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Math and Logic Online Courses for Harvard Students
This course covers mathematical topics in trigonometry. Trigonometry is the study of triangle angles and lengths, but trigonometric functions have far reaching applications beyond simple studies of... more
We live in a world of numbers. You see them every day: on clocks, in the stock market, in sports, and all over the news. Algebra is all about figuring out the numbers you don't see. You might know... more
This is a master course given in Moscow at the Laboratory of Algebraic Geometry of the National Research University Higher School of Economics by Valery Gritsenko, a professor of University LilleWe invite you to a fascinating journey into Graph Theory - an area which connects the elegance of painting and the rigor of mathematics; is simple, but not unsophisticated. Graph Theory gives us,... more
The lectures of this course are based on the first 11 chapters of Prof. Raymond Yeung\'s textbook entitled Information Theory and Network Coding (Springer 2008). This book and its predecessor, A... more
Welcome to Practical Time Series Analysis! Many of us are \"accidental\" data analysts. We trained in the sciences, business, or engineering and then found ourselves confronted with data for which... more
Welcome to Cryptographic Hash and Integrity Protection! This course reviews cryptographic hash functions in general and their use in the forms of hash chain and hash tree (Merkle tree). Building on... more
Data science courses contain math-no avoiding that! This course is designed to teach learners the basic math you will need in order to be successful in almost any data science math course and was... more
Counting is one of the basic mathematically related tasks we encounter on a day to day basis. The main question here is the following. If we need to count something, can we do anything better than... more
Discrete Math is needed to see mathematical structures in the object you work with, and understand their properties. This ability is important for software engineers, data scientists, security and... more
More than 2000 years ago, long before rockets were launched into orbit or explorers sailed around the globe, a Greek mathematician measured the size of the Earth using nothing more than a few facts | 677.169 | 1 |
3D Math Primer for Graphics and Game Development (Wordware Game Math Library) (Paperback) Год выпуска: 2002 Издательство: Wordware Publishing ISBN: 1556229119 476 Описание: 3:
* Explains basic concepts such as vectors, coordinate spaces, matrices, transformations, Euler angles, homogenous coordinates, geometric primitives, intersection tests, and triangle meshes.
* Discusses orientation in 3D, including thorough coverage of quaternions and a comparison of the advantages and disadvantages of different representation techniques.
* Describes working C++ classes for mathematical and geometric entities and several different matrix classes, each tailored to specific geometric tasks.
* Includes complete derivations for all the primitive transformation matrices. Об авторе: Fletcher Dunn is the principal programmer at Terminal Reality, where he has worked on Nocturne and 4x4 Evolution and is currently lead programmer for BloodRayne. He has developed games for Windows, Mac, Dreamcast, Playstation II, Xbox, and GameCube. Ian Parberry is a professor of computer science at the University of North Texas and is internationally recognized as one of the top academics teaching computer game programming with DirectX. He is also the author of Learn Computer Game Programming with DirectX 7.0 and Introduction to Computer Game Programming with DirectX 8.0. Язык: Английский | 677.169 | 1 |
Improvers Maths E2
Ref: C3742374
This course will help students to become more confident with some of the foundation topics of
Maths, such as working with decimals, fractions, algebra and the order of operations (BIDMAS) as well as starting to look at aspects of Data Handling such as using Tally charts.
Students will learn in a relaxed and friendly and supportive environment and gain some of the skills needed to move onto a GCSE course.
Course aim
To provide a 'refresher' course to increase students' fluency & confidence in the basic foundations
of Maths. To begin to prepare students in the skills needed before starting a GCSE Foundation
courseSimplify algebraic expressions
Find equivalent fractions for given fractions
Convert between fractions & decimals
Design & complete tally charts for sets of data
Work out the answers to calculations using mathematical operations in the correct order (BIDMAS)- Students will need pen, paper, ruler, rubber and a basic scientific calculator.
- A geometry set would also be useful.
- Students may be required to purchase a course book as a volunteer for a WEA partner or another organisation | 677.169 | 1 |
These resources have been developed as part of the Mathematics Enhancement Program and were funded principally by the Garfield Weston Charitable Foundation NOTE that these tutorial programs are still under development. Comments and Corrections will be welcomed.
The year is divided into 2 parts - 8A and 8B.
For each part there is a Pupils' Practice Book.
Book 8A covers Units 1 to 11. Book 8B covers Units 12 to 20.
Each Unit will have its own set of interactive tutorials - one for each section within that unit.
Copyright
The copyright in all of this material belongs to the originators who created it.
The material is made available through the CIMT for downloading and dissemination for NON-PROFIT MAKING PURPOSES ONLY. | 677.169 | 1 |
Thanks again!
Terms are separated by addittion or subtraction so $2u^2 - 1$ is 2 terms. Something like $u^3v^5wxy^7$ is only 1 term , whereas $x-y+z$ is 3 termsDo you think I should start with a high school algebra text or a precalculus text? I have gone through high school and college, but it has been some time. Apparently why I am so rusty.... I thought I had a good grip on algebra...
I mention precalc. because it is my understanding that most of those texts cover some basic algebra before advancing.
One week, no... two months, eh... maybe! I really am looking to ferret out the key concepts that I may have forgotten or mislearned.
The problem with "pre-calc" is that there does not seem to be a standard definition for it. In some schools, it is college algebra. In others it is an introduction to analysis and applications of calculus.
Because you previously studied algebra, I'd start with a high school algebra text. It may be that with a bit of practice, it all comes back to you. If, however, it is a struggle, you are not ready for calculus. If you breeze through a high school algebra text getting almost 100% of the problems with answers right, get a college algebra text. In college, they generally cover all the algebra (except maybe for trig functions) you will need for calculus in a single year. See if you get that easily. If not, take a college algebra course before you tackle calculus. | 677.169 | 1 |
In class (midterm)
examinations:Monday,
October 11; Monday, November 15
Final examination: Thursday,
December 16, 8-11 AM. The location is TBA (but I would not bet against
it being the regular classroom).
The course is about the general theory of algebraic operations. You have
studied numbers, polynomials, functions, vectors and matrices and you have
probably observed that computations with them have some broad similarities.
This course goes into the theory common to these examples, a theory that has
evolved at an ever-increasing rate over the last 100 to 200 years. One of our
goals is to show how some much older problems (including these nice samples)
can be solved using these more modern ideas. These simple ancient questions are
pertinent to current problems such as the security of financial transactions on
the internet (Hungerford, Chap. 12).
Most of the course will be devoted to the study of two types of algebraic
objects:rings (which generalize
the integers, i.e., they have operations of addition and multiplication) and
groups (which generalize the set of all permutations of a set, i.e., there is a
single operation generalizing composition).We will begin (Chapters 1 and 2) by
recalling some properties of the integers.Then (Chapter 3 and part of Chapter 6) we will define rings and give
some basic definitions and results about their structure.Next (Chapters 4 and 5) we will consider
other important examples of rings, notably polynomial rings and then study some
further structural properties of rings (in the remainder of Chapter
6). We will then begin the study of groups (Chapter 7). At the end
of the course we will study some additional topics in group theory
(Chapter 8) and ring theory (Chapter 9).
Here it the tentative schedule for the course
September 1 – Lecture on Sections 1.1, 1.2; Workshop #1
September 8 – Lecture on Sections 1.3, 2.1
September 13 – Lecture on Section 2.2, 2.3
September 15 – Lecture on Sections 3.1, 3.2.Workshop #2
September 20 – Lecture on Section 3.3
September 22 – Lecture on Section 6.1; Workshop #3
September 27 – Lecture on Section 6.2
September 29 – Lecture on Section 4.1; Workshop #4
October 4 – Lecture on Sections 4.2, 4.3
October 6 – Lecture on Section 4.4; Workshop #5
October 11 – Exam #1
October 13 – Sections 5.1, 5.2; Workshop #6
October 18 – Lecture on Section 5.3
October 20 – Lecture on Section 6.3; Workshop #7
October 25 – Lecture on Sections 7.1, 7.2
October 27 – Lecture on Section 7.9; Workshop #8
November 1 – Lecture on Sections 7.3, 7.4
November 3 – Lecture on Section 7.5; Workshop #9
November 8 – Lecture on Sections 7.6, 7.7
November 10 – Lecture on Section 7.8; Workshop #10
November 15 – Exam #2
November 17 – Lecture on Section 8.1; Workshop #11
November 22 – Lecture on Section 8.2; Workshop #12
November 29 – Lecture on Section 8.3
December 1 – Lecture on Section 8.4; Workshop #13
December 6 – Lecture on Section 9.1
December 8 – Lecture on Section 9.2; Workshop #14
December 13 – Lecture on Section 9.2
The material to be covered on exams will be announced in class, no later
than two weeks in advance. Review materials will be posted in advance, and
should be used together with workshop and homework assignments for preparation.
Course level:
This is a high-level course. You will be expected to understand the proofs
which are given in the text, and (especially) in lectures, and to construct
your own proofs. You should expect to become more experienced in this as the
term progresses. The course is one of two that satisfies the algebra
requirement for the mathematics major. The alternative is Mathematics 350 (advanced
linear algebra).
As a general rule, undergraduates should expect to spend approximately two
hours outside of class for every hour spent in class. As Mathematics 351 is a
4-credit course, and is one of our more challenging courses,
Students in Mathematics 351 should be prepared to spend 8
to 10 hours per week on the course, in addition to the class meetings.
Writing proofs may be particularly time-consuming at first. If you get stuck
or just need feedback, paying a visit to office hours will probably be
worthwhile. However: for this to be really useful, you should plan to discuss a
couple of specific problems that you have thought about carefully, and have
written as much as you can about. The more specifically you can describe your
line of thinking, and where you are stuck, the more productive the office hours
will be.
Calculator A numerical calculator will certainly be useful (as
will become apparent during the first lecture).However, graphing is irrelevant. When
working with numerical data you will usually need to keep everything in exact
terms (21/2 rather than 1.414 for example), so the usefulness of the
calculator is limited to rather simple calculations. We do need to do some substantial
arithmetic occasionally, of the sort you would not want to do by hand. On
exams, a TI-83 will be permitted, but no calculator with alphabetic keyboard,
large memory or built-in algebra system will be allowed.
Grading:
Two in-class midterm examinations (20% each):
40%
Final examination:
40%
Workshops:
10%
Homework and Quizzes:
10%:
Homework and quizzes:
Regular homework will be assigned in class from the book. Quizzes will be given
based on the homework. They may be announced or unannounced. Questions
concerning the homework should be raised in class.
Workshops:
In the Workshop sessions you will work in groups on more difficult problems, to
be handed out in the workshop, under your professor's supervision. Some of
these problems, but not all, come from the book. Selected workshop problems
will be assigned to be written up and handed in.
Collaboration vs. Plagiarism1:
The workshop involves a mixture of two very different kinds of work: collaborative
and independent. It is important to understand what this means in
practice.
During the workshop
session you are expected to collaborate fully with your group. The
goal here is not only to solve the problems, but to develop a range of
skills relating to technical communication and group work, and to reach a common
understanding of the issues with your classmates.
Outside the classroom
you may also discuss these problems with your classmates, but you should mention
explicitly any such collaboration in your write-up. Unacknowledged
collaboration is considered plagiarism, which is a serious
violation of the principles of academic honesty, and which the professor
takes seriously. If two students' write-ups are identical, that is very
strong evidence of plagiarism. The line between legitimate collaboration
and plagiarism is subtle, but you can steer clear of it by making it a
habit to acknowledge both your collaboration with others and your use of
outside sources such as books and websites. Then you can make maximum use
of collaboration.
The final write-up is your
own.
This means that it is entirely written in your own words, even if you like
someone else's words better, and even if a classmate has come up with the
key idea!
The only person whom you are permitted to consult about the write-up is
the professor. Consult the professor if you have trouble with the
presentation, or whenever the rules concerning write-ups are unclear in
practice. | 677.169 | 1 |
Mr. Tom Carol, NY
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Please contact your nearest Dymocks store to confirm availability
Email store
This book is available in following stores
This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions.
The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression.
In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest.
The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which "e;Fourier Analysis"e; is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability | 677.169 | 1 |
GRE Math Workbook
By Kaplan Test Prep
Book Details Summary
Kaplan's GRE Math Workbook provides hundreds of realistic practice questions and exercises to help you prepare for the Math portion of the GRE. With expert strategies, content review, and realistic practice sets, GRE Math Workbook will help you face the test with confidence.
The Best Review Six full-length Quantitative Reasoning practice setsDiagnostic tool for even more targeted Quantitative practiceReview of crucial math skills and concepts, including arithmetic, algebra, data interpretation, geometry, and probabilityKey strategies for all Quantitative Reasoning question types on the revised GREAn advanced content review section to help you score higherExpert GuidanceWe know the test: The Kaplan team has spent years studying every GRE-related document available.Kaplan's expert psychometricians ensure our practice questions and study materials are true to the test.We invented test prep—Kaplan ( has been helping students for almost 80 years. Our proven strategies have helped legions of students achieve their dreams. | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
Study and exam practice for NCEA Level 1 Achievement Standard 90148 - Sketch and interpret graphs (1.2) Affordable revision booklets for NCEA external exams, tailored to speci?c Achievement Standards and written by experienced authors. Each contains a CD with interactive exercises and answers to help you master the content you needfor Achievement. The booklets are packed with advice on how to study, and how to display your knowledge for Merit and Excellence credits.
Product Information
Table of contents
Preface; 1 Graphing Achievement: Review of linear patterns; Linear graphs using tables of values ; Linear graphs using patterns ; Using tables of values or patterns; Features of linear graphs; Quadratic graphs using tables of values; Quadratic graphs in factored form using patterns; Quadratic graphs using patterns; Using tables of values or patterns; Features of quadratic graphs; 2 Graphing Merit: Review of linear patterns to find a rule; Determining the equation of a linear graph; Other forms of straight-line graphs; Patterns with quadratic graphs; 3 Graphing Excellence: Review of quadratic patterns to find a rule; Finding quadratic equations from a graph; Mixed problems | 677.169 | 1 |
USEFUL LINKS
PREREQUISITES
The prerequisite for Math 30 is completion of Math
20 with a grade of "C" or better,
or a qualifying score on the Math Competency Exam (MCE).
COURSE DESCRIPTION
Designed to prepare students for Intermediate Algebra.
Elementary Algebra
teaches simplifying algebraic expressions involving polynomials and rational
terms; factoring; solving linear equations; solving quadratic equations using
factoring; analyzing graphs of linear equations; and solving applied problems.
This course will also include an introduction to algebraic operations with rational
expressions.
STUDENT LEARNING OUTCOMES
Upon successful completion of Math 30:
Students will be able to make use of factoring techniques, solve quadratic equations by factoring, and make use of factoring to simplify rational expressions.
Students will be able to translate English phrases into algebraic expressions and equations, and solve applied problems.
Students will be able to perform operations related to linear equations in two variables, solve systems of linear equations, and solve applications which can be modeled using systems of linear equations.
COURSE OBJECTIVES
At the end of this course you should be able to…
(1) evaluate and simplify algebraic expressions using the rules of exponents,
order of operations, combining like terms, and the distributive property
(2) add, subtract, multiply and divide using either monomials or polynomials;
(3) solve a linear equation or inequality and check the solution;
(4) analyze verbal problems, model with appropriate equations, substitute the
known values, solve the resulting equations, and interpret the result in the
context of the problem;
(5) factor polynomials;
(6) simplify, multiply and divide rational expressions;
(7) graph first degree equations in two variables;
(8) write an equation for a
given line, identify the slope of a line; and
(9) solve quadratic equations by factoring. | 677.169 | 1 |
Mathematical Problem Solving Alan H Schoenfeld
This book is addressed to people with research interests in the nature of mathematical thinking at any level, topeople with an interest in "higher-order thinking skills" in any domain, and to all mathematics teachers. The focal point of the book is a framework for the analysis of complex problem-solving behavior. That framework is presented in Part One, which consists of Chapters 1 through 5. It describes four qualitatively different aspects of complex intellectual activity: cognitive resources, the body of facts and procedures at one's disposal; heuristics, "rules of thumb" for making progress in difficult situations; control, having to do with the efficiency with which individuals utilize the knowledge at their disposal; and belief systems, one's perspectives regarding the nature of a discipline and how one goes about working in it.
Part Two of the book, consisting of Chapters 6 through 10, presents a series of empirical studies that flesh out the...
Download: | 677.169 | 1 |
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CBSE Class 10 Mathematics Syllabus 2017-18
Download latest cbse class 10 Mathematics syllabus for the session 2017-18. This year CBSE has changed the exam pattern. Maths Board exam will consist of 80 marks and 20 marks will be internal of assessment. In internal assessment Periodical Test will be of 10 Marks, Note Book Submission of 05 Marks and Lab Practical (Lab activities to be done from the prescribed books) of 05 Marks. This year, there is no CCE pattern or division of syllabus for SA1 or SA2 as it used to be before. This year complete syllabus will be there in your final/annual exam. Each year cbse keep on issuing revised syllabus. Although the topics and textbooks remain same, the contents within vary year by year. Along with Mathematics syllabus for class 10 cbse has also issued the chapter-wise breakup of questions along with weightage of each chapter. The package also enlists the learning objectives. That's why it is strongly recommended to download new class 10 maths syllabus 2017-18 this year and be on safe side.
This year is going to be tough for you because you've to cover complete syllabus for your board exam. Earlier it used to be only half the syllabus ie syllabus of SA2. So it is very important for you to prepare for your board exams smartly according to the mathematics syllabus by cbse because it will reduce the topics, will tell the weightage of each chapter so that you can invest more time on important topics.
Here we have presented the CBSE syllabus for Mathematics class 10 in the same format as released by CBSE, without any changes or editing from our side. Students are advised to download the class 10 cbse mathematics syllabus for the session 2017-2018 even if they have the previous year syllabus because there is a lot of changes in the syllabus, question pattern and weightage of each chapter.
Click on the button below to download the syllabus in pdf format. Scroll down to view the detailed syllabus.
UNIT II: ALGEBRA
Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.
2. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES (15 Periods)
Pair of linear equations in two variables and their graphical solution. Geometric representation of different possibilities of solutions/inconsistency.
Algebraic conditions for number of solutions. Solution of a pair of linear equations in two variables algebraically – by substitution, by elimination and by cross multiplication method. Simple situational problems must be included. Simple problems on equations reducible to linear equations.
3. QUADRATIC EQUATIONS (15 Periods)
Standard form of a quadratic equation ax2+bx+c=0, (a ≠ 0). Solution of the quadratic equations (only real roots) by factorization, by completing the square and by using quadratic formula. Relationship between discriminant and nature of roots.
Situational problems based on quadratic equations related to day to day activities to be incorporated.
4. ARITHMETIC PROGRESSIONS (08 Periods)
Motivation for studying Arithmetic Progression Derivation of the nth term and sum of the first n terms of A.P. and their application in solving daily life problems.
UNIT IV: GEOMETRY
(Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
(Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
(Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
(Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
(Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
(Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
(Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.
(Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
(Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right traingle.
2. CIRCLES (08 Periods)
Tangents to a circle motivated by chords drawn from points coming closer and closer to the point.
(Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.
(Prove) The lengths of tangents drawn from an external point to circle are equal.
3. CONSTRUCTIONS (08 Periods)
Division of a line segment in a given ratio (internally).
Tangent to a circle from a point outside it.
Construction of a triangle similar to a given triangle.
UNIT V: TRIGONOMETRY
1 . INTRODUCTION TO TRIGONOMETRY (10 Periods)
Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° and 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° and 60°. Relationships between the ratios.
2. TRIGONOMETRIC IDENTITIES (15 Periods)
Proof and applications of the identity sin2A + cos2A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.
3. HEIGHTS AND DISTANCES (8 Periods)
Simple and believable problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation/depression should be only 30°, 45°, 60°.
UNIT VI: MENSURATION
1. AREAS RELATED TO CIRCLES (12 Periods)
Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas and perimeter/circumference of the above said plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60°, 90° and 120° only. Plane figures involving triangles, simple quadrilaterals and circle should be taken).
2. SURFACE AREAS AND VOLUMES (12 Periods)
(i) Problems on finding surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone.
(ii) Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combination of not more than two different solids be taken).
Class 10 Mathematics Syllabus for Session 2016-17
Download latest cbse class 10 Mathematics syllabus for the session 2016-17. This year CBSE has issued the syllabus as usual. This year, there in CCE pattern we have division of syllabus for SA1 or SA2 as it used to be earlier. Each year cbse keep on issuing revised syllabus. Although the topics and textbooks remain same, the contents within vary year by year. Along with Mathematics syllabus for class 10 cbse has also issued the marking scheme along with weightage of each chapter. The package also enlists the learning objectives. That's why it is strongly recommended to download new class 10 maths syllabus 2016-17 to avoid any confusion | 677.169 | 1 |
On a scale of 1-10, 1 being not at all reliable and 10 being completely reliable, how reliable would you say your textbooks are/were? How does the reliability of a textbook vary between different subjects? What factors do you think influence your likelihood to trust a source presented to you in a classroom?
