirements, each tutor is required to demonstrate proficiency in the subject they tutor by completing carefully selected exercises from the textbook or from professors of the subject they are to tutor. In addition, new tutors are required to attend some classroom sessions of the course they intend to tutor.
Each staff of the center must attend two workshops organized by the director. These workshops are generally scheduled at the beginning of the semester and at the midterm. Issues discussed in such workshops include policies of the center and training on how to address student related questions.
Conclusion:
This article presents Mathematics Learning Centers as one of the tools higher institutions need to address attrition and failure rate in their student population. While having a dedicated MLC is not a magic pill to cure students' failure in Mathematics, the lack of such a center is certain to exacerbate students' frustration, and failure to achieve their educational goals. Creating a Mathematics learning center is a win-win-win solution for the students, the tutors and the institution.
Acknowledgements:
The author is grateful to York College for providing the facilities and sponsoring of students tutors. I also benefitted greatly from being a part of York College's Tutorial Services taskforce lead by Professor Steven Tyson and the author.
References
Adelman, C. (2006). The toolbox revisited: Paths to degree completion from high school through college. Washington, D. C.: Department of Education.
Ishitani, T., & DesJardins, S. (2002). A longitudinal investigation of dropouts from college in the United States. Journal of College Student Retention: Research, Theory & practice, 4(2) 173-201.
Kennedy, P. (1989). Effects of Supplemental Instruction on student performance in a college level mathematics course. University of Texas-Austin. Dissertation Abstracts for Social Sciences, 50, DA8909688.
Kunsch, C., Jitendra, A., & Sood, S. (2007). The effects of peer-mediated instruction in mathematics for students with learning problems: A research synthesis. Learning Disabilities Research & Practice, 22(1), 1-12. | 677.169 | 1 |
Goals: Refresh the notions of linear algebra. Review the concept of vector algebra applied to signal processing. Concepts of projection, subspace, orthogonality as occurring in such applications as approximations, series expansions, prediction, linear filtering, de-noising, estimation. Familiarity with matrix operations, their fundamental spaqces, solutions of large sets of simultaneous linear equations.
Learning Objectives:
At the end of this course, students will be able to:
1. Have a deeper understanding of the theory and relevance of vector space concepts, metricity, and matrices as principal operators in finite dimensional Hilbert spaces;
2. Be able to analyze linear models and subspace solutions.
3. Be capable of reasoning in terms of explanatory variables and dependent variables
4. Be capable of carrying out filtering, estimation, prediction, signal analysis and synthesis algorithms
Course Structure: The class meets for four lectures a week, consisting of four 50-minute sessions. 7-8 sets of homework problems are assigned per semester. There is one in-class mid-term exams, 5 quizzes, and a final exam.
Computer Resources: Students are encouraged to use MATLAB to solve their homework problems.
(a) Apply math, science and engineering knowledge. This course requires probability theory, linear system theory, vector-matrix algebra. Engineering intuition and commonsense is required to translate signal processing instances for the proper application of linear algebra statistical tools.
(b) Design a system, component or process to meet desired needs. The students have to design a few statistical filtering algorithms.
(c) Use of modern engineering tools. Students use Matlab on a few occasions. | 677.169 | 1 |
Product Overview
Taking the Complication out of Complex VariablesComplex variables is an essential mathematical and engineering field, playing a crucial role in signal processing, electromagnetics, image analysis, differential equations, mathematical modeling, fluid flow, astrophysics, and modern analytical science. Written in the bestselling Demystified format, this user-friendly guide offers a straightforward way for you to comprehend this complex topic. Hundreds of examples and worked equations make it easy to understand the material, and end-of-chapter questions and a final exam help reinforce learning. | 677.169 | 1 |
Pearson Australia would like to thank Luke Borda for his assistance in reviewing and writing some of the material in
International
Mathematics for the Middle Years Book 4
. Luke is the Mathematics Coordinator for the Middle and Senior Schools at Mercedes
College in Adelaide, Australia, where he has been teaching MYP mathematics for the last 5 years. Luke has been involved in
submitting year 5 MYP moderation and has focussed on the assessment criteria and their application and interpretation.
Every effort has been made to trace and acknowledge copyright. However, should any infringement have occurred, the publishers
tender their apologies and invite copyright owners to contact them.
IM4_Ch00_Prelims.fm Page ii Monday, March 16, 2009 3:44 PM
iii
Contents
Features of
International Mathematics
for the Middle Years
viii
Using this Book for Teaching MYP
for the IB xii
Metric Equivalents xiii
The Language of Mathematics xiv
8:01 The distance between two points 201
8:02 The midpoint of an interval 206
8:03 The gradient of a line 210
8:04 Graphing straight lines 215
8:05 The gradient–intercept form of a
straight line:
y
=
mx
+
c
220
What does
y
=
mx
+
c
tell us? 220
8:06 The equation of a straight line, given point
and gradient 225
8:07 The equation of a straight line, given
two points 227
8:08 Parallel and perpendicular lines 230
8:09 Graphing inequalities on the number plane 234
Set A Addition and subtraction of integers
Set B Integers: Signs occurring side by side
Set C Multiplication and division of integers
Set D Order of operations
1:02B Fractions
Set A Improper fractions to mixed numerals
Set B Mixed numerals to improper fractions
Set C Simplifying fractions
Set D Equivalent fractions
Set E Addition and subtraction of fractions (1)
Set F Addition and subtraction of fractions (2)
Set G Addition and subtraction of mixed numerals
Set H Harder subtractions of mixed numerals
Set I Multiplication of fractions
Set J Multiplication of mixed numerals
Set K Division of fractions
Set L Division of mixed numerals
1:02C Decimals
Set A Arranging decimals in order of size
Set B Addition and subtraction of decimals
Set C Multiplication of decimals
Set D Multiplying by powers of ten
Set E Division of a decimal by a whole number
Set F Division involving repeating decimals
Set G Dividing by powers of ten
Set H Division of a decimal by a decimal
Set I Converting decimals to fractions
Set J Converting fractions to decimals
1:02D Percentages
Set A Converting percentages to fractions
Set B Converting fractions to percentages
Set C Converting percentages to decimals
Set D Converting decimals to percentages
Set E Finding a percentage of a quantity
Set F Finding a quantity when a part of it is known
Set G Percentage composition
Set H Increasing or decreasing by a percentage
These can be used as a diagnostic tool or
for revision. They include multiple choice,
pattern-matching and fill-in-the-gaps
style questions.
Destinations
Links to useful websites that relate directly to the
chapter content.
Challenge Worksheets
Technology Applications
IM4_Ch00_Prelims.fm Page vii Monday, March 16, 2009 3:44 PM
viii INTERNATIONAL MATHEMATICS 4
Features of International Mathematics for
the Middle Years
International Mathematics for the Middle Years is organised with the international student in mind.
Examples and exercises are not restricted to a particular syllabus and so provide students with a
global perspective.
Each edition has a review section for students who may have gaps in the Mathematics they have
studied previously. Sections on the language of Mathematics and terminology will help students
for whom English is a second language.
Areas of Interaction are given for each chapter and Assessment Grids for Investigations provide
teachers with aids to assessing Analysis and Reasoning, Communication, and Reflection and
Evaluation as part of the International Baccalaureate Middle Years Program (IBMYP). The
Assessment Grids will also assist students in addressing these criteria and will enhance students'
understanding of the subject content.
How is International Mathematics for the Middle Years organised?
As well as the student coursebook, additional support for both students and teachers is provided:
• Interactive Student CD — free with each coursebook
• Companion Website
• Teacher's Resource — printout and CD.
Coursebook
Chapter-opening pages summarise the key content and present the learning outcomes addressed in
each chapter.
Areas of Interaction references are included in the chapter-opening pages to make reporting easier.
For example, Homo faber.
Prep Quizzes review skills needed to complete a topic. These anticipate problems and save time in the
long run. These quizzes offer an excellent way to start a lesson.
Well-graded exercises — Within each exercise, levels of difficulty are indicated by the colour of
the question number.
green . . . foundation blue . . . core red . . . extension
Worked examples are used extensively and are easy for students to identify.
p
r
e
p
quiz
1 4 9
a An equilateral triangle has a side of length 4.68 m. What is its perimeter?
Solve the following equations.
a b
a A radio on sale for $50 is to be reduced in price by 30%. Later, the discounted price is
increased by 30%. What is the final price? By what percentage (to the nearest per cent)
must the first discounted price be increased to give the original price?
2
7
x
2
---
x
3
--- + 5 =
p
6
---
p
2
--- + 8 =
8
worked examples
1 Express the following in scientific notation.
a 243 b 60 000 c 93 800 000
IM4_Ch00_Prelims.fm Page viii Monday, March 16, 2009 3:44 PM
ix
Important rules and concepts are clearly highlighted at regular intervals throughout the text.
Cartoons are used to give students friendly advice or tips.
Foundation Worksheets provide alternative exercises for students who
need to consolidate earlier work or who need additional work at an
easier level. Students can access these on the CD by clicking on the
Foundation Worksheet icons. These can also be copied from the
Teacher's Resource CD or from the Teacher's Resource Centre on the Companion Website.
Challenge activities and worksheets provide more difficult investigations and exercises. They can be
used to extend more able students.
Fun Spots provide amusement and interest, while often reinforcing course work. They encourage
creativity and divergent thinking, and show that Mathematics is enjoyable.
Investigations and Practical Activities encourage students to seek knowledge and develop research
skills. They are an essential part of any Mathematics course. Where applicable, investigations are
accompanied by a set of assessment criteria to assist teachers in assessing criteria B, C and D as
prescribed by the MYP.
Diagnostic Tests at the end of each chapter test students' achievement of outcomes. More
importantly, they indicate the weaknesses that need to be addressed by going back to the section
in the text or on the CD listed beside the test question.
Assignments are provided at the end of each chapter. Where there are two assignments, the first
revises the content of the chapter, while the second concentrates on developing the student's
ability to work mathematically.
The See cross-references direct students to other sections of the coursebook relevant to a particular
section.
The Algebra Card (see p xx) is used to practise basic algebra skills. Corresponding terms in
columns can be added, subtracted, multiplied or divided by each other or by other numbers.
This is a great way to start a lesson.
The table of
values looks
like this!
Grouping symbols
1 a (3 + 2) × 10
2 a (8 − 2) × 3
3 a 10 − (4 + 3)
Foundation Worksheet 4:01A
c
h
alle
n
g
e
f
u
n s
p
o
t
n
o
i
t
a
g i t s e v
n
i
d
i
a
gn
o
s
t
i
c
t
e
s
t
a
s
sign
m
e
n
t
e
e
s
IM4_Ch00_Prelims.fm Page ix Monday, March 16, 2009 3:44 PM
x INTERNATIONAL MATHEMATICS 4
The Language of Mathematics
Within the coursebook, Mathematics literacy is addressed in three specific ways:
ID Cards (see pp xiv–xix) review the language of Mathematics by asking students to identify
common terms, shapes and symbols. They should be used as often as possible, either at the
beginning of a lesson or as part of a test or examination.
Mathematical Terms met during the chapter are defined at the end of each chapter. These terms are
also tested in a Drag and Drop interactive that follows this section.
Reading Mathematics help students to develop maths literacy skills and provide opportunities for
students to communicate mathematical ideas. They present Mathematics in the context of everyday
experiences.
An Answers section provides answers to all the exercises in the coursebook, including the ID Cards.
Interactive Student CD
This is provided at the back of the coursebook and is an important part of the total learning
package.
Bookmarks and links allow easy navigation within and between the different electronic components
of the CD that contains:
• A copy of the student coursebook.
• Appendixes A–D for enrichment and review work, linked from the coursebook.
• Printable copies of the Foundation Worksheets and Challenge Worksheets, linked from the
coursebook.
• An archived, offline version of the Companion Website, including:
• Chapter Review Questions and Quick Quizzes
• All the Technology Applications: activities and investigations and drag-and-drops
• Destinations (links to useful websites)
All these items are clearly linked from the coursebook via the Companion Website.
• A link to the live Companion Website.
Companion Website
The Companion Website contains a wealth of support material for students and teachers:
• Chapter Review Questions which can be used as a diagnostic tool or for revision. These are self-
correcting and include multiple-choice, pattern-matching and fill-in the-gaps-style questions. Results
can be emailed directly to the teacher or parents.
• Quick Quizzes for most chapters.
• Destinations — links to useful websites which relate directly to the chapter content.
• Technology Applications — activities that apply concepts covered in most chapters and are
designed for students to work independently:
d i
s
m r e
t
l
a
c
i
t
a
m
e
h
t
a
m
s
c i t a
m
e
h
t
a
m
g
n
i
d a e r
IM4_Ch00_Prelims.fm Page x Monday, March 16, 2009 3:44 PM
xi
Activities and investigations using technology, such as
Excel spreadsheets and The Geometer's Sketchpad.
Drag and Drop interactives to improve mastery of
basic skills.
Animations to develop key skills by
manipulating visually stimulating and interactive
demonstrations of key mathematical concepts.
• Teacher's Resource Centre — provides a wealth of teacher support material and is password
protected:
— Coursebook corrections
— Topic Review Tests and answers
— Foundation and Challenge Worksheets and answers
Teacher's resource
This material is provided as both a printout and as an electronic copy on CD:
• Electronic copy of the complete Student Coursebook in PDF format
• Teaching Program, including treatment of learning outcomes, in both PDF and editable
Microsoft Word formats
• Practice Tests and Answers
• Foundation and Challenge Worksheets and answers
• Answers to some of the Technology Application Activities and Investigations
Most of this material is also available in the Teacher's Resource Centre of the Companion Website.
Sample Drag and Drop
Sample Animation
IM4_Ch00_Prelims.fm Page xi Monday, March 16, 2009 3:44 PM
xii INTERNATIONAL MATHEMATICS 4
Using this Book for Teaching MYP for
the IB
• Holistic Learning
• Intercultural Awareness
• Communication
These elements of the MYP Mathematics course are integrated throughout the text. Links are made
possible between subjects, and different methods of communicating solutions to problems through
investigations allow students to explore their own ideas.
The Areas of Interaction
• Approaches to Learning
• Community and Service
• Health and Social Education
• Environment
• Homo Faber
Areas of Interaction covered are outlined at the start of each chapter, allowing teachers to develop
links between subjects and formulate their own Interdisciplinary Units with additional assistance in
the Teacher's Resource.
Addressing the Objectives
Assessment grids are provided for Investigations throughout the text to not only help teachers
assess criteria B, C and D of the MYP, but also to assist students in addressing the criteria. The
assessment grids should be modified to suit the student where necessary.
A Knowledge and Understanding
This criterion is addressed in the Diagnostic Tests and Revision Assignments that accompany
each chapter. Teachers can also use the worksheets from the CD to add to material for this
criterion.
B Investigating Patterns
It is possible to address this criterion using the Working Mathematically sections accompanying
each chapter, and also using the Investigations throughout the text.
C Communication
This can be assessed using the Investigations throughout the book.
D Reflection in Mathematics
This can be assessed using the Investigations throughout the book.
Fulfilling the Framework for Mathematics
The content of the text covers the five broad areas required to fulfil the Framework:
• Number
• Algebra
• Geometry
• Statistics
• Discrete Mathematics
Although the material in the text is not exhaustive, it covers the required areas in sufficient depth.
Teachers can use the text as a resource to build on as they develop their own scheme of work
within their school.
IM4_Ch00_Prelims.fm Page xii Monday, March 16, 2009 3:44 PM
xiii
Metric Equivalents
Months of the year
30 days each has September,
April, June and November.
All the rest have 31, except February alone,
Which has 28 days clear and 29 each leap year.
Seasons
Southern Hemisphere
Summer: December, January, February
Autumn/Fall: March, April, May
Winter: June, July, August
Spring: September, October, November
Northern Hemisphere
Summer: June, July, August
Autumn/Fall: September, October, November
Winter: December, January, February
Spring: March, April, May
Length
1 m = 1000 mm
= 100 cm
= 10 dm
1 cm = 10 mm
1 km = 1000 m
Area
1 m
2
= 10 000 cm
2
1 ha = 10 000 m
2
1 km
2
= 100 ha
Mass
1 kg = 1000 g
1 t = 1000 kg
1 g = 1000 mg
Volume
1 m
3
= 1 000 000 cm
3
= 1000 dm
3
1 L = 1000 mL
1 kL = 1000 L
1 m
3
= 1 kL
1 cm
3
= 1 mL
1000 cm
3
= 1 L
Time
1 min = 60 s
1 h = 60 min
1 day = 24 h
1 year = 365 days
1 leap year = 366 days
It is important
that you learn
these facts
off by heart.
IM4_Ch00_Prelims.fm Page xiii Monday, March 16, 2009 3:44 PM
xiv INTERNATIONAL MATHEMATICS 4
The Language of Mathematics
You should regularly test your knowledge by identifying the items on each card.
See page 610 for answers. See page 610 for answers.
ID Card 1 (Metric Units) ID Card 2 (Symbols)
1
m
2
dm
3
cm
4
mm
1
=
2
Ӏ or ≈
3
≠
4
<
5
km
6
m
2
7
cm
2
8
km
2
5
р
6
Ͽ
7
>
8
у
9
ha
10
m
3
11
cm
3
12
s
9
4
2
10
4
3
11 12
13
min
14
h
15
m/s
16
km/h
13 14
||
15 16
|||
17
g
18
mg
19
kg
20
t
17
%
18
∴
19
eg
20
ie
21
L
22
mL
23
kL
24
˚C
21
π
22
∑
23 24
P(E)
d i
2 2
3
x
See 'Maths
Terms' at
the end of
each chapter.
IM4_Ch00_Prelims.fm Page xiv Monday, March 16, 2009 3:44 PM
xv
See page 610 for answers.
ID Card 3 (Language)
1
6 minus 2
2
the sum of
6 and 2
3
divide
6 by 2
4
subtract
2 from 6
5
the quotient of
6 and 2
6
3
2)6
the divisor
is . . . .
7
3
2)6
the dividend
is . . . .
8
6 lots of 2
9
decrease
6 by 2
10
the product
of 6 and 2
11
6 more than 2
12
2 less than 6
13
6 squared
14
the square
root of 36
15
6 take away 2
16
multiply
6 by 2
17
average of
6 and 2
18
add 6 and 2
19
6 to the
power of 2
20
6 less 2
21
the difference
between 6 and 2
22
increase
6 by 2
23
share
6 between 2
24
the total of
6 and 2
d
i
We say
'six squared'
but we write
6
2
.
IM4_Ch00_Prelims.fm Page xv Monday, March 16, 2009 3:44 PM
xvi INTERNATIONAL MATHEMATICS 4
See page 610 for answers.
d i
ID Card 4 (Language)
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
All sides
different
IM4_Ch00_Prelims.fm Page xvi Monday, March 16, 2009 3:44 PM
xvii
See page 610 for answers.
ID Card 5 (Language)
1
A
............
2
............
3
............
4
............
5
............ points
6
C is the ............
7
............
............
8
............
9
all angles less
than 90˚
10
one angle 90˚
11
one angle greater
than 90˚
12
A, B and C are
......... of the triangle.
13
Use the vertices
to name the ∆.
14
BC is the ......... of
the right-angled ∆.
15
a˚ + b˚ + c˚ = .........
16
∠BCD = .........
17
a˚ + b˚ + c˚ + d˚ = .....
18
Which (a) a˚ < b˚
is true? (b) a˚ = b˚
(c) a˚ > b˚
19
a˚ = .............
20
Angle sum = ............
21
AB is a ...............
OC is a ...............
22
Name of distance
around the circle.
.............................
23
.............................
24
AB is a ...............
CD is an ...............
EF is a...............
d
i
A
B
A
B
A
B
P
Q
R
S
A C B –4 –2 0 2 4
A
B
C A
B
C
A
B
C
A B
C
b°
c°
a°
A D
B
C
b°
a°
b°
d°
c°
a°
b° a°
a°
A B
C
O
O
O
B
C
D
F
E
A
IM4_Ch00_Prelims.fm Page xvii Monday, March 16, 2009 3:44 PM
xviii INTERNATIONAL MATHEMATICS 4
See page 610 for answers.
ID Card 6 (Language)
1
..................... lines
2
..................... lines
3
v .....................
h .....................
4
..................... lines
5
angle .....................
6
..................... angle
7
..................... angle
8
..................... angle
9
..................... angle
10
..................... angle
11
.....................
12
..................... angles
13
..................... angles
14
..................... angles
15
..................... angles
16
a˚ + b˚ + c˚ + d˚ = .....
17
.....................
18
..................... angles
19
..................... angles
20
..................... angles
21
b............ an interval
22
b............ an angle
23
∠CAB = ............
24
CD is p.......... to AB.
d i
A
B
C
(less
than
90°)
(90°)
(between
90° and 180°)
(180°)
(between
180° and
360°)
(360°)
a° + b° = 90°
a°
b°
a° + b° = 180°
a° b°
a° = b°
a° b°
a°
b°
c°
d°
a° = b°
a°
b°
a° = b°
a°
b°
a° + b° = 180°
a°
b°
A B
C
D
E
A
B
C
D
A B
C
A B
D
C
IM4_Ch00_Prelims.fm Page xviii Monday, March 16, 2009 3:44 PM
xix
See page 610 for answers.
ID Card 7 (Language)
1
a............ D............
2
b............ C............
3
a............ M............
4
p............ m............
5
area is 1 ............
6
r............ shapes
7
............ of a cube
8
c............-s............
9
f............
10
v............
11
e............
12
axes of ............
13
r............
14
t............
15
r............
16
t............
17
The c............
of the dot are E2.
18
t............
19
p............ graph
20
c............ graph
21
l............ graph
22
s............ graph
23
b............ graph
24
s............ d............
d
i
AD BC am pm
100 m
1
0
0
Students will be able to:
• Compare, order and calculate with integers.
• Operate with fractions, decimals, percentages, ratios and rates.
• Identify special angles and make use of the relationships between them.
• Classify, construct and determine the properties of triangles and quadrilaterals.
Note: A complete review of Book 3 content is found in Appendix A located
on the Interactive Student CD.
Click here
Appendix A
IM4_Ch01_2pp.fm Page 1 Thursday, February 5, 2009 10:10 AM
2
INTERNATIONAL MATHEMATICS
4
This chapter is a summary of the work covered in
International Mathematics 3
. For an explanation
of the work, refer to the cross-reference on the right-hand side of the page which will direct you to
the Appendix on the Interactive Student CD.
1:01
|
Basic Number Skills
Rational numbers:
Integers, fractions, decimals and percentages (both positive and negative) are
rational numbers. They can all be written as a terminating or recurring decimal. The following
exercises will remind you of the skills you should have mastered.
(
g
)
3000
4000
5000
6000
7000
Baby's weight
2
IM4_Ch01_2pp.fm Page 15 Thursday, February 5, 2009 10:10 AM
16 INTERNATIONAL MATHEMATICS 4
Chapter 1 | Working Mathematically
1 Complete a table of values for each matchstick pattern below, and hence find the rule for
each, linking the number of coloured triangles (t) to the number of matches (m).
a
b
c
2 Divide this shape into
three pieces that have
the same shape.
3 Ryan answered all 50 questions in a maths
competition in which he received 4 marks
for each correct answer but lost one mark
for each incorrect answer.
a What is Ryan's score if he answered
47 questions correctly?
b How many answers did he get right if
his score was 135?
4 It takes 3 min 15 s to join two pieces of
pipe. How long would it take to join
6 pieces of pipe into one length?
5 A number of cards can be shared between
4 people with no remainder. When shared
between 5 or 6 people, there are two cards
left over. If there are fewer than 53 cards,
how many are there?
6 From August 2003 to August 2004, the
unemployment rate fell from 6·0% to 5·6%.
a If the number of unemployed in August
2000 was 576 400, how many were
unemployed in August 2003? Answer
correct to four significant figures.
b If 576 400 represents 5·6% of the total
workforce, what is the size of the total
workforce? Answer correct to four
significant figures.
c If the rate of 5·6% is only correct to one
decimal place, the rate could really be
from 5·55% to 5·65%. How many
people does this approximation range of
0·1% represent?
t
n e
m
n
g
i
s
s
a
1
Appendix B
t 1 3 5
m
t 2 4 6
m
t 1 3 6
m
%
6·0
5·5
5·0
6·5
Aug
2003
Nov Feb
2004
May Aug
Unemployment rate
Trend
Source:
Australian
Bureau of
Statistics,
Labour Force,
October 2004.
IM4_Ch01_2pp.fm Page 16 Thursday, February 5, 2009 10:10 AM
Proportion
2
17
I think we'll need more eggs.
Chapter Contents
2:01
Review — Proportional Change
2:02
A New Approach
Investigation: A proportional flip
2:03
Inverse Proportion (Inverse Variation)
Mathematical Terms, Diagnostic Test,
Revision Assignment
Learning Outcomes
Students will be able to:
• Solve problems relating to direct proportion.
• Solve problems relating to inverse proportion.
In Book 3 you learnt that if all parts of an object or shape are increased or decreased by the same
ratio or fraction then the original shape or object and the new shape or object are said to be in
proportion.
150 kg of fertiliser costs $975. How much would 180 kg of the same fertiliser cost?
It costs Ben
3.36 in fuel costs to drive his car 550 km.
a
How much would it cost him to drive 800 km?
b
How far can he drive for $100? (Answer to the nearest whole kilometre.)
Cycling at a steady pace, Bob can cycle 260 km in 4 days.
a
How long would it take him to cycle 585 km?
b
How far would he be able to cycle in 6 days?
worked examples
1 A photograph and its enlargement are in
proportion. One of the photographs measures
10 cm long and 8 cm wide. If the other is 15 cm
long, how wide is it?
2 Three girls can paint 18 m of fence in one day.
How many metres of fence can 5 girls paint if
they all paint at the same rate?
Solutions
1 The two lengths and the two widths go together. Let the unknown width be w.
∴ =
w =
w = 12 cm
2 The number of girls and the amount of fencing go together. Let the unknown amount of
fencing be f.
∴ =
f =
f = 30 m
These are examples of direct proportion because as one item increases (eg: the number of
girls) the other item (the amount of fencing) increases as well. Sometimes these examples
are also called direct variation.
A ratio is the
same as a
fraction.
I The ratio of the widths = the ratio of the lengths
w
8
----
15
10
------
15
10
------ 8 ×
I The ratio of the number of girls = the ratio of the fencing
f
18
------
5
3
---
5
3
--- 18 ×
Exercise 2:01
1
2
3
IM4_Ch02_3pp.fm Page 18 Monday, March 16, 2009 9:25 AM
CHAPTER 2 PROPORTION
19
When making a cake for 4 people, Lotty needs 6 eggs.
a
How many eggs would she need if she were making a cake for 10 people?
b
How big a cake could she make with 9 eggs?
It has been estimated that 38 000-kilojoule survival pack would be enough to sustain 4 people
for a period of 5 days. Keeping this proportion:
a
How many kilojoules would be required for 7 people for 3 days?
b
How many days would a 133 000 kilojoule pack last 4 people?
c
How many people could survive on a 114 000 kilojoule pack for 3 days?
As a general rule a minimum of 60 m
2
is required for every 5 people working in an office.
Keeping this proportion:
a
How much space would be required for 4 people?
b
What would be the maximum number of people that could work in a space of 200 m
2
?
A telephone call lasting 4 minutes 20 seconds cost $6.50. If the cost is proportional to the
length of the call:
a
How much would be the cost of a 5 minute, 10 second call?
b
How long could your call be if you had $9.00 to spend?
A leaky tap leaks 2.25 L of water in 3 hours. At this rate:
a
How much will leak in 5 hours?
b
How long will it take for 5 litres to leak?
Fourteen cans of cat food can feed 10 cats for 3 days. At this rate:
a
How many days would 21 cans feed 10 cats?
b
How many cans would be needed to feed 10 cats for 9 days?
c
How many cans would be needed to feed 15 cats for 3 days?
For a certain species of tree, 5 mature trees can produce the same amount of oxygen inhaled
by 250 people over a 4-year period.
a
Over a 4-year period, how many people inhale the oxygen produced by 7 mature trees?
b
How many trees would be required to produce the oxygen needed for 100 people over
a 4-year period?
c
How many trees would be required to produce the oxygen inhaled by 250 people over
a 10-year period?
4
5
6
7
8
9
10
IM4_Ch02_3pp.fm Page 19 Monday, March 16, 2009 9:25 AM
20
INTERNATIONAL MATHEMATICS
4
2:02
|
A New Approach
The symbol
∝
means 'is proportional to' so if we look again at example 1 from the previous section:
Any length on the enlargement (
L
) is proportional to the corresponding length on the original
photograph (
l
).
We can write this as
L
∝
l
Since, to be in proportion, all the lengths are multiplied by the same factor we can write
L
∝
l
∴
L
=
kl
where
k
=
the
proportional constant
or
constant of variation
So for example 1
L
∝
l
∴ L = kl Now substitute values that are known:
∴15 = k × 10 the lengths of the two photographs.
∴ k =
So now we know that for any length on the original photograph (l), the corresponding length
on the enlargement (L) is given by
L = l
So if the width of the original is 8 cm, the corresponding width of the enlargement is given by
L = × 8 = 12 cm
You can see the same ratio appearing in the working out for both methods.
Now, let's try example 2 from the previous section:
We are looking for the amount of fencing (f) which is proportional to the number of girls (g).
f ∝ g
∴ f = kg
∴18 = k × 3
∴ k = = 6
∴ f = 6g
Now f = 6 × 5
∴ f = 30 m
This may seem like a long way round to get to the answer but it will help in the next section.
15
10
------
15
10
------
15
10
------
18
3
------
IM4_Ch02_3pp.fm Page 20 Monday, March 16, 2009 9:25 AM
CHAPTER 2 PROPORTION 21
Set up a proportion statement using the proportional constant to solve the following questions.
The height (h) of a plant is directly proportional to the number of days (d) it has been growing.
a Write a proportional statement to help you find the height of the plant.
b If the plant is 80 cm high after 3 days
i what is the proportional constant?
ii how high will it be after 5 days?
The mass (m) of a liquid varies directly as its volume (v).
a Write a proportional statement to help you find the volume of the liquid.
b Four litres of the liquid has a mass of 11 kg.
i What is the constant of variation?
ii If the mass of the liquid is 5 kg, what volume is present?
The amount of sealing paint (p) required for a job is proportional to the area (a) to be painted.
a Write a proportional statement to help you find the area to be painted.
b If 5 litres of paint covers an area of 37 m
2
i what is the proportional constant?
ii what area could be covered with 9 litres of paint?
The number of toys (n) produced by a machine is directly proportional to the length of time (t)
it operates.
a Write a proportional statement to help you find the number of toys made.
b If the machine can make 19 toys in 5 hours
i what is the proportional constant?
ii how many whole toys could be made in 12 hours?
The length of a ditch (l) is directly proportional to both the number of men (n) and the amount
of time (t) they spend digging it.
a Write two proportional statements to help you find the length of the ditch.
b If it takes 6 men 4 days to dig a ditch 10 m long, find the length of a ditch that could be dug
by 9 men in 6 days.
c Write the ratio of men:days:length for each case. What do you notice?
Exercise 2:02
1
2
3
4
5
IM4_Ch02_3pp.fm Page 21 Monday, March 16, 2009 9:25 AM
22 INTERNATIONAL MATHEMATICS 4
Investigation 2:02 | A proportional flip
Please use the Assessment Grid on the following page to help you understand what is required
for this Investigation.
Consider the following situations:
1 It takes 5 men 15 hours to build brick wall. How long would it take:
a 1 man? b 3 men?
2 Four women can row a boat 25 km in 3 hours. How long will it take:
a 1 woman to row the same distance?
b 3 women to row the same distance?
3 18 tins of 'Yum' dog food can will last 3 dogs an
average of 5 days.
How long will the same amount of dog food last:
a 1 dog? b 5 dogs?
4 12 boy scouts can paint a fence in a day.
How long would it take:
a 1 scout? b 15 scouts?
Complete the table with answers from the questions.
5 What do you notice about the answers for the time in each case as the number of men,
women, dogs and scouts
i decreases? ii increases?
6 Do you think that these types of problems still refer to proportion? Explain your answer
in as much detail as you can.
7 Try to write two more examples of this type of situation.
i
n
v
e
s
t
igatio
n
2:02
1 Time for 5 men Time for 1 man Time for 3 men
15 hours
2 Time for 4 women Time for 1 woman Time for 3 women
3 hours
3 Time for 3 dogs Time for 1 dog Time for 5 dogs
5 days
4 Time for 12 scouts Time for 1 scout Time for 15 scouts
1 day
IM4_Ch02_3pp.fm Page 22 Monday, March 16, 2009 9:25 AM
CHAPTER 2 PROPORTION 23
Assessment Grid for Investigation 2:02 | A proportional flip problem solving to recognise the patterns.
1
2
c
Mathematical problem solving techniques have been
applied and patterns recognised. A general rule has been
suggested.
3
4
d
The correct techniques have been applied and patterns
recognised are described as a general rule. Conclusions
have been drawn consistent with the results.
5
6
e
All of the above have been achieved and conclusions are
justified with a proof or further examples hard to follow.
1
2
c
There is a sufficient use of mathematical language and
representation. Lines of reasoning are clear but not always
logical or complete.
3
4
d
A good use of mathematical language and representation.
Lines of reasoning draw a
connection to real-life problems.
1
2
c
There is a correct but brief explanation of whether the
results make sense. A description of a real life application
is given in question 7.
3
4
d
There is a critical explanation of whether the results make
sense. Examples given in question 7 provide detailed
applications of inverse proportion.
5
6
IM4_Ch02_3pp.fm Page 23 Monday, March 16, 2009 9:25 AM
24 INTERNATIONAL MATHEMATICS 4
2:03
|
Inverse Proportion (Inverse Variation)
In Investigation 2:02 you found that as the number of one thing in the problem decreased, the other
increased.
For example, if it takes 5 men 15 hours to do a job, it will take 1 man 15 × 5 = 75 hours to do the
same job. So, as the number of men decreases, the time taken to do the job increases.
Likewise, if it takes 1 man 75 hours to do the job, it will take 3 men = 25 hours to do the same
job. So, as the number of men increases, the time taken to do the job decreases.
These are examples of inverse proportion (sometimes referred to as inverse variation).
75
3
------
worked example
If I travel at 50 km/h it will take me 3 hours to
complete my journey. How long will it take to
complete the journey if I travel at 60 km/h?
The faster I travel, the less time it takes, so this is an
example of inverse proportion.
There are 2 methods of solving problems of this type:
Method 1: Unitary method
This requires us to find out how long it
takes when travelling at 1 km/h and then
at 60 km/h.
At 50 km/h it takes 3 hours
At 1 km/h it will take 3 × 50 = 150 hours
At 60 km/h it will take = 2 hours
Method 2: Using a proportional statement
Since this is an example of inverse
proportion, time (t) is inversely
proportional to speed (s).
∴ t ∝
∴ t = this is k ×
Using the information we have been given: 3 =
∴ k = 3 × 50 = 150
∴ k =
So to solve the problem: t =
∴ t = 2 hours
The time taken is inversely
proportional to the speed
travelled. In other words:
hasta la vista baby!
Remember: write the
proportional statement
then work out k.
I'm not that complicated.
150
60
---------
1
2
---
1
s
---
k
s
---
1
s
---
k
50
------
150
s
---------
150
60
---------
1
2
---
IM4_Ch02_3pp.fm Page 24 Monday, March 16, 2009 9:25 AM
CHAPTER 2 PROPORTION 25
A water tank is leaking so that after 4 hours the water in the tank is only 20 cm deep. How
deep will the water be after 10 hours?
Eight people can survive on the provisions in a life raft for 6 days. For how long could 3 people
survive on the same provisions?
If 4 boys can mow 10 lawns in three days, how many days would it take 6 boys?
Five teenage boys can eat 3 jumbo pizzas in 30 minutes. How long would the pizzas last if
there were 4 teenage boys sharing them?
A Boeing 747 jumbo jet with a cruising speed of Mach 0.85 flies from Singapore to London in
14 hours. How long would it take an Airbus A320 with a cruising speed of Mach 0.82 to fly the
same route? Answer to the nearest minute.
A team of 10 fruit pickers can clear 4 hectares
of fruit in a week.
a How many days would it take 14 fruit pickers?
b If the 4 hectares had to be cleared in 2 days,
how many fruit pickers would be needed?
Typing at 30 words per minute, Michael will be
able to finish his essay in 2 hours. His friend
Michelle says that she would be able to finish it
in 1 and a half hours. At what speed must she be
able to type to do this?
If it takes 5 cats 2 days to catch 6 mice, how
many days will it take 3 cats to catch 9 mice?
Hint: It might be useful to set up a table with
the headings shown.
If 3 boys take 2 days to mow 3 lawns, how many boys can mow 6 lawns in 3 days?
If 2 workers can paint 12 metres of fence in 3 hours, how many metres of fence can 4 workers
paint in 2 hours?
Cats Days Mice
Exercise 2:03
1
2
3
4
5
6
7
8
9
10
IM4_Ch02_3pp.fm Page 25 Monday, March 16, 2009 9:25 AM
26 INTERNATIONAL MATHEMATICS 4
Mathematical terms 2
Mathematical Terms 2
Ratio
• A comparison of two quantities
Proportion
• When all aspects of an object or shape
are increased or decreased by the
same amount
Proportional constant
• The numerical factor by which both
objects are increased or decreased in
the given ratio
Direct proportion
• When both variables increase at a given
ratio, ie if one number is multiplied by
a constant then the other is multiplied
by the same value
Inverse proportion
• When one variable increase at a given
ratio as the other decreases, ie if one
number is multiplied by a constant then
the other is divided by the same value
Diagnostic Test 2: | Proportion
• Each section of the test has similar items that test a certain type of example.
• Failure in more than one item will identify an area of weakness.
• Each weakness should be treated by going back to the section listed.
s
m r e
t
l
a
c
i
t
a
m
e
h
t
a
m
2
t
s e t
c
i
t
s
o
n
g
a
i
d
2
1 Which of the following are examples of direct proportion and which are
examples of inverse proportion?
a The ingredients of a cake and the number of people that are going
to eat it.
b The number of people eating a cake and the time taken to eat it.
c The speed at which you travel and the time taken to finish the journey.
d The speed at which you travel and the distance you are from your
starting point.
e The number of horses in a stable and the amount of fodder for
required to feed them.
2 If I am travelling at 50 km/h, I am 40 km from my starting point.
How far would I be if I had been travelling at 80 km/h?
3 The SRC has estimated that 30 pieces of pizza will be needed to feed the
12 people attending a meeting. How many pieces would be needed to
feed 18 people?
4 The 30 pieces of pizza in the previous question will last the 12 people
20 minutes. How long would the same number of pieces last 18 people?
5 The amount of water in a leaking tank is inversely proportional to the
time it has been leaking. If, after 4 hours there are 50 litres left in
the tank, how many litres will be left after 5 hours?
Section
2:01 to 2:03
2:01
2:01
2:03
2:03
IM4_Ch02_3pp.fm Page 26 Monday, March 16, 2009 9:25 AM
CHAPTER 2 PROPORTION 27
6 It is estimated that when travelling from Singapore to Frankfurt,
a distance of approximately 10 000 km, a person is responsible for
releasing 1300 kg of CO
2
into the atmosphere. If emissions are
proportional to the distance flown
a what is the proportional constant and what does it represent?
b how many kilograms of CO
2
is a person responsible for if they fly
from Bangkok to Sydney, a distance of 7500 km? Answer to the
nearest kilogram.
7 At a particular, time Emma's distance from home is directly proportional
to the speed at which she travels. If her proportional constant is 8
a how far is she from home if she travels at 55 km/h?
b at what speed must she be traveling if she is 240 km from home?
c what does the proportional constant represent in the context of
the problem?
8 When swimming the 1500 m freestyle, the time Marko has remaining to
swim is inversely proportional to the speed at which he is swimming.
If his proportional constant is 600:
a What time has he left when swimming at 100 metres/minute?
b How fast must he be swimming if he has 12 minutes left?
c What does the proportional constant represent?
Section
2:02, 2:03
2:01, 2:02
2:02, 2:03
1
2
---
IM4_Ch02_3pp.fm Page 27 Monday, March 16, 2009 9:25 AM
28 INTERNATIONAL MATHEMATICS 4
Chapter 2 | Revision Assignment
1 When cooking rice, the ratio of water to
rice is 4:3. How much water will I need to
cook 500 grams of rice?
2 Jasmine receives a pay increase in the
ratio 9:7. If her weekly income was $550,
what will be her new weekly wage?
3 A 4 litre can of paint will cover 18 m
2
of
wall. Write down a formula connecting the
number of litres, n, and the area covered,
A. How many litres would be needed to
paint a room with wall area 40·5 m
2
?
4 The speed of a falling object is directly
proportional to the time from release. If the
speed after 5 seconds is 49 m per second:
a what will the speed of the object be at
8 seconds?
b how long till the object reaches
100 m/sec?
5 In order to buy goods from overseas you
must work out how many Australian
dollars (AUD$) are equal to the amount
in the overseas currency.
Eg, if AUD$1.00 is worth USD$0.7370,
then USD$15.00 ÷ 0·7370 = AUD$20.35.
Use the information above and below to
answer parts a and b:
AUD$1.00 is worth 6·03 Chinese yuan,
or 78·14 Japanese yen.
a Convert these amounts to AUD$.
USD$50.00
30.00 yuan
2000 yen
b When arriving overseas, Australian
tourists convert Australian dollars into
the local currency. Convert
AUD$100.00 into USD$, Chinese yuan
and Japanese yen.
6 If it takes me 2 hours to drive to my beach
house at 80 km/h, how long would it take
me if I drove at 100 km/h?
7 If 35 workers could do a job in 5 days
a how long would it take 20 workers?
b how many workers are required to
complete the job in 3 days?
t
n e
m
n
g
i
s
s
a
2A
IM4_Ch02_3pp.fm Page 28 Monday, March 16, 2009 9:25 AM
Consumer
Arithmetic
3
29
Well it's about time.
Your pay has arrived!
Mum, can I have some money?
Some people work for themselves and charge a fee for their services or sell for a profit, but most
people work for others to obtain an income. In the chart below, the main ways of earning an
income from an employer are introduced.
Employment
Salary Piece work Casual Commission Wages
Meaning
A fixed amount is
paid for the year's
work even though it
may be paid weekly
or fortnightly.
The worker is paid
a fixed amount for
each piece of work
completed.
A fixed rate is paid
per hour. The
person is not
permanent but is
employed when
needed.
This payment is
usually a percentage
of the value of
goods sold.
Usually paid weekly
to a permanent
employee and based
on an hourly rate,
for an agreed
number of hours
per week.
Advantages
Permanent
employment.
Holiday and sick
pay. Superannuation.
A bonus may be
given as an incentive
or time off for
working outside
normal working
hours.
The harder you
work, the more you
earn. You can
choose how much
work you do and in
some cases the work
may be done in your
own home.
A higher rate of pay
is given as a
compensation for
other benefits lost.
Part time work may
suit some or casual
work may be a
second job.
Superannuation.
The more you sell
the more you are
paid. Some firms
pay a low wage plus
a commission to act
as an incentive.
Permanent
employment.
Holiday and sick
pay. Superannuation.
If additional hours
are worked,
additional money
is earned, sometimes
at a higher hourly
rate of pay.
Disadvantages
During busy
periods, additional
hours might be
worked, without
additional pay. Very
little flexibility in
working times
eg 9 am–5 pm
No holiday or sick
pay. No fringe
benefits. No
permanency of
employment in
most piece work.
No holiday or sick
pay. No permanency
of employment.
Few fringe benefits.
Less job satisfaction.
There may be no
holiday or sick pay.
If you sell nothing
you are paid
nothing. Your
security depends on
the popularity of
your product.
There is little
incentive to work
harder, since your
pay is fixed to time
not effort. Little
flexibility in
working times
eg 9 am–5 pm
Salary Piece work Casual Commission Wages
teachers dressmakers swimming instructors sales people mechanics
IM4_Ch03_3pp.fm Page 30 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC
31
Superannuation, sometimes called a pension fund or pension plan, is an investment
fund usually contributed to by both employer and employee on the employee's behalf.
It provides benefits for employees upon retirement, or for the widow or widower if the
member dies.
worked examples
1 Use the information on the right to answer these
questions.
a How much would an employee earn in a week
if no sales were made?
b If Jane sold $18 000 worth of building products
in one week, how much would she earn?
c If Peter sold $24 000 worth of materials in
one week and $5000 worth in the next,
find his average weekly income for the
two weeks.
2 Luke has a casual job from 4:00 pm till 5:30 pm Monday to Friday. He also works from
9 am till 12:30 pm on Saturdays. Find his weekly income if his casual rate is $8.80 per
hour Monday to Friday, and $11.50 an hour on Saturdays.
Solutions
1 a Week's earnings = $100 + 2% of $0
= $100 + $0
∴ Employee making no sales is paid $100.
b Jane's earnings = $100 + 2% of $18 000
= $100 + 0·02 × $18 000
= $460 in the week
c Week 1
Peter's earnings = $100 + 2% of $24 000
= 100 + 0·02 × 24 000
∴ Earnings week 1 = $580
Week 2
Peter's earnings = $100 + 2% of $5000
= $100 + 0·02 × $5000
∴ Earnings week 2 = $200
∴ Peter's average weekly wage = ($580 + $200) ÷ 2
= $390
2 Luke's weekly income = (hours, Mon–Fri) × $8.80 + (hours, Sat) × $11.50
= (1 × 5) × $8.80 + 3 × $11.50
= 1·5 × 5 × $8.80 + 3·5 × $11.50
= $106.25
POSITIONS VACANT
5 people required to
promote our nationally
known building product in
the suburbs.
Pay: $100 pw and
2% commission.
Please phone YRU-POOR
during business hours.
1
2
---
1
2
---
IM4_Ch03_3pp.fm Page 31 Monday, March 16, 2009 9:33 AM
32
INTERNATIONAL MATHEMATICS
4
Write answers in your own words. Refer to page 30 if necessary.
a
What are the advantages of working for a wage?
b
What is piece work?
c
What is a salary?
d
What form of payment gives the worker a percentage
of the value of goods sold?
e
What advantages are there in casual work?
f
What are the disadvantages of being on a salary?
g
What are wages?
h
Which forms of payment depend on success or the
amount of work completed?
i
What are the disadvantages of casual work?
j
Which two forms of payment are often combined
in determining a worker's pay?
Wages and salaries
a
A man is paid $18.50 an hour for a 35-hour week.
What is his normal weekly wage?
b
A boy is paid a wage based on $9.15 an hour.
How much is he paid for an 8-hour day of work?
c
For a 38-hour working week a woman is paid $731.50.
Find her hourly rate of pay?
d
Adam is paid
16.05 an hour for a 35-hour week.
