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Chapter 4 Test: Functions and Integers
In this functions and integers worksheet, students combine positive and negative integers through addition, subtraction, multiplication and division. They complete function tables and graph the results of linear equations. This six-page worksheet contains 26 problems. Answers are provided on the last two pages.
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Designed as an avenue of communication for mathematics educators concerned with the views, ideas, and experiences of two-year college students and teachers, this journal contains articles on mathematics exposition and education, as well as regular features presenting book and software reviews and math problems. The first of two issues of volume 14 contains the following major articles: "Technology in the Mathematics Classroom," by Mike Davidson; "Reflections on Arithmetic-Progression Factorials," by William E. Rosenthal; "The Investigation of Tangent Polynomials with a Computer Algebra System," by John H. Mathews and Russell W. Howell; "On Finding the General Term of a Sequence Using Lagrange Interpolation," by Sheldon Gordon and Robert Decker; "The Floating Leaf Problem," by Richard L. Francis; "Approximations to the Hypergeometric Distribution," by Chitra Gunawardena and K. L. D. Gunawardena; and "Generating 'JE(3)' with Some Elementary Applications," by John J. Edgell, Jr. The second issue contains: "Strategies for Making Mathematics Work for Minorities," by Beverly J. Anderson; "Two-Year Mathematics Pioneers," an interview with Allyn J. Washington; "Using Linear Programming To Obtain a Minimum Cost Balanced Organic Fertilizer Mix," by Stephen J. Turner; "Problems Whose Solutions Lie on a Hyperbola," by Steven Schwartzman; and "The Shape of a Projectile's Path: Explorations with a Computer Algebra System," by John H. Mathews and Robert Lopez. (BCY) | 677.169 | 1 |
Availability 17Practical Algebra If you studied algebra years ago and now need a refresher course in order to use algebraic principles on the job, or if you're a student who needs an introduction to the subject, here's the perfect book for you. Practical Algebra is an easy and fun-to-use workout program that quickly puts you in command of all the basic concepts and tools of algebra. With the aid of practical, real-life examples and applications, you'll learn: * The basic approach and application of algebra to problem solving * The number system (in a much broader way than you have known it from arithmetic) * Monomials and polynomials; factoring algebraic expressions; how to handle algebraic fractions; exponents, roots, and radicals; linear and fractional equations * Functions and graphs; quadratic equations; inequalities; ratio, proportion, and variation; how to solve word problems, and more Authors Peter Selby and Steve Slavin emphasize practical algebra throughout by providing you with techniques for solving problems in a wide range of disciplines--from engineering, biology, chemistry, and the physical sciences, to psychology and even sociology and business administration. Step by step, Practical Algebra shows you how to solve algebraic problems in each of these areas, then allows you to tackle similar problems on your own, at your own pace. Self-tests are provided at the end of each chapter so you can measure your mastery Practical Algebra: A Self-Teaching Guide, 2nd Edition by Peter H. Selby & Steve Slavin 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.
More About Peter H. Selby & Steve Slavin
PETER SELBY (deceased) was Director of Educational Technology at Man Factors Associates, a human factors engineering consulting firm. He is the author of two other self-teaching guides: Quick Algebra Review: A Self-Teaching Guide and Geometry and Trigonometry for Calculus: A Self-Teaching Guide, both published by Wiley.
STEVE SLAVIN, Ph.D., is Associate Professor of Economics at Union County College, Cranford, New Jersey. He has written over 300 newspaper and magazine articles, and is the author of four other books, including All the Math You ll Ever Need: A Self-Teaching Guide and Economics: A Self-Teaching Guide, both published by Wiley.
I decided I needed a serious review of algebra as I will soon be taking a physics class, not having had a math or any science class in over 20 years, and my level of math skills had deteriorated to the basics needed to balance a checkbook and nothing more. I have almost finished with this terrific book, and now I not only feel comfortable with algebra again, but I feel I truly understand the subject. The authors' excellent instruction and numerous practice problems really help to create a sense of comfort with the subject, something I thought impossible not that long ago. In fact, looking at samples in other books I purchased (but which were relegated to my bookshelves), I actually find myself thinking there are easier ways to reach the answer, or manipulating the equation beyond what the author intended, because of an understanding of the subject I have acquired through self-study...something I still find amazing. This book takes things step-by-step, building gradually on past lessons so readers are seldom (if ever) left scratching their heads in confusion, and the authors are both clear and thorough in their explanations (although it could benefit from acknowledging some of the standard pnemonics one often sees in math, such as PEDMAS or FOIL). I wish the follow-up book for trig/calculus was still readily available! Overall, this is a highly recommended book.
Helping My Son May 19, 2008
My son is now starting algebra and unfortunately, it's been some 25 years since I've had any exposure to it. This book is a nice way to review and polish up so that even if I can't make him an expert, I don't look so foolish.
Brilliant Book Mar 4, 2008
This book is a very good book to learn algebra, if you just started. I personally believe that I have learned much from this book. Plus, there are good problems that help reinforce the knowledge learned. This book explains many concepts and is very easy to understand. This is also good review too if you forget some concepts in algebra.
Excellent, excellent, excellent! Jan 1, 2008
This is the absolute best book I've ever read on Algebra. My teachers made the subject so confusing in High School, but two months with this book and I became an Algebra whiz!
If you need to learn Algebra, BUY THIS BOOK.
Amazing! Nov 8, 2007
I recently returned to school intending to major in Forensic Science. I graduated from high school in 1992 and only had pre-algebra. When I took the placement exam it put me in Algebra I. I suppose that is pretty good since it meant that I hadn't forgotten the stuff I learned in high school, but it meant I would have to spend at least an extra year just getting caught up on math before I could even get into the science classes I needed for my desired major. I purchased several algebra books that weren't helping at all, then I found this one. I love this book. I spent 5 days working through the book from cover to cover. I also used a software program called "College Algebra Solved!" for the areas where I needed a little more help (like simplifying radicals). After spending those five days with the book and software I retook the placement exam and placed into precalulus (my placement score went from 34 to 85. That's a 51 point increase!). So, yes, you can teach yourself algebra with this book. Everything is explained well and there are lots of practice problems and self-tests.
If you are looking to refresh or even teach yourself from scratch, this is | 677.169 | 1 |
Mathematicians are expected to publish their work: in journals, conference proceedings, and books. It is vital to advancing their careers. Later, some are asked to become editors. However, most mathematicians are trained to do mathematics, not to publish it. But here, finally, for graduate students and researchers interested in publishing their work, Steven G. Krantz, the respected author of several 'how-to' guides in mathematics, shares his experience as an author, editor, editorial board member, and independent publisher.This new volume is an informative, comprehensive guidebook to publishing mathematics. Krantz describes both the general setting of mathematical publishing and the specifics about all the various publishing situations mathematicians may encounter. As with his other books, Krantz's style is engaging and frank. He gives advice on how to get your book published, how to get organized as an editor, what to do when things go wrong, and much more.He describes the people, the language (including a glossary), and the process of publishing both books and journals. Steven G. Krantz is an accomplished mathematician and an award-winning author.
He has published more than 130 research articles and 45 books. He has worked as an editor of several book series, research journals, and for the Notices of the AMS. He is also the founder of the ""Journal of Geometric Analysis"". Other titles available from the AMS by Steven G. Krantz are ""How to Teach Mathematics"", ""A Primer of Mathematical Writing"", ""A Mathematician's Survival Guide"", and ""Techniques of Problem Solving" | 677.169 | 1 |
ATHEMATICS: ITS POWER AND UTILITY, Tenth Edition, combines a unique and practical focus on real-world problem solving allowing even the least-interested or worst-prepared student to appreciate the beauty and value of math while mastering basic concepts and skills they will apply to their daily lives. The first half of the book explores the POWER and historic impact of mathematics and helps students harness that POWER by developing an effective approach to problem solving. The second half builds upon this foundation by exploring the UTLITY and application of math concepts to a wide variety of real-life situations: money management; handling of credit cards; inflation; purchase of a car or home; the effective use of probability, statistics, and surveys; and many other topics of life interest. Unlike many mathematics texts, MATHEMATICS: ITS POWER AND UTILITY, Tenth Edition, assumes a basic working knowledge of arithmetic, making it effective even for students with no exposure to algebra. Completely self-contained chapters make it easy to teach to a customized syllabus or support the precise pace and emphasis that students require.
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The first half of the book, "The Power of Math," develops some ideas in arithmetic, algebra, and geometry. The second half of the book, "The Utility of Math," develops the ideas around mathematics that students will use outside of the classroom.
Each chapter opener helps students anticipate upcoming topics. Topics included in "The Power of Math" or "The Utility of Math" section openers are identified. A list of essential ideas tells students what they need to know after they have worked through the chapter.
Each section of the book begins with an introductory box called either "The Power of Math" or "The Utility of Math." These introductory boxes are designed to get students to THINK about how the material in the section relates to their life. The first few problems at the end of every section are also designed to engage students in this way.
Throughout this book, notes explain steps or offer hints about what students should look for as they read.
At the beginning of the book, hints for success set students off on the right foot. If they follow this advice, their chance for success in this course will be greatly increased.
Street signs (e.g., a red "Stop" sign or a yellow "Caution" sign) throughout the text serve as guideposts to help students when they are most likely to need help.
Historical Notes show some of the "humanness" of mathematicians.
Procedure boxes give step-by-step procedures for important processes; definitions and properties are also highlighted in boxes.
Problems are organized by difficulty into Level 1, Level 2, and Level 3. The problems are also separated into essential ideas, drill and practice, applications, and "right or wrong?" problems. There are Sudoku problems, puzzle problems, and even Tergiversation problems.
The end-of-chapter material has been carefully crafted to help students study effectively and time-efficiently, and to clearly point to areas where more review may be required in their preparations.
Individual Projects give students a chance to expand their horizons and research problems on their own.
Team Projects provide students with a chance to interact together in a job situation, and thereby learn to work together as a team.
What's New
The "Power/Utility of Math" section openers (formerly called "In This World: The Power/Utility of Math") have been rewritten. These introductory boxes are designed to engage students by encouraging them to think about how the material in the section relates to their life.
The "Problem of the Day" feature in the end-of-section exercise sets is now called "The Power/Utility of Math" to better reflect its direct connection to the section opener with the same name. These exercises, now boxed and prominently titled, can be found at the beginning of each end-of-section exercise set.
An outline of "Essential Ideas," together with problems that correspond with these ideas, has been moved from the end-of-chapter review material to each chapter opener.
Over 300 new problems have been included.
The Chapter Summary and Review sections have been reorganized and redesigned, making them easier for students to use in identifying their strengths and weaknesses.
Section 7.5, which is included in the chapter on applications of percent, now includes information on 401(K) and retirement.
Section 7.6, which is included in the chapter on applications of percent, now contains information on amortization, including an amortization table.
A new section called "Connectives and Truth Tables" has been added to chapter 8, sets and logic.
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Complement your text and course content with study and practice materials. Cengage Learning's Liberal Arts Mathematics CourseMate brings course concepts to life with interactive learning, study, and exam preparation tools that support the printed textbook. Watch student comprehension soar as your class works with the printed textbook and the textbook-specific website. Liberal Arts Mathematics CourseMate goes beyond the book to deliver what you need!
This CD-ROM (or DVD) provides the instructor with dynamic media tools for teaching, including Microsoft® PowerPoint® lecture slides and figures from the book. You can create, deliver, and customize tests (both print and online) in minutes with ExamView® Computerized Testing Featuring Algorithmic Equations. You'll also find a link to the Solution Builder online solutions manual, allowing you to easily build solution sets for homework or exams.
The Student Survival and Solutions Manual provides helpful study aids and contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
This guide helps students navigate Enhanced WebAssign. It includes instructions on how to use the Assignment page and its Summary, tips on using MathPad for providing easy input of math notation and symbols, an overview of the Graphing Utility's drawing tools for completing graphing assignments, and information on how to access grades and scores summary CourseMate.
The Student Survival and Solutions Manual provides helpful study aids and fully worked-out solutions to all of the odd-numbered exercises in the text. It's a great way to check your answers and ensure that you took the correct steps to arrive at an answer.
If your instructor has chosen to package Enhanced WebAssign with your text, this manual will help you get up and running quickly with the Enhanced WebAssign system so you can study smarter and improve your performance in classKarl Smith | 677.169 | 1 |
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New Data Science Course
By William Schellhorn
Mar 17, 2014
The Simpson College Mathematics Department is excited to offer a new special topics course called Data Science during the Spring 2015 semester. The class will be taught by Dr. Bill Schellhorn, who recently spent his sabbatical leave studying topics in the field.
What is data science? Data science is the study of the extraction of knowledge from data. In practice, it involves learning from data in order to gain insight and make useful predictions. Knowledge in the field is increasingly important for Mathematics and Actuarial Science majors.
Why is data science important? Data is being generated faster than it can be analyzed. Many of the current challenges in science, government, industry, economics, marketing, and sports are "big data" problems. Some examples include:
the Large Hadron Collider experiments;
the Sloan Digital Sky Survey;
the human genome project;
surveillance data collected by the National Security Agency;
social network data collected by Facebook;
marketing data collected by Amazon and NetFlix;
statistics from professional sports leagues.
What topics will be covered in the Math 390 Data Science course? The course will introduce methods used in data science, including techniques in data collection, data management, exploratory data analysis, prediction, and communication of results. Real-world examples will be used to illustrate the methods presented. The analyses and methods will be implemented in a statistical software package (either R or JMP).
What are the prerequisites for the course? Math 152 Calculus II and CmSc 150 Introduction to Programming.
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The Exact Differential
Students investigate the exact differential and apply it to a variety of problems. They practice forming the exact differential when integrating complicated functions. They use sample problems to review the concept and the teacher can use them for instructional purposes.
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Lesson Planet has greatly assisted me in finding appropriate worksheets to accompany my lesson plans. I recently switched from high school to junior high and needed assistance in finding grade appropriate material. | 677.169 | 1 |
Solving 2-Step Equations Booklet of Steps
57 Downloads
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0.07 MB | 1 pages
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This product (a small booklet) was created for the student who has been introduced to solving 2-step equations, but has a difficult time remembering the steps.
As the student has more practice with solving 2-step equations, he or she can refer to the booklet only when the steps have been forgotten. The student should be encouraged to try each step first without without referring to the booklet, and, in most cases, the student should understand that eventually he or she should be able to solve without referring to the booklet.
The booklet is useful for students of special needs.
The booklet is created by cutting the sections apart and stapling them together on one side. The booklet is small (you make two booklets from each sheet of paper), so they are easy for the students to keep accessible | 677.169 | 1 |
Systems of Equations Introduction & Types of Solutions: INB Pages
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Solving Systems of Equations Word Problems:
This resource includes 2 interactive pages. The first page is an introduction to systems of equations. It defines what a system of equations consists of and what a solution to a system of equations is. The second flippable is on types of solutions. It shows students why/how they may find that there are no solutions, infinite solutions or just one solution to a system of equations.Terms | 677.169 | 1 |
C.a.R.
C.a.R.
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C.a.R. simulates constructions with a pair of compasses and
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With functions, curves and expressions, C.a.R goes far
beyond compass and ruler. With the powerful macros very
complicated constructions can be explored. Other geometries,
elliptic or hyperbolic, can be explored too. Graphics can be
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Triangle of Thought
Feynman, Richard, Robert Leighton and Matthew Sands. By Evelyn Lamb on September 12, 2013 On Monday, the Onion reported that the " Nation's math teachers introduce 27 new trig functions ." Adding and subtracting signed numbers worksheets, fraction and mixed number to decimal tool, how to solve 3rd order polynomials, RAtIO FORmula, Square roots in radical form, how to print sum of a cube with java. Right triangles are often used in trigonometry, as seen above.
Pages: 179
Publisher: Amer Mathematical Society (January 1, 2001)
ISBN: 082182614X
Trigonometry Update for Montgomery College
Fundamentals of trigonometry
Manual of Logarithms: Treated in Connection with Arithmetic, Algebra, Plane Trigonometry, and Mensuration, for the Use of Students Preparing
Simply click View Steps in the answer screen to sign up.[/note] Not only can this advanced calculator check answers but it can also provide additional practice problems to help hone your skills in preparation for tests and quizzes Success with trigonometry (My Math Tutor Came Over Yesterday). C.-based trade associations — the National Governors Association (NGA), and the Council of Chief State School Officers (CCSSO). The bulk of the work to create the CCS was done by Achieve, Inc. (also a D Catalogue of stars for the epoch Jan. 1, 1892 from observations by the Great trigonometrical survey. Suppose I just want to integrate a cube. sin^3 x. But I do have an odd power of a trig function, of a sine or cosine. And the procedure that I was suggesting says I want to take out the largest even power that I can, from the sin^3 Second course in algebra, with trigonometry. The user interface is simple and clear with lots of pictures so that problem solving is fun. ★ Learn at your own pace: Splash Math enables your child to learn various math concepts at his or her own pace. After each question, the app appropriately chooses the next question. Each topic starts with an easy level and based on the child's progress, the medium and hard levels are gradually unlocked Four Place Tables of Logarithms and Trigonometric Functions. JMAP resources for the CCSS include Regents Exams in various formats, Regents Books sorting exam questions by CCSS: Topic, Date, Type and at Random, Regents Worksheets sorting exam questions by Type and at Random, an Algebra I Study Guide, and Algebra I Lesson Plans College Algebra and Trigonometry With Applications (Itt Version. Use the interactive whiteboard to draw functions and graphs and review sine and cosine. If you're studying for a test or trigonometry regents, a tutor can help you find worksheets and practice problems and review problems you need extra help with. Our tutors help students master trigonometry concepts and get better grades Trigonometry With Applications. Trigonometry is the branch of mathematics that deals with triangle s, circle s, oscillations and waves; it is absolutely crucial to much of geometry and physics. You will often hear it described as if it was all about triangles, but it is a lot more interesting than that Plane and Spherical Trigonometry and Tables (Large Print Edition).
The sine of a middle part is the product of the cosines of the opposite parts, or the product of the tangents of the adjacent parts. Check this rule with the identities in the preceding paragraph. Vector calculations can be made graphically, but trigonometry is required for numerical solutions. Finding components of a vector in a given direction, and finding the resultant of vectors, are the most common problems download Triangle of Thought pdf. Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships. Trigonometry has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. It is usually taught in secondary schools either as a separate course or as part of a precalculus course Mathematical and Astronomical Tables: For the Use of Students in Mathematics, Practical Astronomers, Surveyors, Engineers, and Navigators; Preceded by ... and Trigonometrical Tables, Plane and Spheri. Short version: Buy a new text bundled with MyMathLab (MML) ISBN 0-321-698630, Use the course ID for the specific semester you are registering. Long version: If you have the time to maneuver the maze then read on: This book is a third of College Algebra with Trigonometry by the same authors. You may use MyMathLab (MML) software with this course. ISBN for text with MyMathLab Access code is: 0-321-698630; This is available in university or KampusKorner bookstores Manual of Plane Trigonometry, by J.a. Galbraith and S. Haughton HBJ Algebra 2 with Trigonometry. More specifically, Rina's research area is at the intersection of algebraic geometry, representation theory, and category theory. "I study the categories that are associated to geometric objects, mostly derived categories of sheaves, and related DG and A-infinity categories," Rina says. "I am especially interested in functors between these objects and some algebraic structures that they generate, such as categorifications of group and algebra representations," she added Plane trigonometry as far as the solution of triangles.
Elements of Plane and Spherical Trigonometry: With Its Applications to the Principles of Navigation and Nautical Astronomy; with Logarithmic and Trigonometrical Tables
College Algebra and Trigonometry
Elements Of Trigonometry, Plane And Spherical.
Trigonometry Update for Montgomery College
Life of Fred 2-Book Set : Trigonometry, Freds Home Companion for Trigonometry
Also investigate De Moivre's theorem, a shortcut for raising complex numbers to any power. x Embark on the first of four lectures on systems of linear equations and matrices. Begin by using the method of substitution to solve a simple system of two equations and two unknowns. Then practice the technique of Gaussian elimination, and get a taste of matrix representation of a linear system. x Deepen your understanding of matrices by learning how to do simple operations: addition, scalar multiplication, and matrix multiplication Algebra & Trigonometry with Analytic Geometry (Classic Edition), 11th (eleventh) Edition. Elementary math formula sheets, grade nine alberta math right angle and tangent ratio, solve algebra equation with division, mixed numbers in decimal form, Do adding radical expression use the ascending and descending order, Math trivias about graduation. Probability quizzes year 8, algebraic processes in factorization of quadratic equation, cross multiplying +frations, howdo i find the y intercept of (-4,-10), unit 6 grade 5 adding and subtracting with unlike fractions, Adding integers worksheets A treatise on surveying, containing the theory and practice; to which is prefixed a perspicuous system of plane trigonometry. The whole clearly ... particularly adapted to the use of scho. Common Core excludes certain Algebra II and Geometry content that is a prerequisite at almost every four-year college. 3. Common Core fails to teach prime factorization. Consequently, CCS does not include teaching about least common denominators or greatest common factors. 4 Plane Trigonometry and Four-Place Tables of Logarithms. The values of sin, cos and tan are related to each other, for instance, In euclidean space the area of a triangle depends on length of 2 sides and one angle: This is half the product of two sides and the sin of the included angle Wiley trigonometric tables. So, because I can't help everyone who contacts me for help I don't answer any of the emails asking for help. Also, when I first started this site I did try to help as many as I could and quickly found that for a small group of people I was becoming a free tutor and was constantly being barraged with questions and requests for help The Civil Engineer's Pocket-Book. And as shown in the figure< C= 90 º than we can easily find out < B which will be 54 º. Now, as according to the steps let's make the unknown side as numerator and make the known side as denominator of fraction, Now, as per step second name the function of angle, Now, as per step 3 use the trigonometric table to evaluate that function, Now, solve the unknown side= a= 10x .588= 5.88 cm Logarithmisch-Trigonometrische Tafeln Mit Funf Dezimalstellen. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education Instructor's annotated exercises to accompany A graphical approach to college algebra & trigonometry. So there's an extra 2 that I have to put in here when I integrate it. And then did a simple trig integral, getting your help to get the sign right Trigonometry: A Complete Introduction: Teach Yourself. Whether you are looking for visually-stunning photographs for your next marketing campaign or eye-catching pictures for your website or product brochures, we�ve got what you need for very low prices. All images are supplied in the popular JPEG file format and are available in both lower resolutions (suitable for on-screen applications) and various higher resolutions (suitable for high-quality print applications) Treatise on Plane and Spherical Trigonometry. | 677.169 | 1 |
Algebra Review - Intermediate Algebra Intermediate Algebra...
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Unformatted text preview: Intermediate Algebra
Intermediate Algebra
by Gustafson and Frisk Chapter 1
A Review of Basic Algebra Section 1.1: The Real Number System
Section 1.1: The Real Number System SETS: collections of objects. Natural Numbers
Whole Numbers
Rational Numbers
Irrational Numbers
Real Numbers Use { } Integers
Positive Numbers
Negative Numbers
Even Numbers
Odd Numbers {x | x > 5}
{x | x > 5}
is read "the set of all x such that x is greater than 5" Section 1.1: The Real Number System
Section 1.1: The Real Number System GRAPHS: plot on the number line. Individual numbers are dots -3 -2 -1 0 1 2 3 4 Section 1.1: The Real Number System GRAPHS: plot on the number line. Intervals including end points [
-3 -2 -1 0 1 2 3 4 [ -3 -2 -1 0 1 2 3 4 Section 1.1: The Real Number System GRAPHS: plot on the number line. Intervals not including end points (
-3 -2 -1 0 1 2 3 4 ( ) -3 -2 -1 0 1 2 3 4 Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers OPERATIONS: Addition
Subtraction (the same as adding a number with the opposite sign)
Multiplication
Division (the same as multiplying by the reciprocal) Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers ADDITION: Addends that have the same signs Add absolute values Keep the sign of the addends
Addends that have opposite signs Subtract absolute values Keep the sign of the addend with the largest absolute value Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers MULTIPLICATION: Multiply absolute values
If the factors have the same signs, the product is positive
If the factors have opposite signs,
If
the product is negative
the Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers STATISTICS: measures of central tendency Mean Median Mode Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers Properties: Associative – addition, multiplication
Commutative – addition, multiplication Distributive – multiplication is distributed over addition
a (b + c) = ab + ac Section 1.2: Arithmetic & Properties of Real Numbers
Section 1.2: Arithmetic & Properties of Real Numbers Identities: Addition – zero
Multiplication – one Inverses: Addition – opposites
Multiplication – reciprocals Section 1.3: Definition of Exponents
Section 1.3: Definition of Exponents EXPONENTS: repeated multiplication
In the expression: an a is the base and n is the exponent Exponents are NOT factors Means to multiply "a" n times Section 1.3: Definition of Exponents
Section 1.3: Definition of Exponents ORDER OF OPERATIONS: If an algebraic expression has more than one operation, the following order applies:
1. Clear up any grouping.
2. Evaluate exponents.
3. Do multiplication and division from left to right.
4. Do addition and subtraction from left to right. Section 1.5: Solving Equations
Section 1.5: Solving Equations Algebraic Expression vs. Equation Expressions: a combination of numbers and operations
Equation: a statement that two
Equation:
expressions are equal
expressions Section 1.5: Solving Equations
Section 1.5: Solving Equations EXPRESSIONS: Terms
Like terms
When multiplying, the terms do not need to be alike
Can only add like terms! Section 1.5: Solving Equations
Section 1.5: Solving Equations TO SOLVE AN EQUATION IN ONE VARIABLE:
If you see fractions, multiply both sides by the LCD. This will eliminate the fractions.
Simplify the algebraic expressions on each side of the equal sign (eliminate parentheses and combine like terms).
Use the addition property of equality to isolate the variable terms from the constant terms on opposite sides of the equal sign.
Use the multiplication property to make the coefficient of the variable equal to one.
Check your results by evaluating. Section 1.5: Solving Equations
Section 1.5: Solving Equations TYPES OF EQUATIONS: CONDITIONAL: if x equals this, then y equals that.
IDENTITY: always true no matter what numbers you use.
CONTRADICTION: never true no matter what numbers you use.
FORMULAS: conditional equations that model a relationship between the variables. Section 1.6 & 1.7: Solving Problems, Applications
Section 1.6 & 1.7: Solving Problems, Applications TYPES OF PROBLEMS: Geometry
Percent
Physics (forces)
Uniform motion
Mixtures
Good 'ole common sense analysis Chapter 1: Basic Algebra Review
Chapter 1: Basic Algebra Review SUMMARY: KNOW YOUR VOCABULARY! You can't follow directions if you don't know what the words in the instructions mean.
Memorize the processes and the properties.
I will provide formulas for your reference.
Ask questions if you are unsure.
Always check your work to make sure that you answered the question, and that your answer is reasonable. This powerpoint was kindly donated to
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ResMore...
Res appreciate that the text has also long been recognized for its careful, thorough explanations and its presentation of mathematics in an informal yet mathematically precise manner. The authors also emphasize the all-important "why?" of mathematics--which is addressed in both the exposition and in the exercise sets by focusing on algebraic, graphical, and numerical perspectives.
Thomas W. Hungerford received his M.S. and Ph.D. from the University of Chicago. He has taught at the University of Washington and at Cleveland State University, and is now at St. Louis University. His research fields are algebra and mathematics education. He is the author of many notable books for undergraduate and graduate level courses. In addition to ABSTRACT ALGEBRA: AN INTRODUCTION, these include: ALGEBRA (Springer, Graduate Texts in Mathematics, #73. 1974); MATHEMATICS WITH APPLICATIONS, Tenth Edition (Pearson, 2011; with M. Lial and J. Holcomb); and CONTEMPORARY PRECALCULUS, Fifth Edition (Cengage, 2009; with D. Shaw | 677.169 | 1 |
521534291
ISBN: 0521534291
Edition: 2
Publication Date: 2003
Publisher: Cambridge Univ Pr
AUTHOR
Ablowitz, Mark J., Fokas, Athanassios S., Crighton, D. G.
SUMMARY
In addition to being mathematically elegent, complex variables provide a powerful tool for solving problems that are very difficult to solve in any other way. This book provides an introduction to complex variables and their applications.Ablowitz, Mark J. is the author of 'Complex Variables Introduction and Applications', published 2003 under ISBN 9780521534291 and ISBN 0521534 | 677.169 | 1 |
Fraleigh and Beauregard's text is known for its clear presentation and writing style, mathematical appropriateness, and overall usability. Its inclusion of calculus-related examples, true/false problems, section summaries, integrated applications, and coverage of Cn make it a superb text for the sophomore or junior-level linear algebra course. This Third Edition retains the features that have made it successful over the years, while addressing recent developments of how linear algebra is taught and learned. Key concepts are presented early on, with an emphasis on geometry. KEY TOPICS:Vectors, Matrices, and Linear Systems; Dimension, Rank, and Linear Transformations; Vector Spaces; Determinants; Eigenvalues and Eigenvectors; Orthogonality; Change of Basis; Eigenvalues: Further Applications and Computations; Complex Scalars; Solving Large Linear Systems MARKET: For all readers interested in linear algebra. | 677.169 | 1 |
This book equips undergraduates with the mathematical skills required for degree courses in economics, finance, management, and business studies. The fundamental ideas are described in the simplest mathematical terms, highlighting threads of common mathematical theory in the various topics. Coverage helps readers become confident and competent in the use of mathematical tools and techniques that can be applied to a range of problems.
Mathematics of Finance
To the student of pure mathematics the term mathematics of finance often seems somewhat of a misnomer since, in solving the problems usually presented in textbooks under this title, the types of mathematical operations involved are very few and very elementary. Indeed, in a first course in the mathematics of finance the development of the most important formulas usually involves no greater difficulties than those encountered in the study of geometric progressions.
"A wonderful display of the use of mathematical probability to derive a large set of results from a small set of assumptions. In summary, this is a well-written text that treats the key classical models of finance through an applied probability approach….It should serve as an excellent introduction for anyone studying the mathematics of the classical theory of finance." –SIAM
An Introduction to Mathematicsfor Economics (repost)
An Introduction to Mathematics for Economics introduces quantitative methods to students of economics and finance in a succinct and accessible style. The introductory nature of this textbook means a background in economics is not essential, as it aims to help students appreciate that learning mathematics is relevant to their overall understanding of the subject | 677.169 | 1 |
iTunes is the world's easiest way to organize and add to your digital media collection.
Hands-On Precalculus precalculus the fun way, with 60 dynamic video lessons and 30 hands-on interactives that let you explore the subject. In Hands-On Precalculus, you'll never go more than a minute without interacting with our Virtual Tutor. We don't do the proofs in this book, you do! So get ready to get schooled!
If you have any questions or comments, let us know at info@schoolyourself.org or visit
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Customer Reviews
Effective way to learn precalculus
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PsyOppa
I loved how precalculus was presented in this book. Everything is explained in a clear manner, and I found all the interactive elements fun to play with and great at teaching the concepts. The content is all very accessible and broken down into short, easy chunks. The hints in the book are also well-designed, and are a great addition when I got stuck on a problem; it really feels like having a familiar person explaining the concepts to me directly! I recommend this book to anyone who wants to learn precalculus in a fun way and I'm looking forward to their next book on calculus!
Unlike anything else out there
by
ggeometry
This is really not an ordinary book. It is very hands on and true to its title. It is really the first book I've seen that truly is interactive, and it actually makes me WANT to learn. It's amazing how they take boring abstract concepts and actually make them understandable and engaging!
Compact, encouraging, and effective
by
MichaelSZL
Keeps even the most diffident of users optimistic and cumulatively more confident about learning math. | 677.169 | 1 |
4
The issue: The Feynman lectures In the early 1960s, Richard Feynman taught a two-year physics course that was recorded for posterity as The Feynman Lectures (Feynman, Leighton, & Sands, 2013). The Feynman Lectures became regarded as classics Feynman was "a great teacher, perhaps the best of his era" (Goodstein & Neugebauer, 1995, p. xix) His lectures were widely praised for their clarity and explanatory value (Davies, 1995).
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Lectures in advanced mathematics: Common perceptions "The teaching of abstract algebra is a disaster and this remains true almost independently of the quality of the lectures. This is especially true for some excellent instructors whose lectures are truly masterpieces" (Leron & Dubinsky, p. 227).
8
Lectures in advanced mathematics: Common perceptions "The teaching of abstract algebra is a disaster and this remains true almost independently of the quality of the lectures. This is especially true for some excellent instructors whose lectures are truly masterpieces" (Leon & Dubinsky, p. 227).
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Lectures in advanced mathematics: Limited research base Speer, Smith, and Horvarth (2010) reported a literature review on studies of collegiate teaching –Research on the teaching of advanced mathematics was limited –Only one study (Weber, 2004) met standards of rigor, including observing instruction and seeking the professor's perspective on the instruction –No studies examined teachers' and students' perceptions of this lesson
14
Blame the mathematicians "A typical lecture in advanced mathematics… consists entirely of definition, theorem, proof, definition, theorem, proof, in solemn and unrelieved concatenation" (Davis & Hersh, p. 151). If examples are presented at all, it is "parenthetical and in brief" (p. 151). Teaching "almost exclusively the one very convenient and important aspect which has been described above, namely the polished formalism, which so often follows the sequence theorem- proof-application" (Dreyfus, 1991, p. 27).
15
Blame the mathematicians This dry limited form of instruction is attributed to poor behavior of mathematicians. Mathematicians are rewarded for research rather than teaching (Kline, 1977) Mathematicians are more interested in establishing truth than in providing explanation (Hersh, 1993) Mathematicians use effortless deduction to appear brilliant or to hide what they do not know (Davis & Hersh, 1991) Mathematicians are indifferent since they assume most students are not capable of learning the material (Leron & Dubinsky, 1995)
16
Blame the mathematicians Hersh (1991) distinguishes between The front of mathematics that the public sees. Formal precise definitions, theorems, and proofs The back of mathematics that is hidden from the public, including sketches of diagrams, guesses, visual arguments, idioms, etc. In short, mathematicians only show the front of mathematics, not the back, in their lectures, giving students in an inaccurate and harmful view of the discipline.
17
Blame the mathematicians These views have no empirical support. In fact, the few studies of mathematicians' teaching advanced mathematics have found that: Mathematicians regularly use diagrams and examples in their lectures (Fukawa-Connelly & Newton, 2014; Mills, 2014) Mathematicians say they are trying to provide methods and insights in their proofs. Conviction is secondary (Weber, 2012; Yopp, 2011) Mathematicians attempt to explain their practices to show how proofs can be written (Fukawa-Connelly, 2012) Mathematicians' instruction is based on coherent beliefs and a good deal of thought (Weber, 2004)
18
Blame the students "We assume that the problem is with the students rather than with communication: that the students either just don't have what it takes, or else just don't care" (Thurston, 1994, p. 166). "To understand syllogism is not something that you can learn; you are either born with the ability or you are not" (Halmos, 1970, p. 124).
19
Blame the students Several interview studies with mathematicians find they feel that some students are mathematically "tone deaf" and will never understand advanced mathematics (Harel & Sowder, 2009; Weber, 2012). M: Basically the class consists of two groups. There are groups that understand it, and probably hardly need it, and then there are those who really need it, and are not learning it. M: Everyone thinks that those things are somehow so obvious that anybody who is not a complete moron and has some ability to do mathematics should be able to see that. Therefore it is a waste of time to teach that because the people who need to be taught are the people who are hopeless anyhow. [... ] And I would have been vehemently opposed to that idea a few years ago. And now I'm less sure because I'm certainly trying to teach that and I can so maybe these people have a point (Alcock, 2010).
21
Blame the format Only trouble is, changing the format does not improve comprehension: E-proofs did no better than text and (not statistically significantly) worse than a lecture (Roy, Alcock, & Inglis, 2010) Similar results for structured proofs (Fuller et al, 2014) and generic proofs (Lew et al, 2012) "Changing the presentation in these ways requires substantial instructor effort, and given the underwhelming empirical results, this may not be effort well spent" (Hodds, Alcock, & Inglis, 2014).
22
Blame the format Maybe it's the lecture itself. No lectures can be effective. "Telling students about mathematical processes, objects and relations is not sufficient to induce mathematical learning. Hence the sorry state of affairs with the best of lecturers" (Leron & Dubinsky, 1995, p. 241) Many have proposed student-centered inquire instruction: In group theory (Leron & Dubinsky, 1995; Larsen, 2013) Real analysis (Swinyard & Larsen, 2012; Roh, 2010).
23
Blame the format Wu (1999) states that lectures are required to cover all the required content. Reform-oriented usually instruction covers less content: The cited abstract algebra instruction (Larsen, 2013; Leron & Dubinsky, 1995) does not reach the first isomorphism theorem Larsen and Swinyard (2012) had students spend ten hours "re- inventing" the definition of limit Most assessments deal with concept understanding, not the ability to write proofs This requires instructors to change their pedagogical beliefs and goals, which is difficult and some are not willing to do this (Johnson et al, 2013)
24
An alternative explanation: Playing different games Skemp (1976, 1978) wrote a classic article about instrumental (procedural) and relational (conceptual) understanding. "Let us imagine that school A sends a team to play school B at a game called 'football', but neither knows that there are two kinds (called 'association' [soccer] and 'rugby'). School A plays soccer and has never heard of rugger, and vice versa. Each team will rapidly decide the others are crazy, or a lot of foul players" "The problem here is one of mismatch… and does not depend on whether A or B's meaning is the right one" (Skemp, 1978, p )
25
An alternative explanation: Playing different games The idea that the instructor and students might perceive different goals (conceptual understanding vs. procedural fluency) and different expectations of students has had a profound influence on mathematics education. –Good inquiry-based instruction activities will not be implemented faithfully or meaningfully engage students if students have different beliefs about math instruction (e.g., Herbst, 2003; Herbst & Brach, 2006). Is there a difference between how mathematics professors and students perceive instruction in advanced mathematics? Can this account for students' failures to learn?
26
Different expectations: Some examples 28 math majors were interviewed. One interview question was what makes a good proof? (Weber, 2010, 2012). 16 of 28 students said a good proof contained all logical details and the reader shouldn't have to infer how new statements were deduced. –It's got to be really detailed. You have to tell every detail. Every step, it is very clear. I like doing things step by step. –It has to cover all the bases so that it is in fact a complete rigorous proof. For me, as a student, what else I would like to see are all the intermediate sorts of steps, things to help along, graphs, and visual things. Things that recalled facts that perhaps I should know but you know, maybe not immediately at the tip of my tongue. That's to me what makes a good mathematical argument.