Generally, I view textbooks as being a less reliable tool for learning, since they emphasize more of memorization of facts, knowledge, or processes, but if I were to give them a rating:
English textbooks-9 (English classes are more based on interpretation, rather than knowledge, and can be reasonably well-rounded in giving students the application of certain skills) Math (algebra, geometry, advanced algebra, &ct.)-8Rating math textbooks (in general) anything less than like 9.5 if not 10 is also nonsense. A math textbooks (and most textbooks) are perfectly reliable for what they are meant to do. If you're expecting a textbook to teach you how to think and take over the role of a teacher or your own thought then you're misinterpreting the question.I wrote, "I view textbooks as being as being a less reliable tool for learning"...I never questioned that they provided a basic grip of facts. But even so, that doesn't make it entirely reliably. If you want, I'll make my score for the math textbook a ten, if you're going to call my posts hilariously idiotic because of how I view textbooksMath (algebra, geometry, advanced algebra, &ct.)-10 English textbooks-9 (English classes are more based on interpretation, rather than knowledge, and can be reasonably well-rounded in giving students the application of certain skills)Leafrod might be right about the math-textbooks, but I wonder if he objects to any of my other critiques on the 'reliability of textbooks'."A stupid despot may constrain his slaves with iron chains; but a true politician binds them even more strongly with the chain of their own ideas" - Michel FoucaultNote that this was inferred, if you believe that the textbook provides basic facts. If I'm wrong about what you believe, then please tell me about the role of the teacher that you perceive as in the education systemAt 9/7/2011 12:51:42 PM, nonentity wrote: I'm surprised History got such a high rating. In terms of reliability I would give History a 2 at most.
lol why? Don't tell me its the old subjective perspective story. From the history books I have read, they supply you with the facts and nothing more. They give in depth and unbiased accounts of historical incidences and their meaning.
You're exaggerating quite a bit don't you think? A history book will tell you that Andrew Johnson was the 17th president of the United States. In his term he did A, B, and C. There is no room for subjective commentary as this is an entirely fact based replay of what indeed happened. Why would you be inclined to believe that Johnson did not do A,B and C? If the book states that he did A,B, and C and that was good,....then that would make the book subjective and thus unreliable. You don't see that good/bad bias in many textbooksReliable? oh. Well math is a 10 on reliable. I guess I was rating on quality.
"A stupid despot may constrain his slaves with iron chains; but a true politician binds them even more strongly with the chain of their own ideas" - Michel Foucault
Revised List: Math-10 [Apart from my disagreements with how math textbooks are formatted, they are often pristine clear and well-written] Science-10 [Cogent and to the point] English-9 History-7intuitive.
"A stupid despot may constrain his slaves with iron chains; but a true politician binds them even more strongly with the chain of their own ideas" - Michel Foucault"A stupid despot may constrain his slaves with iron chains; but a true politician binds them even more strongly with the chain of their own ideas" - Michel Foucault
I agree that history books are unreliable, but to be fair its impossible to interpret history without a strong understanding of economics, sociology, and psychology. I suppose they could try to show history from many different perspectives. The history books tend to be unkind to capitalism.
For, example I looked back at my old book on the Great Depression. Not one mention that the Great Depression was caused by the Federal Reserve and the contraction of the money supply. There's also the fact that Hoover is shown as a do nothing president, which is blatantly false.Alright, I'll concede on that...my views of education are incomplete, though I still give it a lower grade than history due to the complications of language. Language is acknowledged to have complications, and a single book or host of textbooks can't capture all those complications...do you agree?
I'll give history an 8 and foreign languages a 7....And as a second note, how do psychology classes present lessons, other than in the form of studies and surveysI expect that psychology is not as developed as the other sciences. I don't expect it to go retarded on me.
And as a second note, how do psychology classes present lessons, other than in the form of studies and surveys?At 9/7/2011 4:04:17 PM, darkkermit wrote: I expect that psychology is not as developed as the other sciences. I don't expect it to go retarded on me.
I'd expect that too...I consistently consider the opinions of psychologists to be ridiculous. The notorious McMartin Preschool trials comes to mind: therapist Kee MacFarlane somehow managed to interpret that (two examples of their interpretations): drawings of stick figures with hands represent child molest, and a child's dislike of tuna fish=exposure to vaginal odors [1] I wonder which of Freud's theories are fallacious...I do know that his method of interpretation of dreams is famous, but not much about his theories about behavior and sexual orientationWell, according to my Freud, the Oedipus complex "denotes the emotions and ideas that the mind keeps in the unconscious, via dynamic repression, that concentrate upon a boy's desire to sexually possess his mother, and kill his father." (
Criticism of it includes the fact that while parent-child conflict has been recorded, and noted, little to none of it is realized or instigated by 'repression' or stimulation of sexual desire for possession...(Martin Daly, Margo Wilson Homicide (New York: Aldine de Gruyter, 1988), and as you stated, it doesn't really seem to manifest in today's society...
As for the " sexually theories of development (oral, anal, phallical, plutonic,sexual)", I suppose that is the psychosexual development proposed by Freud? And isn't the Oedipus complex part of that developmenthey.. maybe they're sexin' each other.... :P
anyway, if i remember correctly, one of my history books told me that christopher columbus was the first to discover america. BS.
depends on the book, but plenty of them are filled with propaganda. psychology books can also be very bias depending on the viewpoints of the authors. i could say the same most likely about theoretical science, art, economics and plenty of other subjects. the less objective the subject, the less objective the books will be.
I believe that the books reliability is much better than verbal interactions. This is because author usually double checks the facts being printed. However, the content assembly , presentation, relevance target audience are various issues . A true learner should learn from classics, reference books and the other books to supplement and reenforce learning | 677.169 | 1 |
Even in a world where every cell phone is also a calculator, basic math competency is a must! In this book, you'll learn how to efficiently solve common problems and effortlessly perform foundational math operations like addition, subtraction, multiplication, and division. Once you've got that down, we'll go over how to handle the scary stuff—like exponents, square roots, geometry, and algebra. | 677.169 | 1 |
Description
At Desmos, we imagine a world of universal math literacy and envision a world where math is accessible and enjoyable for all students. We believe the key is learning by doing.
To achieve this vision, we've started by building the next generation of the graphing calculator. Using our powerful and blazingly-fast math engine, the calculator can instantly plot any equation, from lines and parabolas up through derivatives and Fourier series. Sliders make it a breeze to demonstrate function transformations. It's intuitive, beautiful math. And best of all: it's completely free.
Features:
Graphing: Plot polar, cartesian, or parametric graphs. There's no limit to how many expressions you can graph at one time - and you don't even need to enter expressions in y= form!
Sliders: Adjust values interactively to build intuition, or animate any parameter to visualize its effect on the graph
Tables: Input and plot data, or create an input-output table for any function
Statistics: Find best-fit lines, parabolas, and more.
Zooming: Scale the axes independently or at the same time with the pinch of two fingers, or edit the window size manually to get the perfect window.
Points of Interest: Touch a curve to show maximums, minimums, and points of intersection. Tap the gray points of interest to see their coordinates. Hold and drag along a curve to see the coordinates change under your finger.
Scientific Calculator: Just type in any equation you want to solve and Desmos will show you the answer. It can handle square roots, logs, absolute value, and more.
Inequalities: Plot cartesian and polar inequalities.
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This app is amazing while I'm in 8th and. We are doing rise over run it works omg and it's great for irrational number. And I don't have to pay $150.00 for a Texas instruments pro graphing calculator
Isaac MellenGreat app with simple user interface. I'm suggesting to give an option to locate the co-ordinates which we want in the graph by a mark( If give enter an x value indicate it in the.graph and give the respective y values ) | 677.169 | 1 |
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Description: 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. | 677.169 | 1 |
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9-12.N Number and Quantity
9-12.N-CN The Complex Number System
9-12. Represent complex numbers and their operations on the complex plane.
9-12.N-CN.1 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
9-12.N-L.1 Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.
9-12.N-VM Vector and Matrix Quantities
9-12. Represent and model with vector quantities.
9-12.N-VM.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
9-12.N-VM.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
9-12.N-VM.4.c Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
9-12.N-VM.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|·||v||. Compute the direction of cv knowing that when |c|v is not equal to 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
9-12.N-VM.9 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
9-12.N-VM.10 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
9-12.F-TF.3.b Solve applied problems that include sequences with recurrence relations.
9-12. Extend the domain of trigonometric functions using the unit circle
9-12.F-TF.4 Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi–x, pi+x, and 2pi–x in terms of their values for x, where x is any real number.
9-12. Explain volume formulas and use them to solve problems
9-12.G-GPE.3 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
9-12.S Statistics and Probability
9-12.S-MD Using Probability to Make Decisions
9-12. Calculate expected values and use them to solve problems
9-12.S-MD.1 Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. | 677.169 | 1 |
pearson pre algebra worksheet answers
This is it, the pearson education algebra 1 worksheet answers.So, it will not make you feel hard to bring the book everywhere. Because, the pearson education algebra 1 worksheet answers that we provided in this website is the soft file forms. [download] ebookspearson education algebra 1 worksheet answers pdf.We will share you a new way to get the best recommended book now. pearson education algebra 1 worksheet answers becomes what you need to make real of your willingness. printable worksheets algebra for year 7algebra pizzazz worksheets answers creative publicationspre algebra with pizzazz Daffynition Decoder answer key This is it, the pearson pre algebra answers that will be your best choice for better reading book. Your five times will not spend wasted by reading this website. You can take the book as a source to make better concept. Many of our Pre-Algebra worksheets contain an answer key and can be downloaded or printed, making them great for Pre-Algebra homework, classwork, or extra math practice. Pre-Algebra is typically taught in grades 6, 7, 8 in middle school math courses. Pearson Algebra 2 Worksheet Answers. New updated! The latest book from a very famous author finally comes out.After reading page by page in only your spare time, you can see how this pearson algebra 2 worksheet answers will work for your life. [download] ebookspearson education algebra 1 worksheet answers pdf.And do you know our friends become fans of pearson education algebra 1 worksheet answers as the best book to read? Yeah, its neither an obligation nor order. math programs pearson k 12 mathematics curriculums. pre algebra practice workbook prentice hall mathematics bass. all worksheets ?? glencoe mcgraw hill algebra 1 answers worksheets. Pearson Algebra 2 Worksheet Answers. Come with us to read a new book that is coming recently.And why dont try this book to read? pearson algebra 2 worksheet answers is one of the most referred reading material for any levels. | 677.169 | 1 |
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TabletClass Math learn the basics of calculus quickly. This video is designed to introduce calculus concepts for all math students and make the topic easy to understand.09 May 2017
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Calculus 1 Lecture 1.1: An Introduction to Limits
Calculus 1 Lecture 1.1: An Introduction to Limits Harrison21:58
Understand Calculus in 10 Minutes
Understand Calculus in 10 MinutesThe Basic Idea of Calculus24:44
The Birth Of Calculus (1986)
The Birth Of Calculus (1986)12:11
Introduction to Calculus (1 of 2: Seeing the big picture)
Introduction to Calculus (1 of 2: Seeing the big picture)
Introduction to Calculus (1 of 2: Seeing the big picture)
6:04
Calculus Rhapsody
Calculus Rhapsody46:17
Calculus: What Is It?
Calculus: What Is It? atWhat is Calculus?
This clip provides an introduction to Calculus. More information can be found at
9:46
Calculus - The basic rules for derivatives
Calculus - The basic rules for derivatives p...
published: 28 Apr 2017W...
published: 29 Sep 2016 I in 20 Minutes (The Original) by Thinkwell 09 May 2017 21 Aug 2013 m...
published: 19 May 2011
Introduction to Calculus (1 of 2: Seeing the big picture)
published: 26 Jun 2015 ...
published: 22 May 2009 ...
What is Calculus?
This clip provides an introduction to Calculus. More information can be found at
published: 17 Feb 2009 geom... inv... pre... jus...... between: What Is It?
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wonderi...
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wondering why you might want to learn calculus, start here! - The basic rules for derivatives
This video will give you the basic rules you need for doing derivatives. This covers taking derivatives over addition and subtraction, taking care of constants... h... Mouthorgan9:05
Calculus I in 20 Minutes (The Original) by Thinkwell
Want to see the ENTIRE Calculus in 20 Minutes for FREE? Click on this...If you are wondering what Calculus is, or what you're teacher was ranting on about, this i...This talk describes the motivation for developing mathematical models, including models th...Aaoge jab tum karaoke Jab we met Devs Music Academy varietyDekho naa Performance at Devs Music Karaoke Night Burden Of Grief
The war is over The last battles are gone Swords laying broken My bloodwork is all done I sit down for calming My breath is lessening I�m starting to tremble My sight is clearing My head is weary A dreadful awakening What has driven me Into insanity Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger on the shelf Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger back on the shelf I look around As I raise from my rest Discover what I�ve done No life I have left My heart is in pieces My soul is laying bare Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger on the shelf Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Lightning McQueen spiral notebook from the movie, "Cars." So my three movement, professional level composition is in a "Cars" themed notebook," Heumann said with a chuckle ... ....
"Now is the time to explore if we're going to change things," said Steve Johnson, deputy superintendent ... "We're leaders in AP now, and we don't have the resources to train all the teachers who want it," TrusteeHeide Arneson said ... More than 30 percent of BozemanHigh students take AP classes, on subjects ranging from world history to calculus, and 84 percent pass the national AP exams, which some colleges accept as college credits ... ....
Von Miller was the No. 1 reason the Broncos won Super Bowl 50. His contract is the No. 1 reason Denver might not get back to the Super Bowl until after Miller leaves town ... My intention here is not to bash Miller ... The No ... How the team expects to allocate in excess of $46 million on Miller and Keenum alone in '19 and still have enough wiggle room under the cap to field a top-flight team figures to be an extremely sticky football calculus ... ....
PresidentDonald Trump launched the next salvo in his widening war on Chinese trade abuses, this time taking aim at China's unfair seizure of US intellectual property ... "We have a tremendous intellectual property theft problem," Trump said ... "Talk is not cheap ... "In terms of the broader calculus of the harm that is done by what is the theft of intellectual property is almost incalculable," the official said. .... | 677.169 | 1 |
Work, Power, & Time Practice Bundle
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Challenge your students with more than 50 practice problems over work, power, and converting between basic time units! All original problems that require students to master solving for all variables in the work and power formulas.
This includes 3 separate assignments. Work practice problems (front and back), Time Conversion practice (front only), and Power practice problems (front and back). All have answer keys!
The work and power practice problem sets have 2 versions. One version has 3 of the practice problems as examples, whereas the other version simply uses all questions as practice problems. | 677.169 | 1 |
Free Complete Course on Algebra – Part 5
We are already in the 5th part of our series of "Complete Course on Algebra". This post deals with expansion of expression with parenthesis. Students must be able include all components that are required so as to get the correct result. Ensure that you practice on the worksheet at least two times to get maximum benefit. If you are still a little slow in completing the worksheets (more than 3 minutes per worksheet), then you should repeat the practice until you get a better feel of the process.
The below video is purposely prepared without voice (for the first time) so as to get students to be able to comprehend what is shown in video without voice. In this way students can concentrate and pay better attention. | 677.169 | 1 |
Mathematics
Subject Overview
Mathematics is a wide-ranging subject that is both practical and theoretical, geared to applications and of intrinsic interest. Its manifestations permeate the natural and constructed world and provide the language and techniques for handling many aspects of everyday and scientific life. It is also an intellectual discipline that deals with abstractions and logical argument.
Junior Certificate Mathematics is offered at three levels: Foundation, Ordinary and Higher, with separate syllabuses for each level.
Content
Subject content is presented at each syllabus level under the headings:
Sets
Number systems
Applied arithmetic and measure
Algebra
Statistics (and data handling at Foundation level)
Geometry
Trigonometry (Ordinary level and Higher level only)
Functions and graphs (and relations at Foundation level)
Assessment
Junior Certificate mathematics is examined at all three levels by means of terminal examination papers. There is one examination paper at Foundation level and two at Ordinary and Higher levels.
Status
The current syllabuses were introduced in 2000 and first examined in 2003. The NCCA is currently conducting a review of post-primary mathematics education. | 677.169 | 1 |
Pages
Do You Need to Know Algebra for Technical School?
Before I became a math teacher I managed chemical production plant operations. Moreover before I worked in chemical plants I was a Surface Warfare Officer in the US Navy managing naval power plants and other complex engineering/technical systems. I say this only to gain your trust that I have a pretty good sense of the math involved in many technical vocations.
So now on to our question- Do You Need to Know Algebra for Technical School? Well the obvious answer is it depends what program you're studying. If you're studying to be an Electronic Technician you will need to know a good amount of algebra and maybe even more advanced math. However for many other technical training programs like HVAC, Automotive Technician and Medical Assistant you will only need a working knowledge of some basic algebra skills.
So, the good news for you out there that suffer from "math phobia" is that you don't need to know everything in algebra. Don't worry you won't have to do a quadratic equation or solve a system of equations to repair an air conditioner. However depending on what your job skills involve you may very well have to perform some level of algebra. For example if you're an electrician you might have to solve a basic equation like Ohm's Law or if you're in HVAC you may need think about the algebraic relationship between pressure, temperature and volume.
Technical jobs are based on science. As you know science has a lot of theory and formulas and the language of these formulas is math. As a student you want to have an understanding of the theories that involve your technical job- it will make you smarter and better. A true vocational master is one that has a total command of the practical knowledge and theoretical knowledge. So as a technical school student you should embrace the "book" stuff as very important to your overall development as a professional.
Lastly let's talk about what to do if you have no clue about algebra. If you have the time you want to try to review basic algebra by studying "pre-algebra". I like to recommend online self-paced video instruction programs for a comprehensive review but a good textbook may be enough depending on your current skill level. Here are the basic core topics that I suggest technical student focus on first:
· Algebra basic terms and concepts (order of operations, working with variables, etc.)
· Fractions
· Positive and Negative numbers (these are called Real Numbers in algebra)
1 comment:
So now on to our question- Do You Need to Know Algebra for Technical School? Well the obvious answer is it depends what program you're studying. If you're studying to be an Electronic technical schools | 677.169 | 1 |
Essential Mathematics for Games and Interactive Applications, Third Edition
4.11 - 1251 ratings - Source
Expert Guidance on the Math Needed for 3D Game Programming Developed from the authorsa€™ popular Game Developers Conference (GDC) tutorial, Essential Mathematics for Games and Interactive Applications, Third Edition illustrates the importance of mathematics in 3D programming. It shows you how to properly animate, simulate, and render scenes and discusses the mathematics behind the processes. New to the Third Edition Completely revised to fix errors and make the content flow better, this third edition reflects the increased use of shader graphics pipelines, such as in DirectX 11, OpenGL ES (GLES), and the OpenGL Core Profile. It also updates the material on real-time graphics with coverage of more realistic materials and lighting. The Foundation for Successful 3D Programming The book covers the low-level mathematical and geometric representations and algorithms that are the core of any game engine. It also explores all the stages of the rendering pipeline. The authors explain how to represent, transform, view, and animate geometry. They then focus on visual matters, specifically the representation, computation, and use of color. They also address randomness, intersecting geometric entities, and physical simulation. An Introduction to Creating Real and Active Virtual Worlds This updated book provides you with a conceptual understanding of the mathematics needed to create 3D games as well as a practical understanding of how these mathematical bases actually apply to games and graphics. It not only includes the theoretical mathematical background but also incorporates many examples of how the concepts are used to affect how a game looks and plays. Web Resource A supplementary website contains a collection of source code, supporting libraries, and interactive demonstrations that illustrate the concepts and enable you to experiment with animation and simulation applications. The site also includes slides and notes from the authorsa€™ GDC tutorials.Numerical Analysis. PWS Publishing Company, Boston, 5th edition, 1993.
Thomas Busser. Polyslerp: A fast and accurate polynomial ... Computers and
Graphics, 3:23a€"28, 1978. Bruce Dawson. Comparing floating point numbers,
2012 edition.
Title
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Essential Mathematics for Games and Interactive Applications, Third Edition
Author
:
Publisher
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CRC Press - 2015 | 677.169 | 1 |
The need for improved mathematics education at the high school and college levels has never been more apparent than in the 1990s. As early as the 1960s, I.M. Gelfand and his colleagues in the USSR thought hard about this same question and developed a style for presenting basic mathematics in a clear and simple form that engaged the curiosity and intellectual interest of thousands of school and college students. These same ideas, this development, are available in the present books to any student who is willing to read, to be stimulated, and to learn. Algebra is an elementary algebra text from one of the leading mathematicians of the world - a major contribution to the teaching of the very first high school level course in a centuries old topic - refreshed by the author's inimitable pedagogical style and deep understanding of mathematics and how it is taught and learned | 677.169 | 1 |
PC.PCN.1 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
PC.PCN.2 Understand and use complex numbers, including real and imaginary numbers, on the complex plane in rectangular and polar form, and explain why the rectangular and polar forms of a given complex number represent the same number.
PC.PCN.4 State, prove, and use DeMoivre's Theorem.
PC.F Functions
PC.F.1 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
PC.F.2 Find linear models by using median fit and least squares regression methods. Decide which among several linear models gives a better fit. Interpret the slope and intercept in terms of the original context.
PC.F.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
PC.F.4 Determine if a graph or table has an inverse, and justify if the inverse is a function, relation, or neither. Identify the values of an inverse function/relation from a graph or a table, given that the function has an inverse. Derive the inverse equation from the values of the inverse.
PC.F.5 Produce an invertible function from a non-invertible function by restricting the domain.
PC.F.6 Describe the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values of k (both positive and negative). Find the value of k given the graph f(x) and the graph of f(x) + k, k f(x), f(kx), or f(x + k). Experiment with cases and illustrate an explanation of the effects on the graph using technology. Recognize even and odd functions from their graphs and algebraic expressions.
PC.F.7 Decide if a function is continuous at a point. Find the types of discontinuities of a function and relate them to finding limits of a function. Use the concept of limits to describe discontinuity and end-behavior of the function.
PC.F.8 Define arithmetic and geometric sequences recursively. Use a variety of recursion equations to describe a function. Model and solve word problems involving applications of sequences and series, interpret the solutions and determine whether the solutions are reasonable.
PC.F.9 Use iteration and recursion as tools to represent, analyze, and solve problems involving sequential change.
PC.F.10 Describe the concept of the limit of a sequence and a limit of a function. Decide whether simple sequences converge or diverge. Recognize an infinite series as the limit of a sequence of partial sums.
PC.QPR Quadratic, Polynomial, and Rational Equations and Functions
PC.QPR.1 Use the method of completing the square to transform any quadratic equation into an equation of the form (x - p)^2 = q that has the same solutions. Derive the quadratic formula from this form.