Luke receives
15.75 for a 38-hour week. Who has
the higher weekly wage and by how much?
e Irene is paid $594.70 for a 38-hour week, while Shireen
is paid $565.25 for a 35-hour week. Who has the higher
rate of pay and by how much?
f A painter works a 38-hour week for an hourly rate of
£19.65. An extra height allowance of 95 pence per hour
is paid. Find his total weekly wage.
g A woman is paid a salary of $46 089 per year. How much
would she receive each week if it is calculated on
52·178 weeks in a year? (Answer to nearest dollar.)
h Find the weekly income (assuming there are 52·178 weeks
in the year) for a salary of:
i 43 000 ii $26 400
iii ¥895 000 iv $58 200
(Give answers to the nearest cent.)
i Find the yearly salary of a person whose monthly income is:
i 4600 ii $3150.50 iii $5305 iv CNY194 750
j Two jobs are advertised: one with a salary of $55 000, the other a salary with a fortnightly
payment of $2165.60. Which is the greater weekly salary and by how much? Use 'one year
equals 52·178 weeks.' (Give your answer correct to the nearest cent.)
Exercise 3:01
1
Use your calculator!
I Salary
It is assumed that each
day of the year, the
salaried person earns
of the salary.
There are 365 days,
on average, in a year.
∴ On average, 52·178
(approximately) weeks
are in each year.
1
365
1
4
--
------------
1
4
---
2
IM4_Ch03_3pp.fm Page 32 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 33
k Two jobs are advertised. One is based on 37 hours per week at $20.15 an hour, the other is
a yearly salary of $39 400. If one year is taken to be 52·178 weeks, which weekly income is
higher and by how much? (Answer to the nearest cent.)
Commission, piece work and casual work
a i Does this job guarantee an income?
ii If you have never heard of the products
of this company, is it likely that you
will sell much of their product?
iii Is any compensation mentioned for
petrol used or provision of a vehicle?
iv Find the commission paid on sales of:
1 300 2 743
3 1658 4 92
b Janice is offered a sales position with a retainer (guaranteed wage) of $140 plus a
commission of 7% on sales.
i How much could she make in one week if her weekly sales total were:
1 $800? 2 $3500? 3 $4865? 4 $5213?
ii She is told that the average weekly sales per person is $6300. What is the average
weekly income?
iii If Janice is a shy person who has no previous sales experience, is it likely that she will
succeed in this job?
c John works as a sales assistant receiving $300 per week plus 10% commission on sales in
excess of $5000. Find his weekly income if, in one week, the amount of his sales was:
i $3400 ii $5700 iii $8424 iv $6129.50
a Heather works in a supermarket on a casual basis. She is paid £16.60 an hour from
Monday to Friday and £20.85 an hour on Saturdays. Find her week's income if she
works from 3:30 pm till 5:30 pm, Monday to Friday, and from 8:30 am till 1:00 pm
on Saturday.
b Edward works as a waiter from 6:00 pm till 1:30 am four days in one week. His hourly rate
of pay is £18.35 and he gets an average of £6.50 as tips per working night. Find his income
for the week. (A 'tip' is a payment in appreciation of good service.)
c An electrician charged £35.80 per hour for labour. Find the charge for labour if he works
from 11:20 am till 1:50 pm.
a A factory worker was paid $2.16 for each garment completed. How much would be earned
if 240 garments were completed?
b A doctor charges each patient $37.50 for a consultation. If she works for 5 hours during one
day and sees an average of six patients per hour, find the amount of money received that
day. Her costs per day are $343. What was her profit for the day?
c Smokey and Smiley were two shearers who were paid $2.15 for each sheep shorn. By how
much was Smokey's pay greater than Smiley's, if Smokey was able to shear 673 sheep while
Smiley was able to shear only 489?
CLEAN-U-UP PTY LTD
Selling cleaning machinery,
equipment and chemicals.
Sales people required to sell on
total commission of 23% of sales.
Great potential!
Excellent reward!
Ring: Ugo Broke. YRU-000.
3
4
5
IM4_Ch03_3pp.fm Page 33 Monday, March 16, 2009 9:33 AM
34 INTERNATIONAL MATHEMATICS 4
d A tiler charges $30.40 per square metre to lay tiles. Find how much he would charge to lay
an area of:
i 9·4 m
2
ii 6·25 m
2
iii 18·2 m
2
iv 15 m
2
e Flo works at home altering dresses for a dress shop. She is paid 14.95 for a major
alteration and 6.80 for a small alteration. In the week before Christmas she completed
13 major alterations and 27 small alterations. Find her income for the week. If she spent
39 hours working on the alterations what was her hourly rate of pay? (Answer to the
nearest cent.)
3:02
|
Extra Payments
There are several additional payments that may add to a
person's income. Terms needed are listed below.
Overtime: This is time worked in excess of a standard
day or week. Often rates of 1 or 2 times the normal
rate of pay are paid for overtime.
Bonus: This is money, or an equivalent, given in addition
to an employee's usual income.
Holiday bonus: A payment calculated as a fixed
percentage of the normal pay over a fixed number
of weeks. It may be paid at the beginning of annual
holidays to meet the increased expenses often
occurring then.
Time card: This card is used to record the number of
hours worked in a week. A time clock is used to stamp
the times onto the card. Therefore a worker 'clocks on'
in the morning and 'clocks off' in the evening.
3
4
---
I'm working
overtime . . .
1
1
2
---
2
3
4
TIME CARD
Whit. Pty Ltd
No. 53
Name:
Week
ending
Fri
21 Jan
Fri
28 Jan
Fri
4 Feb
Fri
11 Feb
Fri
18 Feb
Tom McSeveny
Day
Sat
Sun
Mon
Tues
Wed
Thu
Fri
–
–
7:57
7:58
8:00
8:02
8:00
–
–
4:00
4:02
4:01
4:05
4:00
–
–
8:00
7:55
8:00
7:58
8:00
–
–
4:04
3:59
4:02
7:00
4:00
8:00
–
7:59
7:56
8:03
7:58
8:00
10:02
–
4:00
4:02
4:01
4:03
4:01
8:00
–
8:00
8:00
7:56
8:01
8:02
12:00
–
4:02
4:05
3:02
4:02
6:31
8:02
_
7:57
8:00
7:58
7:55
7:59
11:30
–
3:59
4:05
4:07
6:00
6:30
Hourly rate:
Lunch:
Normal hours:
Overtime:
$16.20
12 noon till 1:00 pm (unpaid)
Mon-Fri, 8:00 a.m. – 4:00 p.m.
'Time-and-a-half' is paid and 'double-time' for overtime in excess of 3 hours (on any one day)
IN OUT IN OUT IN OUT IN OUT IN OUT
IM4_Ch03_3pp.fm Page 34 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 35
Note: 1 In the week ending 21 Jan, no overtime was worked.
Total of hours worked = (8 hours × 5) − 5 hours for lunch
= 35 hours
2 In the week ending 11 Feb, only 34 normal hours were
worked, as Tom left work 1 hour early on Wednesday.
However, 2 hours overtime was worked on Friday
and 4 hours on the Saturday. Three of the hours worked
on Saturday are time-and-a-half and one (that in excess
of 3 h) was at double-time.
I A few minutes
variation from the hour
or half-hour will not be
considered in determining
hours worked.
1
2
---
worked examples
1 During one week Peter worked 35 hours at the normal rate
of $11.60 per hour. He also worked 6 hours overtime:
4 at 'time-and-a-half' and 2 at 'double-time'. How much did
he earn?
2 Use the time card on the previous page to calculate
Tom McSeveny's wage for the week ending Friday,
18 February.
3 Calculate Diane's holiday bonus if she is given 17 % of
four weeks salary and she earns $980 per fortnight.
Solutions
1 Peter's earnings = (35 h at $11.60) + (4 h at $11.60 × 1 ) + (2 h at $11.60 × 2)
= (35 × 11·6) + (4 × 11·6 × 1·5) + (2 × 11·6 × 2) dollars
= $522 (using a calculator)
2 For the week ending Friday, 18 February, Tom worked:
'Normal hours': 35 hours (8–4, Mon–Fri with 1 hour lunch)
Overtime—'Time-and-a-half': 7 hours (8–11 on Sat, 4–6 on Thur, 4–6:30 on Fri)
'Double-time': hour (11–11:30 on Sat as double-time
is paid only after 3 hours)
∴ Tom's earnings = (35 h at $16.20) + (7 h at $16.20 × 1 ) + ( h at $16.20 × 2)
= (35 × 16·2) + (7·5 × 16·2 × 1·5) + (0·5 × 16·2 × 2) dollars
= $765.45 (using a calculator)
3 Diane's holiday loading = 17 % of four weeks salary
= 17 % of ($980 × 2)
= 0·175 × (980 × 2) dollars
= $343
I don't get paid
for lunch . . .
1
2
---
time-and-a-half double-time
1
2
---
1
2
---
1
2
---
1
2
---
1
2
---
1
2
---
I 17 % =
= 17·5 ÷ 100
= 0·175
1
2
---
17·5
100
-----------
1
2
---
1
2
---
IM4_Ch03_3pp.fm Page 35 Monday, March 16, 2009 9:33 AM
36 INTERNATIONAL MATHEMATICS 4
a Bill earns $9.60 per hour. Calculate his wages for
the week if he worked 35 hours at the normal rate
and 5 hours overtime at 'time-and-a-half'.
b At Bigfoot Enterprises a wage rate of $12.70 per hour
is paid on the first 37 hours and 'time-and-a-half' after that.
What is the wage for a 42-hour week?
c Each day Pauline receives $18.10 per hour for the first 7 hours, 'time-and-a-half' for the
next 2 hours, and 'double-time' thereafter. Find her wage for:
i a 6-hour day
ii a 9-hour day
iii an 8 -hour day
iv an 11-hour day
d An electrician earns a wage of 22.40 per hour for a 35-hour week and 'time-and-a-half'
after that. How much would he earn in a week in which he works:
i 30 hours
ii 37 hours
iii 41 hours
iv 45 hours
e A pipe factory asks a labourer to work 8 hours on Saturday at 'time-and-a-half' for the first
3 hours and 'double-time' after that. If his normal rate of pay is $20.40, how much is he
paid for the day's work?
f Brian earns $17.60 an hour, while his boss earns $23.20 an hour. How much more than
Brian is the boss paid for a 7-hour day? If Brian gets
'time-and-a-half for overtime, how many additional
hours would he need to work in a day to get the same
wage as his boss?
g By referring to the time card on page 34,
complete the summary below.
No. 53 Time Card Summary Whit. Pty Ltd.
Name: Tom McSeveny Rate: $16.20 ph
Week
ending
Number of hours at: Wage
normal rates time-and-a-half double-time
21 Jan
28 Jan
4 Feb
11 Feb
18 Feb
Exercise 3:02
Extra payments
1 Calculate:
a $15 × 1 × 4 b $17.50 × 1 × 7
2 Calculate:
a $18 × 35 + $18 × 1 × 4
b $22 × 36 + $22 × 1 × 6
1
2
---
1
2
---
1
2
---
1
2
---
Foundation Worksheet 3:02
1
1
2
---
1
2
---
1
2
---
I The tricky parts of the
time card on page 34 are
in colour.
IM4_Ch03_3pp.fm Page 36 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 37
a If 17 % holiday bonus is given on 4 weeks normal pay, find the holiday bonus for:
i John, who earns $4000 in 4 weeks
ii Mary, who earns $822 in a fortnight
iii Wilkes, who earns $495 a week
iv Conway, who earns $19.80 an hour (for a 35-hour week)
(Assume that in each case no extra payments are included.)
b When June was given her holiday pay she received 4 weeks pay and a 17 % holiday bonus.
If her normal wage is 427 per week, how much holiday pay did she receive?
c Franko works a 35-hour week at a rate of $17.60 per hour. Calculate his holiday bonus
if 17 % is given on 4 weeks wage.
d Mr Bigsuccess earns a salary of $96 000 per year. At the end of the year he is given a bonus
equal to 80% of one month's pay. How much did he earn in the year?
e Alana managed a small business for a salary of $56 400. At the end of a successful year in
which the business made a profit of $211 000, she was given a bonus of 1·2% of the profits.
What was her bonus and what was her income for the year?
f Luke works for a mining company at a wage rate of $22.80 per hour. If he works
underground he is paid a penalty rate of $3.65 per hour in addition to his normal pay.
Find his weekly wage if during the normal 38 hours he works underground for 16 hours.
g Mary works in a food processing plant at a wage rate of $15.95 per hour. From time to time
she is required to work beside ovens where the temperature is uncomfortable. When this is
necessary she is paid an additional 95 cents an hour. Calculate her wage for a normal
working week of 36 hours where 5 hours were beside the ovens.
h Lyn received a 17 % holiday bonus on four weeks normal wages. (She works a 36-hour
week.) Find her normal weekly wage if the total of four weeks wages and the holiday
bonus was:
i $2350 ii $2162
i Sundeep received a holiday bonus payment of £364 that represented 17 % on
four weeks wages. What is his weekly wage? What percentage of his total income for
the year (containing 52 weeks) does the holiday bonus represent? (Answer correct
to 2 decimal places.)
Ayse's salary is 47 300, but she expects it to rise 2% each year to keep pace with inflation.
a After the beginning of the second year, what would Ayse expect her salary to be?
b What would the salary be at the beginning of the third year?
c What would the salary be at the beginning of:
i the sixth year? ii the eleventh year?
Investigation 3:02 | Jobs in the papers
1 Look in the careers section of a newspaper to see what salaries are offered for various
employment opportunities.
2 Examine the positions vacant columns for casual jobs. How many mention the pay offered?
What level of pay is offered in different fields of employment?
3 What other benefits are mentioned in employment advertisements, other than the wage
or salary?
2
1
2
---
1
2
---
1
2
---
1
2
---
1
2
---
3
n
o
i
t
a
g i t s e v
n
i
3:02
IM4_Ch03_3pp.fm Page 37 Monday, March 16, 2009 9:33 AM
38 INTERNATIONAL MATHEMATICS 4
3:03
|
Wage Deductions
A person's weekly wage or salary is referred to as the weekly gross pay. After deductions have been
made the amount actually received is called the weekly net pay.
Possible deductions
Income tax The Commonwealth government takes a part of all incomes earned to
finance federal, state and local government activities. Employers deduct this
tax on the government's behalf at the end of each pay period. It is called
PAYE (pay-as-you-earn). The rate of tax varies according to the amount of
money earned and the number of dependants.
Superannuation This is a form of insurance or investment. Usually both the employee and the
or pension fund employer contribute to this fund on behalf of the employee. It provides for
an income or lump-sum payment on retirement, and in the event of the
member's death, it provides a pension for the family.
Miscellaneous Other deductions could be for medical insurance, life insurance, home
payments, credit union savings and union membership fees.
Here is a pay advice slip representing two weeks' pay.
1
2
3
Peter
Newby
6552750
400.50 40.60 163.80 604.90 1797.10 1797.10
AA 8436 46884.70 113 113
– –
4/11/02 12381 1192.20
Serial
No.
Dept. Location
Taxation S'annuation M'laneous Total Normal
Pay
Adjustments Gross
Earnings
Overtime
Deductions this fortnight Pay this fortnight
Gross
Salary
or Wage
Rate
Entld.
Super Units Fortnight
Ended
Net
Pay
Pay
Advice7 8 6 2 3 4
1 5
IM4_Ch03_3pp.fm Page 38 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 39
Peter Newby has a gross yearly salary or wage of $46 884.70, earns $1797.10 per fortnight
and in the fortnight ending 4/11/02 had no adjustments or overtime . This means that his gross
earnings were $1797.10 . His net pay was only $1192.20 because his total deductions
were $604.90.
Peter paid $400.50 tax on an income of $1797.10. This is about 22%. His contribution to
superannuation was $40.60.
1 2
3
4 5 6
7
8
worked examples
Find the net pay for the week if John earns $423.60, is taxed $67.80, pays $32.10 for
superannuation and has miscellaneous deductions totalling $76.30. What percentage of
his gross pay did he pay in tax?
Solution
Total deductions = Tax + Superannuation + Miscellaneous
= $67.80 + $32.10 + $76.30
= $176.20
∴ John's net pay = $423.60 − $176.20
= $247.40
John's tax payment = $67.80
Tax as a percentage of gross pay
= × 100%
= × 100%
A 16%
I' ve just collected my net pay.
tax
gross pay
-----------------------
$67.80
$423.60
--------------------
IM4_Ch03_3pp.fm Page 39 Monday, March 16, 2009 9:33 AM
40 INTERNATIONAL MATHEMATICS 4
Find the net pay if:
a gross pay is $315.60 and total deductions are 115
b gross pay is $518.20, tax is $99.70, superannuation
is $20.70 and miscellaneous is $94.80
c gross pay is $214.55, tax is $22.20, superannuation
is $4.80 and union fees are $2.60
d gross pay (for a fortnight) is £1030.40,
superannuation is £48.50, miscellaneous is £174.70
and tax is £303.55
e gross salary is $612.10, superannuation is $31.90,
medical insurance is $21.60, life insurance is $4.10,
house payment is $76.50, credit union savings are
$11.00, union fees are $2.45 and tax is $131.60
a Vicki Turner receives a yearly salary of $32 096, pays 16% of her weekly gross
(calculated on 52·18 weeks in the year) in income tax, pays 5% of her weekly gross
to her superannuation fund and has $86 in miscellaneous deductions each week.
Find her:
i weekly gross salary
ii weekly tax deductions
iii weekly superannuation payment
iv weekly net pay
b Use question 2a to complete this pay advice slip.
Exercise 3:03
1
Wow! You must have
some fishing spot!
Where'd you net this
stuff?
At the
river bank!
2
I 'Net pay' is what
you take home.
Turner,
Vicki
6841672 98 98
– –
18/11/02 11364
Serial No.
Taxation S'annuation M'laneous Total Normal
Pay
Adjustments Gross
Earnings
Overtime
Deductions this week Pay this week
Gross Salary
or Wage Rate
Entld.
Super Units Week
Ended
Net
Pay
Pay AdviceIM4_Ch03_3pp.fm Page 40 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 41
Find the net pay and the tax as a percentage of the gross pay for each person.
a Upon retirement, Joe Simmons received annual superannuation payments of 68% of his
final year's salary. If his salary at that time was 62 600, how much is his annual
superannuation (before tax)?
b Ellen's annual superannuation is 63% of her finishing wage of 52 186. How much would
her net monthly income be if she pays 22% of her gross income in taxation payments?
(Answer correct to the nearest cent.)
c John earns 32 492 a year. If he were to die, his widow would receive 65% of this figure in
superannuation payments each year. What would be her weekly income (before tax) taking
1 year to be 52·18 weeks?
d Jim has just retired. He has the option of receiving a monthly payment of 3667 or 330 000
as a 'lump sum' (a final single payment).
i Find the yearly superannuation payment.
ii What yearly income would result if the
lump sum could be invested at 12% p.a.?
iii Which option seems most attractive
and why?
iv What would his yearly income be if he
elected to receive 30% of the monthly
payment, and 70% of the lump sum of
which he invested 40% at 12% p.a.?
e Janice retired in 1995 on a fixed income of
2400 per month. How many toothbrushes
costing 1.80 could she buy with a month's income?
One year later inflation had caused the cost
of toothbrushes to rise by 8%. How many toothbrushes could she buy with a month's
income after the rise? (As years pass, inflation greatly affects the purchasing power of people
on fixed incomes.)
Maryanne's gross pay for a week is $874.20. Her employer must pay an additional 9% of this
amount into a superannuation fund for Maryanne. Maryanne chooses to also pay 5% of her
weekly pay into this fund.
a How much is being paid into the superannuation fund each week?
b What is the total cost to the employer each week of employing Maryanne?
c Maryanne receives a 4% pay rise. By how much will the superannuation contributions
increase?
Name Gross Pay Tax Net Pay Tax as % of
Gross Pay
a R. Collison $ 385.70 $ 56.40
b G. Foster $1450.00 $500.75
c B. Jones $ 947.50 $265.15
d R. Sinclair $ 591.60 $124.65
3
4
I Challenge
1 Expected returns in superannuation,
based on average life span, could be
calculated.
2 Effects of inflation (with or without
indexation) on these expected
returns could be considered.
5
IM4_Ch03_3pp.fm Page 41 Monday, March 16, 2009 9:33 AM
42 INTERNATIONAL MATHEMATICS 4
3:04
|
Taxation
• Many countries have an Income Tax Return form which is filled out each year, to determine the
exact amount of tax that has to be paid, for the preceding 12 months. Since most people have
been paying tax as they have earned their income, this exercise may mean that a tax refund is given.
• Some expenses, such as those necessary in the earning of our income, are classified as tax
deductions and the tax we have paid on this money will be returned to us. On the other hand, if
we have additional income (such as interest on savings) that has not yet been taxed, additional
taxes will have to be paid.
The tax deductions are subtracted from the total income to provide the taxable income.
• The table below gives an example of an index taxation system in which the amount of tax you pay
varies according to your taxable income.
TABLE 1—Resident for full year
Taxable income Tax on this income
$1–$6000 Nil
$6001–$20 000 17 cents for each $1 over $6000
$20 001–$50 000 $2380 + 30 cents for each $1 over $20 000
$50 001–$60 000 $11 380 + 42 cents for each $1 over $50 000
$60 001 and over $15 580 + 47 cents for each $1 over $60 000
worked examples
Alan received a salary of $47 542 and a total from other income
(investments) of $496. His total tax deductions were $1150. During the
year he had already paid tax instalments amounting to $10 710.75. Find:
1 his total income
2 his taxable income
3 the tax payable on his taxable income
4 his refund due or balance payable
5 how much extra Alan would receive each week if he is given a wage rise
of $10 per week
Solution
1 Alan's total income 2 Alan's taxable income
= $47 542 + $496 = total income − tax deductions
= $48 038 = $48 038 − $1150
= $46 888
3 Taxable income = $46 888 (or $20 000 + $26 888)
Tax on $20 000 = $2380.00 (from the table on page 304) . . .
Tax on $26 888 at 30 cents = $8066.40 (30c/$ for amount over $20 000) . . .
∴ Tax on $46 888 = +
= $2380 + $8066.40
= $10 446.40
A
B
A B
IM4_Ch03_3pp.fm Page 42 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 43
Use the table on page 42 to determine the tax payable
on a taxable income of:
a $3963 b $12 587 c $31 460
d $67 346 e $284 914
Mrs Short has a salary of $33 600, receives income from other sources of $342, has tax
deductions of $655, and has paid PAYE tax instalments throughout the year of $6570.
Find:
a her total income
b her taxable income
c tax payable on her taxable income
d her refund or balance payable
When Joy left school she had a weekly wage of $488. During the financial year, there were
52 weeks' pay received. (Note: In a normal year of 52 weeks and one day, there could be
53 paydays.) She had no extra income and calculated her tax deductions to be $217.
Find the tax payable on her taxable income.
Karl earned $52 850 as a tiler in 2002/03. His employer deducted tax payments of $13 110.
However, Karl earned a further $4156 on weekends and during his holidays. His tax deductions
came to $2096 as his expenses in earning the additional income were considerable. Find:
a Karl's total income
b Karl's taxable income
c the tax on his taxable income
d additional tax payable (balance).
Eight workers in a factory were each given a pay rise of $2000 per annum. How much of the
$2000 would each have received after tax?
Their yearly taxable incomes are listed below.
(Use the tax table on page 42.)
a M. Callow: $3900 b A. Smith: $9322
c P. Farmer: $16 112 d M. Awad: $28 155
e R. Sissi: $41 300 f P. Mifsud: $55 788
g R. Ringe: $63 950 h S. Sze: $82 000
4 Tax on $46 888 = $10 446.40
Tax instalments paid = $10 710.75
∴ Refund due = $10 710.75 − $10 446.40
= $264.35
5 For salaries over $20 000 and less than $50 001, for each additional $1 earned you pay
30 cents tax.
∴ Tax on an extra $10 per week = 10 × $0.30
= $3.00
∴ Amount left after tax = $10 − $3.00
= $7.00 per week
Exercise 3:04
Taxation
1 Find:
a 30c × 7300 b 42c × 5600
2 Calculate the tax payable on
a taxable income of:
a $15 000 b $40 000
Foundation Worksheet 3:04
1
2
3
4
5
IM4_Ch03_3pp.fm Page 43 Monday, March 16, 2009 9:33 AM
44 INTERNATIONAL MATHEMATICS 4
Fun Spot 3:04 | What is brought to the table, cut, but never eaten?
Work out the answer to each question and put the letter for that part in the box that is above
the correct answer.
Express as a percentage:
A 0·76 A 0·125 A
O 12% of $81 250 C $13.50 × 38 + $20.25 × 6
C Sid is paid $22.80 an hour for a 40-hour week. What is his normal weekly wage?
D For a 38-hour week Naomi is paid $744.80. Find her hourly rate of pay.
F My salary is $31 306.80 per year. How much would I receive each week if it is calculated
on 52·178 weeks in a year?
S Alana has a casual job from 4:00 pm till 5:30 pm Monday to Friday. What is her weekly
income if her casual rate is $9.20 per hour?
P A tiler charges $28.50 per square metre to lay tiles. How much would he charge to lay an
area of 8 square metres?
K Find the net pay for one week if Anne earns $520.50, is taxed $104.10, pays $8 for
superannuation and has miscellaneous deductions totalling $77.40.
R In the previous question, what percentage of Anne's gross pay was paid in tax?
f
u
n
spo
t
3:04
3
5
---
6
0
%
$
2
2
8
7
6
%
$
6
3
4
.
5
0
$
3
3
1
$
9
7
5
0
$
6
0
0
$
9
1
2
1
2
·
5
%
2
0
%
$
1
9
.
6
0
$
6
9
• It is important to
balance your budget!
IM4_Ch03_3pp.fm Page 44 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 45
3:05
|
Budgeting
A budget is a plan for the use of expected income.
When trying to minimise non-essential expenses it is helpful to group
expenses under the heading 'fixed' and 'variable'. Fixed expenses
would include rent, electricity, water, local government rates,
insurance and hire purchase repayments. Variable expenses would
include food, clothing, fares, entertainment, gifts, petrol and car
repairs, dentist and the like.
Incom
e
Expenditu
re
Job
Gifts
Interest
Other
E
a
r
n
$
10
0
S
p
e
n
d
$
9
9
h
a
p
p
in
e
s
s
!
E
a
r
n
$
1
0
0
S
p
e
n
d
$
1
0
1 m
is
e
r
y
.
.
.
Essential
Non-essential
savings.
Suggested steps for compiling a budget
1 Determine expected income.
2 Analyse present spending habits.
3 Minimise non-essential expenses.
4 Tabulate income versus expenses.
5 Determine savings.
I Budgeting is
money management.
I A budget helps to
keep the balance!
worked examples
Mr Jones analysed the family income to see if 800 could be found for a holiday. He decided to
make up a monthly budget.
Monthly income (net) Fixed expenses Variable expenses
Job (4 weeks) 2480
Investments 83
Casual work 310
Children's board 433
Other 43
House payment 740
Car payment 280
Electricity 80
Water 46
Local government rates 62
Insurance 70
Medical 104
Other 20
Food 740
Clothing 250
Fares 60
Entertainment 190
Petrol & Repairs 185
Telephone 88
Other 360
Total: 3349 1402 1873
Assume that Mr Jones has already paid the tax.
1
3
---
continued §§§
IM4_Ch03_3pp.fm Page 45 Monday, March 16, 2009 9:33 AM
46 INTERNATIONAL MATHEMATICS 4
Rhonda earns $200 per month from odd jobs, $42 as an allowance from her parents and $24
average from other sources. In each month she must spend $50 on food, $36 repaying a loan
and $16 on school needs. She would like to save $20 per month and divide what remains
equally between clothes, entertainment and gifts.
a Make up a budget as in the example above.
b What percentage of her total income does she save?
James earns 428 a week gross. His weekly tax payment is 86. Here is a list of his monthly
expenses. Rent 608; electricity/telephone/water 86; medical expenses 164; food 182;
fares 72; clothing 44; entertainment/sport 84 and other 50.
a Make up a budget as in the example, using 4 weeks for 1 month.
b How much can James save each month? What is this as a percentage of his net income?
c James would like to buy a car. He estimates that payments, including registration, would
amount to 180 per month; repairs and servicing the car, 60 per month; and petrol 36 per
month. He no longer would have to pay fares. Is it possible? Can you suggest a solution?
Using similar headings to those in question 2, make up a budget for a school-leaver with an
income of $500 per week who is:
a living at home
b renting a room for $80 a week
Make up your own monthly budget based on your real income and expenses. Consider all
sources of income, essential expenditure, non-essential expenditure and savings.
1 What is the monthly balance (surplus)? Could they save enough for the holiday in
12 months?
2 How much is left of the yearly balance if holiday money is removed?
3 Is it wise to save so little, or should the variable expenses be reduced?
Where could the reductions most easily be made?
Solutions
1 Balance (surplus) = income − expenses
= 3349 − 3275
∴ Monthly balance = 74
∴ Yearly balance = 74 × 12 or 888
∴ Sufficient money can be saved.
2 Money left = yearly balance − 800
= 888 − 800
= 88
3 It is not wise to save so little unless savings are already substantial. Emergencies are likely
to occur that will destroy your budget unless you have savings on hand. If the holiday is
important, then sacrifices can be made in the areas of entertainment, food, clothing,
telephone and especially 'other'.
Exercise 3:05
1
2
1
3
---
3
4
IM4_Ch03_3pp.fm Page 46 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 47
3:06
|
Best Buy, Shopping Lists
and Change
Let's consider the signs above.
price Save $6 10–50% off
20% off Were $12, now $10.20
Save up to $20 Save over 30% Save 40%
These statements assume that the original price was
reasonable. I could have a pencil for sale for $9, reduce its
price by 90% or $8.10 (whichever sounds best) and attempt
to sell it for 90 cents. If the pencil is worth only 30 cents
this is not a bargain.
Sale Special Value Bargain Low prices
Super Special
Some stores have 'sale' signs displayed all year round. When advertising, stores are more
interested in selling their product than in examining the meaning of words.
Buy 3, get one free Prizes to be won Free trip Free tin of stain
You never really get anything for nothing. The 'free' gift or slim chance at a prize are
alternatives to giving you a lower price. In most competitions, the value of the postage stamps
used to send in the entries would be greater than the total value of the prizes.
From $9.50 Save up to $20
It is amazing how often the lower priced items are those you don't want.
Buy 3, get one free 3 for $30.00 Limit of 3
Do you really need three hammers, or whatever they are? Even when they seem to be
restricting you to three, you may not need them.
Buy direct
Many firms give companies and tradespeople a good discount but they charge the ordinary
customer as much as they can.
1
PRICE
2
SAVE
$6
SALE
10-50%
OFF
SPECIAL
BUY 3
GET ONE
FREE
PRIZES
TO BE
WON
FREE
TRIP
FREE
TIN OF
STAIN
FROM
$9.50
20%
OFF
WERE $12
NOW
$10.20
3 FOR
$30.00
VALUE
BARGAIN
LIMIT
OF 3
SAVE
UP TO
$20
BUY
DIRECT
LOW
PRICES
SUPER
SPECIAL
SAVE
OVER
30%
SAVE
40%
FOR
THE BEST
BUYS
COMPARE
PRICES
a b c
d
e
f
g
h
i
j
k
l
m n
o
p
q
r
s
t
u
v
I When shopping ask:
1 Do I really need this
item?
2 Is this the best price
available?
3 Is it worth the price?
4 Is it good quality?
(Will it last?)
1 a
1
2
--- b d
k l
q s v
2 c e n o t
u
3 f
g
h i
4 j
q
5 f m p
6 r
IM4_Ch03_3pp.fm Page 47 Monday, March 16, 2009 9:33 AM
48 INTERNATIONAL MATHEMATICS 4
a If the marked price of an item is 137 find the price
that must be paid if the discount in sign from
page 47 were applied. Do the same for signs
and . For sign give the range
of prices that might have to be paid.
b An item has a marked price of 85 in two shops. One offers a 14% discount (reduction in
price) and the other a discount of 10.65. Which is the better buy and by how much?
c Paint is advertised: 'Was 12. Now 10.20.'
i What has been the reduction?
ii Express the reduction as a percentage of the original price.
worked examples
1 Determine the best buy on a 'front door'. Assume that the
quality of each door is the same.
Y-Mart: 32% off marked price of $435.
N-Boss Discounts: $120 off marked price of $426.
Cottonworths: Buy 3 doors for $1005 and get one free.
Walgrams: Buy a door for $360 and get a tin of stain free.
2 Smooth coffee costs $14.80 for 500 g, Ringin coffee costs
$9.60 for 300 g. Which brand is the best value?
(Assume similar quality.)
Solutions
1
Cost = $435 − 0·32 × $435 Cost = $426 − $120
= $295.80 = $306
Cost = 4 doors for $1005 Cost = $360 less value of stain
= $251.25 per door Is a tin of stain worth $64.20?
The cheapest cost per door is from Cottonworths but we want to buy one front door not
four, so the best buy is from Y-Mart.
2 500 g of Smooth coffee costs $14.80 300 g of Ringin coffee costs $9.60
∴ 100 g of Smooth coffee costs ∴ 100 g of Ringin coffee costs
$14.80 ÷ 5 = $2.96 $9.60 ÷ 3 = $3.20
Clearly Smooth coffee is less expensive.
Y-MART N-BOSS DISCOUNTS
COTTONWORTHS WALGRAMS
Exercise 3:06
Best buy, shopping lists, change
1 Find:
a 15% of $25 b 20% of $140
2 What percentage is:
a $5 of $25? b $7.50 of $60?
2 Which is cheaper:
a 500 g for $12 or 750 g for $17?
Foundation Worksheet 3:06
1
a
b f k v q
IM4_Ch03_3pp.fm Page 48 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 49
d John travelled to a factory for a 'direct buy' purchase of 60 litres of paint for 690.
He noticed that the same paint was being sold at a local store for a 'sale' price of 46.60 for
4 litres. Which was the less expensive and by how much? If John spent an hour of travelling
and 4.60 worth of petrol, which do you think was the better buy? Why?
e Jane bought a new tyre for 160, Robyn bought one for 130 and Diane bought a retread for
76. If Jane's tyre lasted 32 000 km, Robyn's 27 500 km, and Diane's 16 000 km, which was
the best buy? (Assume that safety and performance for the tyres are the same.)
a Frozen peas come in three brands,
Yip peas at 84c for 200 g, Hap peas
at $1.56 for 440 g and Nap peas at
$1.84 for 600 g. Which is the
best buy?
b If Coco Pops are sold at $4.90 per
750 g box, Rice Bubbles at $3.20
per 500 g box and Corn Flakes at
$1.30 per 250 g box, which is the
cheapest per unit of weight?
c James and Maree go shopping for
ice-cream. James like Strawberry
and Maree likes Vanilla. They decide
to buy the container that represents
the best value. The Strawberry is
$6.90 for 4 L or $3.90 for 2 L, the
Vanilla is $5.20 for 3 L or $10 for 6 L.
Which did they buy? What is their
change from $20?
d Alice wants to buy the least expensive
tea. Which will she buy if Paa tea costs
$2.50 for 250 g, Katt tea costs $6.20 for
600 g, Jet tea costs $11 for 1 kg and
Yet tea costs $7.65 for 800 g?
e Which is the least expensive sauce:
Soy at $1.45 for 200 mL, Barbecue at
$2.30 for 250 mL, Chilli at $2.25
for 300 mL or Tomato sauce at
$1.80 for 750 mL? Which is the
most expensive?
2
IM4_Ch03_3pp.fm Page 49 Monday, March 16, 2009 9:33 AM
50 INTERNATIONAL MATHEMATICS 4
a Use the method in the box to find
the change from $20 if the bill is:
i $2.95 ii $7.80 iii $12.15 iv $17.10 v $8.05 vi $4.30
b Find the total cost of each list below and find the change from $100.
c Make up your own table to find the cost of the following list and change from $100.
• 3 packets of tea at $2.45 • 5 tins of mushrooms at 70 cents
• 2 tins of mixed vegetables at $2.10 • 8 tins of sardines at 70 cents
• a bottle of tomato sauce at $1.80 • 7 tins of cat food at $1.75
• 2 packets of instant pudding at 95 cents • 1 tub of ice-cream at $7.45
• 2 jars of stock cubes at $2.15 • 4 tins of beans at 80 cents
• 3 tins of tuna at $3.95
i Item Price No. Cost ii Item Price No. Cost
Peanut butter $1.40 2 Pickles $0.85 3
Vegetable soup $0.80 5 Cake mix $2.30 1
Vitamin C $1.70 1 Baby food $0.55 15
Mousetrap $0.30 7 Strawberry jam $3.95 1
Icing mix $0.80 2 Gravy mix $2.55 2
100 mL oil $2.40 1 Vitamins $5.30 1
Tin of cream $0.85 3 Honey $1.30 2
Flavouring $2.20 2 Salt $0.75 1
Total =
Change from $100 =
Total =
Change from $100 =
Do you have
any change?
No, but if you read
the box behind you,
you'll find some . . .
I A quick way to find change
eg Change from $20 if $3.75 is spent.
$3.75 up to $4 is 25 cents
$4 up to $20 is $16
+
∴ Change is $16.25
3
• I do my shopping
on the internet!
IM4_Ch03_3pp.fm Page 50 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 51
3:07
|
Sales Tax
A sales tax is a broad-based tax on most goods and services you buy. Usually the sales tax is
included in the price you pay. However, in some countries no sales tax is applied to some items
such as basic food goods. In these cases a bill or shopping receipt may itemise each product
showing whether the sales tax was charged and how much tax was included in the bill.
A sales tax is sometimes called a goods and services tax (GST) or value added tax (VAT).
In the following exercises the sales tax will be referred to as a GST which will be taxed at a rate
of 10%. It is simple to calculate the GST included in a price in these cases by dividing by 11, since
the base price has been increased by 10% or .
Find 10% of: 1 $20 2 $3.50 3 $7.10 4 $14.36
Calculate: 5 $66 ÷ 11 6 $9.24 ÷ 11 7 $5.6 × 1·1 8 $127 × 1·1
9 $66 × 10 $9.24 ×
p
r
e
p
quiz
3:07
10
11
------
10
11
------
1
10
------
worked examples
1 Find the GST that needs to be applied to a price of $325.
2 What is the retail price of a DVD player worth $325 after the GST has been applied?
3 How much GST is contained in a price of $357.50?
4 What was the price of an item retailing at $357.50 before the GST was applied?
Solutions
1 The GST is 10% of the price.
∴GST = $325 × 10%
= $32.50
2 The GST is added on to get the retail price.
∴ Retail price = $325 + $32.50
= $357.50
Note: The retail price can also be calculated by multiplying the original price by 110%
(or 1·1) since 10% is added on
ie Retail price = $325 × 1·1
= $357.50
3 To find the GST contained in a price, we divide it by 11.
(If the original price is increased by , then the retail price,
including the GST, is of the original price.)
∴GST = $357.50 ÷ 11
= $32.50
I To calculate the
GST to add on to a
price, simply find
10% of the price.
I To find the GST
included in a price,
divide the price
by 11.
1
10
------
11
10
------
continued §§§
IM4_Ch03_3pp.fm Page 51 Monday, March 16, 2009 9:33 AM
52 INTERNATIONAL MATHEMATICS 4
Determine the GST that needs to be added to original prices of:
a $70 b $120
c $46.20 d $59.70
e $863 f $12 740
g $97.63 h $142.37
i $124.55 j $742.15
Determine the retail price after the GST of 10% is added to the following.
a 60 b 150 c 220 d 740
e 75 f 112 g 343 h 947
i 8.60 j 27.40 k 56.45 l 19.63
How much GST is contained in the following retail prices?
a $22 b $110 c $165 d $286
e $61.60 f $57.20 g $13.20 h $860.20
i $46.75 j $349.80 k $1004.30 l $1237.50
4 To find the original price, simply subtract the GST from the retail price.
∴ Original price = $357.50 − $32.50
= $325
Note: The original price can also be found by multiplying the retail price by .
ie Original price = $357.50 ×
= $325
10
11
------
10
11
------
Exercise 3:07
Goods and services tax
1 Find 10% of:
a $17 b $23.50 c $1125
2 Find of:
a $13.20 b $26.40 c $62.48
1
11
---
Foundation Worksheet 3:07
1
2
3
• Find the total GST paid on the
camping gear shown if its retail
price is $1130.
IM4_Ch03_3pp.fm Page 52 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 53
What was the original price before the GST of 10% was added to each of these retail prices?
a $77 b $132 c $198 d $275
e $15.40 f $28.60 g $106.70 h $126.50
i $13.86 j $41.25 k $237.05 l $1238.38
When retailers sell a mixture of goods where the
GST is only applied to some, they normally indicate
which items on the bill include the GST. This bill
indicates the items which include the GST with an
asterisk (*). Determine the amount of GST that
would be included in the total on this bill.
The retailer indicates the GST items with a percentage sign (%). Determine the GST included
in these bills.
a b c
a The prices of three items were 10.56, 10.30 and 21.00. The total of the GST included in
the bill was 0.96. Which was the only price that included a 10% GST?
b Four prices on a bill were 5.60, 7.70, 9.30 and 10.56. The total of the bill, excluding
GST, was 31.50. Which of the four prices did not include GST?
c Five articles cost 20.90, 37.40, 52.80, 61.60 and 45.10. GST of 10% was then added to
some of these prices. The total bill was then 230.56. Which prices had GST added to them?
Investigation 3:07 | Shopper dockets
1 Collect 'shopper dockets' from as many different supermarkets you can. Examine them to
see how they indicate the GST charges.
2 Draw up a list of all the items you can that do not include GST.
3 When was the GST introduced in Australia?
4
SHOPQUICK
Sauce $ 1.20
Biscuits* $ 1.35
Bread $ 3.60
Toothbrush* $ 3.10
Soap* $ 1.93
Total $11.18
GST incl = ?
5
6
% Magazine $ 2.85
% Deodorant $ 4.00
Milk $ 5.60
Muffins $ 2.60
% Biscuits $ 2.17
Total $17.22
GST incl = ?
Bread $ 2.70
Flour $ 0.90
% Ice creams $ 4.48
% Soap pads $ 0.98
% Tissues $ 5.10
Total $14.16
GST incl = ?
Bananas $ 3.50
% Soap $ 3.92
% Detergent $ 3.31
Tea bags $ 4.46
Eggs $ 2.89
% Bath salts $ 6.05
Rice $ 1.54
% Biscuits $ 2.56
Total $28.23
GST incl = ?
7
n
o
i
t
a
g i t s e v
n
i
3:07
IM4_Ch03_3pp.fm Page 53 Monday, March 16, 2009 9:33 AM
54 INTERNATIONAL MATHEMATICS 4
3:08
|
Ways of Paying and Discounts
When buying the things we need, we can pay cash (or cheque or use electronic transfer of funds),
buy on terms or use credit cards. The wise buyer will seek discounts wherever possible, comparing
prices at different stores.
Using money
Seeking discount Buying with
credit card
Buying on terms Paying cash or
transferring funds
Meaning
A process of bargaining
to seek a reduced price.
A readily acceptable
method of making
credit purchases.
'Buy now pay later'.
A way of having the
item and spreading the
payment over a period
of time. (Hire-purchase)
An immediate payment
with cheque, electronic
transfer of funds card
or money.
Advantages
You pay less because
you can challenge one
shop to beat the price of
another. Taking time
allows you to compare
the quality of items.
Convenient. Safer than
carrying large sums of
money. Useful in
meeting unexpected
costs. If payment is
made promptly the
charge is small. Many
stores accept credit
cards.
You can buy essential
items and make use of
them as you pay. Buying
a house on terms saves
rent. The item bought
may be used to generate
income. Little
immediate cost.
Paying cash may help
you get a discount.
Money is accepted
anywhere. You own the
item. You keep out of
debt. It doesn't
encourage impulse
buying. With cheque
or electronic transfer
of funds card you don't
have to carry a lot of
money.
Disadvantages
It takes time and energy
to compare prices. To
get the best price you
may have less choice in
things like colour, after
sales service and maybe
condition of the item.
'Specials' are discounts.
There is a tendency
to overspend, buy on
impulse and not out
of need. The interest
charged on the debt is
high. Hidden costs
(stamp duty and charge
on stores) generally lift
prices.
Relies on a regular
income and, if you
cannot continue
payments, the item can
be repossessed, sold
and, if its value has
depreciated (dropped),
you still may owe
money. High interest
rates. You are in debt.
Carrying large sums
of money can be
dangerous (risk of loss)
and many shops won't
accept cheques or
electronic transfer of
funds cards. You may
miss out on a good buy
if you don't carry much
money with you.
IM4_Ch03_3pp.fm Page 54 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 55
Using money
Seeking discount Buying with
credit card
Buying on terms Paying cash or
transferring funds
Charge
it!
The payments are so high,
I can't afford petrol!
Look at the bulging pockets
on that guy!
Nah, that's just
loose change.
SALE
Trousers
50% off
worked examples
1 Brenda bought a car on terms of $100 deposit and 60 monthly repayments of $179.80.
The price of the car was $5000.
a How much did she pay for the car?
b How much interest did she pay on the money borrowed?
c How much money had she borrowed?
2 a Greg was given 12 % discount on a rug with a marked price of $248. How much did
he pay?
b A television marked at $2240 was eventually sold for $2128. What was the discount and
what was the percentage discount given on the marked price?
c After a discount of 14% was given, I paid $5848 for my yellow Holden. What was the
original marked price?
3 Brenda bought a TV priced at $1200 after it was discounted by 10%. Brenda received a
further 5% discount because she was a member of staff. How much did she pay for the TV?
Solutions
1 a Total payments for car = deposit + payments
= $100 + 60 × $179.80
= $10 888
b Interest = extra money paid
= $10 888 − $5000
= $5888
c Amount borrowed = price of car − deposit
= $5000 − $100
= $4900
2 a Discount on rug = 12·5% of $248
= 0·125 × $248
= $31
Amount paid = $248 − $31
= $217
1
2
---
continued §§§
IM4_Ch03_3pp.fm Page 55 Monday, March 16, 2009 9:33 AM
56 INTERNATIONAL MATHEMATICS 4
Use the table on page 54 to answer these questions in your own words.
a What are the disadvantages of buying on terms?
b What are the advantages of paying cash?
c What are the advantages of seeking out discounts?
d What are the disadvantages of buying with credit cards?
a Find the amount John will pay for a fishing line worth $87 if he pays $7 deposit and $5.70
per month for 24 months. How much extra does he pay in interest charges?
b How much will Ingrid pay for her wedding dress worth $1290 if she pays $90 deposit and
$74 per month for two years? How much interest did she pay on the money borrowed?
c When Robyn said that she could buy an item marked at $640 for $610 at another store, the
salesman offered her the item for $601.60 if she bought it immediately. She bought it on
terms, paying a deposit of $20 and monthly repayments of $23.30 for four years. How much
did she pay all together? How much extra did she pay in interest charges?
d Joshua wants to buy a tent with a marked price of $730. He wants to pay it off on terms.