27
28 Math majors:75% Mathematicians:27% * *- A Mann-Whitney test indicated the difference in math majors and mathematicians' responses differed significantly.
29
Different expectations: Some examples Mathematicians' rationale: –The goal of lectures is to provide higher-level ideas. The proof in the textbook can be used for reference (Lai & Weber, 2014). –Going over every detail is not feasible. It's time confusing and students would lose the big idea amid the logical details. –Students could benefit from filling in the gaps themselves (Lai, Mejia- Ramos, & Weber, 2012). Students' rationale: –Students are expected to justify everything, even trivially things, in geometry and transition-to-proof courses (cf., Herbst, 2002). –Students do not appreciate that the proofs they hand in have different epistemological purposes from the ones professors present.
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Different expectations: Some examples 175 math majors and 83 math professors were asked whether they agreed with the following statement on a survey using a five-point Likert scale. If an individual [a mathematics major] can say how each statement in a proof follows logically from previous statements, then that student understands this proof completely. Math majors:75% Mathematicians:23% * *- A Mann-Whitney test indicated the difference in math majors and mathematicians' responses differed significantly.
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Goal of the current study Case study– One professor (Dr. A) with 30 years experience and an excellent reputation as a real analysis instructor Three student pairs– rated collectively as above average by the professor One 11-minute proof that a sequence {x n } with the property that |x n – x n+1 |
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Goal of the current study The purpose is not to present sample-to-population generalization. –I do not wish to claim that all lecturers behave like Dr. A. –Indeed, Dr. A is chosen in part because he is atypical. He's an excellent teacher. The purpose is an analytic generalization –The goal is to develop theory about why high quality lectures might not convey understanding –This includes the creating of relevant constructs and distinctions –A fine-grained analysis of how understanding failed to occur –Hypotheses that could be tested in a larger study
37
Theoretical frames for the study Purposes of proving: Conviction– Students should have increased (or absolute) certainty that the claim is true. Explanation– Students can understand alternative ways of thinking about mathematical concepts that illustrate why the claim is true. Discovery- The students will perceive new methods that they can use to prove other theorems. Communicative– the norms for how a proof should be written and should appear. (adapted from deVilliers, 1990)
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Methods- Lecture analysis by professor Instructor was audiotaped in an interview on lecture. –First asked to describe why he presented this proof to students –Then asked to stop the video recording at every point he thought he was trying to convey mathematical content –We coded each content as explanation, method, or communication.
42
Methods- Lecture analysis by students Three student pairs were interviewed where we made four passes through the data. Pass 1: Students were asked to refer to their notes and state what they thought were the main ideas of the proof. Pass 2: Students watched the lecture again in its entirety, taking notes, and were asked the same question. Pass 3: Students were shown individual clips of the video and asked what they thought the professor was trying to convey. Pass 4: Students were told one thing that you might get from some proofs of this theorem was the content that Dr. A highlighted and asked if they got that from this proof.
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Results: Cauchy to establish convergence At three points in the proof, Dr. A emphasized if you want to show a sequence is convergent when you do not have a limit candidate, you can show it is Cauchy. Dr. A: How can we proceed to show that this is a convergent sequence? Anybody have a guess? Student: [Incomprehensible utterance] Dr. A: Well that's not quite the right term. What kind of sequences do we know converge even if we don't know what their limits are? [pause] It begins in 'c'. Student: Cauchy. Dr. A: Cauchy! We'll show it's a Cauchy sequence.
47
Results: Cauchy to establish convergence At three points in the proof, Dr. A emphasized if you want to show a sequence is convergent when you do not have a limit candidate, you can show it is Cauchy. We will show that this sequence converges by showing that it is a Cauchy sequence [writes this sentence on the board as he says it aloud, then turns around to face class]. A Cauchy sequence is defined without any mention of limit.
48
Results: Cauchy to establish convergence At three points in the proof, Dr. A emphasized if you want to show a sequence is convergent when you do not have a limit candidate, you can show it is Cauchy. And now we'll state what it is we have to show. We will show that there is an N- epsilon for which x_n minus x_m would be less than epsilon when m and n are greater than this number N-epsilon. [Dr. A writes this sentence on the board as he says it aloud] This is how we prove it is a Cauchy sequence. [Turns around and faces class]. See there is no mention of how the terms of the sequence are defined. There is no way in which we would be able to propose a limit L. So we have no way of proceeding except for showing that it is a Cauchy sequence or a contractive sequence. So let's look and see how we proceed.
49
Results: Cauchy to establish convergence Our research team highlighted this as the main point of presenting this proof. The other instructor described this as "the main objective". Dr. A highlighted these three points where he was trying to convey important content No student mentioned this in Pass 1 or Pass 2, but two groups noted this in Pass 3 (when shown particular clips) –S1: We should recognize it, figure out it's a Cauchy, we should know that it's converging, but it's limit is not necessarily given. So that we recognize it instantly S2: Because we have no way of figuring out what the limit is. All we have is them in relation to each other. Cauchy makes sense.
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Results: Cauchy to establish convergence Students did not record any of the professor's oral comments. –One student recorded nearly everything that the professor said. –Four students only wrote down what was on the blackboard. –One student did not take any notes. Unclear why students focused on what they did. –But one student noted that he didn't have time to write anything other than the student wrote at the board. Previous research shows students remember less than 5% of a lecture that they do not record. –Indeed, two groups of students were capable of processing the information, but did not note it when asked what the professor was trying to convey.
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Results: Expanding one's toolbox to work with inequalities Dr. A mentioned real analysis proofs as beginning with quantities that are assumed to be small and showing other quantities are small
56
Results: Expanding one's toolbox to work with inequalities Dr. A mentioned the importance of having things in your toolbox, which we interpreted as consisting of tools to keep something small. On the board: ≤r n (1+r+r 2 +…+r m-n ) Dr. A: Now we know this is small [circling r n ]. Now what can we say about this expression right here [circling (1+r+r 2 +…+r m-n ]? [pause] Anybody have a vague idea? I'll give you a hint. Calculus two... Student: Geometric series? Dr. A: … thirty or forty years ago? [gestures to student who spoke] Student: Geometric series. Dr. A: Geometric series! You have to always keep geometric series in your toolbox.
57
Results: Expanding one's toolbox to work with inequalities In his interview, Dr. A spoke of the importance of working with bounds. Dr. A: Once you get into the area where you're doing approximations, you can't do equal, equal, equal. You have to have bounds, bounds, bounds […] The objective is to show how bounds, using the triangle inequality, can be used to show that something is small using information that they're given is small. And this instance turns out that the information which is small is given in a form that allows us to use the geometric series as a bound.
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Results: Expanding one's toolbox to work with inequalities In Pass 3, students were shown this clip. No student mentioned the word "small" or a synonym when describing what Dr. A was trying to convey here. –"basically manipulating the information that we're given so that we can show that a sequence fits the definition" –"Given on the problem to see like what we could, how we can manipulate the problem statement. Just how we can start the proof in general"
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Results: Expanding one's toolbox to work with inequalities In Pass 4, each group was explicitly asked: "One last thing you might get from this proof is that mathematics students need to have a toolbox of ideas that help them to prove things are small. Is this something that you got from this presentation?" All six students answered yes, five students did not mention the word "small" or a synonym in their responses, referring to to other techniques in their toolbox. –"I think if he structures the way that he does, and you keep seeing it, it stays in your toolbox memory area […] not just in this specific proof itself, but it carries over to any other areas of math when you want to start to prove something"
62
Results: Expanding one's toolbox to work with inequalities The student who did mention the word small gave a revealing response. S5: We can use Mathematica, or like a tool to convert to make something small. I: So right so mathematics students need to have a toolbox of ideas to help them prove things are small. S5: Things are small. Oh you mean that they're not so complicated. When you say that things are small? I: No I mean like in terms of convergent sequences. Is that something that you think you got from this presentation? S5: I mean, in terms of simplifying them and deriving for approximating the answer, I think it's on the path, it's like it's working.
63
Results: Expanding one's toolbox to work with inequalities A plausible mathematician's perspective: Speaking strictly in terms of formal symbols is cumbersome and difficult. It's common to use mathematical idioms like toolbox and small to facilitate communication. "Small" is a loose term, but it does have a (semi-)precise meaning (e.g., it refers to magnitude, not sign, and it doesn't mean less than a particular value like x<10 -6 ). A plausible student's perspective: The work in these proofs involves simplification so what is needed is terms to simplify. As students are focused on the connections between local steps, overarching goals like keeping things small, are ignored. Words like "small" are insignificant hedges (Oehrtman, 2009)
64
Results: Metaphors In his interview, Dr. A stressed the importance of metaphors (although no metaphor appeared in the proof that we studied). Dr. A: And the whole objective is to get them to have in their mind a certain way of approaching problems. That's learning how to do mathematics, is learning structures to carrying out certain types of proofs. Having pictures and a structure for how they develop their understanding of the pictures. Because the proofs relate to the pictures. Let epsilon be greater than zero be given tells us that there's a neighborhood. The epsilon is the error which is the neighborhood in which the approximation will be. It can't be more than epsilon away from where the answer lies. So those are the things they've been seeing over and over again. Repetition, epsilon means error. And if you can make the error smaller than any epsilon, then you know that you have a sequence that's approximating something.
65
Results: Metaphors In pass 4, students' were asked: ""another thing that you might get from this proof is that the epsilon used is the error in approximations. Is that something that you got from this presentation?" All participants said yes, but none described the meaning of error or mentioned the word approximation. S3: Yeah. I think the first day or the second day, he explained what epsilon was so I think every time I see epsilon, I think about the error. S4: Yeah. I: So in this, so this proof is included in every time you see the epsilon? S3: Yeah. S4: [Nodding] Yeah, every time, every time I begin a proof I see let epsilon greater than zero be given. And we've had the same opening for that proof for, ever since we started dealing with like lower bounds and upper bounds.
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Summary We studied an excellent instructor who was clear (to us) in his communication and emphasized important math content. His students learned little of this content, even being asked directly after viewing the proof. We give several accounts for why this happened: –Students focused on the written work and seemed to ignore what was stated orally. –Students viewed the proof as doing algebraic manipulations rather than working to keep quantities small. Indeed, students did not know what the idiom "keeping quantities small" meant. –Students could repeat the metaphors that Dr. A used, but could not map them to a coherent conceptual structure where they could do real work.
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Recommendations Students need to be aware that what is said is as important as what is written in a lecture. Giving an informal argument is only useful if students have an accurate working definition of the semi-formal terms in the argument. –In many cases, ideas like "small" are quite sophisticated. –Students would need activities or instruction to build an understanding of this for these terms to aid comprehension. Students' comprehension will be limited if they view real analysis proofs in terms of manipulations. –Giving proof comprehension tests that highlight the main ideas of the proof is a start, both in emphasizing the importance of global understanding and conceptualizing what it means to understand a proof.
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Thank you Some papers discussed in this talk can be found at: pcrg.gse.rutgers.edu | 677.169 | 1 |
This course is for eighthPre-Algebra This course is for 8th grade level students. A variety of topics (including readings in mathematics, computers and the Internet, and projects for the enrichment of mathematics education) is incorporated.Placement into this class is based upon teacher recommendation and performance on a general mathematics test, which includes basic skills, concepts, and applications.Topics include:
sEvaluating algebraic expressions
sLinear equations and inequalities
sThe real number system
sPercent
sGeometry and measurement
sProbability and statistics including permutations and combination
Pre-Algebra Advanced
This course is designed for sixth and seventh grade students who are advanced in mathematics.It serves as a alternative to the traditional Pre-Algebra course.Instruction will include an overview of topics that are nor4mally covered in Math I and II.Instruction will occur at a pace suitable for advanced learners with high level of rigor.
Algebra I
This class parallels the high school algebra course.It is designed for mathematic students who are capable of thinking at an abstract level.Standard topics from algebra are taught but with an emphasis on algebraic thinking, reasoning, and application. Topic include:
sLinear equations and inequalities
sAbsolute value equations
sConjunctions and disjunctions
sSolving systems of linear equations
sPolynomials
sOperations on rational expressions and equations
sQuadratic functions and the real number system
sTopics and applications from probability and statistics
Geometry
Geometry is the mathematics of the properties, measurement, and relationships of points, lines, angles, surfaces, and solids. The topics that are studied include lines and angles, triangle congruence, quadrilaterals, lines and planes, similar polygons, use of the Pythagorean Theorem, circle and their parts, area, surface area, volume, coordinate geometry, and constructions.
Problem solving strategies, mathematical reasoning, and communications skills both written and verbal, will be used on a regular basis. Students will develop their conceptual understandings of the listed topics through proof, transformations, and constructions | 677.169 | 1 |
Numerical and Analytical Methods for Scientists and Engineers Using Mathematica is written from the perspective of a physicist, not a mathematician, emphasising modern practical applications in the physical and engineering sciences. The book itself is essentially software, written in the language of Mathematica, whichMore...
This work is written from the perspective of a physicist, not a mathematician, emphasising modern practical applications in the physical and engineering sciences. The book itself is essentially software, written in the language of Mathematica, which is widely used in engineering and physics.
Preface
Ordinary Differential Equations in the Physical Sciences
Introduction
Definitions
Exercises for Sec. 1.1
Graphical Solution of Initial-Value Problems
Direction Fields; Existence and Uniqueness of Solutions
Direction Fields for Second-Order ODEs: Phase-Space Portraits
Exercises for Sec. 1.2
Analytic Solution of Initial-Value Problems via DSolve
DSolve
Exercises for Sec. 1.3
Numerical Solution of Initial-Value Problems
NDSolve
Error in Chaotic Systems
Euler's Method
The Predictor-Corrector Method of Order 2
Euler's Method for Systems of ODEs
The Numerical N-Body Problem: An Introduction to Molecular Dynamics
Exercises for Sec. 1.4
Boundary-Value Problems
Introduction
Numerical Solution of Boundary-Value Problems: The Shooting Method
Exercises for Sec. 1.5
Linear ODEs
The Principle of Superposition
The General Solution to the Homogeneous Equation
Linear Differential Operators and Linear Algebra
Inhomogeneous Linear ODEs
Exercises for Sec. 1.6
References
Fourier Series and Transforms
Fourier Representation of Periodic Functions
Introduction
Fourier Coefficients and Orthogonality Relations
Triangle Wave
Square Wave
Uniform and Nonuniform Convergence
Gibbs Phenomenon for the Square Wave
Exponential Notation for Fourier Series
Response of a Damped Oscillator to Periodic Forcing
Fourier Analysis, Sound, and Hearing
Exercises for Sec. 2.1
Fourier Representation of Functions Defined on a Finite Interval
Periodic Extension of a Function
Even Periodic Extension
Odd Periodic Extension
Solution of Boundary-Value Problems Using Fourier Series
Exercises for Sec. 2.2
Fourier Transforms
Fourier Representation of Functions on the Real Line
Fourier sine and cosine Transforms
Some Properties of Fourier Transforms
The Dirac [delta]-Function
Fast Fourier Transforms
Response of a Damped Oscillator to General Forcing. Green's Function for the Oscillator | 677.169 | 1 |
GMAT Math Formulas
Math Formulas You Cannot do Without
In the GMAT Quantitative Section, you will have to solve 37
multiple-choice questions in 75 minutes in a computer adaptive test.
The questions are of two different types: Problem Solving and Data
Interpretation. For success in this section, knowledge of math formulas is crucial.
What are Math Formulas?
Formulas are no different from
other math formulas that one comes across. The only difference that
comes forth is the range of formulas. If you pick up any test prep
book, you'll find the initial few chapters devoted to explaining the
syllabus of the exam. It is from this syllabus that we draw our ideas of studying for the test. There are some concepts that are not asked in the test
while they are given importance in school education and vice-versa.
Hence, prudence lies is going though the syllabus to decide what
formulas classify as important.
Where to get Formulae
One of your initial sources of studying is the GMATPrep Software itself. There is no other free source from the makers of the test,
which can supply you more information than this software can for
preparation. Browse through the subsections, which are Arithmetic,
Algebra, Geometry and Word Problems of the Math Review in the software
to understand the range of topics covered in the exam. The corresponding descriptions shall give you an idea of formulae
that you need to collect. The Official Guide for GMAT Quantitative
Review should be your next bet if you are ready to make a purchase. Of
course, there are other preparation books and software for which you
don't have to spend a dime, but there is no guarantee that they furnish
you with absolute content; they could emphasise on unnecessary details
to make their content seem more competitive and wholesome.
Why are Formulae Important?
The concept of accurate formulas is unique to calculations in
mathematics. These formulas remain unchanged through time, as they do
not depend on changes in society, environment, growth, loss or other
such variables. It is this quality of math formulas that makes them
applicable to a variety of problems. Math formulas for one topic can be
applied to different problems from a variety of topics. Hence, each formula usage gains more importance than is obvious.
Strange as it may sound, each problem in math can be solved purely
on the basis of concepts without the application of any formula.
However, formulas are like tools that help you arrive at the correct
answer without having to go back to the basics. They make calculations easier and you save on time.
How to utilize Math Formulas
For making the best use of formulas, you should follow the following guidelines.
Make a list of all the formulas in one place. There should be one long list in which you categorize the formulas as per the topics.
Memorize them while keeping in mind what the short forms stand for. After memorizing them, practice math problems side-by-side, instead of learning all at once before and applying them later to problems.
The real challenge does not lie in memorizing formulas but in understanding where they are to be applied. This is particularly significant in the word problems. At times two or more formulas need to be applied to arrive at a solution.
Remember that there shall be some in
your list that are not used often. Nonetheless, you should learn such
formulas as well. At times knowledge and application of such concepts are decisive in competition.
At times it is not required that all the forms of the fundamentals
be learnt. For instance, you need not learn how to calculate the speed
when distance and time are given if you are aware of the formula for
calculating the distance when speed and time are given. You can work
around one formula to obtain its other forms. | 677.169 | 1 |
Course Syllabus 2009 2010
1.
Name: ____________________ August 31, 2009
Honors Algebra 2 – Smith Period 6
HONORS ALGEBRA 2, 2009–2010
Course Description:
This course gives students a deeper understanding of the structure of Algebra. It is designed to
expand the students' mathematical ability in applying algebraic concepts and skills in the solving
of problems. This course includes a review of concepts from Algebra 1 and explores new
concepts, such as polynomial and logarithmic functions, matrices, and the binomial theorem.
Course Goals:
1. To provide the student with a deeper understanding of the mathematical content,
concepts, and connections between algebra and geometry.
2. To provide the student with the ability to determine algebraic solutions in various content
areas and apply these solutions to real-world scenarios.
Required Materials:
All students are required to have a textbook (McDougal Litell), two folders, pencils, and loose-
leaf paper on a daily basis. Calculators are optional, although recommended.
Grading:
A preliminary breakdown of percentages for each quarter is:
Homework – 25 % Notebook/Participation – 20 %
Tests/Projects – 35 % Quizzes – 20 %
You will be responsible for keeping track of your grades throughout each quarter.
Guidelines for acceptable work:
1. Homework will be given on a daily basis. All homework MUST be done on loose-leaf
paper. Grading will be based upon completeness, organization, and neatness.
2. Show all work – get used to doing this. Answers alone are insufficient. Partial credit
will be given on tests and quizzes for work.
3. Each assignment should start on a new sheet of paper. Consecutive problems should be
numbered top-to-bottom on a page. You are encouraged to use both sides of the paper.
Procedures for absences:
1. Any work not turned in on time because of an excused absence will be due on the day of
the student's return.
2. Students are responsible for all assignments and notes given during absences. Ask
classmates for missed work; in addition, notes will be available online.
3. Tests that are missed MUST be made up the day that you return.
Further assistance:
I will be available for extra help outside of class from 7:00 to 7:30 and 3:30 to 4:30.
Contact information:
e-mail: smithr@dbcr.org phone: (301) 891-4750 ext. 132 website: edline.net | 677.169 | 1 |
COURSE GOALS: To teach basic numerical methods required for typical engineering and business applications. Give students experience in understanding the properties of different numerical methods so as to be able to choose appropriate methods and interpret the results for engineering problems that they might encounter. Students will implement and study some of the numerical methods using C++, C, FORTRAN, MATLAB or some other high-level language. Emphasis is given to the graphical representation of results.
COMPUTER USAGE: There are five programming assignments. Students can use a PC or a workstation and are required to master a program such as MATLAB OR MATHEMATICA that will allow them to produce graphical representations of their results.
LABORATORY PROJECTS: None
GRADES:
Homework - 55 %
Midterm - 20 %
Final - 25 %
COURSE OBJECTIVES: When a student completes this course, s/he should be able to:
• Understand the type of numerical problem he/she is facing and relate it to one of the problem classes discussed in the course.
• Find a numerical routine that will solve the engineering problem, or use one of the numerical tools such as MATLAB or MATHEMATICA.
• Design a driver that will use a number of numerical routines to perform the desired task. | 677.169 | 1 |
ISBN: 9781107564619 Format: Book with Other Items Published: 6 August 2015 Country of Publication: GB Dimensions (cm): 25.8 x 19.2 x 2.3 Description: CambridgeMATHS GOLD NSW Syllabus for the Australian Curriculum is a complete teaching and learning package for students who may need additional support studying mathematics in Years 7 and 8.
The series provides an accessible approach to the NSW Syllabus and leads students from Stage 3 through Stage 4, helping to develop the knowledge and skills needed to succeed in Stage 4 mathematics while preparing them for Stage 5.
Written by a highly successful team of expert authors, CambridgeMATHS GOLD: Applies the same logical structure as the CambridgeMATHS series with a bright, friendly and uncluttered visual design. Groups carefully-graded questions according to the Working Mathematically components of the NSW Syllabus, covering both Stage 3 and Stage 4 content. Activities designed to improve automaticity, fluency and understanding through hands-on resources, games, competitions, puzzles, investigation activities and sets of closed questions. These are available on Cambridge GO and are fully-integrated with each section of the textbook. < | 677.169 | 1 |
Mathematica
Explore this collection of resources for middle and high school mathematics. Mathematica offers detailed, imaginative visualizations of dozens of advanced math concepts and principles. The collection includes lessons on geometry, algebra, statistics, and more! | 677.169 | 1 |
Cirrito Fabio Editor., Tobin 1. Mathematics, 2. International Baccalaureate. Series Title: International Baccalaureate in Detail ISBN: 1 876659 17 3 (10-digit) 978 1 876659 17 2 (13-digit) All rights reserved except under the conditions described in the Copyright Act 1968 of Australia and subsequent amendments. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior permission of the publishers. While every care has been taken to trace and acknowledge copyright, the publishers tender their apologies for any accidental infringement where copyright has proved untraceable. They would be pleased to come to a suitable arrangement with the rightful owner in each case. This book has been developed independently of the International Baccalaureate Organisation (IBO). The text is in no way connected with, or endorsed by, the IBO. "This publication is independently produced for use by teachers and students. Although references have been reproduced with permission of the VCAA the publication is in no way connected or endorsed by the VCAA." We also wish to thank the Incorporated Association of Registered Teachers of Victoria for granting us permission to reproduced questions from their October Examination Papers. Cover design by Adcore Creative. Published by IBID Press, at For further information contact fabio@ibid.com.au Printed by SHANNON Books, Australia.
ii
PREFACE TO 3RD EDITION The 3rd edition of the Mathematics Standard Level text has been completely revised and updated. Sections of the previous two editions are still present, but much has happened to improve the text both in content and accuracy. In response to the many requests and suggestions from teachers worldwide the text was extensively revised. There are more examples for students to refer to when learning the subject matter for the first time or during their revision period. There is an abundance of well-graded exercises for students to hone their skills. Of course, it is not expected that any one student work through every question in this text - such a task would be quite a feat. It is hoped then that teachers will guide the students as to which questions to attempt. The questions serve to develop routine skills, reinforce concepts introduced in the topic and develop skills in making appropriate use of the graphics calculator. The text has been written in a conversational style so that students will find that they are not simply making reference to an encyclopedia filled with mathematical facts, but rather find that they are in some way participating in or listening in on a discussion of the subject matter. Throughout the text the subject matter is presented using graphical, numerical, algebraic and verbal means whenever appropriate. Classical approaches have been judiciously combined with modern approaches reflecting new technology - in particular the use of the graphics calculator. The book has been specifically written to meet the demands of the Mathematics Standard Level course and has been pitched at a level that is appropriate for students studying this subject. The book presents an extensive coverage of the syllabus and in some areas goes beyond what is required of the student. Again, this is for the teacher to decide how best to use these sections. Sets of revision exercises are included in the text. Many of the questions in these sets have been aimed at a level that is on par with whatFISH PRODUCT | 1981 | 1982 | 1983 | 1984 | 1985 | 1986 | 1987 | 1988 |
Total Mass...
...Cindy Hwang
IB Math 11 SL 1-3
IB Math Portfolio
Gold medal heights
Aim:
The aim of this task is to consider the winning height for the men's high jump in the Olympic Games.
Introduction:
The Olympic Games which are held in every four years have an event called Men's High Jump and usually an athlete tries to jump over a bar which is set up in a certain range from1 meter to 3 meters. The table 1 shows the record of the gold medalists during the...
...Winning Heights in the Men's Olympic High Jumps
Introduction:
The Olympics are an international sporting event that is held every four years where people from around the world send their best athletes to compete and see who the best of the best is when it comes to sports. The Olympics date back to Ancient Greece where their basic events included track and field, which are like men's high jump which is the topic of this report. In this problem we are looking at the data collected from the...
...Jonghyun Choe
March 25 2011
Math IB | 677.169 | 1 |
ISBN 13: 9780030152023
Beginning Algebra
EXCEPTIONALLY CLEAR AND ACCESSIBLE, THIS BEST-SELLING INTRODUCTORY TEXT/WORKBOOK FOR ELEMENTARY ALGEBRA IS APPROPRIATE FOR LECTURE COURSES, LEARNING CENTERS, LABS AND SELF-PACED COURSES. WRITTEN IN A CLEAR AND CONCISE STYLE, THIS BOOK OFFERS ALL THE REVIEW, DRILL AND PRACTICE STUDENTS NEED TO DEVELOP PROFICIENCY IN ALGEBRA. IN A LECTURE-FORMAT CLASS, EACH SECTION OF THE BOOK CAN BE DISCUSSED IN A FORTY-FIVE- TO FIFTY-MINUTE CLASS SESSION. IN A SELF-PACED SITUATION, THE "PRACTICE PROBLEMS" IN THE MARGINS THE STUDENT TO BECOME ACTIVELY INVOLVED WITH THE MATERIAL BEFORE WORKING THE PROBLEMS IN THE "PROBLEM SET | 677.169 | 1 |
Ocean Tides: Mathematical Models and Numerical Experiments
Ocean Tides: Mathematical Models and Numerical Experiments discusses the mathematical concepts involved in understanding the behavior of oceanic tides. The book utilizes mathematical models and equations to interpret physical peculiarities of tidal generation. The text first presents the essential information on the theory of tide, and then proceeds to tackling the studies on the equations of tidal dynamics. Next, the book covers the numerical methods for the solution of the equations of tidal dynamics. Chapter 4 deals with the tides in the World Ocean, while Chapter 5 talks about the bottom boundary layer in tidal flows. The last chapter tackles the vertical structure of internal tidal waves. The book will be of great interest to individuals whose profession involves the direct interaction with tides, such as mariners, marine biologists, and oceanographers | 677.169 | 1 |
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Adaptive Curriculum Launches Windows 8 Apps for Math and Science
By Tim Sohn
01/02/13
Adaptive Curriculum has launched the first four in a series of Windows 8 apps that provide interactive lessons in math and science for middle and high school students. These AC VBook Apps focus on algebra, geometry, and physics.
The apps are based on Adaptive Curriculum's AC Math and AC Science programs. Adaptive Curriculum, which is designed for grades 6-12, provides hundreds of activity objects, such as interactive simulations, graphics, and 3D models. They are aligned to textbook, Common Core, and state standards, and include assessments and printable activity sheets at the end of each activity. Lessons can be presented on interactive whiteboards, and assigned to either groups or individual students.
The physics and algebra AC VBooks can be purchased for $1.49, and the geometry VBook is free. According to Adaptive Curriculum, next it will release VBook Apps for Algebra I & II, geometry, statistics, biology, chemistry, physics, and Earth science | 677.169 | 1 |
Mathematics for Elementary Teachers: A Contemporary Approach
ISBN :
9780470531341
Publisher :
John Wiley & Sons Ltd
Author(s) :
Gary L Musser
Publication Date :
1 Jan 2010
Edition :
9
Overview
When students truly understand the mathematical concepts, it's magic. Students This edition can also be accompanied with WileyPlus, an online teaching and learning environment that integrates the entire digital textbook with the most effective resources to fit every learning style.WileyPLUS is sold separately from the text. | 677.169 | 1 |
This review volume consists of an indispensable set of chapters written by leading scholars, scientists and researchers in the field of Randomness, including related subfields specially but not limited to the strong developed connections to the Computability and Recursion Theory. Highly respected, indeed renowned in their areas of specialization, many... more...
This text is a rigorous introduction to Ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. It describes some recent applications to number theory, and goes beyond the standard texts in this topic. more...
Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books provide a global view of algebra and its role in mathematics. more...
Mathematical modelling modules feature in most university undergraduate mathematics courses. As one of the fastest growing areas of the curriculum it represents the current trend in teaching the more complex areas of mathematics. This book introduces mathematical modelling to the new style of undergraduate - those with less prior knowledge, who require... more... | 677.169 | 1 |
With a strong emphasis on skill-building, applications in the real world, and preparation for further math courses, this text unites the concepts of Elementary Algebra and Intermediate Algebra in one convenient and economical | 677.169 | 1 |
College Algebra and Trigonometry: Graphs and Models with Olc Bi-Card
Mathematical reform is the driving force behind the organization and development of this new text in college algebra and trigonometry. The use of technology, primarily graphing utilities, is assumed throughout the text. The development of each topic proceeds from the concrete to the abstract and takes full advantage of technology, wherever appropriate. The first major objective of this book is to encourage students to investigate mathematical ideas and processes graphically and numerically, as well as algebraically. Proceeding in this way, students gain a broader, deeper, and more useful understanding of a concept or process. Even though concept development and technology are emphasized, manipulative skills are not ignored, and plenty of opportunities to practice basic skills are present. A brief look at the table of contents will reveal the importance of the function concept as a unifying theme. The second major objective of this book is the development of a library of elementary functions, including their important properties and uses. Having this library of elementary functions as a basic working tool in their mathematical tool boxes, students will be able to move into calculus with greater confidence and understanding. In addition, a concise review of basic algebraic concepts is included in Appendix A for easy reference, or systematic review. The third major objective of this book is to give the student substantial experience in solving and modeling real world problems. Enough applications are included to convince even the most skeptical student that mathematics is really useful. Most of the applications are simplified versions of actual real-world problems taken from professional journals and professional books. No specialized experience is required to solve any of the applications.
Book Description McGraw-Hill Science/Engineering/Math. Hardcover. Book Condition: New. 0072916990BMATH080 | 677.169 | 1 |
Linear Algebra and Linear Models comprises a concise and rigorous introduction to linear algebra required for statistics followed by the basic aspects of the theory of linear estimation and hypothesis testing. The emphasis is on the approach using generalized inverses. Topics such as the multivariate normal distribution and distribution of quadratic forms are included. For this third edition, the material has been reorganised to develop the linear algebra in the first six chapters, to serve as a first course on linear algebra that is especially suitable for students of statistics or for those looking for a matrix theoretic approach to the subject. Other key features include: coverage of topics such as rank additivity, inequalities for eigenvalues and singular values; a new chapter on linear mixed models; over seventy additional problems on rank: the matrix rank is an important and rich topic with connections to many aspects of linear algebra such as generalized inverses, idempotent matrices and partitioned matrices. This text is aimed primarily at advanced undergraduate and first-year graduate students taking courses in linear algebra, linear models, multivariate analysis and design of experiments. A wealth of exercises, complete with hints and solutions, help to consolidate understanding. Researchers in mathematics and statistics will also find the book a useful source of results and problems.
This new edition illustrates the power of linear algebra in the study of graphs. The emphasis on matrix techniques is greater than in other texts on algebraic graph theory. Important matrices associated with graphs (for example, incidence, adjacency and Laplacian matrices) are treated in detail. Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph. Such an extensive coverage of the subject area provides a welcome prompt for further exploration. The inclusion of exercises enables practical learning throughout the book. In the new edition, a new chapter is added on the line graph of a tree, while some results in Chapter 6 on Perron-Frobenius theory are reorganized. Whilst this book will be invaluable to students and researchers in graph theory and combinatorial matrix theory, it will also benefit readers in the sciences and engineering.
This book studies algebras and linear transformations acting on finite-dimensional vector spaces over arbitrary fields. It is written for readers who have prior knowledge of algebra and linear algebra. The goal is to present a balance of theory and example in order for readers to gain a firm understanding of the basic theory of finite-dimensional algebras and to provide a foundation for subsequent advanced study in a number of areas of mathematics.
This book, translated from the French, is an introduction to first-order model theory. The first six chapters are very basic: starting from scratch, they quickly reach the essential, namely, the back-and-forth method and compactness, which are illustrated with examples taken from algebra. The next chapter introduces logic via the study of the models of arithmetic, and the following is a combinatorial tool-box preparing for the chapters on saturated and prime models. The last ten chapters form a rather complete but nevertheless accessible exposition of stability theory, which is the core of the subject.
A self-contained introduction to matrix analysis theory and applications in the field of statistics Comprehensive in scope, Matrix Algebra for Linear Models offers a succinct summary of matrix theory and its related applications to statistics, especially linear models. The book provides a unified presentation of the mathematical properties and statistical applications of matrices in order to define and manipulate data. Written for theoretical and applied statisticians, the book utilizes multiple numerical examples to illustrate key ideas, methods, and techniques crucial to understanding matrix algebra's application in linear models. Matrix Algebra for Linear Models expertly balances concepts and methods allowing for a side-by-side presentation of matrix theory and its linear model applications. Including concise summaries on each topic, the book also features: Methods of deriving results from the properties of eigenvalues and the singular value decomposition Solutions to matrix optimization problems for obtaining more efficient biased estimators for parameters in linear regression models A section on the generalized singular value decomposition Multiple chapter exercises with selected answers to enhance understanding of the presented material Matrix Algebra for Linear Models is an ideal textbook for advanced undergraduate and graduate-level courses on statistics, matrices, and linear algebra. The book is also an excellent reference for statisticians, engineers, economists, and readers interested in the linear statistical model.
After a brief introduction reviewing the concepts of principal ideal domains and commutative fields, this book discusses the theory of algebraic numbers. The theorems and definitions are carefully motivated, and the author frequently stops to explain how things fit together and what will come next. Many exercises and useful examples provide a solid background for more recent topics.
Simply put, quantum calculus is ordinary calculus without taking limits. This undergraduate text develops two types of quantum calculi, the q-calculus and the h-calculus. As this book develops quantum calculus along the lines of traditional calculus, the reader discovers, with a remarkable inevitability, many important notions and results of classical mathematics. This book is written at the level of a first course in calculus and linear algebra and is aimed at undergraduate and beginning graduate students in mathematics, computer science, and physics. It is based on lectures and seminars given by Professor Kac over the last few years at MIT. Victor Kac is Professor of Mathematics at MIT. He is an author of 4 books and over a hundred research papers. He was awarded the Wigner Medal for his work on Kac-Moody algebras that has numerous applications to mathematics and theoretical physics. He is a honorary member of the Moscow Mathematical Society. Pokman Cheung graduated from MIT in 2001 after three years of undergraduate studies. He is presently a graduate student at Stanford University. | 677.169 | 1 |
Über diesen Titel:
Inhaltsangabe: This innovative textbook leads students on a carefully guided discovery of introductory number theory. The book has two equally significant goals. The first is to help students develop mathematical thinking skills, particularly theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for independent study, or for a course designed as an introduction to abstract mathematics. It is designed to be used with an instructional technique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). Instructors' materials explain the instructional method, which gives students a totally different experience compared to a standard lecture course. Students develop an attitude of personal reliance and a sense that they can think effectively about difficult problems; goals that are fundamental to the educational enterprise within and beyond mathematics.
Inhaltsangabe:
This innovative textbook leads students on a carefully guided discovery of introductory number theory. The book is designed to develop students' mathematical thinking skills, particularly theorem-proving skills, whilst helping them understand some of the wonderfully rich ideas in the mathematical study of numbers | 677.169 | 1 |
Showing 1 to 30 of 59
33130 Mathematical Modelling 1
Autumn 2015 Lecture Week 6
_
Note: Determinants and Cramers rule (p12) may be useful first
APPLICATIONS OF VECTORS: FORCES
In the following diagram, the forces that act at the point P are in equilibrium.
Find the sizes of th
33130 Mathematical Modelling 1 Autumn 2015 Lecture Week 3
INTEGRATION AND APPLICATIONS
First we recall the idea of Primitive Functions.
Definition: let f be a given function. A function F such that F = f is called a primitive, or an indenite integral o T1
Tutorial 1: Differentiation and
applications
This tutorial covers advanced differentiation and its
applications for modelling, including related rates and
Newtons method.
This version has no spaces for writing your answers
use your own paper. Yo Mathematical Modelling 1 Spring 2014
Lecture Week 5 _
3D Coordinate-Systems, Vectors, Vector Algebra-
7L ,
We are used to the Cartesian plane in two dimensions, with x
and y axes. In three dimensions there are two possibilities for
locatin
33130 Mathematical Modelling 1 AUTUMN 2015
Lecture Week 5
3D Co-ordinate Systems, Vectors, Vector Algebra
We are used to the Cartesian plane in two dimensions, with x
and y axes. In three dimensions there are two possibilities for
locating the third axis,
33130 Mathematical Modelling 1
Autumn 2015
Lecture Week 3
INTEGRATION AND APPLICATIONS
First we recall the idea of Primitive Functions.
Definition: let f be a given function. A function F such that F = f is called a primitive, or an indefinite integral of
33130 Mathematical Modelling 1
Autumn 2015
Lecture week 4
METHODS OF INTEGRATION (Integration techniques)
With the availability of Computer Algebra Systems such as Mathematica, Wolfram Alpha, Derive, Matlab and so on, the importance
of being able to find
33130 Mathematical Modelling 1
Autumn 2015 Lecture Week02
Functions and their derivatives (continued)
-Introduction to inverse functions
Recall the definition of a function:
Let A be a set of numbers. A function f with domain A is a rule or computational
33130 Mathematical Modelling 1
Autumn 2015
Lecture Week 01
What is mathematical modelling?