PC.EL.3 Graph and solve real-world and other mathematical problems that can be modeled using exponential and logarithmic equations and inequalities; interpret the solution and determine whether it is reasonable.
PC.EL.4 Use technology to find a quadratic, exponential, logarithmic, or power function that models a relationship for a bivariate data set to make predictions; compute (using technology) and interpret the correlation coefficient.
PC.PE Parametric Equations
PC.PE.1 Convert between a pair of parametric equations and an equation in x and y. Model and solve problems using parametric equations.
PC.PE.2 Analyze planar curves, including those given in parametric form. | 677.169 | 1 |
All branches of the Ajax Public Library will be closed on Friday, March 30th and Monday, April 2nd for Easter. Regular hours continue on March 31st and April 1st.
Mathematics at Work
Practical Applications of Arithmetic, Algebra, Geometry, Trigonometry, and Logarithms to the Step-by-step Solutions of Mechanical Problems, With Formulas Commonly Used in Engineering Practice and A Concise Review of Basic Mathematical Principles
The new fourth edition retains the original purpose which has made this book such a large success through every one of its previous editions: to effectively help its readers solve a wide array of mathematical problems specifically related to mechanical work. Aside from its unique compilation of mathematical problems, this book is renowned for its ability to duplicate, as far as possible, personal instruction. Its usefulness as a self-learning guide for the mathematics of mechanical problems is therefore unexcelled. | 677.169 | 1 |
Complex Numbers Calculator Using the complex numbers calculator the answers to algebra problems covering this topic is only as far as your mobile phone
mathGame Mathematic game for student and general by indonesiasoftware.comMobileMaths v1.7 Thereas no doubt that Mobilemath is a greatest math apps for symbian s60v5 edition. This math apps really awesome because it has so many function such as equation, Integral, function, Periodic Table, Statistics, Matrics function, Powerful Curve Fitting Regression, Interpolation using Cubic
Maths Terms Mathematics is the study of quantity, structure, space, and change
sodBrain MathEasy sodBrain MathEasy is a brain-trainer, a free java-based mobile application that allows you to accelerate your brain by solving basic mathematical and logical problems | 677.169 | 1 |
[EAN: 9780387908397], Neubuch, Publisher/Verlag: Springer, Berlin | In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for calculating them will help students gain a rounded understanding of the subject. | From the contents:Preface.- Introduction.- Continuity.- Compactness and connectedness.- Identification spaces.- The fundamental group.- Triangulations.- Surfaces.- Simplicial homology.- Degree and Lefschetz number.- Knots and covering spaces.- Appendix: Generators and relations.- Bibliography.- Index. | Format: Hardback | Language/Sprache: english | 540 gr | 251 pp, [PU: Springer]
In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for their calculating help students gain a thorough understanding of the subject. | 677.169 | 1 |
Friday, January 31, 2014
What Makes a Pre-AP Math Course Pre-AP?
Pre-AP courses are designed to prepare students for college. According to The College Board (2014), Pre-AP courses are based on the following premises:
All students can perform at rigorous academic levels
Every student can engage in higher levels of learning when they are prepared as early as possible
As we transition into implementation of the revised Texas Essentials of Knowledge and Skills (TEKS), we have to ensure that Pre-AP courses still fulfill the purpose for which they are intended. The revised TEKS have added a level of academic rigor for ALL students in the general education classroom. Students are expected to deepen their conceptual understanding of math concepts, including reasoning and justifying their solution. This means that students in Pre-AP courses have to be met with challenges that expand their knowledge and skills and push them a notch above, toward the next level. We have to be cautious to avoid students receiving Pre-AP credit for course work that is not Pre-AP.
Pre-AP Math Course Goals:
Teach on grade level but at a higher level of academic rigor
Assess students at a level similar to what is offered in an AP course (rigorous multiple-choice and free-response formats)
Promote student development in skills, habits, and concepts necessary for college success
Encourage students to develop their communication skills in mathematics to interpret problem situations and explain solutions both orally and written
Incorporate technology as a tool for help in solving problems, experimenting, interpreting results, and verifying solutions
This is just a small list of goals for Pre-AP math courses. The College Board has official Pre-AP courses in mathematics (and English language arts) for middle and high school students offered through their SpringBoard program (The College Board, 2014). These courses offer rigorous curriculum and formative assessments consistent with their beliefs and expectations. | 677.169 | 1 |
In this first of 5 articles describing how to produce and use CAI software on a home or school computer, the basics of writing mathematics drill and practice exercises for any computer are considered. (DT) | 677.169 | 1 |
Booklet summarising Edexcel Decision 1 laid out like the textbook including worked through examples and key points. Perfect for students learning at A-Level looking for a booklet to help summarise all of Decision 1.
Full lesson on Dijkstra's algorithm for finding the shortest path in a network. Fully animated powerpoint takes students slowly through the process of labelling the nodes and selecting which nodes will form part of the solution. The lesson resource is fully write-on and will only need printing (answers are included). There is another resource which covers more basic practice questions (with answers given).
Unconstrained Optimization is one of the topic which is covered during Mathematics lectures and seminars in order to enhance students knowledge.
During this Lecture and Seminar we are covering following subtopics (Agenda):
•Intervals of increase and decrease
•Concavity and convexity
•Critical and stationary points
•Optimization of economic functions
In this File you will find:
- 1 Unconstrained Optimization Lecture Power Point Presentation 28 Slides
- 1 Seminar Plan
- 19 Seminar Activities with full answer list for students
All covered materials are taught for bachelor level students Level 3.
Please write your comments once you purchase this lesson in order to have some suggestions for further improvements of teaching materials.
A selection of D1 revision resources for the AQA A-Level Spec. These resources include revision sheets and posters with fill in the gaps to help student remember the information. These sheets cover the sorting algorithms, definitions for D1 and other algorithms.
A series of lesson sheets that take you through the whole of the Edexcel AS Further Decision Maths 1 course. They are intended to be used in the order they are numbered. When used in conjunction with exercises on the topics these deliver the entire content of the course.
Contains presentations and assignments to deliver the unit.
All resources and assignments have passed SV twice and contain all relevant learning criteria for the unit.
Can be used for A Level and GCSE Computer Science lessons as well.
Couple of spreadsheets to aid teaching of Dijkstra's shortest path algorithm and A* algorithm.
Example and step by step explanation included. Tried to make it as self explanatory as possible so can be given straight to the pupils whilst you explain it etc.
A Powerpoint game that can be used to help engage students in D1 and D2.
Although it is made with the travelling salesperson algorithm in mind, it can also be used for Kruskal's, Prim's on a distance matrix and the nearest neighbour algorithm.
PowerPoints written so that a teacher can learn from the PowerPoints and teach from them. Students can use the PowerPoints to learn independently too. Many colleagues have used my lessons on Decision 1 without a strong understanding of Decision themselves.
Includes all chapters: Algorithms, Networks, Route Inspection, Critical paths, Linear Programming and Matchings
Includes all worksheets, answers and assessments | 677.169 | 1 |
KS3 Maths Revision Guide (Foundation)
Description
This fantastic Study Guide explains everything students need to know for Key Stage Three Maths - it's all fully up-to-date for the new curriculum from September 2014 onwards (and still a brilliant resource for the current curriculum)!
It's ideal for students working at a higher level (that's what we would have called Levels 5-8 in the pre-2014 curriculum). Every topic is explained with clear, friendly notes and worked examples, and there's a range of practice questions to test the crucial skills.
KS3 Maths Revision Guide (Foundation)
Product Code
EN1364
Publisher
CGP
ISBN 10
1841460400
ISBN 13
9781841460406
Reviews (2)
finn(08/01/2015)
5/5
Who did you buy the KS3 Maths Revision Guide (Foundation) for and why?
My Mum bought it for me
Has it benefited them?
Yep. It reminds you how to do the things you've forgotten, and shows you how to do the things you haven't been taught yet or you missed. I like looking at the topics just before class so i can look good.
Would you recommend it?
very much so.
Liz Smith(07/10/2014)
5/5
Who did you buy the KS3 Maths Revision Guide (Foundation) for and why?
Purchased to help my son keep up in Year 7 as he is in and out of hospital and misses a lot of school.
Has it benefited them?
Yes. The book is very useful and helps my son understand better.
Would you recommend it?
Yes
Be an Exam Ninja!
At Exam Ninja we are passionate about kids being prepared for their exams. We sell revision guides, study books and practice papers so your ninjas are ready to combat their exams and emerge victorious! | 677.169 | 1 |
This course includes video and text explanations of everything in calculus 1 including:
Pre-calculus Functions Limits & Continuity Derivatives Theorems and Proofs These are the five fundamental chapters in the study of Calculus 1.
One-On-One Assistance: You can ask me for calculus help in the Q&A section any time, any day whether it's related to the video content or another problem you're struggling with at home. Either way, I'm here to help you pass and do the best you possibly can!
Buy Premium From My Links To Get Resumable Support,Max Speed & Support Me | 677.169 | 1 |
Linear Programming Powerpoint
Linear Programming Powerpoint linear programming introduction purplemath explains the terminology and demonstrates the basic techniques for linear programming that is for maximizing or minimizing a linear relation subject to certain powerpoint presentation linear programming a model which is used for optimum allocation of scarce or limited resources to peting products or activities under such assumptions as
Beautiful Linear Programming Powerpoint if you wish to acquire these fantastic graphics regarding Linear Programming Powerpoint, click save icon to save the pics to your computer. There're available for down load, if you want and wish to own it, simply click save logo on the web page, and it'll be instantly saved to your computer.
Thanks for visiting our site, contentabove Linear Programming Powerpoint published by admin. At this time we're excited to announce that we have found an awfullyinteresting contentto be reviewed, that is Linear Programming Powerpoint Some people searching for information about and of course one of them is you, is not it? | 677.169 | 1 |
Maths for Economics 2
This module will help you to bridge the gap between abstract mathematical concepts and economic applications, while developing your overall numeracy, analytical and problem solving skills – key skills sought by graduate recruiters. You will learn how economic models are specified and manipulated through mathematical techniques, applying economic and mathematical theories to 'real world' situations. | 677.169 | 1 |
Discrete Mathematics DeMYSTiFied
If you're interested in learning the fundamentals of discrete mathematics but can't seem to get your brain to function, then here's your solution. Add this easy-to-follow guide to the equation and calculate how quickly you learn the essential concepts.
Written by award-winning math professor Steven Krantz, Discrete Mathematics Demystified explains this challenging topic in an effective and enlightening way. You will learn about logic, proofs, functions, matrices, sequences, series, and much more. Concise explanations, real-world examples, and worked equations make it easy to understand the material, and end-of-chapter exercises and a final exam help reinforce learning.
This fast and easy guide offers:
Numerous figures to illustrate key concepts
Sample problems with worked solutions
Coverage of set theory, graph theory, and number theory
Chapters on cryptography and Boolean algebra
A time-saving approach to performing better on an exam or at work
Simple enough for a beginner, but challenging enough for an advanced student, Discrete Mathematics Demystified is your integral tool for mastering this complex subject.
Read more
About the author
Steven G. Krantz is a Professor of Mathematics and Deputy Director at the American Institute of Mathematics. He is an award-winning teacher, and author of How to Teach Mathematics, Calculus Demystified, and Differential Equations Demystified | 677.169 | 1 |
Khan Academy – Math
You only have to know one thing: You can learn anything!
Take control of your learning by working on the skills you choose at your own pace with free online courses. Sharpen your skills with over 100,000 interactive exercises. Math, science, computer programming,history, art, economics, and evenmore free online classes. Free classes and courses available for online learning at every level: Elementary, High school, and College lessons. | 677.169 | 1 |
When students are ready to write code, in whatever language, to deliver
these unique prime factorizations, that's when CS starts to meet Algebra.
The Sieve of Eratosthenes, Trial By Division, Euclid's Algorithm, start
phasing in here, as things to code.
Yes, we're still doing arithmetic, using the four basic operators plus
modulo (%), but we're also introducing functions, the composition of which
will be our basis for getting work done.
Algebra has much to do with controlling the components of a function, one
might say inputs, arguments or parameters. The specifics are often fixed
with constants, as in:
A sin (Bx + C) + D
the paradigm oscillator. Only x is considered the dependent variable at
the end of the day, as A, B, C, D are used to construct a special case
function.
Polynomials are the same way. The coefficients fix the function, and then
x or t do the heavy lifting. | 677.169 | 1 |
Topology now!
Topology is a branch of mathematics packed with intriguing concepts, fascinating geometrical objects, and ingenious methods for studying them. The authors have written this textbook to make this material accessible to undergraduate students without requiring extensive prerequisites in upper-level mathematics. - Publisher. | 677.169 | 1 |
Many advanced and complex problems in engineering, finance and science cannot be solved analytically and for this reason numerical methods have been developed to approximate the solution of these problems and which can then be implemented in a digital computer. These are many kinds of numerical methods that are used to solve a range of problems and in this course we discuss a number of these techniques. The contents correspond to what is taught in university courses in Numerical Analysis (typically courses 261 and 461) in combination with the implementation of numerical algorithms in C++ using the standard STL and Boost libraries. Finally, we apply all these techniques in numerous examples and applications. To our knowledge, this is the only course that offers this integrated set of features.
Subjects Covered
This course discusses the numerical building blocks that are used in applications. The approach is rigorous and integrated. In general, numerical analysis is concerned with algorithms that operate on various kinds of data structures. To this end, we discuss the following topics in detail:
This course has been created and is supported by Dr. Daniel J. Duffy, an internationally known writer, author, numerical analyst and C++ proponent. He has MSc and Phd theses in Numerical Analysis from Trinity College, Dublin and has used the methods in engineering and finance. He has also been writing C++ applications since 1989.
Prerequisites
Some knowledge of calculus and algebra is assumed. We review the necessary mathematical foundations in the course. Some knowledge of a high-level programming language (ideally C, C++, C# or Java) is also recommended.
Who should attend?
This course is of general applicability. We see three specific groups who can benefit from this course:
Undergraduates in disciplines that use and need mathematical and numerical methods. This course fills the gap in our opinion.
Those working in industry and finance who feel that they need to broaden their knowledge.
As a refresher course for those professionals who would like to learn new skills or upgrade existing ones. | 677.169 | 1 |
Tuesday, January 15, 2013
Mathematics, Science Solve Universal Queries
The Black and Gold 1975
Preparing students for the challenging demands of the future, the Math and Science departments offered a wide variety of elective courses. The two departments shared common foundations of precise and accurate work to provide tools to solve the puzzling questions of our ever expanding universe of human knowledge. Besides their classroom laboratory experiments, the science students journeyed on various excursions to broaden and add to their learning abilities; visiting such places as the Look Laboratory, a Coast Guard Cutter, Air Route Traffic Control Center and cruising onboard the ship, The Machias. A typical example of the new approaches in stimulating students interests, the Oceanography classes, with the cooperation of the Aquarium, embarked on a project whereby students became trained to do practical job duties at the Aquarium. This included serving as tour guides, setting up displays and rearranging showcases. Besides Oceanography, the science department offered Biology, Chemistry, Physical Science and Physics courses for those students interested in broadening their horizons. The enthusiastic students of the Computer Math and Calculus classes made extensive use of their new computer terminal, which drew upon the resources of the University of Hawaii's main computer. Complex problems such as areas of unusual curves were quickly solved b the mathematical brain. A new television screen enabled the students to solve their problems fast and easy. Other elective offerings included General Math, Pre-Algebra, Algebra I & II, III & IV, Unified Geometry, Trigonometry and Analytic Geometry. | 677.169 | 1 |
Junior Certificate Project Maths - Ordinary Level
This course is for students interested in studying the Project Maths Junior Certificate Ordinary Level Course in its entirety. This free online course provides students with tutorial videos on all the ordinary level topics in one location listed by module and topic. In addition, a comprehensive assessment is provided which tests learners on the entire content of the Project Maths Ordinary Level Syllabus. These topics include Probability and Statistics, Geometry and Trigonometry, Numbers and Shapes and Algebra | 677.169 | 1 |
Course info
This 3-week course is designed to address K-5 algebraic thinking standards. Participants in this module will examine learning progressions in Algebraic Thinking with a focus on foundational ideas that promote algebraic understanding. Specific goals include the following:
Recognize that addition, subtraction, multiplication, and division operate under the same properties in algebra as they do in arithmetic
Learn variables are tools that are used to describe mathematical ideas in a variety of ways
Develop strategies for describing relationships among quantities to determine equivalence or inequalities
This module runs from March 19th to April 6th
15 renewal units
Enrollment is open from Feb 26th - March 19th or until the course is full. | 677.169 | 1 |
In a few courses, all it requires to go an examination is take note having, memorization, and recall. Nonetheless, exceeding in a math class can take another sort of hard work. You cannot just display up for the lecture and view your teacher "talk" about calculus and . You understand it by carrying out: being attentive at school, actively studying, and resolving math problems – even though your teacher hasn't assigned you any. For those who find yourself battling to perform effectively inside your math course, then check out ideal internet site for fixing math troubles to see the way you may become a better math student.
Low cost math experts on the web
Math programs adhere to a pure progression – each one builds upon the awareness you've received and mastered within the former training course. If you are acquiring it tricky to abide by new ideas at school, pull out your aged math notes and overview former material to refresh oneself. Make certain that you satisfy the stipulations before signing up for your course.
Evaluate Notes The Evening Just before Course
Dislike each time a instructor calls on you and you have neglected ways to resolve a specific challenge? Stay clear of this moment by reviewing your math notes. This will allow you to decide which concepts or issues you'd like to go over in class the subsequent working day.
The considered performing homework each and every evening could appear annoying, however, if you would like to achieve , it can be essential that you continually exercise and master the problem-solving solutions. Use your textbook or online guides to operate by top math difficulties over a weekly basis – regardless if you have no homework assigned.
Make use of the Health supplements That include Your Textbook
Textbook publishers have enriched present day publications with excess product (for example CD-ROMs or on the internet modules) that may be utilized to aid pupils get excess follow in . Some of these products can also include a solution or rationalization guidebook, which can allow you to with doing the job via math issues all by yourself.
Examine Forward To remain Forward
If you want to reduce your in-class workload or even the time you shell out on research, use your free time soon after university or on the weekends to go through in advance to the chapters and ideas that could be protected the following time you are in school.
Critique Previous Assessments and Classroom Illustrations
The operate you need to do in class, for homework, and on quizzes can offer clues to what your midterm or remaining exam will search like. Use your aged exams and classwork to create a private review guideline for your personal upcoming exam. Appear for the way your trainer frames thoughts – this is certainly likely how they are going to look with your take a look at.
Learn how to Get the job done Through the Clock
This can be a well-liked examine suggestion for people taking timed examinations; primarily standardized exams. In case you only have 40 minutes for your 100-point exam, you'll be able to optimally spend four minutes on every 10-point dilemma. Get info regarding how long the check will be and which forms of questions are going to be on it. Then plan to attack the simpler concerns very first, leaving by yourself adequate time to spend about the far more difficult types.
Improve your Means to receive math research assistance
If you are getting a tough time knowing principles in class, then you should definitely get help beyond class. Question your pals to create a study group and stop by your instructor's office environment hours to go above hard challenges one-on-one. Attend examine and review periods when your instructor announces them, or seek the services of a private tutor if you need one.
Discuss To By yourself
After you are examining difficulties for an exam, try to clarify out loud what tactic and procedures you used to get your solutions. These verbal declarations will appear in helpful throughout a test whenever you have to recall the methods you'll want to acquire to locate a solution. Get extra exercise by hoping this tactic by using a pal.
Use Study Guides For Extra Practice
Are your textbook or course notes not assisting you recognize everything you must be understanding at school? Use study guides for standardized exams, including the ACT, SAT, or DSST, to brush up on old substance, or . Analyze guides normally come equipped with thorough explanations of the best way to address a sample challenge, , and also you can usually obtain the place will be the improved purchase mathcomplications. | 677.169 | 1 |
Vectors and Their Applications
Vectors and Their Applications
Geared toward undergraduate students, this text illustrates the use of vectors as a mathematical tool in plane synthetic geometry, plane and spherical trigonometry, and analytic geometry of two- and three-dimensional space. Its rigorous development includes a complete treatment of the algebra of vectors in the first two chapters. Among the text's outstanding features are numbered definitions and theorems in the development of vector algebra, which appear in italics for easy reference. Most of the theorems include proofs, and coordinate position vectors receive an in-depth treatment. Key concepts for generalized vector spaces are clearly presented and developed, and 57 worked-out illustrative examples aid students in mastering the concepts. A total of 258 exercise problems offer supplements to theories or provide the opportunity to reinforce the understanding of applications, and answers to odd-numbered exercises appear at the end of the book. | 677.169 | 1 |
Mathematics II California
We have developed Tutorials specifically for the California Common Core State Standards to help prepare your students for the SBAC exams.
Math Tutorials offer targeted instruction, practice and review designed to develop computational fluency, deepen conceptual understanding, and apply mathematical practices. Students engage with the content in an interactive, feedback-rich environment as they progress through standards-aligned modules. By constantly honing the ability to apply their knowledge in abstract and real world scenarios, students build the depth of knowledge and higher order skills required to demonstrate their mastery when put to the test.