For each of the following, work out how much he pays altogether and the interest charged.
i Deposit of $100 and monthly payments of $64.50 for one year.
ii Deposit of $100 and monthly payments of $38.30 for two years.
iii Deposit of $100 and monthly payments of $29.60 for three years.
iv Deposit of $100 and monthly payments of $25.20 for four years.
v Deposit of $100 and monthly payments of $22.60 for five years.
vi Deposit of $100 and monthly payments of $17.35 for ten years.
b Discount on TV = $2240 − $2128
= $112
Percentage discount = ($112 ÷ $2240) × 100%
= 5%
c Price paid = (100 − 14)% of marked price
= 86% of marked price
1% of marked price = $5848 ÷ 86
= $68
100% of marked price = $6800
3 Price after original 10% discount = (100 − 10)% of $1200
= 90% of $1200
= $1080
Price after a further 5% discount = (100 − 5)% of $1080
= 95% of $1080
= $1026
(Note: This is not the same as a 15% discount off the original price
ie 85% of $1200 = $1020)
Exercise 3:08
1
2
IM4_Ch03_3pp.fm Page 56 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 57
e A man bought a car for $8250 paying a deposit
of $250, $130 in extra charges, and paying
$280 per month over five years. How much would
he have to pay altogether? After paying four payments
he finds it too expensive. The car is repossessed and
sold for $6400. The finance company then sent him
a bill for $1066.70 being the difference between what
he owed and what they received for the car. He paid
the bill and had no car. How much money had he
paid altogether?
a Rachel was given a discount of 45% on a T-shirt with a marked price of 32. How much did
she pay?
b Katherine was given a 7% discount on all purchases because she was a store employee.
If the total of her purchases came to 85, how much did she pay?
c After bargaining, I purchased a chess set for 240 that was originally advertised for 320.
What discount was I given and what is this as a percentage of the advertised price?
d My airline ticket was to cost me 550 but, because I booked more than six weeks ahead,
I was given a discount price of 385. What percentage discount was I given?
e After a discount of 10% was given, Luke paid 585 for an additional hard disk drive for his
computer. What was the original price of this hard disk drive?
f My mother paid 21.70 to attend the exhibit. She had been given a discount of 65% because
she was a pensioner. What was the original price?
g Because Helen was prepared to pay cash for a yellow car with a marked price of 9200, she
was given a discount of 1012. What percentage was the discount of the marked price?
h Greg was given a discount of 10% on the marked price of a kitchen table. If the discount
was 22, how much did he pay?
Calculate the final price if successive discounts of 20% and 10% were applied to a price of $100.
A bookstore discounted all its books by 15%. A further discount of 10% was given to the local
school. How much did they pay for books which originally sold for a total of $1750?
Apply the following successive discounts
to the given prices.
a 10%, 10%, $500
b 15%, 25%, $720
c 5%, 20%, $260
d 12 %, 15%, $965
e 7 %, 17 %, $2230
a A radio on sale for £50 is to be reduced in
price by 30%. Later, the discounted price is
then increased by 30%. What is the final
price? By what percentage (to the nearest
per cent) must the first discounted price
be increased to give the original price?
I When borrowing money:
1 read the terms
2 work out the costs
3 be prepared to say NO
3
4
5
Does it matter in
which order the
discounts are
applied?
6
1
2
---
1
2
---
1
2
---
7
IM4_Ch03_3pp.fm Page 57 Monday, March 16, 2009 9:33 AM
58 INTERNATIONAL MATHEMATICS 4
b Edna bought a caravan, priced at £10 050,
on terms of £300 deposit and 60 monthly
repayments of £369.30. Find the total
amount paid. Find also the interest paid
expressed as a percentage of the money
borrowed. (Answer correct to 1 decimal place.)
c Stephen bought a computer that had a marked
price of £960. He received a discount of 15%. He
paid a deposit of £81.60 and monthly repayments
of £67.30 for one year. Find the interest paid
expressed as a percentage of the money borrowed.
d John had to pay interest on his credit card. He pays 1·7% per month on the greatest amount
owing during each month. The greatest amount owing in April was £166, in May £294 and,
in June, £408. Find the total interest charged for the three months.
A hardware store was offering multiple discounts of 12% and 15% on a chain saw with a list
price of $480. The first discount is given on the list price, the second on the first net price.
a What is the final purchase price?
b What is the final purchase price, if the 15% discount is applied first?
c Are the answers to a and b the same?
d Is the multiple discount equivalent to a single discount of 27%?
e What single discount is equivalent to multiple discounts of 12% and 15%?
f What single discount is equivalent to multiple discounts of 10% and 20%?
The following formula converts multiple discounts to a single discount. Single discount rate
= [1 − (1 − d
1
) (1 − d
2
) (1 − d
3
) … ] × 100% where d
1
, d
2
, d
3
, … are the successive discounts
expressed as decimals. Use the formula to find a single discount equal to the multiple discounts
of:
a 11% and 8% b 16%, 12% and 7%
c 10%, 9% and 5% d 12 %, 4 % and 2·1%
Fun Spot 3:08 | The puzzle of the missing dollar
Three men had lunch in a busy restaurant.
When it came time to pay their bill, each of
the men gave to the waiter a ten-dollar note.
Thus the waiter received $30 altogether.
But when the waiter added up the bill, he found
it only came to $25. Knowing that $5 would not
divide among the three men evenly, the waiter
decided to give each of the men $1 change and
put the remaining $2 in his pocket.
Now it appears that each of the men have paid
$9 for their meal, and the waiter has $2; a total
of $29.
Where is the missing dollar?
8
9
1
2
---
1
4
---
f
u
n
spo
t
3:08
IM4_Ch03_3pp.fm Page 58 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 59
3:09
|
Working for a Profit
People who work for themselves may charge a fee for their services or sell for a profit. However,
they are not the only people concerned with profit and loss. We all, from time to time, will need to
consider whether our investment of time, money and effort is justified by the results. This may be
in our work for charity, organisations or in our hobbies.
• When buying and selling:
Selling price = Cost price + Profit
or
Profit = Selling price − Cost price
Note: If the profit is negative we have made a loss.
• When calculating money made:
Profit = Money received − Expenses
worked examples
1 Rhonda bought a parachute for
$80 and sold it for $340. Find the
profit as a percentage of the cost.
2 We held a dance at school to
make money to support the
work of a charity organisation.
We charged each of the
400 people who came
an entrance fee of $8.
The band cost $780,
decorations cost $88.50,
prizes $36, food $180,
cleaning $147 and
advertising $385.
How much money did we
make and what percentage
is this of the sum received?
Solutions
1 Rhonda's profit = Selling price − Cost price
= $340 − $80
= $260
Percentage profit =
=
= 325%
Okay, anything!
I'll pay!
Just give me the
parachute!
Profit
Cost price
------------------------- 100% ×
$260
$80
------------ 100% ×
continued §§§
IM4_Ch03_3pp.fm Page 59 Monday, March 16, 2009 9:33 AM
60 INTERNATIONAL MATHEMATICS 4
Complete the tables below.
For parts a to e of question 1, find the profit or loss as a percentage of the cost price.
(Answer correct to 1 decimal place.)
For parts f to j of question 1, find the profit or loss as a percentage of the money received.
(Answer correct to 1 decimal place.)
For parts i to j of question 1, find the profit as a percentage of the expenses
(correct to 1 decimal place).
Luke bought a microphone for $28 and sold it for $50. Find:
a his profit from the sale
b the profit as a percentage of the cost price
c the profit as a percentage of the selling price
When selling products in a store, a percentage mark-up is added to the cost price to obtain the
marked (or selling) price. If sporting gear has a mark-up of 60%, toys 40% and clothing 45%
what will be the marked price of:
a a baseball bat with a cost price of £24?
b a doll with a cost price of £15?
c a T-shirt with a cost price of £8.60?
A discount of 10% is given on the price of a car marked at $32 860. Find the discount price
of the car.
Selling
price
Cost
price
Profit
(or loss)
Money
received
Expenses Profit
(or loss)
a $2146 $1645 f 3816.50 1308.50
b $468 −$179 g 491.80 846.60
c $58.75 $95.50 h 916 8423
d $27 940 $13 650 i 27 648 2494
e $85 420 $36 190 j 7684 15 686
2 Income = Money received − Expenses
Money received = 400 × $8
= $3200
Expenses = $780 + $88.50 + $36 + $180 + $147 + $385
= $1616.50
Income = $3200 − $1616.50
= $1583.50
Percentage of sum received =
=
Ӏ 49·5%
Income
Money received
--------------------------------------- 100% ×
$1583.50
$3200
----------------------- 100% ×
Exercise 3:09
1
2
3
4
5
6
7
IM4_Ch03_3pp.fm Page 60 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 61
Heather works for a toy store where the percentage mark-up is 40% of the cost price. She is
offered 10% discount on any item and can have this on the cost price before mark-up occurs,
or on the marked price. Use each method to find the discount price Heather would have to pay
on a game that has a cost price of $32. What do you notice?
A store marks up everything by 40%. If the selling price of a tennis racquet is $308, what is the
cost price and the profit on the racquet?
A travelling salesman marks up each item by 110%. Find the cost price and the profit on an
item with a selling price of:
a $630 b $8.40 c $127.05
During a clearance sale, clothing was sold for 20% below cost. A dress was sold for $144.
Calculate the cost price and the loss.
Unwanted stock was sold for 70% of the cost. If the amount received from sales was ¥1 092 000,
what was the cost price of the stock sold and what was the loss?
The ratio Cost price : Profit : Selling price is 100 : 30 : 130. If the profit is $24 what is the selling
price?
The ratio Cost price : Profit : Selling price is 8 : 3 : 11. If the cost price is $74.20 what will be the
selling price?
Michael bought DVD players for $320 each. He wants to make a profit of 30% after passing on
a 10% GST to the government. How much should he charge for each DVD player?
Lounge suites initially bought for $1100 each are to be sold at a loss of 40%. However, GST
must still be charged and passed on to the government. What must be the sale price of each
suite?
A machine was sold for a 25% profit of 270 on the cost price.
a What was the selling price if an additional GST of 10% has to be added?
b What was the cost price?
c What percentage was the selling price, including the GST, of the cost price?
8
worked example
A shop owner marks everything up by 30% and the selling price of an article is $78.
What is the cost price and the profit on this article?
Solution
First, find the ratio of Cost : Profit : Selling price.
∴ C: P : SP = 100 : 30 : 130
= =
= =
∴ Cost = × $78 ∴ Profit = × $78
= $60 = $18
Cost
Selling price
------------------------------
100
130
---------
Profit
Selling price
------------------------------
30
130
---------
Cost
$78
-----------
100
130
---------
Profit
$78
--------------
30
130
---------
100
130
---------
30
130
---------
9
10
11
12
13
14
15
16
17
IM4_Ch03_3pp.fm Page 61 Monday, March 16, 2009 9:33 AM
62 INTERNATIONAL MATHEMATICS 4
Mathematical terms 3
Investigation 3:09 | Let's plan a disco
Please use the Assessment Grid on the following page to help you understand what is required
for this Investigation.
Plan a disco to raise money for charity.
Step 1
Estimate the cost of music, decorations, prizes, tickets, food, cleaning, advertising, etc.
Step 2
Estimate the number of people you expect to come.
Step 3
Set a ticket price to provide a profit of $300.
Use these estimates to draw on the same set of axes:
• a graph of costs ($) versus number of tickets sold (n)
• a graph of income ($) versus number of tickets sold (n)
Use the graphs to determine how many tickets need to be sold to:
• break even • make $200 • make $300
Mathematical Terms 3
budget
• A plan for the use of expected income.
commission
• Income usually calculated as a percentage
of the value of the goods sold.
discount
• To reduce the price of goods sold (v).
• The amount or percentage a price is
reduced (n).
GST
• Goods and services tax.
• 10% of a base price is added onto the cost
of most goods and services and included
in the advertised retail price.
gross pay
• The amount of pay before any deductions
such as income tax are subtracted.
income tax
• Tax paid to the government which is
based on the level of income received.
net pay
• The amount of pay an employee receives
after deductions such as income tax have
been subtracted.
overtime
• Time worked by an employee in excess of
a standard day or week.
• Usually rates of pay 1 or 2 times the
normal rate of pay are paid for overtime.
profit
• The gain when a good is sold for a higher
price than its cost price.
• If the selling price is lower a negative
profit, or loss, is made.
salary
• A fixed amount paid for a year's
employment. It may be paid weekly or
fortnightly.
superannuation
• An investment fund usually contributed to
by both employer and employee on the
employee's behalf.
• It provides benefits for employees upon
retirement, or for relatives if the member
dies.
taxable income
• Amount after allowable deductions are
subtracted from the gross pay.
• Income tax is calculated on this amount.
wages
• Pay given to an employee often based on
an agreed hourly rate.
• Usually paid weekly or fortnightly.
i
n
v
e
s
t
igatio
n
3:09
s
m r e
t
l
a
c
i
t
a
m
e
h
t
a
m
3
1
2
---
IM4_Ch03_3pp.fm Page 62 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 63
Assessment Grid for Investigation 3:09 | Let's plan a disco to graph the linear relationships given.
1
2
c
Mathematical problem-solving techniques have been
selected and applied to accurately graph the lines
required.
3
4
d
The student has graphed the required lines and attempted
to use their graphs to determine the ticket sales required.
5
6
e
The student has effectively graphed the required lines and
been able to use and interpret their graphs to determine
the correct ticket sales required, with some reference to
the equation of a line and explanations are clear but not
always logical or complete.
3
4
d
A good use of mathematical language and representation.
Graphs are accurate, to scale and fully labelled.
Explanations aspects raised.
1
2
c
There is a correct but brief explanation of whether results
make sense and how they were found. A description of
the important aspects of the graphs is given along with
their relation to finding ticket sales.
3
4
d
There is a critical explanation of the graphs obtained and
their related equations. The ticket sales are explained with
consideration of the accuracy of the results obtained and
possible further applications discussed.
5
6
IM4_Ch03_3pp.fm Page 63 Monday, March 16, 2009 9:33 AM
64 INTERNATIONAL MATHEMATICS 4
Diagnostic Test 3 | Consumer Arithmetic3
1 a John sells cars for a living. He is paid a retainer (a base wage) of $150
a week as well as 2% commission on sales made. Find his income for
the week, if in one week he sells cars to the value of:
i $8000 ii $21 500
b Luke has a casual job from 4:00 pm till 5:30 pm Monday to Friday.
He also works from 9 am till 12:30 pm on Saturdays. Find his weekly
income if his casual rate is $9.80 per hour Monday to Friday, and
$14.70 an hour on Saturdays.
2 a During one week Petra worked 35 hours at the normal rate of 12.60
per hour. She also worked 6 hours overtime: 4 at 'time-and-a-half'
and 2 at 'double-time'. How much did she earn?
b Calculate Diane's holiday loading if she is given 17 % of four weeks
salary and she earns 860 per fortnight.
3 a Find the net pay for the week if John earns £586.80, is taxed
£107.95, pays £43.50 for superannuation and has miscellaneous
deductions totalling £79.40.
b What percentage of John's gross pay did he pay in tax?
4 Alana received a salary of $38 465 and a total from other income
(investments) of $965. Her total tax deductions were $2804. During the
year she had already paid tax instalments amounting to $13 800.50.
Find:
a her total income
b her taxable income
c the tax payable on her taxable income using the table on page 42
d her refund due or balance payable
e how much extra Alana would receive each week if she is given a wage
rise of $100 per week.
5 a A lawn fertiliser comes in three sizes: 20 kg (for $11.60), 50 kg
(for $24.80) and 110 kg (for $56.60). Which size is the best buy?
b Rich Red strawberry flavouring can be purchased at 240 mL for $1.70,
660 mL for $3.75, or 1 L for $6.25. Which buy represents the best
value?
6 a Determine the GST that needs to be added to a base price of $73.70.
b Determine the retail price after 10% GST is added to a base price
of $53.90.
c How much GST is contained in a retail price of $32.45?
d What was the base price before 10% GST was added to give a retail
price of $21.45?
Section
3:01
3:02
3:03
3:04
3:06
3:07
1
2
---
IM4_Ch03_3pp.fm Page 64 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 65
7 a Jim bought a car with a marked price of 3000. He paid a deposit of
100 and 36 monthly payments of 136.20. How much did he pay?
How much more than the marked price did he pay?
b Pauline bought a lawnmower marked at $692. She paid a deposit of
$40 and 24 monthly payments of $39.20. How much did she pay?
By how far did that exceed the marked price?
8 a Naomi was given 12 % discount on a rug with a marked price of
$460. How much did she pay?
b A television marked at $4200 was eventually sold for $3612. What
was the discount and what was the percentage discount given on the
marked price?
c After a discount of 13% was given, I paid $27 840 for my yellow
Holden. What was the original marked price?
9 a Jane bought a desk with a marked price of $650. She was given a
discount of 10% for paying cash, and then received a further 10%
off the discounted price because it was scratched. How much did
she pay?
b What is the final price if successive discounts of 15% and 20% are
applied to a retail price of $1250?
10 a Rachel bought a painting for $250 and sold it for $575. Find the
profit as a percentage of the cost.
b We held a games night to raise money for The House with no Steps.
We charged each of the 287 people who came an entrance fee of
$17.50. Hire of the hall cost $110, decorations cost $63, prizes
$185.60, food $687, cleaning $96 and advertising $240.
How much money did we make and what percentage is this of the
money received?
Section
3:08
3:08
3:08
3:09
1
2
---
IM4_Ch03_3pp.fm Page 65 Monday, March 16, 2009 9:33 AM
66 INTERNATIONAL MATHEMATICS 4
Chapter 3 | Revision Assignment
1 a A woman works for wages of $16.80
per hour. How much will she earn in a
week in which she works
i 40 hours at normal time?
ii 40 hours of normal time and 5 hours
of overtime if overtime is paid at
1 times the normal rate of pay?
b A salesman works for a wage of $500
per week plus 3% commission. How
much will he earn in a week if he sells
$4500 worth of goods?
c A factory worker is paid a wage of $540
a week. The factory has a special bonus
system which enables a worker to be
paid an extra 25c per article for every
article in excess of the weekly quota of
5000. How much will the worker earn
in a week in which 7200 articles are
made?
d How much holiday pay will a girl
receive if she is to be paid 4 weeks'
holiday pay plus a holiday loading of
17 % of 4 weeks' pay? Her weekly
wage is $452.
2 a Fibreglass resin comes in the following
sizes: 1 kg for $9.80; 5 kg for $26.80,
and 21 kg for $59.60.
i What is the best value for money?
ii What is the most economical way of
buying 17 kg?
b A TV set with a cash price of $680
is bought for a deposit of $68 and
48 monthly payments of $15.68.
Find the difference between the cash
price and the price paid.
c Calculate the amount of GST included
in items which retailed for $736, $245
and $579.
3 Mary-Ann is paid an annual salary of
$53 350. Her allowable tax deductions total
$1340. During the year her employer paid
income tax instalments on her behalf of
$12 732.
a What is Mary-Ann's taxable income?
b How much income tax should she pay
for the year?
c What is the amount of her refund from
the tax office?
4 Jeremy added 10% GST onto the price of a
book valued at $29.90 to get its retail price.
He then discounted the retail price by 10%
to get a sales price.
a What is the retail price?
b What is the sales price?
c Is the sales price the same as the
original value of the book?
d By what percentage should Jeremy have
discounted the retail price to get back
to the original value of the book?
5 a Vicki sold azaleas in her nursery for
$15.90. She bought them for $11.35.
What percentage profit does she make?
b Michael sold a bike he bought for $350
to a friend two years later for $230.
What percentage loss is this?
t
n e
m
n
g
i
s
s
a
3A
1
2
---
1
2
---
Wages
1 Finding the weekly wage
2 Going shopping
3 GST
IM4_Ch03_3pp.fm Page 66 Monday, March 16, 2009 9:33 AM
CHAPTER 3 CONSUMER ARITHMETIC 67
Chapter 3 | Working Mathematically
1 Use ID Card 2 on page xiv to identify:
a 10 b 12 c 17 d 18 e 19
f 20 g 21 h 22 i 23 j 24
2 Use ID Card 4 on page xvi to identify:
a 1 b 2 c 3 d 4 e 5
f 6 g 7 h 8 i 9 j 13
3 Through how many degrees does the hour
hand of a clock turn in half an hour?
4 Tom was given a cheque for an amount
between $31 and $32. The bank teller
made a mistake and exchanged dollars and
cents on the cheque. Tom took the money
without examining it and gave 5 cents to
his son. He now found that he had twice
the value of the original cheque. If he had
no money before entering the bank, what
was the amount of the cheque.
5
This travel graph shows the journeys of
John and Bill between town A and town B.
(They travel on the same road.)
a How far from A is Bill when he
commences his journey?
b How far is John from B at 2:30 pm?
c When do John and Bill first meet?
d Who reaches town B first?
e At what time does Bill stop to rest?
f How far does John travel?
g How far apart are John and Bill when
Bill is at town A?
h How far does Bill travel?
6 A loan of
$1000 is to
be repaid at
an interest
rate of 20% pa.
The faster the
loan is repaid,
the less interest
is charged.
The graph
shows how the
amount to be repaid varies according to the
time taken to repay the loan.
a How much has to be repaid if 3 years
is taken to repay the loan?
b If a person wished to repay the loan in
2 years what amount would have to be
repaid?
c How much must be paid monthly if this
loan is to be repaid in 4 years?
a
s
sign
m
e
n
t
3B
$31.62 . . .
. . . $62.31?
50
40
30
20
10
0 11 noon 1 2 3 4 5
John
B
i
l
l
B
A
Time
D
i
s
t
a
n
c
e
In mathematics, the method of solving a problem is sometimes hard to express in words. In cases
like this, pronumerals are often used. The result could be a simple formula.
• Some numbers in a pattern are known. How can we find the others?
For example: 9, 8, 7, 6, . . . or 3, 5, 7, 9, . . .
Patterns like these can be written in a table of values, where
n
represents the position of the
number in the pattern, and
T
the actual number (or term).
• Two angles of a triangle are known. How can we find the third?
P
a
t
t
e
r
n
s
a None of the following descriptors has been achieved. 0
b
Some help was needed to be able to expand the brackets
and complete the table.
1
2
c
Mathematical techniques have been selected and applied
to complete the table and suggest relationships or general
rules.
3
4
d
The student has completed the table and accurately
described the rules for the square of a binomial.
5
6
e
The above has been completed with justification using the
patterns within the columns of the table and further
examples.
7
8
C
r
i
t
e
r
i
o
n
Students will be able to:
• Apply index laws to evaluate arithmetic expressions.
• Apply index laws to simplify algebraic expressions.
• Use standard (scientific) notation to write small and large numbers.
• Understand the difference between rational and irrational numbers.
• Perform operations with surds and indices.
P
a
t
t
e
r
n
s
a None of the following descriptors has been achieved. 0
b
Some help was needed to be able to write the fractional
indices.
1
2
c
Mathematical techniques have been selected and applied
to write each fractional index and suggest an emerging
pattern.
3
4
d
The student has completed all fractional indices and
accurately described the rules for the square of a binomial.
Some attempt at the final two parts has been made.
5
6
e
The above has been completed with specific justification
using the patterns and index lawns shown and the further
questions have been completed accuratelynotation, with some errors or inconsistencies evident.
Lines of reasoning are insufficient.
1
2
c
There is sufficient use of mathematical language and
notation. Explanations are clear but not always complete.
3
4
d
Correct use of mathematical language and notation has
been shown. Explanations of all rules are complete and
concise.
5
6
IM4_Ch05_3pp.fm Page 107 Monday, March 16, 2009 4:33 PM
108 INTERNATIONAL MATHEMATICS 4
5:04
|
Scientific (or Standard) Notation
The investigation above should have reminded you that:
1 when we multiply a decimal by 10, 100 or 1000, we move the decimal point 1, 2 or 3 places to
the right
2 when we divide a decimal by 10, 100 or 1000, we move the decimal point 1, 2 or 3 places to the left.
• This number is written in scientific notation
(or standard form).
6·1 × 10
5
• The first part is between 1 and 10.
• The second part is a power of 10.
Scientific notation is useful when writing very large or very small numbers.
Numbers greater than 1
5 9 7 0 · = 5·97 × 10
3
To write 5970 in standard form:
• put a decimal point after the first digit
• count the number of places you have to move the
decimal point to the left from its original position.
This will be the power needed.
Investigation 5.04 | Multiplying and dividing by powers of 10
Please use the Assessment Grid on the following page to help you understand what is required
for this Investigation.
• Use the x
y
button on your calculator to answer these questions.
• Look for a connection between questions and answers and then fill in the
rules at the end of the investigation.
Exercise
1 a 1⋅8 × 10
1
b 1⋅8 × 10
2
c 1⋅8 × 10
3
d 4⋅05 × 10
1
e 4⋅05 × 10
2
f 4⋅05 × 10
3
g 6⋅2 × 10
4
h 6⋅2 × 10
5
i 6⋅2 × 10
6
j 3⋅1416 × 10
2
k 3⋅1416 × 10
3
l 3⋅1416 × 10
4
2 a 1⋅8 ÷ 10
1
b 1⋅8 ÷ 10
2
c 1⋅8 ÷ 10
3
d 968⋅5 ÷ 10
2
e 968⋅5 ÷ 10
3
f 968⋅5 ÷ 10
4
i
n
v
e
s
t
igatio
n
5:04
To multiply by 10
n
move the decimal point ______ places to the ______.
To divide by 10
n
move the decimal point ______ places to the ______.
When expressing numbers in scientific (or standard) notation each number is written
as the product of a number between 1 and 10, and a power of 10.
I 'Scientific notation'
is sometimes called
'standard notation'
or 'standard form'.
To multiply 5
.
9 by 10
3
, we
move the decimal point 3 places
to the right - which gives 59 0.
IM4_Ch05_3pp.fm Page 108 Monday, March 16, 2009 4:33 PM
CHAPTER 5 INDICES AND SURDS 109
Assessment Grid for Investigation 5:04 | Multiplying and dividing by
powers of 10
The following is a sample assessment grid for this investigation. You should carefully read the
criteria before beginning the investigation so that you know what is required.
Assessment Criteria (B, C Some help was needed to complete the exercises.
1
2
c The student independently completes the exercises.
3
4
d
The student has correctly done all exercises and
attempted to complete the rules following.
5
6
e
All exercises and rules are completed correctly with
thorough explanation and justification, possibly with
further examples for support.
7
8
C
r
i
t
e
r
i
o
n
Exp
+
/
−
=
I Note: Not all calculators work the same way.
The answers to 1 and 2
are too long to fit on
the screen.
Exercise 5:05
1 Exp
2
I
has 4 significant figures,
as four figures are used
in the decimal part.
1·402 × 10
7
• Equations are number sentences where one or more of the numbers is missing or unknown.
Because it is unknown, the number is represented by a pronumeral.
• When we solve an equation, we are trying to find the numerical value of the pronumeral that
makes the sentence true. With some equations it is easy to find this value or solution. With
harder equations more work has to be done before the solution is found.
• A solution is correct if it gives a true number sentence when it replaces the pronumeral in the
equation. We say that the solution
satisfies
the equation.
6:01
|
Equivalent Equations
• Solving equations is like balancing scales. With equations we know
that one side of the equation is equal to the other side. We could
say that the two sides are
balanced
.
• The solution of the equation is the value of the pronumeral that
balances the equation.
• Solving difficult equations requires us to change the equation into a simpler equation. We change
equations into simpler equations by performing the same operation on both sides of the
equation.
• We may add (
+
), subtract (
−
), multiply (
×
) or divide (
÷
) by any number, provided we do the
same to both sides of the equation.
• If we do not do the same to both sides of the equation, the equation becomes unbalanced and
the sides no longer remain equal.
• If we commence with an equation and do the same thing to both sides of the equation, then
the sides will remain equal or balanced and the new equation will have the same solution
as the original equation.
An equation is a number sentence where one or more of the numbers has been replaced
by a pronumeral.
8
The sides are balanced.
3 + 5
x + 10 = 15 y − 3 = 8
x = 5 balances the scale.
x = 5 is the solution.
y = 11 balances the scale.
y = 11 is the solution.
15 5 + 10
x + 10
15
y – 3 8
8 11 – 3
If one equation can be changed into another by performing the same operation on both
sides, then the equations are said to be equivalent.
To use equations to solve problems we must be able to analyse
a written problem, translate it into an equation and then solve it.
Approach
• Read the problem carefully, examining the wording of the question.
• Establish what is to be found and what information is given.
• Ask yourself whether any other information can be assumed,
eg that a pack of cards mentioned is a standard pack.
• Try to connect the given information to form an
equation. This will often require a knowledge of
a formula or the meaning of mathematical terms.
Draw a
diagram.
3x
3x
x x
worked examples
Example 1
A rectangle is three times longer than it is wide. If it has
a perimeter of 192 m, what are its dimensions?
7
8
IM4_Ch06_3pp.fm Page 168 Friday, March 13, 2009 4:54 PM
CHAPTER 6 EQUATIONS, INEQUATIONS AND FORMULAE 169
The kinetic energy K (in joules) of a particle of mass mkg, moving with a velocity of v m/s, is
given by the formula:
K = mv
2
If the kinetic energy of a particle is 4·6 joules, find:
a its mass, if the velocity is 1·9 m/s
b its velocity, if the mass is 1·26 kg
(Give answers correct to 1 decimal place.)
A cylindrical tank holds 1200 litres of water. Its radius is 0·8 metres,
what is the depth of the water? (Note: 1 cubic metre = 1000 litres.)
Give your answer correct to the nearest centimetre.
The formula for compound interest is:
where A is the amount accumulated after investing P dollars for n years at a rate of r% pa.
Find:
a the amount A after investing $2000 for 8 years at 11% pa.
b the original investment, P, if it accumulated to $11 886 in 12 years at 9 % pa.
(Answer correct to the nearest dollar.)
c At what rate must $10 000 be invested to accumulate to $19 254 in 5 years?
Answer correct to 2 significant figures.
a Construct a formula for the area of this annulus.
b If R = 6·9 cm and r = 4·1 cm, find its area.
c If its area is 45 cm
2
and R = 5·2 cm, find r.
d If its area is 75 cm
2
and r = 3·9 cm, find R.
(Give answers correct to 1 decimal place.)
a Construct a formula for the volume of this solid.
b Find its volume if r = 2·6 m and h = 5·1 m.
c Find h if its volume is 290 m
2
and r = 3·2 m.
(Give answers correct to 3 significant figures.)
(Volume of sphere = πr
3
)
The formula gives the potential V volts, at a distance r metres from a point charge
of q coulombs.
a Find V if q = 1·0 × 10
−8
coulombs, r = 0·2 m and .
b Find r if q = 3·0 × 10
−7
coulombs, V = 54 000 volts and .
The formula gives the force between two point charges of q
1
and q
2
coulombs that
are r metres apart. If two equally charged balls are placed 0·1 m apart and the force between the
balls is 9·8 × 10
−4
newtons, calculate the charge on each ball. K = 9·0 × 10
9
.
9
1
2
---
depth
0·8 m
10
11
A P 1
r
100
--------- +
n
=
1
2
---
O
r
R
12
h
r
13
4
3
---
14 V
q
4πε
0
r
-------------- =
4πε
0
9·0 10
9
× =
4πε
0
9·0 10
9
× =
15 F
Kq
1
q
2
r
2
-------------- =
IM4_Ch06_3pp.fm Page 169 Friday, March 13, 2009 4:54 PM
170 INTERNATIONAL MATHEMATICS 4
Mathematical terms 6
Mathematical Terms 6
equation
• A number sentence where one or more of
the numbers is missing or unknown.
• The unknown number is represented by a
pronumeral.
eg x + 5 = 8,
expression
• An algebraic expression consists of one
or more terms joined together by
operation signs.
eg a + 5, x
2
− x + 4,
• An expression does not have an 'equals'
sign like an equation.
formula (plural: formulae)
• A formula represents a relationship
between physical quantities.
• It will always have more than one
pronumeral.
eg A = L × B represents the relationship
between the area (A) of a rectangle
and its length (L) and breadth (B).
grouping symbols
• The most common types are:
parentheses ( )
brackets [ ]
braces { }
• Used to 'group' a number of terms
together in an expression.
eg 5(x + 3)
inequality signs
• > greater than, < less than
• у greater than or equal to,
р less than or equal to
eg x + 3 < 4 means that
x + 3 is less than 4
inequation
• An equation where the 'equals' sign has
been replaced by an inequality sign.
eg 4x − 1 > 5 or р 4
inverse operation
• The operation that will reverse or 'undo' a
previous operation.
eg addition is the inverse operation of
subtraction; division is the inverse
operation of multiplication
pronumeral
• A symbol used to represent a number.
• Usually a letter such as x.
solution
• Method of finding the answer to a problem
OR
the answer to a problem.
• The solution to an equation is the number
or numbers that satisfy the equation or
make it a true sentence.
eg x = 3 is the solution to x + 2 = 5
solve
• Find the solution or answer to a problem or
equation.
subject
• The subject of a formula is the pronumeral
by itself, on the left-hand side.
eg in the formula v = u + at the subject is v.
substitution
• The replacing of a pronumeral with a
numeral in a formula or expression.
eg to substitute 3 for a in the expression
4a − 2 would give:
4(3) − 2
= 12 − 2
= 10
s
m r e
t
l
a
c
i
t
a
m
e
h
t
a
m
6
3x 1 +
7
---------------
x 5 –
2
----------- =
3m 1 –
7
-----------------
x
3
---
IM4_Ch06_3pp.fm Page 170 Friday, March 13, 2009 4:54 PM
CHAPTER 6 EQUATIONS, INEQUATIONS AND FORMULAE 171
Diagnostic Test 6: | Equations, Inequations and Formulae
• Each part of this test has similar items which test a certain type of question.
• Failure in more than one item will identify an area of weakness.
• Each weakness should be treated by going back to the section listed.
d
i
a
gn
o
s
t
i
c
t
e
s
t
6
Fun Spot: Why did the banana go out with
a fig?
Mathemathical Terms, Diagnostic Test, Revision
Assignment, Working Mathematically
Learning Outcomes
Students will be able to:
• Find the distance between two points.
• Find the midpoint of an interval.
• Find the gradient of an interval.
• Graph straight lines on the Cartesian plane.
• Use the gradient-intercept form of a straight line.
• Find the equation of a straight line given a point and the gradient, or two points on the line.
• Identify parallel and perpendicular lines.
• Graph linear inequalities on the Cartesian plane.
The French mathematician René Descartes first introduced the number plane. He realised that
using two sets of lines to form a square grid allowed the position of a point in the plane to be
recorded using a pair of numbers or coordinates.
Coordinate geometry is a powerful mathematical technique that allows algebraic methods to
be used in the solution of geometrical problems.
In this chapter, we will look at the basic ideas of:
• the distance between two points on the number plane
• the midpoint of an interval
• gradient (or slope)
• the relationship between a straight line and its equation.
We shall then see how these can be used to solve problems.
8:01
|
The Distance Between Two Points
The number plane is the basis of coordinate geometry, an important branch of mathematics.
In this chapter, we will look at some of the basic ideas of coordinate geometry and how they
can be used to solve problems.
1
Which of the following is the correct statement of
Pythagoras' theorem for the triangle shown?
Run
----------
1
–1
2
3
4
5
6
0
1 2 3
y
x
4 5
B(2, 6)
5
1
A(1, 1)
A(0, 6)
1
–1
2
3
4
5
6
0
1 2 3
y
x
4 5
B(3, 2)
5
3
1
–1
–1
2
3
4
5
6
0
1 2 3
y
x
4 5
B(1, 5)
6
2 A(3, –1)
1
2
3
4
5
0 1 2 3 4 5 x
y
A
B
C
D
E
F G
H
3
I If two lines have
the same gradient
they are parallel.
4
If a line has no
slope m = 0.
5 m
y
2
y
1
–
x
2
x
1
–
---------------- =
IM4_Ch08_3pp.fm Page 213 Thursday, April 9, 2009 1:24 PM
214 INTERNATIONAL MATHEMATICS 4
a Find the gradient of the line that passes through
A(3, 1) and B(5, 11).
b Find the slope of the line that passes through
O(0, 0) and B(−1, −2).
c On the graph shown, all of the points A, B, C and D
lie on the same straight line, x + 2y = 6.
Find the gradient of the line using the points:
i A and B ii C and D
iii A and D iv B and C
Conclusion: Any two points on a straight line
can be used to find the gradient of that line.
A straight line has only one gradient.
d Use the gradient of an interval to show that the
points (−2, 5), (2, 13) and (6, 21) are collinear
(ie, lie on the same straight line).
a i Find the gradient of BC and of AD.
ii Find the gradient of AB and of DC.
iii What kind of quadrilateral is ABCD?
Give a reason for your answer.
b Prove that a quadrilateral that has vertices A(2, 3),
B(9, 5), C(4, 0) and D(−3, −2) is a parallelogram.
(It will be necessary to prove that opposite sides
are parallel.)
Use the fact that a rhombus is a parallelogram with a pair of adjacent sides equal to prove that the
points A(−1, 1), B(11, 4), C(8, −8) and D(−4, −11) form the vertices of a rhombus.
1
–1
2
3
4
5
0
1 2 3
y
x
4 5 6 7
A(0, 3)
B(2, 2)
C(4, 1)
C(6, 0)
y
x
B(3, 7)
C(7, 8)
D(5, 3)
A(1, 2)
6
7
8
IM4_Ch08_3pp.fm Page 214 Thursday, April 9, 2009 1:24 PM
CHAPTER 8 COORDINATE GEOMETRY 215
8:04
|
Graphing Straight Lines
A straight line is made up of a set of points, each with its own pair of coordinates.
• Coordinate geometry uses an equation to describe the relationship between the x- and
y-coordinates of any point on the line.
In the diagram, the equation of the line is x + y = 3. From the points shown, it is clear that the
relationship is that the sum of each point's coordinates is 3.
• A point can only lie on a line if its coordinates satisfy the equation of the line. For the points
(−3, 2) and (2, 3), it is clear that the sum of the coordinates is not equal to 3. So they do not
lie on the line.
To graph a straight line we need:
• an equation to allow us to calculate the
x- and y-coordinates for each point on
the line
• a table to store at least two sets of coordinates
• a number plane on which to plot the points.
Two important points on a line are:
• the x-intercept (where the line crosses the x-axis)
This is found by substituting y = 0 into the line's
equation and then solving for x.
• the y-intercept (where the line crosses the y-axis)
This is found by substituting x = 0 into the line's
equation and then solving for y.
3
0
–1 –2 –3 –4 1 2 3 4 5
x
y
5
4
(–2, 5)
(–1, 4)
(0, 3)
(2, 3)
(–3, 2)
(1, 2)
(2, 1)
(4, –1)
(5, –2)
x
P
a
t
t
e
r
n
s
a None of the following descriptors has been achieved. 0
b
Some help was needed to complete the table and identify
the simple patterns in questions 2 and 3.
1
2
c
Mathematical problem-solving techniques have been
selected and applied to accurately graph the lines required
and complete the table, with some suggestion of emerging
patterns.
3
4
d
The student has graphed the required lines and used the
patterns evident in the table to find a connecting rule to
give the equations of the lines in question 4.
5
6
e
The patterns evident between the equations and their
graphs have been explained and summarised as a
mathematical rule. The patterns for the lines in part 6
have been explained, tables and explanations are clear
but not always logical or complete.
3
4
d
A good use of mathematical language and representation.
Graphs are accurate, to scale and fully labelled.
Explanations and answers are complete and concise.
5
6
C
r
i
t
e
r
i
o
n
(1, 2)
To find the equation of a straight line that has a gradient of 2 and passes through (7, 5):
(1) Substitute m = 2, x = 7 and y = 5 into the formula y = mx + c to find the value of c.
(2) Rewrite y = mx + c replacing m and c with their numerical values.
y
x
gradient = m
(x
1
, y
1
)
0
y y
1
–
x x
1
–
-------------- m =
IM4_Ch08_3pp.fm Page 225 Thursday, April 9, 2009 1:24 PM
226 INTERNATIONAL MATHEMATICS 4
For each part, find c if the given point lies on the given line.
a (1, 3), y = 2x + c c (2, 10), y = 4x + c c (−1, 3), y = 2x + c
d (5, 5), y = 2x + c e (3, 1), y = x + c f (−1, −9), y = −2x + c
Find the equation of the straight line (giving answers in the form y = mx + c) if it has:
a gradient 2 and passes through the point (1, 3)
b gradient 5 and passes through the point (0, 0)
c gradient 3 and passes through the point (2, 2)
d slope 4 and passes through the point (−1, 6)
The equation of a line with gradient m, that passes through the point (x
1
, y
1
) is given by:
y − y
1
= m(x − x
1
) or .
y y
1
–
x x
1
–
-------------- m =
worked examples
1 Find the equation of the line that passes through (1, 4) and has
gradient 2.
2 A straight line has gradient − and passes through the point (1, 3).
Find the equation of this line.
Solutions
1 Let the equation of the line be: or 1 y − y
1
= m(x − x
1
)
y = mx + c (x
1
, y
1
) is (1, 4), m = 2
∴ y = 2x + c (m = 2 is given) ∴ y − 4 = 2(x − 1)
4 = 2(1) + c [(1, 4) lies on the line] y − 4 = 2x − 2
4 = 2 + c ∴ y = 2x + 2 is the
∴ c = 2 equation of the line.
∴ The equation is y = 2x + 2.
2 Let the equation be: or 2 y − y
1
= m(x − x
1
)
y = mx + c (x
1
, y
1
) is (1, 3), m = −
∴ y = − x + c (m = − is given) ∴ y − 3 = − (x − 1)
3 = − (1) + c [(1, 3) is on the line] y − 3 = − x +
3 = − + c ∴ y = − x + 3 is the
∴ c = 3 equation of the line.
∴ The equation is y = − x + 3 .
You can use
either formula.
I y − y
1
= m(x − x
1
)
or
y = mx + cExercise 8:06
Point–gradient form
Use y − y
1
= m(x − x
1
)
to find the equation of
a line when:
1 m = 2, (x
1
, y
1
) = (1, 4)
2 m = −2, (x
1
, y
1
) = (−1, 3)
Foundation Worksheet 8:06
1
2
IM4_Ch08_3pp.fm Page 226 Thursday, April 9, 2009 1:24 PM
CHAPTER 8 COORDINATE GEOMETRY 227
e gradient −1 and passes through
the point (−2, 8)
f gradient −2 and passes through
the point (0, 7)
g slope −5 and passes through
the point (1, 0)
h gradient and passes through
the point (4, 5)
i gradient and passes through
the point (6, 3 )
j slope − and passes through
the point (−4, −1)
a A straight line has a gradient of 2 and passes through the point (3, 2). Find the equation of
the line.
b A straight line has a gradient of −1. If the line passes through the point (2, 1), find the
equation of the line.
c What is the equation of a straight line that passes through the point (−2, 0) and has a
gradient of 3?
d A straight line that passes through the point (1, −2) has a gradient of −3. What is the
equation of this line?
e A straight line that has a gradient of 3 passes through the origin. What is the equation of
this line?
f Find the equation of the straight line that has a gradient of 4 and passes through the point
(−1, −2).
g (2, 8) is on a line that has a gradient of 4. Find the equation of this line.
h The point (−6, 4) lies on a straight line that has a gradient of −2. What is the equation of
this line?
i Find the equation of the straight line that has a gradient of 2 and passes through the
midpoint of the interval joining (1, 3) and (5, 5).
j A straight line passes through the midpoint of the interval joining (0, 0) and (−6, 4).
Find the equation of the line if its gradient is .
8:07
|
The Equation of a Straight Line,
Given Two Points
Only one straight line can be drawn through two points. Given two points on a straight line, we can
always find the equation of that line.
Consider the line passing through (1, 1) and (2, 4). Let the equation of the line be:
y = mx + c (formula)
'Slope' is another name
for 'gradient'.
1
2
---
1
4
---
1
2
---
1
2
---
3
1
2
---
IM4_Ch08_3pp.fm Page 227 Thursday, April 9, 2009 1:24 PM
228 INTERNATIONAL MATHEMATICS 4
First find the gradient using the two points.
m =
=
= 3
∴ y = 3x + c (since m = 3)
4 = 3(2 + c) [(2, 4) lies on the line]
∴ c = −2
∴ The equation of the line is y = 3x −2.
y
x
4
3
2
1
–1
–1 0 1 2 3
(1, 1)
(2, 4)
(x
1
, y
1
) = (1, 1)
(x
2
, y
2
) = (2, 4)
y
2
y
1
–
x
2
x
1
–
----------------
4 1 –
2 1 –
------------
To find the equation of a straight line that passes through the two points (1, 2)
and (3, 6):
1 Find the value of the gradient m, using the given points.
2 For y = mx + c, find the value of c by substituting the value of m and the
coordinates of one of the given points.
3 Rewrite y = mx + c replacing m and c with their numerical values.
Another method is to use the formula:
where (x
1
, y
1
) and (x
2
, y
2
) are points on the line.
y y
1
–
y
2
y –
1
x
2
x –
1
----------------- x x
1
– ( ) =
worked example
Find the equation of the line that passes through the points (−1, 2) and (2, 8).
Solution
Let the equation of the line be: or
y = mx + c
Now m =
=
=
∴ m = 2
∴ y = 2x + c (since m = 2)
(2, 8) lies on the line.
∴ 8 = 2(2) + c
∴ c = 4
∴ The equation is y = 2x + 4.
(x
1
, y
1
) = (–1, 2)
(x
2
, y
2
) = (2, 8)
y − y
1
=
(x
1
, y
1
) is (−1, 2), (x
2
, y
2
) is (2, 8)
∴ y − 2 =
y − 2 =
y − 2 = 2(x + 1)
y − 2 = 2x + 2
∴ y = 2x + 4 is the equation of the line.
y
2
y
1
–
x
2
x
1
–
---------------- x x
1
– ( )
8 2 –
2 1 – ( ) –
-------------------- x 1 – ( ) – [ ]
6
3
--- x 1 + ( )
y
2
y
1
–
x
2
x
1
–
----------------
8 2 –
2 1 – ( ) –
--------------------
6
3
---
IM4_Ch08_3pp.fm Page 228 Thursday, April 9, 2009 1:24 PM
CHAPTER 8 COORDINATE GEOMETRY 229
Find the gradient of the line that passes through
the points:
a (2, 0) and (3, 4) b (−1, 3) and (2, 6)
c (3, 1) and (1, 5) d (−2, −1) and (0, 9)
e (−2, 1) and (2, 2) f (5, 2) and (4, 3)
g (0, 0) and (1, 3) h (1, 1) and (4, 4)
i (−1, 8) and (1, −2) j (0, 0) and (1, −3)
Use your answers for question 1 to find the equations
of the lines passing through the pairs of points in
question 1.
a Find the equation of the line that passes through the points (−2, −2) and (1, 4).
b The points A(4, 3) and B(5, 0) lie on the line AB. What is the equation of AB?
c What is the equation of the line AB if A is the point (−2, −4) and B is (2, 12)?
d Find the equation of the line that passes through the points (1, 6) and (2, 8).