Most standard definitions of Mathematical Modelling describe it as a process of making a mathematical model to
represent real world situations. These situations mig
33130 Mathematical Modelling 1 Lecture Week 9
Second order Linear Differential Equations
Solving second order linear DE 5, with constant coefficients
A certain kind of second order DE occurs in mechanical, structural, and electrical systems. These have | 677.169 | 1 |
Are the mathematics books listed in the guide in the sticky up to date ( I mean, obviously the content probably doesn't change much, but books that teach that content more effectively might've been released.
I took some baby calculus courses in college, but I really want to self study and get deeper into it. Not sure what books I should go with.
This is fantastic. You'd probably want to be familiar with algebraic geometry and homotopy theory before hand, but it goes over a lot of the prerequisites in the first half of the book. A lot of it is available online on the various authors websites.
What's a good calculus book that can supplement Stewart? That is the book we are using in class, and we HAVE to do exercises out of that book and turn them in, but what is something I can use alongside it since it seems like Stewart's isn't that great
>>7840308 Stewart's is fine, you will find few books with such quality graphics, it's rigorous enough for an intro calculus text (moreso than most alternatives) which in itself is not a rigorous field and will be useful as a reference. It's a bit long which is why people complain (and because it's more popular than their favourite textbook). People
>>7840326 >People How much retardation can you cram into one post.
>>7840308 First off, you should only be looking for a supplement text if you want a rigorous intro to the subject. If you're only interested in the cookbook-style (i.e. you don't want proofs of everything you read) stuff just watch some youtube videos.
That said, if you do want a rigorous supplement take a look at vol.1 of Courant if you're a physics student and Spivak if you're a maths student.
>>7841677 A bit more basic than lang, but i reaaaaly like Knapp's presentation in his "Basic Algebra" and "advanced algebra." Anyone else have opinions on these?
Speaking of favorite books, I think that all of Milnor's books are great. Bott and Tu is a masterpiece. A lot of other books I find are really useful for specific areas of research/reference, but few that I really love like the above. Maybe Demailly' notes and Griffiths-Harris (find the errata!) are up there. Oh yeah, and anything by Eli Stein (his princeton lecture series are perfect for advanced undergrads...helps you really 'get' what analysis is all about, and his harmonic analysis texts are bibles, don't know TOO much in them though). I also love any well written book on distributions because they're beautiful.
Any tips on how to learn about geometric representation theory? I have basic graduate knowledge of subjects, including basic representation theory and lie theory and in depth knowledge of complex geometry --analytic techniques (Demailly style) and algebraic geometry, but from the more geometric perspective :(
I just really want to read a bit about geometric representation theory because it seems sexy. Book suggestions?
>>7847299 Could you elaborate a little on what you think of Knapp's books?
Is Basic Algebra too terse for a reasonably sophisticated undergrad to use as a first look? (I have a theoretical linear algebra course and a decent amount of analysis behind me)
I've seen a couple of posts saying good things about them but some of the details of the book leave me a bit unsure (categories pop up and fairly early, although I'm not sure how much they're relied on, and the pace in chapter 4 on groups seems absolutely blazing compared to other texts).
>>7847318 Yeah, I think it can be a little hard as a first text. It's not that the material he presents assumes you know a lot already, but his pacing in some places does (as you noticed). It goes through some details a tad quickly for someone who hasn't seen them before. For gentler introductions that aren't "easy" (like Galiian I think is way too easy), I only know of artin and dummit and foote but I don't know many others. Artin has a very unique style that some people dislike (I like it :/). Dummitt and Foote is very detailed and comprehensive yet approachable (does not introduce stuff from categorical perspective). These qualities make it a favorite for a lot but it can get kind of dull. Nice thing is if you're self studying from D&F there's solutions to many of the exercises online. I don't really know of what the "best" introduction is. A lot of people here seem to like Pinter. I've never looked at it, though.
>>7847415 welcome to 2016: >Note: A second printing of the book is projected for the end of ... February, and will incorporate the errata listed here but include no other major changes.
>>7847340 I'm going through Pinter and it gets quite tedious. The theory is well explained and motivated, but usually there's about 5-8 pages per chapter and then there's 30+ exercises, and a lot of them are somewhat trivial, so it gets annoying to do all of them or knowing which of them are harder.
>>7848138 Just start programming. If you are a beginner I assume that you already know all about C++'s rules. Now you have to apply them. First go read some sample code online to see the proper styles people use. Then just #include "windows.h" and try to make something useful.
>>7848173 Any idea of the depth of a project you would like to see in order to hire a junior cpp programmer ?like a parsing engine how deep would you need it to be? A raytracer ? I have a molecular biology degree tho so how to convince you I'm not shite
>>7848203 #1: Something that utilizes the standard library a lot, to show you know your away around C++ #2: Something that uses external libraries that most companies will be using. Again, windows.h is the best option for this #3: Something that connects in one way or another to the internet. Just make a SQL database and have your program retrieve and/or send data to it.
What that could be? Many things, really. A good place to start thinking is your own degree. But if you are stuck with creativity there, then just google intermediate/advanced C++ projects and do | 677.169 | 1 |
9780321228192
032122819770
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Summary
Most students taking this course do so to fulfill a requirement, but the true benefit of the course is learning how to use and understand mathematics in daily life. This quantitative reasoning text is written expressly for those students, providing them with the mathematical reasoning and quantitative literacy skills they' ll need to make good decisions throughout their lives. Common-sense applications of mathematics engage students while underscoring the practical, essential uses of math. | 677.169 | 1 |
The book gives the reader the basis for understanding the way numerical schemes achieve accurate and stable simulations of physical phenomena. It is based on the finite-difference method and simple problems that allow also the analytic solutions to be worked out. ODEs as well as hyperbolic, parabolic and elliptic types are treated. The book builds on simple model equations and, pedagogically, on a host of problems given together with their | 677.169 | 1 |
Circuit Training - Arithmetic & Geometric Sequences (algebra)
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
0.07 MB | 2 pages
PRODUCT DESCRIPTION
I wrote this circuit for my algebra one students as we wrap up the course, although it is structured so that students could work through it cooperatively with very little prior knowledge. It also would be perfect for the beginning of an algebra two course. The 24 questions allow students to take a tour through the landscape of arithmetic and geometric sequences. The questions are progressive in nature (build from easiest to hardest) and the last four are standardized test challenge problems. Calculator use should be minimal (or maybe zero -- you be the judge) on this circuit.
I do not include an answer key since the answers are imbedded in the circuit; it is how students move from one question to the next. This element of self-check is essential because it requires students to ask questions if they can not find their answer. The only thing the teacher needs to do is work the circuit ahead of the students to decide what, if any, vocabulary / notation / concepts should be addressed before the students begin. You will love the sounds of your students as they work through a circuit!
If you suspect an error or get stuck, please do not hesitate to email me at virginia.cornelius@gocommodores.org | 677.169 | 1 |
ALGEBRA 1A Documents
Showing 1 to 4 of 4
Alnita Holland
Math/116 Algebra 1A
Instructor: Gabriel Burns
December 1, 2011
Concept Check Week 4
Explain why the line x = 4 is a vertical line. The line x =4 is a vertical line, and since it has
no slope, or the slope is undefined, it is vertical becaus
Does pursuing a degree help you build on the competencies you need for your
future career? Is the academic work you do for your course assignments related to
those competencies?
There are many ways Microsoft Office, which include- Word, PowerPoint, and E
Percentile ranking = (number of values below given value / total number of values) x 100.
Quartiles refer to the percentiles if the data are divided into 4 groups. The four groups are the first,
second, third, and fourth quartiles. Because there are 100 p
Pure mathematics
Mathematics that studies entirely abstract
concepts.
Pure mathematics is "not necessarily
applied mathematics": it is possible to
study abstract entities and not be
concerned with how they manifest in the
real world.
Applied mathematics
A | 677.169 | 1 |
PreCalculus Conic Sections Real World Applications
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Conic Sections Real World Applications Task Cards - QR - Quiz
This resource is designed for PreCalculus, Algebra 2, or Geometry and includes real world applications to Hyperbolas, Ellipses, and Parabolas. This activity is great with small groups or as a station activity.
Included:
*There are two sets of 12 illustrated task cards related to applications of conics; one set has QR codes with the answers and one set does not. Students do not need and internet connection for the QR codes, but do need a device with the QR code reader app installed.
*Master List of questions which can be used for an alternate class or for an absent student.
*Two quizzes or additional HW assignments with similar problems.
*Answer key
*Student response sheet with room for work and unnumbered so you can choose the cards you want.
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Be the first to hear about my freebies, sales and new products designed to help you teach, save you time, and engage your students. Click on the green star near the top of any page in my store to become a follower.
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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 collegeMore...
With 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
Vernon Barker has retired from Palomar College where he was Professor of Mathematics. He is a co-author on the majority of Aufmann texts, including the best-selling developmental paperback series.
Chapters 26 are followed by Cumulative Review Exercises
Whole Numbers
Introduction to Whole Numbers
Addition of Whole Numbers
Subtraction of Whole Numbers
Multiplication of Whole Numbers
Division of Whole Numbers
Exponential Notation and the Order of Operations Agreement
Prime Numbers and Factoring
Focus on Problem Solving: Questions to Ask Projects and Group Activities: Order of Operations, Patterns in Mathematics, Search the World Wide Web
Fractions
The Least Common Multiple and Greatest Common Factor
Introduction to Fractions
Writing Equivalent Fractions
Addition of Fractions and Mixed Numbers
Subtraction of Fractions and Mixed Numbers
Multiplication of Fractions and Mixed Numbers
Division of Fractions and Mixed Numbers
Order, Exponents, and the Order of Operations Agreement
Focus on Problem Solving: Common Knowledge Projects and Group Activities: Music, Construction, Fractions of Diagrams
Decimals
Introduction to Decimals
Addition of Decimals
Subtraction of Decimals
Multiplication of Decimals
Division of Decimals
Comparing and Converting Fractions and Decimals a fraction
Focus on Problem Solving: Relevant Information Projects and Group Activities: Fractions as Terminating or Repeating Decimals
Ratio and Proportion
Ratio
Rates
Proportions
Focus on Problem Solving: Looking for a Pattern Projects and Group Activities: The Golden Ratio, Drawing the Floor Plans for a Building, The U.S. House of Representatives
Percents
Introduction to Percents
Percent Equations: Part 1
Percent Equations: Part II
Percent Equations: Part III
Percent Problems: Proportion Method
Focus on Problem Solving: Using a Calculator as a Problem-Solving Tool, Using Estimation as a Problem-Solving Tool Projects and Group Activities: Health, Consumer Price Index
Applications for Business and Consumers
Applications to Purchasing
Percent Increase and Percent Decrease
Interest
Real Estate Expenses
Car Expenses
Wages
Bank Statements
Focus on Problem Solving: Counterexamples Projects and Group Activities: Buying a Car | 677.169 | 1 |
Summary and Info
Configurations can be studied from a graph-theoretical viewpoint via the so-called Levi graphs and lie at the heart of graphs, groups, surfaces, and geometries, all of which are very active areas of mathematical exploration. In this self-contained textbook, algebraic graph theory is used to introduce groups; topological graph theory is used to explore surfaces; and geometric graph theory is implemented to analyze incidence geometries.After a preview of configurations in Chapter 1, a concise introduction to graph theory is presented in Chapter 2, followed by a geometric introduction to groups in Chapter 3. Maps and surfaces are combinatorially treated in Chapter 4. Chapter 5 introduces the concept of incidence structure through vertex colored graphs, and the combinatorial aspects of classical configurations are studied. Geometric aspects, some historical remarks, references, and applications of classical configurations appear in the last chapter.With over two hundred illustrations, challenging exercises at the end of each chapter, a comprehensive bibliography, and a set of open problems, Configurations from a Graphical Viewpoint is well suited for a graduate graph theory course, an advanced undergraduate seminar, or a self-contained reference for mathematicians and researchers.
More About the Author
Tomaž (Tomo) Pisanski (born May 24, 1949 in Ljubljana, Slovenia) is a Slovenian mathematician working mainly in discrete mathematics and graph theory. | 677.169 | 1 |
Pre Calculus Analyzing Functions with Graphs and Tables FREEBIE
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In this sample of my Analyzing Functions with Graphs and Tables, students will work with a single graph and answer questions about function values, where a function has a maximum or minimum, what the maximum or minimum value is, and where a function is increasing or decreasing.
(Although an answer key is included, the teacher will have to determine how the students should answer the questions involving interval notation as different textbooks define this in different manners.)
If you like this worksheet, you may wish to view the complete product which contains 7 worksheets...click to see the full product | 677.169 | 1 |
Addition and multiplication rules of probability, conditional probability, Bayes Theorem, independence of events, computation of probability of events using permutations and combinations.
Two dimensions: Cartesian coordinates, distance between two points, section formulae, shift of origin.
Equation of a circle in various forms, equations of tangent, normal and chord.
Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line.
Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal.
Locus Problems.
Three dimensions: Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.
Limit and continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, L'Hospital rule of evaluation of limits of functions.
Even and odd functions, inverse of a function, continuity of composite functions, intermediate value property of continuous functions. | 677.169 | 1 |
Exam Board: Edexcel Level & Subject: AS Maths First teaching: September 2008 First exams: June 2009
Endorsed by Edexcel
Collins Student Support Materials for Edexcel AS Maths Mechanics 1 covers all the content and skills your students will need for their optional Mechanics 1 examination, including:
• Mathematical models in mechanics • Vectors in mechanics • Kinematics of a particle moving in a straight line • Dynamics of a particle moving in a straight line or plane • Statics of a particle • Moments • EXAM PRACTICE • Answers
Clear explanations and worked examples are accompanied by Essential notes and Exam tips. Find practice exam questions with fully worked answers, as well as guidance from examiners on securing top marks. | 677.169 | 1 |
Offering a uniquely modern, balanced approach, Tussy/Gustafson/Koenig's BASIC MATHEMATICS FOR COLLEGE STUDENTS, Fourth Edition, integrates the best of traditional drill and practice with the best elements of the reform movement. To many developmental math students, mathematics is like a foreign language.
To effectively deal with any chemical-based problem, including pollution, environmental, health and safety professionals must have at least a rudimentary understanding of the basic concepts of chemistry.
As photonics and materials science span new horizons, it is paramount that one's mathematical skills be honed. The primary objective of this book is to offer a review of vector calculus needed for the physical sciences and engineering. | 677.169 | 1 |
Quick Math Review to Prep for Algebra 1
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Quick Math Review to Prep for Algebra 1
This is 1 of 4 videos I custom made for an educator in California for an experimental 1-week video homework program. I have only edited the beginning and ending titles. Although not designed for the general public, I figured you may find them useful.This video provides a quick general math review of concepts needed to do well in Algebra 1. Covers: Order of Operations, Fraction Arithmetic, Basic Algebraic Concepts and Terminology, Converting Verbal Phrases to Algebraic Expressions and other Miscellaneous Math Concepts.
Length:
25:17
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Get free advice from education experts and Noodle community members. | 677.169 | 1 |
The goal ofElementary and Intermediate Algebra: Concepts and Applications,4e is to help today's students learn and retain mathematical concepts by preparing them for the transition from "skills-oriented" elementary and intermediate algebra courses to more "concept-oriented" college-level mathematics courses, as well as to make the transition from "skill" to "application." This edition continues to bring your students a best-selling text that incorporates the five-step problem-solving process, real-world applications, proven pedagogy, and an accessible writing style. The Bittinger/Ellenbogen/Johnson series has consistently provided teachers and students with the tools needed to succeed in developmental mathematics. This revision has an even stronger focus on vocabulary and conceptual understanding as well as making the mathematics even more accessible to students. Among the features added are newConcept Reinforcementexercises, Student Notes that help students avoid common mistakes, and Study Summaries that highlight the most important concepts and terminology from each chapter. Introduction to Algebraic Expressions; Equations, Inequalities, and Problem Solving; Introduction to Graphing; Polynomials; Polynomials and Factoring; Rational Expressions and Equations; Functions and Graphs; Systems of Equations and Problem Solving; Inequalities and Problem Solving; Exponents and Radicals; Quadratic Functions and Equations; Exponential and Logarithmic Functions; Conic Sections; Sequences, Series, and The Binomial Theorem; Elementary Algebra Review For all readers interested in elementary and intermediate algebra.
Larson IS student success. ELEMENTARY AND INTERMEDIATE ALGEBRA: ALGEBRA WITHIN REACH, 6E, International Edition owes its success to the hallmark features for which the Larson team is known: learning by example, a straightforward and accessible writing style, emphasis on visualization through the use of graphs to reinforce algebraic and numeric solutions and to interpret data, and comprehensive exercise sets. These pedagogical features are carefully coordinated to ensure that students are better able to make connections between mathematical concepts and understand the content. With a bright, appealing design, the new Sixth Edition builds on the Larson tradition of guided learning by incorporating a comprehensive range of student success materials to help develop students' proficiency and conceptual understanding of algebra. The text also continues coverage and integration of geometry in examples and exercises.
The new edition of BEGINNING & INTERMEDIATE ALGEBRA welcomes two new co-authors Rosemary Karr and Marilyn Massey who along with David Gustafson have developed a learning plan to help students succeed in Beginning Algebra and transition to the next level in their coursework. The new edition has been thoroughly updated with new pedagogical features and a new interior design that make the text both easier to read and easier to use. Based on their years of experience in developmental education, the new accessible approach builds upon the book's known clear writing and engaging style which teaches students to develop problem-solving skills and strategies that they can use in their everyday lives. The authors have developed an acute awareness of students' approach to homework and present a learning plan keyed to new Learning Objectives and supported by a comprehensive range of exercise sets that reinforces the material that students have learned setting the stage for their success. The new edition of BEGINNING & INTERMEDIATE ALGEBRA is an exciting and innovative revision that takes an already successful text and makes it more compelling for today's instructor and student. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
As in previous editions, the focus in INTRODUCTORY ALGEBRA, remains on the Aufmann Interactive Method (AIM). Students are encouraged to be active participants in the classroom and in their own studies as they work through the How To examples and the paired Examples and You Try It problems. The role of active participant is crucial to success. Presenting students with worked examples, and then providing them with the opportunity to immediately work similar problems, helps them build their confidence and eventually master the concepts. To this point, simplicity plays a key factor in the organization of this edition, as in all other editions. All lessons, exercise sets, tests, and supplements are organized around a carefully-constructed hierarchy of objectives. This objective-based approach not only serves the needs of students, in terms of helping them to clearly organize their thoughts around the content, but instructors as well, as they work to design syllabi, lesson plans, and other administrative documents. The Eighth Edition features a new design, enhancing the Aufmann Interactive Method and the organization of the text around objectives, making the pages easier for both students and instructors to follow. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. | 677.169 | 1 |
An iPad® can be used to teach students to graph parabolas with ease and grasp vocabulary quickly. Parabolas come to life for students in this easily implemented activity described in this article. Teachers can use this tool in a fun and interactive way to not only address these graphing and vocabulary concepts but also introduce and explore parabolas and more in the algebra classroom. The Free Graphing Calculator ( Jockusch 2010) app can be downloaded and accessed to introduce graphs of quadratic equations and the relevant vocabulary, including "axis of symmetry," "roots," "maxima," "minima," and "vertices." The app helps students visualize the terms and provides a strong conceptual background for future lessons. After the exploration, students can model their new knowledge by using the free app Skitch (Evernote Corp. 2012) to compare and contrast quadratic equation families and apply vocabulary that they learned during the calculator activity. Both of the free apps, the Free Graphing Calculator and Skitch, are available from the Apple App store. The activity described in this article is based on a graphing calculator investigation found in the "Algebra 1" textbook by Glencoe Mathematics (Holliday et al. 2005). It has been adapted for the iPad and the two free apps. | 677.169 | 1 |
UNDERSTANDING THE DEFINITION OF A LIMIT-1
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Students often struggle to understand the formal definition of a limit. This activity sheet has the students build both tables and graphs to see what the definition is saying. In the tabular form, the students keep refining the table to be convinced that as long as they stay very close to a value of a on the x-axis all y values will stay in a close range of a particular number called a limit. The same idea is explored graphically by trying to keep all y values visible when given a particular window of x values. Eight additional problems are given to the students to | 677.169 | 1 |
Exterior Analysis: Using Applications Of Differential Forms
Hardcover | September 6, 2013
Exterior analysis uses differential forms a mathematical technique to analyze curves, surfaces, and structures. Exterior Analysis is a first-of-its-kind resource that uses applications of differential forms, offering a mathematical approach to solve problems in defining a precise measurement to ensure structural integrity. The book provides methods to study different types of equations and offers detailed explanations of fundamental theories and techniques to obtain concrete solutions to determine symmetry. It is a useful tool for structural, mechanical and electrical engineers, as well as physicists and mathematicians. Provides a thorough explanation of how to apply differential equations to solve real-world engineering problems Helps researchers in mathematics, science, and engineering develop skills needed to implement mathematical techniques in their research Includes physical applications and methods used to solve practical problems to determine symmetry: Exterior Algebra
2: Differential Manifolds
3: LIE Groups
4: Tensor Fields of Manifolds
5: Exterior Differential Forms
6: Homotopy Operator
7: Linear Connections
8: Integration of Exterior Forms
9: Differential Equations
10: Variational Calculus
11: Physical Applications
(i.e. Mechanics, Electromagnetism, Thermodynamics)
Editorial Reviews
"The book is carefully written. It contains a rich material and covers an important part of differential geometry. Applications of the main abstract results can be found frequently. There are many examples and exercises.The book is useful for mathematicians, applied scientists and engineers."--Zentralblatt MATH, 1277.53001 "Suhubi introduces mathematicians, physicists, and engineers to the fundamental concepts and tools of exterior analysis, helps them gain competence in using these tools, and describes the advantages of the approach. He keeps the mathematics as simple as possible without sacrificing rigor. His topics include differential manifolds, tensor fields on manifolds, exterior differential forms, the integration of exterior forms, partial differential equations,."--ProtoView.com, February 2014 | 677.169 | 1 |
Fraction Calculator Free simple Fraction Calculator. It accepts Whole numbers, mixed numbers and fractions. Outputs in in fractions, decimal and mixed number all at the same time. Optional "show work" button for those learning to do fraction problems. Finally, the web has a
Vinny Graphics Vinny Graphics is a novel graphing and data-analysis program for science and engineering students. It is easy to use and accepts and exports data through a variety of sources. Perform math and regression analysis, working on up to 21 independent data sets.
Geometry Calculator Geometry Calculator is an application includes four sections to choose from Circle, Area, Surface Area, and Volume. For the calculations used in this program "Pi" has been rounded off to 10 decimal places. You can calculate area of sector, ellipse, square,Karnaugh Studio Karnaugh Studio is a product line for student of engineer and professional, which offers the possibility of obtaining the best solution to the simplification of boolenas functions using the map of karnaugh to show it. Features include: Map Karnaugh iterati | 677.169 | 1 |
Quantitative Aptitude For The CAT is aimed at aspirants of competitive management exams. This book comprehensively covers the subject matter, and contains questions, solutions, and shortcuts.
Summary Of The Book
How To Prepare For Quantitative Aptitude For The CAT, published in 2012, is now in its fifth edition. This guide has been revised and updated to help Common Admission Test aspirants prepare for this competitive test. This guide has been designed to comprehensively cover each topic that is part of the Quantitative Ability syllabus.
How To Prepare For Quantitative Aptitude For The CAT starts off with an introduction about the format of QA in this online examination. The book is then divided into blocks, and each one has a number of chapters within. Block I contains information about Number Systems, and Progressions. The first chapter explains the Remainder Theorem, the concept of the Great Common Divisor, and number of zeroes in an expression. Block II gives examples of situations where Allegations can be put to use. The chapters included in the following block are Percentages, Interest, Time and Work, Profit and Loss, Time, Speed and Distance, and Ratio, Proportion and Variation. The 11th chapter is a part of Block IV, and it is further divided into two parts. Part I focuses on Geometry, while the second part explains Mensuration. This block also has a chapter on Coordinate Geometry. The next block of the guide has four chapters, which are Functions, Inequalities, Quadratic Equations, and Logarithms. The last block explains to students principles of Permutaftions and Combinations, Probability, and Set Theory.
Each chapter of this guidebook has questions at the end which are listed into three sections as per their level of difficulty. In addition to this, the author has also added many tips and shortcuts to help students in increasing their problem solving skills. Mock and model questions sets have been incorporated at the end of How To Prepare For Quantitative Aptitude For The CAT. Those who are preparing management entrance tests like NMAT, SNAP, XAT, and CMAT will find this book to be very resourceful. Previous editions of these books have been best-sellers
About Arun Sharma
Arun Sharma is a popular writer of books based on preparations of entrance exams like CAT.
Some of the books written by the author are How To Prepare For Verbal Ability And Reading Comprehension For The CAT, CSAT Data Interpretation Logical Reasoning And General Mental Ability For General Studies, Study Package For The CAT Online, and Comprehension, Interpersonal & Communication Skills For General Studies.
Sharma has a BSc. in Electrical Engineering, and B.E. in Tele-Communication Engineering. He also has a MBA from the Indian Institute of Management, Bangalore. He is the CEO of Mindworkzz, and the Director of Kalyanpur Cements Ltd. He was a Major in the Indian Armed Forces. He has also been the President, General Manager, and Managing Director of different organizations. The author holds the record for having cracked the CAT examination eleven times. | 677.169 | 1 |
College algebra : in simplest terms(
Visual
) 3
editions published
in
1991
in
English
and held by
105 WorldCat member
libraries
worldwide
Presents the role of algebra in daily life and demonstrates practical applications in the workplace. Uses symbols, charts,
pictures, and state-of-the-art computer graphics to illustrate basic algebraic techniques. Reviews problems step-by step,
focusing on the methods students find most difficult to grasp
Mathematics : modeling our world(
Book
) 14
editions published
between
1998
and
2010
in
English
and held by
83Algebra : in simplest terms(
Visual
) 4
editions published
in
1991
in
English
and held by
80 WorldCat member
libraries
worldwide
Solving equations is a basic operation of all higher math. This set shows algebra's usefulness to retailers, biologists, and
even anyone who drives a car. Host Sol Garfunkel walks viewers through realistic problems, highlighting the common trouble
spots
For all practical purposes(
Visual
)
in
English
and held by
72 WorldCat member
libraries
worldwide
A series which stresses the connections between contemporary mathematics and modern society. Presents a great variety of problems
that can be modeled and solved by quantitative means
Algebra : in simplest terms(
Visual
) 2
editions published
between
1991
and
2000
in
English
and held by
38 WorldCat member
libraries
worldwide
This is an instructional series of 26 half-hour programs for high school, college, and adult learners, or for teachers seeking
to review the subject matter. Host Sol Garfunkel explains concepts that may baffle many students, while graphic illustrations
and on-location examples demonstrate how algebra is used for solving real-world problems. Algebra is important in today's
world, used in such diverse fields as agriculture, sports, genetics, social science, and medicine. This series helps students
connect algebra's mathematical themes and applications to daily life
Mathematics : modeling our world : Pre-Calculus(
Book
) 2
editions published
in
2000
in
English
and held by
26For all practical purposes(
Visual
) 5
editions published
between
1987
and
1988
in
English
and held by
26 WorldCat member
libraries
worldwide
Deals with how mathematics can be used to make social choices ranging from a fair voting system, determining award winners,
and setting economic and governmental planning priorities. Uses computer graphics, animation sequences, and live action. Intended
for entry-level liberal arts students
Mathematics : modeling our world(
Book
)
in
English
and held by
25 WorldCat member
libraries
worldwide
The authors demonstrate mathematical modeling and using mathematical concepts to solve truly interesting problems about how
our world works. Mathematical modeling is the process of looking at a problem, finding a mathematical core, working within
that core, and coming back to see what mathematics tells you about the problem. Real problems ask such questions as: How do
we create computer animations? Where should we locate a fire station? How do we effectively control an animal population?
This approach integrates a mix of ideas in geometry, algebra, and data analysis with technologies of computers and graphing
calculators. Course 4 (Pre-calculus) is intended as a bridge between MMOW and collegiate mathematics. It introduces students
to a number of new concepts (i.e. matrices and vectors) and teaches them new skills that will help prepare them for entry-level
undergraduate mathemematics courses, including calculus and discrete mathematics
Systems of linear inequalities ; Arithmetic sequences and series(
Visual
) 3
editions published
in
1991
in
English
and held by
23 WorldCat member
libraries
worldwide
Program 21 sets up a problem, finds a solution, develops linear inequalities, graphs these solutions, and forms a region of
feasible solutions. Program 22 explores basic properties and formulas, emphasizing sums of arithmetic series and developing
concepts
Functions ; Composition and inverse functions(
Visual
) 2
editions published
in
1991
in
English
and held by
22 WorldCat member
libraries
worldwide
Program 13 defines a function, develops an equation from real situations, and discusses domain and range. Program 14 uses
graphics to introduce composites and inverses of functions as applied to cost and production level
The prisoner's dilemma(
Visual
) 5
editions published
between
1986
and
1987
in
English
and held by
18 WorldCat member
libraries
worldwide
The games of "chicken" and "prisoner's dilemma" illustrate issues in corporate takeovers and labor relations
For all practical purposes(
Visual
)
in
English
and held by
6 WorldCat member
libraries
worldwide
These five programs from the series For All Practical Purposes explore the nature and use of statistics in the modern world
Statistics overview(
Visual
) 2
editions published
in
1986
in
English
and held by
4 WorldCat member
libraries
worldwide
Program 6 moves from baseball scores and roulette odds to national unemployment figures and quality control testing, to show
that statistics help us to understand information and make better decisions. This overview introduces the subject, featuring
professionals in labor statistics and medicine who use statistical methods to determine probable outcomes in their fields
It grows and grows ; Stand up conic(
Visual
) 1
edition published
in
1986
in
English
and held by
0 WorldCat member
libraries
worldwide
"It grows and grows" describes how population grows mathematically and why it is important to be able to calculate population
growth. Examples include compound interest on money, fish in the sea for harvest, and world human population. "Stand up conic"
describes how conic sections are a fundamental part of mechanical inventions. Discusses varions conic shapes and how they
are used in such things as the circus, astronomy, acoustics, airplanes, bridges, etc | 677.169 | 1 |
About me
1. Sets
Sets and their representations.Empty set.Finite and Infinite sets.Equal sets.Subsets.Subsets of the set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and intersection of sets.Difference of sets. Complement of a set, Properties of Complement sets.Types of relations: Reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function.Binary operations.
2. Relations and Functions
Ordered pairs, Cartesian product of sets. Number of elements in the Cartesian product of two
Process of the proof by induction, motivating the application of the method by looking at natural
numbers as the least inductive subset of real numbers. The principle of mathematical induction
and simple applications.
2. Complex Numbers and Quadratic Equations
Need for complex numbers, especially , to be motivated by inability to solve every quadratic equation. Brief description of algebraic properties of complex numbers.Argand plane and polar representation of complex numbers.Statement of Fundamental Theorem of Algebra, solution of quadratic equation in the complex number system, Square-root of a Complex number.
3. Linear Inequalities
Linear inequalities, Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Solution of system of linear inequalities in two variables - graphically.
Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Non-commutativityof multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
8. Determinants
Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.
Applications of derivatives: Rate of change, increasing/decreasing functions, tangents and normal, approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).
4. Integrals
Integration as inverse process of differentiation.Integration of a variety of functions by substitution, by partial fractions and by parts, to be evaluated.
Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without proof). Basic
properties of definite integrals and evaluation of definite integrals.
5.Applications of the Integrals
Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ellipses (in standard form only), area between the two above said curves (the region should be clearly identifiable).
6:Differential Equations
Definition, order and degree, general and particular solutions of a differential equation.Formation of differential equation whose general solution is given.Solution of differential equations by method of separation of variables, homogeneous differential equations of first order and first degree.
VECTORS AND THREE-DIMENSIONAL GEOMETRY
1. Vectors
Vectors and scalars, magnitude and direction of a vector.Direction cosines/ratios of vectors. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors, scalar triple product.
2. Three-dimensional Geometry
Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane.Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.Distance of a point from a plane.
Mathematically acceptable statements. Connecting words/phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words - difference between contradiction, converse and contrapositive. | 677.169 | 1 |
Online homework, tutorial & testing from Pearson for middle school through college math
Overview
MathXL is an online homework, tutorial and assessment system from publishing giant Pearson. The system includes tutorials that walk students through solving math problems, as well as customizable and editable content for online homework assignments, quizzes and tests correlated to specific standards or objectives as well as to chapters in Pearson's textbooks. Based on those test results, MathXL also creates personalized study plans for students, with recommendations of tutorial exercises they should focus on. Homework and assessments are graded automatically by the system, but teachers can also use MathXL to manage their entire gradebook, including assignments, projects and tests conducted outside MathXL.
Although Pearson says MathXL can be used alongside any math textbook, the system is tightly aligned with Pearson's hundreds of math and statistics textbooks. MathXL also powers Pearson's full math courseware products like MyMathLab and MyStatLab, which include full e-book versions of those texts.
However, MathXL can also be purchased separately as a cheaper alternative for teachers and schools that don't need or want the full sequence of course content nor the e-textbook. Schools (or, in the case of higher education, students themselves) pay about $50 per year per student for a full year of MathXL access as part of their online, blended or face-to-face courses.
(In Summit Reflections, educators review tools that they have seen at an EdSurge Tech for Schools Summit. Summit Reflections can only be completed onsite at the event; reviewers are incentivized to leave reviews.) | 677.169 | 1 |
Explains how a standard electronic computer function helps 2-man student teams solve engineering problems assigned as homework. The MOD function increases students' learning rate, helps them to understand theory and obtain correct answers in a system where a major percentage of their grades is based on homework results. (WM) | 677.169 | 1 |
Download Calculus With Analytic Geometry pdf free written by mathematician George Simmons. Analytic geometry, the study of geometry using coordinate system and calculus, the study of infinitesimal changes are two branch of mathematics required to solve higher physics, mathematics and engineering problem. This books teaches these two branch of mathematics in an integrated manner.
The book has divided the topics chosen into three parts. The first part is the introductory part dealing with the basic but foundation for understanding calculus with analytic geometry. What is calculus, the formula for derivatives, maxima and minima, tangents, meaning of integration and others are topics covered in the first part. The second part is also foundational but little bit harder than first part. Herein trigonometry and laws of trigonometry, power series, mean value problem, convergence etc are covered. Then the third part is on using the 1st and 2nd part learned calculus to the problem of geometry. Here it starts with cone, coordinate system, parametric equation of curve, 3D spaces, vectors and dot products, function of several variables, volume and line integrals. | 677.169 | 1 |
Synopsis
Maths - Foundation Tier: Revision Workbook by
Accessible and motivating, this Maths workbook provides high quality practice examples and is suitable for all ability levels and exams boards. Plenty of practice questions ensure that students are thoroughly prepared for the exam when it comes. All key concepts and skills are covered in this clear and user-friendly Maths foundation tier workbook. GCSE-style practice questions reinforce understanding and help students prepare for the exam with confidence. Included in this book: * Invaluable tips to help students pick up all available marks * Advice to help students avoid common mistakes * Plenty of practice questions * Examiners' advice | 677.169 | 1 |
Remember How To Multiply Exponents
Exponents Multiplied:
When you can't remember how to multiply exponents all you have to
do is cry out MA. your mother won't be able to help you but the
acronym MA will.
M= when Multiplying
A= Add exponents
Therefore: x^2
Showing 1 to 1 of 1
This is a great course to jump start your knowledge of math. It reviews things learned in algebra II and things learned in trigonometry. This course was great for me and I would recommend it for anyone who is not quite ready for harder college math courses. If you have taken algebra II and trigonometry and feel fairly strong on you understanding of their contents, this course will be too easy for you.
Course highlights:
It is a great way to strengthen your understanding of basics of advanced math mathematics.
Hours per week:
3-5 hours
Advice for students:
Be sure to ask a lot of question. Your teacher will likely be able to answer any question regarding math even if it is beyond the scope of the course you are taking
Course Term:Fall 2016
Professor:Mathew, Lennon
Course Required?Yes
Course Tags:Background Knowledge ExpectedGreat Intro to the SubjectMany Small Assignments | 677.169 | 1 |
IGCSE Mathematics Study Guide: Table of Contents
This is the table of contents for IGCSE Mathematics. The material here is relevant and up to date. Here, we will focus more on theory and you can apply the theory to your nice big blue (or whatever color) textbooks. We will, however, have a separate section focusing on questions, after we finish with the theory.
Hey, the published here are all oriented towards the IGCSE Mathematics 0607. ALL the questions come from these chapters. We focus on the theory, and you can reinforce your knowledge by doing more problems on the textbook etc. | 677.169 | 1 |
Elementary Geometry for Teachers is the sequel to Elementary Mathematics for Teachers (EMFT) by Parker and Baldridge. Elementary Geometry for Teachers is designed for the second semester of a mathematics course for prospective elementary teachers that is taught by mathematics faculty. This text takes prospective teachers through the development of measurement and geometry in grades K-8; it also includes material on probability and data analysis.
Elementary Geometry for Teachers covers both the mathematics and other aspects of the K-8 geometry curriculum. For this purpose, this text — like EMFT — is used in conjunction with six school textbooks from Singapore (two of these are also used with EMFT). The homework sets include exercises that ask students to read a section in a Primary Mathematics book, do the problems, and then study the material from a teachers' perspective, thinking about which skills are developed, how the problems are organized, what the prerequisite knowledge is, what order topics are developed, etc.
Features:
The material focuses directly on the mathematics relevant to elementary teachers.
The focus is always on mathematics; this is not a "teaching methods'' course.
The text is divided into short sections, each with a homework set, of a size appropriate for a single class session.
Prospective teachers are asked to write "Teacher's Solutions" to problems and to write "Elementary Proofs". These are special ways of presenting specific geometry content to elementary and middle school students that are standard in some of the world's most highly-regarded curricula. The emphasis is on building and perfecting skill at writing clear, concise solutions.
The
Primary Mathematics textbooks serve as teacher guides. They provide examples and activities that teachers can use in their classrooms and that help teachers understand what is important in K-8 geometry.
The Primary Mathematics books were chosen because of their clarity, organization, low cost and their exceptional fidelity to mathematics. Studying the Primary Mathematics books prepares teachers for teaching from any elementary school materials. Furthermore, as prospective teachers work though these books, they are constantly aware that the pace, the breadth and the difficulty of the problems in the Primary Mathematics books are at a higher level than what they experienced in their own elementary education. They come away with new expectations about the mathematics capabilities of elementary students.