In each module, the Learn It and Try It make complex ideas accessible to students through focused content, modeled logic and process, multi-modal representations, and personalized feedback as students reason through increasingly challenging problems. The Review It offers a high impact summary of key concepts and relates those concepts to students' lives. The Test It assesses students' mastery of the module's concepts, providing granular performance data to students and teachers after each attempt. To help students focus on the content most relevant to them, unit-level pretests and posttests can quickly identify where students are strong and where they're still learning | 677.169 | 1 |
免費玩Calculus APP玩免費
免費玩Calculus App
Calculus APP LOGO
Calculus APP QRCode
Calculus相關介紹
en.wikipedia.org
Calculus - Wikipedia, the free encyclopedia
Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of chan
en.wikipedia.org
Calculus (dental) - Wikipedia, the free encyclopedia p
Calculus-Help.com: Survive calculus class! - Calculus-Help.com
Features a new practice calculus problem every week with complete solutions. Includes an archive of prior weeks' problems and solutions.
stewartcalculus.com
Stewart Calculus h
archives.math.utk.edu
Visual Calculus - Mathematics Archives WWW Server
Pre-Calculus Limits and Continuity Derivatives Applications of Differentiation Integration Applications of Integration Sequences and Series
Calculus [Ron Larson, Bruce H. Edwards] on Amazon.com. *FREE* shipping on qualifying offers. The Larson CALCULUS program has a long history of innovation in the calculus market. It has been widely praised by a generation of students and professors for its
betterexplained.com
A Gentle Introduction To Learning Calculus | BetterExplained
I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education. Calculus relates topics in an elegant, ... I like these sorts of examples for people who have never seen calculus before because, honestly,
Calc101.com Automatic Calculus, Linear Algebra, and Polynomials
Check calculus homework. Enter a function and click for a step-by-step derivative or integral with each step explained. | 677.169 | 1 |
Mathematics for the Million Critical Context - Essay
Critical Context
Mathematics for the Million and its companion volume, Science for the Citizen (1938), can be seen as the centerpiece of Lancelot Hogben's career for a number of reasons: They are his best-known works, they were published at around the middle of his life, and they represent a synthesis of his interests. Hogben began as a zoologist, writing technical books on physiology and genetics, and wound up trying his hand at a number of areas—from language, in Essential World English (1963), to political reform, in Interglossa: A Draft of an Auxiliary for a Democratic World Order (1943). Perhaps his most common theme, however, and the one for which he is best known, is the idea of expressing technical subjects in ways that make them intelligible by, and relevant to the daily concerns of, as many people as possible. Mathematics for the Million was followed by more specialized mathematical and scientific books for adults, on probability, statistics, and document design, and by children's books in the same areas, including The Wonderful World of Mathematics (1955) and Beginnings and Blunders: Or, Before Science Began (1970).
The response to Hogben has been predominantly favorable. Critics praised him for his thoroughness, his expository skills, and his ability to make technical material interesting and understandable, but some have made objections to the polemical nature of his writing. Hogben's attempts to aim Mathematics for the Million at a general adult audience—rather than a specifically juvenile or young adult one—have limited the discussion of this text as a school book, but it is one whose ease of exposition makes it accessible to most adolescents.
Access our Mathematics for the Million Study Guide for Free
Start your 48-hour free trial to access our study guide, along with more than 30,000 other titles. Get help with any book.Start Free Trial | 677.169 | 1 |
Category: Mathematical Programming
Topics that are not product-specific, related to modeling and solving MP problems, including best…Topics that are not product-specific, related to modeling and solving MP problems, including best practices. | 677.169 | 1 |
Geometric Probability. New Topics for Secondary School Mathematics. Materials and Software.
National Council of Teachers of Mathematics, Inc., Reston, VA.
These materials on geometric probability are the first unit in a course being developed by the Mathematics Department at the North Carolina School of Science and Mathematics. This course is designed to prepare high school students who have completed Algebra 2 for the variety of math courses they will encounter in college. Assuming only a knowledge of linear functions and of areas of simple geometric regions, the unit deals with how to determine probabilities of events for which the number of possible outcomes is infinite. Computer software and a user's guide have been written to accompany the unit. The software provides simulations of seven of the experiments discussed in the unit. These simulations are designed to help students understand the random nature of the experiments and appreciate the relationship between empirical and theoretical probabilities. The unit also provides an introduction to mathematical modeling, computer simulations, and the distinction between discrete and continuous phenomena. (PK) | 677.169 | 1 |
Getting Started with Maple by Douglas B. Meade PDF
The aim of this advisor is to provide a brief creation on the best way to use Maple. It essentially covers Maple 12, even though many of the consultant will paintings with prior models of Maple. additionally, all through this consultant, we'll be suggesting information and diagnosing universal difficulties that clients are inclined to come upon. this could make the educational strategy smoother.This consultant is designed as a self-study instructional to benefit Maple. Our emphasis is on getting you quick on top of things. This consultant is usually used as a complement (or reference) for college kids taking a arithmetic (or technology) direction that calls for use of Maple, similar to Calculus, Multivariable Calculus, complex Calculus, Linear Algebra, Discrete arithmetic, Modeling, or facts.
This publication presents someone wanting a primer on random indications and tactics with a hugely available advent to those topics. It assumes a minimum volume of mathematical heritage and specializes in ideas, comparable phrases and fascinating functions to numerous fields. All of this is often encouraged by means of a number of examples carried out with MATLAB, in addition to a number of workouts on the finish of every bankruptcy.
Can be shipped from US. Used books would possibly not comprise better half fabrics, could have a few shelf put on, may possibly comprise highlighting/notes, would possibly not contain CDs or entry codes. a hundred% a reimbursement warrantly.
1. The MATLAB code below generates and plots some basic discrete-time signals. 1. 2. 40 . 2. Real and imaginary parts of a complex discrete-time signal K is a constant amplitude factor and Re{c} sets the attenuation, while Im{c} is related to the dumped signal period (12 points per period). 3. 100 . 3. 4. 005 s. 4. Spectral aliasing illustration A spectral representation provides information about the variation rate of the corresponding signal in the time domain. The more extended the signal spectrum, the faster the signal temporal variation is and the higher the sampling frequency has to be in order to avoid information loss. | 677.169 | 1 |
Mathematics
In high school mathematics students will focus on college and career readiness. The high school mathematics courses follow a sequence targeting post secondary experiences of students. Students will all start on a targeted sequence: Algebra I, Geometry, Algebra II and then have additional options in higher mathematics targeted towards their postsecondary focus.
Elective Courses
Algebra I (3304/3305)
Length: 2 semesters
Credit: 1.0 unit
Grade: 9
Other Info: This course will count toward the Math graduation requirement.
This course is an integrated study of skills and techniques traditionally associated with algebra and elementary geometry. This Algebra course is a rigorous course, which is a prerequisite for accelerated mathematics courses including Honors Geometry. Students enrolling in this class should have strong computational skills with fractions, integers, and decimals. Some topics presented are properties of real numbers, function notation, and evaluation of variable expressions. The student will learn to solve equations and inequalities, graph functions, and solve systems of linear equations. Also, this course contains the study of non-linear relationships, which includes operations with exponents and radicals, polynomial expressions, and solutions to radical equations. Finally, students will learn to solve and graph quadratic relationships along with topics in statistics and probability. Throughout the course, algebraic skills will be linked to problem solving and critical thinking.
Algebra I Extension (3806/3807)
Length: 2 semesters
Credit: 1.0 unit
Grade: 9
Other Info: Students will receive one elective credit for successful completion of this course.
This course is taken in addition to Algebra I The extension period will precede the Algebra 1Geometry (3312/3313)
Length: 2 semesters
Credit: 1.0 unit
Grade: 9, 10, 11, 12
Prerequisite: Algebra IGeometry (3412/3413)
Length: 2 semesters
Credit: 1.0 unit
Grade: 9, 10, 11, 12
Prerequisite: Algebra I with a B or higherAlgebra II (3315/3316)
Length: 2 semesters
Credit: 1.0 unit
Grade: 10, 11, 12
Prerequisite: Geometry (3415/3416)
Length: 2 semesters
Credit: 1.0 unit
Grade: 10, 11, 12
Prerequisite: Geometry with a B or higher Extension (3821/3822)
Length: 2 semesters
Credit: 1.0 unit
Grade: 10, 11, 12
Prerequisite: Teacher recommendation.
Other Info: Students will receive one elective credit for successful completion of this course.
This course is taken in addition to Algebra II. The extension period will precede the Algebra IIPre-Calculus with Trigonometry (3315/3316)
Length: 2 semesters
Credit: 1.0 unit
Grade: 11, 12
Prerequisite: Algebra II (3415/3416) with a CPre-Calculus with Trigonometry (3420/3421)
Length: 2 semesters
Credit: 1.0 unit
Grade: 11, 12
Prerequisite: Algebra II (3415/3416) with a BRobot Engineering and Coding (3844/3845)
Length: 2 semesters
Credit: 1.0 unit
Grade: 10, 11, 12
Prerequisite: AP Computer Science of AP Computer Science Principles.
Other Info: JHS only.
Students will walk through the design and build a mobile robot to play a sport-like game. During the process they will learn key STEM principles, robotics concepts and the RobotC computer language. At the culmination of this course they will compete head-to-head against peers in the classroom.
Expanding Mathematics through Application (3842/3343)
Length: 2 semesters
Credit: 1.0 unit
Grade: 12
Prerequisite: Algebra 2.
Other Info: This course will count toward the Math graduation requirement.
This course is intended for students that are college bound. The objectives of this course are those of the 096 and 098 math curriculum at Elgin Community College. Students will cover these objectives through an application rich course along with study skills that have been identified by ECC faculty. Students will take a pre-test of the ALEKS assessments to assess college readiness.
Intro to Computer Science (3833/3834)
Length: 2 semesters
Credit: 1.0 unit
Grade: 9, 10, 11, 12
Prerequisite: Geometry.
Other Info: JHS only.
This course will count toward the Strand 2 graduation requirement. Students learn the fundamentals of computer programming in this course. Topics studied are looping structures, arrays, files, and incorporation of sound and graphics into programs. Students who take this course should enjoy problem solving and be able to work independently as well as cooperatively. This course does not fulfill any part of the mathematics graduation requirement.
AP Calculus AB (3526/3527Advanced Placement Calculus covers the College Board requirements in preparation for the Advanced Placement Calculus Exam. Topics are typical to those offered in a first semester college course. Pre-calculus mathematics is reinforced. Limit theory, derivatives, anti-derivatives, and integration are studied in relation to their applications in science and mathematics. Broad concepts are emphasized using multiple representations. Upon completion of the class, students are encouraged to take the AP Exam for possible college credit.
AP Calculus BC (3528/3529Calculus BC is a full-year course in the calculus of functions of a single variable. It includes all topics covered in Calculus AB plus additional topics. Both courses represent college-level mathematics for which most colleges grant advanced placement and credit. The content of Calculus BC is designed to qualify the student for placement and credit in a course that is one course beyond that granted for Calculus AB. Upon completion of the class, students are encouraged to take the AP Exam for possible college credit.
AP Statistics (3531/3532)
Length: 2 semesters
Credit: 1.0 unit
Grade: 10, 11, 12
Prerequisite: Algebra 2 with a B or better.
Other Info: A graphing calculator is required for this class. The calculator model must be a Tl-84-Plus. This course will count toward the Math graduation requirement.
Advanced Placement Statistics covers the College Board requirements in preparation for the Advanced Placement Statistics Exam. Topics are typical to those offered in a first semester college course. This course will expose students to four broad conceptual themes: Exploring Data, Planning a Study, Anticipating Patterns, and Statistical Inference. Upon completion of the class, students are encouraged to take the AP Exam for possible college credit.
AP Computer Science A (3533/3534)
Length: 2 semesters
Credit: 1.0 unit
Grade: 10, 11, 12
Prerequisite: Algebra II with a B or better.
Other Info: This course will count toward the Strand 2 graduation requirement.
AP Computer Science covers the College Board requirements in preparation for the Advanced Placement Computer Science A Exam. Topics are typical to those offered in a first semester college course. The topics include designing and implementing solutions to problems by writing programs, using and implementing commonly used algorithms and data structures, coding fluently in an object oriented paradigm and utilize the standard AP Java subset, and to read and understand the AP Computer Science case study. Upon completion of the class, students are encouraged to take the AP Exam for possible college credit.
AP Computer Science Principles (3538/3539)
Length: 2 semesters
Credit: 1.0 unit
Grade: 9, 10, 11, 12
Prerequisite: Algebra 1.
Other Info: JHS only.
The AP Computer Science Principles course is designed to be equivalent to a first semester introductory collegecomputing course. The key sections of this course framework are computational thinking practices, abstraction, data and information, algorithms, programming, the internet and the global impact of computers. Upon completion of the class, students are encouraged to take the AP Exam for possible college credit.
ECC Calculus with Analytic Geometry II (ECC MTH 134)
Length: May be taken in the fall or spring semester (Scheduled as a double period for one semester)
Credit: 2.0 units per semester
Grade: 11, 12
Prerequisite: Completion of AP CalculusECC Calculus with Analytic Geometry Ill (ECC MTH 201)
Length: May be taken in the spring semester (Scheduled as a double period for one semester)
Credit: 2.0 units per semester
Grade: 11, 12
Prerequisite: ECC Calculus with Analytic Geometry IIThird and final course in the calculus sequence. Topics include the following: vectors in 2 and 3 dimensions; planes and lines in space, surfaces and quadric surfaces, space curves; cylindrical and spherical coordinates; vector valued functions and their graphs; functions of two or more variables; partial derivatives, directional derivatives, gradients; double and triple integrals; applications involving functions of several variables; vector fields, line integrals and Green's Theorem; parametric surfaces, surface integrals, the Divergence Theorem and Stokes' Theorem.
Computer Science Innovations (3856/3857)
Length: 2 semesters
Credit: 1.0 unit
Grade: 12
Prerequisite: AP Computer Science or AP Computer Science Principles.
Other Info: This course will be part of the computer science pathways and also for students planning on taking AP Computer Science.
This course is a rigorous study computer science and will include creating Chrome Webstore apps, Google Add-ons, apps for multiple platforms, website design and security. Students will be able to obtain different computer certifications.
Quantitative Literacy (3850/3851)
Length: 2 semesters
Credit: 1.0 unit
Grade: 12
Prerequisite: Algebra II.
Other Info: This course is a college readiness math course designed to engage students in real world math.
Emphasis is placed on real world math situations and will introduce the concepts of numeracy, proportional reasoning, dimensional analysis, rates of growth, personal finance, consumer statistics, practical probabilities, and mathematics for citizenship. Upon completion, students should be able to utilize quantitative information as consumers and to make personal, professional, and civic decisions in their everyday life.
Advanced Statistics (3854/3855)
Length: 2 semesters
Credit: 1.0 unit
Grade: 12
Prerequisite: Algebra II.
Other Info: This course is a 4th year math course for students looking to be engaged in mathematical experience of statistics.
As a result of this course, the students will be able to interpret categorical and quantitative data, make inferences and justify their conclusions, use conditional probability and the rules of probability to solve problems and make decisions, they will create equations and reason with those equations to solve problems in a statistical context. | 677.169 | 1 |
4 Mathematics For Engineering Trigonometry Tutorial …
MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL 1 – TRIGONOMETRIC RATIOS, TRIGONOMETRIC TECHNIQUES AND GRAPHICAL METHODS This is the one of a series of basic ...
7 Lines Of Fit: Prac Tice D - Grade A Math Help
Created by GradeAmathhelp.com, all rights reserved. 7. The scatter plot below shows the relationship between games played and tickets won. Which of the graphs below ...
8 Year 9 Mathematics Test - Free Resources For …
Paper 1 Calculator not allowed First name Last name Class Date Please read this page, but do not open your booklet until your teacher tells you to start.
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Assessment Notes The Night Right before Class
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The thought of performing research each night time could seem frustrating, but when you would like to achieve , it is actually important that you continually practice and master the problem-solving procedures. Use your textbook or on the net guides to operate via best math problems on the weekly foundation – even if you may have no homework assigned.
Utilize the Dietary supplements That include Your Textbook
Textbook publishers have enriched modern publications with more material (such as CD-ROMs or on the net modules) which will be utilized to assistance students acquire excess exercise in . Some of these resources can also include things like an answer or rationalization information, which often can assist you to with functioning as a result of math issues by yourself.
Read In advance To stay In advance
If you want to lessen your in-class workload or perhaps the time you commit on homework, make use of your spare time soon after university or on the weekends to read in advance to your chapters and ideas that can be protected the following time that you are in school.
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Maximize your Resources to have math homework assistance
If you're possessing a hard time being familiar with ideas at school, then be sure to get help outside of class. Check with your friends to produce a review team and visit your instructor's business hours to go over challenging problems one-on-one. Go to examine and review sessions when your instructor announces them, or employ the service of a personal tutor if you want a single.
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Tables of Functions
What's Included
Tables of Functions
Students are introduced to linear functions in the form y = mx + c by breaking them down using function machines with input and outputs. An interactive Excel file and Jigsaw puzzle are included with the lesson for additional consolidation and practice.
Differentiated Learning Objectives
All students should be able to use a function machine to determine an output of a linear function given its x value.
Most students should be able to create a table of results showing x and y for any linear function.
Some students should be able to derive a linear function given corresponding x and y values | 677.169 | 1 |
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Free Homeschool Math Courses
(Basic Versions)
Start a free version of a math course by following the links below. However, before starting any course please take a moment to read the description of a basic course and a full upgrade version on
Free Basic Version vs. Full Version:
Because the free courses on HomeschoolMathOnline use my TabletClass lesson videos I want to summarize the difference between the two platforms. The free courses on this website serve as a great way to sample my instruction to determine if you would like to invest in a full TabletClass Math membership.
HomeschoolMathOnline FREE Courses
TabletClass Math Complete Courses
FREE lesson videos to all course topics- about 20% of the videos available on TabletClass
Great free way to learn or supplement your child's education
Complete course curriculum
Full lesson videos
Videos that explain all practice problems
Worksheets
Tests/keys
Notes
Software
100+ hours of great video instruction designed to get students to master a complete course level | 677.169 | 1 |
Honors Math
The Honors Math Program begins in Grade Four so that students will have adequate time to complete all elementary math skills prior to taking Algebra I in Grade Eight. Challenging activities will be provided in an effort to make learning math an active, meaningful process for those students who exhibit talent, ability, achievement and interest in math. Students who successfully complete the program will be able to begin high school with a mathematics course beyond Algebra I. The following is the summary of the curriculum covered in this subject area.
Grade Four
Text – Go Math, Houghton Mifflin Harcourt, 2015
In Grade 4, instructional time will focus on three critical areas:, perpendicular sides, particular angle measures, and symmetry.
Grade Five
Text – Go Math, Houghton Mifflin Harcourt, 2015
In Grade 5, instructional time will focus on three critical areas: (1) developing fluency with addition, subtraction, and multiplication of fractions, plus with division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.
Grade Six
Text – Glencoe Math (Course 1). Glencoe McGraw-Hill, 2013.
In Grade 6, instructional time willGrade Seven
Text – Glencoe Pre-Algebra. Glencoe McGraw-Hill, 2012.
In Grade 7, instructional time will focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.
Grade Eight
Text – Algebra 1. Glencoe McGraw-Hill, 2014.
Students will build on their basic understanding of mathematics to explore relationships shown in a variety of ways. They will use integers and fractions to represent relationships in graphs and with variables in equations and matrices. Functions will be used to model real-world situations with graphs, tables and equations. Students will write and graph systems of equations and inequalities, as well as compound inequalities, absolute and quadratic equations. They will work with polynomials and learn how to solve for their roots by factoring and with the quadratic formula. They will work with exponents and exponential functions, as well as right triangles, radical expressions, irrational numbers, conjunctions and disjunctions. As these concepts develop, the students will learn to incorporate graphing calculators as a technology tool for success | 677.169 | 1 |
This paper, the second of a two part article, expands on an idea that appeared in literature in the 1950s to show that by restricting the domain to those complex numbers that map onto real numbers, representations of functions other than the ones in the real plane are obtained. In other words, the well-known curves in the real plane only depict part of a bigger whole. This expanded representation brings new insight into visualising complex roots. The suggestion is that this new approach be introduced to students firstly through relating the path in history and secondly by imparting the visual presentation as exposed in the paper to offer a richer teaching and learning approach to the topic. Furthermore this approach provides a new way of employing technology to visualise concepts and curves that were previously not noticed. (Contains 8 figures.) [For Part 1, see EJ775687.] | 677.169 | 1 |
The Great Mathematical Problems [EPUB]
There are some mathematical problems whose significance goes beyond the ordinary - like Fermat's Last Theorem or Goldbach's Conjecture - they are the enigmas which define mathematics.
The Great Mathematical Problems explains why these problems exist, why they matter, what drives mathematicians to incredible lengths to solve them and where they stand in the context of mathematics and science as a whole. It contains solved problems - like the Poincaré Conjecture, cracked by the eccentric genius Grigori Perelman, who refused academic honours and a million-dollar prize for his work, and ones which, like the Riemann Hypothesis, remain baffling after centuries.
Stewart is the guide to this mysterious and exciting world, showing how modern mathematicians constantly rise to the challenges set by their predecessors, as the great mathematical problems of the past succumb to the new techniques and ideas of the present.
Painless Algebra, 4th Edition [EPUB]
01 February 2018, 06:10
Painless Algebra, 4th Edition by Lynette Long
2016 | EPUB | 60.66MB
Defines algebraic terms, shows how to avoid pitfalls in calculation, presents painless methods for understanding and graphing equations, and makes problem-solving fun. Titles in Barron's extensive Painless Series cover a wide range of subjects, as they are taught at middle school and high school levels. Perfect for supporting Common Core Standards, these books are written for students who find the subjects somewhat confusing, or just need a little extra help. Most of these books take a lighthearted, humorous approach to their subjects, and offer fun exercises including puzzles, games, and challenging "Brain Tickler" problems to solve. Bonus Online Component: includes additional games to challenge students, including Beat the Clock, a line match game, and a word scramble.
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The course provides an introduction to using the computer as a tool to solve problems in physics. Students will learn to analyze problems, select appropriate numerical algorithms, implement them using Python, a programming language widely used in scientific computing, and critically evaluate their numerical results. Problems will be drawn from diverse areas of physics. | 677.169 | 1 |
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Algebra Touch is Genius!