By substitution in this equation, show that (3, 10) also lies on this line.
e What is the equation of the line CD if C is the point (2, 3) and D is the point (4, 5)?
A is the point (−2, 1), B is the point (1, 4) and
C is the point (3, −2).
a Find the gradient of each side of ∆ABC.
b Find the equation of each of the lines
AB, BC and AC.
c Find the y-intercept of each of the lines
AB, BC and AC.
d Find the equation of the line passing through
point A and the midpoint of interval BC.
e Find the gradient and y-intercept of the line passing
through point A and the midpoint of interval BC.
a Find the equation of the line joining
A(1, 2) and B(5, −6). Hence show that
C(3, −2) also lies on this line.
b A(−2, 2), B(1, −4) and C(3, −8) are points
on the number plane. Show that they are
collinear.
c Show that the points (−2, −11), (3, 4)
and (4, 7) are collinear.
Find the equation of the lines in general form
that pass through the points:
a (3, −2) and (−4, 1)
b (−2, −4) and (3, 2)
c (1·3, −2·6) and (4, −7·3)
d (1 , − ) and (−2 , )
Exercise 8:07
m =
y
2
– y
1
x
2
– x
1
1
2
3
A
B
C
5
4
3
2
1
–1
–2
–3
–3 –2 –1 0 1 2 3 4
x
y
4
I Collinear
points lie on
the same
straight line.
Recipe for
question 5a
1. Find the
equation of AB
2. Substitute
C into this
equation.
5
6
1
2
---
2
3
---
1
3
---
1
2
---
IM4_Ch08_3pp.fm Page 229 Thursday, April 9, 2009 1:24 PM
230 INTERNATIONAL MATHEMATICS 4
8:08
|
Parallel and Perpendicular Lines
Questions 1 to 4 in the Prep Quiz remind us that:
• two straight lines are parallel if their gradients are equal
• the gradients of two lines are equal if the lines are parallel.
Questions 5 to 10 of the Prep Quiz suggest that a condition for two lines to be perpendicular might
be that the product of their gradients is equal to −1.
We do not intend to prove this here, but let us look at several pairs of lines where the product of
the gradient is −1 to see if the angle between the lines is 90°.
A Line r has gradient −1. ∴ m
1
= −1
Line s has gradient 1. ∴ m
2
= 1
Note that m
1
m
2
= −1.
B Line t has gradient . ∴ m
1
=
Line u has gradient −2. ∴ m
2
= −2
Note that m
1
m
2
= −1.
C Line v has gradient − . ∴ m
1
= −
Line w has gradient . ∴ m
2
=
Note that m
1
m
2
= −1.
• By measurement, or use of Pythagoras' theorem, we can
show that the angle between each pair of lines is 90°.
• If two lines are perpendicular, then the product of their gradients is −1.
• m
1
m
2
= −1 (where neither gradient is zero).
In the diagram, AB is perpendicular to BC,
and DE is perpendicular to EF.
1 What is the gradient of each line? 5 Find the gradient of AB. Call this m
1
.
2 Are the lines parallel? 6 Find the gradient of BC. Call this m
2
.
3 If EF was drawn parallel to AB, 7 Using your answers to 5 and 6, find
what would its gradient be? the product of the gradients, m
1
m
2
.
4 Is it possible for two lines with 8 Find the gradient of DE. Call this m
3
.
with different gradients to be 9 Find the gradient of EF. Call this m
4
.
parallel? 10 Using your answers to 8 and 9, find
the product of the gradients, m
3
m
4
.
p
r
e
p
quiz
8:08
A
C
D
B
5
4
3
2
1
–1
–2
–3
–3 –2 –1 0 1 2 3 4
x
y
A
B
C
F
E
D
5
4
3
2
1
–1
–1 0 1 2 3 4 5 6
x
y
6
5
4
3
2
1
–1
–2
–3 –1 0 1 2 3 4
x
y
v
w
u
t
r s
1
2
---
1
2
---
2
5
---
2
5
---
5
2
---
5
2
---
IM4_Ch08_3pp.fm Page 230 Thursday, April 9, 2009 1:24 PM
CHAPTER 8 COORDINATE GEOMETRY 231
• If the product of the gradients of two lines is −1, then the lines are perpendicular.
Two lines with gradients of m
1
and m
2
are:
• parallel if m
1
= m
2
• perpendicular if m
1
m
2
= −1
(or ) where neither m
1
nor m
2
can equal zero. m
1
1 –
m
2
------- =
worked examples
1 Which of the lines y = 4x, y = 3x + 2 and y = x is perpendicular to x + 4y + 2 = 0?
2 Find the equation of the line that passes through the point (2, 4) and is perpendicular to
y = 3x − 2.
3 Find the equation of the line that passes through the point (1, 4) and is parallel to
y = 3x − 2.
Solutions
1 Step 1: Find the gradient of x + 4y + 2 = 0.
Writing this in gradient form gives:
y = − x − 2
∴ The gradient of this line is − .
Step 2: Find the gradients of the other lines.
The gradient of y = 4x is 4.
The gradient of y = 3x + 2 is 3.
The gradient of y = x is 1.
Step 3: Find which gradient in step 2 will
multiply − to give −1.
Conclusion: − × 4 = −1
∴ x + 4y + 2 = 0 is perpendicular to y = 4x.
2 Let the equation of the line be y = mx + b.
Now the gradient of y = 3x − 2 is 3.
∴ m = − (since − × 3 = −1)
∴ y = − x + c
4 = − (2) + c [since (2, 4) lies on line]
4 = − + c
∴ c = 4
∴ The equation of the line is y = − x + 4 .
If m
1
m
2
= –1,
then m
1
=
–1
m
2
3 Let the equation of the line be y = mx + c.
y = 3x − 2 has gradient 3
∴ m = 3 (Parallel lines have equal
gradients.)
∴ y = 3x + c
4 = 3(1) + c, [(1, 4) lies on the line]
∴ c = 1
∴ The equation of the line is y = 3x + 1.
1
4
---
1
4
---
1
4
---
1
4
---
1
3
---
1
3
---
1
3
---
1
3
---
2
3
---
2
3
---
1
3
---
2
3
---
IM4_Ch08_3pp.fm Page 231 Thursday, April 9, 2009 1:24 PM
232 INTERNATIONAL MATHEMATICS 4
Are the following pairs of lines parallel or not?
a y = 3x + 2 and y = 3x − 1
b y = 5x − 2 and y = 2x − 5
c y = x + 7 and y = x + 1
d y = x − 3 and y = 1x + 2
e y = 3x + 2 and 2y = 6x − 3
f y = 2x + 1 and 2x − y + 3 = 0
g 3x + y − 5 = 0 and 3x + y + 1 = 0
h x + y = 6 and x + y = 8
Are the following pairs of lines perpendicular or not?
a y = x + 3, y = −5x + 1 b y = 3x − 2, y = − x + 7
c y = 2x − 1, y = − x + 3 d y = x + 4, y = − x − 5
e y = 4x, y = x − 3 f y = x − 1, y = − x
g y = 3x − 1, x + 3y + 4 = 0 h x + y = 6, x − y − 3 = 0
a Which of the following lines are parallel to y = 2x + 3?
y = 3x + 2 2x − y + 6 = 0 2y = x + 3 y = 2x − 3
b Two of the following lines are parallel. Which are they?
y = x − 3 x + y = 3 y = 3x 3y = x y = −x + 8
c A is the point (1, 3), B is (3, 4), C is (6, 7) and D is (8, 8).
Which of the lines AB, BC, CD and DA are parallel?
a Which of the following lines are perpendicular to y = 2x?
y = 3x y = 2x − 3 x + 2y = 4 y = −0·5x + 5
b Two of the following lines are perpendicular. Which are they?
y = −1 x + 2 y = x − 1 y = x
c A is the point (2, −1), B is the point (3, −2) and C is (4, −1). Prove that AB ⊥ BC.
a Find the equation of the line that has y-intercept 3 and is parallel to y = 5x − 1.
b Line AB is parallel to y = 3x − 4. Find the equation of AB if its y-intercept is −1.
c Line EF is parallel to y = x + 5. Its y-intercept is 3. What is the equation of EF?
d A line has a y-intercept of 10 and is parallel to the line x + y = 4. What is the equation
of this line?
a Find the equation of the line that has y-intercept 5 and is perpendicular to y = − x + 1.
b The line AB is perpendicular to y = −x + 4. Its y-intercept is 3. What is the equation of AB?
c Find the equation of CD if CD is perpendicular to the line y = − x and has a y-intercept of 0.
d A line has a y-intercept of 1·5 and is perpendicular to the line y = −2x + 1. Find the
equation of the line.
Exercise 8:08
Parallel and
perpendicular lines
1 Use y = mx + c to find the
slope of the lines in columns
A and B.
2 Which lines in columns
A and B are:
a parallel? b perpendicular?
Foundation Worksheet 8:08
1
2
1
5
---
1
3
---
1
2
---
2
3
---
3
2
---
1
4
---
3
4
---
4
3
---
3
4
1
2
---
1
2
---
2
3
---
5
6
1
3
---
1
2
---
IM4_Ch08_3pp.fm Page 232 Thursday, April 9, 2009 1:24 PM
CHAPTER 8 COORDINATE GEOMETRY 233
a AB is a line which passes through the point (2, 3). What is the equation of AB if it is parallel
to y = 5x + 2?
b Find the equation of the line that passes through (1, 0) and is parallel to y = −3x − 1.
c A is the point (0, 0) and B is the point (1, 3). Find the equation of the line that has
y-intercept 5 and is parallel to AB.
d Find the equation of the line that has y-intercept −3 and is parallel to the x-axis.
e What is the equation of the line that is parallel to the x-axis and passes through the
point (−2, −3)?
a If AB passes through the point (2, 3) and is perpendicular to y = 2x − 7, find the equation
of AB in general form.
b Find the equation of the line that passes through (1, 0) and is perpendicular to y = −3x − 1.
Write your answer in general form.
c A is the point (0, 0) and B is the point (1, 3). Find the equation of the line that has
y-intercept 5 and is perpendicular to AB. Give the answer in general form.
d Find the equation of the line that has y-intercept −3 and is perpendicular to the y-axis.
e What is the equation of a line that is perpendicular to the x-axis and passes through (3, −2)?
a Find the equation of the line that is parallel to the line 2x − 3y + 6 = 0 and passes through
the point (3, −4). Give the answer in general form.
b A line is drawn through (−1, 2), perpendicular to the line 4x + 3y − 6 = 0. Find its equation
in general form.
c A line is drawn through the point (−1, −1), parallel to the line 2x − 3y + 9 = 0. Where will
it cross the x-axis?
d A line is drawn parallel to 4x − 3y + 1 = 0, through the points (1, 3) and (6, a). What is the
value of a?
In the diagram, the line 5x + 2y + 5 = 0 cuts
the x-axis and y-axis at E and C respectively.
BD is the line x = 2, AB is parallel to the x-axis
and BE and CD are perpendicular to AC. Find the
coordinates of the points A, B, C, D and E.
7
8
9
1 2
x
y
A B
C
D
E
10
IM4_Ch08_3pp.fm Page 233 Thursday, April 9, 2009 1:24 PM
234 INTERNATIONAL MATHEMATICS 4
8:09
|
Graphing Inequalities on the
Number Plane
In Prep Quiz 8:09 questions 1, 2 and 3, we see that, once x = 3 is graphed on the number line,
all points satisfying the inequation x > 3 lie on one side of the point and all points satisfying the
inequation x < 3 lie on the other side.
On the number plane, all points satisfying the equation y = 2x + 1 lie on one straight line.
All points satisfying the inequation y < 2x + 1 will lie on one side of the line.
All points satisfying the inequation y > 2x + 1 will lie on the other side of the line.
A y = 2x + 1 B y < 2x + 1 Note: • Inequations B, C and D are often
called 'half planes'.
• In D, the line is part of the solution
set. In B and C, the line acts as a
boundary only, and so is shown as
a broken line.
• Choose points at random in each
of the half planes in B, C and D
to confirm that all points in each
half plane satisfy the appropriate
C y > 2x + 1 D y р 2x + 1 inequation.
For each number line graph, write down the appropriate equation or inequation.
1 2 3
4 5 6
7 8 9
10 On a number line, draw the graph of x < −2 where x is a real number.
p
r
e
p
quiz
8:09
0 1 2 3 4 5 6
x
0 1 2 3 4 5 6
x
0 1 2 3 4 5 6
x
–2 –1 0 1 2 3 4
x
–2 –1 0 1 2 3 4
x
–2 –1 0 1 2 3 4
x
–5 –4 –3 –2 –1 0 1
x
–5 –4 –3 –2 –1 0 1
x
–5 –4 –3 –2 –1 0 1
x
3
2
1
–1
–1 0 1 2 3 4 5
x
y
y Ͻ 3x – 1
continued §§§
IM4_Ch08_3pp.fm Page 235 Thursday, April 9, 2009 1:24 PM
236 INTERNATIONAL MATHEMATICS 4
By testing a point from each side of the line, write down
the inequation for each solution set graphed below.
a b
c d e
a The union of the two half planes b The intersection is the region that
is the region that is part of one or belongs to both half planes. It is the
the other or both graphs. part that the graphs have in common.
The union is written: The intersection is written:
{(x, y): x + 2y у 2 ∪ y < 3x − 1} {(x, y): x + 2y у 2 ∩ y < 3x − 1}
Note: • Initially draw the boundary lines as broken lines.
• Part of each region has a part of the boundary broken and a part unbroken.
5
4
3
2
1
2
x
1 2 -
- -
1 2 -
- -
1
3
1 2 -
- -
1 2 -
- -
1 2 -
- -
1 5 -
- -
1
0
1 2 -
- -
2 3 -
- -
4 3 -
- -
3
2
IM4_Ch08_3pp.fm Page 238 Thursday, April 9, 2009 1:24 PM
CHAPTER 8 COORDINATE GEOMETRY 239
Mathematical terms 8
Mathematical Terms 8
coordinates
• A pair of numbers that gives the position
of a point in a number plane relative to
the origin.
• The first of the coordinates is the
x-coordinate. It tells how far right
(or left) the point is from the origin.
• The second of the coordinates is called
the y-coordinate. It tells how far the
point is above (or below) the origin.
distance formula
• Gives the distance between the points
(x
1
, y
1
) and (x
2
, y
2
).
general form
• A way of writing the equation of a line.
• The equation is written in the form
ax + by + c = 0.
where a, b, c are integers and a > 0.
gradient
• The slope of a line or interval. It can be
measured using the formula:
Gradient =
gradient formula
• Gives the gradient of the interval joining
(x
1
, y
1
) to (x
2
, y
2
).
gradient–intercept form
• A way of writing the equation of a line.
eg y = 2x − 5, y = x + 2
When an equation is rearranged and
written in the form y = mx + c then m is
the gradient and c is the y-intercept.
graph (a line)
• All the points on a line.
• To plot the points that lie on a line.
interval
• The part of a line between two points.
midpoint
• Point marking the middle of an interval.
midpoint formula
• Gives the midpoint of the interval joining
(x
1
, y
1
) to (x
2
, y
2
).
Midpoint =
number plane
• A rectangular
grid that
allows the
position of
points in a
plane to be
identified by
an ordered
pair of
numbers.
origin
• The point where the x-axis and y-axis
intersect, (0, 0). See under number
plane.
plot
• To mark the position of a point on the
number plane.
x-axis
• The horizontal number line in a number
plane. See under number plane.
x-intercept
• The point where a line crosses the x-axis.
y-axis
• The vertical number line in a number
plane. See under number plane.
y-intercept
• The point where a line crosses the y-axis.
s
m r e
t
l
a
c
i
t
a
m
e
h
t
a
m
8
d x
2
x
1
– ( )
2
y
2
y
1
– ( )
2
+ =
Rise
Run
y
x
rise
run
--------
m
y
2
y
1
–
x
2
x
1
–
---------------- =
1
2
---
x
1
x
2
+
2
----------------
y
1
y
2
+
2
---------------- ,
C(–3, 2) A(3, 2)
E(0, –1)
B(3, –2)
D(–2, –3)
3
2
1
0
–1
–3
–1 –2 –3 1 2 3 x
y
–2
4
3
1
2
3
1
2
IM4_Ch08_3pp.fm Page 239 Thursday, April 9, 2009 1:24 PM
240 INTERNATIONAL MATHEMATICS 4
Diagnostic Test 8: | Coordinate Geometry8
1 Find the length of the interval AB in each of the following.
(Leave answers in surd form.)
a b c
2 Use the distance formula to find the distance between the points:
a (1, 2) and (7, 10) b (3, 0) and (5, 3) c (−3, −2) and (1, −3)
3 Find the midpoint of the interval joining:
a (1, 2) and (7, 10) b (3, 0) and (5, 3) c (−3, −2) and (1, −3)
4 What is the gradient of each line?
a b c
5 Find the gradient of the line that passes through:
a (1, 3), (2, 7) b (−2, 8), (4, 5) c (0, 3), (3, 5)
6 a Does the point (3, 2) lie on the line x + y = 5?
b Does the point (−1, 3) lie on the line y = x + 2?
c Does the point (2, −2) lie on the line y = x − 4?
7 Graph the lines:
a y = 2x + 1 b 2x − y = 3 c 3x + 2y = 6
8 State the x- and y-intercepts of the lines:
a 2x − y = 3 b x + 3y = 6 c x + 2y = 4
Section
8:01
8:01
8:02
8:03
8:03
8:04
8:04
8:04
1
−1 −2
B
A
2
4
−1
3
1 2 3
y
x
−2
−2
B
A
2
4
0
2
y
x
−4
−2
−1 −2
B
A
1
−1
2
1 2 3
y
x
−2
−3
−4IM4_Ch08_3pp.fm Page 240 Thursday, April 9, 2009 1:24 PM
CHAPTER 8 COORDINATE GEOMETRY 241
9 Graph the lines:
a x = 2 b y = −1 c x = −2
10 Write down the equation of the line which has:
a a gradient of 3 and a y-intercept of 2
b a gradient of and a y-intercept of −3
c a y-intercept of 3 and a gradient of −1
11 Write each of the answers to question 10 in general form.
12 What is the gradient and y-intercept of the lines:
a y = 2x + 3? b y = 3 − 2x? c y = −x + 4?
13 Rearrange these equations into gradient–intercept form.
a 4x − y + 6 = 0 b 2x + 3y − 3 = 0 c 5x + 2y + 1 = 0
14 Find the equation of the line that:
a passes through (1, 4) and has a gradient of 2
b has a gradient of −3 and passes through (1, 3)
c has a gradient of and passes through (−2, 0)
15 Find the equation of the line that:
a passes through the points (1, 1) and (2, 3)
b passes through the points (−1, 2) and (1, −4)
c passes through the origin and (3, 4)
16 Find the equation of the line that:
a has a y-intercept of 2 and is parallel to y = 4x − 1
b passes through (1, 7) and is parallel to y = −3x + 4
c is perpendicular to y = x + 1 and passes through (−1, 4)
d is perpendicular to y = 1 − 2x and passes through (−1, 4)
17 Write down the inequation for each region.
a b c
18 Graph a the union and b the intersection of the half planes representing
the solutions of x + 2y у 2 and y < 3x − 1.
8:04
8:05
8:05
8:05
8:05
8:06
8:07
8:08
8:09
8:09
1
2
---
1
2
---
2
3
---
3
2
1
–1
–2
–3
x
y
–3 –2 –1 0 1 2 3
x
=
–
1
3
2
1
–1
–2
–3
x
y
–3 –2 –1 0 1 2 3
y
=
–
2
x
3
2
1
–1
–2
–3
x
y
–3 –2 –1 0 1 2 3
2
x
–
y
+
2
=
0
IM4_Ch08_3pp.fm Page 241 Thursday, April 9, 2009 1:24 PM
242 INTERNATIONAL MATHEMATICS 4
Chapter 8 | Revision Assignment
1 Find:
a the length
AB as a surd
b the slope
of AB
c the midpoint
of AB.
2 A is the point (2, 5) and B is the point
(7, 17).
a What is the length AB (as a surd)?
b What is the slope of AB?
c What is the midpoint of AB?
3 A is the point (6, 5) and B is the point
(2, −2).
a What is the equation of the line AB?
b The line AB passes through the point
(100, b). What is the value of b?
c AC is perpendicular to AB. Find its
equation in general form.
4 a A line has an x-intercept of 3 and a
gradient of 1. Find where the line
crosses the y-axis and hence write down
its equation.
b A line has a slope of − and a
y-intercept of 6. What is its equation?
What is its x-intercept?
c A line has an x-intercept of 3 and a
y-intercept of 6. What is its equation?
5 The points X(2, 2), Y(−2, 4) and Z(−4, 0)
form a triangle. Show that the triangle is
both isosceles and right-angled.
6 A line is drawn perpendicular to the line
2x − 3y + 4 = 0 through its y-intercept.
What is the equation of the line? Give the
answer in general form.
7 A median of a triangle is a line drawn from
a vertex to the midpoint of the opposite
side. Find the equation of the median
through A of the triangle formed by the
points A(3, 4), B(−2, −4) and C(−6, 8).
8 What inequalities
describe the region
shown?
8A
t
n e
m
n
g
i
s
s
a
1
−1
−2
−1
2
3
4
5
6
1 2 3
y
x
4 5 6
A(6, 5)
B(2, −2)
1
2
---
1 (1, 1)
2
x
y
1 2
The coordinate system for
locating points on the earth
is based on circles.
1 x and y intercept and graphs
2 Using y = mx + c to find the gradient
3 General form of a line
4 Parallel and perpendicular lines
5 Inequalities and regions
Linear graphs and equations
IM4_Ch08_3pp.fm Page 242 Thursday, April 9, 2009 1:24 PM
CHAPTER 8 COORDINATE GEOMETRY 243
Chapter 8 | Working Mathematically
1 The diagram shows a
4-minute timer.
a If this timer was started
with the pointer on
zero, what number
would it be pointing
to after 17 minutes?
b At what time between 30 minutes and
1 hour will the pointer be pointing at
number 3?
2 The faces of a cube are
divided into 4 squares.
If each square on
each face is to be painted,
what is the minimum
number of colours
needed if no squares that
share an edge can be the same colour?
3 Brendan and Warwick wish to use a
photocopier to reduce drawings.
a Brendan's drawing is 15 cm high but
must be reduced to 8 cm to fit into the
space he has left in his project. What
percentage setting must he choose on
the photocopier to achieve the required
reduction?
b Warwick thinks the machine is
malfunctioning so he decides to check
it by reducing his drawing, which is
20 cm long. He chooses the 60%
setting. If the machine is functioning
properly, what would you expect the
length of his picture to be?
c The setting button jams on 68%. What
sized copies are possible by repeated
use of this button? (Give all answers
above 20%.)
4 Four friends decide to play tennis.
Find out how many different:
a singles matches can be played.
(A singles match is one player against
another player.)
b doubles matches can be played.
(A doubles match is two players
against two players.)
5 What is the smallest whole number that, if
you multiply by 7, will give you an answer
consisting entirely of 8s?
6 A 4 × 4 grid is drawn
and the numbers 1, 2, 3
and 4 are placed in the
grid so that every number
occurs only once in each
row and only once in
each column.
a Find the missing numbers in the grid
shown.
b Now place the numbers in a 4 × 4 grid,
following the rules above, so that the
sums of the diagonals are 16 and 4.
a
s
sign
m
e
n
t
8B
0
1 3
2
1 2
2
3
4
4
IM4_Ch08_3pp.fm Page 243 Thursday, April 9, 2009 1:24 PM
Simultaneous
Equations
9
244
3x + 5y = 14 ........ (1)
7x - 2y = 19 ........ (2)
x = 3
y = 1
Chapter Contents
Investigation: Solving problems by
'guess and check'
9:01
The graphical method of solution
Investigation: Solving simultaneous
equations using a graphics calculator
Fun Spot: What did the book say to
the librarian?
9:02
In this chapter, you will learn how to solve problems like those in Investigation 9:01A more
systematically. Problems like these have two pieces of information that can be represented by
two equations. These can then be solved to find the common or 'simultaneous' solution.
Investigation 9:01A | Solving problems by 'guess and check'
Consider the following problem.
A zoo enclosure contains wombats and emus. If there are 50 eyes and 80 legs, find the
number of each type of animal.
Knowing that each animal has two eyes but a wombat has 4 legs and an emu has two legs, we
could try to solve this problem by guessing a solution and then checking it.
Solution
If each animal has two eyes, then, because there are
50 eyes, I know there must be 25 animals.
If my first guess is 13 wombats and 12 emus, then
the number of legs would be 13
×
4
+
12
×
2
=
76.
Since there are more legs than 76, I need to increase
the number of wombats to increase the number of
legs to 80.
I would eventually arrive at the correct solution of
15 wombats and 10 emus, which gives the correct
number of legs (15
×
4
+
10
×
2
=
80).
Try solving these problems by guessing and then
checking various solutions.
1
Two numbers add to give 86 and subtract to
give 18. What are the numbers?
2
At the school disco, there were 52 more girls
than boys. If the total attendance was 420,
how many boys and how many girls attended?
3
In scoring 200 runs, Max hit a total of 128 runs as boundaries. (A boundary is either 4 runs
or 6 runs.) If he scored 29 boundaries in total, how many boundaries of each type did he
score?
4
Sharon spent $5158 buying either BHP shares or ICI shares. These were valued at $10.50
and $6.80 respectively. If she bought 641 shares in total, how many of each did she buy?
n
o
i
t
a
g i t s e v
n
i
9:01A
IM4_Ch09_3pp.fm Page 245 Monday, March 16, 2009 11:13 AM
246
INTERNATIONAL MATHEMATICS
4
9:01
|
The Graphical Method of Solution
There are many real-life situations in which we wish to find when or where two conditions come or
occur together. The following example illustrates this.
If
y
=
2
x
−
1, find
y
when:
1
x
=
1
2
x
=
0
3
x
=
−
1
4
x
=
−
5
If
x
−
2
y
=
5, find
y
when:
5
x
=
0
6
x
=
1
7
x
=
2
8
x
=
−
4
9
If 3
x
−
y
=
2, complete the table below.
10
Copy this number plane and
graph the line 3
x
−
y
=
2.
p
r
e
p
quiz
9:01
x
y
2 –2
–2
2
4
–4
–4 4
x 0 1 2
y
worked example
A runner set off from a point and maintained a speed of 9 km/h. Another runner left the same
point 10 minutes later, followed the same course, and maintained a speed of 12 km/h. When,
and after what distance travelled, would the second runner have caught up to the first runner?
We have chosen to solve this question graphically.
First runner
Second runner
t = time in minutes after the first runner begins
d = distance travelled in kilometres
• From the graph, we can see that the lines cross
at (40, 6).
• The simultaneous solution is t = 40, d = 6.
• The second runner caught the first runner
40 minutes after the first runner had started
and when both runners had travelled
6 kilometres.
t 0 30 40 60
d 0 4·5 6 9
t 10 30 40 70
d 0 4 6 12
From these tables
we can see that
the runners meet
after 6 km and
40 minutes.
0 10 20 30 40 50 60 70
12
10
8
6
4
2
Time (in min)
D
i
s
t
a
n
c
e
(
i
n
k
m
)
d
t
After the second
runner has run
for 30 minutes,
t = 40.
IM4_Ch09_3pp.fm Page 246 Monday, March 16, 2009 11:13 AM
CHAPTER 9 SIMULTANEOUS EQUATIONS
247
Often, in questions, the information has to be written in the
form of equations. The equations are then graphed using a
table of values (as shown above). The point of intersection
of the graphs tells us when and where the two conditions
occur together.
It is sometimes difficult to graph accurately either or both lines, and it is often difficult to read
accurately the coordinates of the point of intersection.
Despite these problems, the graphical method remains an extremely useful technique for solving
simultaneous equations.
worked example
Solve the following equations simultaneously.
x + y = 5
2x − y = 4
Solution
You will remember from your earlier work on coordinate geometry that, when the solutions to
an equation such as x + y = 5 are graphed on a number plane, they form a straight line.
Hence, to solve the equations x + y = 5 and 2x − y = 4
simultaneously, we could simply graph each line and
find the point of intersection. Since this point lies on
both lines, its coordinates give the solution.
• The lines x + y = 5 and 2x − y = 4 intersect at (3, 2).
Therefore the solution is:
x = 3
y = 2
x + y = 5 2x − y = 4
x 0 1 2 x 0 1 2
y 5 4 2 y −4 −2 0
0 2 4 6 –2
6
4
2
Ϫ2
Ϫ4
y
x
(3, 2)
x
1
Explain why
(g) and (h)
above are
unusual.
2
3
The graphical method
doesn't always give exact
answers.
4
5
IM4_Ch09_3pp.fm Page 248 Monday, March 16, 2009 11:13 AM
CHAPTER 9 SIMULTANEOUS EQUATIONS 249
A car passed a point on a course at exactly 12 noon and maintained a speed of 60 km/h.
A second car passed the same point 1 hour later, followed the same course, and maintained
a speed of 100 km/h. When, and after what distance from this point, would the second car
have caught up to the first car? (Hint: Use the method shown in the worked example on
page 438 but leave the time in hours.)
Mary's salary consisted of a retainer of $480 a week plus $100 for each machine sold in that
week. Bob worked for the same company, had no retainer, but was paid $180 for each machine
sold. Study the tables below, graph the lines, and use them to find the number, N, of machines
Bob would have to sell to have a wage equal to Mary (assuming they both sell the same number
of machines). What salary, S, would each receive for this number of sales?
Mary
Bob
N = number of machines
S = salary
No Frills Car Rental offers new cars for rent at
38 per day and 50c for every 10 km travelled
in excess of 100 km per day. Prestige Car Rental
offers the same type of car for 30 per day plus
1 for every 10 km travelled in excess of
100 km per day.
Draw a graph of each case on axes like those
shown, and determine what distance would
need to be travelled in a day so that the rentals
charged by each company would be the same.
Star Car Rental offers new cars for rent at $38 per day and $1 for every 10 km travelled in
excess of 100 km per day. Safety Car Rental offers the same type of car for $30 per day plus 50c
for every 10 km travelled in excess of 100 km per day.
Draw a graph of each on axes like those in question 8, and discuss the results.
N 0 4 8
S 480 880 1280
N 0 4 8
S 0 720 1440
6
7
R
D 100 180 260 340
30
40
50
Distance in kilometres
R
e
n
t
a
l
i
n
d
o
l
l
a
r
s
8
9
IM4_Ch09_3pp.fm Page 249 Monday, March 16, 2009 11:13 AM
250 INTERNATIONAL MATHEMATICS 4
Investigation 9:01B | Solving simultaneous equations using
a graphics calculator
Using the graphing program on a graphics calculator
complete the following tasks.
• Enter the equations of the two lines y = x + 1 and
y = 3 − x. The screen should look like the one shown.
• Draw these graphs and you should have two straight
lines intersecting at (1, 2).
• Using the G-Solv key, find the point of intersection
by pressing the F5 key labelled ISCT.
• At the bottom of the screen, it should show x = 1,
y = 2.
Now press EXIT and go back to enter other pairs of
equations of straight lines and find their point of
intersection.
Fun Spot 9:01 | What did the book say to the librarian?
Work out the answer to each part and
put the letter for that part in the box
that is above the correct answer.
Write the equation of:
A line AB C line OB
U line BF A line EB
I the y-axis O line AF
U line OF K line AE
E line CB T the x-axis
T line EF N line OD
Y line CD O line OA
i
n
v
e
s
t
igatio
n
9:01B
x = 1
y
1
= x +1
y
2
= 3 – x
y = 2 ISECT
Graph Func : y =
y
1
= x +1
y
2
= 3 – x
y
3
:
y
4
:
y
5
:
y
6
:
Note: You can change the scale
on the axes using the V-Window
option.
f
u
n
spo
t
9:01
x
y
0
2
6
4
–2
–4
–2 –4 2 4
EE FF
CC DD
AA B B
y
1
IM4_Ch09_3pp.fm Page 260 Monday, March 16, 2009 11:13 AM
CHAPTER 9 SIMULTANEOUS EQUATIONS 261
Chapter 9 | Revision Assignment
1 Solve the following simultaneous equations
by the most suitable method.
a x + y = 3 b 4x − y = 3
2x − y = 6 2x + y = 5
c 4a + b = 6 d 6a − 3b = 4
5a − 7b = 9 4a − 3b = 8
e a − 3b = 5 f 2x − 3y = 6
5a + b = 6 3x − 2y = 5
g p = 2q − 7 h 4x − y = 3
4p + 3q = 5 4x − 3y = 7
i 7m − 4n − 6 = 0
3m + n = 4
2 A man is three times as old as his daughter.
If the difference in their ages is 36 years,
find the age of father and daughter.
3 A theatre can hold 200 people. If the price
of admission was $5 per adult and $2 per
child, find the number of each present if
the theatre was full and the takings were
$577.
4 A man has 100 shares of stock A and
200 shares of stock B. The total value of
the stock is $420. If he sells 50 shares of
stock A and buys 60 shares of stock B,
the value of his stock is $402. Find the
price of each share.
5 Rectangle A is 3 times longer than
rectangle B and twice as wide. If the
perimeters of the two are 50 cm and 20 cm
respectively, find the dimensions of the
larger rectangle.
6 A rectangle has a perimeter of 40 cm. If the
length is reduced by 5 cm and 5 cm is
added to the width, it becomes a square.
Find the dimensions of the rectangle.
7 A canoeist paddles at 16 km/h with the
current and 8 km/h against the current.
Find the velocity of the current.
a
s
sign
m
e
n
t
9A
IM4_Ch09_3pp.fm Page 261 Monday, March 16, 2009 11:13 AM
262 INTERNATIONAL MATHEMATICS 4
Chapter 9 | Working Mathematically
1 You need to replace
the wire in your
clothes-line. Discuss
how you would
estimate the length
of wire required.
a On what
measurements
would you base your estimate?
b Is it better to overestimate or
underestimate?
c What level of accuracy do you feel is
necessary? The diagram shows the
arrangement of the wire.
2 What is the last digit of the number 3
2004
?
3 Two smaller isosceles
triangles are joined to
form a larger isosceles
triangle as shown in the
diagram. What is the
value of x?
4 In a round-robin competition each team
plays every other team. How many games
would be played in a round-robin
competition that had:
a three teams? b four teams?
c five teams? d eight teams?
5 How many different ways are there of
selecting three chocolates from five?
6 A school swimming coach has to pick a
medley relay team. The team must have
4 swimmers, each of whom must swim one
of the four strokes. From the information
in the table choose the fastest combination
of swimmers.
t
n e
m
n
g
i
s
s
a
9B
A
B C
x°
AB = AC
Name Back Breast Fly Free
Dixon 37·00 44·91 34·66 30·18
Wynn 37·17 41·98 36·59 31·10
Goad 38·88
Nguyen 41·15 49·05 39·07 34·13
McCully 43·01 32·70
Grover 43·17
Harris 37·34 34·44
• What is the fastest medley relay?
IM4_Ch09_3pp.fm Page 262 Monday, March 16, 2009 11:13 AM
Graphs of
Physical
Phenomena
As covered in Book 3:
• A distance–time graph (or travel graph) can be a type of line graph used to describe one or more
trips or journeys.
• The vertical axis represents distance from a certain point, while the horizontal axis
represents time.
• The formulae that connect distance travelled (
D
), time taken (
T
) and average speed (
S
) are
given below.
D S T S ×
D
T
---- T
D
S
---- = = =
worked examples
Example 1
The travel graph shows the journey made by
a cyclist.
a How many hours was the journey?
b How far did the cyclist travel?
c What was the cyclist's average speed?
d Between what times did the cyclist stop to rest?
e Between what times was the cyclist travelling
fastest?
Solution 1
a The cyclist began at 8 am and finished at 7 pm. That is a total of 11 hours.
b He travelled a total of 45 km.
c Average speed = Total distance travelled ÷ the time taken
= 45 ÷ 11
= 4 km/h
d Between 12 am and 1:30 pm the line is flat which means no distance was travelled during
that time. This must have been the rest period.
e When the graph is at its steepest, more distance is being covered per unit of time. So this
is when the cyclist is travelling fastest. This would be from 6 pm to 7 pm.
10
0
20
30
40
50
10 8 4 2 6 8
D
i
s
t
a
n
c
e
(
k
m
)
am am pm pm pm pm
Time of day
noon
12
1
11
------
IM4_Ch10_3pp.fm Page 264 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA
265
Two women, Tamara and Louise, are travelling
along the same road. Their progress is shown on
the graph.
a
Who started first?
b
Who stopped for a rest?
c
Who had the fastest average speed?
d
When did they meet on the road?
e
What was the fastest speed for either of them?
Example 2
The graph shows the journeys of two friends,
Mike and Mal.
They each leave their own house at 8 am and
walk to the other's house to see who can walk
the fastest.
a Which graph shows Mike's journey?
b When do the two friends pass one another?
c Who has the longest rest?
d What is the speed of each friend?
e Who walks the fastest at any time in their
journey?
Solution 2
a Because the graph shows the distance from Mike's house, the person who starts 0 km from
there must be Mike. Therefore the blue graph is Mike and the red graph is Mal.
b They pass one another when they are the same distance from Mike's house. So they pass one
another where the graphs cross — at 2:15 pm.
c Mike rests for 1 hour, Mal rests for 2 hours (where the graphs are flat) so Mal has the
longest rest.
d Mike's speed = Mal's speed =
= 2·5 km/h = 2·1875 km/h
e Mal walks the fastest from 5 pm to 8 pm as the gradient is the steepest of all.
Gradient = . At no other time is the gradient steeper than this.
5
0
10
15
20
25
10 8
noon
4 2 12 6 8
D
i
s
t
a
n
c
e
The graph shows the journey taken by Max
as he went for a training run on his bicycle.
a
When was Max travelling fastest?
What was the speed at this time?
b
Did he stop? If so, for how long?
c
How many kilometres did Max cycle?
d
What was his average speed for the entire trip?
e
Max's brother Thilo did the same trip, but
cycled at a constant speed all the way and did
not stop. Show his journey on the same graph.
A family left Hamburg by car at 9 am. They drove 200 km in the first 2 hours then stopped
for half an hour for lunch. Then they drove 150 km along the autobahn in the next hour.
They then left the autobahn in the next hour. They then left the autobahn and drove at
an average speed of 50 km/h for the last 1·5 hours of their journey.
a
Draw a graph showing their journey.
b
What was their average speed for the whole trip?
The graph shows the progress of a group of
bushwalkers hiking in bush over a number
of days. It shows their distance from the start
of the hike which is at the ranger's station.
a
How far did they hike?
b
Did they hike every day? If not, on which
day did they rest?
c
On which day did they hike the least distance?
d
On average, how far did they hike per day?
e
How much more than the average did they
hike on the last day?
0
20
40
60
80
0 2 4 8 10
100
6
Time taken
D
i
s
t
a
n
c
e
f
r
o
m
h
o
m
e
(
k
m
)
2
3
0
5
10
15
20
0 1 2 4 5
25
3
Day
D
i
s
t
a
n
c
e
f
r
o
m
R
a
n
g
e
r
S
t
a
t
i
o
n
(
k
m
)
4
IM4_Ch10_3pp.fm Page 266 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA
267
The graph shows the journey taken by two motorists
— one represented in blue and the other in red.
a
If the speed limit on all the roads travelled is
80 km/h, did either motorist break the speed
limit? If so which one?
b
Apart from when they stopped, what was the
slowest speed for each motorist?
c
Which motorist drove most consistently?
d
What can be said about the average speed
of the two motorists?
e
If both motorists were in the same make of car,
and the fuel consumption is 8 L/100 km under
70 km/h and 10 L/100 km over 70 km/h, which
motorists will use the least fuel?
Investigation 10:01 | Graphing coins
By rolling a coin along a wooden ruler it is possible to
graph the position of a coloured mark as the coin moves.
The mark on the coin is highest when it is at the top.
The mark is actually touching the ground when it is
at the bottom.
• The greatest height is equal to the diameter of the coin.
• The smallest height is zero, which occurs when the
mark is on the ground. Because the coin is rolling,
the height of the mark will oscillate between these
two positions as the distance rolled increases.
1
Choose three coins of different sizes. Produce a separate graph for each, similar to the
one above.
• The distance–time graphs in 10:01A were all composed of straight line segments. In reality, these
might only be an average representation of the motion.
• Consider this simple graph which shows a car's
journey from A to B.
• The straight red line shows the car arriving at B,
5 km away, in 5 minutes. The average speed of
1 km/min is shown by this line.
• However, the curved green line would more
accurately represent the motion of the car
during its journey.
• At A and B, the curve is flat, or horizontal,
showing that its speed is zero, since the car
has stopped at these points.
• At point C, the car would be going fastest as
the curve is steepest at this point.
• An indication of the speed at any point on the
curve can be worked out by noting the slope of the tangent to the curve at this point. Three
tangents have been drawn in blue at points X, Y and C. Since the slope of the tangents at X and
Y is the same as the red line, the speed of the car at X and Y would be 1 km/min.
• Can you see that the speed of the car between X and Y would be greater than 1 km/min since
the curve is steeper in this section?
• Can you see that from A to X and from Y to B the car's speed would be less than 1 km/min?
1
2
3
4
5
0 1 2
Time (minutes)
3 4 5
A
B
C
Y
X
D
i
s
t
a
n
c
e
(
k
m
)
worked examples
Example 1
1 A ball is thrown from ground level and
lands on a roof.
a To what height did the ball rise and
how high was the roof?
b What was the average speed of the ball
until it reached its maximum height?
c What was the speed of the ball at its
maximum height?
d When was the ball travelling at its fastest
speed?
e When was the height decreasing?
0 1 2
Time (seconds)
3 4 5 6
H
e
i
g
h
t
(
m
e
t
r
e
s
)
6
12
18
24
IM4_Ch10_3pp.fm Page 268 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 269
Solution 1
a The maximum height (H) is 24 m.
The height of the roof (R) is 18 m.
b The average speed from the ground
G to H is indicated by the straight
red line. The ball travelled 24 m in
4 seconds, so the average speed
was 6 m/s.
c The speed of the ball at H is zero
because the graph is flat at this point.
That is, the slope of the tangent to the
curve is zero.
d The fastest speed was at G, when
the ball was first thrown. After that,
the ball was slowing down. Its height
was increasing, but at a decreasing
rate. (It goes up, but the same increase
will take a longer time.)
e The height was obviously decreasing
after 4 seconds. This is shown on the
graph by the line going down or
decreasing. The slope of the tangent at
any point between H and R would be
negative.
Example 2
This graph shows the height of a particular seat on a ferris wheel as it rotates to a height of 40 m.
0 1 2
Time (seconds)
3 4 5 6
H
e
i
g
h
t
(
m
)
20
40
A
R S Q P
C
B D
O
continued §§§
IM4_Ch10_3pp.fm Page 269 Monday, March 16, 2009 11:25 AM
270 INTERNATIONAL MATHEMATICS 4
Amanda cycled from her home (H) to
her friend's house (F), 30 km away.
The black curved line shows her actual
distance from H at any time.
a What is her average speed for the
whole ride?
b What is the average speed from
H to point P?
c What is the average speed from
P to F?
d If the blue line XY is the tangent
to the curve at point P, what is
Amanda's actual speed at point P?
a At what point is the rate of change in the height zero?
b When is the seat rising at its fastest speed?
c When is the seat falling at its fastest speed?
d Describe the rate of change in height of the seat from O to A.
Solution 2
a At the top and bottom of each rotation, the rate of change in height is zero; ie at points
O, A, B, C, D on the graph.
b The seat is rising its fastest when the graph is increasing at the greatest rate. This is at
points P and R, where the slope of the curve is steepest; ie at a height of 20 metres.
c The seat is also falling at its fastest rate when the height is 20 m, but at points Q and S,
where the slope of the curve is negative.
d The height of the seat increases from zero at O at an increasing rate until it reaches
point P. The height continues to increase, but at a decreasing rate, until it reaches its
maximum height at point A, where the rate of change is zero.
Exercise 10:01B
9 am
Time
10 am 11 am
D
i
s
t
a
n
c
e
f
r
o
m
h
o
m
e
(
k
m
)
5
10
20
15
25
30
H
Y
P
X
F
1
IM4_Ch10_3pp.fm Page 270 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 271
Jono drove from O to S as shown by this
graph, in 3 hours. His speed varied due
to traffic and road conditions.
a What was Jono's average speed from
O to Q?
b Was Jono driving faster at point P or
point Q?
c Was Jono driving faster at point Q or
point R?
d Use the blue tangent drawn through
point P to determine the speed at which
Jono was travelling at this time.
e Similarly calculate Jono's speed at
point R.
This graph shows Benny's journey in
blue and Robyn's journey in red as
they made their way from A to B via
different routes. They both arrived
at the same time.
a What was their average speed for
the journey?
b At 2 pm they were both 40 km
from A, but who was travelling at
the greater speed at this point?
c When they were each 10 km
from B, who was travelling at the
greater speed?
d At approximately what two times
during their journeys was Benny's
and Robyn's speed the same?
10 am 11 am
Time
12 noon 1 pm
D
i
s
t
a
n
c
e
(
k
m
)
50
100
120
O
P
Q
R
S
1 pm
Time
2 pm 2.45 pm
D
i
s
t
a
n
c
e
f
r
o
m
h
o
m
e
(
k
m
)
10
20
40
30
50
70
60
A
P
B
How did she
pass me?
2
1
2
---
3
IM4_Ch10_3pp.fm Page 271 Monday, March 16, 2009 11:25 AM
272 INTERNATIONAL MATHEMATICS 4
Briony drove from home to a friend's house
100 km away. After staying a short while,
she then drove home.
a What was Briony's average speed from
home to her friend's house?
b Is Briony's speed greater or less than this
average speed at:
i point A? ii point B?
c After point D, the slope of the graph is
negative, indicating that Briony is travelling
in the opposite direction, ie towards home.
At what point on the journey home does
Briony's speed appear the greatest?
d Between which two points on the journey
to Briony's friend's house was Briony's
distance from home increasing at a
decreasing rate?
A projectile was fired 90 metres into the air and returned
to the ground after 6 seconds.
a What was the average speed of the projectile from the
start to its maximum height H?
b Determine the speed of the projectile at point P, when
the height was 80 m.
c Determine the speed at point Q, when the projectile
was at a height of 50 m on its return journey.
d The projectile's height increased for the first
3 seconds. Did it do so at an increasing or decreasing
rate?
e The height then decreased until the projectile reached
the ground. Did it do so at an increasing or decreasing
rate?