Elementary Geometry for Teachers pays special attention to two themes:
Developing skills at solving problems involving measurements. International comparisons indicate that U.S. students are especially weak at solving problems involving measurements. These skills are important prerequisites for middle and high school science. Elementary Geometry for Teachers builds teachers' facility at solving such problems by following the Primary Mathematics curriculum through the grades. The problem below is one of a sequence of Grade 5 "tank problems."
A rectangular tank, 40 cm long and 20 cm wide, originally contains water to a depth of 9 cm. When a stone is added, the water rises to a depth of 15 cm, covering the stone. What is the volume of the stone in liters?
Unknown angle problems. One reason for studying geometry is to acquire skill at logical reasoning. The Primary Mathematics books develop geometric reasoning in depth. In grades 4-6, students are introduced to a specific collection of geometric facts (e.g. the sum of interior angles of a triangle is 180° and opposite angles in a parallelogram are equal. These are used to solve entertaining puzzles like the one below. As they work through Elementary Geometry for Teachers, teachers solve such problems and learn to write Teacher Solutions that display the reasoning.
Supplementary Texts
The Elementary Geometry for Teacher textbook is designed to be used in conjunction with the following five Primary Mathematics books (all are U.S. Edition) and one New Elementary Mathematics book.
Primary Mathematics 3B Textbook (ISBN 9789810185039)
Primary Mathematics 4A Textbook (ISBN 9789810185060)
Primary Mathematics 5A Textbook (ISBN 9789810185107)
Primary Mathematics 5B Textbook (ISBN 9789810185144)
Primary Mathematics 6B Textbook (ISBN 9789810185152)
New Elementary Textbook 1 (ISBN 978981274114)
The Primary Mathematics books were developed by the Curriculum Planning and Development Division of Singapore's Ministry of Education, and published by Marshall Cavendish. While these books were initially created for Singapore elementary students, they have been adapted for use in the United States and other countries. We will refer to them as "Primary Math 3B", and so on.
The Primary Mathematics series is printed as one course book per semester, each with an accompanying workbooks. The semesters are labeled 'A' and 'B' , so '5A' refers to the first semester of Grade 5. In each grade, the first semester focuses mainly on numbers and arithmetic, while the second semester focuses more on measurement and geometry.
For teachers using Primary Mathematics Standards Edition textbooks and workbooks, here is a link to the EGFT homework adaption for the Standards Edition. This homework adaption may be printed out and used at no cost by teachers using the EGFT textbook. They may not be sold or incorporated into any other document. List of universities and colleges using Elementary Mathematics and Elementary Geometry for Teachers:
Our recommendation: Elementary Geometry for Teachers is written primarily for elementary teachers. A number of universities use this book as course material in classes for students taking mathematics education. This book is also suitable for individuals who would like to learn more about teaching elementary mathematics. | 677.169 | 1 |
Sample records for absolute values lavWhat is the meaning of absolutevalue? And why do teachers teach students how to solve absolutevalue equations? Absolutevalue is a concept introduced in first-year algebra and then reinforced in later courses. Various authors have suggested instructional methods for teaching absolutevalue to high school students (Wei 2005; Stallings-Roberts…
This article explores how conceptualization of absolutevalue can start long before it is introduced. The manner in which absolutevalue is introduced to students in middle school has far-reaching consequences for their future mathematical understanding. It begins to lay the foundation for students' understanding of algebra, which can change…
Presents an approach to the concept of absolutevalue that alleviates students' problems with the traditional definition and the use of logical connectives in solving related problems. Uses a model that maps numbers from a horizontal number line to a vertical ray originating from the origin. Provides examples solving absolutevalue equations and…
Making connections between various representations is important in mathematics. In this article, the authors discuss the numeric, algebraic, and graphical representations of sums of absolutevalues of linear functions. The initial explanations are accessible to all students who have experience graphing and who understand that absolutevalue simply…
The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolutevalue equation presented, for an interesting way to form an overall understanding of the concept.
This paper gives an account of a teaching experiment on absolutevalue inequalities, whose aim was to identify characteristics of an approach that would realize the potential of the topic to develop theoretical thinking in students enrolled in prerequisite mathematics courses at a large, urban North American university. The potential is…
The absolutevalue learning objective in high school mathematics requires students to solve far more complex absolutevalue equations and inequalities. When absolutevaluevalue. The initial point of reference should help students successfully evaluate numeric problems involving absolutevalue. They should also be able to solve absolutevalue equations and inequalities that are typically found in…values of the choice alternatives is not necessary for choosing the best alternative, it may nevertheless hold valuable information about the context of the decision. To test the sensitivity of human decision making to absolutevalues, we manipulated the intensities of brightness stimuli pairs while preserving either their differences or their ratios. Although asked to choose the brighter alternative relative to the other, participants responded faster to higher absolutevalues. Thus, our results provide empirical evidence for human sensitivity to task irrelevant absolutevaluesvalues subject to the temporal dynamics of lateral inhibition. The potential adaptive role of such choice mechanisms is discussed. PMID:26022836value, graphs of absolutevaluevalue, addition and multiplication in terms of absolutevalue, graphs of absolute…
The research accomplished during this period is reported. Published abstracts and technical reports are listed. Articles presented include: boundedness of absolutevalues of generalized Fourier coefficients, propagation in stochastic media, and stationary conditions for stochastic differential equations.
In this paper, we suggest and analyze an improved generalized Newton method for solving the NP-hard absolutevaluevalue inequalities,…value concept but generates a lack of stimulation during problem solving in further phases of the instruction. This could be explained as a result of current examination system which requires a habituation of the analytical process in solving mathematical questions resistor resistor. The output current through the load resistor is proportional to the absolutevalue of the input voltage difference between the bipolar input terminals. A third differential amplifier has its plus input terminal connected to the load resistor. A second gain determining resistor value…
The divergent integral ∫a b f ( x ) ( x - x 0 ) - n - 1 d x , for -∞ < a < x0 < b < ∞ and n = 0, 1, 2, …, is assigned, under certain conditions, the value equal to the simple average of the contour integrals ∫C±f(z)(z - x0)-n-1dz, where C+ (C-) is a path that starts from a and ends at b and which passes above (below) the pole at x0. It is shown that this value, which we refer to as the analytic principal value, is equal to the Cauchy principal value for n = 0 and to the Hadamard finite-part of the divergent integral for positive integer n. This implies that, where the conditions apply, the Cauchy principal value and the Hadamard finite-part integral are in fact values of absolutely convergent integrals. Moreover, it leads to the replacement of the boundary values in the Sokhotski-Plemelj-Fox theorem with integrals along some arbitrary paths. The utility of the analytic principal value in the numerical, analytical, and asymptotic evaluations of the principal value and the finite-part integral is discussed and demonstrated.
Understanding graphical representations of algebraic equations, particularly graphical representations of absolutevalue equations, significantly improves students' mathematical comprehension and ignites within them an appreciation of the beauty and aesthetics of mathematics. In this paper, we focus on absolutevalue equations of linear and quadratic expressions, by examining various cases, presenting different methods of solving them by graphical representation, exhibiting the advantage of using dynamic software such as GeoGebra in solving them, and illustrating some examples of interesting graphical solutions. We recommend that teachers take advantage of the rapid development in technology to help learners tangibly visualize the solutions of absolutevalue equations before proceeding to the analytical solutionsvalue expressions are involved, the definition that is most helpful is the one involving solving by intervals and evaluating critical points. In point of fact, application of this technique is one reason that the topic of absolutevalue is important in mathematics in general and in mathematics teaching in particular. We present here an authentic practical problem that is solved using absolutevalues and the 'intervals' method, after which the solution is generalized with surprising results. This authentic problem also lends itself to investigation using educationalThe ECP Working Group on AIDS has evaluated data on seropositivity to LAV/HTLV-III supplied by members in II Western European countries. The period covered is 1981-84. The rise in LAV/HTLV seropositivity parallels the incidence of cases of AIDS in the different countries. LAV/HTLV now spreads freely within Europe and spread has become less dependent upon promiscuity. The epidemic is about to enter Eastern Europe. Intravenous drug abusers appear to be the risk group experiencing the most rapid spread at present. Furthermore, seropositivity in males and females outside the traditional risk groups seems on the rise, and as in the US the percentage seronegative in individuals with PGL is quite high. AIDS is rapidly becoming a major cause of cancer in young adults. A coordinated European preventive effort is urgently needed. PMID:3474150
Modules in Computer Algebra Systems (CAS) make Mathematics interesting and easy to understand. The present study focused on the implementation of the algebraic, tabular (numerical), and graphical approaches used for the construction of the concept of absolutevalue function in teaching mathematical content knowledge along with Maple 9. The study…
The study aimed to investigate students' conceptions on the notion of absolutevalue and their abilities in applying the specific notion in routine and non-routine situations. A questionnaire was constructed and administered to 17-year-old students. Data were analysed using the hierarchical clustering of variables and the implicative method,…
Multiphase flow imaging is a very challenging and critical topic in industrial process tomography. In this article, simulation and experimental results of reconstructing the permittivity profile of multiphase material from data collected in electrical capacitance tomography (ECT) are presented. A multiphase narrowband level set algorithm is developed to reconstruct the interfaces between three- or four-phase permittivity values. The level set algorithm is capable of imaging multiphase permittivity by using one set of ECT measurement data, so-called absolutevalue ECT reconstruction, and this is tested with high-contrast and low-contrast multiphase data. Simulation and experimental results showed the superiority of this algorithm over classical pixel-based image reconstruction methods. The multiphase level set algorithm and absolute ECT reconstruction are presented for the first time, to the best of our knowledge, in this paper and critically evaluated. This article is part of the themed issue 'Supersensing through industrial process tomography'. PMID:27185966
Most existing deconvolution techniques are incapable of determining phase properties of wavelets from time series data; to assure a unique solution, minimum phase is usually assumed. It is demonstrated, for moving average processes of order one, that deconvolution filtering using the absolutevalue norm provides an estimate of the wavelet shape that has the correct phase character when the random driving process is nonnormal. Numerical tests show that this result probably applies to more general processes.In a companion paper in this issue we presented a review of the current state of (17)O-corrections for CO(2) mass spectrometry and considered an approach (including algebraic formulae) of how to determine absolutevalues for (17)R(VPDB-CO2) and (17)R(VSMOW). Here we present the results of experiments conducted to determine these values. Two oxygen gases (one depleted in heavy isotopes and the other isotopically normal oxygen) were analysed to obtain the relative (17)O content. Samples of both gases were converted into CO(2), and the resulting CO(2) samples were analysed as well. Possible experimental and analytical errors are carefully considered and eliminated as far as feasible. Much attention was paid to understanding and dealing with cross-contamination effects occurring in the mass spectrometer. Based on the data obtained, the absolutevalues are calculated to be: (17)R(VPDB-CO2) = 0.00039511 +/- 0.00000094 and (17)R(VSMOW) = 0.00038672 +/- 0.00000087 (expanded uncertainties). Both values are on the original scale of Craig (Geochim. Cosmochim. Acta 1957; 12: 133-149) with (13)R(VPDB-CO2) = 0.0112372. A (17)O-correction algorithm incorporating the newly determined value for (17)R(VPDB-CO2) and lambda = 0.528 by Meijer and Li (Isot. Environ. Health Stud. 1998; 34: 349-369) is constructed. A computational test is performed to demonstrate the degree of delta(13)C bias relative to the previously known correction algorithms. delta(13)C values produced by the constructed algorithm are in the middle of the values produced by the other algorithms. We refrain, however, from giving any recommendation concerning which (17)O-correction algorithm to use in order to obtain delta(13)C data in the most accurate way. The present work illuminates the need to reconsider recommendations concerning the correction algorithm. PMID:12720281This work presents a study aimed at the theoretical prediction of pK(a) values of aminopyridines, as a factor responsible for the activity of these compounds as blockers of the voltage-dependent K(+) channels. To cover a large range of pK(a) values, a total of seven substituted pyridines is considered as a calibration set: pyridine, 2-aminopyridine, 3-aminopyridine, 4-aminopyridine, 2-chloropyridine, 3-chloropyridine, and 4-methylpirydine. Using ab initio G1, G2 and G3 extrapolation methods, and the CPCM variant of the Polarizable Continuum Model for solvation, we calculate gas phase and solvation free energies. pK(a) values are obtained from these data using a thermodynamic cycle for describing protonation in aqueous and gas phases. The results show that the relatively inexpensive G1 level of theory is the most accurate at predicting pK(a) values in aminopyridines. The highest standard deviation with respect to the experimental data is 0.69 pK(a) units for absolutevalues calculations. The difference increases slightly to 0.74 pK(a) units when the pK(a) is computed relative to the pyridine molecule. Considering only compounds at least as basic as pyridine (the values of interest for bioactive aminopyridines) the error falls to 0.10 and 0.12 pK(a) units for the absolute and relative computations, respectively. The technique can be used to predict the effect of electronegative substituents in the pK(a) of 4-AP, the most active aminopyridine considered in this work. Thus, 2-chloro and 3-chloro-4-aminopyridine are taken into account. The results show a decrease of the pK(a), suggesting that these compounds are less active than 4-AP at blocking the K(+) channel. PMID:16844281
The prevalence of antibody to LAV/HTLV-III among homosexual men attending a community clinic and a sexually transmitted disease clinic in Seattle, Washington in early 1985 was 42 per cent and 32 per cent, respectively. Seropositivity was apparently not related to age or number of sexual partners. The high prevalence of LAV/HTLV-III seropositivity in an area where overt AIDS (acquired immune deficiency syndrome) is still relatively uncommon suggests that public health measures to prevent acquisition and transmission of LAV/HTLV-III should be a high priority even in areas with low incidences of AIDS. PMID:3008580
This paper illustrates the role of a "Thinking-about-Derivatives" task in identifying learners' derivative conceptions and for promoting their critical thinking about derivatives of absolutevalue functions. The task included three parts: "Define" the derivative of a function f(x) at x = x[subscript 0], "Solve-if-Possible" the derivative of f(x) =…
The massively parallel computation of absolute binding free energy with a well-equilibrated system (MP-CAFEE) has been developed [H. Fujitani, Y. Tanida, M. Ito, G. Jayachandran, C.D. Snow, M.R. Shirts, E.J. Sorin, V.S. Pande, J. Chem. Phys. 123 (2005) 084108]. As an application, we perform the binding affinity calculations of six theophylline-related ligands with RNA aptamer. Basically, our method is applicable when using many compute nodes to accelerate simulations, thus a parallel computing system is also developed. To further reduce the computational cost, the adequate non-uniform intervals of coupling constant λ, connecting two equilibrium states, namely bound and unbound, are determined. The absolute binding energies Δ G thus obtained have effective linear relation between the computed and experimental values. If the results of two other different methods are compared, thermodynamic integration (TI) and molecular mechanics Poisson-Boltzmann surface area (MM-PBSA) by the paper of Gouda et al. [H. Gouda, I.D. Kuntz, D.A. Case, P.A. Kollman, Biopolymers 68 (2003) 16], the predictive accuracy of the relative values ΔΔ G is almost comparable to that of TI: the correlation coefficients ( R) obtained are 0.99 (this work), 0.97 (TI), and 0.78 (MM-PBSA). On absolute binding energies meanwhile, a constant energy shift of ˜-7 kcal/mol against the experimental values is evident. To solve this problem, several presumable reasons are investigated.
The presence of core antigens of retrovirus HTLV-III/LAV, referred to as "AIDS-related virus" (AV), has been sought in lymph node samples of patients with persistent generalized lymphadenopathy (PGL, 28 patients), prodromal AIDS (1 patient) and AIDS with Kaposi sarcoma (3 patients). In 30 patients the deposition of viral antigens, detected by monoclonal antibodies to HTLV-III and LAV, could be observed within the germinal centers (GCs) primarily within the extracellular network of immune complexes, and the two patients who were negative were atypical. No AV could be found in normal tonsil or in samples with follicular hyperplasia of unknown etiology (20 cases). These findings, taken together with the ultrastructural identification of typical retrovirus particles in all 9 PGL and 2 AIDS cases studied, indicates that the network of follicular dendritic (FD) cells is an important reservoir of AV virus antigen at this site. The persistence of this retrovirus inside the GCs helps explain how the follicular hyperplasia affecting FD cells and B blasts in PGL may in progressive cases be accompanied by destruction of FD cells and gradual development of T4+ lymphopenia. T4+ T cells may circulate through the GCs and become infected with AV there. In addition, the identification of retrovirus antigen in situ may be of diagnostic value. ImagesFigure 1Figure 2 PMID:3008562
The Global Initiative for Chronic Obstructive Lung Disease (GOLD) severity criterion for COPD is used widely in clinical and research settings; however, it requires the use of ethnic- or population-specific reference equations. We propose two alternative severity criteria based on absolute post-bronchodilator FEV1 values (FEV1 and FEV1/height2) that do not depend on reference equations. We compared the accuracy of these classification schemasto those based on % predicted values (GOLD criterion) and Z-scores of post-bronchodilator FEV1 to predict COPD-related functional outcomes or percent emphysema by computerized tomography of the lung. We tested the predictive accuracy of all severity criteria for the 6-minute walk distance (6MWD), St. George's Respiratory Questionnaire (SGRQ), 36-item Short-Form Health Survey physical health component score (SF-36) and the MMRC Dyspnea Score. We used 10-fold cross-validation to estimate average prediction errors and Bonferroni-adjusted t-tests to compare average prediction errors across classification criteria. We analyzed data of 3772 participants with COPD (average age 63 years, 54% male). Severity criteria based on absolute post-bronchodilator FEV1 or FEV1/height2 yielded similar prediction errors for 6MWD, SGRQ, SF-36 physical health component score, and the MMRC Dyspnea Score when compared to the GOLD criterion (all p > 0.34); and, had similar predictive accuracy when compared with the Z-scores criterion, with the exception for 6MWD where post-bronchodilator FEV1 appeared to perform slightly better than Z-scores (p = 0.01). Subgroup analyses did not identify differences across severity criteria by race, sex, or age between absolutevalues and the GOLD criterion or one based on Z-scores. Severity criteria for COPD based on absolutevalues of post-bronchodilator FEV1 performed equally as well as did criteria based on predicted values when benchmarked against COPD-related functional and structural outcomes, are simple to use
All isotope amount ratios (hereafter referred to as isotope ratios) produced and measured on any mass spectrometer are biased. This unfortunate situation results mainly from the physical processes in the source area where ions are produced. Because the ionized atoms in poly-isotopic elements have different masses, such processes are typically mass dependent and lead to what is commonly referred to as mass fractionation (for thermal ionization and electron impact sources) and mass bias (for inductively coupled plasma sources.) This biasing process produces a measured isotope ratio that is either larger or smaller than the "true" ratio in the sample. This has led to the development of numerous fractionation "laws" that seek to correct for these effects, many of which are not based on the physical processes giving rise to the biases. The search for tighter and reproducible precisions has led to two isotope ratio measurement systems that exist side-by-side. One still seeks to measure "absolute" isotope ratios while the other utilizes an artifact based measurement system called a delta-scale. The common element between these two measurement systems is the utilization of isotope reference materials (iRMs). These iRMs are used to validate a fractionation "law" in the former case and function as a scale anchor in the latter. Many value assignments of iRMs are based on "best measurements" by the original groups producing the reference material, a not entirely satisfactory approach. Other iRMs, with absolute isotope ratio values, have been produced by calibrated measurements following the Atomic Weight approach (AW) pioneered by NBS nearly 50 years ago. Unfortunately, the AW is not capable of calibrating the new generation of iRMs to sufficient precision. So how do we get iRMs with isotope ratios of sufficient precision and without bias? Such a focus is not to denigrate the extremely precise delta-scale measurements presently being made on non-traditional and tradition
Currently, spectrophotometric standard reference materials are calibrated only by using the illumination and viewing geometries recommended by the Commission Internationale de l'Eclairage, and for some geometries the spectral range is limited to the visible wavelengths. A need exists for procedures that calibrate standards at many other geometries and for a broader spectral range. Two methods for calibrating the spectral bidirectional reflectance factor are described. The absolute bidirectional reflectance factor of a sintered polytetrafluoroethylene (PTFE) sample is determined for nearly all the possible illumination and viewing geometries from 400 nm to 2500 nm. The references are a 45/0 reflectance standard calibrated by the National Institute of Standards and Technology and a sintered PTFE sample with a directional, hemispherical reflectance factor traceable to the Institute. The results of the two methods agree to within 0.01 in reflectance factor values. With this PTFE sample as a transfer standard, the instrument described can also be used to measure the absolute bidirectional reflectance factor at nearly all the illumination and viewing geometries from 400 nm to 2500 nm. PMID:20820258 P<0.0001 for both genders and remained statistically significant after adjusting for confounding factors. Hazard ratios of type 2 diabetes associated with 1 standard deviation increase in waist circumference were 1.7 (95%CI 1.3 to 2.2) for males and 2.1 (95%CI 1.7 to 2.6) for females. At 45 years of age with baseline waist circumference of 100 cm, a male had an absolute diabetic risk of 10.9%, while a female had a 14.3% risk of the disease. Conclusions The constructed model predicts the 10-year absolute diabetes risk in an Aboriginal Australian communityIt is well known that gravity values have been decreasing in Southeast Alaska, mainly due to glacier mass changes from the end of the Little Ice Age to the present. For example, absolute gravity gravity gravity values at 6 deviations that are too large to render statistically significant results. This work presents a novel correction method of a very low mathematical and numerical complexity that can reduce the standard deviation of such results and increase their statistical significance. Two conditions are to be met: the inter-system variations of matter while its absolutevalue8E5/LAV cells harbor a single HIV provirus, and are used frequently to generate standards for HIV genome quantification. Using flow cytometry-based in situ mRNA hybridization validated by qPCR, we find that different batches of 8E5 cells contain varying numbers of cells lacking viral mRNA and/or viral genomes. These findings raise concerns for studies employing 8E5 cells for quantitation, and highlight the value of mRNA FISH and flow cytometry in the detection and enumeration of HIV-positive cells. PMID:27515378
Stored blood samples from 164 intravenous drug abusers who attended a Scottish general practice were tested for HTLV-III/LAV (human T cell lymphotropic virus type III/lymphadenopathy associated virus) infection. Of those tested, 83 (51%) were seropositive, which is well above the prevalence reported elsewhere in Britain and Europe and approaches that observed in New York City. The timing of taking samples of negative sera and continued drug use suggest that as many as 85% of this population might now be infected. The infection became epidemic in late 1983 and early 1984, thereafter becoming endemic. The practice of sharing needles and syringes correlated with seropositivity, which, combined with the almost exclusive intravenous use of heroin and other behavioural patterns, may explain the high prevalence of HTLV-III/LAV infection in the area. Rapid and aggressive intervention is needed to control the spread of infection. PMID:3081158
Unbiased symmetric metrics provide a useful measure to quickly compare two datasets, with similar interpretations for both under and overestimations. Two examples include the normalized mean bias factor and normalized mean absolute error factor. However, the original formulations...
Unbiased symmetric metrics provide a useful measure to quickly compare two datasets, with similar interpretations for both under and overestimations. Two examples include the normalized mean bias factor and normalized mean absolute error factor. However, the original formulations of these metrics are only valid for datasets with positive means. This paper presents a methodology to use and interpret the metrics with datasets that have negative means. The updated formulations give identical results compared to the original formulations for the case of positive means, so researchers are encouraged to use the updated formulations going forward without introducing ambiguity.We made a systematic separation of both the neutral phases using the atlases of 21-cm profiles of Heiles & Habing (1974) and Colomb et al. (1980), complemented with other data. First, we fitted the emission of the warm neutral medium (WNM) by means of a broad Gaussian curve (velocity dispersion sigma approximately 10-14 km/s). We derived maps of the column densities NWH and the radial velocities VW of the WNM. Its overall distribution appears to be very inhomogeneous with a large hole in the range b greater than or equal to +50 deg. However, if the hole is excluded, the mean latitude-profiles admit a rough cosec absolutevalue of b-fit common to both hemispheres. A kinematical analysis of VW for the range 10 deg less than or equal to absolutevalue of b less than or equal to 40 deg indicates a mean differential rotation with a small nodal deviation. At absolutevalue of b greater than 50 deg VW is negative, with larger values and discontinuities in the north. On the mean, sigma increases for absolutevalue of b decreasing, as is expected from differential rotation. From a statistical study of the peaks of the residual profiles we derived some characteristics of the cold neutral medium (CNM). The latter is generally characterized by a single component of sigma approximately 2-6 km/s. Additionally we derived the sky-distribution of the column densities NCH and the radial velocities VC of the CNM within bins of 1.2 deg sec b x 1 deg in l, b. Furthermore, we focused on the characteristics of Linblad's feature A of cool gas by considering the narrow ridge of local H I, which appears in the b-V contour maps at fixed l (e.g. Schoeber 1976). The ridge appears to be the main component of the CNM. We suggest a scenario for the formulation and evolution of the Gould belt system of stars and gas on the basis of an explosive event within a shingle of cold dense gas tilted to the galactic plane. The scenario appears to be consistent with the results found for both the neutralvaluesvalue V(ub) by combining the partial branching fractions measured in ranges of the momentum transfer squared and theoretical calculations of the form factor. Using a recent lattice QCD calculation, we find absolutevalueCrucial to lava flow hazard assessment is the development of tools for real-time prediction of flow paths, flow advance rates, and final flow lengths. Accurate prediction of flow paths and advance rates requires not only rapid assessment of eruption conditions (especially effusion rate) but also improved models of lava flow emplacement. Here we present the LAV@HAZARD web-GIS framework, which combines spaceborne remote sensing techniques and numerical simulations for real-time forecasting of lava flow hazards. By using satellite-derived discharge rates to drive a lava flow emplacement model, LAV@HAZARD allows timely definition of parameters and maps essential for hazard assessment, including the propagation time of lava flows and the maximum run-out distance. We take advantage of the flexibility of the HOTSAT thermal monitoring system to process satellite images coming from sensors with different spatial, temporal and spectral resolutions. HOTSAT was designed to ingest infrared satellite data acquired by the MODIS and SEVIRI sensors to output hot spot location, lava thermal flux and discharge rate. We use LAV@HAZARD to merge this output with the MAGFLOW physics-based model to simulate lava flow paths and to update, in a timely manner, flow simulations. Thus, any significant changes in lava discharge rate are included in the predictions. A significant benefit in terms of computational speed was obtained thanks to the parallel implementation of MAGFLOW on graphic processing units (GPUs). All this useful information has been gathered into the LAV@HAZARD platform which, due to the high degree of interactivity, allows generation of easily readable maps and a fast way to explore alternative scenarios. We will describe and demonstrate the operation of this framework using a variety of case studies pertaining to Mt Etna, Sicily. Although this study was conducted on Mt Etna, the approach used is designed to be applicable to other volcanic areas around the worldHigh-energy gamma-ray (energy above 35 MeV) data from the SAS 2 satellite have been used to compare the intensity distribution of gamma rays with that of neutral hydrogen (H I) density along the line of sight, at high galactic latitudes (absolutevalues greater than 30 deg). A model has been constructed for the case where the observed gamma-ray intensity has been assumed to be the sum of a galactic component proportional to the H I distribution plus an isotropic extragalactic emission. A chi-squared test of the model parameters indicates that about 30% of the total high-latitude emission may originate within the GalaxyvalueThe human immunodeficiency virus type 2 (HIV-2) strain LAV-2/B is able to infect a variety of human cell lines via a CD4-independent pathway. We have used the glycosylation inhibitors tunicamycin, swainsonine, and deoxymannojirimycin to further characterize this putative alternative receptor for HIV-2 (LAV-2/B). These antibiotics resulted in an increase (5- to 30-fold) in the susceptibility of a variety of CD4- human cell lines to infection by LAV-2/B (RD, HeLa, HT29, Rsb, Heb7a, Hos, and Daudi). Several nonprimate cell lines (mink Mv-1-lu, rabbit SIRC, hamster a23, mouse NIH 3T3, cat CCC, and rat HSN) remained resistant to infection by LAV-2/B after treatment with glycosylation inhibitors, suggesting that they do not express the HIV-2 CD4-independent receptor. Two of these nonprimate cell lines are readily infected by HIV-2 when they express CD4 (Mv-1-lu and CCC). Treatment of human cells with neuraminidase had no effect on subsequent infection by LAV-2/B, suggesting that the increase in susceptibility to infection of deglycosylated cells is not due to a change in the electrostatic charge of the cell surface. Treatment of RD CD4- cells and HeLa CD4+ cells with a variety of proteases resulted in a 75 to 90% decrease in infection by LAV-2/B when compared with untreated cells. Taken together, all these data suggest that HIV-2 can utilize a membrane glycoprotein other than CD4 to attach and fuse with a variety of human cells. PMID:7745686
Yb(3+)-doped Ba2LaV3O11 vanadate phosphors with near-infrared (NIR) emission are synthesized via the sol-gel method. The phase purity and structure of samples are characterized by X-ray diffraction (XRD). The electronic structure of the self-activated phosphor host Ba2LaV3O11 is estimated by density functional theory (DFT) calculation, and the host absorption is mainly ascribed to the charge transition from the O-2p to V-3d states. Photoluminescence emission (PL) and excitation (PLE) spectra, decay curves, absorption spectra and theoretical quantum yields of samples are also investigated. Results indicate that Ba2LaV3O11:Yb(3+) phosphors have strong broad band absorption and efficient NIR emission, which matches well with the spectral response of the Si-based solar cells. The energy transfer processes from [VO4](3-) to Yb(3+) and possible transfer models are proposed based on the concentration of Yb(3+) ions. Results demonstrate that Ba2LaV3O11:Yb(3+) phosphors might act as a promising NIR DC solar spectral converter to enhance the efficiency of the silicon solar cells by utilizing broad band absorption of the solar spectrum. PMID:26388401 aim of this study is to compare a two-wavelength light emitting diode-based tissue oximeter (INVOS), which is designed to show trends in tissue oxygenation, with a four-wavelength laser-based oximeter (FORE-SIGHT), designed to deliver absolutevalues of tissue oxygenation. Simultaneous values of cerebral tissue oxygenation (StO2) are measured using both devices in 15 term and 15 preterm clinically stable newborns on the first and third day of life. Values are recorded simultaneously in two periods between which oximeter sensor positions are switched to the contralateral side. Agreement between StO2 values before and after the change of sensor position is analyzed. We find that mean cerebral StO2 values are similar between devices for term and preterm babies, but INVOS shows StO2 values spread over a wider range, with wider standard deviations than shown by the FORE-SIGHT. There is relatively good agreement with a bias up to 3.5% and limits of agreement up to 11.8%. Measurements from each side of the forehead show better repeatability for the FORE-SIGHT monitor. We conclude that performance of the two devices is probably acceptable for clinical purposes. Both performed sufficiently well, but the use of FORE-SIGHT may be associated with tighter range and better repeatability of data.
Observations of the Hubble Space Telescope (HST) between 1993 May 31 and 1993 July 19 in 20 epochs in the F555W passband and five epochs in the F785LP passband have led to the discovery of 14 Cepheids in the Amorphous galaxy NGC 5253. The apparent V distance modulus is (m-M)(sub AV) = 28.08 +/- 0.10 determined from the 12 Cepheids with normal amplitudes. The distance modulus using the F785LP data is consistent with the V value to within the errors. Five methods used to determine the internal reddening are consistent with zero differential reddening, accurate to a level of E(B-V) less than 0.05 mag, over the region occupied by Cepheids and the two supernovae (SNe) produced by NGC 5253. The apparent magnitudes at maximum for the two SNe in NGC 5253 are adopted as B(sub max) = 8.33 +/- 0.2 mag for SN 1895B, and B(sub max) = 8.56 +/- 0.1 and V(sub max) = 8.60 +/- 0.1 for SN 1972E which is a prototype SN of Type Ia. The apparent magnitude system used by Walker (1923) for SN 1859B has been corrected to the modern B scale and zero point to determine its adopted B(sub max) valueDeterminations of the local correction (ΔR) to the globally averaged marine radiocarbon reservoir age are often isolated in space and time, derived from heterogeneous sources and constrained by significant uncertainties. Although time series of ΔR at single sites can be obtained from sediment cores, these are subject to multiple uncertainties related to sedimentation rates, bioturbation and interspecific variations in the source of radiocarbon in the analysed samples. Coral records provide better resolution, but these are available only for tropical locations. It is shown here that it is possible to use the shell of the long-lived bivalve mollusc Arctica islandica as a source of high resolution time series of absolutely-dated marine radiocarbon determinations for the shelf seas surrounding the North Atlantic ocean. Annual growth increments in the shell can be crossdated and chronologies can be constructed in a precise analogue with the use of tree-rings. Because the calendar dates of the samples are known, ΔR can be determined with high precision and accuracy and because all the samples are from the same species, the time series of ΔR values possesses a high degree of internal consistency. Presented here is a multi-centennial (AD 1593 - AD 1933) time series of 31 ΔR values for a site in the Irish Sea close to the Isle of Man. The mean value of ΔR (-62 14C yrs) does not change significantly during this period but increased variability is apparent before AD 1750Probing absolutevaluesvalues quantitiesvalues 162 and 266 A. The short-wavelength limit is determined only by the sensitivity of the current-measuring apparatus and by present knowledge of the photoionization processes that occur in the rate gases corresponding Mo Kα data. PMID:19461838valuevalues of damage can be compared with that of films of any thickness under different experimental conditions. PMID:22562941Geodetic GNSS applications routinely demand millimeter precision and transmission channeling msec differenceIn today's world, traffic jams waste hundreds of hours of our life. This causes many researchers try to resolve the problem with the idea of Intelligent Transportation System areSpectral reflectance measurements absolute measurements and most of them are co-located with continuous GPS. We give an overview of the sites, instrumentation and campaigns, and show examples of results achieved so farvaluevalue PMID:26382502value correlationsThe absolute instability is a subject of considerable physics interest as well as a major source of self-oscillations in the gyrotron traveling-wave amplifier (gyro-TWT). We present a theoretical study of the absolute instabilities in a TE01 mode, fundamental cyclotron harmonic gyro-TWT with distributed wall losses. In this high-order-mode circuit, absolute instabilities arise in a variety of ways, including overdrive of the operating mode, fundamental cyclotron harmonic interactions with lower-order modes, and second cyclotron harmonic interaction with a higher-order mode. The distributed losses, on the other hand, provide an effective means for their stabilization. The combined configuration thus allows a rich display of absolute instability behavior together with the demonstration of its control. We begin with a study of the field profiles of absolute instabilities, which exhibit a range of characteristics depending in large measure upon the sign and magnitude of the synchronous value of the propagation constant. These profiles in turn explain the sensitivity of oscillation thresholds to the beam and circuit parameters. A general recipe for oscillation stabilization has resulted from these studies and its significance to the current TE01 -mode, 94-GHz gyro-TWT experiment at UC Davis is discussed. PMID:15600760
There is currently no reliable method for early characterization of breast cancer response to neoadjuvant chemotherapy (NAC) [1,2]. Given that disruption of normal structural architecture occurs in cancer-bearing tissue, we hypothesize that further structural changes occur in response to NAC. Consequently, we are investigating the use of modalityindependent elastography (MIE) [3-8] as a method for monitoring mechanical integrity to predict long term outcomes in NAC. Recently, we have utilized a Demons non-rigid image registration method that allows 3D elasticity reconstruction in abnormal tissue geometries, making it particularly amenable to the evaluation of breast cancer mechanical properties. While past work has reflected relative elasticity contrast ratios [3], this study improves upon that work by utilizing a known stiffness reference material within the reconstruction framework such that a stiffness map becomes an absolute measure. To test, a polyvinyl alcohol (PVA) cryogel phantom and a silicone rubber mock mouse tumor phantom were constructed with varying mechanical stiffness. Results showed that an absolute measure of stiffness could be obtained based on a reference value. This reference technique demonstrates the ability to generate accurate measurements of absolute stiffness to characterize response to NAC. These results support that `referenced MIE' has the potential to reliably differentiate absolute tumor stiffness with significant contrast from that of surrounding tissue. The use of referenced MIE to obtain absolute quantification of biomarkers is also translatable across length scales such that the characterization method is mechanics-consistent at the small animal and human application with Standards series process service sectors are a major driver for the growth of the world economy, we are challenged to implement service-oriented infrastructure as e-Gov platform to achieve further growth and innovation for both developed and developing countries. According to recent trends in service industry, it is clarified that main factors for the growth of service sectors are investment into knowledge, trade, and the enhanced capacity of micro, small, and medium-sized enterprises (MSMEs). In addition, the design and deployment of public service platform require appropriate evaluation methodology. Reflecting these observations, this paper proposes macro-micro simulation approach to assess public values (PV) focusing on MSMEs. Linkage aggregate variables (LAVs) are defined to show connection between macro and micro impacts of public services. As a result, the relationship of demography, business environment, macro economy, and socio-economic impact are clarified and their values are quantified from the behavioral perspectives of citizens and firms.values aboutabsoluteIn fringe projection profilometry, a simplified method is proposed to recover absolute phase maps of two-frequency fringe patterns by using a unique mapping rule. The mapping rule is designed from the rounded phase values to the fringe order of each pixel. Absolute phase can be recovered by the fringe order maps. Unlike the existing techniques, where the lowest frequency of dual- or multiple-frequency fringe patterns must be single, the presented method breaks the limitation and simplifies the procedure of phase unwrapping. Additionally, due to many issues including ambient light, shadow, sharp edges, step height boundaries and surface reflectivity variations, a novel framework of automatically identifying and removing invalid phase values is also proposed. Simulations and experiments have been carried out to validate the performances of the proposed methodvalues absolute calibration of the instruments for this flight was accomplished at the National Bureau of Standards Synchrotron Radiation Facility which significantly improves calibration of solar measurements made in this spectral region. atmosphere fluxes for both early-type stars and DA white dwarfs, will be used for Voyager astronomical observationsA reciprocating quantum refrigerator is analyzed with the intention to study the limitations imposed by external noise. In particular we focus on the behavior of the refrigerator when it approaches the absolute zero. The cooling cycle is based on the Otto cycle with a working medium constituted by an ensemble of noninteracting harmonic oscillators. The compression and expansion segments are generated by changing an external parameter in the Hamiltonian. In this case the force constant of the harmonic oscillators mω2 is modified from an initial to a final value. As a result, the kinetic and potential energy of the system do not commute causing frictional losses. By proper choice of scheduling function ω(t) frictionless solutions can be obtained in the noiseless case. We examine the performance of a refrigerator subject to noise. By expanding from the adiabatic limit we find that the external noise, Gaussian phase, and amplitude noises reduce the amount of heat that can be extracted but nevertheless the zero temperature can be approached performedBackground The extent of immunosuppression and the probability of developing an AIDS-related complication in HIV-infected people is usually measured by the absolute number of CD4 positive T-cells. The percentage of CD4 positive cells is a more easily measured and less variable number. We analyzed sequential CD4 and CD8 numbers, percentages and ratios in 218 of our HIV infected patients to determine the most reliable predictor of an AIDS-related event. Results The CD4 percentage was an unsurpassed predictor of the occurrence of AIDS-related events when all subsets of patients are considered. The CD4 absolute count was the next most reliable, followed by the ratio of CD4/CD8 percentages. The value of CD4 percentage over the CD4 absolute count was seen even after the introduction of highly effective HIV therapy. Conclusion The CD4 percentage is unsurpassed as a parameter for predicting the onset of HIV-related diseases. The extra time and expense of measuring the CD4 absolute count may be unnecessary. PMID:16916461 atStatus-dependent strategies represent one of the most remarkable adaptive phenotypic plasticities. A threshold value for individual status (e.g., body size) is assumed above and below which each individual should adopt alternative tactics to attain higher fitness. This implicitly assumes the existence of an "absolute" best threshold value, so each individual chooses a tactic only on the basis of its own status. However, animals may be able to assess their status on the basis of surrounding individuals. This "relative" assessment considers a threshold value to be changeable depending on individual situations, which may result in significant differences in ecological and evolutionary dynamics compared with absolute assessment. Here, we incorporated Bayesian decision-making and adaptive dynamics frameworks to explore the conditions necessary for each type of assessment to evolve. Our model demonstrates that absolute assessment is always an evolutionarily stable strategy (ESS) in a stable environment, whereas relative assessment can be an ESS in stochastic environments. The consequences of future environmental change differ considerably depending on the assessment chosen. Our results underscore the need to better understand how individuals assess their own status when choosing alternative tactics. PMID:27322126value of 238U 204Pb = 9.09 ?? 0.06 for stratiform leads is little different from the value 8.99 ?? 0.05 originally computed by Ostic, Russell and Stanton. Absolutevalues for lead isotope ratios for all interlaboratory standard samples presently available from the literature are tabulated. ?? 1969873 K using a differential scanning calorimeter (DSC) and calculated the absolute entropyClinicians' work depends on sincere and complete disclosures from their patients; they honour this candidness by confidentially safeguarding the information received. Breaching confidentiality causes harms that are not commensurable with the possible benefits gained. Limitations or exceptions put on confidentiality would destroy it, for the confider would become suspicious and un-co-operative, the confidant would become untrustworthy and the whole climate of the clinical encounter would suffer irreversible erosion. Excusing breaches of confidence on grounds of superior moral values introduces arbitrariness and ethical unreliability into the medical context. Physicians who breach the agreement of confidentiality are being unfair, thus opening the way for, and becoming vulnerable to, the morally obtuse conduct of others. Confidentiality should not be seen as the cosy but dispensable atmosphere of clinical settings; rather, it constitutes a guarantee of fairness in medical actions. Possible perils that might accrue to society are no greater than those accepted when granting inviolable custody of information to priests, lawyers and bankers. To jeopardize the integrity of confidential medical relationships is too high a price to pay for the hypothetical benefits this might bring to the prevailing social order. PMID:3761330Eight pairs of enantiomeric neolignans, norlignans, and sesquineolignans (1a/1b-8a/8b), together with five known neolignans (9a/9b and 10-'-epoxy-8'-8''/7'-2''-Chaperones are fundamental to regulating the heat shock response, mediating protein recovery from thermal‐induced misfolding and aggregation. Using the QconCAT strategy‐peptides per protein to improve confidence in protein quantification. In contrast to the global proteome profile of S. cerevisiae in response to heat shock heat shock spatial, viscous stability analysis of Poiseuille pipe flow with superimposed solid body rotation is considered. For each value of the swirl parameter (inverse Rossby number) L>0, there exists a critical Reynolds number Rec)(L above which the flow first becomes convectively unstable to nonaxisymmetric disturbances with azimuthal wave number n=-1. This neutral stability curve confirms previous temporal stability analyses. From this spatial stability analysis, we propose here a relatively simple procedure to look for the onset of absolute instability that satisfies the so-called Briggs-Bers criterion. We find that, for perturbations with n=-1, the flow first becomes absolutely unstable above another critical Reynolds number Ret)(L>Rec)(L, provided that L>0.38, with Ret[right arrow]Rec as L[right arrow]infinity. Other values of the azimuthal wave number n are also considered. For Re>Ret)(L, the disturbances grow both upstream and downstream of the source, and the spatial stability analysis becomes inappropriate. However, for Ret, the spatial analysis provides a useful description on how convectively unstable perturbations become absolutely unstable in this kind of flow.