By Cindy Downeson Sun, 08/05/2012
Algebra Touch is the perfect app for visual and kinesthetic learners. Here's how it works. There are three stages: Explain, Practice and My Problems. In the Explain stage, you can either work your way through the lessons in order or select a lesson at random. Once you have chosen your lesson, you scroll through the instructions at the bottom of the screen and tap out your answers at the top. If you make a mistake, the problem won't solve. If you're right, you see the problem solve on the screen. You really have to watch this in action to see how genius this is. Check out this videoAfter you understand how the problem works, you can go to the Practice stage to practice the concept as many times as you want. I wasn't able to determine if there was a limit to the number of practice problems or not. There were more than enough for me. The last stage is My Problems. In this stage, you can type in your own problems and solve them using the same intuitive features as in the practice problems. You can type in simple fractions and up to three unknowns, but no exponents or square roots.
Algebra Touch 1.4.4 covers:
· Addition
· Like Terms
· Negatives
· Multiplication
· Order of Operation
· Pairs
· Replacement
· Prime Numbers
· Convenience
· Substitution
· Cross Out
· Equals
· Division
· Products Only
· Variables
· Isolate
· Split
· Basic Equations
· Distribution
· Factoring Out
· Advanced (combination of above) | 677.169 | 1 |
Connected Mathematics 2 Homework Answers
Lesson 8-1 Answer Key Preview the document View in a. Here you will find activities and tools to help you with the ProblemsInvestigations in each unit. Need math homework help. Nightly homework. Algebra 1 Common Core.
Get help now. CMP is a problem-centered mathematics curriculum, which means that students spend a significant amount of class time working on problem solving tasks. Math Videos For Grade 6. Parents, Use These Helpful Videos To Bolster Your Math Knowledge As You Help Your Student With Homework.
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Javascript and Cookies MUST be enabled for this site to function properly. Integrated Mathematics 1 Volume 1.
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Handwritten notes that I made for my HL students on Topic 6 in the IB - Calculus. It is one of the biggest and most important topics in the course, so despite being 20 pages long, these are concise, rich notes. (Note: This is just the regular Calculus section of the syllabus, not the option). It includes:
6.1 - Limits, First Principles, Equation of Tangent/Normal, Increasing/Dec. Regions, Higher Derivatives
6.2 - Derivative Rules, Chain/Product/Quotient Rules, Implicit Differentiation, Related Rates
6.3 - Stationary/Inflection Points, Applications of Differentiation
6.4 - Basic Integration Rules
6.5 - Definite Integrals, Area (x & y-axis), Volume
6.6 - Kinematics
6.7 - Integration By Parts/By Substitution
If you are an HL teacher (or student) and you found these helpful, please 'follow' me, then you will see as soon as I get around to uploading the remaining topics of the HL course.
6-minute video leading students through the derivation of the four Equations of Motion for objects moving with constant acceleration (SUVAT), starting from a velocity-time graph and then using the gradient and the area under the graph (and a little algebraic manipulation) to arrive at the set of four. Great for some flipped learning, a little revision or some consolidationLearning Objectives and Formulas that are used for each chapter for the New A-level Mechanics Maths Curriculum. These could be print on a card and students could fill in their understanding over the lessons as well as teacher could give feedback on students learning.
A revision sheet (with answers) containing IGCSE exam-type questions, which require the students to differentiate to work out equations for velocity and acceleration.
This sheet is designed for International GCSE revision (IGCSE), but could also be used as a homework for first-year A-level students.
👍If you like this resource, then please rate it and/or leave a comment💬.
If the rate-resource button on this page does not work, then go to your ratings page by clicking here 👉 | 677.169 | 1 |
Algebra I
Honors
Algebra I is a comprehensive course that provides an in-depth exploration of key algebraic concepts. Through a "Discovery-Confirmation-Practice"-based exploration of these concepts, students are challenged to work toward a mastery of computational skills, to deepen their understanding of key ideas and solution strategies, and to extend their knowledge in a variety of problem-solving applications.
Course topics include an Introductory Algebra review; measurement; an introduction to functions; problem solving with functions; graphing; linear equations and systems of linear equations; polynomials and factoring; and data analysis and probability.
Within each Algebra I lesson, students are supplied with a post-study Checkup activity that provides them the opportunity to hone their computational skills in a low-stakes, 10-question problem set before moving on to a formal assessment. Additionally, many Algebra I lessons include interactive-tool-based exercises and math explorations to further connect lesson concepts to a variety of real-world contexts.
To assist students for whom language presents a barrier to learning, this course includes audio resources in both Spanish and English.
The content is based on the National Council of Teachers of Mathematics (NCTM) standards and is aligned with | 677.169 | 1 |
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In a few classes, all it takes to move an test is notice taking, memorization, and recall. Nevertheless, exceeding inside a math course takes a different kind of energy. You can not merely exhibit up for a lecture and check out your teacher "talk" about geometry and . You master it by doing: paying attention in school, actively learning, and solving math challenges – regardless if your instructor hasn't assigned you any. Should you find yourself battling to perform very well as part of your math class, then go to most effective website for resolving math challenges to see how you can become an improved math pupil.
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Math courses follow a pure development – each builds on the information you have obtained and mastered from the former class. For those who are getting it hard to follow new principles in school, pull out your old math notes and review previous content to refresh yourself. Make certain that you satisfy the prerequisites just before signing up for just a class.
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Loathe when a teacher phone calls on you and you've neglected how to clear up a certain difficulty? Keep away from this minute by examining your math notes. This tends to make it easier to decide which ideas or issues you'd want to go about in school the next working day.
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Textbook publishers have enriched fashionable publications with added materials (which include CD-ROMs or on-line modules) which can be used to support students get extra exercise in . A few of these components may incorporate an answer or rationalization guideline, which often can allow you to with doing work as a result of math complications on your own.
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If you want to reduce your in-class workload or perhaps the time you invest on homework, make use of your spare time soon after faculty or around the weekends to browse ahead on the chapters and concepts that may be included the next time you will be in school.
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Hours
Chavez Student Center
Math 32
Formats of Service for Math 32 (Spring 2018)
Faculty
Lec. Time
Adjunct
Study Group
Review
Drop-In
STAFF
MWF 3-4
No
No
No
Yes
For more detailed information about our support services for your course, please click on the headers below:
Drop-In
Drop-In Tutoring is a collaborative space designed to encourage students to work with tutors or each other on problem sets, concepts, theory, or anything else related to the material in their courses. The tutors may provide a conceptual framework on a given concept or create parallel problems to assess and develop content mastery. Typically there are 10 to 15 tutors available every hour to work with students in small groups or individually.
Drop-In Tutoring is free for all students and starts the 3rd week of instruction and continues until the end of instruction. Students should sign in at the Math/Stat Drop-In computer in the Cesar Chavez Atrium by room 103. Drop-In Tutoring is open Mon-Thu 10-4. | 677.169 | 1 |
FUNDAMENTALS OF MATHEMATICS, 9th Edition offers a comprehensive review of all basic mathematics concepts and prepares students for further coursework. The clear exposition and the consistency of presentation make learning arithmetic accessible for all. Key concepts are presented in section objectives and further defined within the context of How and Why; providing a strong foundation for learning. The predominant emphasis of the book focuses on problem-solving, skills, concepts, and applications based on "real world" data, with some introductory algebra integrated throughout. The authors feel strongly about making the connection between mathematics and the modern, day-to-day activities of students. This textbook is suitable for individual study or for a variety of course formats: lab, self-paced, lecture, group or combined formats. Though the mathematical content of FUNDAMENTALS OF MATHEMATICS is elementary, students using this textbook are often mature adults, bringing with them adult attitudes and experiences and a broad range of abilities. Teaching elementary content to these students, therefore, is effective when it accounts for their distinct and diverse adult needs. Using Fundamentals of Math meets three needs of students which are: students must establish good study habits and overcome math anxiety; students must see connections between mathematics and the modern day-to-day world of adult activities; and students must be paced and challenged according to their individual level of understanding.
Synopsis
This text offers a comprehensive review of all basic mathematics concepts and prepares students for further coursework. The arithmetic is presented with an emphasis on problem-solving, skills, concepts, and applications based on "real world" data, with some introductory algebra integrated throughout.
Synopsis
Succeed in math with FUNDAMENTALS OF MATHEMATICS! By offering a comprehensive review of all basic mathematics concepts, this mathematics text takes the intimidation out of arithmetic and makes learning accessible to everyone. Studying is made easy with tools found throughout the text such as objectives, vocabulary definitions, calculator examples, good advice for studying, concept reviews, and chapter tests. Through caution remarks that alert you to common pitfalls and how and why segments that explain and demonstrate concepts and skills in a step-by-step format, you will easily build confidence in your own skills.
About the Author
James Van Dyke has been an instructor of mathematics for over 30 years, teaching courses at both the high school and college levels. He is the co-author of eight different mathematics textbooks, including Fundamentals of Mathematics, published by Cengage Learning.James Rogers has been teaching high school and college-level mathematics courses for more than 30 years. Though retired, he currently teaches part-time at Portland Community College. He is the co-author of nine mathematics textbooks, including Fundamentals of Mathematics, published by Cengage Learning.Hollis Adams teaches mathematics at Portland Community College, a position she has been in for nearly 20 years. She is the co-author of two mathematics textbooks, including Fundamentals of Mathematics, published by Cengage Learning. | 677.169 | 1 |
Introductory number theory is relatively easy. When I took it we covered primes, quadratic reciprocity, algebraic numbers, and lots of examples and relatively easy theorems. Most of the proofs we did in the class were very straightforward (wilsons & fermat's little theorem, etc) and was not difficult at all. The 'next level' of number theory, Algebraic number theory, involves upper level algebra and can be difficult at first glance, but if you have done any studying in field theory or a related subject you will recognize some stuff.
Number theory may not seem like the most practical thing to learn but it gets used in group theory, discrete math, and other typical third year math courses.
The biggest thing is that Number theory is different; it simply doesn't have the same flavor as more continuous subjects.
It is a harder subject, but that's offset by the fact an introductory course is going to be working mostly with the simplest things: modular arithmetic, divisibility, multiplicative functions, and the like.
I assume you mean number theory as a first-year, standard number theory course.
If you haven't taken a math course that requires you to write proofs, then you might feel number theory is a little challenging, but not too demanding, and it is also a good place to start seeing/writing proofs. On the other hand, if you have an experience with writing mathematical proofs, then I think you have no problem with number theory.
As Hurkyl mentioned, it is different from courses like calculus or linear algebra, which might make the subject harder. | 677.169 | 1 |
Limits Day 2 and 3 (PP)
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this file type before downloading and/or purchasing.
432 KB|16 pages
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Product Description
I use this powerpoint for my day 2 and day 3 lessons on limits.
The first day I use charts and graphs.
This has 22 questions mixed in with the basic buliding blocks of limits (theorems and hints).
It includes problems with piecewise functions, factoring with synthetic division, square roots, determining functions going to numbers or infinity, etc.
All steps and answers are given. | 677.169 | 1 |
Description: Guided by the premise that solving some of the world's most important mathematical problems will advance the field, this book offers a fascinating look at the seven unsolved Millennium Prize problems. This work takes the unprecedented approach of describing these important and difficult problems at the professional level.
Mathematics for the Physical Sciences by Leslie Copley - De Gruyter Open A text on advanced mathematical methods with numerous applications. The book begins with a thorough introduction to complex analysis, which is then used to understand the properties of ordinary differential equations and their solutions. (930 views)Problems with and Without ... Problems! by Florentin Smarandache - viXra This book is addressed to College honor students, researchers, and professors. It contains 136 original problems published by the author in various journals. The problems could be used to preparing for courses, exams, and Olympiads in mathematics. (7170 views) | 677.169 | 1 |
Math Tutorial
Math Tutorial
Objectives
Assistance in understanding concepts covered in grade 1 to grade 12 math as outlined in the Alberta Education Curriculum.
Content
Work will be provided by the instructor. Students should also bring their own books and materials that they are currently using in math class. Individual help will be given on specific topics or homework questions brought forth by each student. | 677.169 | 1 |
pocket calculator to do your basic arithmetic. The calculator is fast and accurate as long, of course, as you punch in the right numbers. So what could be bad about a tool that saves you so much work and gives you the right answers? Let me be brutally frank. You know why you bought this book, and it's not for the story. By working your way through this book, problem by problem, you will be amazed by how much your math skills will improve. But—and this is a really big BUT—I don't want you to use your calculator at all. So put it away for the time you spend working through this book. And who knows—you may never want to use it again. Your brain has its own built-in calculator, and it, too, can work quickly and accurately. But you know the saying, "Use it or lose it."
vii
MATH ESSENTIALS
The book is divided into four sections—a review of basic arithmetic, and then sections on fractions, decimals, and percentages. Each section is subdivided into four to eight lessons, which focus on building specific skills, such as converting fractions into decimals, or finding percentage changes. You'll then get to use these skills by solving word problems in the applications section. There are 21 lessons plus four review lessons, so if you spend 20 minutes a day working out the problems in each lesson, you can complete the entire book in about a month. One thing that distinguishes this book from most other math books is that virtually every problem is followed by its full solution. I don't believe in skipping steps. You, of course, are free to skip as many steps as you wish, as long as you keep getting the right answers. Indeed, there may well be more than one way of doing a problem, but there's only one right answer. When you've completed this book, you will have picked up some very useful skills. You can use these skills to figure out the effect of mortgage rate changes and understand the fluctuations in stock market prices or how much you'll save on items on sale at the supermarket. And you'll even be able to figure out just how much money you'll save on a lowinterest auto loan. Once you've mastered fractions, decimals, and percentages, you'll be prepared to tackle more advanced math, such as algebra, business math, and even statistics. At the end of the book, you'll find my list of recommended books to further the knowledge you gain from this book (see Additional Resources). If you're just brushing up on fractions, decimals, and percentages, you probably will finish this book in less than 30 days. But if you're learning the material for the first time, then please take your time. And whenever necessary, repeat a lesson, or even an entire section. Just as Rome wasn't built in a day, you can't learn a good year's worth of math in just a few weeks. While I'm doing clichés, I'd like to note that just as a building will crumble if it doesn't have a strong foundation, you can't learn more advanced mathematical concepts without mastering the basics. And it doesn't get any more basic than the concepts covered in this book. So put away that calculator, and let's get started.
viii
SECTION I
R EVIEWING B ASICS
THE
O
n every page of this book you're going
to be playing with numbers, so I want you to get used to them and be able to manipulate them. In this section you'll review the basic operations of arithmetic—addition, subtraction, multiplication, and division. Your skills may have grown somewhat rusty. Or, as the saying goes, if you don't use it, you'll lose it. This section will quickly get you back up to speed. Of course different people work at different speeds, so when you're sure you have mastered a particular concept, feel free to skip the rest of that lesson and go directly to the next. On the other hand, if you're just not getting it, then you'll need to keep working out problems until you do.
1
REVIEWING THE BASICS
Indeed, the basic way most students learn math is through repetition. It would be great if you could get everything right the first time. Of course if you could, then this book and every other math book would be a lot shorter. Once you get the basics down, there's no telling how far you'll go. So what are we waiting for? Let's begin.
2
P RETEST
T
he first thing you're going to do is take a
short pretest to give you an idea of what you know and what you don't know. This pretest covers only addition, subtraction, multiplication, and division; all of which are necessary for learning the other concepts we will be studying later on in this book. Remember, you must not use a calculator. The solutions—completely worked out so you can see exactly how to do the problems—follow immediately after the pretest for you to check your work.
N EXT S TEP
If you got all 18 problems right, then you probably can skip the rest of this section. Glance at the next four lessons, and, if you wish, work out a few more problems. Then go on to Section II. If you got any questions wrong in addition, subtraction, multiplication, or division, then you should definitely work your way through the corresponding lessons.
7
REVIEW LESSON 1
This lesson reviews how to add whole numbers. If you missed any of the addition questions in the pretest, this lesson will guide you through the basic addition concepts.
A DDITION
A
ddition is simply the totaling of a
column, or columns, of numbers. Addition answers the following question: How much is this number plus this number plus this number?
Did you get 37? Good. A trick that will help you add a little faster is to look for combinations of tens. Tens are easy to add. Everyone knows 10, 20, 30, 40. Look back at the problem you just did, and try to find sets of two or three numbers that add to ten. What did you find? I found the following sets of ten. Solution: 6 8 = 10 4 3 5 =10 2 +9 37 Here's another column to add. Again, see if you can find sets of tens. Problem: 3 8 2 5 7 4 1 8 6 3 4 8 +4
Did you get 63? I certainly hope so. Look at the tens marked in the solution.
10
ADDITION
Solution: 10
10
10 10
3 8 2 5 7 4 1 8 6 3 4 8 +4 63
10
10
10 10 10
As you can see, there are a lot of possibilities, some of them overlapping. Do you have to look for tens when you do addition? No, certainly not. But nearly everyone who works with numbers does this automatically.
I'll bet you got 270. You carried a 2 into the second column because the first column totaled 20. Solution: 24 63 43 18 52 + 70 270
11
2
REVIEWING THE BASICS
HOW TO CHECK YOUR ANSWERS
When you add columns of numbers, how do you know that you came up with the right answer? One way to check, or proof, your answer is to add the figures from the bottom to the top. In the problem you just did, start with 0 + 2 in the right (ones) column and work your way up. Then, carry the 2 into the second column and say 2 + 7 + 5 and work your way up again. Your answer should still come out to 270.
Did you get the correct answer? You'll know for sure if you proofed it. If you haven't, then go back right now and check your work. I'll wait right here. Solution: 196 312 604 537 578 943 + 725 3,895
23
Did you get it right? Did you get 3,895 for your answer? If you did the problem correctly, then you're ready to move on to subtraction. You may skip the rest of this section, pass GO, collect $200, and go directly to subtracting in the next section.
12
ADDITION
P ROBLEM S ET
If you're still a little rusty with your addition, then what you need is some more practice. So I'd like you to do this problem set.
1.
209 810 175 461 334 520 312 685 + 258
2.
175 316 932 509 140 462 919 627 + 413
3.
119 450 561 537 366 914 838 183 + 925
Solutions Did you check your answers for each problem? If so, you should have gotten the answers shown below.
1.
209 810 175 461 334 520 312 685 + 258 3,764
33
2.
175 316 932 509 140 462 919 627 + 413 4,493
24
3.
119 450 561 537 366 914 838 183 + 925 4,893
34
N EXT S TEP
Now that you've mastered addition, you're ready to tackle subtraction. But if you still need a little more practice, then why not redo this lesson? If you've been away from working with numbers for a while, it takes some getting used to.
13
REVIEW LESSON 2
This lesson reviews how to subtract whole numbers. If you missed any of the subtraction questions in the pretest, this lesson will guide you through basic subtraction concepts.
S UBTRACTION
S
ubtraction is the mathematical opposite
of addition. Instead of combining one number with another, we take one away from another. For instance, you might ask someone, "How much is 68 take away 53?" This question is written in the form of problem 1.
SIMPLE SUBTRACTION
First let's start off by working out some basic subtraction problems. These problems are simple because you don't have to borrow or cancel any numbers.
15
REVIEWING THE BASICS
P ROBLEM S ET
Try these two-column subtraction problems.
1.
68 – 53 94 – 41
3.
77 – 36 82 – 50
2.
4.
Solutions How did you do? You should have gotten the following answers.
1.
68 – 53 15 94 – 41 53
3.
77 – 36 41 82 – 50 32
2.
4.
C HECKING Y OUR S UBTRACTION
There's a great way to check or proof your answers. Just add your answer to the number you subtracted and see if they add up to the number you subtracted from.
1.
15 + 53 68 53 + 41 94
3.
41 + 36 77 32 + 50 82
2.
4.
SUBTRACTING WITH BORROWING
Now I'll add a wrinkle. You're going to need to borrow. Are you ready?
16
SUBTRACTION
P ROBLEM S ET
Okay? Then find answers to these problems.
5.
54 – 49 63 – 37
7.
86 – 58 97 – 49
6.
8.
Solutions
5.
54 – 49 5 63 – 37 26
51
4 1
7.
86 – 58 28 97 – 49 48
81
71
6.
8.
If you got these right, please go directly to the next section, multiplication. And if you didn't? Well, nobody's perfect. But you'll get a lot closer to perfection with a little more practice. We need to talk about borrowing. In problem 5, we needed to subtract 9 from 4. Well, that's pretty hard to do. So we made the 4 into 14 by borrowing 1 from the 5 of 54. Okay, so 14 – 9 is 5. Since we borrowed 1 from the 5, that 5 is now 4. And 4 – 4 is 0. So 54 – 49 = 5. Next case. In problem 6, we're subtracting 37 from 63. But 7 is larger than 3, so we borrowed 1 from the 6. That makes the 6 just 5, but it makes 3 into 13. 13 – 7 = 6. And 5 – 3 = 2.
SUBTRACTING WITH MORE THAN TWO DIGITS
Now we'll take subtraction just one more step. Do you know how to do the two-step, a dance that's favored in Texas and most other western states? Well now you're going to be doing the subtraction three-step, or at least the three-digit.
P ROBLEM S ET
Try these three-digit subtraction problems.
13.
532 – 149 714 – 385
15.
903 – 616 840 – 162
14.
16.
Solutions
13.
5 32 – 149 383 71 4 – 385 329
6 10 1
4 12 1
15.
90 3 – 616 287 840 – 162 678
7 13 1
891
14.
16.
Did you get the right answers? Proof them to find out. If you haven't already checked yours, go ahead, and then check your work against mine.
18
SUBTRACTION
Answer Check
13.
383 + 149 532 329 + 385 714
11
11
15.
287 + 616 903 678 + 162 840
11
11
14.
16.
N EXT S TEP
How did you do? If you got everything right, then go directly to Review Lesson 3. But if you feel you need more work subtracting, please redo this lesson.