'Wow! I was increasing
at a decreasing rate.'
0
20
40
60
80
100
11 am 12
noon
1 pm 2 pm
Time
D
i
s
t
a
n
c
e
(
m
)
P
H
Q
IM4_Ch10_3pp.fm Page 272 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 273
Grain is poured into a conical-shaped silo. The graph shows
the height of grain as the grain is poured in after t minutes.
The equation for this graph is
(that is, 1·56 × )
a Determine the height of grain in the silo when the elapsed
time is:
i 10 min ii 2 min
b Describe the rate of change of the height as the silo is filled.
c By drawing tangents to the curve, estimate the rate of
change in the height of the grain in metres/min when:
i t = 1 ii t = 2
10:01 World record times
6
h 1·56 = 100
3
t 100
3
t
0
5
15
10
5 10
Time (minutes)
H
e
i
g
h
t
(
m
)
t
h
IM4_Ch10_3pp.fm Page 273 Monday, March 16, 2009 11:25 AM
274 INTERNATIONAL MATHEMATICS 4
Challenge 10:01 | Rolling down an inclined plane
• When an object is dropped, the force of gravity causes
it to increase its speed by about 9·8 m/s every second;
ie acceleration due to gravity Ӏ 9·8 m/s per second.
• When a ball rolls down an inclined plane the
acceleration will be much less.
This investigation involves finding the acceleration of a ball (shot put or marble) as it rolls
down an inclined plane of your choice.
If the ball starts from rest (ie speed = 0), then the formula s = at
2
describes the motion.
Here: s is distance travelled,
t is time taken,
a is acceleration
Steps
1 Make a long inclined plane (over 2 m long) with an angle of inclination of about 10°.
2 Use a stop watch to time the ball as it
rolls 0·5, 1, 1·5 and 2 m down the
inclined plane. Complete the table below
as you go.
3 Plot s against t and draw the curve of best fit.
4 Plot s against t
2
(ie the value of s on the
vertical axis and the value of t
2
on the
horizontal axis).
5 Draw the line of best fit. The gradient of
this line (rise divided by run) will be an
approximation to a. Double the gradient
to find a. (If friction were not present,
9·8 sin θ would be the acceleration of the
ball, where θ is the angle of inclination
of the inclined plane.)
c
h
a
lleng
e
10:01
inclined plane
tape measure
angle of inclination
1
2
---
1
2
0 t
2
s
1
2
0 t
s
s 0·5 1 1·5 2
t
t
2
1
2
---
IM4_Ch10_3pp.fm Page 274 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 275
10:02
|
Relating Graphs to Physical
Phenomena
Much of a mathematician's work is concerned with finding a relationship between two quantities
that can change or vary their values.
For example, we could study a person's height at different times of their life. As one quantity
changes (eg time), the other changes or varies (eg height). The features that change, and the
pronumerals used to represent them, are called variables.
Graphs provide an excellent means of exploring the relationship between variables.
They give an immediate 'picture' of the relationship, from which we can see such things as:
• whether a variable is increasing or decreasing with respect to the other variable
• when a variable has its highest or lowest value
• whether a variable is increasing quickly or slowly with respect to the other variable.
Graphs can be used to show relationships between data such as:
• temperature and time of day (or year) • distance and speed
• height and weight • light brightness and proximity
• water level before, during and after a bath • tidal movements over time, and many more.
In the graphs on the right,
in which graph does:
1 M increase as t increases?
2 M decrease as t increases?
3 M stay unchanged?
4 5 In each diagram on the left,
for which line is the mass
increasing more quickly,
I or II?
In which part of this graph does:
6 the height increase slowly?
7 the height increase quickly?
In which of the graphs does:
8 M increase slowly at
first and then quickly?
9 M increase quickly at
first, then slowly?
10 M increase at the same rate?
p
rep
q
u
i
z
10:02
Graph A
M
t
Graph B
M
t
Graph C
M
t
Time
M
a
s
s
I
II
Time
M
a
s
s
I
II
Time
H
e
i
g
h
t
I
II
Graph X
M
N
Graph Y
M
N
Graph Z
M
N
IM4_Ch10_3pp.fm Page 275 Monday, March 16, 2009 11:25 AM
276 INTERNATIONAL MATHEMATICS 4
worked examples
Example 1
A person is driving a car at a certain speed and then increases that speed.
Which graph represents this?
Solution 1
Since two speeds are involved, the second one greater than the first, the graph must have two
sections, one for each speed. Graph A is unsuitable because it consists of only one section.
Graph B is unsuitable because the first section is a horizontal section, which indicates that the
car was not moving. But the question indicates that the car was moving and then changed its
speed. Graph C best illustrates the information given.
Example 2
Water is added to the tank shown at a steady rate. Which graph best represents the increase
in the water level h?
Solution 2
Looking at the tank we notice that the middle part is skinnier than
the other parts. Therefore, if water is poured in at a steady rate it
will fill up faster in the middle part than in the other two sections.
Hence in our graph the water level, h, will increase more quickly
for this section of the tank than for the others. Hence the correct
graph must consist of three sections, with the steepest section in
the middle. Hence graph A is the best representation.
Example 3
A point A is on the circumference of a wheel. If this wheel
is rolled, make a graph to show the height of this point
above the ground.
Solution 3
The highest point on the wheel above the ground
is when it is at the top of the wheel. Therefore the greatest
height above the ground is the diameter of the wheel.
The smallest height above the ground is zero, which occurs
when the point is actually on the ground.
Time
D
i
s
t
a
n
c
e
A
Time
D
i
s
t
a
n
c
e
B
Time
D
i
s
t
a
n
c
e
C
h
t
A
h
t
B
h
t
C
The skinny one will
fill up faster than
the wide one.
Once around
is called a
'revolution'.
IM4_Ch10_3pp.fm Page 276 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 277
Because the wheel is rolling,
the height of point A will oscillate
between these positions.
Example 4
An automatic pump is used to fill a cylindrical tank. The tank, which is empty to start with,
is filled at a steady rate.
The tank remains full for a period before it is emptied at a steady rate. If it is emptied faster
than it is filled, make a sketch which shows the variation in water level in the tank for one
complete pumping cycle.
Solution 4
At the start of the cycle the tank is empty
and so the water level is zero.
The tank is then filled at a steady rate,
which means that the water level will
rise at a steady rate.
After it is filled, the tank remains full for
a period. For this period, the water level
remains the same.
The tank is then emptied at a steady rate
but one which is faster than the rate at
which it was filled. This means that the
line in this section of the graph must be
steeper than the line in the first section.
It is also important to be able to describe and interpret information presented in graphs.
Example 5
Describe what this graph is showing.
Solution 5
A flat basketball was inflated, probably with a
motorised pump. It was then used for a while
before a puncture developed.
Distance rolled
H
e
i
g
h
t
o
f
p
o
i
n
t
A
Time
Tank full.
Water level constant.
Tank
fills and
empties at a steady
rate. Empties faster
than it fills.
Tank is empty at start
and end of cycle.
W
a
t
e
r
l
e
v
e
l
Time
Pressure in a basketball
P
r
e
s
s
u
r
e
IM4_Ch10_3pp.fm Page 277 Monday, March 16, 2009 11:25 AM
278 INTERNATIONAL MATHEMATICS 4
Choose the graph above that best represents each situation below.
a Population growth as time passes.
b The number of people waiting for a train on a station as time passes.
c The maximum daily temperature in Sydney throughout a year.
d The depth of water in a bath as time passes as it is filled, a person gets in, has a bath, the
person gets out and then the water is let out.
e The height of a ball as time passes as it is dropped from a window and continues to bounce.
f The height of a stone as time passes if the stone is thrown into the air.
A lady in a car drives for three hours
averaging 80 km/h, 60 km/h and 90 km/h
respectively in each hour. Which graph
best represents her trip?
A boy rode his bike down the road.
He said he rode quickly at the start
and then slowed down. Which graph
best represents his journey?
The diagram on the right shows
the water level of a tank which is
filled and emptied periodically.
Give an interpretation of this
graph by describing what happens
in the first 120 minutes.
Exercise 10:02
1
t
A
t
B
t
C
t
D
t
E
t
F23
Water level in a tank
20
0
40 60 80 100 120 140
W
a
t
e
r
l
e
v
e
l
(
m
e
t
r
e
s
)
Time (min)
1
2
3
4
IM4_Ch10_3pp.fm Page 278 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 279
Give a reasonable story or explanation for the information shown in each graph below.
a b
c d
X is a point on the circumference of
a roller, as shown in the diagram.
The roller can be moved in either a
clockwise or an anticlockwise direction.
The graphs below give the height of
X above the ground. Which graph
represents the clockwise rotation?
5
Time
Jill's pulse rate
P
u
l
s
e
Time
Air in a balloon
V
o
l
u
m
e
Distance
A car's journey
F
u
e
l
Time
Pizza in selfish Sam's
stomach
A
m
o
u
n
t
X
or
ground
Hmm ... will X move
up or down first?
6
A
Distance rolled
H
e
i
g
h
t
o
f
X
B
Distance rolled
H
e
i
g
h
t
o
f
X
IM4_Ch10_3pp.fm Page 279 Monday, March 16, 2009 11:25 AM
280 INTERNATIONAL MATHEMATICS 4
Shock absorbers in a car are designed to reduce the bouncing up and down that is caused when
the car hits a bump in the road.
The following three graphs show the vertical distance moved by a point on the front bonnet of
the car, after hitting a bump.
Which graph best represents a well-designed shock absorber?
Each of the four containers pictured is filled with water at a steady rate. When the level of
water in each container was plotted, the graphs I to IV were obtained. Match each container
to its graph.
The following is known about the solubility of four chemical salts.
Salt A: As the temperature increases the solubility increases, slowly at first and then at a much
faster rate.
Salt B: Shows very little increase in solubility as the temperature increases.
Salt C: Increases its solubility at a steady rate as the temperature increases.
Salt D: As the temperature increases the solubility decreases and then increases.
Match each salt to the graphs I to IV.
7
I
Time
V
e
r
t
i
c
a
l
d
i
s
t
a
n
c
e
II
Time
V
e
r
t
i
c
a
l
d
i
s
t
a
n
c
e
III
Time
V
e
r
t
i
c
a
l
d
i
s
t
a
n
c
e
8
A
B
C
D
Time
W
a
t
e
r
l
e
v
e
l I
Time
W
a
t
e
r
l
e
v
e
l
II
Time
W
a
t
e
r
l
e
v
e
l
III
Time
W
a
t
e
r
l
e
v
e
l
IV
9
Temperature
S
o
l
u
b
i
l
i
t
y
I
Temperature
S
o
l
u
b
i
l
i
t
y
II
Temperature
S
o
l
u
b
i
l
i
t
y
III
Temperature
S
o
l
u
b
i
l
i
t
y
IV
IM4_Ch10_3pp.fm Page 280 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 281
A point A is on the circumference of a wheel. The wheel
completes one revolution where the starting position of A
can be at any of the points 1, 2, 3 or 4 marked on the
diagram on the right.
Which one of the graphs below best represents the
height of A above the ground when A starts at:
a position 1? b position 3? c position 2?
You will need to know that the wheel is rolled from left to right.
Two boys are riding on a see-saw. Make a sketch representing the height of a boy above the
ground as time passes if he starts on the ground and goes up and down twice.
When boiling water is allowed to cool,
it is known that it loses heat quickly at
the start and, as time goes on, it loses
heat at a slower rate.
Make a sketch to show the shape of a
graph which would support this
information. Show water temperature
on the vertical axis and time on the
horizontal axis.
A mass of 1 kg is hanging from a vertical spring as
shown in the diagram. If this mass is pulled downwards
and then released, the mass will move.
Make a sketch to show how the position of the mass
will vary from its equilibrium position as time passes.
1
2
3
4
F14.82
10
Distance rolled
H
e
i
g
h
t
o
f
A
I
Distance rolled
H
e
i
g
h
t
o
f
A
II
Distance rolled
H
e
i
g
h
t
o
f
A
III
Distance rolled
H
e
i
g
h
t
o
f
A
IV
11
12
1 kg
Equilibrium
Pulled down
and released
position
13
IM4_Ch10_3pp.fm Page 281 Monday, March 16, 2009 11:25 AM
282 INTERNATIONAL MATHEMATICS 4
Sketch a line graph to represent each of the following situations. Choose axis labels wisely.
Time, when mentioned, should always appear on the horizontal axis.
a how water level varies in a bath as it is being filled
b the relationship between speed and time when a car travels at a constant speed
c the relationship between speed and time when a car travelling at a constant speed brakes
slowly before coming to a halt
d a similar relationship as in part c, but the car brakes quickly
e how the brightness of a red traffic light varies against time
f how the water level in a bath varies against time when you take a bath
g how the water level in a leaking tank varies against time just after the leak starts
h how the temperature of boiled water varies against time as it cools
i how the fluid level in a cup of tea varies against time as it is consumed
j how the level of fuel in a fuel tank varies against the number of kilometres travelled by a car
k how the water level in a glass containing water and floating ice changes as the ice melts
l how your excitement level changes as you work through this exercise
A parachutist jumps from a plane. Before she opens her chute, her speed increases at a constant
rate. On opening her chute her speed falls rapidly, approaching a constant terminal value.
Make a sketch showing how her speed varies with time.
A car is approaching a set of traffic lights at a constant speed when the driver sees the lights
change and immediately applies the brakes. The car comes to a stop. After waiting for the lights
to change the car accelerates away until it reaches the same constant speed at which it had
approached the lights. Make a sketch showing:
a how the car's speed relates to time
b how the distance travelled by the car relates to time
14
15
16
IM4_Ch10_3pp.fm Page 282 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 283
A person walking in the desert attempts to walk in a straight
line. However, her legs are not exactly the same length and so
she walks in a circle.
The table below shows the relationship between the difference
(d) between the right and left leg in regards to length of step
and the radius (r) of the circle in which the person walks.
a If her left leg is longer than her right leg, would she
turn to the left or to the right?
b As the difference between the lengths of steps
increases, what happens to the radius of the circle?
c Plot the information in the table onto the number
plane to the right. Draw a curve joining these points.
d Write a formula to describe this relationship.
e If the difference in steps is 1·5 mm, what would
be the radius of the circle in which she walked?
f What happens when there is no difference in steps?
Gary discovered that, when draining his pool, the volume
of water (V litres) remaining in the pool is related to the
time (t minutes) that the water has been draining.
The formula relating V and t is:
V = 20(30 − t)
2
.
a When t = 0 (initially), the pool was full.
What volume of water can the pool hold?
b Copy and complete this table using the formula.
c Graph the relationship using the axes shown.
d How long does it take to drain 9000 L from the pool?
e Does the water drain from the pool at a steady rate? Explain your answer.
10:02 Filling tanks
Difference in
millimetres, d
1 2 3 4 5
Radius of circle
in metres, r
180 90 60 45 36
t 0 5 10 20 25 30
V 18 000
r
1 2 3 4 5 0 d
r
M
e
t
r
e
s
50
100
150
200
Millimetres
10 20 30 0 t
V
5000
10000
15000
20000
17
18
IM4_Ch10_3pp.fm Page 283 Monday, March 16, 2009 11:25 AM
284 INTERNATIONAL MATHEMATICS 4
Investigation 10:02 | Spreadsheet graphs
The data contained in
a spreadsheet can be
presented in the form
of a graph or chart.
A baby's weight in grams
was recorded every two
weeks. The results for the
first ten weeks were
entered on a spreadsheet
as shown.
This data was then
displayed as a graph by
'inserting a chart' on
the spreadsheet. Three
different samples are
shown here.
• Which graph do you
think shows the data
most clearly?
• Investigate the 'chart'
option in a spreadsheet
program such as Excel
by using this data or
other data of your own.
i
n
v
e
s
t
igatio
n
10:02
1
age weight
3
4
5
A B B D C E
F14
2
7
8
9
6
fx
0w
2w
4w
6w
8w
10w
3000
4000
4500
5200
6100
7400
0w
8000
6000
4000
2000
0
2w 4w 6w 8w 10w
Weight in grams
Weight in grams
8000
6000
4000
2000
0
0w 2w 4w 6w 8w 10w
Weight in grams
8000 6000 4000 2000 0
0w
4w
8w
IM4_Ch10_3pp.fm Page 284 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 285
Fun Spot 10:02 | Make words with your calculator
Use the sums and the clues below to complete the crossword. You read the answers to
the sums by turning your calculator around and looking at the numbers upside down.
(Ignore any decimal points.)
f
u
n s
p
o
t
10:02
Across
1 A light worker (4231 × 9)
5 Santa sound (2 ÷ 5)
6 Dirty stuff (45 × 1269)
8 Not the stinging type
(9·5 × 4)
9 Water tube (12 × 4 × 73)
11 Some people open theirs too
much
(1970 − 43 × 27)
12 A fowl animal
(210 × 165 + 359)
13 One of these is always right
(11·62 − 2·25)
14 Not profitable
(92·1 − 37·03)
19 Peas or the sea
(278
2
+ 61)
21 Good to duck
23 Spots (97 × 5 × 11)
24 Don't buy!
(7103 + 829 − 197)
25 Grain store (0·5 × 1·43)
26 Top person
(9634 − 4126)
28 More of 12 across
(40 000 − 4661)
Down
1 Yukky muck (0·3
2
)
2 Give it the O.K.
(2768·9 × 20)
3 Don't stand around!
(2·37 − 1·47)
4 Exists
(47·3 − 28·6 + 32·3)
6 Come in pairs (10 609 × 5)
7 This word is slack
(190
2
− 1093)
8 A shocking sound
(1·7 − 1·62)
9 What a pig! (8 × 113)
10 A bit fishy
(100·6 − 93·27)
12 Set substance
15 Slippery stuff
(7·1 × 25 × 4)
16 Cry, etc (35 × 23)
17 A manly title
18 'Well?' – 'Not exactly.'
(800 − 29)
20 Was it an adder?
(6325 + 1607 − 2418)
22 Ice house (0·6903 − 0·6112)
23 Opposite to 17 down
(15 × 23)
24 Only one of these
(2470 × 1·5)
27 Therefore
(2·451 + 1·63 − 3·581)
1 2 3 4
5 6 7
8
9 10 11
12
13 14 15 16
17 18
19 20 21 22
23 24
25
26 27
28
5329 4·18 + ( )
752 169 – ( )
1296 2 – ( )
IM4_Ch10_3pp.fm Page 285 Monday, March 16, 2009 11:25 AM
286 INTERNATIONAL MATHEMATICS 4
Challenge 10:02 | Curves and stopping distances
The approximate stopping distance
of a car is given by the formula:
d = v + 0·073v
2
while the approximate stopping
distance of a truck is given by the
formula:
d = v + 0·146v
2
• For both formulae, d is measured in
metres and v is measured in metres per
second.
• This table relates stopping distance
and the speed of a car for various speeds.
d = v + 0·073v
2
1 Complete the table below, which relates stopping distance and the speed of a truck for
various speeds.
d = v + 0·146v
2
2 The graph below is a model of the distance between cars after stopping. The cars are
100 m apart, heading towards one another, when the drivers sense danger. The drivers
brake at the same instant.
• The right axes and graph refer to the car coming from the right. The left axes and graph
refer to the car coming from the left.
• In the first table above, the reading on the vertical axis (speed) is mentioned first.
c
h
a
lleng
e
10:02 0 m 13·4 m 36·9 m 58·3 m 84·1 m
30
20
10
S
p
e
e
d
o
f
c
a
r
1
(
m
/
s
)
S
p
e
e
d
o
f
c
a
r
2
(
m
/
s
)
0
30
20
10
0
0 20 40 60 80 100
100 80 60 40 20 0
At 100 km/h (27·8 m/s)
stopping distances overlap
80 km/h (22·2 m/s)
60 km/h (16·7 m/s)
B
A
Distance (m)
IM4_Ch10_3pp.fm Page 286 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 287
Examples
• If both cars are travelling at 60 km/h (16·7 m/s), the interval A gives the distance apart
when they stop.
From the graph, this is 26 m.
• If the stopping distances of the cars have overlapped, then a collision has occurred.
At 100 km/h (27·8 m/s) the readings on the distance axis have overlapped, indicating
a collision.
• If the red car is travelling at 60 km/h (16·7 m/s) and the green car at 80 km/h
(22·2 m/s), the interval B gives the distance apart when they stop.
From the graph, this is about 4 m.
1 a Do the cars collide if they are both travelling at 80 km/h?
At what speed (in m/s) would a collision just be avoided if the cars are travelling at
the same speed?
b What would be the distance between the cars after stopping, if they had both been
travelling at:
i 10 m/s? ii 20 m/s?
c At what speed would both cars be travelling if they stop:
i 20 m apart? ii 60 m apart?
d What would be the distance between the cars after stopping, if the speeds of the
cars were:
i red, 10 m/s; green, 27 m/s?
ii red, 23 m/s; green, 12 m/s?
2 Draw a graph, similar to the one above, to model the distance between vehicles after
stopping if the vehicles are the car and the truck referred to in the tables on page 521.
• Use the questions in 1 above as a guide to list similar findings using your new graph.
Mathematical Terms 10 | Graphs of physical phenomena
axis (pl. axes)
• Each graph has two axes, a horizontal
axis and a vertical axis, which show the
two quantities that are being compared
on the graph.
graph
• A representation of numerical data in the
form of a diagram.
• A graph provides a quick way of
analysing patterns in numerical data.
phenomenon (pl. phenomena)
• An object or occurrence that is observed.
• A physical phenomenon is one that can
be measured.
scale
• Set of marks at measured distances on
an axis.
• Used in measuring or making proportional
reductions or enlargements.
speed
• Relative rate of motion or action.
• Defined as:
• Measuring in units such as km/h or m/s.
travel graph
• A line graph where distance travelled is
plotted against time taken.
• The gradient (or slope) of the line is an
indication of the speed of the motion.
m
a
them
a
t
i
c
a
l
t
e
r
m
s
10
distance
time
--------------------
IM4_Ch10_3pp.fm Page 287 Monday, March 16, 2009 11:25 AM
288 INTERNATIONAL MATHEMATICS 4
Diagnostic Test 10: | Graphs of physical phenomena10
1 a At what time is Ms Jonas 5 km
from home?
b How far from home is Ms Jonas
at 1:30 pm?
c What is the furthest distance
from home?
d At what time did she rest?
For how long?
e How far has she travelled on
her trip?
2 This graph shows the distance of two
brothers, Joe and Jacky, from home.
a How far does Joe start from home?
b What is Jacky's average speed from:
i 10 am to 12 noon?
ii 12 noon to 1:30 pm?
iii 2 pm to 3 pm?
c What is Joe's average speed for the
entire journey from 10 am to 3 pm?
d What is Joe's greatest speed and
between which times is it recorded?
3 This graph shows the journey of
a car from A to B, 200 km away.
a What is the average speed
from A to B?
b Is the car driving faster at
point P or point Q?
c Use the red tangent line
drawn at R to determine the
speed of the car at this point.
Section
10:01A
10:01A
10:01B
0
10
20
30
40
10 9 11 noon 1 2
Time
D
i
s
t
a
n
c
e
(
k
m
)
t
A
P
Q
R
B
IM4_Ch10_3pp.fm Page 288 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 289
4 A tennis ball was hit 75 m into the air
and landed back on the ground after
4 seconds.
a Was the speed increasing or
decreasing for the first 2 seconds?
What tells you this?
b What was the average speed of the
ball over the first 2 seconds?
c Use a tangent to find the speed
at point H. What does this point
represent?
d Determine the speed at point R, when the ball is at 40 m on the
way down.
5 a How long does it take the water
to reach a temperature of 80°C?
b What is the temperature
of the water after 10 min?
c What is the temperature
of the water after 20 min?
d From the graph it can be
seen that the water doesn't
cool at a constant rate.
The dotted line represents
a constant cooling rate.
If the water had cooled at
a constant rate, what would
its temperature have been
after 10 min?
6 The graph shows the variation in
solubility of three salts in water with
change in temperature.
a How much of salt II will dissolve at
40°C?
b What temperature is needed to
dissolve 40 g of salt III?
c Which salt would have the greatest
solubility at 40°C?
d Will 50 g of salt II dissolve at 30°C?
7 Draw a line graph to represent:
a how the water level in a tank varies
against time as it is being constantly filled
b how the intensity of light changes as the sun rises and falls during
the day
c the average temperature each month for a year in Sydney
Section
10:01B
10:02
10:02
10:02
0
20
40
60
80
1 2 3 4
Time (s)
H
e
i
g
h
t
w
a
t
e
r
Solubility curves
I
II
III
IM4_Ch10_3pp.fm Page 289 Monday, March 16, 2009 11:25 AM
290 INTERNATIONAL MATHEMATICS 4
Chapter 10 | Revision Assignment
1 The graph shows the trips of Ms Chew and Ms Travers
and gives their distance from town A at different times.
a At 10 am how far is Ms Chew from A?
b At 10 am how far is Ms Travers from A?
c At what time are they the same distance from A?
d Who completes the trip by returning to A?
e How far is Ms Chew from A when Ms Travers is
20 km from A? (Note: There are two possible answers.)
2 Draw a travel graph which shows the information given.
Briony left home at 9 am, walking at 5 km/h until
11 am when she accepted a lift from a friend. The car travelled at an average speed of
40 km/h for 2 hours. Briony then stopped for lunch until 1:30 pm. She then caught a train
and arrived home at 3 pm.
The horizontal axis should show the time and the vertical axis the distance from home.
3 Choose the heading from the list to the right that would best fit each graph.
a b
c d
4 Describe what is happening on each graph in question 3 as time passes, using the heading
most appropriate for that graph.
5 Give a reasonable story or explanation for the information shown on this graph.
t
n e
m
n
g
i
s
s
a
10A
0
A
20
40
60
80
8 9 10 11 noon 1
Time
D
i
s
t
a
n
c
e
f
r
o
m
A
(
k
m
)
C
h
e
w
T
r
a
v
e
r
s
A The hook of
a fishing line
while fishing.
B An arrow fired
into the air.
C Flying a kite.
D Position of
my head while
pole vaulting.
E A parachute
jump.
F Position of my
foot as I kick
a ball.
Time
H
e
i
g
h
t
Time
H
e
i
g
h
t
Time
H
e
i
g
h
t
Time
H
e
i
g
h
t
Andrew and Helen go bungy jumping
Andrew
Helen
H
e
i
g
h
t
a
b
o
v
e
g
r
o
u
n
d
Time
IM4_Ch10_3pp.fm Page 290 Monday, March 16, 2009 11:25 AM
CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA 291
Chapter 10 | Working Mathematically
1 Use ID Card 7 on page xix to identify:
a 5 b 8 c 9 d 10 e 11
f 12 g 18 h 22 i 23 j 24
2 Naomi bought a computer system for Luke. The marked price was $2300.
She paid a deposit of $1200 and 12 monthly payments of $115.
a How much did she pay?
b How much more than the marked price did she pay?
c What percentage was the extra money paid of the amount owing after the deposit was
paid? (Give the percentage correct to 1 decimal place.)
3 For this cylinder find:
a the area of the base
b the curved surface area
c the total surface area
4 A salesman's wages are $230 per week plus a commission of 4 % on his sales. How much
will he earn in a week when he sells $7000 worth of goods?
5 a What month has the highest rainfall?
b What month has the lowest rainfall?
c Which month recorded a rainfall of
180 mm?
d What rainfall was recorded in May?
e How much rain fell in the year?
f How much rain fell during winter
(June, July, August)?
g How much less rain fell in autumn
than in spring?
a
s
sign
m
e
n
t
10B
12.6 cm
15 cm
1
2
---
20
60
100
140
180
220
260
J J J A S O N D F M M A
Months
R
a
i
n
f
a
l
l
(
m
m
)
Rainfall graph
IM4_Ch10_3pp.fm Page 291 Monday, March 16, 2009 11:25 AM
292 INTERNATIONAL MATHEMATICS 4
6
The graph above shows the 'normal' weight for girls aged 0 to 3 years. The numbers on
the right side of the graph are percentages. (Only 3% of girls 3 years old have a weight
less than 11·3 kg.)
a Why are there two 3s on the horizontal axis?
b What is the median weight for girls of age:
i 3 months? ii 1 year? iii 19 months?
iv 2 years 3 months? v 1 year 2 months?
c What percentage of 3-year-old girls have a weight between 13·9 kg and 16 kg?
d What weights would be considered 'normal' for a girl of age 2 months?
e What weights would be considered 'normal' at birth for a girl?
0 1 3 6 9 12 15 18 21 2 2
Age in months Age in years
W
e
i
g
h
t
P
a
t
t
e
r
n
s
a None of the following descriptors has been achieved. 0
b
Some help was needed to apply mathematical techniques
and answer questions 1–3.
1
2
c
Mathematical techniques have been selected and applied to
use the properties of polygons and tessellations to effectively
answer questions 1–3, and attempt questions 4 and 5.
3
4
d
Questions 1–3 are correctly answered with a reasonable
explanation for questions 4 and 5.
5
6
e
The student has completely answered all questions,
explained all patterns evident and justified their results Working out and explanations are
insufficient.
1
2
c
There is satisfactory use of mathematical language and
representation. Working and explanations are clear and
the student has been able to use the properties of polygons
well to explain results.
3
4
d
There is good use of mathematical language and
representation. Answers are correct and explanations are
complete connection
to the real-life applications.
1
2
c
There is a correct but brief explanation of whether results
make sense and how they were found. A description of the
important aspects of polygons and tessellations is given.
3
4
d
There is a critical explanation of the results obtained and
their relation to real life. The answers to questions 4 and 5
are fully explained with consideration of the accuracy of the
results obtained and possible further applications of
tessellations.
5
6
IM4_Ch11_3pp.fm Page 303 Monday, March 16, 2009 12:11 PM
304 INTERNATIONAL MATHEMATICS 4
Investigation 11:02B | Spreadsheet
The spreadsheet below is used to calculate:
• the angle sum of a polygon
• the size of an interior angle in a regular polygon.
The following steps show how the spreadsheet was reproduced.
1 Entering text
Open the Microsoft Excel program, move the cursor to cell A1 and type Number of Sides then
press ENTER. Move the cursor to cells B1 and C1 and add the other headings.
2 Adjusting the column width
The width of the columns can be adjusted by selecting FORMAT from the top menu bar
followed by COLUMN and then WIDTH. The number shown is the present width. Typing a
larger or smaller number in the box followed by clicking the OK will increase or decrease the
column width.
The column width can also be adjusted in other ways.
3 Using a formula
Move the cursor to cell A2 and press 3 followed by ENTER. The cursor will move to A3. Enter
a formula by typing =A2+1 followed by ENTER. The number 4 will now appear in cell A3.
In cells B2 and C2 enter the formulae =(A2-2)*180 and =B2/A2 respectively. The numbers
180 and 60 should appear in cells B2 and C2 respectively.
4 Copying a formula
Move the cursor to cell A3. Now select EDIT on the top menu bar and then choose COPY.
Highlight the cells A3 to A19 by holding down SHIFT and then pressing the down arrow .
Cells A4 to A19 should be blackened. Choose EDIT from the menu followed by PASTE. The
numbers in cells A3 to A19 will now appear as shown. (To remove the flashing cursor from cell
A3 move the cursor to A3 and press ENTER.)
i
n
v
e
s
t
igatio
n
11:02B
1
2
3
4
5
6
7
8
9
11
12
A
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
B
10
D E
13
14
15
16
17
19
20
18
F
Number of Sides Interior Angle Sum Size of Interior Angle
C8 =B8/A8 =
C
180
360
540
720
900
1080
1260
1440
1620
1800
1980
2160
2340
2520
2700
2880
3060
3240
60
90
108
120
128.5714286
135
140
144
147.2727273
150
152.3076923
154.2857143
156
157.5
158.8235294
160
161.0526316
162
File Edit Format Tools Data Window Help Insert View
↓
IM4_Ch11_3pp.fm Page 304 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 305
11:02 Spreadsheet
Challenge worksheet 11:02 Regular polygons and tessellations
Move the cursor to cell B2 and then select EDIT followed by COPY. Highlight cells B2 to B19
using the SHIFT and down arrow key as before. Select EDIT followed by PASTE and the
numbers in cells B2 to B19 will appear.
Move the cursor to cell C2 and select EDIT followed by COPY. Then highlight cells C2 to C19.
Select EDIT followed by PASTE and the numbers in cells C2 to C19 will appear.
QUESTIONS
1 Move the cursor to cell A2 and type the number 10 followed by ENTER. What are the
numbers in cells A16, B16 and C16 and what do they represent?
2 By changing the number in cell A2 find the angle sum and size of an interior angle in a
regular polygon with 90 sides.
3 Move to cell A2 and type 3 followed by ENTER. Now extend the table to row 49 using the
copying and highlighting skills used above.
From the numbers you have produced read off:
a How many sides has the first polygon with an angle sum greater than 5000°?
b How many sides has the first polygon with an angle sum greater than 6000°?
c How many sides must a regular polygon have for its interior angle to be larger than 172°?
Fun Spot 11:02 | The game of Hex
Hex is played on a board of hexagons like the
one shown. This board has 7 hexagons along
each edge, but bigger boards may be used.
The game is played between two players,
one having a supply of white counters, the
other a supply of black counters. Each takes
it in turn to place a counter anywhere on
the board, with the object being to form a
continuous chain of one's own counters
from one side of the board to the other.
The player with the white counters must
form a chain connecting the two 'white'
edges of the board before black can join
the two 'black' edges.
(This game was invented by a Dane named
Piet Hein. He introduced the game in 1942
under the name of 'Polygon'.)
f
u
n s
p
o
t
11:02
Black
Black
White
White
IM4_Ch11_3pp.fm Page 305 Monday, March 16, 2009 12:11 PM
306 INTERNATIONAL MATHEMATICS 4
11:03
|
Deductive Reasoning in
Non-Numerical Exercises
Many problems in geometry are non-numerical. In these problems, the reasoning process becomes
more involved. As there are no numbers involved, pronumerals are used to represent unknown
quantities. With the use of pronumerals, the reasoning will involve algebraic skills learned in other
parts of the course.
Because exercises do not involve specific numbers, the results we obtain will be true irrespective
of the numbers used. The results obtained are called generalisations or, more commonly, proofs.
Copy the diagram.
1 Mark ∠ABC with 'x'. 2 Mark ∠BAC with '•'.
In the diagram, which angle is equal to:
3 ∠ABC? 4 ∠BAC?
5 Which angle is adjacent to ∠DCE?
6 Which two adjacent angles make ∠ACE?
7 If a + b = 180, then 180 − a = . . . .? 8 If a + c = 180, then c = . . . .?
9 If b = 180 − a and c = 180 − a, 10 If a = b and b = c, what can we say
then b = . . . .? about a and c?
p
r
e
p
quiz
11:03
A
C B
D
E
worked examples
I'll have to be
wide awake for
this work!
1 In the diagram, prove that x = y. 2 In the diagram, prove that ∠ABD = ∠CEF.
Solutions
1 ∠CFE = x° (vert. opp. ∠s) 2 ∠ABD = ∠BDE (alt. ∠s, AC//DF)
∠CFE = y° (corresp. ∠s, AB//CD) ∠CEF = ∠BDE (corresp. ∠s DB//EC)
∴ x = y (both equal to ∠CFE) ∴ ∠ABD = ∠CEF (both equal to ∠BDE)
A
C
G
D
E
H
B
F
x °
y°
A
C
D
E
B
F
IM4_Ch11_3pp.fm Page 306 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 307
Prove that x = y in each of the following.
a b
3 In the diagram, prove that: 4 ∆ABC is isosceles with AB = BC. AB is
a ∠ABD = ∠DAC produced to D and BE is drawn through
b ∠BAD = ∠ACD B parallel to the base AC. Prove that
BE bisects ∠CBD.
Solutions
3 4
Let ∠ABD = x°, ∠BAD = y°, Let ∠BAC = a°
∠DAC = w° and ∠ACD = z°. ∴ ∠BCA = a° (base ∠s of isos. ∆)
a x + y = 90 (comp. ∠s in ∆ABD) Now ∠EBC = a° (alt. ∠s, BE//AC)
w + y = 90 (comp. ∠s) ∠DBE = a° (corresp. ∠s, BE//AC)
∴ x = w ∴ ∠DBE = ∠EBC
∴ ∠ABD = ∠DAC ∴ BE bisects ∠CBD
b x + y = 90 (comp. ∠s in ∆ABD)
x + z = 90 (comp. ∠s in ∆ABC)
∴ y = z
∴ ∠BAD = ∠ACD
A
C
D B
A
C
D B
x°
y°
w°
z°
A C
E
B
D
a° a°
Exercise 11:03
Non-numerical proofs
1 Prove ∠ABC = 2 × ∠BDC.
2 Prove ∠FAB = ∠ECD.
A
B
C
D
F
A
C
G
E B
D
Foundation Worksheet 11:03
1
C
B A
x°
y°
D
E
F
A
x° B C
D
y°
E
Y
X
IM4_Ch11_3pp.fm Page 307 Monday, March 16, 2009 12:11 PM
308 INTERNATIONAL MATHEMATICS 4
∠ABC is a straight angle. BE ∆ABC and ∆ADC are isosceles.
and BF bisect ∠ABD and AC is the base of both
∠DBC respectively. Prove that triangles. Prove that ∠BAD
EB is perpendicular to BF. = ∠BCD.
A line drawn parallel to the ∆ABC has AB and BC equal. ∆ABC and ∆BDC are
base AC of an isosceles ∆ABC D is any point on AC, and DE isosceles. AB = BC and
cuts the equal sides at D and and DF are perpendicular to BD = DC. BC is a common
E. Prove that ∆DBE is isosceles. AB and BC respectively. Prove side, while A and D lie on
that ∠EDA = ∠FDC. opposite sides of BC. Prove
that ∠BCD = 2 × ∠BCA.
D is a point on the side AC A, B and C are collinear. AB is a diameter and CD is a
of ∆ABC. D is equidistant AD = AB, BC = EC and chord of a circle which has
from the three vertices of the AD//EC. Prove that ∠DBE centre O. CD produced meets
triangle. Prove that ∠ABC is is a right angle. AB produced at E and
a right angle. DE = OD. Prove that
∠AOC = 3 × ∠DOB.
In ∆ABC, AE and CD are AB is the diameter of a circle
perpendicular to BC and AB centred at O. Prove that
respectively. Prove that ∠ACB = 90°.
∠BAE = ∠BCD.
The word 'respectively'
means 'in the order given'.
2
A B C
E
D
F
3
A
B
C
D
4
B
A C
E D
5 B
A C
E
F
D
6
B
A
C
D
7
A
D
C
B 8
A B C
D
E
9
A
B
C
O
D
E
10
A
D
E
C
B
11
C
A B
O
12
O
A
B
C
O is the centre of the
circle. Prove that
∠AOB = 2 × ∠ACB.
I Hint!
Join CO.
IM4_Ch11_3pp.fm Page 308 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 309
11:04
|
Congruent Triangles
In Years 7 and 8, congruent figures would have been studied.
• Congruent figures are the same shape and size. When one is superimposed on the other, they
coincide exactly.
• Congruent figures are produced by translations, reflections and rotations.
• When congruent figures are placed on top of each other so that they coincide exactly, the
matching sides and angles are obviously equal. The word corresponding is often used instead
of matching.
Congruent triangles
• In geometry, we are often asked to show that two sides or two angles are equal. A common way
of doing this is by showing that they are the matching sides or angles of congruent triangles.
• To check that two triangles are congruent, we would normally need to compare six pieces of
information (three sides and three angles).
• In the next exercise we will investigate the minimum conditions for congruent triangles.
A minimum condition is the smallest amount of information that we need to know about
two triangles before we can say they are congruent.
1 Which figure is congruent to figure A?
2 Which figure is congruent to figure B?
The figures are congruent.
3 Name the angle that matches ∠A.
4 Name the side that matches FE.
The figures are congruent.
5 Name the angle that matches ∠B.
6 Name the side that matches AB.
7 Name the angle that matches ∠N.
Are the following pairs of triangles congruent?
8
p
rep
q
u
i
z
11:04
A B C D
A
E
F
B C M N
O P
Q L
D
A E
O
N
M L
P
B C
D
9
10
IM4_Ch11_3pp.fm Page 309 Monday, March 16, 2009 12:11 PM
310 INTERNATIONAL MATHEMATICS 4
Sketch two possible triangles with:
a a side of 5 cm b an angle of 60°
c sides of 4 cm and 5 cm d angles of 50° and 60°
e a 5 cm side and a 60° angle.
Are two triangles congruent if they:
a have only one side equal? b have only one angle equal?
c have only two sides equal? d have two angles equal?
e have one side and one angle equal?
Can two triangles be congruent if we can compare
only two pieces of information on each one?
To compare three pieces of information
we could compare:
• 3 sides
• 2 sides and 1 angle
• 1 side and 2 angles
• 3 angles.
a When a photograph is enlarged, are:
i the angles in the photo and the
enlargement the same?
ii the photo and the enlargement
congruent?
b If two triangles have their three angles
equal, does it mean they are congruent?
In Years 7 and 8 you would have been shown how to construct a triangle given its three sides
(eg 3 cm, 4 cm and 5 cm). The diagrams show the four possible shapes, starting with a side AB.
a Is triangle I a reflection of triangle II?
b Is triangle II a reflection of triangle IV?
Exercise 11:04
1
2
3
Same angles—
Different sizes...
4
5
4 cm
4 cm
3 cm
3 cm
5 cm
A
D
C
I
II
B
3 cm
3 cm
4 cm
4 cm
5 cm
A
D
C
III
IV
B
IM4_Ch11_3pp.fm Page 310 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 311
c Are the four triangles congruent?
d When you construct a triangle given the lengths of its three sides, can you obtain solutions
that are not congruent?
e If two triangles have all their sides equal, are they congruent?
a Do the triangles above have two sides and one angle equal?
b Are the triangles congruent? (Check by measuring the third side.)
c Do the triangles above have two sides and one angle equal?
d Are the triangles congruent? (Check the length of the third side.)
e Where is the angle in relation to the sides?
f Explain why placing the angle between the two sides
automatically fixes the length of the third side and the
sizes of the other angles.
a If a triangle has angles of 60° and 70°, what is the size of the third angle?
b If you are given two angles of a triangle, do you also know the size of the third angle?
c In any triangle, is the largest side opposite the largest angle?
d In any triangle, where is the smallest side in relation to the angles?
e The two triangles drawn below have a 50° angle, 60° angle and a 4 cm side. Are they
congruent?
6
3 cm
4 cm
60°
3 cm
4 cm
60°
3 cm
4 cm
60°
3 cm
4 cm
60°
4 cm
5 cm
40°
7
4 cm
60° 50°
70°
4 cm
60° 50°
70°
IM4_Ch11_3pp.fm Page 311 Monday, March 16, 2009 12:11 PM
312 INTERNATIONAL MATHEMATICS 4
f The two triangles below also have a 50° angle, a 60° angle and a 4 cm side. Are they
congruent?
g You are told that two triangles have two angles (and, hence, the third angle) and one side
equal. Where must the side be placed in relation to the angles if the triangles are to be
congruent?
Right-angled triangles have a special set of minimum conditions.
a If two triangles have all their sides equal, are they congruent?
b Which theorem allows us to calculate the third side of a right-angled triangle given the
other two?
c If two right-angled triangles have their hypotenuses and one other pair of sides equal, are
they congruent?
The results from questions 5 to 8 are summarised below.
60° 50°
70°
4 cm
60° 50°
4 cm
8
4 cm
5 cm
4 cm
5 cm
• SSS means
'side, side, side'.
• SAS means
'side, angle, side'.
• AAS means
'angle, angle, side'.
• RHS means
'right angle,
hypotenuse, side'.
Summary
• Two triangles are congruent if three sides of one
triangle are equal to three sides of the other. (SSS)
• Two triangles are congruent if two sides and the
included angle of one triangle are equal to two sides
and the included angle of the other. (SAS)
• Two triangles are congruent if two angles and a side
of one triangle are equal to two angles and the
matching side of the other. (AAS)
• Two right-angled triangles are congruent if the
hypotenuse and one side of one triangle are equal to
the hypotenuse and one side of the other triangle.
(RHS)
IM4_Ch11_3pp.fm Page 312 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 313
11:05
|
Proving Two Triangles Congruent
The minimum conditions deduced in the last section are used to prove that two triangles are
congruent. Special care must be taken in exercises that involve overlapping triangles.
Name the side that corresponds to:
1 AC 2 AB 3 BC
Name the angle that corresponds to:
4 ∠A 5 ∠B 6 ∠C
Name the side that corresponds to:
7 LM 8 MN
9 Find the value of x.
10 Are the 2 cm sides
corresponding?
p
rep
q
u
i
z
11:05
B
A
C
5
3
6
F
E
D
6
5
3
M
L
N Q R
P
Y
X
Z B C
A
70°
60°
50° 50° 60°
x°
2 cm 2 cm
'Corresponding'
can be used
instead of
'matching' when
describing position.
worked examples
Examples
1 Prove that ∆ABC ≡ ∆DFE and list the 2 Show that ∆ABC ≡ ∆DCB.
pairs of matching sides and angles.
Solutions
1 ∆ABC and ∆DFE have all their 2 In ∆ABC and ∆DCB
sides equal. i ∠ABC = ∠DCB
∴ ∆ABC ≡ ∆DFE (SSS) (alt. angles, DC//BA)
The pairs of matching angles are: ii AB = CD
∠A and ∠D, ∠B and ∠F, ∠C and ∠E. iii BC = BC
The pairs of matching sides are: ∴ ∆ABC ≡ ∆DCB (SAS)
AB and DF, AC and ED, BC and EF.
A B
D C
B
A
C
F D
E
8 cm
6 cm
6 cm 5 cm
8 cm
5 cm
When working with congruent figures, the term
'corresponding' is often used instead of the term 'matching'
to refer to angles or sides in the same position.
'ϵ' means
'is congruent to'.
If ⌬ABC is
congruent
to ⌬DEF,
we write
⌬ABC ϵ ⌬DEF
continued §§§
IM4_Ch11_3pp.fm Page 313 Monday, March 16, 2009 12:11 PM
314 INTERNATIONAL MATHEMATICS 4
The following pairs of triangles are congruent. State the
congruence condition used to establish the congruence.
All side lengths are in the same units.
a
b c
d e
Examples
3 Are these two triangles congruent? 4 Prove that ∆ABC ≡ ∆EDC.
Solutions
3 • Because we are only given the length 4 In ∆s ABC and EDC:
of one side we cannot use SSS, SAS 1 ∠ABC = ∠ECD (vert. opp. ∠s)
or RHS conditions. 2 ∠CAB = ∠CED (alt. ∠s AB//DE)
• Hence we can only look at the AAS AB = DE (given)
condition. 3 ∴ ∆ABC ≡ ∆EDC (AAS)
• We can see that both triangles have
the same angles, as the missing angle
in ∆XYZ must be 50° because the
angle sum of a triangle is 180°.