Chaperones are fundamental to regulating the heat shock-peptides per protein to improve confidence in protein quantification. In contrast to the global proteome profile of S. cerevisiae in response to heat shock heat shockA handful of events, such as the condensation of refractory inclusions and the formation of chondrules, represent important stages in the formation and evolution of the early solar system and thus are critical to understanding its development. Compared to the refractory inclusions, chondrules appear to have a protracted period of formation that spans millions of years. As such, understanding chondrule formation requires a catalog of reliable ages, free from as many assumptions as possible. The Pb-Pb chronometer has this potential; however, because common individual chondrules have extremely low uranium contents, obtaining U-corrected Pb-Pb ages of individual chondrules is unrealistic in the vast majority of cases at this time. Thus, in order to obtain the most accurate 238U/235U ratio possible for chondrules, we separated and pooled thousands of individual chondrules from the Allende meteorite. In this work, we demonstrate that no discernible differences exist in the 238U/235U compositions between chondrule groups when separated by size and magnetic susceptibility, suggesting that no systematic U-isotope variation exists between groups of chondrules. Consequently, chondrules are likely to have a common 238U/235U ratio for any given meteorite. A weighted average of the six groups of chondrule separates from Allende results in a 238U/235U ratio of 137.786 ± 0.004 (±0.016 including propagated uncertainty on the U standard [Richter et al. 2010]). Although it is still possible that individual chondrules have significant U isotope variation within a given meteorite, this value represents our best estimate of the 238U/235U ratio for Allende chondrules and should be used for absolute dating of these objects, unless such chondrules can be measured individually sourcevaluesvaluesRelative sea level curves contain coupled information about absolute sea level change and vertical lithospheric movement. Such curves may be constructed based on, for example tide gauge data for the most recent times and different types of geological data for ancient times. Correct account for vertical lithospheric movement is essential for estimation of reliable values of absolute sea level change from relative sea level data and vise versa. For modern times, estimates of vertical lithospheric movement may be constrained by data (e.g. GPS-based measurements), which are independent from the relative sea level data. Similar independent data do not exist for ancient times. The purpose of this study is to test two simple inversion approaches for simultaneous estimation of lithospheric uplift rates and absolute sea level change rates for ancient times in areas where a dense coverage of relative sea level data exists and well-constrained average lithospheric movement values are known from, for example glacial isostatic adjustment (GIA) models. The inversion approaches are tested and used for simultaneous estimation of lithospheric uplift rates and absolute sea level change rates in southwest Scandinavia from modern relative sea level data series that cover the period from 1900 to 2000. In both approaches, a priori information is required to solve the inverse problem. A priori information about the average vertical lithospheric movement in the area of interest is critical for the quality of the obtained results. The two tested inversion schemes result in estimated absolute sea level rise of ˜1.2/1.3 mm yr-1 and vertical uplift rates ranging from approximately -1.4/-1.2 mm yr-1 (subsidence) to about 5.0/5.2 mm yr-1 if an a priori value of 1 mm yr-1 is used for the vertical lithospheric movement throughout the study area. In case the studied time interval is broken into two time intervals (before and after 1970), absolute sea level rise values of ˜0.8/1.2 mm yr-1 (beforea role of adjuster to keep the tectonic stress at a constant level. The spatial range of stress rotation extends to 55 km from BBS. From comparison of the characteristics of the inverted stress field with the results of numerical simulation, we can conclude the friction coefficient of BBS is 0.3, which is a half of the standard value expected from rock experiments. However, this does not mean a weak SAF. In this case the absolute strength of BBS itself reaches 140 MPa at the intermediate depth (6 km) of the seismogenic zone, because of the high normal stress due to plate convergence at BBS Nivalues absolute calibration capability has not yet been fully developed. Because of this, PICS are primarily limited to providing only long term trending information for individual sensors or cross-calibration opportunities between two sensors. This paper builds an argument that PICS can be used more extensively for absolute calibration absolute calibration model that can be Système international d'unités (SI) traceable. These initial concepts suggest that absolute calibration using PICS is possible on a broad scale and can lead to improved on-orbit calibration capabilities for optical satellite sensorsFour new pyrrolidone substituted bibenzyls, dusuanlansins A-D (1-4) were isolated from the pseudo bulbs of Pleione bulbocodioides, along with 19 known compounds (5-23). Compounds 1-4 are two pairs of epimers of pyrrolidone substituted bibenzyls, which were separated successfully by a Chiralcel OD-RH C18 column. Their absolute configurations were elucidated by calculated ECD. Biological investigations showed that compounds 5 and 7 exhibited potent anti-inflammatory activities on LPS-stimulated NO production in BV-2 microglial cells, with IC50 values of 2.46 and 3.14μM, respectively. PMID:25647325 useAn experimental examination of the absolute intensity, polarization, and relative line intensities of rotational Raman scattering (RRS) from N2, O2, and CO2 is reported. The absolute scattering intensity for N2 is characterized by its differential cross section for backscattering of incident light at 647.1 nm, which is calculated from basic measured values. The ratio of the corresponding cross section for O2 to that for N2 is 2.50 plus or minus 5 percent. The intensity recent for N2, O2, and CO2 are shown to compare favorably to values calculated from recent measurements of the depolarization of Rayleigh scattering plus RRS. Measured depolarizations of various RRS lines agree to within a few percent with the theoretical value of 3/4. Detailed error analyses are presented for intensity and depolarization measurements. Finally, extensive RRS spectra at nominal gas temperatures of 23 C, 75 C, and 125 C are presented and shown to compare favorably to theoretical predictions measurementsvaluesEight pairs of enantiomeric neolignans, norlignans, and sesquineolignans (1a/1b–8a/8b), together with five known neolignans (9a/9b and 10–′-epoxy-8′-8′′/7′-2′′– sensorAbsolutevalues the of that autos Therefore capacitive absolute pressure calculatedA sizable body of evidence has shown that the brain computes several types of value-related signals to guide decision making, such as stimulus values, outcome values, and prediction errors. A critical question for understanding decision-making mechanisms is whether these value signals are computed using an absolute or a normalized code. Under an absolute code, the neural response used to represent the value of a given stimulus does not depend on what other values might have been encountered. By contrast, under a normalized code, the neural response associated with a given value depends on its relative position in the distribution of values. This review provides a simple framework for thinking about value normalization, and uses it to evaluate the existing experimental evidence. PMID:22939568 of 39 target proteins upon antibiotic treatment, which correlate well with literature values. The described method is generally applicable and exploits the inherent performance advantages of SRM theAbstract The neutrophil/lymphocyte ratio (NLR), lymphocyte/monocyte ratio (LMR), and absolute lymphocyte count/absolute monocyte count prognostic score (ALC/AMC PS) have been described as the most useful prognostic tools for patients with diffuse large B-cell lymphoma (DLBCL). We retrospectively analyzed 148 Taiwanese patients with newly diagnosed diffuse large B-cell lymphoma under rituximab (R)-CHOP-like regimens from January 2001 to December 2010 at the Tri-Service General Hospital and investigated the utility of these inexpensive tools in our patients. In a univariate analysis, the NLR, LMR, and ALC/AMC PS had significant prognostic value in our DLBCL patients (NLR: 5-year progression-free survival [PFS], P = 0.001; 5-year overall survival [OS], P = 0.007. LMR: PFS, P = 0.003; OS, P = 0.05. ALC/AMC PS: PFS, P < 0.001; OS, P < 0.001). In a separate multivariate analysis, the ALC/AMC PS appeared to interact less with the other clinical factors but retained statistical significance in the survival analysis (PFS, P = 0.023; OS, P = 0.017). The akaike information criterion (AIC) analysis produced scores of 388.773 in the NLR, 387.625 in the LMR, and 372.574 in the ALC/AMC PS. The results suggested that the ALC/AMC PS appears to be more reliable than the NLR and LMR and may provide additional prognostic information when used in conjunction with the International Prognostic Index dynamics radiox photons standards concentration. Our results demonstrate that FSCAV can be effectively used in brain slices to measure prolonged changes in extracellular level of endogenous DA expressed as absolutevalues, complementing studies conducted in vivo with microdialysis. PMID:26322962Clinical risk assessment involves absolute risk measures, but information on modifying risk and preventing cancer is often communicated in relative terms. To illustrate the potential impact of risk factor factors in risk assessment and in sharing information with patients of their absolute risks with and without modifiable risk factors. PMID:26012643 changes gravityOn orbit measurements starting in the late 1970's, have revealed the 11 year cycle of the Total Solar Irradiance (TSI). However, the absolute results from individual experiments differ although all instrument teams claim to measure an absolutevalueThe absolute configuration of the naturally occurring isomers of 6β-benzoyloxy-3α-tropanol (1) has been established by the combined use of chiral high-performance liquid chromatography with electronic circular dichroism detection and optical rotation detection. For this purpose (±)-1, prepared in two steps from racemic 6-hydroxytropinone (4), was subjected to chiral high-performance liquid chromatography with electronic circular dichroism and optical rotation detection allowing the online measurement of both chiroptical properties for each enantiomer, which in turn were compared with the corresponding values obtained from density functional theory calculations. In an independent approach, preparative high-performance liquid chromatography separation using an automatic fraction collector, yielded an enantiopure sample of OR (+)-1 whose vibrational circular dichroism spectrum allowed its absolute configuration assignment when the bands in the 1100-950 cm(-1) region were compared with those of the enantiomers of esters derived from 3α,6β-tropanediol. In addition, an enantiomerically enriched sample of 4, instead of OR (±)-4, was used for the same transformation sequence, whose high-performance liquid chromatography follow-up allowed their spectroscopic correlation. All evidences lead to the OR (+)-(1S,3R,5S,6R) and OR (-)-(1R,3S,5R,6S) absolute configurations, from where it follows that samples of 1 isolated from Knightia strobilina and Erythroxylum zambesiacum have the OR (+)-(1S,3R,5S,6R) absolute configuration, while the sample obtained from E. rotundifolium has the OR (-)-(1R,3S,5R,6S) absolute configuration. PMID:27214755 Furthermore modification absolute calibrations as the defacto standard. The IGS now distributes a catalog of absolute calibr absolute calibrations, and if/when it is valid to combine the NGS and IGS catalogs. Therefore, in this study, we compare the NGS catalog of relative calibrations against the IGS catalog of absolute calibrvaluesption spect Spectroscopy experimentResults are presented on measurements of absolute state-selected total cross sections for O2(+), CO2(+), CO(+), and C(+) produced in the reaction between O(+)(4S) and CO2, which were conducted in the center-of-mass collision energy (Ecm) range of 0.2-150 eV. It was found that, with increasing collisional energy, the cross section of O2(+) dropped off rapidly and became essentially zero at Ecm above 3 eV. The dependence of O2(+) cross section on the Ecm is consistent with a collision complex mechanism for the reaction between O(+)(4S) and CO2 yielding CO2(+) + O. The values for O2(+) obtained in this experiment were significantly higher than those reported by Rutherford and Vroom (1976). test | 677.169 | 1 |
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Algebra & Geometry: An Introduction to University Mathematics provides a bridge between high school and undergraduate mathematics courses on algebra and geometry. The author shows students how mathematics is more than a collection of methods by presenting important ideas and their historical ori...
Choose the Correct Solution Method for Your Optimization Problem
Optimization: Algorithms and Applications presents a variety of solution techniques for optimization problems, emphasizing concepts rather than rigorous mathematical details and proofs.
The book covers both gradient and stochastic me tex...
The new edition of this popular text is revised to meet the suggestions of users of the previous edition. A readable yet rigorous approach to an essential part of mathematical thinking, this text bridges the gap between classic theoretical texts and less rigorous ones, providing a smooth transition ...
Convex Optimization for Signal Processing and Communications: From Fundamentals to Applications provides fundamental background knowledge of convex optimization, while striking a balance between mathematical theory and applications in signal processing and communications.
In addition to...
What Is Combinatorics Anyway?
Broadly speaking, combinatorics is the branch of mathematics dealing
with different ways of selecting objects from a set or arranging objects. It
tries to answer two major kinds of questions, namely, counting questions: how many ways can a selection or arrangement be...
Uncertainty quantification in composite materials and structures has gained immense attention from the research community over the last few decades. This book presents efficient uncertainty quantification schemes following meta-model-based approaches for stochasticity in material and geometric...
This second edition of The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance has been fully updated and revised to reflect recent developments in the theory and practical applications of wavelet transform methods.
The book is...
This book covers the development of methods for detection and estimation of changes in complex systems. These systems are generally described by nonstationary stochastic models, which comprise both static and dynamic regimes, linear and nonlinear dynamics, and constant and time-variant structures...
Linear and Complex Analysis for Applications aims to unify various parts of mathematical analysis in an engaging manner and to provide a diverse and unusual collection of applications, both to other fields of mathematics and to physics and engineering. The book evolved from several of the author's...
Advanced Engineering Mathematics with MATLAB, Fourth Edition builds upon three successful previous editions. It is written for today's STEM (science, technology, engineering, and mathematics) student. Three assumptions under lie its structure: (1) All students need a firm grasp of the traditional...
If you know a little bit about financial mathematics but don't yet know a lot about programming, then C++ for Financial Mathematics is for you.
C++ is an essential skill for many jobs in quantitative finance, but learning it can be a daunting prospect. This book gathers together everything you need...
Actions and Invariants of Algebraic Groups, Second Edition presents a self-contained introduction to geometric invariant theory starting from the basic theory of affine algebraic groups and proceeding towards more sophisticated dimensions." Building on the first edition, this book provides an...
Abstract Algebra: A Gentle Introduction advantages a trend in mathematics textbook publishing towards smaller, less expensive and brief introductions to primary courses. The authors move away from the 'everything for everyone' approach so common in textbooks. Instead, they provide | 677.169 | 1 |
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Back to school with a great Review!!! Help your students to organize, refresh, and remember important topics and concepts with this Review Foldable Flip Book - Study Guide which covers over 50 topics.
Now includes 2 versions!!!
By popular request, I have designed a second version of this popular resource to align to the STAAR standards for Texas. If you teach in Texas, or if your state or district is aligned with STAAR, then this version is perfect for you as an end of course review for Algebra 1 or a back to school Review for Algebra 2. Both versions come both with all the notes included and also with a student fill-in version. All 4 Flip Books kits in this one resource!!!
Easy to make - No mess, no cutting, no glue, and a paper saver, too. Three sheets of paper make one book.
Perfect for End of year Review for Algebra 1 and also for the beginning of the course for Algebra 2 and Geometry students.
Please click on the green star next to my name to follow me and be the first hear of sales, and new, engaging resources for all of your classes.
This purchase is for one teacher only. Additional teachers must purchase their own license. You may not upload this resource to the internet in any form. Multi - user licenses available. Custom Bundle requests gladly accepted. Note: This resource is not editable3.00. | 677.169 | 1 |
Guide to using Khan Academy in higher education
College students who have gaps in knowledge can struggle in math, especially within courses that build on previous learning and on placement exams . These students need relevant, high-quality resources that can be accessible at any time.
Khan Academy is a free, online tool, which houses thousands of instructional math videos, exercises, and a personalized learning platform in which students can learn at their own pace through an entire subject.
Here's what you can do to help students get started using Khan Academy.
Encourage your students to use missions for supplemental practice
Missions are individualized math experiences within Khan Academy which are grouped by subjects. Students can use our personalized missions to guide students within their relevant, standards-aligned math content.
Students can use missions to:
fill in gaps at their own pace within a certain subject,
master skills that are challenging and appropriate for their level, and
use hints and videos immediately when they need help
Our most popular mission for college math is Algebra Basics. This subject is ideal for anyone looking to prepare for a high school or college placement exam. It covers all of the foundational ideas in algebra and related topics in pre-algebra and geometry. If you're looking for more exhaustive coverage, then the Algebra I and Algebra II subjects may be better for you. Learn about best practices when using Khan Academy with your students here.
Let students know that they have a free tutor (who doesn't ever sleep!)
If students need help with a particular mathematical concept, missed a lecture and want to catch up, or just need a refresher, they can search our site to find relevant math videos and practice exercises.
Letting students know that they have the option to get help at any time, without fear of judgment about what they don't yet know, can be empowering and can bring them one step closer to success. | 677.169 | 1 |
This book leads readers through a progressive explanation of what mathematical proofs are, why they are important, and how they work, along with a presentation of basic techniques used to construct proofs. The Second Edition presents more examples, more exercises, a more complete treatment of mathematical induction and set theory, and it incorporates suggestions from students and colleagues. Since the mathematical concepts used are relatively elementary, the book can be used as a supplement in any post-calculus course.
This title has been successfully class-tested for years. There is an index for easier reference, a more extensive list of definitions and concepts, and an updated bibliography. An extensive collection of exercises with complete answers are provided, enabling students to practice on their own. Additionally, there is a set of problems without solutions to make it easier for instructors to prepare homework assignments.
* Successfully class-tested over a number of years * Index for easy reference * Extensive list of definitions and concepts * Updated biblography
"synopsis" may belong to another edition of this title.
Book Description:
This simple, updated guidebook shows how to read, understand and construct proofs
From the Back Cover:
Well-regarded in its first edition, The Nuts and Bolts of Proofs leads readers through an explanation of mathematical proofs, the basic rules of logic that makes them work and basic techniques used to construct them.
The Nuts and Bolts of Proofs can be used to supplement upper-level mathematics courses. It can also be used effectively for self-learning. The mathematical concepts are elementary, the presentation is accessible and exercises with solutions are included.
This second edition presents more examples, more exercises and a more complete treatment of mathematical induction and set theory. This book has been class tested with success for many years.
"I particularly like the 'Warning' given just before the solutions. This provides excellent instructions on how to work at the writing of a proof." --Patricia R. Allaire, Queensborough Community College
"The author puts a lot of care into explaining the components and logic of mathematical proofs. The style allowed a slow yet methodical reading of the various explanations and examples." --Joel Zinn, Texas A&M University | 677.169 | 1 |
This book looks at the kind of mathematics required in the construction of software - particularly that used in the early stages of design. It provides an introduction to the subject, carefully explaining the need for mathematics in the software development process and showing in clear and simple terms what this mathematics is and how it can be used. The authors use the current Z notation, which is now an industry standard. The ideas and notation that underpin Z are addressed, thus forming a solid basis for the user to move on to more advanced texts at a later stage. Both authors have had considerable experience in this area, and the book anticipates and avoids learning difficulties. An integrated and progressive style is adopted and each new topic is illustrated by relevant examples. The book will provide both experienced programmers and beginners with a thorough insight into the advantages to be gained from using mathematics in software programming.
Book Description Ellis Horwood Ltd , Publisher, 1990. Paperback. Book Condition: Very Good. Mathematics of Software Construction (Ellis Horwood Series in Mathematics & Its Applications35633885 | 677.169 | 1 |
lied Partial Differential Equations
Buy Now orders cannot be placed without a valid Australian shipping address.
This textbook is for the standard, one-semester, junior-senior course that often goes by the title "Elementary Partial Differential Equations" or "Boundary Value Problems." The audience consists of students in mathematics, engineering, and the physical sciences. The topics include derivations of some of the standard models of mathematical physics (e.g., the heat equation, the wave equation, and Laplace 's equation) and methods for solving those equations on unbounded and bounded domains (transform methods and eigenfunction expansions). Prerequisites include multivariable calculus and elementary differential equations. The text differs from other texts in that it is a brief treatment; yet it provides coverage of the main topics usually studied in the standard course as well as an introduction to using computer algebra packages to solve and understand partial differential equations. The many exercises help students sharpen their computational skills by encouraging them to think about concepts and derivations. The student who reads this book carefully and solves most of the problems will have a sound knowledge base for a second-year partial differential equations course where careful proofs are constructed or for upper division courses in science and engineering where detailed applications of partial differential equations are introduced.To give this text an even wider appeal, the second edition has been updated with a new chapter on partial differential equation models in biology, and with various examples from the life sciences that have been added throughout the text. There are more exercises, as well as solutions and hints to some of the problems at the end | 677.169 | 1 |
In real-world problems related to finance, business, and management, mathematicians and economists frequently encounter optimization problems. In this classic book, George Dantzig looks at a wealth of examples and develops linear programming methods for their solutions. He begins by introducing the basic theory of linear inequalities and describes the powerful simplex method used to solve them. Treatments of the price concept, the transportation problem, and matrix methods are also given, and key mathematical concepts such as the properties of convex sets and linear vector spaces are covered.
George Dantzig is properly acclaimed as the "father of linear programming." Linear programming is a mathematical technique used to optimize a situation. It can be used to minimize traffic congestion or to maximize the scheduling of airline flights. He formulated its basic theoretical model and discovered its underlying computational algorithm, the "simplex method," in a pathbreaking memorandum published by the United States Air Force in early 1948. "Linear Programming and Extensions" provides an extraordinary account | 677.169 | 1 |
Algebra II Topics by Design
Overview
Description
Algebra II Topics by Design contains 42 activity pages covering topics such as finding the slopes of lines, simplifying radical expressions, working with complex numbers, and working with simple matrices. The book employs a search-and-shade technique that rewards students for their efforts, while allowing them to self-check their work. Each page contains exercises with shading codes that students use to shade a grid labeled with the answers. If the answers are correct, a symmetrical design emerges. Teachers have permission to copy the pages for classroom use and an answer key is provided. | 677.169 | 1 |
Math 1426 Midterm 2 – Version A Fall 2008 Page 1of 5The square brackets following an exam question number refer to a section/problem number in the text. Problem numbers preceded by the symbol ~ are modeled on that problem from the text, but not identical to it. Problem numbers without the symbol are identical to or very close to the problem from the text. INSTRUCTIONS FOR PART I:Write your answers for these questions on a scantron (form 882-E or 882-ES) and mark only one answer per question. Scantrons will not be returned so mark your
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Flintridge, CA AlgebraAnna M.
...The math sections measure a student?s ability to reason quantitatively, solve mathematical problems, and interpret data presented in graphical form. These sections focus on four areas of mathematics that are typically covered in the first three years of American high school education: Arithmetic...
Michael TMax M | 677.169 | 1 |
Download Presentation
AlgGeolies Needed Every Day
Writing Utensil – preferably pencil, but please NEVER use red ink in this class
Paper that will fit in your notebook
Supplies Needed Occasionally (Keep in your locker)
Compass
Protractor
Ruler with inches & centimeters
Basic Calculator (Trig keys will help!)
(I have a class set of these items for you to use in class, but you may not remove them from the classroom. i.e.: without them, how do you finish your homework if it is incomplete at the end of class?)
Classroom Rules – Ideas?
Classroom Rules
Use common sense before you say or do anything in this classroom. If you do not believe Mrs. St.Clair will appreciate what you say or do, then please do not say or do it.
Please be on time to class.
2 tardies to class = 15 min. detention after school.
Tardy = Distraction
If you must be late – thank you in advance for being quiet, polite and non-disruptive.
May you chew gum in my class?
What do you think?
You may chew gum in my class as long as:
I do not see it or hear it.
It does not end up stuck to anything or anyone in the room.
It is disposed of properly in the trash can when not in use.
If the above requests are not met, then there will be no gum in this classroom.
Feel free to NARC on your peers in order to continue chewing gum. I never reveal my sources.
Grades
Progress Book is a phenomenal tool – use it!
I may have pop quizzes to find out if you are using it or not.
I will not print out your grade for you. You have Internet access at the school – do not abuse it – use it wisely. Keep yourself informed of your grade daily or at least weekly.
You may request to use the student computers in my room in order to check your grades on Progress Book at the end of class, time permitting. I will have a sign up sheet on my desk. You will need my permission first, then you will need to sign the log. Please do not abuse this privilege.
Grades are based on the General Scale listed in your Student Handbook.
Homework
Homework matters! It accounts for 40% of your grade.
It is your practice for the course and it will be assigned almost every day.
It is usually worth 5-10 points.
You MUST show your work for full credit.
If you copy the problem and show your work on your homework, you have created a perfect study tool to use on your quizzes and tests. Use common sense when deciding whether work is required or not.
Homework in Class
Time will be given at the end of each class period to work on homework as long as:
All students cooperate with minimum distractions.
We complete the whole lesson in time.
All students utilize the given time to work on the HW.
Time will not be given at the end of each class period as long as:
All students are not cooperative and we have multiple distractions.
We do not complete the whole lesson in time.
Students abuse the time given by not starting their HW.
Homework Incentive
HW % may replace your lowest test score.
ex: Your lowest test score for the 1st grading period is a 59%. Your Homework grade is an 89%.
I will replace that 59% with the 89%.
Notebooks Matter
Notebooks are your reference tool in place of a textbook.
Notebooks are required each day in class.
Notebooks should be organized daily.
Notebooks will be graded at least 1 time during each grading period.
I will check for notebooks within the 1st 4 weeks of school and award bonus points for being prepared and organized.
Notebook Incentive
Notebooks will be worth 25 points at the end of each quarter.
Depending on the completeness and organization of your notebook, you may earn 25 points toward your current Quarter grade.
Low HW score? Notebook will help.
Low Quiz/Test score? Notebook will help.
Quizzes/Test
Quizzes & Tests are worth 40% of your grade
You must show your work. Use common sense to decide when this is necessary. If you're not sure – show it. Partial credit can only be earned if you show your work.
Absolute quiet during tests. Do not talk or distract or you will earn a zero. You may quietly ask Mrs. St.Clair questions that pertain to the test. Please do not ask to leave the room during an assessment period. Mrs. St.Clair will tell you when you may talk again, but it may not be until the next day.
High Test Scores, but Low Homework Scores?
Many students do well on tests, but fail to complete their daily homework assignments.
Proposal (Provided you PASS the OGT): If you earn a C or higher on your quiz/test scores for one quarter, but your HW percentage is much lower, I will give you 20 bonus points in your Homework category.
Reason: If you score well on the Q/T (remember to show your work!) without doing the homework, then you have proven to me that you understand the material.
Downfall: If you allow your HW score to fall too low, the bonus points will not help you one bit. This is not to be considered as a "get out of doing homework" proposal.
Late Homework
When absent, mark "absent" at the top of your HW and turn it in. You are allowed 1 day late for each day of excused absence.
When you are forgetful, lazy, not feeling well, etc. and you do not turn your HW in on time - Please do it and show your work. Write "late" at the top of your HW and turn it in for partial credit. I will accept late work for the current grading period only. Cut off dates for late work will be posted on the chalkboard.
Copied HW
Do not copy someone else's HW, this will not prepare you for the Quiz/Test.
Do not allow someone to copy your HW, this will not prepare them for the Quiz/Test, you are enabling them to fail.
Any violation of the above will result in a zero for both people and a detention after school; peer-tutoring the copied assignment.
Projects/Participation
Worth 20% of your grade
Projects are usually group work and a group grade as well as an individual grade will be awarded.
There is one participation point for each day of the school year.
I will not wake you up if you fall asleep in class, however, you will lose your participation point for the day. Progress book with have a notation for your parents to see that you were sleeping, had your head down, not participating, etc.
If you choose to misbehave, distract, break classroom rules, earn a detention, have unexcused absences, sleep, not participate, have your head down, etc. you will lose your participation point for the day.
Student Accountability
Check Progress Book and be aware of your daily, weekly, monthly grade.
Know when you are missing HW assignments by checking Progress Book.
When you are absent, check on Progress Book to see what you missed and turn it in.
Check your student folder before you sit down daily to get information you need for class. | 677.169 | 1 |
Computational GeometrySo far as calculus is concerned, this book attaches primary importance to basic about analytic geometry, vectors, and calculus that students normally. Ralph Palmer Agnew Calculus: Analytic Geometry and Calculus, with Vectors calculus and analytic geometry
Differentiate between deductive and inductive reasoning to students by linking to the previous lessons. Deductive reasoning begins with a conjecture lesson 2.3 practice a using deductive reasoning to verify conjectures | 677.169 | 1 |
More PARCC Style Problems
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PRODUCT DESCRIPTION
Area: Algebra 1
This is a collection of PARCC Style Questions on a variety of topics. It would work well to help prepare students for PARCC tests.
Teachers could use this on their own, as an extension activity for students who excel while you remediate with other students, as a Friday group work problems set, or assign it to be completed with a substitute.
You could also use the questions to help you make your own algebra test, quiz, or exam | 677.169 | 1 |
Description of the book "A Level Mathematics for Edexcel: Statistics S1":
Oxford A Level Mathematics for Edexcel takes a completely fresh look at presenting the challenges of A Level. It specifically targets average students, with tactics designed to offer real chance of success to more students, as well as providing more stretch and challenge material. This Statistics 1 book includes a background knowledge chapter to help bridge the gap between GCSE and A level study.
Reviews of the A Level Mathematics for Edexcel: Statistics S1
Up to now regarding the ebook we've A Level Mathematics for Edexcel: Statistics S1 feedback consumers have not yet eventually left their overview of the experience, or otherwise make out the print still. But, for those who have already look at this guide and you are ready to produce their studies convincingly have you spend time to depart an overview on our website (we can easily release equally negative and positive evaluations). Put simply, "freedom regarding speech" We all wholeheartedly helped. Your responses to book A Level Mathematics for Edexcel: Statistics S1 - different followers are able to make a decision of a guide. Such help is likely to make people far more United!
James Nicholson
However, at this time do not have got specifics of the particular designer James Nicholson. Even so, we will value when you have just about any details about this, and so are willing to give that. Post this to all of us! We've got all the check out, in case everything are generally genuine, we're going to distribute on the internet site. It is vital for many people that each one correct concerning James Nicholson. We all thanks a lot in advance for being happy to visit meet you! | 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.
3.94 MB | 8 pages
PRODUCT DESCRIPTION
This set of activities will focus on the basic concepts of elementary Algebra. The student will apply the basic definitions and theory related to graphing data. Terms such as independent and dependent variable, slope of a line, line of best fit, plotting points on a coordinate graph will be discussed.
The teacher should introduce and discuss the basic concepts to their students prior to each student completing each activity. As an optional activity when calculating the "line of best fit", the student may use a spreadsheet software such as Excel to generate the graph and equation for the "line of best fit". The "How to Incorporate a Spreadsheet" teacher's manual will be included in this packet of activities for the teacher to present to their class.
Concepts to be discussed prior to assigning this activity booklet include:
1. Definition of dependent and independent variable.
2. Calculating the slope of a line.
3. Graphing data on a coordinate axis.
4. Calculating an equation for the "line of best fit".
5. Projecting future outcomes applying your equation for the "line of best fit".
Common core standards covered in this activity are 6.NS.C.8, 6.EE.C.9, 7.EE.B.4a, 7.EE.B.4b.
Some other activities you might wish to check out. Just click on the title of each | 677.169 | 1 |
This revised version of "Mathematics for Horticulture" was developed to meet the needs of educators faced with teaching a mathematics curriculum based on real-life applications. The manual includes a wide range of topics, some remedial in nature, but all very basic to success in the industry. The manual contains seven chapters that cover the following topics: (1) measurement; (2) geometry for the landscape; (3) sales; (4) construction; (5) grass seed mixtures, sod, fertilizers, and chemicals; (6) using drawing scales and writing estimates for landscape plans; and (7) producing a crop for market. Each chapter includes information sheets, examples, and a variety of practice sets that can be used for student review or testing. Sample blank forms are included for writing sales receipts and for writing estimates for landscape projects. A separately bound answer key is included with the student manual. (KC) | 677.169 | 1 |
Algebra 1
Course description:
*MA 27 Algebra I
Difficulty: Average
2 Sem.—1 Cr
This first year algebra course prepares a student
for more advanced
study of mathematics. The curriculum includes symbolic manipulation, data analysis,
patterns and functions represented in multiple forms, linear and quadratic equations.
This course will be taught using software called ALEKS. We will meet in a computer lab for the majority of classtime. Students will be expected to access ALEKS outside of class for a period of time each week. It can be accessed from any computer with an internet connection, including the school and public libraries.
The rest of class will be spent completing projects that explore various Algebra topics that we will be covering throughout the year. Students will be expected to work with a partner or in groups for most projects. Some projects will require work outside of the classroom.
We will also be learning how to take notes, study and utilize test taking strategies.
Obviously this is not your average Algebra class. Extensive amounts of time will be spent on the computer, and the rest of it completing projects and learning how to learn.This is not a lecture and drill class. For some of you this is exactly what you need. For others, maybe not. Either way, I am here to help you work through it.
I want every student in my classes to succeed, and I think you all can. While I love Algebra, I know it is not easy for everyone. You must earn your grade, and that will take hard work. It will take time outside of class. You will need to study. You will need to try. | 677.169 | 1 |
The Student Merit Award Program was designed to motivate, stimulate and reward students for their study and achievement outside the mathematics classroom by providing enrichment material on a variety of mathematical topics. In general, these topics are either not found in the standard curriculum or represent a more in-depth study of standard topics. The topics, presented in 18 units, include: programming languages; microcomputer applications; the fourth dimension; finite groups; transfinite numbers; Archimedes; Pythagoras and his theorem; topology; geometric inversion; mathematics of flight; paradoxes in mathematics; continued fractions; finding equations from tables of values; applications of mathematics in nursing; tessellations; transformations and matrices with applications; ancient numeration systems; and statistics. Each unit consists of a teacher's section and a student section (which may be duplicated) composed of largely independent "requirements" to be completed. Although the units vary in length, most require between 10 and 30 hours of outside research and writing. Teachers can either assign units to be done or allow students to select those that interest them. A sample certificate that can be reproduced and awarded to students on completion of each unit is included. (JN) | 677.169 | 1 |
Listing Detail Tabs
This title is In Stock in the Booktopia Distribution Centre. We will send you a confirmation email with a Tracking Code to follow the progress of your parcel when it ships.