19
REVIEW LESSON 3
Multiplication is one of the most important building blocks in mathematics. Without multiplication, you can't do division, elementary algebra, or very much beyond that. The key to multiplication is memorizing the multiplication table found at the end of this lesson.
M ULTIPLICATION
M
ultiplication is addition. For instance,
how much is 5 × 4? You know it's 20 because you searched your memory for that multiplication fact. There's nothing wrong with that. As long as you can remember what the answer is from the multiplication table, you're all right. Another way to calculate 5 × 4 is to add them: 4 + 4 + 4 + 4 + 4 = 20. We do multiplication instead of addition because it's shorter. Suppose you had to multiply 395 × 438. If you set this up as an addition problem, you'd be working at it for a couple of hours.
SIMPLE MULTIPLICATION
Do you know the multiplication table? You definitely know most of it from 1 × 1 all the way up to 10 × 10. But a lot of people have become so
21
REVIEWING THE BASICS
dependent on their calculators that they've forgotten a few of the multiplication solutions—like 9 × 6 or 8 × 7. Multiplication is basic to understanding mathematics. And to really know how to multiply, you need to know the entire multiplication table by memory. So I'll tell you what I'm going to do. I'll let you test yourself. First fill in the answers to the multiplication problems in the table that follows. Then check your work against the numbers shown in the completed multiplication table that appears at the end of the lesson. If they match, then you know the entire table. But if you missed a few, then you'll need to practice doing those until you've committed them to memory. Just make up flash cards (with the problem on one side and the answer on the other) for the problems you missed. Once you've done that, they're yours.
LONG MULTIPLICATION
We've talked about using calculators before, so remember the deal we made. You should not use calculators for simple arithmetic calculations unless the problems are so repetitive that they become tedious. So I want you to keep working without a calculator.
P ROBLEM S ET
I'd like you to do this problem set.
1.
46 × 37
2.
92 × 18
3.
83 × 78
Solutions
1.
46 × 37 322 _138_ 1,702
2.
92 × 18 736 __92_ 1,656
3.
83 × 78 664 _5 81_ 6,474
You still have to carry numbers in these problems, although they are not shown in the solutions. To show you how it works, I'm going to talk you through the first problem, step-by-step. First we multiply 6 × 7, which gives us 42. We write down the 2 and carry the 4: 46 × 37 2 Then we multiply 4 × 7, which gives us 28. We add the 4 we carried to the 28 and write down 32: 46 × 37 322
23
4 4
REVIEWING THE BASICS
Next we multiply 6 × 3, giving us 18. We write down the 8 and carry the 1: 46 × 37 322 8_ Then we multiply 4 × 3, giving us 12. We add the 1 we carried to the 12, and write down 13: 46 × 37 322 138_ After that we add our columns: 46 × 37 322 _1 38_ 1,702 Did you get the right answers for the whole problem set? Want to see how to check your answers? Read on for an easy checking system.
1 1
HOW TO CHECK YOUR ANSWERS
To prove your multiplication, just reverse the numbers you're multiplying.
1.
37 × 46 222 _1 48_ 1,702
2.
18 × 92 36 _1 62_ 1,656
3.
78 × 83 234 _6 24_ 6,474
If you got these right, then you can skip the section below entitled, "Multiplication: Step-by-Step." But if you're still a little shaky about multiplying, then you should definitely read it.
24
MULTIPLICATION
MULTIPLICATION: STEP-BY-STEP
Long multiplication is just simple multiplication combined with addition. Let's multiply 89 by 57. Here is a step-by-step list describing how to get the answer.
a.
Will you get the same answer multiplying 111 × 532 as you will if you multiply 532 × 111? Let's find out. Please work out both problems: 111 × 532 532 × 111
Solutions 111 × 532 222 333_ 555__ 59,052
532 × 111 532 532_ 532__ 59,052
We get the same answer both ways. So you always have a choice when you multiply. In this case, is it easier to multiply 111 × 532 or 532 × 111? Obviously it's much easier to multiply 532 × 111, because you don't really do any multiplying. All you do is write 532 three times, and then add.
You can see that multiplying 749 × 222 is quite a bit easier than multiplying 222 × 749. As you do more and more problems, you'll recognize shortcuts like this one.
N EXT S TEP
Okay, no more Mr. Nice Guy. Because mastering multiplication is so important, I must insist that you really have this down before you go on to division. After all, if you can't multiply, then you can't divide. It's as simple as that. So if you got any of the problems in this section wrong, go back and work through them again. And memorize your multiplication table!
REVIEW LESSON 4
This lesson will help you master both short and long division. It will also show you how to check, or proof, your answers, so you can know for certain that the answer you came up with is indeed correct.
D IVISION
A
s you'll see, division is the opposite of
multiplication. So you really must know the multiplication table from the previous lesson to do these division problems. In this lesson, you'll learn the difference between short and long division and how to use trial and error to get to the solution.
SHORT DIVISION
We'll start you off with a set of short division problems.
P ROBLEM S ET
Work out the answers to the problems on the next page.
31
REVIEWING THE BASICS
1.
5
140
2.
9
189
3.
7
2,114
Solutions
1.
28 5 14 0
4
2.
9
21 189
3.
7
302 2,114
Let's take a closer look at problem 3. We divide 7 into 21 to get the 3: 3__ 2,114
7
Then we try to divide 7 into 1. Since 7 is larger than 1, it doesn't fit. So we write 0 over the 1: 30_ 2,114
7
And then we ask how many times 7 goes into 14. The answer is 2: 302 2,114
7
HOW TO CHECK YOUR ANSWERS
The answers to each of these problems can be checked, or proven. I'll do the first proof below.
1.
28 ×5 140 Now you do the next proofs. Did your answers check out? Here are my proofs.
32
DIVISION
2.
21 ×9 189
3.
302 ×7 2,114
Each of these came out even. But sometimes there's a remainder. You'll find that that's the case in the next problem set. When you learn about decimals in Section III, you'll find out you can keep dividing until it comes out even, or you can round off the answer.
P ROBLEM S ET
Now try these division problems that don't come out even. They all have remainders.
4.
9
413
5.
8
321
6.
6
501
Solutions
4.
45 R8 9 413
5
5.
8
40 R1 321
6.
83 R3 6 501
2
LONG DIVISION
Long division is carried out in two steps: • Trial and error • Multiplication The process of long division is identical to short division, but it involves a lot more calculation. That's why it's so important to have memorized the multiplication table. Let's work out the next problem together. Problem: 37 596
Solution: How many times does 37 go into 59? Just once. So we put a 1 directly over the 9 and write in 37 directly below 59. 1_ 37 596 – 37_ 22_
33
REVIEWING THE BASICS
Then we subtract 37 from 59, leaving us with 22. Next, we bring down the 6, giving us 226. How many times does 37 go into 226? We need to do this by trial and error. We finally come up with 6, since 6 × 37 = 222. 16 37 596 – 37X 226 – 222 When we subtract 222 from 226, we are left with 4, which is our remainder. 16 596 37 – 37X 226 – 222 4 The proper notation for the answer is 16 R4. Can you check this answer? Yes! Just multiply 16 × 37 and add 4. Go ahead and do it now. Did you get 596? Good. Then you proved your answer, 16 R4, is correct. Here's another problem. Find the answer and then check it. Problem: 43 985
N EXT S TEP
Have you been tempted to reach for your calculator to do some of the problems in this section? Remember that the less you rely on your calculator, and the more you rely on your own mathematical ability, the better off you'll be. The more you rely on your ability, the more your ability will be developed. If you've mastered addition, subtraction, multiplication, and division of whole numbers, you're ready to tackle fractions.
36
SECTION II
F RACTIONS
H
ow many times a day do you hear ads
on television—especially on the home shopping channels— offering you some pretty amazing products at just a fraction of what you would have to pay for them in a store? Of course you need to 9 ask just what kind of fraction they're talking about. Is it 1 , 1 , 1 , or 10 ? 2 3 4 We'll start with the fraction 1 . The top number is called the numera2 tor and the bottom number is called the denominator. So in the fraction 1 2 2 , the numerator is 1 and the denominator is 2. In the fraction 3 , the numerator is 2 and the denominator is 3. In a proper fraction the denominator is always greater than the numerator. We already saw that 1 and 1 are proper fractions. How about 2 3 4 3 19 5 , 8 , and 20 ? These too, are proper fractions.
37
FRACTIONS
What do you think improper fractions look like? They look like these fractions: 2 , 17 , and 7 . So if the numerator is greater than the denomina1 14 5 tor, then it's an improper fraction. What if the numerator and the denominator are equal (making the fraction equal to 1), as is the case with these fractions: 2 , 9 , 20 ? Are these 2 9 20 proper or improper fractions? A while back someone decided that when the numerator and denominator are equal, we must call that an improper fraction. That's the rule, but it's not really all that important. What is important is to recognize the relationship between the numerator and the denominator. Let's take the improper fraction 4 . What 2 are you supposed to do with it? Should we just leave it sitting there? Or maybe do a little division? Okay, you do a little division. Now what do you divide into what? You divide the 2 into the 4, which gives you 2. So the relationship of the numerator to the denominator of a fraction is that you're supposed to divide the denominator (or bottom) of the fraction into the numerator (or top). There's one more term I'd like to introduce, and then we can stop talking about fractions and start using them. The term is mixed number, which consists of a whole number and a proper fraction. Examples would include numbers like 3 3 , 1 5 , and 4 2 . 4 8 3 Do you really have to know all these terms? Not necessarily. Just remember numerator and denominator. If you can also remember proper fraction, improper fraction, and mixed number, then you will have enriched your vocabulary, but you'll still have to get out of bed every morning, and you probably won't notice any major changes in the quality of your life. When you have completed this section, you will be able to convert improper fractions into mixed numbers and convert mixed numbers into improper fractions. You'll also be able to add, subtract, multiply, and divide proper fractions, improper fractions, and mixed numbers.
38
LESSON 1
In this lesson, you'll learn the basic fraction conversion procedures. These procedures will be used when you move on to the more complicated fraction problems, so be sure to read this lesson carefully.
F RACTION C ONVERSIONS
B
y convention, answers to fraction prob-
lems are expressed in terms of mixed numbers, rather than in terms of improper fractions. But when you add, subtract, multiply, and divide mixed numbers—which you'll be doing later in this section—you'll find it a lot easier to work with improper fractions. So you need to be able to convert improper fractions into mixed numbers and mixed numbers into improper fractions.
CONVERTING IMPROPER FRACTIONS INTO MIXED NUMBERS
To convert an improper fraction into a mixed number, you divide the denominator (bottom number) into the numerator (top number). Any remainder becomes the numerator of the fraction part of the mixed number.
39
You generally need to reduce your fractions to the lowest possible terms. In other words, get the denominator as low as possible. You do this by dividing both the numerator and the denominator by the same number. In this case, I divided both 6 and 9 by 3 to change 6 into 2 . 9 3
CONVERTING MIXED NUMBERS INTO IMPROPER FRACTIONS
We've converted improper fractions into mixed numbers, so for our next trick, we're going to convert mixed numbers into improper fractions. You need to follow a two-step process: 1. Multiply the whole number by the denominator of the fraction. 2. Add that number (or product) to the numerator of the fraction.
N EXT S TEP
Converting improper fractions into mixed numbers and mixed numbers into improper fractions are skills you'll be using for the rest of this section. When you're confident that you've mastered these skills, go on to the next lesson. But any time you're not sure you've really got something down, just go back over it. Remember that you're covering a whole lot of math in just 30 days.
42
LESSON 2
First, you're going to be adding fractions with the same denominators, and then you'll move on to fractions with different denominators. When you have completed this lesson, you'll be able to add any fractions and find the right answer.
A DDING F RACTIONS
D
o you have any loose change? I'd like to
borrow a quarter. Thanks. Do you happen to have another quarter I could borrow? Don't worry, it's just a loan. And while you're at it, let me borrow still another quarter. All right, then, how many quarters do I owe you? If I borrowed one quarter from you, then another quarter, and then still another quarter, I borrowed three quarters from you. In other words I borrowed 1 + 1 + 1 , or a total of 3 . 4 4 4 4 Now before I forget, let me return those three quarters.
43
FRACTIONS
ADDING FRACTIONS WITH COMMON DENOMINATORS
Here's another question: How much is
1 10
+
1 10
+
1 10 ?
It's
3 10 .
And how
much is 2 + 2 + 2 + 2 ? Go ahead and add them up. It's 8 . When you add 9 9 9 9 9 fractions with the same denominator, all you have to do is add the numerators. How much is 1 + 1 + 1 ? It's 3 . But we can reduce that to 1 . What did 6 6 6 6 2 we really do just then? We divided the numerator (3) by 3 and we divided the denominator (6) by 3. There's a law of arithmetic that says when you divide the top of a fraction by any number, you must also divide the bottom of that fraction by the same number. Now add together
1 2
Did you reduce all your fractions to their lowest possible terms? If you left problem 1 at 5 , is it wrong? No, but by convention we always reduce 5 our fractions as much as possible. Indeed, there are mathematicians who can't go to sleep at night unless they're sure that every fraction has been reduced to its lowest possible terms. Now I'm sure that you wouldn't want to keep these poor people up all night, so always reduce your fractions.
ADDING FRACTIONS WITH UNLIKE DENOMINATORS
So far we've been adding fractions with common denominators—halves, quarters, sixths, tenths, and so forth. Now we'll be adding fractions that don't have common denominators. Have you ever heard the expression, "That's like adding apples and oranges?" You can add apples and apples—3 apples plus 2 apples equal 5 apples. And you can add oranges—4 oranges plus 3 oranges equal 7 oranges. But you can't add apples and oranges. Can you add
1 2
and 1 ? Believe it or not, you can. The problem is that 3
they don't have a common denominator. In the last problem set the fractions in each problem had a common denominator. In problem 1 you added
1 5
+
you added + 1 + 2 + 2 . 8 8 8 What we need to do to add 1 and 1 is to give them a common denom2 3 inator. Do you have any ideas? Think about it for a while. All right, time's up! Did you think of converting 1 into 3 ? And 1 into 2 6 3 2 1×3 1×2 3 2 5 6 ? Here's how you could do it: 2 × 3 + 3 × 2 = 6 + 6 = 6 . Remember that old arithmetic law: What you do to the bottom of a fraction (the denominator), you must also do to the top (the numerator). Once the fractions have a common denominator, you can add them: 3 2 5 6 + 6 = 6. Try your hand at adding the following two fractions.
45
In problem 9, if you did it the way I did it below, it's okay. By not finding the lowest common denominator, you needed to do an extra step— which doesn't matter if you ended up with the right answer.
1 6
+
1 4
=
1×4 6×4
+
1×6 4×6
=
4 24
+
6 24
=
10 24
=
5 12
46
ADDING FRACTIONS
ADDING SEVERAL FRACTIONS TOGETHER
So far we've been adding two fractions. Can we add three or four fractions the same way? We definitely can—and will. See what you can do with this one: Problem: Solution:
1 4 1 4
SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS
Let's step back for a minute and take stock. When we added fractions with different denominators, we found their common denominators and added. We do the same thing, then, when we do subtraction with fractions having different denominators. Problem: How much is
50
1 3
Remember the shortcut we took a few pages ago when we added fractions? We can apply that same shortcut when we subtract fractions. Let's use it for problem 12:
1 6
–
1 8
==
4 24
–
3 24
=
1 24
MORE SUBTRACTION PRACTICE
Now let's try some more complicated subtraction problems. Since you can now subtract fractions which have the number 1 as the numerator, you're ready to try fractions that don't have 1 as the numerator. You'll see that it's the same procedure, but it just takes a few more steps. Problem: Subtract Solution:
2 5 1 6
N EXT S TEP
Believe it or not, you've done all the heavy lifting in this section. As long as you're sure you know how to add and subtract fractions, multiplying and dividing fractions should be a walk in the park.
53
LESSON 4
If you know how to multiply, then you know how to multiply fractions. Basically, all you do is multiply the numerators by the numerators and the denominators by the denominators.
M ULTIPLYING F RACTIONS
Y
ou'll find that multiplying fractions is
different from adding and subtracting them because you don't need to find a common denominator before you do the math operation. Actually, this makes multiplying fractions easier than adding or subtracting them.
EASY MULTIPLICATIONS
How much is one-eighth of a quarter? This is a straightforward multiplication problem. So let's set it up. Problem: Write down one-eighth as a fraction. Then write down one-quarter.
55
FRACTIONS
Solution: Your fractions should look like this: 1 , 1 . 8 4 Problem: The final step is to multiply them. Give it a try and see what you come up with. Solution:
1 8
×
1 4
=
1 32
A nice thing about multiplying fractions is that it's not necessary to figure out a common denominator, because you'll find it automatically. But is the 32 in the previous problem the lowest common denominator? It is, in this case. Later in this lesson, you'll find that when you multiply fractions, you can often reduce your result to a lower denominator. But for now, let's try another problem that doesn't require you to reduce. Problem: How much is one-third of one-eighth? Solution:
1 3
×
1 8
=
1 24
You can see by the way I'm asking the question that of means multiply, or times. The question would be the same if I said, "How much is one-third times one-eighth?"
P ROBLEM S ET
Try these problems, keeping in mind what the word of means in the following questions.
1. 2. 3. 4. 5. 6. 56
MORE CHALLENGING PROBLEMS
Now that you can multiply fractions that have the number 1 as the numerator, you are ready to tackle these more complicated problems. Problem: How much is three-fifths of three-quarters? Solution:
3 5
×
3 4
=
9 20
Problem: How much is two-thirds of one-quarter? Solution:
2 3
×
1 4
=
2 12
=
1 6
Problem: How much is a quarter of two-thirds? Solution:
1 4
×
2 3
=
2 12
=
1 6
Did you notice what you just did in the last two problems? You just did the same problem and came up with the same answer. So two-thirds of one-quarter comes out the same as one-quarter of two-thirds. When you multiply proper fractions, you get the same answer regardless of the order in which you place the numbers. This is true of any type of multiplication problem.
57
FRACTIONS
P ROBLEM S ET
These problems are a bit more complicated than the first problem set in this lesson.
7. How much is four-fifths of one-half? 8. How much is nine-tenths of one-eighth? 9. How much is four-sevenths of two-thirds? 10. How much is eight-ninths of three-quarters? 11. How much is five-sixths of four-fifths? 12. How much is three-eighths of four-ninths?
Solutions
7. 8. 9. 10. 11. 12.
4 5 9 10 4 7 8 9 5 6 3 8
× × ×
1 2 1 8 2 3
= = =
4 10 9 80 8 21 24 36 20 30 12 72
=
2 5
×3 = 4 × ×
4 5 4 9
= = =
2 3 2 3 1 6
= =
SHORTCUT: CANCELING OUT
When you multiply fractions, you can often save time and mental energy by canceling out. Here's how it works. Problem: How much is Solution:
58
5 6 5 6
× 3? 4
5 8
×
3 4
= 25 × 3 = 6 4
1
MULTIPLYING FRACTIONS
In this problem we performed a process called canceling out. We divided the 6 in 5 by 3 and we divided the 3 in 3 by 3. In other words, the 6 4 3 in the 6 and the 3 in the 3 canceled each other out. Try to cancel out the following problem. Problem: How much is Solution:
2 3 2 3
× 1? 2
1
×
1 21
=
1 3
Canceling out helps you reduce fractions to their lowest possible terms. While there's no law of arithmetic that says you have to do this, it makes multiplication easier, because it's much easier to work with smaller numbers. For example, suppose you had this problem: Problem:
17 20
×
5 34
=
Solution: There are two ways to solve this:
1. 2.
17 20
×
1
5 34
=
1
85 680
=
1 8
17 136
=
1 8
17 4 20
5 × 2 34 =
Obviously, the second version is much easier.
P ROBLEM S ET
In this problem set, see if you can cancel out before you multiply. If you're not comfortable doing this, then carry out the multiplication without canceling out. As long as you're getting the right answers, it doesn't matter whether or not you use this simplification tool.
13. 14. 15. 16.
7 16 4 9 14 32 15 24
× × × ×
8 21 3 20 16 21 9 10
= = = =
59
FRACTIONS
17. 18.
13 20 9 42
× ×
5 39 7 18
= =
Solutions
13. 14. 15. 16. 17. 18.
7 2 16 4 39
1
8 × 3 21 =
1
1 6 1 15 2 6
1
3 × 5 20 =
1
2 14 2 32
15 8 24 13 4 20 9 6 42
× 3 16 = 21
9 × 2 10 = 5 × 3 39 = 7 × 2 18 =
1
=
1 3
3
3
9 16
1
1
1 12 1 12
1
1
N EXT S TEP
So I didn't lie. Multiplying fractions is pretty easy. Dividing fractions, which we take up next, is virtually the same as multiplying fractions, except for one added step.
60
LESSON 5
This lesson explains what a reciprocal is and shows you how to use it to solve fraction division problems. You'll also learn how to do division problems in the proper order so you get the right answer the first time.
D IVIDING F RACTIONS
T
he division of fractions is just like multi-
plication, but with a twist. You'll find the trick is to turn a division problem into a multiplication problem. Let's get right into it. How much is one-third divided by one-half? Don't panic! The trick to doing this is to convert it into a multiplication problem. Just multiply one-third by the reciprocal of one-half. What did I say? The reciprocal of a fraction is found by turning the fraction upside down. So 1 becomes 2 . With all this information, see if you can figure out 2 1 the following problems. Problem: How much is one-third divided by one-half? Solution:
1 3
÷
1 2
=
1 3
×
2 1
=
2 3
61
FRACTIONS
Problem: How much is Solution:
1 4
1 4
divided by 1 ? 6 =
3 2
÷
1 6
= 21 × 4
6 1
3
= 11 2
Let's just stop here for a minute. In the last problem we converted an improper fraction, 3 , into a mixed number, 1 1 . I mentioned earlier how 2 2 mathematicians just hate fractions that are not reduced to their lowest terms— 4 must be reduced to 2 , and 4 must be reduced to 1 . Another 6 3 8 2 thing that really bothers them is leaving an improper fraction as an answer instead of converting it into a mixed number. See if you can work out this problem: Problem: Solution:
1 3 1 3
÷ ÷
1 4 1 4
= =
1 3
×
4 1
=
4 3
= 11 3
Remember that, whenever you need to convert an improper fraction into a mixed number, you just divide the denominator (bottom number) into the numerator (top number). If you need to review this procedure, turn back to Lesson 1, near the beginning of this section.