• Now, the 4 cm side is opposite the
50° angle in ∆ABC and opposite the
60° angle in ∆XYZ.
• Hence the sides are not corresponding.
• Therefore the triangles are not congruent.
A B
C
E D
B C
A
Z
Y X
4 cm 4 cm
70°
60° 50° 60° 70°
I When writing congruent
triangle proofs, write the
vertices in matching order
as shown in the examples.
Exercise 11:05
Congruent triangles
1 Why is each pair of triangles congruent?
a
2 Is each pair of triangles congruent?
a
60° 60° 10 10
60°
60° 70°
50°
8
8
Foundation Worksheet 11:05
1
6
10
6
10
8
7
6
7 8
6
60°
10
10
60°
50°
5
4
4
5
50°
80°
5
40°
40°
80°
5
IM4_Ch11_3pp.fm Page 314 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 315
State whether the triangles in the following pairs are congruent. For those that are, state the
congruence condition used.
a b
c d
In each of the following, prove that the two triangles are congruent.
a b c
d e f
O is the centre O is the centre
of the circle. of both circles.
a Prove that b Prove that
∆ABC ≡ ∆DEC. ∆ABD ≡ ∆BAC.
2
8
10
6
10
8
6
8
10
5
6
10
8
8
5
50°
8
5
50°
10
70°
60°
50°
10
50°
70° 60°
3
A
B
C
D
5
40°
4
4
3
3
A
B
C
D
E
3 cm
3 cm
L M
N
Q P
50°
50°
O
P
Q
S
T
A
B
C
D
O
A
E
D
C
You will have to find
3 facts about each
pair of triangles.
Put reasons in
parentheses.
I Setting Out Proofs
In ∆s ABC and DEF:
1 AB = DE (Given)
2 ..... = ..... (............)
3 ..... = ..... (............)
∴ ∆ABC ≡ ∆DEF (............)
4
A
B
E
D
C
D B
C A
IM4_Ch11_3pp.fm Page 315 Monday, March 16, 2009 12:11 PM
316 INTERNATIONAL MATHEMATICS 4
c Prove that d Prove that e Prove that
∆ABD ≡ ∆ACD. ∆ABC ≡ ∆ADC. ∆ABD ≡ ∆ACD.
f g
O is the centre of the ABCD is a square.
circle. Prove that ∠AFB = ∠CED.
∆AOB ≡ ∆COD. Prove that ∆ABF ≡ ∆CDE.
a b
∠DBC = ∠ACB, and ∠ABC ≡ ∠ACB.
BD = CA. Prove that Prove that
∆ABC ≡ ∆DCB. ∆DBC ≡ ∆ECB.
B D C
A B
C
D
A
B D C
A
A
C
D
B
O
A E D
C F B
They're not tricky.
Just follow
these hints!
I Hints
1 Write down the three
sides of each triangle.
2 Match up the ones
that are equal.
3 Repeat the above for
the angles.
5
B C
D A
D E
C B
A
IM4_Ch11_3pp.fm Page 316 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 317
11:06
|
Using Congruent Triangles to
Find Unknown Sides and Angles
If two triangles can be shown to be congruent, then, of course, all matching sides and angles are
equal.
Using congruent triangles to find the values of unknown angles and sides or to prove relationships
is very important in geometry.
Note that in questions involving circles, O is the centre of the circle.
In each of the following, state why ∆ABC ≡ ∆DEF and hence find the value of DE.
a b
worked examples
1 Show that ∆ABC ≡ ∆DFE and hence 2 AC = CD, ∠ACB = ∠BCD. Prove that
find the length of DE. ∆ACB ≡ ∆DCB, and hence that BD = 15 cm.
Solutions
1 ∆ABC ≡ ∆DFE (SAS) 2 In ∆s ABC and DBC:
∴ DE = AC 1 AC = DC (given)
(matching sides of cong't ∆s) 2 ∠ACB = ∠DCB (given)
∴ DE = 8 cm 3 BC is common to both ∆s.
∴ ∆ABC ≡ ∆DBC (SAS)
∴ AB = DB (corresp. sides of cong't ∆s)
∴ BD = 15 cm
A
B
8 cm
8 cm
7 cm
8 cm
7 cm
C
E
F
D
A
B C
D
110°
1
5
c
m
110°
Exercise 11:06
1
10 cm
1
5
c
m
1
5
c
m
A B
C F
D E
27 m
31 m
60°
A C
B
F E
D
2
0
m
2
0
m
3
1
m
60°
IM4_Ch11_3pp.fm Page 317 Monday, March 16, 2009 12:11 PM
318 INTERNATIONAL MATHEMATICS 4
Find the value of the pronumeral in each of the following, giving reasons for your answers.
a b c
By proving two triangles are congruent, find the value of the pronumeral in each of the
following.
a b c
d e f
a b c
Prove that ∠BAC = ∠DAC. Prove that ∠ABC = ∠CBD. Prove that ∠BAX = ∠CDX,
and hence that AB//CD.
d e f
Prove that AD = DB. Prove that Prove that
∠OCA = ∠OCB = 90°. AC = DB and AC//DB.
2
A
C
B
2
0
m
D
C
B
A
∠EAB = ∠DBA = 70°, AE = BD
a°
85°
D
E
F
B A
4
C
B
D
A
C
B
D
A
C
B
D
A
X
D
O
B A
O
A
C
B
O
A B
D
C
IM4_Ch11_3pp.fm Page 318 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 319
11:07
|
Deductive Geometry and Triangles
If you were asked to define an equilateral triangle, you could say:
'it is a triangle with all its sides equal'
or
'it is a triangle with all its angles equal'
or
'it is a triangle with all its sides equal and all its angles equal'.
Geometrical figures have many properties and it is not practicable
to mention them all when defining the figure.
In deductive geometry, the definitions serve as starting points. The properties of the figures can
then be proved using basic geometrical facts.
The proved result is known as a theorem and this can then be used to produce other theorems.
The definitions imply that an equilateral triangle
must also be an isosceles triangle. Hence, any
property of an isosceles triangle must also be
a property of an equilateral triangle.
• Isosceles triangles are often found in architecture.
Hmm...don't say more
than you have to!
A definition is the minimum amount of information needed to identify
a particular figure.
Observation
A triangle with 3 equal
sides has at least
2 sides equal.
Conclusion
A triangle that is
equilateral must
also be isosceles.
Definitions
• A scalene triangle is a triangle with
no two sides equal in length.
• An isosceles triangle is a triangle
with at least two sides equal in
length.
• An equilateral triangle is a triangle
with all sides equal in length.
IM4_Ch11_3pp.fm Page 319 Monday, March 16, 2009 12:11 PM
320 INTERNATIONAL MATHEMATICS 4
worked examples
1 Use congruent triangles to prove that the angles opposite equal sides in an isosceles triangle
are equal.
Solution
Data: ∆ABC is isosceles with AB = AC.
Aim: To prove that ∠ABC = ∠ACB.
Construction: Draw AD perpendicular to BC,
meeting BC in D.
Proof: In ∆s ABD and ACD:
1 AB = AC (data)
2 AD is common.
3 ∠ABD = ∠ADC (AD ⊥ BC)
∴ ∆ABD ≡ ∆ACD (RHS)
∴ ∠ABD = ∠ACD (corresponding ∠s
of congruent ∆s)
∴ ∠ABC = ∠ACB
2 Prove that the sum of the interior angles of a triangle is 180°.
Solution
Data: ∆ABC is any triangle with angles
α, β and γ.
Aim: To prove that α + β + γ = 180°
(ie the angle sum is 180°).
Construction: Extend AC to D. Draw CE parallel to AB.
Proof: ∠BCE = β (alternate to ∠ABC, AB//CE)
∠ECD = α (corresponding to ∠BAC, AB//CE)
∠BCA = γ (given)
∴ γ + β + α = 180° (∠ACD is a straight angle)
∴ the angle sum of a triangle is 180°. Q.E.D.
3 Use isosceles triangles to prove that any triangle drawn in a semicircle is right-angled.
Solution
Data: ∆ABC is any triangle drawn on the
diameter AB. O is the centre of the circle.
Aim: To prove that ∠ACB = 90°.
Construction: Join CO.
A
B D C
B E
A C D


␣ ␣ ␥
A B
C
O
y° x°
y° x°
IM4_Ch11_3pp.fm Page 320 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 321
It is a well-known result that 'the exterior angle of a triangle is equal to the sum of the interior
opposite angles'.
Complete the proof started below.
Aim: To prove that the exterior angle ∠BCD is equal to the
sum of the interior opposite angles (ie ∠BCD = α + β).
Construction: Draw CE parallel to AB.
Proof: ∠BCE = β (..........)
∠ECD = α (..........)
∴ ∠BCD = ..........
Use the result above to prove that the sum of the
exterior angles of a triangle is 360°.
Use isosceles triangles and the exterior
angle theorem to prove that the reflex
angle AOB = 2 × ∠ACB.
Use the fact that an equilateral triangle is also an isosceles triangle to prove that each angle
of an equilateral triangle is equal to 60°.
Proof: Now AO = BO = CO (radii of a circle)
∴ ∆s OBC and OAC are isosceles.
Let ∠OBC = x° and ∠OAC = y°
∴ ∠OCB = x° (base ∠s of isos. ∆OBC)
∠OCA = y° (base ∠s of isos. ∆OAC)
∴ x + x + y + y = 180 (angle sum of ∆ABC)
∴ 2x + 2y = 180
∴ x + y = 90
But ∠ACB = ∠OCB + ∠OCA
= x° + y°
= 90°
∴ ∆ABC is right-angled.
Exercise 11:07
1
B E
C D A

␣
x°
c°
z°
a°
y°
b°
2
O is the centre of the circle.
A B
C
O
3
4
IM4_Ch11_3pp.fm Page 321 Monday, March 16, 2009 12:11 PM
322 INTERNATIONAL MATHEMATICS 4
a Use congruent triangles to prove that if two angles
of a triangle are equal then the sides opposite those
angles are equal.
b Use the result in a to prove that a triangle that is
equiangular must be equilateral.
Use congruent triangles to prove the following
properties of isosceles triangles.
a A line drawn at right angles to the base of
an isosceles triangle through the third
vertex bisects the base.
b A line drawn from the midpoint of the base
of an isosceles triangle to the third vertex is
perpendicular to the base.
Regular polygons can be inscribed in circles,
as shown on the right.
a Find the size of x.
b For which regular polygon would the side length equal the
radius of the circle? Why is there only one regular polygon
for which this can occur?
A well-known property of triangles is that the
perpendicular bisectors of the sides are concurrent.
The 'Data', 'Aim' and 'Construction' for the
congruence proof of the above result are given
below. Answering the questions will give an
outline of the proof.
Data: ∆ABC is any triangle. E, F, G are the
midpoints of AB, AC and BC respectively.
Perpendiculars drawn from E and F meet
at S.
Aim: To show that SG ⊥ BC (ie that the
perpendicular drawn from G passes through S).
Construction: Join SA, SB, SC.
a Why is ∆AES ≡ ∆BES? b Why does AS = BS?
c Why is ∆AFS ≡ ∆CFS? d Why does AS = CS?
e Why does BS = CS? f What type of triangle is ∆ASC?
g What result proved in question 6 can be used to justify that SG ⊥ BC?
A
B C
5
A
B D C
A
B D C
base
In isosceles ⌬s
the base is the
unequal side.
6
x°
7
A
F
E
C
S
G
B
8
IM4_Ch11_3pp.fm Page 322 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 323
11:08
|
Deductive Geometry and
Quadrilaterals
As we have seen with triangles, the definitions of the quadrilaterals are minimum definitions.
Many people find the definitions above a little unusual at first.
• They start from the simplest shape and by adding more and more constraints end up at the most
complex shape.
• The definitions are hierarchical. Each new shape is defined in terms of a simpler shape which has
already been defined.
eg a rhombus is a parallelogram . . .
This saves repetition and states that a rhombus is in fact a special type of parallelogram.
It has all the properties of a parallelogram and
some extra properties as well.
• They are minimum definitions. Not every
property of the shape is given. By using the
definition and other geometrical techniques
the other properties can be deduced.
eg a rectangle is a parallelogram with a right angle.
There is no need to say that it has four right angles
as this can be derived using the fact that it is a
parallelogram and our knowledge of co-interior
angles and parallel lines.
Definitions
• A trapezium is a quadrilateral with at least one pair of opposite sides parallel.
• A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
• A rhombus is a parallelogram with two adjacent sides equal in length.
• A rectangle is a parallelogram with one angle a right angle.
• A square is a rectangle with two adjacent sides equal
OR
A square is a rhombus with one angle a right angle.
If one angle is
a right angle,
they all must be.
worked examples
1 Prove that the sum of the interior angles of a quadrilateral is 360°.
Solution
Data: ABCD is a quadrilateral.
Aim: To prove that the angle sum of a quadrilateral is 360°.
Construction: Draw in the diagonal DB.
Proof: Let a, b, c, d, e, f be the sizes of angles on the figure.
Now a + b + c = 180° (angle sum of ∆ABD)
and d + e + f = 180° (angle sum of ∆BCD)
∠A + ∠B + ∠C + ∠D = b + (c + d) + e + (a + f)
= (a + b + c) + (d + e + f)
= 180° + 180°
= 360° Q.E.D.
A
B
C
D
a
f
b
c
d
e
continued §§§
IM4_Ch11_3pp.fm Page 323 Monday, March 16, 2009 12:11 PM
324 INTERNATIONAL MATHEMATICS 4
Follow the flowchart below and choose the correct names from the list, to be inserted into the
boxes to .
2 Prove that a quadrilateral is a parallelogram if its opposite angles are equal.
Solution
Data: ABCD is a quadrilateral with ∠A = ∠C and ∠B = ∠D.
Aim: To prove that AB//DC and AD//BC.
Proof: Let ∠A = ∠C = b° and ∠B = ∠D = a°.
2(a + b) = 360 (∠ sum of quad.)
∴ a + b = 180
∴ ∠ADC + ∠DAB = 180°
∴ AB//DC (co-int. ∠s are supp.)
Also, ∠ADC + ∠DCB = 180°
∴ AD//BC (co-int. ∠s are supp.)
∴ ABCD has opposite sides parallel.
∴ ABCD is a parallelogram.
A
B
C
D
a
a
b
b
Exercise 11:08
1
1 6
START
NO
NO
NO
NO
NO
YES
YES
YES
YES
YES
1
2
3
4
5
6
Are any
sides parallel?
Does it
have 2 pairs of parallel
sides?
Are all
the sides equal?
Does it have
a right angle?
Does it have
a right angle?
SQUARE
RECTANGLE
RHOMBUS
PARALLELOGRAM
TRAPEZIUM
QUADRILATERAL
Reading the definitions
again will help.
IM4_Ch11_3pp.fm Page 324 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 325
ABCD is a parallelogram.
a Why does ∠BAC equal ∠DCA?
b Why does ∠BCA equal ∠DAC?
c Does ∠BAD equal ∠BCD?
d Prove that ∆ABC ≡ ∆CDA. Hence prove that:
i ∠ABC = ∠CDA
ii AB = DC and BC = AD
You have proved that the opposite sides and opposite
angles of a parallelogram are equal.
Using the fact that the opposite sides of a parallelogram
are equal, prove that ∆ABE ≡ ∆CDE and hence that
AE = EC and EB = ED.
This question proves that the diagonals of a parallelogram bisect each other.
A rhombus is a parallelogram with a pair of adjacent sides equal. Using this definition and the
properties of the parallelogram already proven, answer the following.
a Why does AB = DC?
b Why does BC = AD?
c Show that all the sides of the rhombus are equal.
d Using your answer to c, what type of triangle is:
i ∆ABD?
ii ∆CBD?
e Why does ∠ABD = ∠ADB? f Why does ∠CBD = ∠CDB?
g Why does ∠ABD = ∠CDB? h Which angles are equal to ∠ABD?
i Prove that the diagonal AC bisects the angles DAB and DCB.
a Why does AE = EC?
b Prove that ∆ABE ≡ ∆CBE and hence that
∠AEB = ∠CEB = 90°.
The last two questions have proved that the diagonals of a rhombus bisect each other at right
angles, and that they bisect the angles through which they pass.
A rectangle is a parallelogram with a right angle.
a Prove that all the angles must be right angles.
b Assuming the answer to a and the properties of a
parallelogram, prove that ∆ABD ≡ ∆DCA and hence
AC = DB.
This question proves that all the angles of a rectangle are right angles and that its diagonals are
equal in length.
2
A B
C D
A B
C D
E
3
4
A B
C D
5
D C
B A
E
6
D C
B A
IM4_Ch11_3pp.fm Page 325 Monday, March 16, 2009 12:11 PM
326 INTERNATIONAL MATHEMATICS 4
To show that a quadrilateral is a parallelogram, we could of course show that both pairs of opposite
sides are parallel (ie use the definition).
There are other tests which can be used to show that a given quadrilateral is a parallelogram.
These are very useful and are given below.
ABCD is a quadrilateral that has opposite sides equal.
Prove that it is a parallelogram (ie that its opposite
sides are parallel).
The 'Data' and 'Aim' for the congruence
proof of Test 4 above are given below.
Answering the questions will give an
outline of the proof.
Data: ABCD is any quadrilateral where
diagonals AC and BD bisect each
other at E.
Aim: To show that ABCD is a parallelogram
(ie AD//BC and AB//CD).
a Why does ∠AED equal ∠CEB?
b Why is ∆AED congruent to ∆CEB?
c Which angle in ∆CBE is equal to
∠ADE? Why?
d How does your answer to c prove that
AD//BC?
e Why is ∆AEB congruent to ∆CED?
f Why is ∠BAE equal to ∠DCE?
g Why is AB parallel to CD?
Use congruent triangles to prove that if
one pair of sides in a quadrilateral is
both equal and parallel, then the quadrilateral
is a parallelogram.
Tests for parallelograms
A quadrilateral is a parallelogram if any one of the following is true.
1 Both pairs of opposite sides are equal.
2 Both pairs of opposite angles are equal.
3 One pair of sides is both equal and parallel.
4 The diagonals bisect each other.
D
C B
A
7
A D
B
E
C
A B
D C
If alternate angles
are equal, then the
lines are parallel.
8
9
IM4_Ch11_3pp.fm Page 326 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 327
Use the tests for parallelograms and the properties of parallelograms to do questions 10 and 11.
In the diagram, ABCD and ABEF are parallelograms.
Prove that DCEF is a parallelogram.
ABCD is a parallelogram with diagonals produced
so that EA = CG and DH = BF. Prove that EFGH
is a parallelogram.
There are also tests for a rhombus and a rectangle. These are given below.
Show how Test 1 for a rhombus could be proved using Test 1 for a parallelogram.
Prove Test 1 for rectangles.
a Does a square have all the properties of a rectangle?
b Is a square a special rectangle?
c Is a rectangle a square? Give a reason for your answer.
d Does a square have all the properties of a rhombus?
e Is a square a special rhombus?
f Is a rhombus a square? Give a reason for your answer.
Carol and Sharon thought that a good test for a square would be 'equal diagonals that bisect
each other at right angles'. Do you agree with their test?
11:08 Quadrilaterals
A B
C D
F E
A
B
C
D
E F
G H
K
10
11
• Builders often use the properties
of quadrilaterals in building.
Tests for a rhombus
1 All sides are equal. OR
2 Diagonals bisect each other at right angles.
Tests for a rectangle
1 All angles are equal. OR
2 Diagonals are equal and bisect each other.
12
13
14
15
IM4_Ch11_3pp.fm Page 327 Monday, March 16, 2009 12:11 PM
328 INTERNATIONAL MATHEMATICS 4
Investigation 11:08 | Theorems and their converses
A theorem usually connects two pieces of information and can be written in the form
'If A then B'.
A is usually called the 'supposition' or 'assumption' while B is called the 'conclusion'.
If A and B are interchanged then we have the statement 'If B then A'. This is called the converse
of the theorem.
Even if a theorem is true, its converse may not be, as shown by the following example.
Theorem: If (two angles are vertically opposite) then (the angles are equal). This is true.
Converse: If (two angles are equal) then (they are vertically opposite). This is false.
For the following theorems, state their converse
and whether the converse is true.
1 If a triangle has all its sides equal then it has
all its angles equal.
2 If a quadrilateral is a square then its diagonals
are equal.
3 If a quadrilateral is a parallelogram then its
opposite angles are equal.
Fun Spot 11:08 | What do you call a man with a shovel?
Work out the answer to each part and put the letter for that part in the box above the correct
answer.
Which congruence test can be used to state why the triangles in each pair are congruent?
i
n
v
e
s
t
igatio
n
11:08
If I have given good service
then I get a tip or conversely...
If I get a tip then I have
given good service.
f
u
n
spo
t
11:08
30 30
D U O G
S
S
S
R
H
S
S
A
S
A
A
S
IM4_Ch11_3pp.fm Page 328 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 329
11:09
|
Pythagoras' Theorem and
its Converse
During your Stage 4 studies, you would have encountered the most famous of all geometric
theorems — Pythagoras' theorem. Both the theorem and its converse are true.
The theorem states that:
Furthermore, the converse states that:
Investigation 11:09 | Proving Pythagoras' theorem
1 How could the two squares above be used to prove Pythagoras' theorem?
2 Investigate other proofs of Pythagoras' theorem.
• Pythagoras' theorem is still used to check that buildings are square.
If a triangle is right-angled
then the square on the longest
side is equal to the sum of
the squares on the two
smaller sides.
For the triangle shown,
this means that c
2
= a
2
+ b
2
.
a
b
c
If the square on the longest side is equal to the sum of the squares on the two smaller
sides then the triangle is right-angled.
n
o
i
t
a
g i t s e v
n
i
11:09
D
B
C
b
a
a b
b a
A
H
F
G
a
b
a
b
a b
E
a
b
O
M
L
N
y°
x°
c
IM4_Ch11_3pp.fm Page 329 Monday, March 16, 2009 12:11 PM
330 INTERNATIONAL MATHEMATICS 4
worked examples
1 Calculate the perpendicular height of an 2 A rhombus has diagonals 8 cm and 4 cm
equilateral triangle if its sides are 6 cm in length. What is the side length of the
long. rhombus?
Solutions
1 2
BD = 3 cm (D is midpoint of BC) AE = 2, BE = 4
6
2
= h
2
+ 3
2
(Pythag. Thm) (Diagonals bisect at rt. ∠s)
36 = h
2
+ 9 x
2
= 2
2
+ 4
2
(Pythag. Thm)
h
2
= 27 = 20
h = x =
= =
3 4
Find the value of DC. Find the value of x.
Solutions
In ∆ABD, In ∆ABD,
25
2
= 15
2
+ BD
2
AB
2
= x
2
+ 9
625 = 225 + BD
2
In ∆ADC,
BD
2
= 400 AC
2
= x
2
+ 36
BD = 20 In ∆ABC,
In ∆ABC, BC
2
= AB
2
+ AC
2
20
2
= 15
2
+ BC
2
9
2
= (x
2
+ 9) + (x
2
+ 36)
400 = 225 + BC
2
81 = 2x
2
+ 45
BC
2
= 175 2x
2
= 36
BC = x
2
= 18
Now DC = BD − BC x =
= =
=
A
B D C
h
6 cm
3 cm 3 cm
D
A
C
B
E
2 cm
4 cm
x
c
m
27 20
3 3 2 5
C
A
B D
15 cm 2
5
c
m
2
0
c
m
C
A
B
D 6 cm 3 cm
x cm
175
18
20 175 – 3 2
20 5 7 – ( ) cm
IM4_Ch11_3pp.fm Page 330 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 331
Find the value of the pronumerals in each
of the following.
a b
Use the converse of Pythagoras' theorem to find which of the following are rectangles.
a b c
Find the values of x and y in each of the following. (All measurements are in cm.)
a b c
a b c
Find the length of AB. Find x. Find CD.
a b c
What must x be if Find AB. O is the centre of a
∠ABC = 90°? semicircle of radius 6·5 cm.
EB = 3 cm. Find AC.
Exercise 11:09
Pythagoras' theorem
1 Find the value of the pronumerals in the following.
a b
2 Find the value of the pronumerals in the following.
a b
6
10
h
2
5
x
6
8 10
x y
6
10
12
x
y
Foundation Worksheet 11:09
1
x
m A B E
C
O
IM4_Ch11_3pp.fm Page 331 Monday, March 16, 2009 12:11 PM
332 INTERNATIONAL MATHEMATICS 4
Mathematical terms 11
Mathematical Terms 11
adjacent angles
• Share a common arm and vertex.
• Lie on opposite sides of the common arm.
alternate angles
• A pair of angles on
opposite sides of the
transversal between
the other two lines.
• In the diagram, the alternate
angles are 1 and 3, 2 and 4.
• Are equal when AB//CD.
co-interior angles
• A pair of angles on
the same side of the
transversal and between
the other two lines.
• In the diagram the
co-interior angles are
1 and 2, 3 and 4.
• Are supplementary when AB//CD.
complementary angles
• Angles that add up to give 90°.
congruent triangles
• Triangles that are identical in shape
and size.
converse (of a theorem)
• If a theorem is stated in the form
'If A then B', the converse is the statement
'If B then A'.
corresponding angles
• Angles that are in
corresponding positions
at each intersection.
• In the diagram, the
corresponding angles
are: 1 and 5, 2 and 6, 3 and 7, 4 and 8.
• Are equal when AB//CD.
deductive
• A system in which results called
theorems are produced from a set of basic
facts that are accepted to be true.
definition
• A statement that describes the essential
properties of something.
exterior angle
• An angle formed when the side of a convex
polygon is produced.
matching angles (or sides)
• Sides (or angles) that are in the same
(or corresponding) positions in congruent
figures.
polygon
• A plane figure with straight sides.
• Regular polygons have all sides and angles
equal.
• Convex polygon has all its angles either
acute or obtuse.
• Some polygons have special names.
(See Investigation 11:02A.)
proof
• A series of steps that establishes the truth
of a result.
quadrilateral
• A polygon with 4 sides.
• There are six special quadrilaterals.
(See 11:08 or ID Card 4.)
supplementary angles
• angles that add up to give 180°.
theorem
• The statement of a result that has been
proved by reasoning.
• Usually stated in an 'If A then B' form.
triangle
• A polygon with 3 sides.
• Equilateral, isosceles and scalene triangles
have 3 sides, 2 sides and no sides equal
in length respectively.
• Acute-angled triangle has three acute angles.
• Right-angled triangle has one right angle.
• Obtuse-angled triangle has one obtuse angle.
vertically opposite angles
• Two pairs of equal angles
formed when two straight
lines cross.
s
m r e
t
l
a
c
i
t
a
m
e
h
t
a
m
11
∠DAC and ∠BAC
are adjacent angles.
A
B
C
D
B
C
A
D
1
2
3
4
B
C
A
D
1
2
3
4
B
C
A
D
1 2
3 4
5
6
8
7
IM4_Ch11_3pp.fm Page 332 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 333
Diagnostic Test 11: | Deductive Geometry
• Each part of this test has similar items that test a certain type of question.
• Errors made will indicate areas of weakness.
• Each weakness should be treated by going back to the section listed.
d
i
a
gn
o
s
t
i
c
t
e
s
t
11
1 For each figure find the value of x, giving reasons.
a b c
2 a Find the angle sum of a polygon with 15 sides.
b What is the size of the interior angle in a regular octagon?
c A regular polygon has an exterior angle of 20°. How many sides does
it have?
3 a b c
Prove that x = y. Prove AB = AD. Given AB = AC,
prove ∠ABC = ∠FTD.
4 State why the two triangles are congruent.
a b c
Section
11:01
11:02
11:03
11:04
B
C
A
D
H
E
F
G
130°
x°
D E
C
B
A
40°
x°
M
N
O
P
L
70°
85°
x°
D C
B A
y°
x°
D B
A
C
D B C
A
E
F
T
D C
A B
E
B A
C
O
A B
C
D 30°
30°
IM4_Ch11_3pp.fm Page 333 Monday, March 16, 2009 12:11 PM
334 INTERNATIONAL MATHEMATICS 4
5 a Prove that b Prove that c Prove that
∠ABD = ∠ACD. ∠AOB = ∠COD. AX = AY.
6 Use congruent triangles to find the value of the pronumerals.
a
∠ABC = ∠DCB = 65°
7 a Prove that b Use congruent c Use congruent
∠ACD = 3 × ∠CAD. triangles to prove triangles to prove
that AD ⊥ BC. that ∠CAB = ∠CAD.
8 Prove the following.
a If all the angles of a quadrilateral are equal, then it is a rectangle.
b If the diagonals of a quadrilateral are equal in
length and bisect each other at right angles,
then it is a square.
c What test for parallelograms can be used to
prove that ABCD is a parallelogram?
9 Find the value of the pronumeral in each of the following.
a b c
ABCD is a square.
11:05
11:06
11:07
11:08
11:09
D
C B
E
A
D
O
C
A
B
B C
A
X Y
a°
55°
14 cm
14 cm
8 cm
9 cm
10 cm
a cm
b
x°
85°
A D
B C
c
A D
B
C A
B C
D
A
B D
E
C
A
B
C
D
x cm
7 cm 18 cm
2
0
c
m
x
4
c
m
A D
B C
x cm
15 cm
9 cm
15 cm
17 cm
IM4_Ch11_3pp.fm Page 334 Monday, March 16, 2009 12:11 PM
CHAPTER 11 DEDUCTIVE GEOMETRY 335
Chapter 11 | Revision Assignment
1 In each of the following, find the value of x.
Give reasons for your answer.
a
b
c
d
2 a A special hexagon is
made with four of its
angles equal and the
remaining two angles
are both half the size
of the others. This
is shown in the
diagram. Find the
size of the angles.
b Sharon makes a
regular pentagon
from three isosceles
triangles as shown.
Find the sizes of
the angles in the
triangles.
3
∆ABC is any triangle. D is the midpoint
of BC, and BE and CF are perpendiculars
drawn to AD, produced if necessary. Prove
that BE = CF.
4
From a point D on the base AC produced
of the isosceles triangle ABC, a straight line
is drawn perpendicular to the base cutting
AB produced at E and BC produced at F.
Prove that ∆BEF is isosceles.
5
AC = DE = 50 cm, EC = 10 cm, DB = 14 cm.
Find AD.
6
∠CAB = ∠CBA and ∠CBD = ∠CDB.
Prove that ∠CAD = ∠CDA.
a
s
sign
m
e
n
t
11A
20°
A
C D
B
F
E
G
x°
70°
A
C
D
B
E
x°
52°
A C
D
B
x°
74°
138°
96°
B C
D
E
F
A
x°
x°
x°
2x° 2x°
2x° 2x°
b° b°
b° b°
d° d°
c°
a°
a°
B
A
D F
C
E
E
B
A
C D
F
E
C
B
D A
10 cm
14 cm
A
C
D
B
IM4_Ch11_3pp.fm Page 335 Monday, March 16, 2009 12:11 PM
336 INTERNATIONAL MATHEMATICS 4
Chapter 11 | Working Mathematically
1 Describe mathematically the shape of the
lamp-shade in the photograph.
2 The ISCF group has 20 members. The
school choir has 16 members. Only Sue
and Graham are members of both groups.
How many different people belong to these
two groups altogether?
3 The petrol tank of my car holds 45 litres.
I drove into a petrol station and filled the
tank. It cost me $9.90. If the petrol cost
82·5 cents per litre, how many litres did
I buy? How much was in the tank before
it was filled?
4 In the card game of cribbage, two points are scored when any combination of cards totals 15.
An Ace counts as 1 and the Jack, Queen and King each counts as 10. For example, for the
hand below, the score is 6.
5 A block of land (as shown in the diagram)
is to be enclosed with a fence of the type
shown.
a Calculate the amount of wire needed
to complete the fence.
b If posts are to be placed at 3 m intervals
(as a maximum), calculate the number
of posts needed for the fence.
c Calculate the cost of the fence if the
posts are $8.50 each and the wire is
95 cents a metre.
6 The graph shows the height of an object
fired into the air.
a How high is the object above ground
level after 1 second?
b At what times is the object 10 m above
ground level?
c What is the greatest height reached by
the object?
d How long does it take for the object to
fall from its maximum height to ground
level?
e For how long is the object above
a height of 17·5 m?
t
n e
m
n
g
i
s
s
a
11B
5
5
K
K
7
7 3
3
2
2 2 + 3 + K = 15
5 + K = 15
3 + 5 + 7 = 15
What would the score be for these hands?
a J, K, Q, 5, 5 b 4, 5, 6, 5, 6
A
D
B
C
18 m
4
0
Students will be able to:
• Use formulae and Pythagoras' theorem to calculate the perimeter and area of figures
composed of rectangles, triangles and parts of circles.
• Calculate the volume and surface area of right prisms including cylinders.
• Apply formulae to find the volume and surface areas of pyramids, cones and spheres.
• Calculate the capacity of a solid using fluid measure.
To calculate the perimeter:
• find the lengths of all the sides
• add the lengths together.
The geometrical properties of some figures allow the perimeter to be calculated using a
simple formula.
The perimeter of a plane figure is the length of its boundary.
Remember! The
perimeter of a
circle is called the
'circumference'.
P = 4s
s
Square
P = 2L + 2B
L
B
Rectangle
P = 2A + 2B
B
A
Parallelogram
P = 4s
s
Rhombus
d
C = πd or 2πr
Circle
IM4_Ch12_3pp.fm Page 338 Thursday, April 9, 2009 4:19 PM
CHAPTER 12 MEASUREMENT
339
• To find the arc length of a sector,
l
, first find what fraction
the sector is of the circle by dividing the sector angle
θ
by 360°.
Then find this fraction of the circumference.
• Composite figures are formed by putting simple figures together or by
removing parts of a figure. The calculation of the perimeter of composite
figures is shown in the examples below.
Challenge 12:01 | Staggered starts
When athletes run around a track with circular ends, they have a 'staggered start', since the
perimeter of the outer lanes is greater.
If the width of a lane is 1 metre, how much start should a runner in lane 1 give to the runner
in lane 2, if the runner in lane 1 is to complete exactly one lap of the field?
Can you find out how much start the inside runner appears to give the outside runner on an
official Olympic track for a 400 m event?
Gee, I feel like a
sector of pizza.
l
θ
360°
----------- 2πr × =
r
r
l
θ
c
h
alle
n
g
e
12:01
?
d d + 2
Try different values for d.
Does it make any difference
to the answer?
Assume the athletes run
along the inside edge of
their lanes.
IM4_Ch12_3pp.fm Page 339 Thursday, April 9, 2009 4:19 PM
340
INTERNATIONAL MATHEMATICS
4
Investigation 12:01A | Skirting board and perimeter
Please use the Assessment Grid on the following page to help you understand what is required
for this Investigation.
• The diagram is a scale drawing of a rumpus room in a house. All measurements are in
millimetres. Rob Young, a local builder, has been asked to fit skirting board to the room.
(Skirting board is used to cover the gap between the wall and the floor of a building.)
• The skirting board is to be placed around the perimeter of the room except for the doorways.
It can be ordered in lengths from 300 mm to 6·6 m at increments of 300 mm (ie 300 mm,
600 mm, 900 mm, 1·2 m and so on, up to 6·6m).
Exercises
1
Rob has been asked to do the job without any joins (except at corners). Is this possible?
Give reasons for your answer.
2
What is the total length of skirting board required?
3
Rob has nine 3·3 m lengths of skirting left from earlier jobs. Show how he could use these
to do the job. What is the smallest number of joins he could have? clear
though not always complete.
3
4
d
There is good use of mathematical language and
representation. Answers are correct and explanations
are applications.
1
2
c
There is a correct but brief explanation of whether results
make sense and how they were found. A description of
the relevance of perimeter in the context of the problem
is given.
3
4
d
There is a critical explanation of the results obtained and
their relation to real life. The answers to questions 3 and 4
are fully explained with consideration of the accuracy of
the results.
5
6
IM4_Ch12_3pp.fm Page 341 Thursday, April 9, 2009 4:19 PM
342
INTERNATIONAL MATHEMATICS
4
Measurement of area
• The area of a plane figure is the amount of space it occupies.
• Area is measured by calculating how many squares it would take to cover the figure.
Small squares are used to measure small areas and large squares are used to measure large
areas. It should not be surprising then that the units for measuring area are called square units.
• 1 cm
2
is the area within a square with 1 cm sides.
1 m
2
is the area within a square with 1 m sides.
1 ha is the area within a square with 100 m sides.
1 km
2
is the area within a square with 1 km sides.
• Area is calculated using a formula.
Area formulae
Square
A
=
s
2
Rectangle
A
=
LB
Triangle
A
=
bh or
Trapezium
A
=
h
(
a
+
b
)
Parallelogram
A
=
bh
Rhombus and kite
A
=
xy
Circle
A
=
π
r
2
Quadrilateral
My square is 1 m
2
.
1 m
1 m
one
square
metre
(1 m
2
)
s B
L
h
b
1
2
--- A
bh
2
------ =
h
a
b
1
2
---
h
b
x
y
y
x
1
2
---
r
There is no formula. The area is found
by joining opposite corners to form
two triangles.
The area of each triangle is calculated
and the two areas added to give the
area of the quadrilateral.
IM4_Ch12_3pp.fm Page 342 Thursday, April 9, 2009 4:19 PM
CHAPTER 12 MEASUREMENT
343
• To find the area of a sector, first find what fraction the sector is of the circle by dividing the
sector angle
θ
by 360°. Then find this fraction of the area of the circle.
• The area of composite figures can be calculated by either of the two methods.
Method I (by addition of parts)
We imagine that smaller figures have been
joined to form the figure, as in Figures 1 and 2.
1
Copy the figure.
2
Divide the figure up into simpler parts. Each part
is a shape whose area can be calculated directly,
eg square or rectangle.
3
Calculate the area of the parts separately.
4
Add the area of the parts to give the area of the figure.
Method II (by subtraction)
We imagine the figure is formed by cutting away simple shapes
from a larger complete figure, as shown.
1
Copy the figure and mark in the original larger figure from
which it has been cut.
P
a
t
t
e
r
n
s
a None of the following descriptors has been achieved. 0
b
Some help was needed to complete the table and identify
the simple patterns.
1
2
c
Measurement techniques have been selected and applied
to complete the table with some suggestion of patterns
and conversion rules.
3
4
d
The student has used the patterns evident in the table to
calculate similar areas of other shapes and find appropriate
conversion rules.
5
6
e
The patterns evident between the different measurements
have been explained, shown to be consistent for different
shapes, summarised as mathematical rules and justified using
all working Tables and explanations are clear but not
always complete.
3
4
d
There is a good use of mathematical language and
representation. Tables are complete. Explanations and rules
are logical and concise.
5
6
C
r
i
t
e
r
i
o
n
m
m
AC = 14 cm
BD = 8 cm
A
B
C
D
3
IM4_Ch12_3pp.fm Page 349 Thursday, April 9, 2009 4:19 PM
350 INTERNATIONAL MATHEMATICS 4
Investigation 12:01C | Covering floors
When covering a floor with tiles or carpet it is not just a matter of calculating the area of the
floor. Other practical considerations alter the problem.
The following examples illustrate some of the factors that need to be considered.
Laying tiles
When laying tiles, an exact number may not cover an area, or a whole number may not lie
along each edge. Look at this diagram.
If the tiles are 10 cm by 10 cm, we can
see that 15 tiles are needed, presuming
that the pieces of tile cut off are not
good enough to be used elsewhere.
(This is true even though the area is
28 cm × 45 cm, ie 1260 cm
2
. Divide this
by 100 cm
2
(the tile area) and this would
suggest that only 12·6 or 13 tiles might
be needed.)
1 How many tiles 10 cm × 10 cm would
be needed to cover an area 3·25 m
by 2·17 m?
2 How many tiles 300 mm by 300 mm
would be needed to cover an area
2·5 m by 3·8 m?
Laying carpet
Carpet comes in rolls, approximately 3·6 m wide. So when we buy a 'metre of carpet' we are
getting a rectangular piece 3·6 m wide by 1 m long. The diagram represents a room 2·9 m wide
and 4·25 m long.
When laying carpet, a carpetlayer can 'run' it along the room or
across the room. The aim is to avoid joins in the carpet and
reduce waste. The way the carpet is run will determine how many
'metres of carpet' must be bought.
1 How many metres of carpet must be bought if it is run lengthways?
How much waste would there be? Would there be any joins?
2 Repeat question 1 for the carpet if it is run across the room.
i
n
v
e
s
t
igatio
n
12:01C
I think
I've found an
easy way to
do these!
45 cm
2
8
m
IM4_Ch12_3pp.fm Page 355 Thursday, April 9, 2009 4:19 PM
356 INTERNATIONAL MATHEMATICS 4
The following solids have been formed from a cylinder. Calculate the surface area of each,
correct to 3 significant figures.
a b c
AB is a diameter. AOB is a quadrant. O is the centre of both
semicircles.
Investigate prisms or other solids that have a uniform cross-sectional area. How is the surface
area of the solid related to the cross-sectional area? Can you write a formula to express the
relationship?
Calculate the surface area of each of these solids.
a b
AB = BC = CD = DE = 5 cm Arc AC is drawn from B.
LM = PQ = 5 cm Arc BC is drawn from A.
Calculate the surface area of the following solids. Give the answer correct to 1 decimal place.
(Measurements are in metres.)
a b c
Calculate the surface area of the following solids. Give the answer correct to 1 decimal place.
(Measurements are in metres.)
a b c
9
A B
12 cm
2
0
c
m
A B
O
8
c
m
5
·
6
c
m
2 cm 2 cm
2 cm
7
c
m
A B
O
10
11
A
L M P Q
E
B D
C
1
0
c
m
A B
C
50 cm
5
0
c
m
12
1
.
5
1
3
3
0
.
6
1
.
2
2
.
8
3
.
2
0
.
8
1
.
2
1
.
6
0
.
8
2
.
4
13
3
1
.
2
2
.
8
2
0
.
6
2
0
.
3
1
.
8
2
.
4
0
.
2
3
3
5
3
IM4_Ch12_3pp.fm Page 356 Thursday, April 9, 2009 4:19 PM
CHAPTER 12 MEASUREMENT 357
12:03
|
Volume of Prisms and Cylinders
Previously, you would have calculated the volume of simple prisms and cylinders. You should
remember that:
• volume is the amount of space occupied by a three-dimensional figure
• volume is measured using cubic units; that is, cubic centimetres (cm
3
) or cubic metres (m
3
).
A prism is a solid with a uniform cross-sectional area. This means it can be 'sliced' parallel to one
of its faces so that you always get the same shape.
The fact that a prism can be cut into identical layers makes the calculation of its volume simple.
The cross-section is shaded blue.
Use the prisms above to check the following relationships.
1 The number of cubic units in each layer is the same as the cross-sectional area, A.
2 The number of layers is the same as the height of the prism, h.
3 The volume of the prism obtained by counting the cubic units is the same as the product Ah.
The above relationships give rise to the following formula.
When you cut a loaf of bread,
you are cutting parallel to the
cross-section, but not when you
cut a piece of cake.
2
2
3
2
4
2
2
3
2
1
1
The volume of all prisms is given by the formula V = Ah, where V = volume,
A = cross-sectional area and h = height of the prism.
IM4_Ch12_3pp.fm Page 357 Thursday, April 9, 2009 4:19 PM
358 INTERNATIONAL MATHEMATICS 4
If we think of a cylinder as a circular prism, the same formula can be
applied as for prisms.
Since the cross-section is a circle, the cross-sectional area A is the area
of a circle. Hence, for a cylinder, the formula V = Ah becomes V = πr
2
h.
The volume of a cylinder is given by the formula:
V = πr
2
h, where r = radius of the cylinder
h = height of the cylinder
worked examples
Find the volume of the following solids. Give answers correct to one decimal place.
1 2 3
Solutions
1 V = Ah 2 V = Ah 3 V = πr
2
h
A = 6·5 × 3·2 cm
2
A =
r = 22·6 ÷ 2
h = 2·6 cm = 11·3 cm
∴ V = 6·5 × 3·2 × 2·6 = 31·49 cm
2
h = 12·3 cm
= 54·1 cm
3
h = 12·7 cm ∴ V = π × (11·3)
2
× 12·3
∴ V = 31·49 × 12·7 = 4934·1 cm
3
= 399·9 cm
3
2·6 cm
3·2 cm
6·5 cm
9·4 cm
12·7 cm
6·7 cm
22·6 cm
12·3 cm
6·7 9·4 ×
2
-----------------------
IM4_Ch12_3pp.fm Page 358 Thursday, April 9, 2009 4:19 PM
CHAPTER 12 MEASUREMENT 359
Investigation 12:03 | Tank sizes
1 Use the fact that a tank with a volume of 1 m
3
has a capacity of 1 kL (or 1000 L) to calculate
the capacity in litres of the three tanks shown.
a b c
2 A tank is to hold 1000 L.
Give the dimensions of
three tanks with this capacity.
Assume that all the tanks are
rectangular prisms.
n
o
i
t
a
g i t s e v
n
i
12:03
0·6 m
0·3 m
1 m
50 cm
50 cm
1·8 m
1 m
800 mm
200 mm
Convert all measurements
to metres before
calculating volumes.
IM4_Ch12_3pp.fm Page 359 Thursday, April 9, 2009 4:19 PM
360 INTERNATIONAL MATHEMATICS 4
12:04
|
Volume of Prisms, Cylinders
and Composite Solids
The Prep Quiz should have reminded you that for solids with a uniform cross-section, such as
prisms and cylinders, the following relationships are true.
• The number of cubic units in each layer is the same as the cross-sectional area, A.
• The number of layers is the same as the height of the prism, h.
• The volume of the prism obtained by counting the cubic units is the same as the product Ah.
The exercise above suggests two ways in which the volume could be calculated.
Volume = (number of cubic units in each layer) × (number of layers)
or
Volume = (area of cross-section, A) × (height of prism, h)
It is the second of these methods that is the most widely applicable.
The prism shown has been made
from layers of cubes. Each cube
has a volume of 1 cm
3
.
1 How many cubes are there in each layer?
2 How many layers are there?
3 Calculate the volume of the prism by
counting cubes.
4 How could the answers to 1 and 2,
be used to calculate the volume?
5 The cross-sectional area, A, has been
shaded. What is the value of A?
6 What is the height, h, of the prism?
7 What is the value of Ah?
8 Are the answers to questions 1 and 5
the same?
9 Are the answers to questions 2 and 6
the same?
10 Are the answers to questions 3 and 7
the same?
p
r
e
p
quiz
12:04
This loaf of bread is like a
prism. It can be thought of
as a series of identical
layers of equal volume.
3
3
5
The volume of all prisms, cylinders and
prism-like solids is given by the formula
V = Ah
where:
V = volume
A = cross-sectional area
h = height of the prism.