New Century Maths Advanced 9 Author: Klaas Bootsma
ISBN: 9780170193085 Format: Book with Other Items Number Of Pages: 648 Published: 17 October 2013 Country of Publication: AU Description: New Century Maths for the Australian Curriculum Years 7-10 is specifically written to meet the requirements of the NSW Mathematics 7-10 syllabus for the Australian Curriculum. These new titles retain all of the successful features of the New Century Maths series, which has been in schools since 1994.
Also available as an interactive NelsonNetBook, either as a supplement to the printed text or as a standalone option for schools seeking a digital-only resource solution.
Features include: Explicit coverage of Australian Curriculum clearly labelled; SkillCheck revises prerequisite skills while Power Plus promotes extension work; Graded exercises are linked to worked examples that reflect the five components of Working Mathematically; Understanding, Fluency, Problem Solving, Reasoning and Communication. The exercises include multiple-choice questions, longer exam-style problems and real life applications; Technology feature promotes ICT in the classroom, including spreadsheets, GeoGebra software, graphics calculators and the Internet; Nel | 677.169 | 1 |
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For your added protection in the case of copyright
inspection, please complete the form below. Retain
this form, the complete original document and the
invoice or receipt as proof of purchase.
Name of Purchaser:
Date of Purchase:
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This master may only be reproduced by the
original purchaser for use with their class(es). The
publisher prohibits the loaning or onselling of this
master for the purposes of reproduction.
In some cases, websites or specific URLs may be recommended. While these are checked and rechecked at the time of publication,
the publisher has no control over any subsequent changes which may be made to webpages. It is strongly recommended that the class
teacher checks all URLs before allowing students to access them.
View all pages online
PO Box 332 Greenwood Western Australia 6924
Website:
Email: mail@ricgroup.com.au
FOREWORD
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Problem-solving does not come easily to most people,
so learners need many experiences engaging with
problems if they are to develop this crucial ability. As
they grapple with problem, meaning and find solutions,
students will learn a great deal about mathematics
and mathematical reasoning; for instance, how to
organise information to uncover meanings and allow
connections among the various facets of a problem
to become more apparent, leading to a focus on
organising what needs to be done rather than simply
looking to apply one or more strategies. In turn, this
extended thinking will help students make informed
choices about events that impact on their lives and to
interpret and respond to the decisions made by others
at school, in everyday life and in further study.
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Books A–G of Problem-solving in mathematics have been developed to provide a rich resource for teachers
of students from the early years to the end of middle school and into secondary school. The series of problems,
discussions of ways to understand what is being asked and means of obtaining solutions have been built up to
improve the problem-solving performance and persistence of all students. It is a fundamental belief of the authors
that it is critical that students and teachers engage with a few complex problems over an extended period rather than
spend a short time on many straightforward 'problems' or exercises. In particular, it is essential to allow students
time to review and discuss what is required in the problem-solving process before moving to another and different
problem. This book includes extensive ideas for extending problems and solution strategies to assist teachers in
implementing this vital aspect of mathematics in their classrooms. Also, the problems have been constructed and
selected over many years' experience with students at all levels of mathematical talent and persistence, as well as
in discussions with teachers in classrooms, professional learning and university settings.
ensure appropriate explanations, the use of the
pages, foster discussion among students and suggest
ways in which problems can be extended. Related
problems occur on one or more pages that extend the
problem's ideas, the solution processes and students'
understanding of the range of ways to come to terms
with what problems are asking.
The student pages present problems chosen with a
particular problem-solving focus and draw on a range
of mathematical understandings and processes.
For each set of related problems, teacher notes and
discussion are provided, as well as indications of
how particular problems can be examined and solved.
Answers to the more straightforward problems and
detailed solutions to the more complex problems
R.I.C. Publications®
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that highlights the particular thinking that the
problems will demand, together with an indication
of the mathematics that might be needed and a list
of materials that could be used in seeking a solution.
A particular focus for the page or set of three pages
of problems then expands on these aspects. Each
book is organised so that when a problem requires
complicated strategic thinking, two or three problems
occur on one page (supported by a teacher page with
detailed discussion) to encourage students to find
a solution together with a range of means that can
be followed. More often, problems are grouped as a
series of three interrelated pages where the level of
complexity gradually increases, while the associated
teacher page examines one or two of the problems in
depth and highlights how the other problems might be
solved in a similar manner.
Problem-solving in mathematics
iii
FOREWORD
both challenges at the point of the mathematics
that is being learned as well as provides insights
and motivation for what might be learned next. For
example, the computation required gradually builds
from additive thinking, using addition and subtraction
separately and together, to multiplicative thinking,
where multiplication and division are connected
conceptions. More complex interactions of these
operations build up over the series as the operations
are used to both come to terms with problems'
meanings and to achieve solutions. Similarly, twodimensional geometry is used at first but extended
to more complex uses over the range of problems,
then joined by interaction with three-dimensional
ideas. Measurement, including chance and data, also
extends over the series from length to perimeter, and
from area to surface area and volume, drawing on
the relationships among these concepts to organise
solutions as well as giving an understanding of the
metric system. Time concepts range from interpreting
timetables using 12-hour and 24-hour clocks while
investigations related to mass rely on both the concept
itself and practical measurements.
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Each teacher page concludes with two further aspects
critical to successful teaching of problem-solving. A
section on likely difficulties points to reasoning and
content inadequacies that experience has shown may
well impede students' success. In this way, teachers
can be on the look out for difficulties and be prepared
to guide students past these potential pitfalls. The
final section suggests extensions to the problems to
enable teachers to provide several related experiences
with problems of these kinds in order to build a rich
array of experiences with particular solution methods;
for example, the numbers, shapes or measurements
in the original problems might change but leave the
means to a solution essentially the same, or the
context may change while the numbers, shapes or
measurements remain the same. Then numbers,
shapes or measurements and the context could be
changed to see how the students handle situations
that appear different but are essentially the same
as those already met and solved. Other suggestions
ask students to make and pose their own problems,
investigate and present background to the problems
or topics to the class, or consider solutions at a more
general level (possibly involving verbal descriptions
and eventually pictorial or symbolic arguments).
In this way, not only are students' ways of thinking
extended but the problems written on one page are
used to produce several more problems that utilise
the same approach.
The difficulty of the mathematics gradually increases
over the series, largely in line with what is taught
at the various year levels, although problem-solving
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The language in which the problems are expressed is
relatively straightforward, although this too increases
in complexity and length of expression across the books
in terms of both the context in which the problems
are set and the mathematical content that is required.
It will always be a challenge for some students
to 'unpack' the meaning from a worded problem,
particularly as problems' context, information and
meanings expand. This ability is fundamental to the
nature of mathematical problem-solving and needs to
be built up with time and experiences rather than be
Problem-solving in mathematics
R.I.C. Publications®
FOREWORD
successfully solve the many types of problems, but
also to give them a repertoire of solution processes
that they can consider and draw on when new
situations are encountered. In turn, this allows them
to explore one or other of these approaches to see
whether each might furnish a likely result. In this way,
when they try a particular method to solve a new
problem, experience and analysis of the particular
situation assists them to develop a full solution.
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An approach to solving problems
Analyse
Try
the problem
an approach
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diminished or left out of the problems' situations. One
reason for the suggestion that students work in groups
is to allow them to share and assist each other with
the tasks of discerning meanings and ways to tackle
the ideas in complex problems through discussion,
rather than simply leaping into the first ideas that
come to mind (leaving the full extent of the problem
unrealised).
Not only is this model for the problem-solving process
helpful in solving problems, it also provides a basis for
students to discuss their progress and solutions and
determine whether or not they have fully answered
a question. At the same time, it guides teacher
questions of students and provides a means of seeing
underlying mathematical difficulties and ways in
which problems can be adapted to suit particular
needs and extensions. Above all, it provides a common
framework for discussions between a teacher and
group or whole class to focus on the problem-solving
process rather than simply on the solution of particular
problems. Indeed, as Alan Schoenfeld, in Steen L (Ed)
Mathematics and democracy (2001), states so well, in
problem-solving:
means to a solution
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The careful, gradual development of an ability to
analyse problems for meaning, organising information
to make it meaningful and to make the connections
among them more meaningful in order to suggest
a way forward to a solution is fundamental to the
approach taken with this series, from the first book
to the last. At first, materials are used explicitly to
aid these meanings and connections; however, in
time they give way to diagrams, tables and symbols
as understanding and experience of solving complex,
engaging problems increases. As the problem forms
expand, the range of methods to solve problems
is carefully extended, not only to allow students to
getting the answer is only the beginning rather than
the end … an ability to communicate thinking is
equally important.
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We wish all teachers and students who use these
books success in fostering engagement with problemsolving and building a greater capacity to come to
terms with and solve mathematical problems at all
levels.
How many?..................................................................... 33
vi
Problem-solving in mathematics
R.I.C. Publications®
INTRODUCTION
Problem-solving and mathematical thinking
By learning problem-solving in mathematics,
students should acquire ways of thinking,
habits of persistence and curiosity, and
confidence in unfamiliar situations that will
serve them well outside the mathematics
classroom. In everyday life and in the
workplace, being a good problem solver can
lead to great advantages.
Problem-solving lies at the heart of mathematics.
New mathematical concepts and processes have
always grown out of problem situations and students'
problem-solving capabilities develop from the very
beginning of mathematics learning. A need to solve a
problem can motivate students to acquire new ways
of thinking as well as to come to terms with concepts
and processes that might not have been adequately
learned when first introduced. Even those who can
calculate efficiently and accurately are ill prepared for
a world where new and adaptable ways of thinking
are essential if they are unable to identify which
information or processes are needed.
Problem-solving
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NCTM principles and standards for school
mathematics
(2000, p. 52)
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Well-chosen problems encourage deeper exploration
of mathematical ideas, build persistence and highlight
the need to understand thinking strategies, properties
and relationships. They also reveal the central role of
sense making in mathematical thinking—not only to
evaluate the need for assessing the reasonableness
of an answer or solution, but also the need to consider
the inter-relationships among the information provided
with a problem situation. This may take the form of
number sense, allowing numbers to be represented
in various ways and operations to be interconnected;
through spatial sense that allows the visualisation of
a problem in both its parts and whole; to a sense of
measurement across length, area, volume and chance
and data.
A problem is a task or situation for which there is
no immediate or obvious solution, so that problemsolving refers to the processes used when engaging
with this task. When problem-solving, students engage
with situations for which a solution strategy is not
immediately obvious, drawing on their understanding
of concepts and processes they have already met, and
will often develop new understandings and ways of
thinking as they move towards a solution. It follows
that a task that is a problem for one student may not
be a problem for another and that a situation that
is a problem at one level will only be an exercise or
routine application of a known means to a solution at
a later time.
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On the other hand, students who can analyse problem
meanings, explore means to a solution and carry
out a plan to solve mathematical problems have
acquired deeper and more useful knowledge than
simply being able to complete calculations, name
shapes, use formulas to make measurements or
determine measures of chance and data. It is critical
that mathematics teaching focuses on enabling all
students to become both able and willing to engage
with and solve mathematical problems.
A large number of tourists visited Uluru during 2007.
There were twice as many visitors in 2007 than in
2003 and 6530 more visitors in 2007 than in 2006. If
there were 298 460 visitors in 2003, how many were
there in 2006?
For a student in Year 3 or Year 4, sorting out the
information to see how the number of visitors each
year are linked is a considerable task and then there
R.I.C. Publications®
Problem-solving in mathematics
vii
INTRODUCTION
is a need to use multiplication and subtraction with
large numbers. For a student in later primary years,
an ability to see how the problem is structured and
familiarity with computation could lead them to use
a calculator, key in the numbers and operation in an
appropriate order and readily obtain the answer:
298460 x 2 – 6530 = 590390
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590 390 tourists visited Uluru in 2006
However, many students feel inadequate when they
encounter problem-solving questions. They seem to
have no idea of how to go about finding a solution and
are unable to draw on the competencies they have
learned in number, space and measurement. Often
these difficulties stem from underdeveloped concepts
for the operations, spatial thinking and measurement
processes. They may also involve an underdeveloped
capacity to read problems for meaning and a tendency
to be led astray by the wording or numbers in a problem
situation. Their approach may then simply be to try a
series of guesses or calculations rather than consider
using a diagram or materials to come to terms with
what the problem is asking and using a systematic
approach to organise the information given and
required in the task. It is this ability to analyse problems
that is the key to problem-solving, enabling decisions
to be made about which mathematical processes to
use, which information is needed and which ways of
proceeding are likely to lead to a solution.
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As the world in which we live becomes ever more
complex, the level of mathematical thinking and
problem-solving needed in life and in the workplace
has increased considerably. Those who understand
and can use the mathematics they have learned will
have opportunities opened to them that those who do
not develop these ways of thinking will not. To enable
students to thrive in this changing world, attitudes
and ways of knowing that enable them to deal with
new or unfamiliar tasks are now as essential as the
procedures that have always been used to handle
familiar operations readily and efficiently. Such
an attitude needs to develop from the beginning of
mathematics learning as students form beliefs about
meaning, the notion of taking control over the activities
they engage with and the results they obtain, and as
they build an inclination to try different approaches.
In other words, students need to see mathematics as
a way of thinking rather than a means of providing
answers to be judged right or wrong by a teacher,
textbook or some other external authority. They need
to be led to focus on means of solving problems rather
than on particular answers so that they understand
the need to determine the meaning of a problem
before beginning to work on a solution.
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In order to solve this problem, it is not enough to
simply use the numbers that are given. Rather, an
analysis of the race situation is needed first to see
that when Jordan started there were 3 cars ahead of
him. When another 6 cars passed him there were now
9 ahead of him. If he is to win, he needs to pass all
9 cars. The 4 and 6 implied in the problem were not
used at all! Rather, a diagram or the use of materials
is needed first to interpret the situation and then see
how a solution can be obtained.
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Making sense in mathematics
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In a car race, Jordan started in fourth place. During
the race, he was passed by six cars. How many
cars does he need to pass to win the race?
Making sense of the mathematics being developed
and used needs be seen as the central concern of
learning. This is important, not only in coming to
terms with problems and means to solutions, but also
in terms of putting meanings, representations and
relationships in mathematical ideas to the forefront of
thinking about and with mathematics. Making sensible
interpretations of any results and determining which
of several possibilities is more or equally likely is
critical in problem-solving.
Number sense, which involves being able to
work with numbers comfortably and competently,
R.I.C. Publications®
INTRODUCTION
Reading the problem carefully shows that each table
seats five couples or 10 people. At first glance, this
problem might be solved using division; however, this
would result in a decimal fraction, which is not useful
in dealing with people seated at tables:
10 317 is 31.7
In contrast, a full understanding of numbers allows
317 to be renamed as 31 tens and 7 ones:
This provides for all the people at the party and
analysis of the number 317 shows that there needs
to be at least 32 tables for everyone to have a seat
and allow party goers to move around and sit with
others during the evening. Understanding how to
rename a number has provided a direct solution
without any need for computation. It highlights how
coming to terms with a problem and integrating this
with number sense provides a means of solving the
problem more directly and allows an appreciation of
what the solution might mean.
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is important in many aspects of problem-solving,
in making judgments, interpreting information and
communicating ways of thinking. It is based on a
full understanding of numeration concepts such
as zero, place value and the renaming of numbers
in equivalent forms, so that 207 can be seen as 20
tens and 7 ones as well as 2 hundreds and 7 ones (or
that 52, 2.5 and 2 12 are all names for the same fraction
amount). Automatic, accurate access to basic facts
also underpins number sense, not as an end in itself,
but rather as a means of combining with numeration
concepts to allow manageable mental strategies and
fluent processes for larger numbers. Well-understood
concepts for the operations are essential in allowing
relationships within a problem to be revealed and
taken into account when framing a solution.
The following problem highlights the importance of
these understandings.
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There were 317 people at the New Year's Eve party
on 31 December. If each table could seat 5 couples,
how many tables were needed?
R.I.C. Publications®
to be interpreted and processed, while the use of
diagrams is often essential in developing conceptual
understanding across all aspects of mathematics.
Using diagrams, placing information in tables or
depicting a systematic way of dealing with the various
possibilities in a problem assist in visualising what is
happening. It can be a very powerful tool in coming
to terms with the information in a problem and it
provides insight into ways to proceed to a solution.
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• a capacity to calculate and estimate
mentally
Spatial sense involves:
• a capacity to visualise shapes and their
properties
• determining relationships among shapes and
their properties
• linking two-dimensional and three-dimensional
representations
• presenting and interpreting information in tables
and lists
• an inclination to use diagrams and models to
visualise problem situations and applications in
flexible ways.
Problem-solving in mathematics
ix
INTRODUCTION
The following problem shows how these
understandings can be used.
A small sheet of paper
has been folded in half
and then cut along
the fold to make two
rectangles.
The perimeter of each
rectangle is 18 cm.
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Which of these arrangements of squares forms a
net for the dice?
Reading the problem carefully and analysing the
diagram shows that the length of the longer side
of the rectangle is the same as the one side of the
square while the other side of the rectangle is half
this length. Another way to obtain this insight is to
make a square, fold it in half along the cutting line and
then fold it again. This shows that the large square is
made up of four smaller squares:
Greengrocers often
stack fruit as a
pyramid.
How many oranges
are in this stack?
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What was the
perimeter of the
original square sheet
of paper?
Many dice are made in the shape of a cube with
arrangements of dots on each square face so that
the sum of the dots on opposite faces is always 7.
An arrangement of squares that can be folded to
make a cube is called a net of a cube.
Since each rectangle contains two small squares, the
side of the rectangle, 18 cm, is the same as 6 sides of
the smaller square, so the side of the small square is
3 cm. The perimeter of the large square is made of 6
of these small sides, so is 24 cm.
comparison. Many measurements use aspects of
space (length, area, volume), while others use numbers
on a scale (time, mass, temperature). Money can be
viewed as a measure of value and uses numbers more
directly, while practical activities such as map reading
and determining angles require a sense of direction
as well as gauging measurement. The coordination
of the thinking for number and space, along with an
understanding of how the metric system builds on
place value, zero and renaming, is critical in both
building measurement understanding and using it to
come to terms with and solve many practical problems
and applications.
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Similar thinking is used with arrangements of twodimensional and three-dimensional
shapes and in visualising how they can
fit together or be taken apart.
Measurement sense includes:
•
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understanding how numeration and computation underpin measurement
extending relationships from number understandings to the metric system
appreciating the relative size of measurements
a capacity to use calculators, mental or written processes for exact and approximate calculations
an inclination to use understanding and facility with measurements in flexible ways.
Problem-solving in mathematics
R.I.C. Publications®
INTRODUCTION
The following problem shows how these
understandings can be used.
A city square has an
area of 160 m2. Four
small triangular garden
beds are constructed at
from each corner to the
midpoints of the sides
of the square. What is
the area of each garden
bed?
• understanding how numeration and
computation underpin the analysis of data
• appreciating the relative likelihood of
outcomes
• a capacity to use calculators or mental and
written processes for exact and approximate
calculations
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• presenting and interpreting data in tables and
graphs
• an inclination to use understanding and facility
with number combinations and arrangements
in flexible ways.
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Reading the problem carefully shows that there 4
garden beds and each of them takes up the same
proportions of the whole square. A quick look at the
area of the square shows that there will not be an
exact number of metres along one side. Some further
thinking will be needed to determine the area of each
garden bed.
There are six possibilities for placing the scoops of icecream on a cone. Systematically treating the possible
placements one at a time highlights how the use of a
diagram can account for all possible arrangements.
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An understanding of the problem situation given by a
diagram has been integrated with spatial thinking and
a capacity to calculate mentally with simple fractions
to provide an appropriate solution. Both spatial sense
and number sense have been used to understand the
problem and suggest a means to a solution.
Data sense is an outgrowth of measurement sense
and refers to an understanding of the way number
sense, spatial sense and a sense of measurement
work together to deal with situations where patterns
need to be discerned among data or when likely
outcomes need to be analysed. This can occur among
frequencies in data or possibilities in chance.
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on a cone?
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If the midpoints of each side are drawn across the
square, four smaller squares are formed and each
garden bed takes up 41 of a small square. Four of the
garden beds will have the same area of one small
square. Since area of the small square is 41 the area of
the large square, the area of one small square is 40 m2
and the area of each triangular garden bed is 10 m2.
Patterning is another critical aspect of sense
making in mathematics. Often a problem calls on
discerning a pattern in the placement of materials,
the numbers involved in the situation or the possible
arrangements of data or outcomes so as to determine
a likely solution. Being able to see patterns is also
very helpful in getting a solution more immediately or
understanding whether or not a solution is complete.
Problem-solving in mathematics
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INTRODUCTION
A farmer had emus and alpacas in one
paddock. When she counted, there were 38
heads and 100 legs. How many emus and
how many alpacas are in the paddock?
There are 38 emus and alpacas. Emus have 2
legs. Alpacas have 4 legs.
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Number of
alpacas
emus
4
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84 – too few
8
30
92 – too few
10
28
96 – too few
26
100
Number of legs
There are 12 alpacas and 26 emus.
As more experience in solving problems is gained, an
ability to see patterns in what is occurring will also
allow solutions to be obtained more directly and help
in seeing the relationship between a new problem and
one that has been solved previously. It is this ability to
relate problem types, even when the context appears
to be quite different, that often distinguishes a good
problem-solver from one who is more hesitant.
Looking at a problem and working through what is
needed to solve it will shed light on the problemsolving process.
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Number of
12
of the problem itself—what is being asked, what
information might be used, what answer might be
likely and so on—so that a particular approach is used
only after the intent of the problem is determined.
Establishing the meaning of the problem before any
plan is drawn up or work on a solution begins is
critical. Students need to see that discussion about
the problem's meaning, and the ways of obtaining a
solution, must take precedence over a focus on 'the
answer'. Using collaborative groups when problemsolving, rather than tasks assigned individually, is an
approach that helps to develop this disposition.
On Saturday, Peta went to the shopping
centre to buy a new outfit to wear at her
friend's birthday party. She spent half of her
money on a dress
and then onethird of what she
had left on a pair
of sandals. After
her purchases,
she had $60.00
left in her purse.
How much money
did she have to
start with?
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While the teaching of problem-solving has often
centred on the use of particular strategies that could
apply to various classes of problems, many students
are unable to access and use these strategies to solve
problems outside of the teaching situations in which
they were introduced. Rather than acquire a process
for solving problems, they may attempt to memorise
a set of procedures and view mathematics as a set of
learned rules where success follows the use of the
right procedure to the numbers given in the problem.
Any use of strategies may be based on familiarity,
personal preference or recent exposure rather than
through a consideration of the problem to be solved.
A student may even feel it is sufficient to have only
one strategy and that the strategy should work all of
the time; and if it doesn't, then the problem 'can't be
done'.
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In contrast, observation of successful problem-solvers
shows that their success depends more on an analysis
By carefully reading the problem, it can be determined
that Peta had an original amount of money to spend.
She spent some on a dress and some on shoes and
then had $60.00 left. All of the information required
to solve the problem is available and no further
information is needed. The question at the end asks
how much money did she start with, but really the
problem is how much did she spend on the dress and
then on the sandals.
The discussion of this problem has served to identify
the key element within the problem-solving process;
it is necessary to analyse the problem to unfold its
meanings and discover what needs to be considered.
R.I.C. Publications®
INTRODUCTION
What the problem is asking is rarely found in the
question in the problem statement. Instead, it is
necessary to look below the 'surface level' of the
problem and come to terms with the problem's
structure. Reading the problem aloud, thinking of
previous problems and other similar problems,
selecting important information from the problem
that may be useful, and discussion of the problem's
meaning are all essential.
Materials could also have been used to work with
backwards: 6 tens represent the $60 left, so the
sandals would cost 3 tens and the dress 9 tens—she
took 18 tens or $180 shopping.
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Ways that may come to mind during the analysis
include:
Another way to solve this problem is with a diagram.
If we use a rectangle to represent how much money
Peta took with her, we can show by shading how much
she spent on a dress and sandals:
Total amount available to spend:
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The next step is to explore possible ways to solve the
problem. If the analysis stage has been completed,
then ways in which the problem might be solved will
emerge. It is here that strategies, and how they might
be useful to solving a problem, can arise. However,
most problems can be solved in a variety of ways,
using different approaches, and a student needs to be
encouraged to select a method that make sense and
appears achievable to him or her.
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Together, these are half of what Peta took, which is
also the cost of the dress. As the dress cost $90, Peta
took $180 to spend.
She spent half of her money on a dress.
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• Try and adjust – Select an amount that Peta
might have taken shopping, try it in the context of
the question, examine the resulting amounts, and
then adjust them, if necessary, until $60.00 is the
result.
• Backtrack using the numbers – The sandals
were one-third of what was left after the dress,
so the $60.00 would be two-thirds of what was
left. Together, these two amounts would match
the cost of the dress.
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At this point she had $60 left, so the twounshaded parts must be worth $60 or $30 per
part—which has again minimised and simplified
the calculations.
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• Use a diagram to represent the information in
the problem.
• Think of a similar problem – For example, it
is like the car race problem in that the relative
portions (places) are known and the final result
(money left, winning position) are given.
Now one of the possible means to a solution can be
selected to try. Backtracking shows that $60 was twothirds of what she had left, so the sandals (which are
one-third of what she had left) must have cost $30.
R.I.C. Publications®
• Materials – Base 10 materials could be used to
represent the money spent and to help the student
work backwards through the problem from when
Peta had $60.00 left.
$30
$30
Each of the six equal parts represents $30, so
Peta took $180 to spend.
Having tried an idea, an answer needs to be analysed
in the light of the problem in case another solution is
required. It is essential to compare an answer back
to the original analysis of the problem to determine
whether the solution obtained is reasonable and
answers the problem. It will also raise the question
as to whether other answers exist, and even whether
there might be other solution strategies. In this
Problem-solving in mathematics
xiii
INTRODUCTION
way the process is cyclic and should the answer be
unreasonable, then the process would need to begin
again.
We believe that Peta took $180 to shop with. She
spent half (or $90) on a dress, leaving $90. She spent
one-third of the $90 on sandals ($30), leaving $60.
Looking again at the problem, we see that this is
correct and the diagram has provided a direct means
to the solution that has minimised and simplified the
calculations.
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Thinking about how the various ways this problem
was solved highlights the key elements within the
problem-solving process. When starting the process,
it is necessary to analyse the problem to unfold its
layers, discover its structure and what the problem
was really asking. Next, all possible ways to solve the
problem were explored before one, or a combination
of ways, was/were selected to try. Finally, once
something was tried, it was important to check
the solution in relation to the problem to see if the
solution was reasonable. This process highlights the
cyclic nature of problem-solving and brings to the fore
the importance of understanding the problem (and
its structure) before proceeding. This process can be
summarised as:
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Further, returning to an analysis of any answers
and solution strategies highlights the importance
of reflecting on what has been done. Taking time to
reflect on any plans drawn up, processes followed and
strategies used brings out the significance of coming
to terms with the nature of the problem, as well as
the value and applicability of particular approaches
that might be used with other problems. Thinking of
how a related problem was solved is often the key to
solving another problem at a later stage. It allows the
thinking to be 'carried over' to the new situation in a
way that simply trying to think of the strategy used
often fails to reveal. Analysing problems in this way
also highlights that a problem is not solved until any
answer obtained can be justified. Learning to reflect
on the whole process leads to the development of
a deeper understanding of problem-solving, and time
must be allowed for reflection and discussion to fully
build mathematical thinking.
This model for problem-solving provides students with
a means of talking about the steps they engage with
whenever they have a problem to solve: Discussing
how they initially analysed the problem, explored
various ways that might provide a solution, and then
tried one or more possible solution paths to obtain a
solution—that they analysed for completeness and
sense making—reinforces the very methods that will
give them success on future problems. This process
brings to the fore the importance of understanding the
problem and its structure before proceeding.
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Analyse
the problem
on their own can be led to see ways of proceeding
when discussing a problem in a group. Therefore
building greater confidence in their capacity to solve
problems and learning the value of persisting with
a problem in order to tease out what is required.
What is discussed with their peers is more likely to
be recalled when other problems are met while the
observations made in the group increase the range of
approaches that a student can access. Thus, time has
to be allowed for discussion and exploration rather
than ensuring that students spend 'time on task' as
for routine activities.
Problem-solving in mathematics
Correct answers that
fully solve a problem are
always important, but
developing a capacity to
use an effective problemsolving process needs to
be the highest priority.
A student who has an
R.I.C. Publications®
INTRODUCTION
answer should be encouraged to discuss his or her
solution with others who believe they have a solution,
rather than tell his or her answer to another student
or simply move on to another problem. In particular,
explaining to others why he or she believes an answer
is reasonable, as well as why it provides a solution,
gets other students to focus on the entire problemsolving process rather than just quickly getting an
answer.
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Building a problem-solving process
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Expressing an answer in a sentence that relates to
the question stated in the problem also encourages
reflection on what was done and ensures that the
focus is on solving the problem rather than providing
an answer. These aspects of the teaching of problemsolving should then be taken further, as particular
groups discuss their solutions with the whole class and
all students are able to participate in the discussion of
the problem. In this way, problem-solving as a way of
thinking comes to the fore, rather than focusing on the
answers to a series of problems that some students
see as the main aim of their mathematical activities.
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A teacher also may need to extend or adapt a given
problem to ensure the problem-solving process is
understood and can be used in other situations, instead
of moving on to another different problem in the way
that one example or topic shifts to another in other
parts of mathematics learning. This can help students
to understand the significance of asking questions of
a problem, as well as seeing how a way of thinking
can be adapted to other related problems. Having
students engage in this process of problem posing is
another way of both assessing and bringing them to
terms with the overall process of solving problems.
The cyclical model, Analyse–Explore–Try, provides
a very helpful means of organising and discussing
possible solutions. However, care must be taken that
it is not seen simply as a procedure to be memorised
and then applied in a routine manner to every new
problem. Rather, it needs to be carefully developed
over a range of different problems, highlighting
the components that are developed with each new
problem.
• As students read a problem, the need to first read
for the meaning of the problem can be stressed.
This may require reading more than once and can
be helped by asking students to state in their own
words what the problem is asking them to do.
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Questions need to encourage students to explore
possible means to a solution and try one or more of
them, rather than point to a particular procedure. It
can also assist students to see how to progress their
thinking, rather than get in a loop where the same
steps are repeated over and over. However, while
having too many questions that focus on the way to
a solution may end up removing the problem-solving
aspect from the question, having too few may cause
students to become frustrated with the task and think
that it is beyond them. Students need to experience
the challenge of problem-solving and gain pleasure
from working through the process that leads to a full
solution. Taking time to listen to students as they try
out their ideas, without comment or without directing
them to a particular strategy, is also important.
Listening provides a sense of how students' problem
solving is developing, as assessing this aspect of
mathematics can be difficult. After all, solving one
problem will not necessarily lead to success on the
next problem, nor will a difficulty with a particular
problem mean that the problems that follow will also
be as challenging.
• Further reading will be needed to sort out which
information is needed and whether some is not
needed or if other information needs to be gathered
from the problem's context (e.g. data presented
within the illustration or table accompanying the
problem), or whether the students' mathematical
understandings need to be used to find other
relationships among the information. As the
form of the problems becomes more complex,
this thinking will be extended to incorporate
further ways of dealing with the information; for
example, measurement units, fractions and larger
numbers might need to be renamed to the same
mathematical form.
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Problem-solving in mathematics
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INTRODUCTION
• Thinking about any processes that might be
needed and the order in which they are used, as
well as the type of answer that could occur, should
also be developed in the context of new levels of
problem structure.
• Developing a capacity to see 'through' the
problem's expression—or context to see
similarities between new problems and others
that might already have been met—is a critical
way of building expertise in coming to terms with
and solving problems.
• When a problem is being explored, some problems
will require the use of materials to think through the
whole of the problem's context. Others will demand
the use of diagrams to show what is needed.
Another will show how systematic analysis of the
situation using a sequence of diagrams, on a list or
table, is helpful. As these ways of thinking about
the problem are understood, they can be included
in the cycle of steps.
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Expanding the problem-solving process
A fuller model to manage problem-solving
can gradually emerge:
• Read carefully.
• What is the problem
asking?
• What is the meaning
of the information? Is
it all needed? Is there
too little? Too much?
• Which operations
will be needed and in
what order?
• What sort of answer
is likely?
• Have I seen a
problem like this
before?
Try
• Many students often try to guess a result. This can
even be encouraged by talking about 'guess and
check' as a means to solve problems, Changing to
'try and adjust' is more helpful in building a way
of thinking and can lead to a very powerful way of
finding solutions.
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Explore
• When materials, a diagram or table have been
used, another means to a solution is to look for a
pattern in the results. When these have revealed
what is needed to try for a solution, it may also
be reasonable to use pencil and paper or a
calculator.
• Use
materials or
a model.
• Use a
calculator.
• Use pencil
and paper.
• Look for a
pattern.
xvi
Analyse
• The point in the cycle where an answer is assessed
for reasonableness (e.g. whether it provides
a solution, is only one of several solutions or
whether there may be another way to solve the
problem) also needs to be brought to the fore as
different problems are met.
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• Put the solution back
into the problem.
• Does the answer
make sense?
• Does it solve the
problem?
• Is it the only answer?
• Could there be
another way?
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Explore
means to a solution
• Use a diagram
or materials.
• Work
backwards or
backtrack.
• Put the
information
into a table.
• Try and adjust.
Problem-solving in mathematics
The role of calculators
When calculators are used, students devote less time
to basic calculations, providing time that might be
needed to either explore a solution or find an answer
to a problem. In this way, attention is shifted from
computation, which the calculator can do, to thinking
about the problem and its solution—work that the
calculator cannot do. It also allows more problems (and
more realistic problems) to be addressed in problemsolving sessions. In these situations, a calculator
serves as a tool rather than a crutch, requiring
students to think through the problem's solution in
R.I.C. Publications®
INTRODUCTION
order to know how to use the calculator appropriately.
It also underpins the need to make sense of the steps
along the way and any answers that result, as keying
incorrect numbers, operations or order of operations
quickly leads to results that are not appropriate.
When problems are selected, they need to be examined
to see if students already have an understanding of
the underlying mathematics required and that the
problem's expression can be meaningfully read by
the group of students who will be attempting the
solution—though not necessarily by all students in
the group. The problem itself should be neither too
easy (so that it is just an exercise, repeating something
readily done before), nor too difficult (thus beyond the
capabilities of most or all in the group), and engages
the interests of the students. A problem should also
be able to be solved in more than one way.
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language and mathematics. Within a problem, the
processes that need to be used may be more or less
obvious, the information that is required for a solution
may be too much or too little, and strategic thinking
may be needed in order to come to terms with what
the problem is asking.
complex expression, complex mathematics
The varying levels of problem
structure and expression
(i) The processes to be used are
relatively obvious as: these problems
are comparatively straightforward
and contain all the information necessary to find a
solution.
(iii) The problem contains more information than is
needed for a solution as these problems contain
not only all the information needed to find a
solution, but also additional information in the form
of times, numbers, shapes or measurements.
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Problem structure and expression
When analysing a problem it is also possible to
discern critical aspects of the problem's form and
relate this to an appropriate level of mathematics
and problem expression when choosing or extending
problems. A problem of first-level complexity uses
simple mathematics and simple language. A 'secondlevel' may have simple language and more difficult
mathematics or more difficult language and simple
mathematics; while a third-level has yet more difficult
R.I.C. Publications®
(ii) The processes required are not immediately obvious
as these problems contain all the information
necessary to find a solution but demand further
analysis to sort out what is wanted and students
may need to reverse what initially seemed to be
required.
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As a problem and its solution is reviewed, posing
similar questions—where the numbers, shapes or
measurements are changed—focuses attention back
on what was entailed in analysing the problem and
in exploring the means to a solution. Extending these
processes to more complex situations enables the
particular approach used to extend to other situations
and shows how to analyse patterns to obtain more
general methods or results. It also highlights the
importance of a systematic approach when conceiving
and discussing a solution and can lead to students
asking themselves further questions about the
situation, thus posing problems of their own as the
significance of the problem's structure is uncovered.
(iv) Further information needs to be gathered and
applied to the problem in order to obtain a solution.
These problems do not contain first-hand all the
necessary information required to find a solution
but do contain a means to obtain the required
information. The problem's setting, the student's
mathematical understanding or the problem's
wording need to be searched for the additional
material.
Problem-solving in mathematics
xvii
INTRODUCTION
(v) Strategic thinking is required to analyse the
question in order to determine a solution strategy.
Deeper analysis, often aided by the use of diagrams
or tables, is needed to come to terms with what
the problem is asking so as to determine a means
to a solution.
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Assessing problem-solving
Assessment of problem-solving requires careful and
close observation of students working in a problemsolving setting. These observations can reveal the
range of problem forms and the level of complexity
in the expression and underlying mathematics that
a student is able to confidently deal with. Further
analysis of these observations can show to what
Observations based on this analysis have led to a
categorisation of many of the possible difficulties that
students experience with problem-solving as a whole,
rather than the misconceptions they may have with
particular problems.
This analysis of the nature of problems can also serve
as a means of evaluating the provision of problems
within a mathematics program. In particular, it can
lead to the development of a full range of problems,
ensuring they are included across all problem forms,
with the mathematics and expression suited to the
level of the students.
extend the student is able to analyse the question,
explore ways to a solution, select one or more methods
to try and then analyse any results obtained. It is the
combination of two fundamental aspects—the types
of problem that can be solved and the manner in which
solutions are carried out—that will give a measure
of a student's developing problem solving abilities,
rather than a one-off test in which some problems are
solved and others are not.
• needs to create diagram or use materials
• needs to consider separate parts of question, then bring parts
together
Student is unable to translate
a problem into a more familiar
process.
• cannot see interactions between operations
• lack of understanding means he/she unable to reverse situations
• data may need to be used in an order not evident in the problem
statement or in an order contrary to that in which it is presented
Problem-solving in mathematics
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INTRODUCTION
A final comment
If an approach to problem-solving can be built up
using the ideas developed here and the problems in
the investigations on the pages that follow, students
will develop a way of thinking about and with
mathematics that will allow them to readily solve
problems and generalise from what they already
know to understand new mathematical ideas. They
will engage with these emerging mathematical
conceptions from their very beginnings, be prepared
to debate and discuss their own ideas, and develop
attitudes that will allow them to tackle new problems
and topics. Mathematics can then be a subject that is
readily engaged with, and become one in which the
student feels in control, instead of one in which many
rules devoid of meaning have to be memorised and
(hopefully) applied at the right time and place. This
enthusiasm for learning and the
ability to think mathematically
will then lead to a search for
meaning in new situations
and processes that will
allow mathematical ideas
to be used across a range
of applications in school and
everyday life.