THE ORDER OF THE NUMBERS
When you multiply two numbers, you get the same answer regardless of their order. For example, 1 × 1 gives you the same answer as 1 × 1 . 2 3 3 2
1 2 1 3
× ×
1 3 1 2
= =
1 6 1 6
When you divide one number by another, does it matter in which order you write the numbers? Let's find out. Problem: How much is Solution:
1 3 1 3
divided by 1 ? 4
4 3
÷
1 4
=
1 3
×
4 1
=
= 11 3
1 4
Problem: Now how much is Solution:
1 4
divided by 1 ? 3
÷
1 3
=
1 4
×
3 1
=
3 4
There is no way that 3 can equal 1 1 . So when you do division of frac4 3 tions, you must be very careful about the order of the numbers. The number that is being divided always comes before the division sign, and
63
FRACTIONS
the number doing the dividing always comes after the division sign. Here are some examples. Problem: How much is Solution:
1 5 1 5
MATHEMATICAL OBSERVATIONS
Take a look at the problem set you just did and make a few observations. In problem 9 you divided 5 by 5 and got an answer of 1. Any number 8 8 divided by itself equals 1. Problem: Try dividing Solution:
3 4 3 4
Our answer was 1 3 . You can generalize: When you divide a number by a 5 smaller number, the answer (or quotient) will be greater than 1. Final observation: In problem 10 you divided 2 by a larger number, 4 . 7 5 Your answer was Your answer was
5 1 5 14 . In problem 11 you divided 4 by a larger number, 8 . 2 5 . Here's the generalization: When you divide a number
by a larger number, the number (or quotient) will be less than 1.
N EXT S TEP
At this point you should be able to add, subtract, multiply, and divide fractions. In the next lesson, we're going to throw it all at you at the same time.
65
LESSON 6
In this lesson, you'll use everything you've learned so far in this entire section. So before going any further, make sure you know what you need to know about improper fractions and mixed numbers (Lesson 1), and about adding, subtracting, multiplying, and dividing proper fractions (Lessons 2, 3, 4, and 5, respectively).
W ORKING WITH I MPROPER F RACTIONS
D
o you really need to know how to add,
subtract, multiply, and divide improper fractions? Yes! In the very next lesson, you'll need to convert mixed numbers into improper fractions before you can do addition, subtraction, multiplication, and division. I know that you can hardly wait.
ADDING WITH COMMON DENOMINATORS
Let's start by adding two improper fractions with common denominators. Problem: How much is Solution:
12 5 12 5
+ 7? 5
+
7 5
=
19 5
= 34 5
67
FRACTIONS
You'll notice I converted the improper fraction, 19 , into a mixed 5 4 number, 3 5 . By convention, you should make this conversion with your answers. Now try another problem. Problem: Solution:
5 3 5 3
N EXT S TEP
Now that you know how to add, subtract, multiply, and divide improper fractions, you'll be using that skill to perform the same tricks with mixed numbers. The only additional trick you'll need to do is to convert mixed numbers into improper fractions and improper fractions into mixed numbers. If you don't remember how to do this, you'll need to go back and look at Lesson 1 again.
73
LESSON 7
Remember mixed numbers? (Right, a mixed number is a whole number plus a fraction.) This lesson will show you how to add, subtract, multiply, and divide mixed numbers. You'll have to convert mixed numbers into improper fractions first, so make sure you're up on your "Fraction Conversions" (Lesson 1).
W ORKING WITH M IXED N UMBERS
I
n this lesson you'll put together everything
you've learned so far about fractions. In order to perform operations on mixed numbers, you'll be following a three-step process: 1. Convert mixed numbers into improper fractions. 2. Add, subtract, multiply, or divide. 3. Convert improper fractions into mixed numbers. You'll add one step here to what you did in the previous lesson. Before you can add, subtract, multiply, or divide mixed numbers, you need to convert them into improper fractions. Once you've done that, you can do exactly what you did in Lesson 6.
It might have occurred to you that there's another way to do these problems. You could add whole numbers, add fractions, and then add them together, carrying where necessary. In other words, you could do problem 1 like this: Problem: 3 4 + 2 7 = 5 8 Solution: 3 + 2 +
4 5
+
7 8
=5+
32 40
+
35 40
= 5 67 = 6 27 40 40
The problem with this method is that you might forget to carry. So stick with my method of converting to improper fractions.
Sometimes we don't need to know the exact answer. All we really need is a fast estimate. In problem 10, we can quickly estimate our answer as between 2 and 3. In problem 11, our answer will be just a bit over 3. And in problem 12, the answer is going to be a little less than 7.
N EXT S TEP
You may have been doing most or all of these problems more or less mechanically. In the next lesson, you're going to have to think before you add, subtract, multiply, or divide. In fact, you're going to have to think about whether you're going to add, subtract, multiply, or divide. You'll get to do this by applying everything that you have learned so far.
80
LESSON 8
This lesson gives you the opportunity to put your fraction knowledge to work. It's now time to apply everything you've learned so far in this section to realworld problems. Consider this lesson to be a practical application of all the principles you've learned in the fractions section.
A PPLICATIONS
W
hile no new material will be covered
in this lesson, the math problems will be stated in words, and you'll need to translate these words into addition, subtraction, multiplication, and division problems, which you'll then solve. In the mathematical world, this type of math problem is often called a word problem.
P ROBLEM S ET
Do all of the problems on the next pages, and then check your work with the solutions that follow.
81
FRACTIONS
1. One morning you walked 4 7 miles to town. On the way home, 8
you stopped to rest after walking 1 1 miles. How far do you still 3 need to walk to get home?
2. To do an experiment, Sam needed
1 12
of a gram of cobalt. If Eileen
gave him
1 4
of that amount, how much cobalt did she give Sam?
3. In an election, the Conservative candidate got one-eighth of the
votes, the Republican candidate got one-sixth of the votes, and the Democratic candidate got one-third of the votes. What fraction of the votes did the three candidates receive all together?
4. When old man Jones died, his will left two-thirds of his fortune to
his four children, and instructed them to divide their inheritance equally. What share of his fortune did each of his children receive?
5. Kerry is 4 feet 4 1 inches tall, and Mark is 4 feet 2 7 inches tall. How 4 8
much taller is Kerry than Mark?
6. If you want to fence in your square yard, how much fencing
would you need if your yard is 21 2 feet long? Remember that a 3 square has four equal sides.
7. Ben and seven other friends bought a quarter share of a restau-
rant chain. If they were equal partners, what fraction of the restaurant chain did Ben own?
8. If it rained 1 1 inches on Monday, 2 1 inches on Tuesday, 2 8
of an inch on Wednesday, and 2 5 inches on Thursday, how much did it 8 rain over the four-day period?
3 4
9. If four and a half slices of pizza were divided equally among six
people, how much pizza does each person get?
10. If four and three-quarter pounds of sand can fit in a box, how
many pounds of sand can fit in six and a half boxes?
82
APPLICATIONS
11. Ben Wallach opened a quart of orange juice in the morning. If he
drank 1 of it with breakfast and 2 of it with lunch, how much of 5 7 it did he have left for the rest of the day?
12. If Kit Hawkins bought
of 1 share in a company, what fraction of 8 the company did she own?
1 3
13. At Elizabeth Zimiles' birthday party, there were four cakes. Each
guest ate 1 of a cake. How much cake was left over if there were 8 20 guests?
14. Suppose it takes 2 1 yards of material to make one dress. How 4
many dresses could be made from a 900-yard bolt of material?
15. Max Krauthammer went on a diet and lost 4 1 pounds the first 2
week, 3 1 the second week, 3 1 the third week, and 2 3 the fourth 2 4 4 week. How much weight did he lose during the four weeks he dieted?
16. Sam Retchnick is a civil servant. He earns a half day of vacation
time for every two weeks of work. How much vacation time does he earn for working 6 1 weeks? 2
17. Goodman Klang has been steadily losing 1 1 pounds a week on his 2
diet. How much weight would he lose in 10 1 weeks? 2
18. Karen, Jeff, and Sophie pulled an all-nighter before an exam. They
ordered 2 large pizzas and finished them by daybreak. If Karen had 2 of a pie and Jeff had 3 , how much did Sophie have? 3 4
19. If four dogs split 6 1 cans of dog food equally, how much would 2
hours on Wednesday, 9 hours on Thursday, and took Friday off, how many hours did he work that week?
83
FRACTIONS
21. Sal and Harry are drinking buddies. On Saturday night they
chipped in for a fifth of bourbon. They shared the bottle for the next two hours. If Harry consumed 3 of it and Sam consumed 2 , 8 5 how much of the bottle of bourbon was left?
Solutions
1. 4 7 – 1 1 = 8 3 2. 3. 4.
1 12 1 8 2 3 39 8
–
4 3
=
39 × 3 8×3
–
4×8 3×8
=
117 24
–
32 24
=
85 24
= 3 13 miles 24
×
1 4
=
1 48
of a gram
1×3 8×3
+1+1= 6 3 ÷
4 1
+
1 6
1×4 6×4
+
1×8 3×8
=
3 24
+
4 24
+
8 24
=
15 24
= 5 of the votes 8
=
2 3
1
× 21 = 4
17 4 4 1
of the fortune
17 × 2 4×2 260 3
5. 4 1 – 2 7 = 4 8 6. 4 × 21 2 = 3 7.
1 4
– ×
23 8 65 3
= =
–
23 8
=
34 8
–
23 8
=
11 8
= 1 3 inches 8
= 86 2 feet 3
×
1 8
=
1 32
of the restaurant chain
3 4
8. 1 1 + 2 1 + 2 8
+ 25 = 8
6 8
3 2
+
56 8
17 8
+
3 4
+
21 8
=
3×4 2×4
+
17 8
+
3×2 4×2
+
21 8
=
12 8
+
17 8
+
9 2
+
6 1
21 8 9 2
=
3
= 7 inches
3 4
9. 4 1 ÷ 6 = 2
÷
19 4
=
13 2
× 21 = 6
247 8
of a slice
10. 4 3 × 6 1 = 4 2
×
=
= 30 7 pounds of sand 8
2×5 7×5) 7 = 1 – ( 35 + 10 35 )
11. 1 – ( 1 + 2 ) = 1 – ( 1 × 7 + 5 7 5×7
=1–
17 35
=
12.
1 3
35 35
–
1 8
17 35
=
18 35
quart
×
=
1 24
of the company
20 8
13. 4 – (20 × 1 ) = 4 – ( 20 × 1 ) = 4 – 8 1 8
=
32 8
–
20 8
=
12 8
= 1 4 = 1 1 cakes 8 2
84
APPLICATIONS
14. 900 ÷ 2 1 = 4
900 1
÷
9 4
=
900 1 9 2
100
× 1 4 = 400 dresses 9
13 4
15. 4 1 + 3 1 + 3 1 + 2 3 = 2 2 4 4
+
7 2
+
+
11 4
=
9×2 2×2
+
7×2 2×2
+
13 4
+
11 4
=
18 4
+
14 4
+
13 4 1 2
+
11 4
=
56 4
= 14 pounds
16. If Sam earns
1 4
day for 2 weeks, then he earns 1 day for one week. 4 =
21 2 13 8
× 61 = 2
1 4
×
3 2
13 2
= 1 5 of a day 8
63 4
17. 1 1 × 10 1 = 2 2
×
=
= 15 3 pounds 4
3×3 4×3) 8 = 2 – ( 12 + 9 12 )
18. 2 – ( 2 + 3 ) = 2 – ( 2 × 4 + 3 4 3×4
=2–
17 12
=
2 × 12 1 × 12
–
17 12
=
24 12
–
17 12
=
13 2
7 12
of a pie
4 1
19. 6 1 ÷ 4 = 2
÷
=
13 2
×
1 4
=
33 4
13 8
= 1 5 cans 8
31 4 × + 9 = 19× 22 + 2 33 4
20. 9 1 + 8 1 + 7 3 + 9 = 2 4 4
19 2
+
138 4
+
+
31 4
+
36 4
=
38 4
+
33 4
+
31 4
+
36 4
=
= 34 2 hours 4
2×8 5×8)
21. 1 – ( 3 + 2 ) = 1 – ( 3 × 5 + 8 5 8×5
= 1 – ( 15 + 40
16 40 )
=1–
31 40
=
40 40
–
31 40
=
9 40
of the bottle
If you bought 100 shares of Microsoft at 109 3 and sold them at 116 3 , 4 8 how much profit would you have made? (Don't worry about paying stockbrokers' commissions.)
Solution
When a stock has a price of 109 3 , it is selling at $109.75, or 109 and 4
3 4
dollars. A fast way of working out this problem is to first look at the difference between 109 3 and 116 3 . Let's ask ourselves the question, how 4 8 much is 16 3 – 9 3 ? (We'll add on the 100 later.) 8 4
85
FRACTIONS
16 3 – 9 3 = 8 4
131 8
–
39 4
=
131 8
– 78 = 8
53 8
= 6 5 , or $6.625. 8
That's the profit you made on one share. Since you bought and sold 100 shares, you made a profit of $662.50. This problem could also be worked out with decimals, which we'll do at the end of Lesson 13.
N EXT S TEP
How are you doing so far? If you're getting everything right, or maybe just making a mistake here and there, then you're definitely ready for the next section. Two of the things you'll be doing are converting fractions into decimals and decimals into fractions. So before you start the next section, you need to be sure that you really have your fractions down cold. If you'd be more comfortable reviewing some or all of the work in this section, please allow yourself the time to do so.
86
SECTION III
D ECIMALS
W
hat's a decimal? Like a fraction, a
decimal is a part of one. One-half, or 1 , can be written as 2 the decimal 0.5. By convention, decimals of less than 1 are preceded by 0. Now let's talk about the decimal 0.1, which can be expressed as one1 tenth, or 10 . Every decimal has a fractional equivalent and vice versa. And as you'll discover in this section, fractions and decimals also have percent equivalents. Later in the section, you'll be converting tenths, hundredths, and thousandths from fractions into decimals and from decimals into fractions. And believe it or not, you'll be able to do all of this without even using a calculator.
87
DECIMALS
When you have completed this lesson, you will know how to add, subtract, multiply, and divide decimals and convert fractions into decimals and decimals into fractions. You'll also see that the dollar is based on fractions and decimals.
88
LESSON 9
In this lesson, you'll learn how to add and subtract numbers that are decimals. You'll also discover the importance of lining up the decimal points correctly before you begin to work a decimal problem.
A DDING AND S UBTRACTING D ECIMALS
I
f you spent $4.35 for a sandwich and $0.75
for a soda, how much did you spend for lunch? That's a decimal addition problem. If you had $24.36 in your pocket before lunch, how much did you have left after lunch? That's a decimal subtraction problem. Adding and subtracting decimals is just everyday math. When you're adding and subtracting decimals, mathematically speaking, you're carrying out the same operations as when you're adding and subtracting whole numbers. Just keep your columns straight and keep track of where you're placing the decimal in your answers.
ADDING DECIMALS
Remember to be careful about lining up decimal points when adding decimals. These first problems are quite straightforward.
89
Problem: Suppose you drove across the country in six days. How much was your total mileage if you went these distances: 462.3 miles, 507.1 miles, 482.0 miles, 466.5 miles, 510.8 miles, and 495.3 miles? Solution: 462.3 507.1 482.0 466.5 510.8 + 495.3 2,924.0
32 2
Problem: It rained every day for the last week. You need to find the total rainfall for the week. Here's the recorded rainfall: Sunday, 1.22 inches; Monday, 0.13 inches; Tuesday, 2.09 inches; Wednesday, 0.34 inches; Thursday, 0.26 inches; Friday, 1.88 inches; and Saturday, 2.74 inches.
90
ADDING AND SUBTRACTING DECIMALS
Solution:
1.22 0.13 2.09 0.34 0.26 1.88 + 2.74 8.66
2 3
In this last problem, you probably noticed the recorded rainfall for Monday (0.13), Wednesday (0.34), and Thursday (0.26) began with a zero. Do you have to place a zero in front of a decimal point? No, but when you're adding these decimals with other decimals that have values of more than 1, placing a zero in front of the decimal point not only helps you keep your columns straight, but it also helps prevent mistakes. Here's a set of problems to work out.
P ROBLEM S ET
Add each of these sets of numbers. Two sets are printed across, so you can practice aligning the decimal points in the correct columns.
1.
SUBTRACTING DECIMALS
Are you ready for some subtraction? Subtracting decimals can be almost as much fun as adding them. See what you can do with this one. Problem: 4.33 – 2.56 4. 3 3 – 2.56 1.77
3 12 1
Problem: The population of Mexico is 78.79 million, and the population of the United States is 270.4 million. How many more people live in the United States than in Mexico? Solution:
1 16 9 13 1
2 7 0.40 – 78.79 191.61
That was a bit of a trick question. I wanted you to add a zero to the 270.4 million population of the United States. Why? To make the subtraction easier and to help you get the right answer. Adding the zero makes it easier to line up the decimal points—and you have to line up the decimal points to get the right answer. You are allowed to add zeros to the right of decimals. You could have made 270.4 into 270.40000 if you wished. The only reason you add zeros is to help you line up the decimal points when you do addition or subtraction. Let's try one more. Problem: Kevin scored 9.042 in gymnastics competition, but 0.15 points were deducted from his score for wearing the wrong sneakers. How much was his corrected, or lowered, score? Solution: 9.0 42 – 0.150 8.892
8 91
Again, you added a 0 after 0.150 so you could line it up with 9.042 easily.
93
DECIMALS
P ROBLEM S ET
Carry out each of these subtraction problems. You'll have to line up the last problem yourself.
5.
121.06 – 98.34 709.44 – 529.65
7.
812.71 – 626.78 Subtract 39.48 from 54.35.
6.
8.
Solutions
5.
1 2 1.06 – 98.34 22.72 70 9.44 – 529.65 179.79
6 9 18 13 1
11 10 1
7.
8 12.71 – 626.78 185.93 54.3 5 – 39.48 14.87
4 13 12 1
7 10 11 16 1
6.
8.
N EXT S TEP
Before you go on to the next lesson, I want you to ask yourself a question: "Self, am I getting all of these (or nearly all of these) problems right?" If the answer is yes, then go directly to the next lesson. But if you're having any trouble with the addition or subtraction, then you need to go back and redo Review Lessons 1 and 2 in Section I. Once you've done that, start this chapter over again, and see if you can get everything right.
94
LESSON 10
You'll learn how to multiply decimals in this lesson. You'll find out that the big trick is to know where to put the decimal point in your answer. If you can count from 1 to 6, then you can figure out where the decimal goes in your answer.
M ULTIPLYING D ECIMALS
W
hen you multiply two decimals that
are both smaller than 1, your answer, or product, is going to be smaller than either of the numbers you multiplied. 1 1 Let's prove that by multiplying the two fractions, 10 × 10 . Our answer is 1 100 . Similarly, if we multiply 0.1 × 0.1, we'll get 0.01, which may be read as one one-hundredth. When you have completed this lesson, you'll be able to do problems like this in your sleep. The only difference between multiplying decimals and multiplying whole numbers is figuring out where to place the decimal point. For instance, when you multiply 0.5 by 0.5, where do you put the decimal in your answer?
95
DECIMALS
You know that 5 × 5 = 25. So how much is 0.5 × 0.5? Is it 0.025, 0.25, 2.5, 25.0, or what? Here's the rule to use: When you multiply two numbers with decimals, add the number of decimal places to the right of the decimal point for both numbers, and then, starting from the right, move the same number of places to find where the decimal point goes in your answer. That probably sounds a lot more complicated than it is. Let's go back to 0.5 × 0.5. How many numbers are after the decimal points? There are two numbers after the decimal points: .5 and .5. Now we go to our answer and place the decimal point two places from the right, at 0.25. When you get a few more of these under your belt, you'll be able to do them automatically. Problem: How much is 0.34 × 0.63? Solution: .34 × .63 102 204_ .2142
( (
( ( ( (
How many numbers follow the decimals in 0.34 and 0.63? The answer is four. So you start to the right of 2142. and go four places to the left: 0.2142. Problem: How much is 0.6 × 0.58? Solution: .58 × .6 .348
How many numbers follow the decimals in 0.6 and 0.58? The answer is three. So you start to the right of 348. and go three places to the left: 0.348. Here's one that may be a little harder.
( ( (
Problem: Multiply 50 by 0.72.
96
MULTIPLYING DECIMALS
Solution:
50 × .72 1 00 35 0_ 36.00
Again, how many numbers follow the decimal in 0.72? Obviously, two. There aren't any numbers after the decimal point in 50. Starting to the right of 3600. we move two places to the left: 36.00. This next one is a little tricky. Just follow the rule for placing the decimal point and see if you can get it right. Problem: .17 × .39 Solution:
( (
.17 × .39 153 51_ 663
It looks like I'm stuck. The decimal point needs to go four places to the left. But I've got only three numbers in my answer. So what do I do? What I need to do is place a zero to the left of 663 and then place my decimal point: 0.0663. (I also added the zero that ends up to the left of the decimal point.) Let's try one more of these. Problem:
( ( ( (
.22 × .36
Solution:
.22 × .36 132 __66_ .0792
P ROBLEM S ET
You can easily get the hang of multiplying decimals by working out more problems. So go ahead and do this problem set.