A
h
IM4_Ch12_3pp.fm Page 360 Thursday, April 9, 2009 4:19 PM
CHAPTER 12 MEASUREMENT 361
For a cylinder, the cross-section is a circle and A = πr
2
.
The formula is then rewritten as
V = πr
2
h
r
h
worked examples
Find the volumes of the following solids.
1 2 3
Solutions
1 V = Ah 2 3
A =
=
= 39·1 − 4·2
= 34·9 cm
2
h = 3·2 cm
∴ V = 34·9 × 3·2
= 111·68 cm
3
2·8 cm
3 cm
8·5 cm
3·2 cm
4
·
6
m
600 mm
Make sure all
measurements are in
the same units.
6
12·5 cm
22·3 cm
16·7 cm
4·2 cm
9·3 cm
16·1 cm
1·8 m
1·5 m
1·5 m
2·9 m
3·6 m
2·4 m
0·5 m
0·5 m
0·5 m
0·9 m
2·1 m
2·5 m
1·5 m
7
35·5
25·1
12·5
15·3
10·4
1·2
4·6
3·5
5·5
7·5
2
2
5·4
5·4
5·4
5·4
8·6
2·6
8·4
7·6
6
8
11·4
4·6
35·7
9·5
9·5
6·8
31·5
5·2
5·2
8·3
15·3
12:04 Greatest volume
IM4_Ch12_3pp.fm Page 363 Thursday, April 9, 2009 4:19 PM
364 INTERNATIONAL MATHEMATICS 4
Investigation 12:04 | Perimeter, area and volume
Please use the Assessment Grid on the following page to help you understand what is required
for this Investigation.
1 A piece of wire 60 cm long is bent to form a rectangle.
a Give the dimensions of four rectangles that could be formed.
b Use the dimensions for the rectangles in a to complete this table.
c What happens to A as L − B becomes smaller?
d Predict the largest area that could be obtained.
e What is the area of the largest rectangle that can be formed from a piece of wire 100 m long?
2 A rectangular piece of cardboard 60 cm long and 20 cm wide is bent to form a hollow
rectangular prism with a height of 20 cm.
a From the results of question 1, predict the maximum volume of a rectangular prism
formed from this piece of cardboard.
b If the piece of cardboard were bent to form a cylinder, what would be the volume of the
cylinder? Will the volume of the cylinder be greater than the maximum volume obtained
in part a?
3 Suppose a farmer has 40 m of fencing to make a chicken pen. What dimensions would you
recommend to give the chickens the maximum living area? Give support for your answer.
What if the farmer built against an existing fence?
i
n
v
e
s
t
igatio
n
12:04
Length (L) Breadth (B) Area (A) L − B
Rectangle 1
Rectangle 2
Rectangle 3
Rectangle 4
60 cm
20 cm 20 cm
• Composite solids of many types
are present in these buildings.
How would you describe them?
IM4_Ch12_3pp.fm Page 364 Thursday, April 9, 2009 4:19 PM
CHAPTER 12 MEASUREMENT 365
Assessment Grid for Investigation 12:04 | Perimeter, area and volume complete the table and identify the
simple patterns.
1
2
c
Measurement techniques have been selected and applied to
complete the table with some suggestion of emerging patterns.
3
4
d
The student has used the measurements in the table to
describe the patterns and make predictions about area.
5
6
e
The patterns evident in the table have been explained and
justified, and used to answer questions 2 and 3 appropriately Lines of reasoning are insufficient.
1
2
c
There is satisfactory use of mathematical language and
representation. Tables and explanations are clear but not
always complete.
3
4
d
There is a good use of mathematical language and
representation. Tables are complete and well set out.
Explanations and answers are logical, with mention of possible real-life applications.
1
2
c
There is a correct but brief explanation of whether results
make sense and how they were found. A good attempt has
been made to answer questions 3 and 4 with working shown.
3
4
d
There is a complete explanation of the results obtained and
their application to other real-life situations. All answers are
fully explained with appropriate accuracy considered.
5
6
IM4_Ch12_3pp.fm Page 365 Thursday, April 9, 2009 4:19 PM
366 INTERNATIONAL MATHEMATICS 4
12:05
|
Practical Applications
of Measurement
A knowledge of perimeter, area and volume is extremely
useful in dealing with many everyday activities.
Many tradesmen require a knowledge of area and volume
to carry out their work, but it is also useful for everyday
people who do their own painting, concreting or tiling.
Measurement is clearly the basis of all building activities.
When calculating the capacity of a container the
following relationship is used.
Find the volume and capacity of the water tanks pictured below. Give the answers correct to
three significant figures.
a b c
A concrete slab is to be 3000 mm by 4000 mm by 100 mm. How many cubic metres of
concrete would be needed for this slab? What would be the cost of the concrete if it costs
$130 a cubic metre?
Calculate the capacity of the swimming pools pictured below. Give your answer correct
to three significant figures.
a b
Every
cubic metre
is 1000 L.
1m
3
ϭ 1kL
ϭ 1000 L
The amount of liquid needed to occupy a volume
of 1 cm
3
is 1 mL. Hence:
1cm
3
= 1mL
This converts to:
1m
3
= 1kL
Exercise 12:05
1
8 m
8 m
1·8 m
2·1 m
5·3 m
12·5 m
7 m
2·4 m
2
3
1 m
12 m
1·6 m
3·1 m
3·1 m
50 m
0·9 m
16·5 m
IM4_Ch12_3pp.fm Page 366 Thursday, April 9, 2009 4:19 PM
CHAPTER 12 MEASUREMENT 367
An above-ground pool has semi-circular ends
joined by straight sides.
a Calculate the volume of the pool if it is filled
to a depth of 1·2 m. Give the answer correct
to one decimal place.
b Find the capacity of the pool to the nearest
kilolitre.
A swimming pool is 25 m long, 10 m wide and 1·8 m high. Assuming that the pool is a
rectangular prism, find:
a the surface area of the pool and the cost of tiling it at $30 per m
2
b the capacity of the pool in litres if it is filled to a depth of 1·5 m (1 m
3
= 1000 L)
c the number of special edging tiles needed to go around the perimeter of the pool if the tiles
are 25 cm long
A roller is cylindrical in shape. It has a diameter
of 0·6 m and a width of 1·2 m.
a Find the area covered by the roller when it
makes one revolution. (Give your answer
correct to 1 decimal place.)
b Find the volume of the roller correct to
3 decimal places.
c Find the mass of the roller to the nearest
kilogram, if 1 m
3
weighs 1200 kg.
(The handle weighs 25 kg.)
a Calculate the volume of metal in the pipe
shown, correct to 1 decimal place.
b Calculate the weight of the pipe if 1 cm
3
of
metal weighs 5·8 g. Answer correct to
1 decimal place.
The solid pictured is formed by filling a mould
with molten metal. Calculate the mass of this
object if 1 cm
3
of metal weighs 11·4 g. Give the
answer correct to 2 significant figures.
A swimming pool has the shape of a trapezoidal
prism as shown in the diagram. Find:
a the cost of tiling the pool at $45 per m
2
b the volume of the pool in cubic metres
c how far the water level will be from the
top of the pool if it is three-quarters full.
(Answer to the nearest centimetre.)
1
1
·
2
m
3
·
6
m
4
5
0·6 m
1
·
2
m
2·3 cm 2·5 cm
1·5 m
3·2 cm
3·6 cm
4·2 cm
7·5 cm
12 cm
50·3 m
3·1 m
1 m 15 m
50 m
6
7
8
9
IM4_Ch12_3pp.fm Page 367 Thursday, April 9, 2009 4:19 PM
368 INTERNATIONAL MATHEMATICS 4
Calculate the area of shade cloth needed for the greenhouse.
When rainwater falls on the roof of a garage it is
drained by the gutter into a tank. Only one side
of the roof is drained.
a If 100 mm of rain falls, find how many litres
of water are drained into the tank.
b By how much would the water level in the
tank rise when the water from a was added?
c How many millimetres of rain would need
to fall on the roof to fill the tank?
Investigation 12:05 | Wallpapering rooms
Please use the Assessment Grid on the page 370 to help you understand what is required for
this Investigation.
Figure 1 shows a wall that is 3·6 m long and
2·4 m high. It is to be covered with wallpaper
that comes in rolls 52 cm wide and 15 m long.
To calculate the number of rolls needed,
follow the steps below.
Step 1 Work out the number of drops needed.
Note: Drops can be full or partial.
Partial drops occur when a wall contains
a door or window.
10
1·2 m
1·2 m
2
.
4 m
10
.
8 m
2·4 m
1·8 m
3·3 m
10·5 m
5
·
5
m
I A rainfall of 100 mm means the rain
would cover the horizontal area on
which it fell to a depth of 100 mm.
11
i
n
v
e
s
t
igatio
n
12:05
If the wallpaper
is patterned
we would need
more wallpaper
to allow for pattern
matching.
52 cm
2
·
4
m
D
r
o
p
1
D
r
o
p
2
3·6 m
Figure 1
Number of drops
Length of room
Width of wallpaper
------------------------------------------------ =
IM4_Ch12_3pp.fm Page 368 Thursday, April 9, 2009 4:19 PM
CHAPTER 12 MEASUREMENT 369
Step 2 Calculate the length of wallpaper needed.
Step 3 Determine the number of rolls by comparing the length of a roll to the length of
wallpaper required.
Use the room shown in Figure 2 to answer these questions.
1 How many rolls, correct to 1 decimal place, would be used to wallpaper:
a the western wall? b the southern wall? c the northern wall?
2 How many rolls, correct to 1 decimal place, would be used to wallpaper the three walls
on the eastern side?
3 Using your answers from above, how many rolls are needed to paper the whole room?
How many rolls would you need to buy?
4 a Calculate the perimeter of the room.
b Ignoring windows and doors, use the perimeter to calculate the number of drops of
wallpaper needed for the whole room.
c Use your answer to b to estimate the number of rolls needed. Compare this with your
answer to question 3.
Length of wallpaper
Number of
full drops
Length of a
full drop
×
length of partial drops + =
Number of rolls
Length of wallpaper
Length of a roll
------------------------------------------------- =
1
·
2
m
2
·
8
m
0·3 m
1
·
2
m
3·2 m
5·8 m
1
·
0
m
1
·
0
m
0
·
8
m
0·4 m
0·9 m
0·8 m
2·6 m
2
·
0
m
2
·
0
m
1
·
2
m
2
·
4
m
East
North
Sourth
West
Figure 2
IM4_Ch12_3pp.fm Page 369 Thursday, April 9, 2009 4:19 PM
370 INTERNATIONAL MATHEMATICS 4
Assessment Grid for Investigation 12:05 | Wallpapering rooms
The following is a sample assessment grid for this investigation. You should carefully read the
criteria before beginning the investigation so that you know what is required.
Assessment Criteria (C, D) for this investigation Achieved well set out
and easy to follow. The student is able to apply the given
steps easily.
3
4
d
There is good use of mathematical language and
representation. Answers are correct and explanations
are thorough, complete and concise. All parts are linked
together well in the context of the information given.
1
2
c
There is a correct but brief explanation of whether results
make sense and how they were found. A description of
the relevance of the findings to real life is given.
3
4
d
There is a critical explanation of the results obtained
and their relation to real life. The answers in questions 3
and 4 are fully explained with consideration of the
accuracy of the results.
5
6
IM4_Ch12_3pp.fm Page 370 Thursday, April 9, 2009 4:19 PM
CHAPTER 12 MEASUREMENT 371
Mathematical terms 12
Mathematical Terms 12
area
• The amount of space inside a
two-dimensional shape.
• Units of area:
square millimetre (mm
2
)
square centimetre (cm
2
)
square metre (m
2
)
hectare (ha)
square kilometre (km
2
)
• Formulae are used to calculate the area of
the common plane figures.
circumference
• The length of a circle's
boundary.
• The circumference is
calculated using either
the formula:
C = πD or C = 2πr
composite figure
• A figure that is formed by joining simple
figures.
composite solid
• A solid that is formed
by joining simple
solids.
cross-section
• The shape on the face
where a solid has been
sliced.
cylinder
• A prism-like solid with
a circular cross-section.
• It has two circular ends and a curved
surface.
hectare
• An area of 10 000 m
2
.
• A square with a side of 100 m.
perimeter
• The length of a plane figure's boundary.
prism
• A solid with two identical ends
joined by rectangular faces.
sector
• A part of a circle bounded
by two radii and an arc.
surface area
• The sum of the areas of the faces
(or surfaces) of a three-dimensional
figure (or solid).
volume
• The amount of space (cubic units) inside
a three-dimensional shape.
m
a
them
a
t
i
c
a
l
t
e
r
m
s
12
C
c
i
r
c
u
m
fe
r
e
n
c
e
rectangle
trapezium
semicircle
rectangle
cross-section
O
IM4_Ch12_3pp.fm Page 371 Thursday, April 9, 2009 4:19 PM
372 INTERNATIONAL MATHEMATICS 4
Diagnostic Test 12: | Measurement12
7 Calculate the surface area of the following prisms.
a c
8 Calculate the surface areas of these cylinders.
a b
9 Calculate the surface area of these solids.
10 Calculate the volumes of the prisms in question 7.
11 Calculate the volumes of the cylinders in question 8 to the nearest m
3
.
12 Calculate the volumes of the solids in question 9.
13 Pierre needs to use a water-filled roller to help compact his soil for lawn.
The roller is 90 cm wide with a diameter of 50 cm.
a find the area covered by one revolution
b find the capacity of water needed, to the nearest litre, if Pierre has to
fill the roller three-quarters full.
14 For the swimming pool shown:
a find the cost of tiling at $51 per m
2
b find the capacity of the pool in kL
Section
12:02
12:02
12:02
12:03
12:03
12:04
12:05
12:05
7·3 m
4·6 m
2·1 m
b
2·4 m
4 m
3 m
8 m
21 m
15 m
10 m
7 m
5·5 m
15 m
1·8 m
7
·5
m
12 cm
4 cm
3 cm
8 cm
7 cm
a
11 cm
11 cm
8 cm
5 cm
5 cm
b
11 cm
9 cm
3 cm
3 cm
2 cm
c
8 m
14·05 m
14 m
2·7 m
1·5 m
IM4_Ch12_3pp.fm Page 373 Thursday, April 9, 2009 4:19 PM
374 INTERNATIONAL MATHEMATICS 4
Chapter 12 | Revision Assignment
1 A floor is as shown in
the diagram. Find the
area of this floor and
the cost of covering it
with cork tiles if the cost
of the tiles is $40 per m
2
.
2 A pentagon is made by
placing an equilateral
triangle on top of a
rectangle. What is the
area of the pentagon?
3 A tent has the shape of a triangular prism
with dimensions as shown in the diagram.
a Find the area of material needed to
make this tent. (Include the floor area.)
b If the material comes in rolls which are
3·7 m wide, what length of material
must be purchased so that the tent can
be made without any joins except those
at the edges? (Hint: Consider the net of
the solid.)
c If special joining tape is needed to
strengthen each join, what length of
tape will be needed?
4 The inside and
outside of this
container are painted.
Calculate the area that
has to be painted.
5
a Calculate the perimeter of the figure
correct to 1 decimal place.
b Calculate the area of the figure correct
to 1 decimal place.
6 a Calculate the
surface area
of the solid.
Measurements
are in metres.
b Calculate the
volume of the
solid.
7
a Find the value of x correct to
1 decimal place.
b Calculate the area of the cross-section
of the prism.
c Calculate the surface area of the
pentagonal prism.
d Calculate the volume of the prism.
8
a Calculate the volume of the solid.
b Calculate the surface area of the solid.
t
n e
m
n
g
i
s
s
a
12A
7·1 m
5·3 m
3·6 m
4·5 m
10 cm
6 cm
1·2 m
1·6 m
2· 5 m
1
·
4
m
2 m
3 m
1 Perimeter
2 Area of sectors and composite figures
3 Surface area
4 Volume
10 cm
16 cm
5 cm 5 cm
120° 120°
0·8
0·6
1·2
0·2 0·2
5·4
9·4
6·5
x
7·3
2·2
20
16
12
25
8·3
10·5
6·5 6·5
IM4_Ch12_3pp.fm Page 374 Thursday, April 9, 2009 4:19 PM
CHAPTER 12 MEASUREMENT 375
Chapter 12 | Working Mathematically
1 Use ID Card 6 on page xviii to give the
correct mathematical term for:
a 13 b 14 c 15 d 16 e 18
f 19 g 20 h 9 i 10 j 11
2 Heather is 7 years younger than Rachel.
Ester is six times as old as Heather. Kuan is
7 years older than Ester. If Kuan is 43, how
old is Rachel?
3 Four different playing cards are dealt into two piles: left first, then right, then left, and then
right. The left pile is then placed on top of the right pile. How many times must this process
be repeated before the cards return to their original positions? How many times would the
process need to be repeated if there had been eight cards?
4 Every male bee has only one parent, a
female. Every female bee has two parents, a
male and a female. In the 8th generation
back, how many ancestors has a male bee?
(Assume that no ancestor occurs more
than once.)
5 If 3! = 3 × 2 × 1 (pronounced 3 factorial),
5! = 5 × 4 × 3 × 2 × 1 and
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1,
how many zeros are there on the end of 20!?
6 George Junkiewicz has prepared the timeline below to show when his employees will take
their holidays. He has designed it so that no more than two employees are on holidays at
the same time. The dots at the end of each line are explained below.
a What is the first day of Conway's holiday? b On what day does Conway return to work?
c Which two employees are on holidays in the week starting on the 8th of March?
d Which two employees are also on holidays during McKellar's holiday?
e George has to take four weeks of holidays. He is prepared to fit in wherever he can.
When must he take his holidays?
a
s
sign
m
e
n
t
12B
8 15 22 1 8 15 22 29
Hall
Raine
Conway
Harris
Browning
Bagnell
Robson
McKellar
Scully
Feb. March April May June
'Fine Flooring' holidays for employees
5 12 19 26 3 10 17 24 31 7 14 21 28
Included in holiday Not included in holiday
IM4_Ch12_3pp.fm Page 375 Thursday, April 9, 2009 4:19 PM
Trigonometry
13
376
I thought you said you were
doing your
trigonometry
work . . .
I am! I'm
working on
a tan!
Chapter Contents
13:01 Right-angled triangles
13:02 Right-angled triangles: the ratio of sides
13:03 The trigonometric ratios
13:04 Trig. ratios and the calculator
Practical Activity: The exact values for the
trig. ratio of 30°, 60° and 45°
13:05 Finding an unknown side
13:06 Finding an unknown angle
13:07 Miscellaneous exercises
13:08 Problems involving two right triangles
Fun Spot: What small rivers flow into
the Nile?
Mathematical Terms, Diagnostic Test, Revision
Assignment, Working Mathematically
Learning Outcomes
Students will be able to:
• Apply trigonometry to solve problems including those with angles of elevation and
depression.
• Apply trigonometry to problems involving compass bearings.
• Apply the sine rule, cosine rule and area of a triangle rule to the solution of problems.
Areas of Interaction
Approaches to Learning (Knowledge Acquisition, Problem Solving, Communication,
Logical Thinking, IT Skills, Collaboration, Reflection), Human Ingenuity, Environments
IM4_Ch13_3pp.fm Page 376 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 377
Trigonometry is a branch of geometry that is very important in fields such as navigation, surveying,
engineering, astronomy and architecture. Basic trigonometry is used to find unknown sides and
angles in right-angled triangles. The word trigonometry is actually derived from two Greek words:
'trigonon', which means triangle, and 'metron', which means measurement.
Measurement of unknown sides in right-angled triangles has been met before when using Pythagoras'
theorem. However, this could only be used to find one side when the other two were known.
13:01
|
Right-angled Triangles
Before introducing trigonometry, we need to be aware of some
further information concerning right-angled triangles.
From Pythagoras' theorem we know that the longest side in a
right-angled triangle is called the hypotenuse. The other two
sides also have names that refer to one of the acute angles in
the triangle. The side furthest from the angle is the opposite
side, whereas the side next to the angle is the adjacent side.
A
C B
adjacent
hypotenuse
opposite
worked examples
1 Name the sides in these two right-angled
triangles with reference to the angle marked.
2 Find the value of these ratios in ∆PQR.
a
b
Solutions
1 a BC = opposite side b XY = opposite side
AB = adjacent side XZ = adjacent side
AC = hypotenuse YZ = hypotenuse
2 a Side opposite angle R = 4 b Side opposite angle Q = 3
Hypotenuse = 5 Side adjacent angle Q = 4
∴ Ratio = ∴ Ratio =
a b
A B
C
Y
Z
X
P
4
3
5
Q
R
side opposite angle R
hypotenuse
----------------------------------------------------
side opposite angle Q
side adjacent angle Q
-----------------------------------------------------
4
5
---
3
4
---
IM4_Ch13_3pp.fm Page 377 Monday, March 16, 2009 1:10 PM
378 INTERNATIONAL MATHEMATICS 4
Name the side opposite the marked angle in each triangle.
a b c
d e f
Name the adjacent side in each of the triangles in question 1.
Name the hypotenuse in each triangle in question 1.
In triangle ABC to the right:
a which side is opposite angle B?
b which side is adjacent to angle C?
c which angle is opposite side AB?
d which angle is adjacent to side AC?
Find the value of the ratio in these triangles.
a b c
Find the value of the ratio for each triangle in question 5.
Exercise 13:01
1
A B
C
D
F E K
L M
P
Q
R
S T
U
X
Y
Z
2
3
4
A
B
C
5
side opposite angle P
side adjacent angle P
----------------------------------------------------
R
Q
12
13
5
P
R Q
P
6
8
10
P
R Q
15
17
8
6
side adjacent angle P
hypotenuse
----------------------------------------------------
• Trigonometry is used in the
building industry to determine
the length of sides and the
size of angles.
IM4_Ch13_3pp.fm Page 378 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 379
13:02
|
Right-angled Triangles:
the ratio of sides
Complete the table on the next page using
these three triangles.
For each triangle, state whether AB is opposite the angle marked, adjacent to the angle marked,
or is the hypotenuse.
1 2 3
Triangle I was enlarged to produce triangle II.
For triangles I and II, find the value of the
following ratios:
4
5
6
7 Is the pair of ratios in 4 equal?
8 Is the pair of ratios in 5 equal?
9 Is the pair of ratios in 6 equal?
10 The triangles are similar.
Does ?
p
rep
q
u
i
z
13:02
A
B C
A
C
B
A
C
B
P
I
5
4
3
P
II
10
8
6
side opposite angle P
side adjacent angle P
----------------------------------------------------
side opposite angle P
hypotenuse
----------------------------------------------------
side adjacent angle P
hypotenuse
----------------------------------------------------
x
6 12
9
x
6
---
9
12
------ =
Exercise 13:02
30°
1
opposite
hypotenuse
adjacent
h
a
o
θ
30°
3
h
a
o
θ
30°
2
h
a
o
θ
1
IM4_Ch13_3pp.fm Page 379 Monday, March 16, 2009 1:10 PM
380 INTERNATIONAL MATHEMATICS 4
Complete the table below using these three triangles.
(Give your answer correct to one decimal place.)
θ
1
2
3
θ
1
2
3
Give these
results correct
to one decimal
place.
The triangles
in each set
are the
same shape.
o
h
---
a
h
---
o
a
--
2
o
h
---
a
h
---
o
a
--
50°
θ
h
o
a
1
2
50°
θ
3
50°
θ
IM4_Ch13_3pp.fm Page 380 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 381
Complete the table below, giving answers correct to 1 decimal place.
a In question 1, are the respective ratios , , the same for each triangle?
b In question 2, are the respective ratios , , the same for each triangle?
a Using a protractor and ruler, construct a right-angled triangle with an angle of 35° and a
base of 5 cm. By measurement, calculate the value of each of the ratios given, correct to
1 decimal place.
i ii iii
b Construct another right-angled triangle with an
angle of 35° and a base of 10 cm and calculate
again the value of the ratios:
i ii iii
c What conclusion can you draw from the results in a and b?
13:03
|
The Trigonometric Ratios
In questions 1 and 2 of Exercise 13:02, each of the three triangles had angles of the same size.
When this happens the triangles are said to be similar.
Similar right-angled triangles can always be superimposed to produce a diagram like the one below.
θ
opposite side
(o)
adjacent side
(a)
hypotenuse
(h)
∆BAC 30°
∆BAD 45°
∆BAE 60°
3
A
E
D
C
B h
30°
∠ BAC = 30°
∠ BAD = 45°
∠ BAE = 60°
o
h
---
a
h
---
o
a
--
4
o
h
---
a
h
---
o
a
--
o
h
---
a
h
---
o
a
--
5
35°
base
o
h
---
a
h
---
o
a
--
o
h
---
a
h
---
o
a
--
IM4_Ch13_3pp.fm Page 381 Monday, March 16, 2009 1:10 PM
382 INTERNATIONAL MATHEMATICS 4
Check the results in the table below.
• Each of the right-angled triangles in this diagram (ie AOB, COD, EOF) is similar to the others,
since the corresponding angles in each are the same. The angle at O is obviously the same for
each triangle.
• The ratios , , are equal in all of the triangles.
• These ratios are called the trigonometric ratios (abbreviated to trig. ratios) and are given special
titles.
The ratio is called the sine ratio. It is abbreviated to sin θ.
The ratio is called the cosine ratio. It is abbreviated to cos θ.
The ratio is called the tangent ratio. It is abbreviated to tan θ.
• The three ratios have constant values for any particular angle irrespective of how big the
right-angled triangle may be.
• For any angle, the values of the ratios can be obtained from a calculator.
• As the lengths of right-angled triangles can often be surds, many trig. ratios are irrational
numbers.
θ
∆AOB 26·5° 0·4 0·9 0·5
∆COD 26·5° 0·4 0·9 0·5
∆EOF 26·5° 0·4 0·9 0·5
These similar
triangles overlap!
1
d
ec. p
l.
I Answers are
given correct to
one decimal place.
θ
O
A
E
F D B
C
o
h
---
a
h
---
o
a
--
o
h
---
a
h
---
o
a
--
o
h
---
side opposite angle θ
hypotenuse
----------------------------------------------------
a
h
---
side adjacent to angle θ
hypotenuse
----------------------------------------------------------
o
a
--
side opposite angle θ
side adjacent to angle θ
----------------------------------------------------------
IM4_Ch13_3pp.fm Page 382 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 383
• Because , we also have that:
opp.
adj.
hyp.
sin θ
opposite
hypotenuse
--------------------------- = cosθ
adjacent
hypotenuse
--------------------------- = tan θ
opposite
adjacent
-------------------- =
o
h
---
a
h
--- ÷
o
h
--- =
sin θ ÷ cos θ = tan θ or tan θ
sin θ
cos θ
------------ =
worked examples
1 Find sin θ, cos θ and tan θ for each triangle,
and express each as a decimal correct to
three decimal places.
2 Find sin α, cos β and tan α.
Solutions
1 a sin θ = cos θ = tan θ =
= = =
Ӏ 0·385 Ӏ 0·923 Ӏ 0·417
b First the hypotenuse must be calculated using Pythagoras' theorem. So, then:
sin θ = tan θ =
Ӏ 0·814 Ӏ 1·400
cos θ =
Ӏ 0·581
2 ABCD is a rectangle.
Hence DC = 8, BC = 5
Also BD = (Pythagoras' theorem)
∴ sin α = cos β = tan α =
= = =
a b
13
12
5
θ
5
7
θ
8
5
β
α
C D
A B
opp.
hyp.
-----------
adj.
hyp.
-----------
opp.
adj.
-----------
5
13
------
12
13
------
5
12
------
7
74
----------
7
5
---
5
74
----------
89
BC
BD
-------
AD
BD
--------
BC
DC
--------
5
89
----------
5
89
----------
5
8
---
I h
2
= 5
2
+ 7
2
= 25 + 49
= 74
ie h =
7
5
h
74
IM4_Ch13_3pp.fm Page 383 Monday, March 16, 2009 1:10 PM
384 INTERNATIONAL MATHEMATICS 4
Find sin θ, cos θ and tan θ in these triangles (as a simple fraction).
a b c
Evaluate sin A, cos A and tan A for each triangle. Give your answers in decimal form correct to
3 decimal places.
a b c
Find the unknown side using Pythagoras' theorem and then find sin θ and cos θ in decimal
form.
a b c
Use Pythagoras' theorem to find side YZ, then state the value of tan X.
a b c
sin
↓
S
↓
Some
opp./hyp.
↓
O
↓
Old
↓
H
↓
Hobos
cos
↓
C
↓
Can't
adj./hyp.
↓
A
↓
Always
↓
H
↓
Hide
tan
↓
T
↓
Their
opp./adj.
↓
O
↓
Old
↓
A
↓
Age
= = =
This should help you remember!
Exercise 13:03
1
3
4
5
θ
13
12
5
θ
7
24
25
θ
2
10
A C
B
6
8
A
B
C
17
15
8
A
B C
2·5
6·0
6·5
3
12
9
θ
6
3
θ
2
2
θ
4
10
X
Y
Z 8
Z
X
Y
61
5
Z
X
Y
2
13
IM4_Ch13_3pp.fm Page 384 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 385
Complete the statements below.
a b c
sin θ = . . . cos θ = . . . sin 60° = . . .
cos (90° − θ) = . . . sin (90° − θ) = . . . cos 30° = . . .
For the triangle on the right, complete the following:
a sin θ = . . . b cos θ = . . .
cos (90° − θ) = . . . sin (90° − θ) = . . .
Find the value of x, given that:
a cos 25° = sin x° b sin 60° = cos x° c cos 10° = sin x°
a For the triangle shown, write down the value of:
i sin A ii cos A
iii tan A iv sin A ÷ cos A
b Does ?
a Use Pythagoras' theorem to find the value of the missing
side as a surd. Hence find the value of sin 30°, cos 30°
and tan 30°. (Leave your answer as a surd.)
b By rationalising the denominator, arrange the values for
sin 30°, cos 30° and tan 30° in ascending order.
a If sin A = , find the values of cos A and tan A.
b It is known that cos θ = . What is the value of sin θ ?
a Find i sin A ii sin C b Find i sin θ ii cos α
5
3
4
5
θ
θ (90°– )
3
5
θ θ (90°– )
34
4
2
30°
60°
12
b
c
a
(90°– )
A
B
C
6
7
A
C
B
3
4
5
8
tan A
sin A
cos A
------------- =
1 cm
30°
2 cm
9
10
1
4
---
3
4
---
11
8
A
B
C
D 6
4
20
30
IM4_Ch13_3pp.fm Page 385 Monday, March 16, 2009 1:10 PM
386 INTERNATIONAL MATHEMATICS 4
a By finding tan θ in two different
triangles, find the value of m.
b Find x and y.
(Note: DE = x and
CE = y.)
c Find:
i sin θ ii cos θ
iii m iv n
v sin 2θ
vi Show that sin 2θ = 2 × sin θ × cos θ
Challenge worksheet 13:03 The range of values of the trig. ratios
4
3
5
m
12
10
5
x y
A B
E
D C
n
m
5
4
2
sin θ = cos (90° − θ)
cos θ = sin (90° − θ)
(90°– )
a
c
b
• Trigonometry is used in
surveying to calculate
lengths and areas.
IM4_Ch13_3pp.fm Page 386 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 387
13:04
|
Trig. Ratios and the Calculator
As we have already found, the values of the trig. ratios are constant for any particular angle and
these values can be found from a calculator. You can also use the calculator to find an angle when
you are given the ratio.
Finding a ratio given the angle
To find tan 31°, ensure your calculator is operating in 'degrees' and then press:
31
The calculator should give tan 31° = 0·600 860 6, correct to 7 decimal places.
Degrees and minutes
So far the angles have all been in whole degrees.
One degree, however, can be divided into
60 minutes.
For example, 31 ° would equal 31 degrees and
30 minutes. This would be written as:
31°30′.
We can now find the trigonometric ratios of angles
given to the nearest minute by using the calculator
as shown in the examples below.
tan =
1 degree = 60 minutes
1º = 60′, [ 1′ = º]
60
1
1
2
---
worked examples
Find:
1 sin 25°41′ 2 tan 79°05′
Give your answers correct to 4 decimal places.
Solutions
Two methods are shown, one for each solution. Choose the one that best suits your calculator.
Method 1:
1 For calculators with a Degrees/Minutes/Seconds button.
This is usually marked in either of two ways.
or
Press: 25 41
The calculator gives 0·433 396 953.
Method 2:
2 We convert 79°05′ into decimal degrees by realising that 05′ is of one degree.
Press 79 5 60
The calculator gives 5·184 803 521.
DMS
°
′ ″
sin DMS =
5
60
------
tan
(
+ ÷
)
=
I Warning:
Your calculator may
work differently to
the one used here.
IM4_Ch13_3pp.fm Page 387 Monday, March 16, 2009 1:10 PM
388 INTERNATIONAL MATHEMATICS 4
Finding an angle, given the ratio
If the value of the trigonometric ratio is known and you want to find the size of the angle to the
nearest minute, follow the steps in the examples below.
Using the degrees/minutes/seconds button on your calculator, write each of the following in
degrees and minutes, giving answers correct to the nearest minute.
a 16·5° b 38·25° c 73·9° d 305·75°
e 40·23° f 100·66° g 12·016° h 238·845°
Write in degrees, correct to 3 decimal places where necessary.
a 17°45′ b 48°16′ c 125°43′ d 88°37′
e 320°15′ f 70°54′ g 241°29′ h 36°53′
Use your calculator to find the value of the following, correct to 4 decimal places.
a sin 30° b cos 30° c tan 30° d sin 71°
e cos 58° f tan 63° g sin 7° h cos 85°
Find the size of θ (to the nearest degree) where θ is acute.
a sin θ = 0·259 b sin θ = 0·934 c sin θ = 0·619
d cos θ = 0·222 e cos θ = 0·317 f cos θ = 0·9
g tan θ = 1·2 h tan θ = 0·816 i tan θ = 3
worked examples
1 If sin θ = 0·632, find θ to the nearest minute.
2 If cos θ = 0·2954, find θ to the nearest minute.
Solutions
Note: One minute may be divided further, into
60 seconds, and this fact will be used to
round off answers to the nearest minute.
Again two methods are shown that correspond to the
two methods on the previous page.
1 If sin θ = 0·632 press: 0·632
The calculator now displays 39·197 833 53°. To
convert this to degrees/minutes/seconds
mode, press . The calculator gives 39°11′52·2″.
∴ θ = 39°12′ (to the nearest minute)
2 If cos θ = 0·2954, press 0·2954
The answer on the screen is 72·818 475 degrees. The alternative method of converting
this to degrees and minutes is to find what 0·818 475 of one degree is, in minutes;
ie 0·818 475 × 60′, which gives an answer of 49·1085 minutes, ie 49′ (to the nearest minute).
∴ θ = 72°49′.
What if I
want to
find the
angle?
2nd F sin =
DMS
2nd F cos =
Exercise 13:04
1
2
3
4
IM4_Ch13_3pp.fm Page 388 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 389
Find, correct to 3 decimal places, the following ratios.
a sin 30°10′ b sin 62°45′ c cos 52°30′ d cos 83°03′
e tan 61·25° f tan 79·36° g sin 17·8° h tan 72·57°
Find θ, to the nearest minute, given that θ is acute.
a sin θ = 0·6 b sin θ = 0·43 c sin θ = 0·645
d cos θ = 0·2 e cos θ = 0·031 f cos θ = 0·5216
g tan θ = 1·3 h tan θ = 0·625 i tan θ = 2·67
Redo question 6, this time giving answers in degrees correct to 2 decimal places.
What is the value of for each of the following triangles, correct to 3 dec. pl.?
a b c
Find the value of for each of the following, correct to 3 decimal places.
a b c
a If , find the value of x.
b If a = 3 sin 40° + 4 cos 30°, find the value of a correct to 3 decimal places.
c By substituting values for A and B, find if sin A + sin B = sin (A + B).
d If sin A = and sin B = find A + B.
e Jim thinks that if you double the size of an angle you double its sine, that is
sin 2A = 2 × sin A. Is Jim correct?
5
6
7
8
o
a
--
o
h
---
a
h
--- , ,
30°
58°
28°15'
9
x
10
------
60°
10
x
28°
10
x
47°10'
10 x
10
x
10
------ cos 60° =
1
2
---
1
3
---
• Trigonometry is used
in many branches of
science.
IM4_Ch13_3pp.fm Page 389 Monday, March 16, 2009 1:10 PM
390 INTERNATIONAL MATHEMATICS 4
Practical Activity 13:04 | The exact values for the trig. ratios 30°, 60° and 45°
∆ABC is an equilateral triangle of side 2 units.
AD is perpendicular to BC.
1 Copy the diagram and write in the size of BD and ∠BAD.
2 Using Pythagoras' theorem, calculate the length of AD
as a surd.
3 Now, from ∆ABD, write down the values of sin, cos and
tan for 30° and 60°.
∆DEF is a right-angled isosceles triangle.
The two equal sides are 1 unit in length.
4 Why is ∠EDF equal to 45°?
5 What is the length of DF as a surd?
6 Write down the values of sin 45°,
cos 45° and tan 45°.
2
A
B C D
2
60°
1
1
F
D
E
I Leave surds in
your answers.
Leave your answers in
surd form. Do not approximate.
sin 60° = , sin 30° = sin 45° =
cos 60° = , cos 30° = cos 45° =
tan 60° = , tan 30° = tan 45° = 1
60°
30°
3
2
1
45°
2
1
1
3
2
-------
1
2
---
1
2
-------
1
2
---
3
2
-------
1
2
-------
3
1
3
-------
IM4_Ch13_3pp.fm Page 390 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 391
13:05
|
Finding an Unknown Side
Pythagoras' theorem is used to find an unknown side in a right-angled triangle when the other two
sides are known. Trigonometry is used when only one side and one of the acute angles are known.
For the triangle given, state:
1 the hypotenuse
2 the side opposite the marked angle
3 the side adjacent to the marked angle
Write true or false for these triangles:
4 5 6
Find correct to 3 decimal places:
7 sin 75° 8 tan 25°30′ 9 If tan 25° = 10 If tan 25° =
then x = . . . then x = . . .
p
rep
q
u
i
z
13:05
A B
C
a
c
b
e
f
d
h g
i
sin θ
c
a
-- = cos θ
e
f
-- = cos θ
g
i
-- =
x
4
---
4
x
---
worked examples
1 Find a in these triangles, correct to 1 decimal place.
a b c
2 A ladder that is 8 metres long leans against a wall, and makes an angle of 21° with the wall.
How far does the ladder reach up the wall, to the nearest centimetre?
3 Find the length of a guy rope that must be used to secure a pole 12·5 m high, if the angle
the guy rope makes with the ground is 56°.
Solutions
Use the trig. button on your calculator.
1 a = sin 29°
∴ a = (sin 29°) × 15 ↔ 29 15
= 7·272 144 3
So a = 7·3 (to 1 decimal place)
a m
15 m
29°
a m
38°
9·6 m
x m
28°
9·2 m
a
15
------
sin × =
Make sure your calculator
is operating in 'degrees' mode.
continued §§§
IM4_Ch13_3pp.fm Page 391 Monday, March 16, 2009 1:10 PM
392 INTERNATIONAL MATHEMATICS 4
b = cos 38°
∴ a = (cos 38°) × 9·6 ↔ 38 9·6
= 7·564 903 2
= 7·6 (to 1 decimal place)
c = sin 28°
=
∴ x = ↔ 9·2 38 28
= 19·6 (to 1 decimal place)
2 From the information in the question, a diagram like the one
to the right can be drawn. Let the height up the wall be h m.
So: = cos 21°
h = 8 × cos 21° ↔
= 7·468 643 4
= 7·47 (to the nearest centimetre)
∴ The ladder reaches 7·47 m up the wall.
3 Let the length of the rope be x metres.
Then: = sin 56°
so: =
x =
Ӏ 15·08
∴ The rope is 15·08 metres long (to the nearest centimetre).
If x is the hypotenuse
you'll need to invert
each side of the equation.
x
x
9·2
28°
a
9·6
--------
cos × =
(Note that x is the
denominator of the
fraction, not the
numerator.)
9·2
x
--------
x
9·2
--------
1
sin 28°
-----------------
9·2
sin 28°
----------------- ÷ sin =
h
m
8 m 21°
8 21
× cos =
h
8
---
12·5 m
56°
x m
12·5
x
-----------
x
12·5
-----------
1
sin 56°
-----------------
12·5
sin 56°
-----------------
IM4_Ch13_3pp.fm Page 392 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 393
Find the value of the pronumeral in each triangle,
correct to 1 decimal place.
a b
c d
e f
g h i
j k l
Determine the value of each pronumeral, correct to 1 decimal place.
a b c
d e f
Exercise 13:05
Using trigonometry to find side lengths
1 Find x correct to 1 decimal place:
a b
2 Find x correct to 1 decimal place:
a
3 In each of the following state which trig.
ratio needs to be used to find x and then
find it correct to 1 decimal place.
a b
4 In each of the following state which trig.
ratio needs to be used to find the length
of the hypotenuse and then find it
correct to 1 decimal place.
a b
x
8
--- sin 15° =
y
6·8
-------- tan 38° . . . =
15
x
------ cos 40° . . . =
x
33°
10
x
62°
10
x
40°
8
x 60° 10
Foundation Worksheet 13:05
1
x
10
36°
y
31°
6
p
29°
20
a
3·6
45°
d
60°
9·2
x
39°
15·6
x m
25° 30′
8 m
a m
31° 15′
12 m
y cm
39° 52′
6 cm
n m
42° 45′
21·2 m
21° 49′
9·4 cm
y cm
x m
62° 10′
4·6 m
2
31°
10
a
41°
x
7
p
59°
25
52°
9·2
k
43°
d
5·3
q
65°
11·3
IM4_Ch13_3pp.fm Page 393 Monday, March 16, 2009 1:10 PM
394 INTERNATIONAL MATHEMATICS 4
g h i
j k l
For questions 3 to 11 the diagrams relate to the questions below them.
Find out everything you can about the triangle.
A ladder leans against a wall so that the angle it makes with the ground is 52° and its base is
4 m from the wall. How far does the ladder reach up the wall (to the nearest centimetre)?
A ladder leaning against a wall reaches 5·3 m up the wall when the angle between the ground
and the ladder is 73°. How long, to the nearest centimetre, is the ladder?
The diagonal of a rectangle is 16·3 cm long and makes an angle with the longest side of 37°.
Find the length of the rectangle, to the nearest centimetre.
A ship out at sea observes a lighthouse on the top of a 70 m cliff at an angle of 3°. How far out
to sea is the ship (to the nearest metre)?
A boat is anchored in a river that is 3·2 m deep. If the anchor rope makes an angle of 52° with
the surface of the water, how long is the rope from the surface of the water? (Answer to the
nearest centimetre.)
43° 20′
7 m
x m
27° 50′
12 m
x m
53° 09′
8 m
x m
35° 42′
5·1 m
x m
63° 25′
5·9 m
x m
37° 12′
9·7 m
x m
3 4 5
27°
C
A B
7 m
52°
4 m
73°
5
·
3
m
3
4
5
6 7 8
37°
16·3 cm
3°
70 m 52°
6
7
8
IM4_Ch13_3pp.fm Page 394 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 395
The equal sides of an isosceles triangle are 16 metres long and the apex angle is 80°.
Find, to the nearest centimetre, the length of the base.
The base of an isosceles triangle is 9·6 cm long and each of the base angles is 38°42′.
Find the length of each of the equal sides. (Answer correct to 3 significant figures.)
If the length of a child's slippery-dip is 3·4 m and one end makes an angle of 38°42′ with the
ground, how high above the ground is the other end? (Answer to the nearest centimetre.)
For questions 12 to 20, draw a diagram first!
a In ∆ABC, ∠A = 90°, ∠B = 63°25′ and BC = 6 m. Find AC, correct to the nearest centimetre.
b In ∆XYZ, ∠Z = 90°, ∠X = 42°34′ and XZ = 9·2 m. Find YZ, correct to the nearest centimetre.
c In ∆ABC, ∠B = 90°, ∠A = 52° and AB = 2·7 cm. Find AC, to 1 decimal place.
d In ∆XYZ, ∠X = 90°, ∠Y = 31°20′ and XZ = 10·3 cm. Find XY, to 1 decimal place.
The diagonal of a square is 21·2 cm. Find the length of each side (to the nearest millimetre).
Find the length of the diagonal of a rectangle if the length of the rectangle is 7·5 cm and the
diagonal makes an angle of 25° with each of the longer sides. (Answer correct to the nearest
millimetre.)
Find the length of a rectangle if its diagonal is 34 cm long and the angle the diagonal makes
with the length is 27°50′. (Answer correct to the nearest centimetre.)
Find the base of an isosceles triangle if the height is 8·2 cm and the base angles are each 39°.
(Answer correct to the nearest millimetre.)
When the altitude of the sun is 51°47′, a vertical stick casts a shadow 45 cm long. How high,
to the nearest millimetre, is the stick?
A painting is hung symmetrically by means of a string passing over a nail with its ends attached
to the upper corners of the painting. If the distance between the corners is 55 cm and the angle
between the two halves of the string is 105°, find the length of the string, correct to the nearest
millimetre.
The vertical rise from the bottom to the top of a track that slopes uniformly at 6°54′ with the
horizontal is 36 m. Find, to 1 decimal place, the length of the track.
9 10 11
80°
16 m
38° 42′
9·6 cm
9
10
11
12
13
14
15
16
17
18
19
IM4_Ch13_3pp.fm Page 395 Monday, March 16, 2009 1:10 PM
396 INTERNATIONAL MATHEMATICS 4
A road rises steadily at an angle of 6°45′. What will be the vertical rise of the road for
a horizontal distance of 300 m? (Answer correct to the nearest metre.)
At noon a factory chimney casts a shadow when the sun's altitude
is 85°24′. If the chimney is 65 m high, what is the length of the
shadow, to the nearest centimetre?
Calculate the sloping area of this roof that needs to be tiled,
given that the width of the roof is 5·4 m and its length is 9·2 m.
Each roof section is pitched at an angle of 23°. (Answer correct
to the nearest square metre.)
A plane is flying at an altitude (height) of 750 metres.
A boy on the ground first observes the plane when it is
directly overhead. Thirty seconds later, the angle of
elevation of the plane from the boy is 24°14′.
a Through what distance did the plane fly in 30 seconds,
to the nearest metre?
b Calculate the speed of the plane in km/h, correct to 3 significant figures.
Calculate the area of a right-angled triangle that has a hypotenuse 8 cm long and an
angle of 50°.
A regular hexagon of side a units is made by joining six
equilateral triangles together, as shown in the diagram.
We want to find a formula for the area of the hexagon in
terms of its side length, a.