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Other possible difficulties result from a focus on being
quick, which leads to:
• no attempt to assess the reasonableness of an
answer
• little perseverance if an answer is not obtained
using the first approach tried
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A major cause of possible difficulties is the lack of
a well-developed plan of attack, leading students
to focus on the surface level of problems. In such
cases, students:
• locate and manipulate numbers with little or no
thought as to their relevance to the problem
• try a succession of different operations if the first
ones attempted do not yield a (likely) result
• focus on keywords for an indication of what might
be done without considering their significance
within the problem as a whole
• read problems quickly and cursorily to locate the
numbers to be used
• use the first available word cue to suggest the
operation that might be needed.
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developed in this series are followed and the specific
suggestions for solving particular problems or types
of problems are discussed with students, these
difficulties can be minimised, if not entirely avoided.
Analysing the problem before starting leads to an
understanding of the problem's meanings. The cycle
of steps within the model means that nothing is tried
before the intent of the problem is clear and the
means to a solution have been considered. Focussing
on a problem's meanings, and discussing what needs
to be done, builds perseverance. Making sense of the
steps that need to be followed and any answers that
result are central to the problem-solving process that
is developed. These difficulties are unlikely among
those who have built up an understanding of this way
of thinking.
A NOTE ON CALCULATOR USE
Many of the problems in this series demand the use of a number of consecutive calculations, often adding, subtracting,
multiplying or dividing the same amount in order to complete entries in a table or see a pattern. This demands (or
will build) a certain amount of sophisticated use of the memory and constant functions of a simple calculator.
4. To divide by a number such as 8 repeatedly, enter a
number (e.g. 128).
• Then press ÷ 4 = = = = to divide each result by 4.
• 32, 16, 8 , 2, 0.5, …
• These are the answers when the given number is
divided by 8.
• To divide a range of numbers by 8, enter the first
number (e.g. 90) and ÷ 4 = 128 ÷ 4 = 32, 64 = gives 4,
32 = gives 8, 12 = gives 1.5, …
• These are the answers when each number is divided
by 8.
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2. To subtract a number such as 5 repeatedly, it is
sufficient on most calculators to enter an initial
number (e.g. 92) then press – 5 = = = = to subtract 5
over and over.
• 92, 87, 82, 77, 62, …
• To subtract 5 from a range of numbers, enter the first
number (e.g. 92) then press – 5 = 95 – 5 = 37, 68 =
gives 63, 43 = gives 38, 72 = gives 67, …
• These are the answers when 5 is subtracted from each
number.
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1. To add a number such as 9 repeatedly, it is sufficient
on most calculators to enter an initial number (e.g.
30) then press + 9 = = = = to add 9 over and over.
• 30, 39, 48, 57, 66, …
• To add 9 to a range of numbers, enter the first number
(e.g. 30) then press + 9 = 30 + 9 = 39, 7 = gives 16, 3 =
gives 12, 21 = gives 30, …
• These are the answers when 9 is added to each
number.
5. Using the memory keys M+, M– and MR will also
simplify calculations. A result can be calculated and
added to memory (M+). Then a second result can be
calculated and added to (M+) or subtracted from (M–)
the result in the memory. Pressing MR will display
the result. Often this will need to be performed for
several examples as they are entered onto a table or
patterns are explored directly. Clearing the memory
after each completed calculation is essential!
Page 3
Students need to be able to visualise the arrangements of
cubes, oranges and cans stacked in several layers.
Some will see the first drawing as consisting of one layer
of 16 cubes, another of nine cubes, another of four cubes
and then one cube on top. Others will see slices going
down the staircase—10, then 9, 7 and 4—or see these
building upwards.
Possible difficulties
• Unable to visualise the cubes in the representations
on the two-dimensional page
• Only considers the cubes that can be readily seen on
the outside of the shapes
Extension
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These pages explore arrangements and dissections of
three-dimensional shapes in order to determine how
particular outcomes are formed. Spatial as well as
logical thinking and organisation are involved as students
investigate all likely arrangements to ensure that the final
forms match the given criteria or visualise a given shape
in terms of its component parts.
Page 5
The problems on this page extend the thinking needed
to visualise and see patterns. When the large cube was
cut into smaller cubes, some of the smaller cubes would
have been hidden inside the whole and careful analysis
is needed to determine the way individual cubes would
be painted. A table of possibilities is one way of keeping
track of what happened.
• Have students make stacks using different
arrangements of cubes and others work out how
many were used.
• Use isometric paper to draw the stacks they make and
have other students see how many cubes are used.
• Extend the problems by asking what would happen
if more blocks had been used—if each arm of the
'T' had one, two or more additional cubes before
painting. What if the large block had been cut into
cubes with sides of one centimetre? What if the cube
was cut into 64 or 125 smaller cubes?
The stacks of oranges and cans use either a triangular
base or square base and have the shapes stacked on the
intersection of the ones below rather than one on top of
another.
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The final shape does not follow a simple pattern and
students will need to visualise how the blocks are stacked
and find a way to organise their count. This will influence
the building of their own structures and how they visualise
and keep track of the individual cubes.
They will need 30 cubes to build the arrangement. The
different ways that the diagram can be visualised and the
staircase built should be discussed with the class.
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Page 4
Seeing how cubes make a larger structure is now extended
to include applying paint to a completed shape and
visualising the effect on the individual cubes. Students
may need to build the structures and manipulate blocks
in order to see what happens. Some students may need
to place labels on certain faces of the cube to help them
come to terms with what the questions are asking.
Encourage the systematic analysis of the shapes when
discussing the results: Could any cubes have all sides
painted? Where would cubes with six, four, three, two or
one side(s) painted be found? Which of these are possible
and which are not possible on each shape.
2
Problem-solving in mathematics
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STACKING SHAPES
1. A set of steps was made by stacking one
cube on top of another. How many cubes
were needed to make the staircase?
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make the staircase.
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2. Greengrocers often stack fruit as a
pyramid. How many oranges are in
this stack?
4. These cubes have been stacked one
on top of another. Some of the cubes
are hidden behind or beneath others
and can not be seen. How many
cubes were used to build the shape?
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Get some cubes and build your own stack. Write the number of cubes
you used and challenge a friend to work out how many you used
without pulling your stack apart.
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Problem-solving in mathematics
3
PAINTED CUBES
This shape was made using eight cubes. After
they were joined together, the final shape was
painted green on all of its sides. When it was
taken apart again, some of the faces of the
cubes have green paint and others do not.
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1. How many of the individual
cubes would have green paint
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2. Would any of the individual cubes have five faces painted? How many?
3. Would any have three faces painted? How many?
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and 2 cm high. The block is painted blue
on all six faces and then cut into 16 cubes,
each with sides of 2 cm.
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5. How many of the cubes would have only three faces painted?
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4. How many of the cubes would have blue paint on four faces?
6. How many of the cubes would have only two faces painted?
7. Why would there be no cubes with only one, five or six faces painted?
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Problem-solving in mathematics
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CUBE PAINTING
In the art gallery, a large polystyrene cube
was hung from a string and spray-painted red
on all six faces. It was then cut into twentyseven smaller cubes to be used by children
to make shapes of their own.
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red paint on only three faces?
2. Would any of the cubes have no red paint?
3. How many?
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1. With the large cube cut, how many of the individual cubes would have
5. When you count all of the cubes you have described, are all 27 included?
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Problem-solving in mathematics
5
TEACHER NOTES
Problem-solving objective
Possible difficulties
To use strategic thinking to solve problems.
Materials
0–99 number boardr
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SFocus
These pages explore students' understanding of the
number system and their ability to solve questions about
numbers. Students need to coordinate the reading and
writing of numerals with the symbols involved in writing
the numbers 400–600.
Extension
• Try other numbers ranges such as 300–500, 400–700
etc.
• Investigate the number of times 'five' is said and
written when counting by fives or 25s.
• Make a table to display the results and present a
description of the problems and their solution to
another class.
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• Unable to keep track of the number of times they
determine a digit or word
• Confusion between saying and writing the digits
• Not understanding that we say 'four' in numbers
which contain 'forty', even though it is written
differently to 'four' or 'fourteen'
• Thinking that 'five' is said for numbers with 'fifty'
• Confusion with 44, 55, etc. including only seeing the
digit once when it actually occurs in both the ones
and tens place
When pronounciating certain numbers—for example, with
numbers which include 'four'—it is important for students
to include numbers which include the pronunciation of the
word part 'four' within 'fourteen', 'forty', 'forty-one' etc.
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he number system and keep track of the possibilities they
find. Students need to discuss what it means to 'say' a
number as opposed to 'writing' a digit.
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If students take the problem further and try other number
ranges, as suggested, they will find a different pattern
altogether for the three hundreds as opposed to the four
hundreds. After the five hundreds the patterns begin to
repeat. When students notice this, they will have really
come to terms with the strategic thinking needed to
organise and solve problems with several interacting
conditions.
Trying different number ranges or counting and writing in
fives and 25s will provide different patterns as students
coordinate what happens to the ones, tens and hundreds
digits.
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Problem-solving in mathematics
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HOW MANY DIGITS?
1. (a) How many times do you say 'four'
when you count from four hundred
to six hundred?
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(b) How many times do you write '4'
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when you write all of the numbers
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he404 , 401
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Four hundred,
four hundred
and one ...
from 400 to 600?
2. (a) If counting from four hundred to six
hundred, would you say 'five' more
or fewer times than you would say
'four'?
(c) How many times would your write '5'
when writing all of the numbers from
400 to 600?
3. Try other 1-digit numbers. Can you see a pattern?
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Problem-solving in mathematics
7
TEACHER NOTES
Problem-solving objective
To identify and use concepts about numeration.
Materials
counters, blocks or a calculator
Focus
These pages explore solving problems involving number
sense, magic squares and logic. Students need to carefully
analyse the problems to locate information necessary to
find the magic number or the arrangement of numbers.
Counters, blocks or a calculator can be used to assist as
these problems focus on the concepts of number sense
and number logic rather than basic facts.
• Considering only rows or columns rather than rows,
columns and diagonals in the magic squares or the
smaller grids in the sudoku puzzles
Extension
• Investigate other magic squares and magic numbers.
��� Explore sudoku games in magazines, newspapers
and on the Internet that involve 4-by-4 grids as well
as 9-by-9 grids. The geoshape koala sudoku pack
available from educational suppliers provides a
concrete way for children to enter sudoku puzzles.
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This activity investigates possible combinations of
numbers to make a total. Students have four scores which
add together to make a total score and a list of numbers
which make up the scores. Number 'sense' and the 'try
and adjust' strategy are needed to find solutions.
For example, the score of 32, using the digits 5 and 9,
requires the combination of 9, 9, 9 and 5; while a score
of 57, using 24, 7 and 13 requires the combination of 24,
7, 13, 13.
The Dürer magic square has a number of points of interest
aside from being a magic square. For example, not only
does each row, column or diagonal add to the same number,
so do the four corners, the four middle numbers and the
four corner mini (2-by-2) grids. Also, it uses the numbers 1
to 16 and shows the year in which it was constructed.
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the idea that each row, column and diagonal adds to the
same magic number. In this case, the concept is further
explored as students investigate a famous magic square.
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This page introduces students to the concept of sudoku.
The word 'sudoku' roughly means 'digits must only occur
once'. In this case a 4-by-4 grid has been used and every
row, column and mini-grid must contain one of each of
the numbers 1, 2, 3 and 4. More commonly a 9-by-9 grid
is used, using the digits 1 to 9. No addition or basic facts
are involved and students need to use logical reasoning to
find solutions.
8
Problem-solving in mathematics
R.I.C. Publications®
STAR GAZE
Star gaze is a new computer game.
Each time you win, you collect stars.
Total the four stars to find your score.
1. Work out these scores.
(a)
8. What digits have been used?
9. Can you see anything to do with the date it was
first constructed?
10
Problem-solving in mathematics
R.I.C. Publications®
SUDOKU
Sudoku puzzles are made up of numbers and to solve them you
must use logic to work out where the numbers go.
Every row, column and mini-grid must contain one of each of
the numbers 1, 2, 3 and 4.
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4 Row –
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numbers
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and 4.
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4 Column
1 – has the
numbers
2 1, 2, 3
3 and 4.
3
2
4 Mini-grid
1 – has the
numbers
1, 2, 3
and 4.
2
1
3
4
3
4
2
1
4
3
1
2
1
2
4
3
Complete each sudoku.
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1
2.
2
3
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The completed sudoku has the numbers 1, 2, 3 and 4 in every
row, column and mini-grid.
These pages explore word problems that mostly require
addition or subtraction. Students need to determine what
the problem is asking and, in many cases, calculate more
than one step in order to find solutions. Analysis of the
problems reveals that some questions contain additional
information that is not needed.
Again, there are a number of ways to find a solution and
students should be encouraged to explore and try different
possibilities of arriving at an answer. The last question
involves students understanding that perch and carp are
varieties of fish, while yabbies are crustaceans.
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Discussion
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These problems involve more than one step and may
involve addition as well as subtraction. The wording
has been kept simple to assist with the problem-solving
process.
Possible difficultiesExtension
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If necessary, materials can be used to assist with the
calculation as these problems are about reading for
information and determining what the problem is asking
rather than computation or basic facts.
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A careful reading of each problem is needed to determine
what the question is asking. In some cases, there is more
information than needed and some problems contain
numbers that are not needed to find a solution. Most
problems require more than one step and both addition
and subtraction are needed at times.
• Explore the possibilities as to whether the dry season
would be before or after spring.
• Discuss how some problems can have more than one
answer depending on different interpretations.
• Students could write their own problems and give
them to others to solve.
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This investigation relates to information about a lake with
lily pads and frogs. The scenario begins with a certain
number of frogs and lily pads. As new information is
introduced, the numbers change to meet the new criteria;
lily pads flower, grow and die while the frogs move from
one lake to another.
Students may choose a number of different ways to find
a solution. For example, the second problem about people
getting out (43) could be subtracted from the people going
swimming (79) and then this number (36) could be added
to 397 or, alternatively, 79 could be added to 397 and then
43 subtracted to obtain a solution. Students should be
encouraged to explore and try different ways of arriving
at a solution.
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Students are required to keep track of the new information
and use it to answer the subsequent questions. The
last question has two possible answers as it depends
on whether spring is before or after the dry season.
Students should be encouraged to explore the different
possibilities.
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Problem-solving in mathematics
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1. 361 adults and 173 children go
through the gates before lunch,
and 219 adults and 106 children
enter after lunch. How many
more adults than children are
there?
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THE WATER PARK
4. In the water, 93 people are floating
on swimming mats, 134 are
swimming and 83 are wading in
the shallow water. Soon, another
21 people with swimming mats
arrive, but 14 also get out. How
many people are now floating on
swimming mats?
2. A total of 397 people are
swimming in the six pools.
Another 79 people go swimming 5. At the cafeteria, 143 people are
while 43 people get out. How
sitting eating lunch and 31 are
many people are now swimming
standing in line waiting to order
in the pools?
lunch. Two large tables of 12 finish
their lunch and leave. How many
people are now in the cafeteria?
3. 248 people are lying on their
towels. Later, 78 people go
swimming, 26 people go for a
6. In the wave pool, 73 surfers are
walk and 36 people leave and
waiting to catch a wave. A large
get something to eat. How many
wave comes and 36 surfers catch
people are still lying on their
and ride it to the beach. How
towels?
many did not catch the wave?
Problem-solving in mathematics
13
LILY PADS AND FROGS
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A lake has 279 lily pads and 372 frogs.
1. 87 of the lily pads are in flower. If each lily pad has three flowers, how
many flowers are there altogether?
5. During which weekend were the least amount of fish and yabbies
caught?
6. During which weekend were the most fish caught?
R.I.C. Publications®
Problem-solving in mathematics
15
TEACHER NOTES
Problem-solving
To solve problems involving money and make decisions
based on particular criteria.
Materials
counters, play money or a calculator
• Unfamiliarity with the '$' symbol
• Not taking into account that they may need two or
more of some items
Extension
Focus
This page explores reading for information, obtaining
information from another source (the takeaway menu) and
using it to find solutions. The problems are about using
money, making decisions based on money and comparing
amounts of money, rather than adding or subtracting.
Solutions can be obtained using materials and comparison
of amounts. Counters, blocks, play money or a calculator
can be used if needed.
• In pairs, students write their own questions based on
the takeaway menu and give them to other pairs to
solve.
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Discussion
Page 17
Students read the items in the menu and note how
much each costs. Students who are not familiar with
money can still do the activity with a calculator. In most
cases, students need to buy two or more of an item and
then determine the amount spent and the change that
remains.
The first two problems involve determining how much
is spent on particular orders. Students need to take into
consideration that more than one of some items have
been ordered. The remaining problems analyse particular
orders, with an additional focus on how much change
would be received from $50.
counters in two different colours (1 of one colour, 17 of
another colour)
Focus
Discussion
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There are several ways these problems can be solved.
One way is to work backwards or 'backtrack' from the
final position; for example: Lucy finishes in eighth position,
must pass 11 cars, then be passed by five cars to get back
to her original position. Counters can be used to model
the process of cars passing and being passed. Using a
counter of one colour to represent Lucy's car and counters
of a second colour to represent the other 17 cars helps
students keep track of who's passing who. Other students
might prefer to base their solution on the diagram on the
page to model what has happened.
Possible difficulties
• Using only the 11 cars that Lucy passed to determine
her starting position
• Not taking into consideration all the criteria for the
serial numbers
• Considering only some aspects of the puzzle scrolls
Extension
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These pages explore more complex problems in which the
most difficult step is to find a way of coming to terms
with the problem and what the question is asking. Using
materials to explore the situation is one way in which
this can be done. Another is to use a diagram to assist in
thinking backwards or making trials and adjusting to find
a solution that matches all of the conditions.
Page 21
The puzzle scrolls contains a number of different problems,
all of which require strategic thinking to find the solutions.
In most cases, students will find tables, lists and diagrams
helpful when exploring the different possibilities. Certain
problems may have different answers, depending on the
criteria the students use; for example, the investigation
about the length of rope the size of the ruler is not stated
so it may be necessary to consider both of the common
ruler sizes of 30 cm and 1 m. The investigation about
each combination of digits that total to 17 each can be
rearranged to give six possibilities, giving 36 different
numbers.
• Students write their own car race problems based on
the questions.
• Students could use a different context other than
racing cars for their stories.
• Students write their own criteria for working out a
serial number.
• Explore other shapes and how many triangles,
squares or rectangles there are in them.
Alternatively, students may choose a position for Lucy and
work through each of the events in the race. If Lucy does
not end up in eighth place, an adjustment can be made to
determine the original starting position
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These problems challenge students understanding as they
investigate the ways the digits can be placed according to
set criteria. Some students will use the listed information
in order to discard combinations until only the correct
number remains, while other students may prefer to try
each number in turn against all of the criteria until they
find one number that answers all conditions. In each case,
students need to take into consideration the information
given about how large or small a number is.
5. Robbie bought 60 m of
rope but the ruler used to
measure the rope was 1cm
too long.
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were the first and last
days of the month?
6. My house number has
three digits that add to 17.
My number could be 935.
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935
How much rope did he
really get?
R.I.C. Publications®
What else
could it be?
Problem-solving in mathematics
21
TEACHER NOTES
Problem-solving
To analyse and use information in word problems.
Materials
Base 10 materials, place value chart, calculator
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Discussion
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These problems involve addition, subtraction and
multiplication. In most cases more than one step is
needed to find a solution. The wording has been kept fairly
simple to assist with the problem-solving process. The
last problem requires students to work backwards to find
a solution: If Samantha needs 82 lilies and only orders an
extra 47, then she would have had 35 left over on Friday.
When the delivery is short by seven lilies, she will have 75
lilies rather than the 82 she needs.
Extension
• Students could write their own problems and give
them to other students to solve.
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These pages explore word problems that require addition,
subtraction or multiplication. The wording has been kept
fairly simple to help with the problem-solving process.
Students need to determine what the problem is asking
and in many cases carry out more than one step in order
to find solutions. Materials can be used to assist with
the calculation if necessary as these problems are about
reading for information and determining what the problem
is asking rather than computation or basic facts.
Possible difficulties
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These investigations involve addition, subtraction and
multiplication. Students may find it helpful to draw a
diagram to work out what is happening in the story and to
determine what needs to be multiplied to find a solution.
The problems about selling the bunches of parsley in the
morning and afternoon (questions 5 and 6) also involves
information from Problem 4. This problem explores how
many bunches of parsley Simon has not sold and is used
in the next problem.
Page 25
The wording and the steps involved in these problems have
been kept fairly simple to assist with the problem- solving
process. Students may find it helpful to draw a picture in
order to work out what is happening in each story. In most
cases more than one step is needed to find a solution.
Problem 2 contains information about white loaves and
multi-grain loaves which is not needed to find a solution.
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Problem-solving in mathematics
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SAMANTHA'S FLOWER SHOP
1. Samantha has 38 pots of chrysanthemums left over from Friday. On
Saturday, she receives a delivery of another 75 pots of chrysanthemums
and sells 59 pots. How many pots of chrysanthemums does she have
available to sell on Sunday?
2. (a) The delivery truck delivers 384 roses. 186 roses are sold in the
morning. In the afternoon, she receives an order for 26 bunches of five
roses. Does Samantha have enough roses for this order?
(b) How many roses does she still have available to sell?
3. Samantha has 120 red roses and 100 yellow roses. She makes 14
bunches, using 9 roses in each bunch. Does she have enough roses to
make another 8 bunches?
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4. (a) Samantha has
some lilies left at the end of the
day
on Friday. She has
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orders for 82 liliese
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customers on
Saturday.
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to make up the shortfall. How many lilies did she have left over on
Friday?
(b) When she goes to work on Saturday, she finds that only 40 lilies were
delivered. How many lilies does she have for her orders?
R.I.C. Publications®
Problem-solving in mathematics
23
HERB MARKET
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Sstall. He sold 373 bunches of fresh herbs and 218 pots
1. Simon has a herb
of herb plants. How many more bunches of fresh herbs than pots were
sold?
2. Simon has six trays of herb pots. How many pots does he have if each
tray holds 24 pots?
2. On Saturday, 83 white loaves, 147 wholemeal loaves and 163 multi-grain
loaves were sold. On Sunday, 132 white loaves, 169 wholemeal loaves
and 178 multi-grain loaves were sold. How many wholemeal loaves were
sold over the weekend?
3. (a) The bakery bakes 15 trays of pies. Each tray holds 12 pies. How many
pies does the bakery have available to sell?
4. The bakery has 14 trays of lamingtons. Each tray has 16 lamingtons. It
sold 23 lamingtons during the first hour, 84 lamingtons during the second
hour and 73 lamingtons during the third hour. How many lamingtons can
it sell for the rest of the day?
R.I.C. Publications®
Problem-solving in mathematics
25
TEACHER NOTES
Problem-solving
To use patterns and logical reasoning to determine
numbers in a table.
Materials
calculator
or 5 in the ones place; the one after for the numbers with
6, 7, 8 or 9 in the ones place ...
For example, a number with 10 tens must be in row 20, 21
or 22—108 will be in row 22.
For Question 5, the patterns for the ones digit is:
Focus
This page explores problems ordering of numbers
to discern patterns that allow larger numbers to be
determined without laboriously writing or counting all of
the numbers up to the point asked for. It also highlights
the value of using factors and multiples when thinking
about numbers.
Column
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With questions 1–4, most students will observe that any
number with a 5 in the ones place occurs in Column E.
However, they may be surprised that a number with a 0
in the ones place does not also occur in this column. Both
87 and 34 are placed in Column D—this may lead them to
observe how each column contains two different types of
numbers. Any number with 2 in the ones place occurs in
Column B, so this is where 92 is; meanwhile, any number
with 8 in the ones place, including 108, is in Column C.
1 or 6
A
5 or 0
B
2 or 7
C
4 or 9
D
3 or 8
E
Numbers with
0 tens are in rows 1–4
1 ten are in row 4–7 etc.
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Discussion
Digit in ones place
With Question 6, comparing the two arrangements, shows
that only the digits 4 and 1 remain in the same columns.
Some students may be able to observe that 20% of the
numbers are unchanged.
Possible difficulties
• Thinking that writing out all of the numbers is the
only way to be sure of a solution
• Only considering the ones place when searching for a
pattern
• Unable to verbalise a mathematical description of
how the numbers are placed
The pages explore concepts of place value and number
sense. The relationships among numbers and place value
are analysed and students are encouraged to not only find
numbers that are possible but also to disregard numbers
that are not possible. Place value and number sense are
needed rather than addition or multiplication.
Discussion
With the first problem, each person should receive nine
chocolate frogs. The combinations are eight and one,
seven and two, six and three and five and four. Some
problems involve a number of possible combinations. The
second problem has three different possible combinations,
while the last problem has many different combinations.
The two children can be given anywhere from 11 to
17 chocolate frogs each and each total offers several
possible combinations. However, since the question asks
for a total of 12 frogs, the boxes could be 138 and 246 or
237 and 156.
Possible difficulties
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These problems involve a list of information about numbers
which students need to read, interpret and enter into a
calculator to find a given number. An understanding of
place value is needed to enter the information. Students
may use a number expander to assist them with the
place value if needed. No formal addition or subtraction
is needed. With Question 1, students need to start with
a number 100 less than 4086. Using an understanding of
place value we know that the number has 40 hundreds
and 1 hundred less would be 39 hundreds, so the starting
number is 3986.
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In this investigation, students need to read and interpret
the information and use it to find combinations that
match specific criteria. Students need to think of possible
combinations as well as discarding combinations that
don't work.
• Poor understanding of place value
• Wanting to add, subtract or multiply rather than using
place value or number sense
• Not considering all of the criteria
problems and write the criteria to match.
• Work out the different possibilities for two children
to have three boxes, each with totals of 11, 13, 14,
15, 16 and 17.
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value. As 10 cones fit into one box and 10 boxes fit one
carton an understanding of place value can be used to
solve each problem. For example, Question 1 involves 9
boxes and 24 cartons. It can be solved by thinking in tens
or by thinking in tens and hundreds. By understanding
place value, it is known that there will be 90 cones in the
boxes and 2400 cones in the cartons, giving a total of 2490
cones. No formal multiplication or addition is required. A
number expander can be used to assist.
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Problem-solving in mathematics
R.I.C. Publications®
CALCULATOR PROBLEMS
Solve the problems. Use your calculator to help you.
2. Enter the
number with
562 tens and
9 ones. Add
23 hundreds,
4 tens and
7 ones. Take
away 10
hundreds.
What number
am I?
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3. Enter the
number 1000
before 8293.
Take away 24
tens. Add 3
ones and 634
tens. What
number am I?
4. Enter the
number 100
more than
6958. Make
it 1000 more.
Make it 100
less. Add 17
hundreds.
What number
am I?
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1. Enter the
number one
hundred less
than 4086.
Take away 317
and add 2006.
What number
am I?
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5. Enter the
number with
82 hundreds
and 5 ones.
Add 34 tens.
Take away 8
hundred and
six. What
number am I?
6. At the end of the day, the ice-cream shop has 840 cones. If there are 6
full cartons, how many extra boxes does it have?
30
Problem-solving in mathematics
R.I.C. Publications®
CHOCOLATE FROGS
Trudy is having a party. She has wrapped chocolate frogs in
eight boxes. The first box has one chocolate frog, the second
box has two chocolate frogs, the third box has three chocolate
frogs and so on.
Box 8
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Box 2 Box 3
Box 4 Box 5 Box 6 kBox 7
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1. How many chocolates frogs has Trudy wrapped?
Trudy wants to give the same number of chocolates frogs to each person.
2. If she gives four children 2 boxes each, which boxes could she give to
each person?
(a)
To interpret and organise information found in a series of
interrelated statements and to use logical thinking to find
solutions.
Focus
These pages explore interrelated statements within a
problem situation concepts of averages, distance and
payments. Students need to read the stories carefully
in order to take into consideration a number of different
criteria. Tables and lists can be used to help manage the
various criteria.
Discussion
Problems 2 and 4 present two different ways of payment
and both options need to be analysed to work out which
option is best. Again, a table showing both options can be
used to assist with the problem-solving process.
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Possible difficulties
• Not using a table or list to mange the data
• Not understanding the term 'average'
• Confusion when dealing with approximate times and
distances
Extension
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Each problem requires students to consider the information
provided in a series of interrelated statements, all of
which needs to be taken into consideration in order to find
a solution. The use of a table or list may be very helpful
to manage the data. For example, in the first problem, a
table listing the various years can be used as a starting
point. The problem states how many visitors came in 2003
and this information can be used to work out the number
of visitors in 2007 (twice as many), and in turn this can be
used to work out the numbers for 2006.
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Each problem needs to be read carefully to determine
what is being asked. In the first question, it is necessary
to work out how many towels were purchased (four) in
order to determine that eight are bought the next day
(twice as many).
• Construct a table to show the running distance and
how it varies from month to month.
• Write other problems using the same form of
complex reasoning for other students to solve.
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These problems explore the concept of average distance
travelled over a period of time. In many cases the solution
is not necessarily exact but rather an approximate time or
distance; for example, Problem 2 states that Susie swims
100 m in 'about 2 minutes'. This is not an exact time and
would vary from lap to lap, so the solution of how far she
has swum would again be an approximate distance.
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Problem 4 deals with the concept of distance travelled over
a month. Discussion could centre around how this would
vary from month to month. A table could be constructed
to show the distance Brian runs during each month of the
year. Some students may reason that he runs 156 km per
week and that it takes a little less than four and a half
weeks per month to travel around 700 kms. As an extra
dimension, Problem 5 contains additional information
about lunch as well as stopping and starting times which
is not needed to find a solution.
A similar table or list can be used for the other problems.
The last problem contains additional information about
the number of caves and the most visited caves, none of
which is needed to find a solution.
Problem-solving in mathematics
R.I.C. Publications®
HOW MANY?
1. A large number of tourists visited
Uluru during 2007. There were twice
as many visitors in 2007 than in 2003.
There were 6530 more visitors in
2007 than in 2006.
If there were 298 460 visitors in 2003,
how many were there in 2006?
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2. During September, 258 000
tourists visited the Great Barrier
Reef. April had twice as many
visitors as January, but 4000 less
than August. August had 8000
more visitors than September.
3. The Jenolan Caves Touring
Company, in the Blue Mountains,
currently offers tours to 11 different
caves. Many of the visitors tour
the Lucas, River and Chiefly caves.
Due to storms, May was a poor
month for visitors and only 19 970
people visited. June had 6230
more tourists than July and July
had 3150 more than March. March
had 4020 more visitors than May.
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How many visitors were there in
June?
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Problem-solving in mathematics
33
HOW FAR?
1. Erin caught a bus from Melbourne to Sydney. The bus left at 4 pm and,
due to traffic, averaged 52 km for the first 2 hours. Once on the highway,
the bus averaged 96 km per hour for the next 6 hours.
How far had Erin travelled?
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Approximately how far does she
swim?
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2. Susie swims each morning in a
50 m pool. She can swim 100 m
in about 2 minutes. She usually
swims for an hour, has a short
break and then swims for another
hour.
3. The train from Brisbane to Rockhampton travels at an average speed of
105 km per hour. If Rockhampton is about 655 km from Brisbane, how
long will it take for the train to arrive?
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4. Brian trains each day for the marathon. During the
week he runs 8 km in the morning and 12 km in
the afternoon. On the weekend he runs 28 km each
day.
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5. Kim-Ly drove from Perth to Albany. She left at 9 am and
averaged a speed of 94 km for 3 hours. She stopped for
lunch and started again at 1pm. She drove for four hours
and averaged 92 km per hour.
How far had she travelled?
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Problem-solving in mathematics
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HOW MUCH?
1. Derek paid $32 for towels that cost $8 each. The next day he saw the
same towels for $6 each, so he bought twice as many as the day before.
How much did he spend on towels?
2. Michael delivered 652 newspapers in
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seven
hours. He can be paid by the
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number of
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is 5.5c and the hourly rate is $4.25
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for the first hour and $3.50 for each
other hour. Which option pays more
money?
3. Alison bought apples from the market at a price of 6 apples for $2. She
then sold them at her fruit shop at a price of 4 for $2 and made a profit of
$10. How many apples did she sell?
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4. The hardware shop sells shelf
brackets both with and without
screws. The brackets with screws
cost $9.90 each and the ones
without screws cost $6.50 each.
Screws cost 80c each. If four screws
are needed for each bracket, which
is the cheapest option and by how
much?
5. Wendy can be paid either by the day or by the number of trees she
plants. She gets $1.20 per tree or $34.70 per day on weekdays and $48
per day on weekends. On average, she can plant about 37 trees per day.
If she works Sunday to Sunday, which is the best payment option?
R.I.C. Publications®
Problem-solving in mathematics
35
TEACHER NOTES
Problem-solving
To solve problems involving time and make decisions
based on particular criteria.
Materials
Conversion table from 12-hour to 24-hour times, clock
Focus
This page explores reading for information, obtaining
information from a number of sources (information about
the plane, the timetable and the shuttle bus) and using it
to find solutions. The problems involve thinking about and
working with time. Decisions are about being 'too early' or
'too late' rather than an exact time.
• Unfamiliarity with a timetable
• Confusion with 24-hour time
• Thinking that an exact flight is needed rather than
flights that fit within the time frame
Extension
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Students read the information on the page and use it to
find a number of solutions. Students who are not familiar
with 24-hour time can still complete the activity by using
a conversion table. The investigation can be used to
introduce the concept of 24-hour time. Students need to
read for information by using a number of sources and
compare it against set criteria.
• Use the information and timetable with other
criteria; for example: If you need to be in Cairns for a
lunchtime meeting, what flights can you take?
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Discussion
Possible difficulties
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Similar thinking can be used for the flights that are too
early; for example, catching the Red Airlines plane at 0830
would get to you to the hotel at around 1100, which is
three hours too early. As such, the flights at 0900 and 1015
are also too early and can be automatically excluded.
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Students need to think in terms of 24-hour time for the
flight information; however, the before and after times of
2 pm and 6.30 pm are in 12-hour time. Students need to
take into consideration that 2 pm and 6.30 pm are also
1400 and 1830 respectively.
There are a number of flights that fit the criteria, with other
flights arriving at Cairns either 'too early' or 'too late'.
Once some flights are deemed to be too late, they can be
automatically excluded; for example, if the Red Airlines
plane at 1700 would not arrive at Cairns in time for dinner,
then the flights at 1730 and 1800 can also be ruled out.
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The information regarding the waiting time at the airport
is not needed. Some students may try to include this in
their calculations.
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TROPICAL CAIRNS
Imagine you have decided to travel from
Brisbane to Cairns for a holiday.
1. I can catch the following flights:
2. These flights are too early:
3. These flights are too late:
R.I.C. Publications®
Problem-solving in mathematics
37
TEACHER NOTES
Problem-solving
To use spatial visualisation and measurement to solve
problems.
Materials
Paper to make and fold squares and equilateral triangles,
triangle and square grid paper
Focus
The perimeter of the large square is 24 cm.
Folding the square in Problem 2 to show 9 small squares
and the square in Problem 3 to show 16 small squares
allows them to solved in the same way.
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Discussion
Page 39
Students need to be able to understand that the perimeter
of the first shape (made from five squares) consists of
12 sides. This can be done by counting all of the sides
in a systematic way or by seeing the shape made up of
symmetric parts—two sections with five sides on the top
and bottom and two in the middle, for example. Since the
perimeter is 36 cm, the side of each small square must be
3 cm long. It may need to be explained to students that the
shape is not to scale.
Page 41
This page continues the students' investigation of
perimeters. While the first problem can be solved by
thinking of the rectangle as being made up of two whole
(9 cm) sides and two half (4.5 cm) sides, this does not readily
generalise to the other ways the square is partitioned.
Thinking of it as six half-sides allows the next shapes to
be seen as, respectively, eight third-sides and 10 fourthsides to readily solve the perimeter of each of the original
squares. Note that thinking of these as fraction symbols
(2/3 and 2/4) makes the solution considerably more difficult
to determine as it requires working with fractions rather
than whole numbers. Another way to visualise solutions is
to divide the squares and rectangles into smaller squares
and see the results directly, as on the preceding pages.
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These pages explore ideas of perimeter by using their
knowledge of squares and equilateral triangles to visualise
shapes and to determine the lengths of sides within or
composed of the shapes. Spatial and logical thinking,
as well as numerical reasoning and organisation, are
involved as students investigate the relationships among
the shapes to determine the required distances.
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The side of the rectangle, 18 cm, is the same as 6 sides of
the small square, so the side of the small square is 3 cm.
• Uncertain of definition of perimeter
• Does not understand that the sides in a square or
equilateral triangle are of equal length
• Unable to visualise the sides of the smaller shapes
within the large shapes
• Can not keep track of the number of sides that need
to be used
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the number of sides multiplied by 3 cm. For questions 2
and 3, there are several possible shapes that can be made
with a perimeter of 48 cm. The only criterion is that each
perimeter must use 16 sides of the smaller squares; for
example, a 3-by-5 rectangle or a 4-by-4 square:
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Students need to be able to visualise the way in which the
squares or triangle are folded and how the perimeter of the
final shape would relate to the whole shape if unfolded.
Students may need to use paper cut into squares:
Folding the square in
Problem 1 in half and
then half again shows
4 small squares when
unfolded.
Extension
• Have students investigate shapes made from small
equilateral triangles in the same way as those made
from small squares.
• Ask students to create their own examples of
perimeters in squares where the square is folded into
5, 6 or more rectangles.
Each triangle contains 2 small squares.
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R.I.C. Publications®
SQUARES AND PERIMETERS
Perimeter is the distance around the boundary of a shape.
Small squares, all of the same size, have been used to
make these shapes.
Shape
A
1. Shape A has a perimeter of 36 cm, what is the
perimeter of each of these shapes?
The perimeter of each rectangle is 50 m.
What was the perimeter of the original square
of plastic?
R.I.C. Publications®
Problem-solving in mathematics
41
TEACHER NOTES
Problem-solving
To analyse and use information in word problems.
Materials
paper to fold
Focus
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Discussion
Page 43
The problems mostly require multiplication, with
some addition and subtraction. Each problem is fairly
straightforward, with all the information needed to find
a solution readily available. The numbers have been kept
simple to assist with the problem-solving process and
there is no additional information that is not needed.