Consider the area of one of the equilateral triangles.
a Using the exact trig. ratios on page 469, find the
exact length of DC.
b What is the area of ∆ABC?
c What is the area of a hexagon of side a units?
d Find the area of a hexagon with a side length of:
i 2 cm ii 5 cm iii 10 cm
20
85° 24′
21
22
23° 23°
5·4 m
9·2 m
750 m
24° 14′
23
24
A
C
B a
A
C
B D
a
60°
25
IM4_Ch13_3pp.fm Page 396 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 397
13:06
|
Finding an Unknown Angle
We have already seen in 13:04 that a calculator can be used to find the size of an angle if the value
of the trigonometric ratio is known.
Complete the ratios below for each triangle.
1 sin θ = 3 tan θ =
2 cos θ = 4 sin θ =
Given that θ is acute, find θ to the nearest degree, if:
5 tan θ = 0·635 6 sin θ = 0·2135 7 cos θ = 0·0926
If 0° р A р 90°, find A to the nearest minute if:
8 sin A = 0·52 9 tan A = 2·673 10 cos A = 0·7231
p
rep
q
u
i
z
13:06
13
12
5
8
17
15
worked examples
1 Find the size of angle θ. 2 What angle, to the nearest minute, does
Answer to the nearest degree. the diagonal of a rectangle make with its
length, if the dimensions of the rectangle
are 12·6 cm by 8·9 cm?
Solutions
1 In the triangle,
tan θ =
= 0·4 ↔ 0·4
∴ θ = 21·801 409°
so θ = 22° (to the nearest degree).
2 Let the required angle be θ. Then:
tan θ = 8·9 12·6
∴ θ = 35°14′7·59″
∴ θ = 35°14′ (to the nearest minute).
5
2
Remember '2nd F' may
be called 'SHIFT' on
some calculators.
2
5
---
2nd F tan =
12·6 cm
8·9 cm
8·9
12·6
-----------
2nd F tan
(
÷
)
= 2nd F DMS
IM4_Ch13_3pp.fm Page 397 Monday, March 16, 2009 1:10 PM
398 INTERNATIONAL MATHEMATICS 4
Find the size of the angle marked θ in each triangle. Give your answers correct to the nearest
degree.
a b c
d e f
g h i
For each, find the size of θ correct to the nearest minute.
a b c
d e f
Use trigonometry to find x in three different ways.
a In ∆LMN, ∠M = 90°, LN = 9·2 m and LM = 8·2 m.
Find ∠L, to the nearest degree.
b In ∆PQR, ∠R = 90°, PR = 6·9 m and QR = 5·1 m.
Find ∠P, to the nearest minute.
a A ladder reaches 9 m up a wall and the foot of the ladder is 2 m
from the base of the wall. What angle does the ladder make with
the ground? (Answer correct to the nearest degree.)
b What angle will a 5 m ladder make with the ground if it is to
reach 4·4 m up a wall? (Answer correct to the nearest degree.)
Exercise 13:06
1
5
2
2
3
10
7
10
6
6
5
14
20
4·7
8·9
6·3
8·9
1·7
2·6
2
9 m
12 m
7 m
5 m
7 m
12 m
6·2 m
4·6 m
6·9 m
11·5 m
8·2 m
10·1 m
4
3
5
x°
2 m
9 m
3
4
5
IM4_Ch13_3pp.fm Page 398 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 399
The beam of a see-saw is 4·2 m long. If one end is 1·2 m above
the ground when the other end is resting on the ground, find
the angle the beam makes with the ground, correct to the nearest
degree.
A road is inclined so that it rises 1 m for each horizontal
distance of 8 m. What angle does the road make with the
horizontal? (Answer correct to the nearest minute.)
At a certain time of the day, a tree 25 m high
casts a shadow 32 m long. At this time of day,
what angle do the rays of the sun make with the
ground? (Answer correct to the nearest minute.)
What angle does a diagonal of a rectangle make with each of the sides if the dimensions of the
rectangle are 4·7 m by 3·2 m? (Answer correct to the nearest minute.)
Find the angle θ in each of the following. (Answer correct to the nearest minute.)
a b c
The cross-section of a roof is an isosceles triangle.
Find the pitch of the roof (the angle it makes with
the horizontal) if the width of the roof is 9·6 m and
the length of one of the pitched sections is 5·1 m.
Give your answer correct to the nearest minute.
Find the size of the base angles of an isosceles triangle if the length of the base is 10 cm and the
height is 8·4 cm. (Answer to the nearest minute.)
Find the apex angle of an isosceles triangle, if the
length of each of the equal sides is 14·3 cm and the
length of the base is 20·8 cm. Give your answer to
the nearest minute.
The diagram shows a trapezium.
a If BC = 8, find θ.
b If CE = 8, find θ.
Challenge worksheet 13:06 Trigonometry and the limit of an area
13:06 Shooting for a goal
4
·2
m
6
7
8 m
1 m
32 m
25 m
8
9
10
3
(3, 2)
x
y
2
−1
x
y
2
(5, 3)
x
y
9·6 m
5·1 m
11
12
20·8 cm
14·3 cm
apex angle
13
14
50°
6
A B E
C D
15
IM4_Ch13_3pp.fm Page 399 Monday, March 16, 2009 1:10 PM
400 INTERNATIONAL MATHEMATICS 4
13:07
|
Miscellaneous Exercises
Before continuing with further trigonometric examples there is some general information that
should be mentioned.
Angles of elevation and depression
When looking upwards towards an object, the angle of
elevation is defined as the angle between the line of sight
and the horizontal.
When looking downwards towards an object, the angle of
depression is defined as the angle between the line of sight
and the horizontal.
angle of elevation
horizontal
lin
e
o
f
s
ig
h
t
angle of depression
l
i
n
e
o
f
s
i
g
h
t
horizontal
worked examples
1 The angle of elevation of the top of a vertical cliff is observed to be 23° from a boat
180 m from the base to the cliff. What is the height of the cliff? (Answer correct to
1 decimal place.)
2 An observer stands on the top of a 40-metre cliff to observe a boat that is 650 metres out
from the base of the cliff. What is the angle of depression from the observer to the boat?
(Answer to the nearest minute.)
Solutions
1 For this example, the diagram would look like
the one on the right.
Let the height of the cliff be h metres.
Then: = tan 23°
ie h = (tan 23°) × 180
= 76·405 467 (from calculator)
∴ Height of cliff = 76·4 m (to 1 decimal place).
2 Note: The angle of depression ∠DAB = ∠ABC
(alternate angles and parallel lines).
tan θ = 40 650
ie θ = 3°31′17·23″
= 3°31′ (to the nearest minute).
h metres
180 metres
23°
h
180
---------
angle of
depression =
650 m
40 m
A D
C B
40
650
---------
2nd F tan
(
÷
)
= DMS
IM4_Ch13_3pp.fm Page 400 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 401
Compass bearings
The direction of a point Y from an original point X
is known as the bearing of Y from X. This is mainly
expressed in one of two ways. Examine the diagram
below.
The bearing of Y from X
can be given as:
1 150° (the angle between
the interval XY and
the north line measured
in a clockwise direction),
or,
2 south 30° east (S30°E).
Sometimes, only letters are used. So SE (or south-east)
is halfway between south (180°) and east (90°); that is,
135° or S45°F.
Other examples would look like these.
060° or N60°E 245°09′ or S65°09′W 319°45′ or N40°15′W
This has great
bearing on
trigonometry!
150°
30°
X
Y
N
S
W E
60°
X
Y
N
S
W E
X
Y
N
S
W E
245° 09′
65° 09′
X
Y
N
S
W E
319° 45′
40° 15′
worked examples
1 If the town of Bartley is 5 km north and 3 km
west of Kelly Valley, find the bearing of Bartley
from Kelly Valley.
2 Two people start walking from the same point.
The first walks due east for 3·5 km and the
second walks in the direction 123° until the
second person is due south of the first person.
How far did the second person walk (to the
nearest metre)?
continued §§§
IM4_Ch13_3pp.fm Page 401 Monday, March 16, 2009 1:10 PM
402 INTERNATIONAL MATHEMATICS 4
The angle of elevation of the top of a
tower from a point 35 m from the base
of the tower was measured with a
clinometer and found to be 63°. Find
the height of the tower, correct to
1 decimal place.
The angle of depression of a boat 800 m out to sea from the top of
a vertical cliff is 9°. Find the height of the cliff, to the nearest metre.
Solutions
1 The diagram for this question
would look like the one on the
right.
Let the angle indicated in the
diagram be θ.
Thus: tan θ =
= 0·6
So: θ = 31° (to the nearest
degree)
So the bearing of Bartley from Kelly Valley would be N31°W or simply 329°.
2 This diagram shows the information in the
question above.
Since ∠SAB = ∠CBA
(alternate angles, AS // CB)
then ∠CBA = 57°
So: = sin 57°
ie =
x =
= 4·173 km
Press: 3·5 57
N
S
W E
329°
Bartley
Kelly Valley
5 km
3 km
3
5
---
57°
3·5 km A C
B
S
South
North
East West
Finish
x
123°
3·5
x
--------
x
3·5
--------
1
sin 57°
-----------------
3·5
sin 57°
-----------------
÷ sin =
Check out this step!
Exercise 13:07
Angles of elevation and
depression, and bearings
1 Find the bearing of B from A
if B is 6 km north and 3 km
east of A.
2 From a lighthouse 105 m
above the sea the angle of
depression of a boat is 2°.
How far is the boat from
the shore?
6
3
A
B
North
East
105 m
2°
x
LH
boat
Foundation Worksheet 13:07
63°
35 m
1
2
800 m
9°
IM4_Ch13_3pp.fm Page 402 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 403
From the top of a cliff 72 m high, the angle of depression of a boat is 12°47′. How far is the
boat from the base of the cliff? (Answer to the nearest metre.)
A vertical shadow stick has a height of 1·8 m. If the angle of elevation of the sun is 42°, what is
the length of the shadow at that time, correct to 1 decimal place?
Find the angle of elevation of the top of a vertical tower from a point 25 m from its base, if the
height of the tower is 40 m. (Answer to the nearest degree.)
From a lighthouse 70 m above sea level a ship, 1·2 km out to sea, is observed. What is the
angle of depression from the lighthouse to the ship? (Answer to the nearest minute.)
A kite is on the end of a string 80 metres long. If the vertical height of the kite, above the
ground, is 69 metres, find the angle of elevation of the kite from the person holding the string.
(Assume the string is a straight line, and answer to the nearest minute.)
A cyclist travels 15 km in the direction N15°27′E. How far has
he travelled in a northerly direction (to the nearest metre)?
A ship sails from P to Q a distance of 150 km on a course of
120°30′. How far is P north of Q? Also, how far is Q east of P?
(Answer to the nearest kilometre.)
Two towns, A and B, are 9 km apart and the bearing of B from
A is 320°. Find how far B is west of A (to the nearest kilometre).
Two cars leave from the same starting point, one in a direction due west, the second in a
direction with a bearing of 195°. After travelling 15 km, the first car is due north of the second.
How far has the second car travelled (to the nearest kilometre)?
An aircraft flew 10 km south and then 6 km west. What is its bearing from its starting point?
(Answer to the nearest degree.)
A, B and C are three towns. A lies 7 km north-east of B, and B lies 12·5 km
north-west of C. Find the bearing of A from C. Also, how far is A from C?
(Answer to the nearest metre.)
A ship is 5 nautical miles from a wharf on a bearing of 321°, and a
lighthouse is 11·5 nautical miles from the wharf on a bearing of 231°.
Find the bearing of the ship from the lighthouse. (Answer correct to
the nearest minute.)
The bearings from a point P of two landmarks X and Y are 35° and 125° and their distances
from P are 420 m and 950 m respectively. Find the bearing of Y from X (to the nearest minute).
X is due north of Y and 2 km distant. Z is due east of Y and has a bearing of S35°12′E from X.
How far, to the nearest metre, is Z from X?
3
4
5
6
7
8
9
10
11
12
45°
A
B
N
C
13
14
15
16
IM4_Ch13_3pp.fm Page 403 Monday, March 16, 2009 1:10 PM
404 INTERNATIONAL MATHEMATICS 4
A wire is stretched from point A on the top of
a building 21·3 m high, to point B on the top of
a shorter building, 15·6 m high. The angle of
depression from A to B is 20°15′.
a What is the horizontal distance between the
buildings (to the nearest centimetre)?
b How long is the wire (to the nearest centimetre)?
PQ is a diameter of the circle, centre O, as shown with
∠PRQ = 90°. If the radius of the circle is 6 cm, find,
to the nearest millimetre, the length of the chord PR,
given that ∠PQR = 40°.
A tangent of length 16 cm is drawn to a circle
of radius 7·5 cm from an external point T.
What is the angle, marked θ in the diagram,
that this tangent subtends at the centre of
the circle?
The diagonals of a rhombus are 11 cm and 7·6 cm. Find the angles, to the nearest degree,
of the rhombus.
Find the acute angle, to the nearest minute, between the diagonals of a rectangle that has sides
of 8 cm and 14 cm.
The eaves of a roof sloping at 23° overhang the walls, the edge of the
roof being 75 cm from the top of the wall. The top of the wall is
5·4 metres above the ground. What is the height above the ground
of the edge of the roof, to the nearest centimetre?
The arms of a pair of compasses are each 12 cm long. To what angle (to the nearest minute)
must they be opened to draw a circle of 4 cm radius? How far from the paper will the joint be,
if the compasses are held upright? (Answer to the nearest millimetre.)
Find the exact value of x in each of the following.
a b c
20° 15′
21·3 m
15·6 m
A
B
17
O
R
P Q
18
O
T
19
20
21
22
23
24
30°
10
x
x
2
45°
30°
12
x
IM4_Ch13_3pp.fm Page 404 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 405
a A rectangle is 10 cm long. The angle between the
diagonal and the length is 30°. What is the exact
area of the rectangle?
b A pole is to be supported by three guy wires.
The wires are to be fixed 10 m from the base of
the pole and must form an angle of 60° with the
ground (which is horizontal). What will be the
exact length of each guy wire?
c Find the exact value of x in the diagram.
13:08
|
Problems Involving Two
Right Triangles
Some problems can be solved by the consideration
of two right-angled triangles within the problem.
Examine the following two problems carefully and
then attempt Exercise 13:08.
60°
10 m
25
60° 45°
5 cm
4 3 cm
x cm
worked examples
Example 1
1 A pole PT stands on the top of a building BT. From a
point A, located 80 m from B, the angles of elevation of
the top of the building and the top of the pole are 43°
and 52° respectively. Find the height of the pole, PT,
correct to the nearest metre.
Solution 1
1 To find the length of the pole PT, the lengths PB and TB are calculated and of course
PT = PB − TB.
In ∆PBA, = tan 52° In ∆TBA, = tan 43°
PB = 80 tan 52° TB = 80 tan 43°
Now PT = PB − TB
= 80 tan 52° − 80 tan 43°
= 80 (tan 52° − tan 43°)
= 28 m (to the nearest metre)
43°
52°
A B
P
T
80 m
PB
80
-------
TP
80
-------
continued §§§
No, you're wrong!
I'm right!
I'm more
right than you!
IM4_Ch13_3pp.fm Page 405 Monday, March 16, 2009 1:10 PM
406 INTERNATIONAL MATHEMATICS 4
The top of a 20-metre tower is observed from two positions,
A and B, each in line with, but on opposite sides of, the
tower. If the angle of elevation from A is 27° and from
B is 35°, how far is point A from point B (to the nearest metre)?
In triangle ABC, BD is perpendicular to
AC. Given that AB = 13 m, BD = 11 m
and DC = 10 m, find, to the nearest
degree, the size of angle ABC.
Two points, P and Q, are in line with the foot of a tower 25 m high.
The angle of depression from the top of the tower to P is 43° and to
Q is 57°. How far apart are the points? (Answer to the nearest metre.)
Example 2
P, Q and R are three villages. Q is 5 km and N25°E from P.
R is east of Q and is 6·7 km from P. What is the bearing of
R from P, to the nearest degree?
Solution 2
To find the bearing of R from P, we need to find the size of
angle NPR. In ∆NPR we know the length of PR, but we need
to know one of the other sides, either NR or NP. Side NP
can be calculated using ∆NPQ.
In ∆NPQ: = cos 25°
ie NP = 5 cos 25°
In ∆NPR: cos ∠NPR =
=
= 0·676 349
∴ ∠NPR = 47° (to the nearest degree)
∴ The bearing of R from P is N47°E.
25° 25°
5
k
m
6
·
7
k
m
N
Q R
P
NP
5
-------
NP
6·7
--------
5 cos 25°
6·7
----------------------
Exercise 13:08
Problems with more than
one triangle
1
a Use ∆ABD to find x.
b Use ∆ADC to find y.
2
Use the fact that a = y − x to
find the value of a.
B
D
C
A
20° 30°
x
y
20
30°
25°
x
a
y
Foundation Worksheet 13:08
1
35° 27°
B A
20 m
A
B
C
D
2
43°
57°
T
B
25 m
Q P
3
IM4_Ch13_3pp.fm Page 406 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 407
A plane is flying at an altitude of 900 m. From a
point P on the ground, the angle of elevation to the
plane was 68°30′ and 20 seconds later the angle of
elevation from P had changed to 25°12′. How far had
the plane flown in that time, and what was its speed,
to the nearest kilometre per hour? (Find the distance
to the nearest metre.)
Find x in each diagram. Give answers correct to 2 decimal places.
a b
c d
a b
Find ∠CAD to the nearest minute. Find θ to the nearest minute.
a From ∆XWY, show that
XW = z cos X.
b From ∆ZWY, show that
ZW = x cos Z.
c Hence show that
y = z cos X + x cos Z.
a Show that AM = c sin B.
b Hence show that the area
of ∆ABC = ac sin B.
Two ladders are the same distance from the base of a wall.
The longer ladder is 15 m long and makes an angle of 58°
with the ground. If the shorter ladder is 12·6 m long, what
angle does it make with the ground? (Answer to the nearest
degree.)
4
25° 12′
68° 30′
900 m
Find a different
side first.
5
50
57° 40°
x
62°
x
3·6
25° 40′
x
2·5
40° 20′
x
3·7
6
A
B C D
5
3 4
2·5
3·5
50° 12′
x
y
z
W Z X
Y
7
b c
a
C M B
A
8
1
2
---
15
12·6
9
IM4_Ch13_3pp.fm Page 407 Monday, March 16, 2009 1:10 PM
408 INTERNATIONAL MATHEMATICS 4
A, B and C are three towns where A and B are due north of C.
From a position X on a map, A has a bearing of N27°E and B
has a bearing of N67°E. Town C is due east of X and 7·5 km
from it. Find the distance, correct to 1 decimal place, between
A and B.
Find the exact value of CE given that AE = 16.
In ∆ABC, AB = 12,
∠CAB = 60° and
∠CBA = 75°. Find
as exact values:
a AC
b BC
c area of ∆ABC
Challenge worksheet 13:08 Three-dimensional problems
Fun Spot 13:08 | What small rivers flow into the Nile?
Work out the answer to each question and put the letter for that part in the box that is above
the correct answer.
What are the exact values of:
I cos 60° E (cos 30°)
2
L (sin 30°)
2
J tan 60°
V sin 60° N (tan 30°)
2
S sin 45° E
U (sin 30°)
2
+ (cos 30°)
2
10
N
X
C
B
A
N
D
E
C
B
A
30°
16
45°
11
C
B A
60°
12
75°
12
f
u
n
spo
t
13:08
3
cos 45°
------------------
1
3
3
3
2
-
-
-
-
-
- -
3 4 -
- -
1 3 -
- -
1 2 -
- -
1 4 -
- -
2
1
2
-
-
-
-
-
- -
IM4_Ch13_3pp.fm Page 408 Monday, March 16, 2009 1:10 PM
CHAPTER 13 TRIGONOMETRY 409
Mathematical Terms 13
adjacent side (to a given angle)
• The side of a triangle which together with
the hypotenuse forms the arms of a given
angle.
angle of depression
• When looking down, the angle between
the line of sight and the horizontal.
angle of elevation
• When looking up, the angle between the
line of sight and the horizontal.
bearing
• An angle used to measure
the direction of a line from
north.
• Bearings can be recorded in
two ways.
eg 120° or S60°E
cosine ratio (of an angle θ)
• The ratio
• Abbreviated to cos θ.
hypotenuse
• The longest side in a right-angled triangle.
• The side which is not one of the arms of
the right-angle in a right-angled triangle.
opposite side (to a given angle)
• The side of a triangle which is not one of
the arms of the given angle.
similar triangles
• Two triangles that have the same shape but
a different size.
• Triangles that can be changed into each
other by either an enlargement or
reduction.
• Triangles that have matching angles equal.
• Triangles where the ratio of matching sides
is constant.
sine ratio (of an angle θ)
• The ratio
• Abbreviated to sin θ.
tangent ratio (of an angle θ)
• The ratio
• Abbreviated to tan θ.
trigonometric (trig.) ratios
• A collective name for different ratios of the
side lengths of right-angled triangles.
• The ratios have constant values for any
particular angle.
trigonometry
• A branch of mathematics, part of which
deals with the calculation of the sides and
angles of triangles.
m
a
them
a
t
i
c
a
l
t
e
r
m
s
13
h
y
p
o
t
e
n
u
s
e
adjacent side
l
i
n
e
m
x
4°
IM4_Ch13_3pp.fm Page 411 Monday, March 16, 2009 1:10 PM
412 INTERNATIONAL MATHEMATICS 4
b A plane flies at a speed of 650 km/h.
It starts from town A and flies on a
bearing of 120° for 3 hours. At that
time, how far is it
i south of A?
ii east of A?
4 From the top, T, of a 135-metre cliff, the
angles of depression of two cabins at A and
B are 23° and 42° respectively. How far
apart are A and B, assuming that A, B and
X, the foot of the cliff, are collinear?
(Answer to the nearest metre.)
5 From A, the
bearing of a
tower, T, is
330°. From B,
which is 10 km
north of A, the
bearing of the
tower is 290°.
a In the
diagram
show that
i x = h tan 70°
ii x = (h + 10)tan 30°
b Use the equations above to find x.
c Find the distance of the tower from B.
Chapter 13 | Working Mathematically
1 Use ID Card 6 on page xviii to identify:
a 2 b 3 c 4 d 10 e 11
f 12 g 14 h 15 i 16 j 17
2 Use ID Card 7 on page xix to identify:
a 5 b 8 c 9 d 10 e 11
f 12 g 18 h 22 i 23 j 24
3 Why is the
diagram
shown
impossible?
4 a If 6 men can do a piece of work in 8
days, in what time will 18 men do it,
working at the same rate?
b If 14 men can do a piece of work in 12
days, how many men will be needed to
do the work in 21 days, working at the
same rate?
5 A solid is formed
from a cube by
cutting off the
corners in such
a way that the
vertices of the
new solid will be
at the midpoints
of the edges of the
original cube.
If each of the new edges is a units long,
what is the surface area of the solid?
6 Two shops sell the same drink for the same
price per bottle. Shop A offers a 10%
discount, while shop B offers 13 bottles for
the price of 12. Which shop offers the
better discount if 12 bottles are bought?
23°
42°
T
X B A
T
B
A
h
x
10 km
Trigonometry ratios
t
n e
m
n
g
i
s
s
a
13B
8
40° 20°
IM4_Ch13_3pp.fm Page 412 Monday, March 16, 2009 1:10 PM
Vectors
14
413
What's our vector Victor?
How should I know and who are
all these other birds following me?
Chapter Contents
14:01
What is a vector?
14:02
Column vectors and vector operations
Investigation: Operating on vectors
14:03
Magnitude of a vector
Practical Activity: Magnitude of a vector
14:04
Solving problems using vectors
Mathematical Terms, Diagnostic Test,
Revision Assignment
Learning Outcomes
Students will be able to:
• Understand the definition of a vector.
• Identify relationships between vectors.
• Perform operations with vectors.
• Calculate the magnitude of a vector.
• Solve problems using vectors.
In Chapter 8 you studied different aspects of intervals on the Cartesian plane.
Although the lengths of these intervals are the same, they are all distinct from one another.
In order to represent this, an arrow is used to show the direction of a vector.
Some points to note on the vectors shown in the grid above:
• vector
=
vector since they both have the same magnitude and direction.
• vector
=
2
×
vector since has the same direction as vector but twice the magnitude.
• vector
=
−
vector since it has the same magnitude as vector but goes the opposite
direction.
• vector
=
−
×
vector since it has half the magnitude of vector and goes the opposite
direction.
So:
=
,
=
2 ,
=
−
and
=
−
Of course, you can use the other forms of notation instead if you wish.
C
E
F
A
B
G H
D All the intervals shown on this Cartesian plane
have one thing in common:
They all have the same length:
AB =
= 5
CD =
= 5
EF =
= 5
GH = 5
5 1 – ( )
2
5 2 – ( )
2
+
6 3 – ( )
2
−5 1 – – ( )
2
+
1 2 – – ( )
2
−3 1 – ( )
2
+
I A vector is different from an interval because it not only has length, which is called its magnitude,
but also direction.
C
E
F
A
B G
H
I
K
D
a
→
b
→
c
→
d
→
e
→
On the grid, five vectors are shown.
All have magnitude represented by their length.
All have direction, represented by the arrow.
Vectors can either be named by their endpoints
For example, , , , ,
(Note how the arrow above the letters gives the
direction of the vector.)
Or by a single lower case letter either written
with an arrow above it, or in bold type.
For example, , , , , or a, b, c, d, e
AB
→
CD
→
EF
→
GH
→
IK
→
a
→
b
→
c
→
d
→
e
→
a
→
b
→
c
→
a
→
c
→
a
→
e
→
a
→
a
→
d
→
1
2
--- a
→
a
→
a
→
b
→
c
→
a
→
e
→
a
→
d
→
1
2
--- a
→
IM4_Ch14_3pp.fm Page 414 Thursday, April 9, 2009 4:28 PM
CHAPTER
14
VECTORS
415
Express each of the following in terms of , and .
a b c
Write the relationship between the
following pairs of vectors.
a
and
b
and
c
and
d
and
e
and
f
and
On a grid, draw the following vectors from those given in the diagram in question 2.
a b
−
2
c
−
d
Exercise 14:01
1 w
→
x
→
y
→
y
→
w
→
x
→
m
→
n
→
l
→
l
→
m
→
n
→
These vectors
have magnitude
and direction.
2
a
→
b
→
c
→
d
→
e
→
f
→
g
→
h
→
k
→
l
→
a
→
h
→
a
→
f
→
b
→
l
→
b
→
k
→
e
→
c
→
d
→
g
→
3
a
2
---
→
e
→
3
2
--- h
→
2f
3
-----
→
IM4_Ch14_3pp.fm Page 415 Thursday, April 9, 2009 4:28 PM
416 INTERNATIONAL MATHEMATICS 4
14:02
|
Column Vectors and
Vector Operations
When we want to refer to a vector without drawing it, we can write it as a column vector.
To do this we must count how many units the vector goes horizontally and how many it goes
vertically.
For example, consider the vectors shown in the diagram.
For the vector to travel from the start of the vector to
the finish it goes 1 horizontally and 3 vertically.
So =
The numbers in the brackets are called the components.
Vector goes 4 units horizontally and −1 unit vertically
So =
Vector goes −2 units horizontally and −3 unit vertically
So = which can be written −
Vector goes −5 units horizontally and 0 unit vertically
So =
Vector goes −2 units horizontally and −4 unit vertically
So = which can be written − or −2
Investigation 14:02 | Operating on vectors
Please use the Assessment Grid on the page 418 to help you understand what is required for
this Investigation.
Each grid shows pairs of vectors, and joined end to end. This can represent a journey
from A to C. Complete the table showing the column vectors for both and and the
vector (not drawn) which represents a shortcut from A to C bypassing B.
1 2
A
B
m
→
n
→
k
→
l
→
AB
→
AB
→
1
3
horizontal units on top
vertical units on the bottom
m
→
m
→
4
1 –
n
→
n
→
2 –
3 –
2
3
k
→
k
→
5 –
0
l
→
l
→
2 –
4 –
2
4
1
2
i
n
v
e
s
t
igatio
n
14:02
AB
→
BC
→
AB
→
BC
→
AC
→
A
B
C A
B
C
IM4_Ch14_3pp.fm Page 416 Thursday, April 9, 2009 4:28 PM
CHAPTER 14 VECTORS 417
3 4
5 6
Test out your general rule with three pairs of vectors of your own.
Now consider the two vectors and in this
grid. Suppose represents the direction and
distance an aeroplane travelled on one part of
a trip and the direction and distance it
travelled on the next part of its trip.
Unfortunately the navigator has been sloppy
and has not started the second part from
where the first finished.
How would you go about representing the
direction and distance they would have
travelled if it went from the start to the finish
in a straight line (ie taking a shortcut)?
Once you have a vector that represents what the journey would have been in a straight line, it
is possible to calculate how far from the starting point the plane finishes. It is also possible to
calculate in what direction it could have flown to get there in a straight line.
Assuming that 1 unit on the grid represents 100 km, calculate these two values.
In what other real-life situations could vectors be used?
A
B
C
A
B
C
A
B
C
A
B
C
1
2
3
4
5
6
AB
→
BC
→
AC
→
Is there a pattern forming between the
values in the column vectors for and
and the values in the column vector
for ?
Write this pattern down as a general rule if
= and =
AB
→
BC
→
AC
→
AB
→
a
b
BC
→
c
d
a
→
b
→
a
→
b
→
a
→
b
→
IM4_Ch14_3pp.fm Page 417 Thursday, April 9, 2009 4:28 PM
418 INTERNATIONAL MATHEMATICS 4
Assessment Grid for Investigation 14:02 | Operating on vectors descriptors below has been achieved. 0
b
With some help, mathematical techniques have been applied
and the patterns in the table have been recognised.
1
2
c
Mathematical techniques have been selected and applied and
the patterns in the table have been recognised and a general
rule suggested.
3
4
d
Mathematical techniques have been selected and applied and
the patterns in the table have been recognised and described
as a general rule. Further examples are given in an effort to
provide justification.
5
6
e
Further to (d), the rule that is identified is explained fully
and its application in the final problem is used as part of the
justificationThere is a basic use of mathematical language and
representation. Lines of reasoning are hard to follow.
1
2
c
There is sufficient use of mathematical language and
representation. Lines of reasoning are clear but not always
logical. Moving between diagrams and column vectors is
done with some success.
3
4
d
There is good use of mathematical language and
representation. Lines of reasoning are clear, logical and
concise. Moving between diagrams and column vectors is
done effectivelyAttempts have been made to explain whether the results in
the table make sense and the importance of the findings in
a real-life context using the final problem.
1
2
c
A correct but brief explanation whether the results in the
table make sense is given. A description of the importance
of the findings in a real-life context using the final problem
is given.
3
4
d
A critical explanation whether the results in the table make
sense is given. A detailed description of the importance of
the findings in a real-life context using the final problem
is given. The significance of the findings has been
demonstrated in the solution to the final problem.
5
6
IM4_Ch14_3pp.fm Page 418 Thursday, April 9, 2009 4:28 PM
CHAPTER 14 VECTORS 419
Vector Operations and Vector Geometry
In Investigation 14:02, you should have seen that to combine two vectors you place them end to
end, so that it is possible to travel from the beginning of the first vector to the end of the last vector.
Combining them in this way and drawing the shortcut from the start to the finish in a straight line
is the same as adding the components of the column vectors.
For example, if the original vectors and are to
be combined, we move so that it starts where
ends; shown in blue.
When the start of the two vectors is joined to the
end of the two vectors, the resultant vector is the
sum of the two original vectors and , shown
in red.
When adding vectors on a diagram, they are
placed head to tail. The resultant vector, which
represents the sum, goes from the tail of the first
vector to the head of the last.
Alternatively the column vectors can be used:
For example, = and = , so + = =
Check this result from the diagram.
Resultant vector
a
→
b
→
+
a
→
b
→
b
→
a
→
b
→
b
→
a
→
a
→
b
→
a
→
4
6 –
b
→
4
2
a
→
b
→
4 4 +
6 – 2 +
8
4 –
worked examples
1 Vectors , and are shown in the diagram.
By using the diagram, draw the resultant vector for + + .
Solution
Move and so that all three vectors are
arranged head to tail.
The resultant vector starts at the tail of and
goes straight to the end of .
So + + =
To check, use the column vectors:
+ + = + + =
Add vectors
head to tail.
a
→
b
→
c
→
a
→
b
→
c
→
a
→
b
→
c
→
a
→
b
→
c
→
+ +
b
→
c
→
a
→
c
→
a
→
b
→
c
→
3
1 –
a
→
b
→
c
→
1
4
5
1 –
3 –
4 –
3
1 –
continued §§§
IM4_Ch14_3pp.fm Page 419 Thursday, April 9, 2009 4:28 PM
420 INTERNATIONAL MATHEMATICS 4
2 The vectors and are shown in the diagram.
Use another vector diagram to show the resultant
vector of − .
Solution
We can only add vectors so to do this problem,
instead of subtracting we add − .
Since − = + (− ).
So and − are placed head to tail to get the
resultant vector.
3 Use the diagram to express the following in
terms of , and .
a b c
d e f
Solution
a = −12
Alternatively, to get from D to A we could
travel − then − then − so that
= − − −
b = −
so = −(−12 ) = 12
or = −(− − − ) = + +
c = + , in other words, to get from A to C you must travel along then
d = − = −( + ) = − −
e = − −
f = − = −(− − ) = +
a
→
b
→
a
→
b
→
a
→
b
→
a
→
a
→
b
→
b
→
b
→
b
→
a
→
b
→
a
→
b
→
a
→
b
→
C
A
B
D
a
→
b
→
c
→
a
→
b
→
c
→
DA
→
AD
→
AC
→
CA
→
DB
→
BD
→
DA
→
b
→
c
→
b
→
a
→
DA
→
c
→
b
→
a
→
AD
→
DA
→
AD
→
b
→
b
→
AD
→
c
→
b
→
a
→
a
→
b
→
c
→
AC
→
a
→
b
→
a
→
b
→
CA
→
AC
→
a
→
b
→
a
→
b
→
DB
→
c
→
b
→
BD
→
DB
→
c
→
b
→
b
→
c
→
IM4_Ch14_3pp.fm Page 420 Thursday, April 9, 2009 4:28 PM
CHAPTER 14 VECTORS 421
Represent the following vectors on a grid.
a = b = c d =
d = e = f f =
From the vectors in question 1, calculate the resultant column vector for the following:
a + b + d c + d −
e + f − f g − h −
i + + f j 3( + )
By using the vectors shown in the grid and writing the vectors in column form, find the
resultant vector of the following.
a + m b −
c + d d +
e m − d f −
Express the vector in terms of other vectors
shown on the grid.
Given that X and Y are the midpoints of AC and AB respectively,
express the following in terms of the vectors and .
a b c d e
Looking at the results for (d) and (e) from question 1, what conclusions can you draw about
the lines XY and CB?
Exercise 14:02
1
a
→
5
3
AB
→
2 –
1
3 –
4 –
m
→
6
4 –
YX
→
0
5
3
0
2
a
→
m
→
a
→
AB
→
YX
→
AB
→
YX
→
AB
→
XY
→
YX
→
AB
→
m
→
BA
→
m
→
a
→
m
→
AB
→
m
→
3
X
Q
P
W
a
→
d
c
→
m
a
→
PQ
→
c
→
WX
→
QP
→
c
→
d
→
a
→
a
→
b
→
x
→
v
→
t
→
4 v
→
A
X Y
B C
5
CX
→
AB
→
CA
→
XA
→
AY
→
XY
→
BC
→
6
IM4_Ch14_3pp.fm Page 421 Thursday, April 9, 2009 4:28 PM
422 INTERNATIONAL MATHEMATICS 4
Using the diagram, express the following in terms of and .
a b c
Using the diagram, express the following in terms of and .
a b c d
The quadrilateral ABCD is made up of three equilateral triangles.
Express the following in terms of the vectors x and y.
a b c d
Show, with the aid of a diagram, how the vector = can be formed by combining
the following vectors:
= = =
= =
A B
C D
7 AB
→
CA
→
CB
→
AD
→
DB
→
8 a
→
b
→
A
C D
B
E
a
→
b
→
AB
→
CD
→
BD
→
BE
→
Same direction
and same size
gives equal
vectors.
B C
D A
E
y
x
9
AE
→
EB
→
CD
→
BD
→
10 r
→
3 –
5
p
→
3
1
s
→
1 –
2
q
→
0
2 –
u
→
6
1 –
t
→
4
2
IM4_Ch14_3pp.fm Page 422 Thursday, April 9, 2009 4:28 PM
CHAPTER 14 VECTORS 423
14:03
|
Magnitude of a Vector
Practical Activity 14:03 | Magnitude of a vector
1 For each of the vectors shown on the grid, complete a right triangle as in the example
vector .
2 Complete the table below as for the example vector .
You should have discovered that for any vector = the magnitude of the vector is given
by .
When writing the magnitude of the vector we write sometimes called the modulus of
the vector.
a
→
a
→
a
→
b
→
c
→
f
g
E
D
Vector Column
vector
Magnitude
units
c
g
a
→
2
3
2
2
3
2
+ 13 =
b
→
DE
→
f
→
v
→
a
b
a
2
b
2
+
v
→
v
→
Pythagoras
strikes again.
I So the magnitude or modulus of the
vector = is given by =
v
→
v
→
a
b
v
→
a
2
b
2
+
IM4_Ch14_3pp.fm Page 423 Thursday, April 9, 2009 4:28 PM
424 INTERNATIONAL MATHEMATICS 4
Find the magnitude of the vectors shown in the grid.
Find the magnitude of the following vectors.
a = b = c = d = e =
Evaluate the following if = , = , = .
a | + | b | + | c | + | d |2 − | e | − |
Exercise 14:03
1
a
→
b
→
c
→
d
→
e
→
f
→
2
a
→
15
8
b
→
6 –
8
c
→
20 –
21 –
d
→
12
5 –
e
→
2 –
10
3 AB
→
5
3 –
t
→
1 –
6
u
→
1
11
AB
→
t
→
u
→
AB
→
t
→
u
→
u
→
t
→
t
→
AB
→
IM4_Ch14_3pp.fm Page 424 Thursday, April 9, 2009 4:28 PM
CHAPTER 14 VECTORS 425
14:04
|
Solving Problems Using Vectors
Vectors are often used to represent objects that are in motion. The length of the vector represents
the magnitude of the motion (distance or speed), and the direction of the vector, the direction of
the object.
You will need to use trigonometry to solve these problems.
worked examples
1 A hiker walks on a bearing of 10° for a distance of 5 km and then on a bearing of 60° for
another 5 km. How far, and in what direction is he, from his starting point?
Solution
Let the vectors and represent the first and second legs of the
hike respectively.
To add the vectors to get the resultant vector we need to
calculate the vertical and horizontal components of and .
If = , then using trigonometry
cos 80 = and sin 80 = since | | = | | = 5
∴ = 5 cos 80 and = 5 sin80
∴ =
Similarly, if = then = 5 cos 30 and = 5 sin30
∴ =
The resultant vector = +
=
=
The distance from the starting position is given by | | =
= 9·06 km
The direction from the starting point can be found by considering the components of the
resultant vector as a triangle.
tanθ =
θ = tan
−1
θ = 55°
As a result, after his hike, the hiker is 9·06 km from his starting point on a bearing of 45°.
80°
30°
a
→
b
→
r
→
a
→
b
→
r
→
a
→
b
→
a
→
A
x
A
y
A
x
5
-----
A
y
5
----- a
→
b
→
A
x
A
y
a
→
5 80 cos
5 80 sin
b
→
B
x
B
y
B
x
B
y
b
→
5 30 cos
5 30 sin
r
→
a
→
b
→
5 80 cos 5 30 cos +
5 80 sin 5 30 sin +
5·20
7·42
r
→
5·20 ( )
2
7·42 ( )
2
+
5.20
7.42
r
→
7·42
5·20
----------
7·42
5·20
----------
continued §§§
IM4_Ch14_3pp.fm Page 425 Thursday, April 9, 2009 4:28 PM
426 INTERNATIONAL MATHEMATICS 4
A yacht sails on a bearing of 100° for a distance of 54 nautical miles
and then on a bearing of 40° for a distance of 80 nautical miles.
How far (to the nearest nautical mile) and in what direction
(to the nearest degree), is the yacht from its starting point?
(Remember: from the diagram, the first y component
is negative.)
2 An aeroplane is flying at a ground speed of 200 km/h on a bearing of 300°. A crosswind is
blowing at 50 km/h on a bearing of 50°. Use vectors to calculate the resultant velocity and
direction of the aeroplane.
Solution
Here the magnitude of the vector represents the speed.
The vectors and must be resolved into their vertical and horizontal parts.
= and =
= and =
where = 200 km/h and = 50 km/h
∴ = and =
So the resultant vector which is + =
=
Therefore the resulting velocity is given by =
= 188·8 km/h
The direction of the plane is given by 270° − θ, where tanθ =
So that θ = tan
−1
= 44·4°
In conclusion, the aeroplane is flying at 188·8 km/h on a bearing of 314·4°.
40°
30°
p
→
w
→
W
x
W
y
P
x
P
y
p
→
w
→
p
→
P
x
P
y
w
→
W
x
W
y
p
→ p 30° cos –
p 30° sin
→
→
w
→
w 40° cos
w 40° sin
→
→
p
→
w
→
p
→
200 30° cos –
200 30° sin
w
→
50 40° cos
50 40° sin
132.14
θ
134.91
Note: It is important that these
signs are correct so that the final
direction can be worked out.
r
→
p
→
w
→
200 30° 50 40° cos + cos –
200 30° sin 50 40° sin +
134·91 –
132·14
r
→
134·91 – ( )
2
132·14 ( )
2
+
132·14
134·91
-----------------
132·14
134·91
-----------------
Exercise 14:04
1
50°
20°
N
IM4_Ch14_3pp.fm Page 426 Thursday, April 9, 2009 4:28 PM
CHAPTER 14 VECTORS 427
Crazy Ivan has set off to cross the Gobi desert. He first walked for
35 km on a bearing of 200° and realised he was heading the wrong
way so he then walked for 20 km on a bearing of 310°. How far,
and in what direction (to the nearest whole degree) is he now from
his starting point?
(Remember: from the diagram, the first x and y components are
negative and the second x component is negative.)
A plane flew first on a bearing of 320º for 150 km and
then due east for 150 km. How far, and in what
direction, is the plane from its starting point?
(Remember: due east only has an x component.)
A balloonist is at the mercy of the wind. When he first
takes off, the wind blows him on a bearing of 100º for
80 km and then on a bearing of 200º for 100 km.
What is th | 677.169 | 1 |
Mathematics
A number of learning theories have been applied to the domain of mathematics. ACT* has been used to develop a computer tutoring program for
geometry. Repair theory provides a detailed analysis of the
cognitive proceses involved in subtraction. Conversation theory served as the basis for studies in learning probability.
Schoenfeld has developed a comprehensive theory of
mathematical problem solving that suggests four kinds of skills are necessary to be
successful in mathematics: resources, heuristics, control processes, and beliefs. The Gestalt theory outlined by Wertheimer suggests some general
mechanisms of problem-solving that are relevant to mathematics.
The structural learning theory of Scandura has been applied
extensively to mathematics. According to this theory, the most fundamental aspect of
learning is the acquistion of higher-order rules that describe mathematical procedures. Bruner applies his constructivist framework to mathematics. The algo-heuristic theory of Landa also emphasizes the importance of
rules in mathematics learning.
In addition, theories of intelligence such as Gardner and Guilford. Research on mathematics instruction is reported in
Charles & Silver (1989), Cocking & Mestre (1988), and Grouws & Cooney (1988). | 677.169 | 1 |
Product Description
Join Fred Gauss, child prodigy teaching at Kittens University, as he guides students through conversion factors, the coefficient of friction, square roots, and much more! Get the entire Life of Fred College Prep series and save even more!
Middle school and up. This is the first book in the Life of Fred Pre-Algebra series and the third in the overall College Prep series. Students will be ready after they have completed Decimals and Percents.
What is included?
288 pages (hardcover) of fun stories to get students excited about math and physics!
Subjects covered:
Ordered pairs
Hooke's law
Solving d = rt for r
Numerals
Making models
Definition of pi
Continuous and discrete variables
...and much more!
Why buy?
Teachers and parents love Life of Fred because it actually gets students interested in math. Instead of the usual "drill and kill" approach, Fred uses entertaining stories that students remember forever. The author, Dr. Stanley F. Schmidt, developed this method through years of teaching in high school and college. Your students will never ask, "When are we ever going to use this stuff?"
Reviews
Submit Review
Leave a Review!
Please select a rating (1-5)
Life of Fred ROCKS!
May 17, 2017
The Life of Fred series has made learning math engaging and fun for my kids - they both jump right into math first thing each morning. Now that we're getting into these higher levels, we're all interested and excited to see what happens next with Fred and Kingie and their adventures at KITTENS U!
Stacey M - Homeschooler, Parent - Member Since December 2015
Great product, great price!
May 10, 2017
This is engaging and yes, entertaining! It is a great way to learn math as it relates to the world.
Guest
Loving Life of Fred
May 03, 2017
Haven't dug into yet because we are not there yet. I'm buying ahead so we can keep on rolling. I flipped through and liked what I saw. When we get into I will update my review. Life of Fred has made doing math a more pleasant experience for my son.
Kimberly R - Homeschooler - Member Since November 2015 | 677.169 | 1 |
Help In Mathematics For The Visually-Impaired
Mathematics is a very visual subject, dependant on one's ability to read/manipulate symbols, analyze diagrams, and interpret graphs. So, what happens if you happen to be visually-impaired or blind? That is, are there technology tools to help?
First, I would suggest spending some time on Susan Osterhaus' web site. She has been teaching secondary mathematics for 29 years at the Texas School for the Blind and Visually Impaired in Austin, Texas. Follow her links to read information about:
Project Math Access, which focuses on creating opportunities for blind or visually impaired students to succeed in their study of mathematics
Second,one might conside products auch as Math Player, as described on Maria Anderson's blog Teaching College Math Technology Teaching College Math Technology. Maria providers a demo and explains how Math Player is available as a free download from the company Design Science. One big problem is the text being read must be written in MathML, an Explorer plug-in.
Also, for graphic purposes, NASA offers two products called Math Trax and Math Description Engine. Available in both a visual and text version, Math Description Machine is a FREE graphing tool for secondary school students that allows graphing equations, graphing physics simulations, or plotting data files. Each graph includes descriptions and sound so a student can hear and read about the graph, allowing visually-impaired students access to visual mathematics data and graphs. Similarily, software developers can use the Math Description Engine Software Development Kit to make computer-rendered graphs more accessible to blind and visually-impaired users, by adding alternative text and sound descriptions to each graph.
These are just some initial suggestions. If you know of...or better yet, use....other or better options for helping the visually-impaired, please let me know so that the information can be shared. | 677.169 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.