Possible difficulties
• Inability to identify the need to add, subtract or
multiply
• Not using place value concepts to solve the problems
• Confusion over the need to carry out more than one
step to arrive at a solution
• Not understanding the concept of 'capacity'
Extension
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These pages explore word problems that require addition,
subtraction, multiplication and an understanding of
place value. The wording is more complex than in the
previous problems that involved a combination of the
three operations. Students need to determine what the
problem is asking and, in many cases, carry out more
than one equation in order to find solutions. Materials
can be used to assist with the calculation if necessary,
as these problems are about reading for information
and determining what the problem is asking rather than
computation or basic facts.
Page 45
A careful reading of each problem is needed to determine
what the problem is asking. These problems focus around
trains, carriages, people and seats. In some problems
people are getting on and getting off and in others it is
necessary to determine how many people are sitting or
standing in a carriage. Each problem requires more than
one step and two or more operations are needed to find a
solution. There are a number of ways to find solutions and
students should be encouraged to explore and try different
possibilities of arriving at solutions.
• Students could write their own problems and give
them to other students to solve.
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Students may find it helpful to draw a diagram in order
to work out what is happening in the problem and to
determine what needs to be multiplied to provide? A
solution. Addition can be used at times rather than
multiplication.
Page 44
This investigation provides information about a library,
with a number of interrelated questions arising from
it. The questions begin with a set number of books,
magazines and CD-ROMs. Using this information as a
basis, the numbers of each item are changed to meet new
criteria, with some books, magazines and CD-ROMs being
borrowed, returned, shelved and dusted. Students are
required to keep track of the new information and use it to
answer the subsequent questions.
Computation is not always needed to find a solution; for
example, Problem 1 states that each shelf holds 100 books
and, as there are 9472 books in the library, 95 shelves are
needed. No division is necessary as place value tells us
that 9472 has 94 hundreds, so 95 shelves must be needed.
Similarly with Problem 2, 48 shelves are being dusted,
which means 48 hundreds. Therefore, 4800 books have
been dusted.
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BEADING
1. Judy has 6 bags of coloured beads, with 42
beads in each bag. If she uses 29 beads to
make a necklace, how many beads does she
have left?
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2. Manu has 7 bags of beads. He buys 2 more bags and uses 25 beads to
make some bracelets. If each bag contains 32 beads, how many beads
does he have now?
3. Clarence bought 3 bags of beads on
Monday, 5 bags on Tuesday and 4 bags
on Wednesday. Each bag contains 48
beads. How many beads did he buy?
5. Ned has 4 large bags of beads and 7
small bags of beads. The large bags
hold 25 beads and the small bags hold
15 beads. He uses 24 beads to make
a necklace and 18 beads to make two
bracelets. How many beads does he have
now?
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6. Zena has 7 bags of beads, with 30 beads in each bag. She used 36 beads
to make 2 necklaces and 54 beads to make 6 bracelets. She wants to
make the same set of jewellery tomorrow. Does she have enough beads?
R.I.C. Publications®
Problem-solving in mathematics
43
LIBRARY
The library has books, magazines and CDROMs. It has a total stock of 9472 books, 315
magazines and 143 CD-ROMs.
1. In the library, each set of shelves holds 100 books. How many sets of
shelves are needed to hold all of the books?
3. (a) At the start of the day, the library's
computer showed there were 6841
books currently in the library. During
the morning, 275 books were taken out
and 97 books returned and during the
afternoon, 166 books were taken out
and 134 returned. How many books are
now in the library?
(b) Later in the day, six boxes of books, two boxes of magazines and four
boxes of CD-ROMs were delivered to the library. If each box of books
holds 65 books, how many books will the library now have?
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4.
Every student in Year Four borrowed a
book from the library. If there are 4 classes
with 26 students in each class, how many
books were borrowed?
5. During stocktaking, 284 old and damaged books were removed from
the shelves to be packed into boxes. If each box can hold 10 books, how
many boxes are needed to pack all of the books?
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AT THE STATION
1. There were 604 people on the train. At
the first station, 58 people got on and 129
people got off. At the next station, 143
people got on and 72 people got off. How
many people are now on the train?
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(a) How many people can sit on the train?
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(b) How many people can travel on the train?
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2. The train has 12 carriages. Each carriage
has 47 seats and can hold 65 people.
3. There are 36 people in the first carriage, 27 people in the second carriage
and 46 people in the third carriage. Each carriage is able to hold 60
people. How many more people are needed to fill the train to capacity?
5. The train has 8 carriages and each carriage has 47 seats. There are 411
people on the train. If every seat has one person on it, how many people
are not sitting down?
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6. At the first station, 63 people
got
on
and
138 people got off. At the next
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station, 94 people got on and 86 people got off. There are now 261
people on the train. How many people were on the train to begin with?
R.I.C. Publications®
Problem-solving in mathematics
45
TEACHER NOTES
Problem-solving objective
To read, interpret and analyse information.
Materials
calculator, number expander
Possible difficulties
• Not considering place value to solve the questions
• Not taking into consideration the starting page of
each chapter
Extension
Focus
Discussion
• Change the criteria involving the number of pages in
the book and the page number read to and explore the
problems again based on the new criteria.
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Students need to read and interpret information and
use it in the context of place value. As the book has 10
equal-length chapters, an understanding of place value
can be used to solve each problem. Given that the book
has 690 pages we know that this number has a place
value of 69 tens, so each chapter must be 69 pages long.
This information can be used to calculate the various
problems.
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Students explore concepts of place value and number
sense. The relationships among numbers and place value
are analysed and students are encouraged to find suitable
numbers and disregard numbers that are not possible.
Place value and number sense are needed rather than
division.
The page number that Walter has read to is given at the
beginning and this information is needed to answer some
of the problems. For example, in order to determine about
how many pages Walter has read past the middle of the
book and needs read to finish the book, it is necessary to
keep in mind that there are 690 pages in the book and he
has read up to page 437.
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starts on page 1, so Chapter 2 must start on page 70 and
so on.
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Problem-solving in mathematics
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BOOKWORMS
Walter's book starts on page 1 and has 690 pages. There are 10 equallength chapters in the book and he has read up to page 437.
1. How many pages are in each
chapter?
2. Walter's favourite page is
409. Is it in Chapter 7?
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morning and 10 pages each
night. How long has he been
reading this book?
5. Walter's favourite chapter
is Chapter 3. From starting
page to finish, what pages
are in Chapter 3?
7. How many more pages does
he have to read to finish the
book?
8. Walter's sister is also
reading this book. She has
read eight pages of Chapter
6. What page is she up to?
9. How many more pages does
she need to read to finish
the book?
10. If she reads 10 pages a day,
how long will it take her to
finish the book?
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R.I.C. Publications®
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Problem-solving in mathematics
47
TEACHER NOTES
Problem-solving
To use spatial visualisation, logical reasoning and
measurement to solve problems.
One approach is to create a table of possible values using
a process of 'try and adjust':
6 rolls and 1 loaf costs $4.20
roll
Materials
paper to fold
Focus
3.00
1.20
2.00
2.40
5.40 – too little
0.40
2.40
1.80
1.60
3.60
5.20 – too little
0.35
2.10
2.10
1.40
4.20
5.60 – too much
0.30
1.80
2.40
1.20
4.80
6.00
Page 49
In these problems, students need to visualise the paths
that the farmer and dog take as they travel around the
outside of each shape. Some of the lengths around the
paddock and garden need to be determined from the
diagrams and an ability to rename from kilometres to
metres or centimetres to metres is required.
A loaf of bread costs $1.80. A bread roll costs 30 c.
Page 51
This page extends the thinking about perimeters explored
on page 39. With Problem 1, when the paper is folded
four squares will result. Since the length of each side
of the small square is 3 cm, each will have an area of
9 cm2, so the area of the original square must be 36 cm2.
Alternatively, the large square has sides of 6 cm in length,
giving an area of 36 cm2. The second problem requires
students to visualise the relationships among the squares
and determine the area after the length of its sides is
known.
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Discussion
Total
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These pages explore different ways of visualising
problems and analysing the possibilities that make up
the whole solution. Logical reasoning is required, as well
as an understanding of measurement (kilometres, metres
and centimetres). In each situation, diagrams can be used
to organise, sort and explore the data.
Page 50
The problems on this page are essentially solved the same
way; however, the provision of the image of a balance for
the first problem makes it is easier to see how the mango
from the top picture can be substituted using the scales
in the second picture. Problem 2 requires more complex
thinking.
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• Not using the given data to determine the sides
whose length is not stated
• Unable to see how the information in the problems
can balance
• Unsure of the area of a square and confusing it with
perimeter
Extension
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In the first problem, the farmer rides around the paddock
more than two times, passing by corners B, C and D three
times. The distance around the paddock is 5300 m which can
be used, along with the distance from A to D, to calculate
the distance travelled to return to D. For Problem 2,
interpreting the diagram to determine the lengths is more
complex, although some students may realise that finding
the distance all around three times and then subtracting
the distance from G to A is simpler.
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Problem-solving in mathematics
• Students could write their own problems involving
distance around an irregular shape where some of the
lengths have to be worked out from the information
in the diagrams. Other problems could involve items
on a balance or areas cut from inside an arrangement
of shapes. The problems are then given to other
students to solve.
R.I.C. Publications®
FARM TRAILS
1. A farmer started at corner A of the paddock and rode his trail bike around
the perimeter to see if there were any gaps in the fence. He found one
hole at D and continued all the way around to A, where he picked up his
tools to fix the fence. He then rode to D and fixed the hole.
1km 200 m
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400 m
650 m
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550 m
To be sure the fence was completely free of holes, he rode all the way
around once more until he arrived at D again.
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3. What is the price of a bread roll?
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SQUARES AND AREA
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Area is the amount of surface covered.
Imagine that you fold the paper so that corner A folds onto corner D and
corner B folds onto corner C.
Now imagine that you fold it again so that A folds onto C.
The length of one side of the folded shape is 3 cm.
This shape is made up of four small squares (each
with sides of 5 cm) and two large squares (with
sides of 12 cm).
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hole?
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A square hole is cut out of the shape as shown.
3. What is the area of the L shape around the hole?
R.I.C. Publications®
Problem-solving in mathematics
51
TEACHER NOTES
Problem-solving
To solve problems involving time or coordinates and to
make decisions based on particular criteria
Materials
digital clock, 0–99 number board
Focus
• Use the information and timetable to write other
questions.
• Construct a similar timetable for two pizza shops
where one opens for lunch and dinner and the other
just for dinner.
• Have students call out the coordinates they have
drawn to other students to construct the path and
compare results.
Discussion
Page 53
This worksheet requires students to read for information
and decide what information answers different criteria.
The concepts of 'earliest' and 'latest' are used in a number
of the problems rather than exact time. The ideas of the
'longest opening day' as well as the 'longest opening
times' are explored.
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These pages explore focus on reading for information,
obtaining information from a number of sources and
using it to find solutions. The problems involve thinking
and working with time and coordinates. Decisions based
on times being 'earliest' or 'latest' are needed as well as
exact time.
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Page 55
This page explores students understanding of 24-hour
digital time as they investigate the ways in which the digits
can be placed to show possible times and determine the
time closest to midday and midnight. The way in which
zero is used on a digital clock also needs to be considered.
There are 24 possibilities as there cannot be 90, 91 or 92
minutes, but '0' is used to show the hour after midnight.
For Problem 2 an understanding of two-digit numbers
needs to be coordinated with an understanding of how
and when the digits change on a digital clock. Thinking
about the two-digit numbers suggest where '2' will occur
in the ones or tens place and how long, the '2' will remain
displayed until it changes to a '3'. For example, when
'2' occurs in the hour display (e.g. 2:00, 12:00, 20:00) it
remains unchanging for the whole of the hour so it should
be counted just once.
Page 54
These problems involve following directions and using
coordinates. With Question 1, students plot the path of a
car using the given coordinates. They need to keep in mind
that the first digit in each pair is across (x axis) while the
second digit is up (y axis). Question 2 requires the students
to provide the coordinates that describe a given path. The
final question requires them to coordinate drawing a path
and recording the coordinates.
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11:00 am to 9:00 pm
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Monday
Total hours
1. What is the difference between
the amount of time both stores
are open in one week?
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(c) What time isc
closest to midnight?
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2. If you watched this clock
all
day,
how
many
times
would the digit '2' be
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displayed:
(b) What time is closest to midday?
(a) to show minutes each hour?
(b) to show minutes in 24 hours?
(c) to show hours in 24 hours?
3. Total number of times '2' is displayed in 24 hours.
R.I.C. Publications®
Problem-solving in mathematics
55
TEACHER NOTES
Problem-solving objective
To use strategic thinking to solve problems.
Materials
counters
Focus
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Discussion
Page 57
There are several ways these problems can be solved.
Emus have two legs and alpaca have four legs. Since there
are 38 heads, there must be 38 animals altogether. If all of
the animals were emus, there would only be 76 legs. The
remaining 24 legs must belong to the alpacas. Since an
alpaca has two more legs than an emu, there would be 12
alpacas and 26 emus (a total of 100 legs).
Possible difficulties
• Not using a table or diagram to manage the data
• Not considering all the possible answers—there may
be more than one possibility
• Only keeping one condition in mind when there are
two aspects to consider and reconcile
Extension
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These pages explore problems that may have several
answers and further analysis of the connections among
the data is needed to see whether this is the case or
whether there is only one solution. A process of 'try and
adjust' could be used; however, using logical reasoning
to think about possibilities and using a table, diagram
or materials to organise them will be more productive.
These ways of thinking can then be used to solve other
complex problems.
Page 59
These problems require careful reading to see how the
information needs to be used. Some children may simply
try to subtract the smaller number from the larger for the
first two problems. However, it is the expression 'more
than' that is critical in each case. Subtraction gives the
difference between the two prices and the actual amounts
need to be found to match both the total and the difference.
In Problem 3, many students may at first just share the
money among the four, whereas these amounts need to
match both the condition of equalling $1200 and keeping
a difference of $100 between each amount.
• Discuss the various methods used by students to
solve the problem. Include the ones discussed above.
Ask them to solve each problem using a different
method to that they used or first tried. Encourage
them to use a diagram rather than simply calculate.
w
ww
a table or diagram and systematically check the remaining
numbers until a solution is reached. Counters could also
be used to model the problem, again focussing on groups
of two and four.
The second problem can be solved in the same way, while
a table or counters will also assist with Problem 3. Since
she sells twice as many emus as alpacas, she must receive
$72 for each alpaca and emu. Since 15 x 72 is 1080, she
sold 15 alpacas.
Page 58
The first two problems can be solved in the same way as
on page 57. However, Question 3 does not give a second
condition and there are several possibilities. As the
animals have either two or four legs, an odd number of
chickens will not be possible.
56
Problem-solving in mathematics
R.I.C. Publications®
ON THE FARM
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1. A farmer had a number of emus and alpacas in one paddock. When she
counted, there were 38 heads and 100 legs. How many emus and how
many alpacas were in the paddock?
She paid $60 for each piglet and $95 for each calf. She paid for the 10
animals she bought with a cheque for $740. How many calves and how
many piglets did she buy?
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3. Her neighbour needed to buy more stock, so the farmer sold him some
alpacas for $48 and twice as many emus for $12 each. She received a
total of $1080. How many alpacas and how many emus did she sell?
R.I.C. Publications®
Problem-solving in mathematics
57
IN THE BARN
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1. The farm workers keep their farm
bikes in the barn. Some of the
workers have bikes with 2 wheels
and some have bikes with 3 wheels,
but all of the bikes have 2 handles.
Peter, one of the farmhands, counts
the handles on the bikes and gets
a total of 50. He also counts a
total number of 64 wheels on the
bikes. How many of the bikes have
3 wheels and how many have 2
wheels?
2. There are many spiders and
beetles in the barn. One of the
workers collects some of each.
She notices that there are 200 legs
and 29 bodies. How many spiders
and how many beetles are in her
collection?
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chcold,
e
3. When the weather turns
the
r
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farmer puts her young calves and
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chickens in the barn to keep warm.
As she puts them in the warmth,
she notices that there are a total
of 28 legs. How many calves and
how many chickens are there?
58
Problem-solving in mathematics
R.I.C. Publications®
MARKET DAY
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1. At the market, the farmer bought a pig and a piglet for $300. If the pig
cost $250 more than the piglet, what did she pay for the pig?
they took the vegetables they grew to the market and sold them for
$1200. Since they did not all work as hard as each other in the vegetable
garden, they decided to divide the money so that each brother got $100
more than his next younger brother. How much did the youngest brother
get?
R.I.C. Publications®
Problem-solving in mathematics
59
TEACHER NOTES
Problem-solving objective
To use logical reasoning and spatial visualisation to solve
problems.
With the third left, he spent 1 third of the amount on
wrapping and a card. This is represented by the shading
in the final block.
Money spent on
wrapping and card
Materials
counters, base 10 materials
Focus
$18
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Discussion
Page 61
These problems can be used as a consolidation of the
'analyse-explore-try' model of problem-solving that has
evolved over the varied number, spatial and measurement
situations posed in this book. This is discussed in detail in
the introduction (pages xiv – xvi). In these problems the
information needs to be carefully analysed to determine
how much money was available at the beginning rather
than at the end of a situation. In this way, some students
may think of what they did for the questions on page 19 as
a guide to these. Students could use counters or base 10
materials to represent the questions as they are worked
through either forwards or backwards. Using 'try and
adjust' is another possible way towards a solution.
The $18 he could spend on himself is shown by the two
unshaded parts of the diagram. This means that each of
the two parts must be $9 and he took 9 x $9, or $81 in
total, to shop with.
Possible difficulties
• Not confident working with fractions
• Thinking that 1 third and 1 half gives 1 fifth
• Thinking that 2 thirds of Peter's initial money, together
with 1 third of what is left, must account for all of his
money
• Not being able to consider using materials or a
diagram and attempting a solution on the basis of
calculations
ew
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Pr
These pages explore problems based on a conceptual
understanding of fractions. Writing the fractions using
numbers and words is designed to help students focus
on the number of parts as well as their comparative sizes
and to lead them to consider other ways of solving the
problems other than by fraction calculations. One way of
solving them is by backtracking from the answers. Counters
can be useful as they allow the parts to be considered
while the whole problem is also kept in mind. Using a
diagram is another method and is probably a different way
of thinking about fractions than is used by many students
and teachers.
• Have students make up problems of their own and
challenge others to solve them using diagrams.
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super
Another way of thinking that can be used is to use a
diagram as on page xiii.
Placing the amounts involved in Question 2 in a diagram
(as shown below) can help. Peter originally spent twothirds of his total amount on the present.
This is represented by the two shaded blocks.
ent
y sp t
e
n
Mo presen
60
on
Problem-solving in mathematics
R.I.C. Publications®
SHOPPING
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1. On Saturday, Peta went to the
shopping centre to buy a new
outfit to wear to her friend's
birthday party. She spent half
of her money on a dress and
then 1 third of what she had
left on a pair of sandals. She
was left with $60.00 in her
purse. How much money did
she have to start with?
2. On Sunday, her brother, Peter,
went to the shopping centre
to buy a birthday present for
the party. He spent 2 thirds
of his money on the present
and then spent 1 third of what
he had left for a card and
wrapping paper. That left him
with $18 to buy something for
himself. How much money
did he originally have when he
went shopping?
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s
super
R.I.C. Publications®
Problem-solving in mathematics
61
SOLUTIONS
Note: Many solutions are written statements rather than just numbers. This is to encourage teachers and students to solve
problems in this way.
95 and 96
100 cards
eight triangles
Sunday is the first day of the month. Tuesday is the last
day of the month.
5. If he used a 30cm ruler—62 m
If he used a 1m ruler—60 m 60 cm or 60.6 m
R.I.C. Publications®
1.
2.
3.
4.
E
(a) D
(b) A
(c) D
(d) B
(e) C
(a) 7
(b) 13
(c) 18
(d) 19
(e) 22
Answers will vary—Each ones place value amount is
repeated in the same column. With the rows, numbers
with the same tens place value appear in rows of threes,
overlapping with the previous tens place value.
5. Answer will vary—Each ones place value amount is
repeated in the same column.
6. Yes, numbers with 4 in the ones place are still in Column
D; numbers with 1 in the ones place will be in Column A.
Derek spent $80 on towels.
Payment per paper is $35.86. Hourly payment is $25.25.
Alison sold 60 apples.
Brackets without screws are 20¢ cheaper.
Payment per tree is $355.20, payment per day is $317.50.
604 people are on the train.
(a) 564 people are seated.
(b) 780 people can travel on the train.
71 more people are needed to fill the train.
256 more people can board the train.
35 people are standing.
There were 328 people on the train to begin with.
69 pages in each chapter.
No, it is in Chapter 6.
Walter has been reading his books for 22 days.
Walter has read 92 pages past the middle of the book.
Pages 139–206
Chapter 5
He needs to read 253 pages to finish the book.
His sister is up to page 422.
She needs to read 268 pages to finish the book.
It will take her 27 days.
R.I.C. Publications®
SOLUTIONS
Note: Many solutions are written statements rather than just numbers. This is to encourage teachers and students to solve
problems in this way.
IN THE BARN......................................................Page 58
1. three wheels = 14, two wheels = 11
2. 13 spiders and 16 beetles
3. Answers will vary as there are multiple answers.
Answers are displayed in the table:
Primary Problem-solving in Mathematics: Book D - Ages 8-9
Problem solving in mathematics is a series that aims to develop problem-solving and mathematical thinking in primary students. Buy now: | 677.169 | 1 |
Dictionary of Mathematics (McGraw-Hill Dictionary of)
Derived from the content of the respected McGraw-Hill Dictionary of Scientific and Technical Terms Sixth Edition, each title provides thousands of definitions of words and phrases encountered in a specific discipline. All include:
* Pronunciation guide for every term * Acronyms, cross-references, and abbreviations * Appendices with conversion tables; listings of scientific, technical, and mathematical notation; tables of relevant data; and more * A convenient, quick-find format
"synopsis" may belong to another edition of this title.
Review:
Excerpts of review by T.R. Faust, Burlington College
McGraw-Hill derives these inexpensive subject-specific dictionaries from its Dictionary of Scientific and Technical Terms, covering 110,000 terms. Libraries could not go wrong purchasing this recognized standard reference--either the parent or its offspring. Choosing which to purchase will probably present the greater challenge. Students may be more likely to favor these more focused titles, whereas librarians may be more enthralled with the larger, more encompassing mother work. ...The offspring reproduce the definitions of terms exactly as they appear in the mother work, with pronunciation but without illustrations. The appropriate appendixes are retained in the smaller volumes, but biographical entries are dropped.
McGraw-Hill tends to include more appendixes [than competition], such as geological time scales and electronic symbols...libraries will be well served by the McGraw-Hill titles. Summing Up: Highly recommended. (Choice 2003-10-01)
From the Back Cover:
THE LANGUAGE OF MATHEMATICS AT YOUR FINGERTIPS
Derived from the world-renowned McGraw-Hill Dictionary of Scientific and Technical Terms, Sixth Edition, this vital reference offers a wealth of essential information in a portable, convenient, quick-find format. Whether you're
Written in clear, simple language understandable to the general reader, yet in-depth enough for scientists, educators, and advanced students, The McGraw-Hill Dictionary of Mathematics, Second Edition: * Has been extensively revised, with 4,000 entries encompassing the language of mathematics and statistics * Includes synonyms, acronyms, and abbreviations * Provides pronunciations for all terms * Covers such topics as algebra, analysis, arithmetic, logic and set theory, number theory, probability and statistics, topology, and trigonometry * Includes an appendix containing tables of useful data and information * Is based on the McGraw-Hill Dictionary of Scientific and Technical Terms – for more than a quarter-of-a-century THE standard international reference
Carefully reviewed for clarity, completeness, and accuracy, the McGraw-Hill Dictionary of Mathematics, Second Edition, offers a standard of excellence unmatched by any similar publication.
Book Description McGraw-Hill Education - Europe, United States, 2008. Paperback. Book Condition: New. 2nd Revised edition. 211 x 137 mm. Language: English . Brand New Book ***** Print on Demand *****.This10496
Book Description McGraw-Hill Education - Europe, United States, 2008. Paperback. Book Condition: New. 2nd Revised edition. 211 x 137 mm. Language: English . Brand New Book ***** Print on Demand *****. This10496
Book Description McGraw-Hill Professional. Paperback. Book Condition: New. Paperback. 307 pages. Dimensions: 8.3in. x 5.4in. x 0.7in.Derived from the content of the respected McGraw-Hill Dictionary of Scientific and Technical Terms Sixth Edition, each title provides thousands of definitions of words and phrases encountered in a specific discipline. All include: Pronunciation guide for every term Acronyms, cross-references, and abbreviations Appendices with conversion tables; listings of scientific, technical, and mathematical notation; tables of relevant data; and more A convenient, quick-find format This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Paperback. Bookseller Inventory # 9780071410496 | 677.169 | 1 |
Ncert solutions for class 6 mathematics
Scholar learning is answer of maths search chapter is provided in the list on our visit. This page will help in finding those NCERT solutions for class 6 mathematics of book in class 6. These are providing to solution cover important concepts and good study for all students on our scholarslearning. It is provide the syllabus on our site. That is a more information about syllabus to check on our visit. | 677.169 | 1 |
This book gives a comprehensive introduction to complex analysis in several variables. It clearly focusses on special topics in complex analysis rather than trying to encompass as much material as possible. Many cross-references to other parts of mathematics, such as functional analysis or algebras, are pointed out in order to broaden the view and the understanding of the chosen topics. A major focus is extension phenomena alien to the one-dimensional theory, which are expressed in the famous Hartog's Kugelsatz, the theorem of Cartan-Thullen, and Bochner's theorem. The book primarily aims at students starting to work in the field of complex analysis in several variables and teachers who want to prepare a course. To that end, a lot of examples and supporting exercises are inserted throughout the text, which will help students to become acquainted with the subject | 677.169 | 1 |
Gráficos en acción (Graphs in Action) (Spanish Version)
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Learn all about graphing in this informative Spanish-translated title! This book helps readers understand how to graph a clear visual representation of data, encouraging mathematical and STEM skills. See strong examples of bar graphs, circle graphs, and line graphs, learning the best way to present data using these tools. Full of vibrant images, easy-to-read informational text, clear mathematical diagrams, and text features including a glossary and index, this title will help Spanish readers improve their graphing skills, creating their best graphs yet | 677.169 | 1 |
Found 1 course by Karim Premji
The skills and tips you need to quickly master how to solve the 4 major types of linear equations in 5 easy steps.
Normal price $20NOW $10
About Karim Premji
A graduate of the University of Toronto's Engineering Science Program, Karim has worked with hundreds of students of all levels to help them discover that Math is a subject that can be mastered so long as they have the right teacher to steer them along the journey. While studying Aerospace Engineering, Karim acquired a deep understanding of mathematics and science and has practiced engineering for 18 years. Karim operates a Math Learning Centre in the Northern suburbs of Toronto where he has mentored and guided hundreds of students to success in math with his patient and endearing teaching approach. He has a knack for explaining difficult concepts in simple terms and his clear and engaging lecture videos will give you the tips, tricks and strategies to solve difficult math problems. Karim wasn't always a Math Wiz and success with Math did not come easily. He struggled with fractions and long division in middle school until a caring and patient math teacher helped him to discover that math was a subject not to be feared and could be mastered with patience and discipline. Now students all over the globe can share Karim's math wisdom and passion and discover their own Math Wiz within. Join Karim on Udemy and propel your Math knowledge to the next level | 677.169 | 1 |
Optimization
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0.5 MB | 14 pages
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Optimizations problems tend to be difficult for calculus students. These notes bring them back to finding the maximum of a parable using the axis of symmetry, and then using their calculus knowledge to solve a similar problem and find the same solution. This lesson optimizes revenue functions, a rectangle bounded by a function on the coordinate plane, the sum/product of two numbers, and the volume of a box. Notes and homework with answer keys are included. Some questions require the use of a graphing calculator, a calculator icon is next to each of those | 677.169 | 1 |
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Summary
Thomas'Calculus Part Two Media Upgrade, Eleventh Edition,responds to the needs of today's readers by developing their conceptual understanding while strengthening their skills in algebra and trigonometry, two areas of knowledge vital to the mastery of calculus. This book offers a full range of exercises, a precise and conceptual presentation, and a new media package designed specifically to meet the needs of today's readers. The exercises gradually increase in difficulty, helping readers learn to generalize and apply the concepts. The refined table of contents introduces the exponential, logarithmic, and trigonometric functions in Chapter 7 of the text.Infinite Sequences and Series, Vectors and the Geometry of Space, Vector-Valued Functions and Motion in Space, Partial Derivatives, Multiple Integrals, Integration in Vector Fields.For all readers interested in Calculus. | 677.169 | 1 |
Mathematics Documents
Showing 1 to 9 of 9
Tan inverse and Cotan
Introduction :In this project we are required to study and graph the indicated function which is sin x and its inverse and the reciprocal of it so we are going to graph it using the graphic calculator. The point of this project is le
Individual Assessment
The Fantastic Burgers restaurant is considering two alternatives to increase profits:
either open a drive-through or introduce a vegetarian burger selection (but not both).
The costs for both options are as follows:
Drive-through
V
Introduction:
In mathematics, there are too much
functions which can be useful for us. A lot of
functions used in other subjects not specially
math. For example Sometimes functions could
hint for science problems and you not solve
without understanding th | 677.169 | 1 |
An entertaining and captivating way to learn the fundamentals of using algorithms to solve problems
The algorithmic approach to solving problems in computer technology is an essential tool. With this unique book, algorithm guru Roland Backhouse shares his four decades of experience to teach the fundamental principles of using algorithms to solve problems. Using fun and well–known puzzles to gradually introduce different aspects of algorithms in mathematics and computing. Backhouse presents you with a readable, entertaining, and energetic book that will motivate and challenge you to open your mind to the algorithmic nature of problem solving.
Provides a novel approach to the mathematics of problem solving focusing on the algorithmic nature of problem solving
Uses popular and entertaining puzzles to teach you different aspects of using algorithms to solve mathematical and computing challenges
Features a theory section that supports each of the puzzles presented throughout the book
Assumes only an elementary understanding of mathematics
Let Roland Backhouse and his four decades of experience show you how you can solve challenging problems with algorithms curious to see manuals e academic texts that are currently used in English Universities (Nottingham, UK, in this case) as I still have fresh memories of the subject from my years at the Politecnico of Milan, and I must say that I was impressed, almost envious, by this book written by Professor Backhouse. The aesthetic comparison alone, compared to my old manuals, is impressive. Here we have a book which has focused on the very finer details: excellent layout, enjoyable font, written and indented formulas and perfectly clear pictures. The smooth paper is very nice and it has a sturdy binding but not too stiff.
Since the topic itself has complexity in its DNA, it is presented well, makes it interesting (almost intriguing) for students in the first year of Computer Science ( the course on which the book is based).
The approach is a bit outside the box, since it presents examples first - almost all the typical courses in Italian universities do this after the mathematical theory/algebraic.
The book is divided into 2 parts practically of the same length and number of pages. In the first part the reader is exposed to: examples from games at the Tower of Hanoi, all with resolution explained in detail, both in words and through equations and algorithms. They are also offered exercises to be solved and ideas for projects. The first, fortunately, have solutions and answers in Appendix.
The second is the mathematical theory which acts as a glue to algorithms - from algebra to number theory of quantifiers.
Everything is explained in English but it is easy to understand by non-native English speakers (just a decent knowledge of language is necessary).Read more ›
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a lot of exercises so you can practise what you've learned. i bought it a few months ago and only read the first 100 pages or so. did help me with my uni work but should've bought it earlier to have enough time to read all the content. but overall its a good book which covers interesting APS concepts
this book is hard to understand, the layout is bad and it jumps a lot! the logic could be better explained, rather than using boolean, and at times even the simplest things are said in the hardest way!!!
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Description
This updated edition presents ten strategies that are effective tools for teaching students how to solve problems, both in mathematics and in real-life situations. The authors demonstrate how the strategies can be used to solve a wide range of problems and provide about 200 examples that illustrate how teachers can include these techniques in their mathematics curriculum. In many cases, the methods presented make the solution of a problem easier, neater, and more understandableA'and thereby more enjoyable. This new edition includes references to current standards, revisions and clarifications throughout the text, and a number of new problems that can be used to teach the different strategies.
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About Author
Alfred S. Posamentier is professor of mathematics education and dean of the School of Education at the City College of the City University of New York. He has authored and co-authored several resource books in mathematics education for Corwin Press. Stephen Krulik is professor of mathematics education at Temple University in Philadelphia, where he is responsible for the undergraduate and graduate preparation of mathematics teachers for Grades K-12, as well as in the inservice training of mathematics teachers at the graduate level. He teaches a wide variety of courses, among them the History of Mathematics, Methods of Teaching Mathematics, and the Teaching of Problem Solving. Before coming to Temple University, he taught mathematics in the New York City public schools for 15 years, where he created and implemented several courses designed to prepare students for the SAT examination. Nationally, Krulik has served as a member of the committee responsible for preparing the Professional Standards for Teaching Mathematics of the National Council of Teacher of Mathematics (NCTM). He was also the editor of the NCTM's 1980 yearbook Problem Solving in School Mathematics. He is the author or co-author of more than 20 books for teachers of mathematics, including Assessing Reasoning and Problem Solving: A Sourcebook for Elementary School Teachers. He has served as a consultant to and has conducted many workshops for school district throughout the United States and Canada, as well as delivering major presentations in Austria, Hungary, Australia, and international professional meetings, where his major focus is on preparing all students to reason and problem-solve in their mathematics classroom, as well as in their lives. Krulik received his BA degree in mathematics from Brooklyn College of the City University of New York, and his MA and EdD in mathematics education from Columbia University's Teachers College.
Contents
Preface About the Authors 1. Introduction to Problem-Solving Strategies 2. Working Backwards The Working Backwards Strategy in Everyday Life Problem-Solving Situations Applying the Working Backwards Strategy to Solve Mathematics Problems Problems Using the Working Backwards Strategy 3. Finding a Pattern The Finding a Pattern Strategy in Everyday Life Problem-Solving Situations Applying the Finding a Pattern Strategy to Solve Mathematics Problems Problems Using the Finding a Pattern Strategy 4. Adopting a Different Point of View The Adopting a Different Point of View Strategy in Everyday Life Problem-Solving Situations Applying the Adopting a Different Point of View Strategy to Solve Mathematics Problems Problems Using the Adopting a Different Point of View Strategy 5. Solving a Simpler Analogous Problem The Solving a Simpler Analogous Problem Strategy in Everyday Life Problem-Solving Situations Applying the Solving a Simpler Analogous Problem Strategy to Solve Mathematics Problems Problems Using the Solving a Simpler Analogous Problem Strategy 6. Considering Extreme Cases The Considering Extreme Cases Strategy in Everyday Life Problem-Solving Situations Applying the Considering Extreme Cases Strategy to Solve Mathematics Problems Problems Using the Considering Extreme Cases Strategy 7. Making a Drawing (Visual Representation) The Making a Drawing (Visual Representation) Strategy in Everyday Life Problem-Solving Situations Applying the Making a Drawing (Visual Representation) Strategy to Solve Mathematics Problems Problems Using the Making a Drawing (Visual Representation) Strategy 8. Intelligent Guessing and Testing (Including Approximation) The Intelligent Guessing and Testing (Including Approximation) Strategy in Everyday Life Problem-Solving Situations Applying the Intelligent Guessing and Testing (Including Approximation) Strategy to Solve Mathematics Problems Problems Using the Intelligent Guessing and Testing (Including Approximation) Strategy 9. Accounting for All Possibilities The Accounting for All Possibilities Strategy in Everyday Life Problem-Solving Situations Applying the Accounting for All Possibilities Strategy to Solve Mathematics Problems Problems Using the Accounting for All Possibilities Strategy 10. Organizing Data The Organizing Data Strategy in Everyday Life Problem-Solving Situations Applying the Organizing Data Strategy to Solve Mathematics Problems Problems Using the Organizing Data Strategy 11. Logical Reasoning The Logical Reasoning Strategy in Everyday Life Problem-Solving Situations Applying the Logical Reasoning Strategy to Solve Mathematics Problems Problems Using the Logical Reasoning Strategy Afterword by Herbert A. Hauptman Sources for Problems Readings on Problem Solving Index | 677.169 | 1 |
Visualizing Quaternions
Other | February 1, 2006 for efficient interpolation of rotations. Despite this, many practitioners have avoided quaternions because of the mathematics used to understand them, hoping that some day a more intuitive description will be available. The wait is over. Andrew Hanson's new book is a fresh perspective on quaternions. The first part of the book focuses on visualizing quaternions to provide the intuition necessary to use them, and includes many illustrative examples to motivate why they are important-a beautiful introduction to those wanting to explore quaternions unencumbered by their mathematical aspects. The second part covers the all-important advanced applications, including quaternion curves, surfaces, and volumes. Finally, for those wanting the full story of the mathematics behind quaternions, there is a gentle introduction to their four-dimensional nature and to Clifford Algebras, the all-encompassing framework for vectors and quaternions.
* Richly illustrated introduction for the developer, scientist, engineer, or student in computer graphics, visualization, or entertainment computing. * Covers both non-mathematical and mathematical approaches to quaternions. * Companion website with an assortment of quaternion utilities and sample code, data sets for the book's illustrations, and Mathematica notebooks with essential algebraic utilities.
Pricing and Purchase Info fo...
Andrew J. Hanson is a professor of computer science at Indiana University in Bloomington, Indiana, and has taught courses in computer graphics, computer vision, programming languages, and scientific visualization. He received a BA in chemistry and physics from Harvard College and a PhD in theoretical physics from MIT. Before coming to ... | 677.169 | 1 |
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Students will understand how to identify arithmetic and geometric sequences through the use of error analysis.If you like this product, consider buying the Algebra Error Analysis bundle. This product and other error analysis problems are included in the bundle. It is discounted at a great price. Click on this link for the Algebra Error Analysis Works | 677.169 | 1 |
Getting Started with the TI-Navigator™ System: What's My Rule?
Learners explore linear relationships. In this sixth through eighth grade mathematics lesson, students explore the symbolic representation of verbal descriptions of mathematical rules defining the relationship between two variables. | 677.169 | 1 |
Warming Up to SimCalc MathWorlds
Students investigate features of MathWorlds. In this secondary mathematics instructional activity, students explore coordinate graphs and graphs associated with linear motion such as Position vs. Time graphs | 677.169 | 1 |
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