You may have noticed in problem 4 that your answer had a couple of excess zeros, 61.2900. These zeroes can be dropped without changing the value of the answer. So the answer is written as 61.29.
98
CHAPTER TITLE
A very common mistake is putting a decimal point in the wrong place. One shortcut to getting the right answer, while avoiding this mistake, is to do a quick approximation of the answer. For example, in problem 4, we're multiplying 6.75 by 9.08. We know that 6 × 9 = 54, so we're looking for an answer that's a bit more than 54. Does 6.129 look right to you? How about 612.900? Clearly, the answer 61.2900 looks the best.
LESSON 11
In this lesson, you'll learn how to divide decimals. You'll find out that the only difference between dividing decimals and dividing whole numbers is figuring out where to place the decimal point.
D IVIDING D ECIMALS
O
4.0 0.5
ne thing to remember when you're
dividing one number by another that's less than 1 is that your answer, or quotient, will be larger than the number divided. For example, if you were to divide 4.0 by 0.5, your quotient would be more than 4.0. We'll come back to this problem in just a minute. Instead of applying an arithmetic rule as we did when we multiplied decimals, we'll just get rid of the decimals in the divisor (the number by which we divide) and do straight division. I'll work out the first problem to show you just how easy this is. How much is 4.0 divided by 0.5? Let's do it. Let's set it up as a fraction to start:
101
DECIMALS
Next we'll move the decimal of the numerator one place to the right, and we'll move the decimal of the denominator one place to the right. We can do this because of that good old law of arithmetic that I mentioned earlier: Whatever you do to the top (numerator) you must also do to the bottom (denominator) and vice versa. So we'll multiply the numerator by 10 and the denominator by 10 to get the decimal place moved over one place to the right. 4.0 × 10 40. 0.5 × 10 = 05. Then we do simple division: 40 5 =8 You'll notice that 8 (the answer) is larger than 4 (the number divided). Whenever you divide a number by another number less than 1, your quotient, or answer, will be larger than the number you divided. How much is 1.59 divided by 0.02? Would you believe that that's the same question as: How much is 159 divided by 2? The problem can be written this way: 1.59 .02 Then let's multiply the top and bottom of this fraction by 100. In other words, move the decimal point of the numerator two places to the right, and move the decimal point of the denominator two places to the right: 1.59 × 100 159 .02 × 100 = 2 We've just reduced the problem to simple division: 79.5 159.0
2
You'll notice that I added a 0 to 159. By carrying out this division one more decimal place, I avoided leaving a remainder. However, it would have been equally correct to have an answer of 79 with a remainder of 1, or, for that matter, 79 1 . 2
102
DIVIDING DECIMALS
Here's one for you to work out. Problem: How much is 10.62 divided by 0.9? Solution: .9 10.62 = 11.8 9.
(
106.2
(
1 7
In this problem, you needed to multiply by 10, so you moved the decimal point one place to the right. You multiplied 0.9 by 10 and got 9. Then you multiplied 10.62 by 10 and got 106.2. Then you divided. Very good! See how you can handle this one. Problem: Divide 0.4 by 0.25. Solution: .25 .4 = 25.
( (
40. = 25
40
= 5
1.6 3 8.0
Here's one that may be a bit harder. Problem: How much is 92 divided by 0.23? Solution: .23 92 = 23.
( (
( (
9200.
( (
400 = 23 9200 – 92XX 0
P ROBLEM S ET
Since practice makes perfect in math, let's get in some more practice dividing with decimals. See if you can get all these problems right.
1.
LESSON 12
After you convert fractions into decimals and then decimals into fractions, you'll be ready to add, subtract, multiply, and divide tenths, hundredths, and thousandths.
D ECIMALS AND F RACTIONS AS T ENTHS , H UNDREDTHS , AND T HOUSANDTHS
D
ecimals can be expressed as fractions,
and fractions can be expressed as decimals. For example, one1 tenth can be written as a fraction, 10 , or as a decimal, .1 (or 0.1). I'll show you how to do these conversions. We'll start out with tenths and hundredths; then we'll move into the thousandths.
TENTHS AND HUNDREDTHS
Can you express the number three-tenths as a fraction and as a decimal? How about forty-five one-hundredths? I'll tell you that forty-five one45 hundredths = 100 = 0.45. How much is three one-hundredths as a fraction and as a decimal? 3 Three one-hundredths = 100 = 0.03. Now see if you can do the problem set on the following page.
107
DECIMALS
P ROBLEM S ET
Express each of these numbers as a fraction and as a decimal.
1. 2. 3. 4. 5. 6.
If you didn't put a zero before the decimal point as shown in the above solutions, were your answers wrong? No. It's customary to put the zero before the decimal point for clarity's sake, but it's not essential to do so.
THOUSANDTHS
Let's move on to thousandths. See if you can write the number three hundred seventeen thousandths as a fraction and as a decimal. Yes, it is 317 1000 = 0.317. Problem: Write the number forty-one thousandths as a fraction and as a decimal. Solution:
41 1000
MULTIPLYING THOUSANDTHS
Multiplying thousandths is really the same as multiplying tenths and hundredths. Let's see if you remember. Work out this problem and be very careful where you place the decimal point. Problem: 1.375 × 9.084 Solution: 1.375 × 9.084 5500 11000_ 12 3750__ 12.490500
The multiplication gives you a product of 12490500. Since there are three numbers after the decimal point of 1.375 and three numbers after the decimal point of 9.084, you need to place the decimal point of your answer six places from the right of 12490500. Moving six places to the left, you get an answer of 12.490500, or 12.4905.
Again, you need to move your decimal point six places to the left of your product. Start at the extreme right and count six places to the left, which gives you an answer of 160.113973. Now I'd like you to do the following problem set.
I'd like you to take another look at the problem set you just did. It's very easy to put the decimal point in the wrong place in your answer, so I'd like to give you a helpful hint. That hint is to estimate your answer before you even do the multiplication. In problem 19, you would expect an answer that's somewhat more than 4 because you're multiplying a number somewhat larger than 4 by a number a bit larger than 1. So if you ended up with 55.72350 or 0.5572350, you can see that neither of those answers make sense. In problem 20, you would estimate your answer to be somewhat larger than 10 (since 5 × 2 = 10). We ended up with 11.702304, which certainly looks right. Go ahead and carry out this reality check on the answers to problems 21 through 24. And remember that when you're multiplying
113
DECIMALS
decimals, it really pays to estimate your answer before you even do the problem.
One day, when you fill up at a gas station, your car's odometer reads 28,106.3. The next time you fill up, your odometer reads 28,487.1. If you just bought 14.2 gallons of gas, how many miles per gallon did you get, rounded to the tenths place?
Solution
28,487.1 – 28,106.3 380.8 380.8 miles 14.2 gallons = 142
142
26.8 miles per gallon 3808.0 2840.0 968.0 –852.0 1160. –1136.
N EXT S TEP
How are you doing? If you're getting everything—or almost everything— right, then go directly to Lesson 13. But if you're having any trouble at all, then you'll need to review some of the material you've already covered. For instance, if you're having trouble adding or subtracting decimals, you'll need to review Lesson 9 as well as the second part of this lesson. If you're not doing well multiplying decimals, then you'll need to rework your way through Lesson 10 and the third part of this lesson. And if you're at all shaky on dividing decimals, then you'll need to review Lesson 11 and the last part of this lesson before moving on to the next lesson.
116
LESSON 13
When you've finished this lesson, you'll be able to convert a decimal into a fraction, which involves getting rid of the decimal point. You've already done some conversion of fractions into decimals. When you convert a fraction into a decimal, you're dividing the denominator into the numerator and adding a decimal point.
C ONVERTING F RACTIONS INTO D ECIMALS AND D ECIMALS INTO F RACTIONS
I
n the last lesson you expressed tenths and
hundredths as fractions and as decimals. Tenths and hundredths are easy to work with, but some other numbers are not as simple. You'll learn to do more difficult conversions in this lesson. Let's start by converting fractions into decimals. Then we'll move into expressing decimals as fractions.
FRACTIONS TO DECIMALS
How would you convert 17 into a decimal? There are actually two ways. 20 Remember the arithmetic law that says what we do to the top (numerator) of a fraction, we must also do to the bottom (denominator)? That's one way to do it. Go ahead and convert 17 into hundredths. 20
117
DECIMALS
17 × 5 20 × 5
=
85 100
= 0.85
Now let's use the second method to convert the fraction 17 into a deci20 mal. Are you ready? Every fraction can be converted into a decimal by dividing its denominator (bottom) into its numerator (top). Go ahead and divide 20 into 17. .85 20 17.00 – 16 0X 1 00 – 1 00 Problem: Use both methods to convert Solution:
19 × 2 50 × 2 19 50
into a decimal.
=
38 100
= 0.38
or
.38 19.00 50 – 15 0X 4 00 – 4 00 We've done two problems so far where we could convert the denominator to 100. But we're not always that lucky. See if you can convert the following fraction into a decimal. Problem: Convert 3 into a decimal. 8 Solution:
3 8
.375 = 8 3.0 0 0
64
Sometimes we have fractions that can be reduced before being converted into decimals. See what you can do with the next one. Problem: Convert
9 12
into a decimal.
118
CONVERTING FRACTIONS AND DECIMALS
Solution:
9 12
.75 =
3 4
4
3.00
2
It often pays to reduce a fraction to its lowest possible terms because that will simplify the division. It's easier to divide 4 into 3 than to divide 12 into 9. By the way, when I divided 4 into 3, I placed a decimal point after the 3 and then added a couple of zeroes. The number 3 may be written as 3.0, and we may add as many zeroes after the decimal point as we wish. Now let's see if you can handle this problem set.
DECIMALS TO FRACTIONS
You're going to catch a break here. Decimals can be converted into fractions in two easy steps. If it were a dance, we'd call it the easy two-step. First I'll do one. I'm going to convert the decimal 0.39 into a fraction. All I have to do is get rid of the decimal point by moving it two places to the 39 right, and then placing the 39 over 100: 0.39 = 100 .
119
DECIMALS
Here's a couple for you to do. Problem: Convert 0.73 into a fraction. Solution: 0.73 =
73 100
Problem: Now convert 0.4 into a fraction. Solution: 0.4 =
4 10
Since we like to convert fractions into their lowest terms, let's change into 2 . For tenths and hundredths, however, you don't necessarily have 5 to do this. It's the mathematical equivalent of crossing your t's and 4 dotting your i's. The fraction 10 is mathematically correct, but there are some people out there who will insist that every fraction be reduced to its lowest terms. Luckily for you, I am not one of them, at least when it comes to tenths and hundredths. Are you ready for a problem set? Good, because here comes one now.
4 10
Do you remember your profitable transaction with Microsoft stock? You bought 100 shares at 109 3 and sold them at 116 3 . Let's calculate your 4 8 profit, this time using decimals.
Solution: You paid $198.75 for each share, which you sold at $116.375.
$116.375 –109.750 $6.625 So you made a profit of $6.625 on each of 100 shares, or a total of $662.50.
N EXT S TEP
Now that you can convert fractions into decimals and decimals into fractions, you're ready to do some fast multiplication and division. Actually, in the next lesson, all you'll need to do is move around some decimal points and add or subtract some zeros.
122
LESSON 14
Doing fast multiplication and division can be a whole lot of fun. When you've completed this lesson, you'll be able to multiply a number by 1,000 in a fraction of a second and divide a number by 1,000 just as quickly.
FAST M ULTIPLICATION AND F AST D IVISION
W
ouldn't it be great to know some
math tricks, so you could amaze your friends with a speedy answer to certain math questions? There are shortcuts you can use when multiplying or dividing by tens, hundreds, or thousands. Let's start with some multiplication problem shortcuts.
MULTIPLYING WHOLE NUMBERS BY 10, 100, AND 1,000
Try to answer the next question as quickly as possible before you look at the solution. Problem: Quick, how much is 150 × 100?
123
DECIMALS
Solution: The answer is 15,000. What I did was add two zeros to 150. Problem: How much is 32 × 1,000? Solution: I'll bet you knew it was 32,000. So one way of doing fast multiplication is by adding zeros. Before we talk about the other way of doing fast multiplication, I'd like you to do this problem set.
MULTIPLYING DECIMALS BY 10, 100, AND 1,000
Multiplying decimals by 10, 100, and 1,000 is different from multiplying whole numbers by them because you can't just add zeros. Take a stab at the following questions. Problem: How much is 1.8 × 10? Solution: You can't add a zero to 1.8, because that would leave you with 1.80, which has the same value as 1.8. But what you could do is move the decimal point one place to the right, 18., which gives you 18. Problem: How much is 10.67 × 100? Solution: Just move the decimal point two places to the right, 1067. and you get 1,067.
( ( ( (
Now here's one that's a little tricky. Problem: How much is 4.6 × 100?
125
DECIMALS
Solution: First we add a zero to 4.6, making it 4.60. We can add as many zeros as we want after a decimal, because that won't change its value. Once we've added the zero, we can move the decimal point two places to the right: 460. By convention, we don't use decimal points after whole numbers like 460, so we can drop the decimal point. Problem: How much is 9.2 × 100? Solution: 9.2 × 100 = 9.20 × 100 = 920 Problem: How much is 1.573 × 1,000? Solution: The answer is 1,573. All you needed to do was move the decimal point three places to the right. To summarize: When you multiply a decimal by 10, move the decimal point one place to the right. When you multiply a decimal by 100, move the decimal point two places to the right. When you multiply a decimal by 1,000, move the decimal point three places to the right. Problem: Now multiply 10.4 × 1,000. Solution: 10.4 × 1,000 = 10.400 × 1,000 = 10,400
( (
FAST DIVISION
Fast division is the reverse of fast multiplication. Instead of adding zeros, you take them away. And instead of moving the decimal point to the right, you move it to the left.
127
DECIMALS
D IVIDING
BY
T EN
Start off by taking zeros away from the first number in the next question. Problem: How much is 140 divided by 10? Solution: The answer is 14. All you did was get rid of the zero. Problem: How much is 1,300 divided by 10? Solution: The answer is 130. So far, so good. But what do you do if there are no zeros to get rid of? Then you must move the decimal point one place to the left. Problem: For instance, how much is 263 divided by 10? Solution: 263 ÷ 10 = 26.30 = 26.3 Problem: How much is 1,094 divided by 10? Solution: 1,094 ÷ 10 = 109.40 = 109.4
Now we'll move on to dividing by 100. Problem: How much is 38.9 divided by 100? Solution: The answer is 0.389—all you need to do is move the decimal point two places to the left. Problem: How much is 0.4 divided by 100? Solution: 0.4 ÷ 100 = 00.4 ÷ 100 = 0.004. You can place as many zeros as you wish to the left of a decimal without changing its value. So 00.4 = 0.4. Problem: How much is 0.06 ÷ 100? Solution: 0.06 ÷ 100 = 00.06 ÷ 100 = 0.0006 Problem: How much is 4 divided by 100? Solution: 4 ÷ 100 = 04.0 ÷ 100 = 0.04 You probably remember that you can add zeros after a decimal point without changing its value. I placed zeros to the left of 4 because if you move the decimal point to the left of a whole number, it's understood that you'll need to add zeros. Problem: How much is 56 ÷ 100? Solution: 56 ÷ 100 = 56.0 ÷ 100 = 0.56
Here are some problems for you to practice dividing by 1,000. Problem: How much is 6,072.5 divided by 1,000? Solution: The answer is 6.0725. All you needed to do was move the decimal point three places to the left. Problem: How much is 400,000 divided by 1,000? Solution: The answer is 400. All you needed to do here was to drop three zeros. Problem: How much is 752 divided by 1,000? Solution: 752 ÷ 1,000 = 0.752 Problem: How much is 39 ÷ 1,000? Solution: 39 ÷ 1,000 = 0.039 Problem: How much is 0.2 divided by 1,000? Solution: Just move the decimal point three places to the left: 0.0002.
( ( (
SUMMARY
So far you've multiplied and divided by 10, 100, and 1,000. Let's summarize the procedures you've followed.
• To multiply by 10, you add a zero or move the decimal point one • • • • •
place to the right. To divide by 10, you drop a zero or move the decimal point one place to the left. To multiply by 100, you add two zeros or move the decimal point two places to the right. To divide by 100, you drop two zeros or move the decimal point two places to the left. To multiply by 1,000, you add three zeros or move the decimal point three places to the right. To divide by 1,000, you drop three zeros or move the decimal point three places to the left.
Don't worry, you won't have to memorize all these rules. All you'll need to do when you want to multiply a number by 10, 100, or 1,000 is ask yourself how you can make this number larger. Do you do it by tack131
DECIMALS
ing on zeros or by moving the decimal point to the right? As you get used to working with numbers, doing this will become virtually automatic. Similarly, when you divide, just ask yourself how you can make this number smaller. Do you do it by dropping zeros, or by moving the decimal point to the left? Again, with experience you'll be doing these problems instinctively. Another way of expressing a division problem is to ask: How much is one-tenth of 50? Or how much is one one-hundredth of 7,000? Onetenth of 50 obviously means how much is 50 divided by 10, so the answer is 5. And one one-hundredth of 7,000 means how much is 7,000 divided by 100, which is 70. Problem: How much is one-tenth of 16,000? Solution: The answer is 1,600. Problem: How much is one-tenth of 1.3? Solution: The answer is 0.13. Problem: How much is one one-hundredth of 9? Solution: The answer is 0.09. Problem: And how much is one one-thousandth of 8.6? Solution: The answer is 0.0086.
N EXT S TEP
I told you this lesson would be a whole lot of fun. In the next lesson, you'll get a chance to apply everything you've learned about decimals in this entire section.
134
LESSON 15
After introducing coins as decimals of a dollar, this lesson will help you apply what you've learned about decimals to real-world problems that involve addition, subtraction, multiplication, and division.
A PPLICATIONS
S
ome of the applications of the math
you've done in this section are money problems. So before you actually do any problems, let's talk a little about the U.S. dollar. The dollar can be divided into fractions or decimals. There are 100 cents in a dollar. If a dollar is 1, or 1.0, then how much is a half dollar (50 cents) as a fraction of a dollar and as a decimal of a dollar? It's 1 or 0.5 2 (or 0.50). Problem: Write each of these coins as a fraction of a dollar and as a decimal of a dollar: a. A penny b. A nickel
135
Before you begin the problem set, let me say a few words about rounding your answers. Suppose your answer came to $14.9743. Rounded to the nearest penny, your answer would be $14.97. If your answer were $30.6471, rounded to the nearest penny it would come to $30.65. So whenever this applies, round your answers to the nearest penny.
P ROBLEM S ET
Do all of these problems, and then check your work with the solutions that follow.
1.
If you had a half dollar, three quarters, eight dimes, six nickels, and nine pennies, how much money would you have all together? If your weekly salary is $415.00, how much do you take home each week after deductions are made for federal income tax ($82.13), state income tax ($9.74), Social Security and Medicare ($31.75), and retirement ($41.50)? You began the month with a checking account balance of $897.03. During the month you wrote checks for $175.00, $431.98, and $238.73, and you made deposits of $300.00 and $286.17. How much was your balance at the end of the month? Carpeting costs $7.99 a yard. If Jose buys 12.4 yards, how much will it cost him? If cashews cost $6.59 a pound, how much would it cost to buy two and a quarter pounds?
2.
3.
4.
5.
136
APPLICATIONS
6.
Sheldon Chen's scores in the diving competition were 7.2, 6.975, 8.0, and 6.96. What was his total score? If gasoline cost $1.399 a gallon, how much would it cost to fill up a tank that had a capacity of 14 3 gallons? 4 The winners of the World Series received $958,394.31. If this money was split into 29.875 shares, how much would one share be worth? If you bought 3 1 pounds of walnuts at $4.99 a pound and 1 1 2 4 pounds of peanuts at $2.39 a pound, how much would you spend all together?
7.
8.
9.
10. The Coney Island Historical Society had sales of $3,017.93. After
paying $325 in rent, $212.35 in advertising, $163.96 in insurance, and $1,831.74 in salaries, how much money was left in profits?
11. If gold cost $453.122 an ounce, how much would
3 8
of an ounce
cost?
12. Jessica owned 1.435 shares, Karen owned 2.008 shares, Jason
owned 1.973 shares, and Elizabeth owned 2.081 shares. How many shares did they own in total?
13. Wei Wong scored 9.007 in gymnastics. Carlos Candellario scored
8.949. How much higher was Wei Wong's score?
14. On Tuesday Bill drove 8.72 hours, averaging 53.88 miles per hour.
On Wednesday he drove 9.14 hours, averaging 50.91 miles per hour. How many miles did he drive on Tuesday and Wednesday?
15. One meter is equal to 39.37 inches. How many inches are there in
70.26 meters?
16. Michael studied for 17.5 hours over a period of 4.5 days. On aver-
age, how much did he study each day?
137
DECIMALS
17. All the people working at the Happy Valley Industrial Park pooled
their lottery tickets. When they won $10,000,000, each got a 0.002 part share. How much money did each person receive?
18. Daphne Dazzle received 2.3 cents for every ticket sold to her
movie. If 1,515,296 tickets were sold, how much money did she receive?
19. A cheese store charged $3.99 a pound for American cheese, $3.49
a pound for Swiss cheese, and $4.99 a pound for brie. If it sold 10.4 pounds of American, 16.3 pounds of Swiss, and 8.7 pounds of brie, how much were its total sales?
20. A prize of $10,000,000 is awarded to three sisters. Eleni receives
one-tenth, Justine receives one-tenth, and Sophie receives the rest. How much are their respective shares?
21. Elizabeth and Daniel received cash bonuses equal to one one-
hundredth of their credit card billings. If Elizabeth had a billing of $6,790.22 and Daniel had a billing of $5,014.37, how much cash bonus did each of them receive?
22. Con Edison charges 4.3 cents per kilowatt hour. How much does
Suppose you wanted to compare your cost per mile using Mobil regular gasoline, which costs $1.199 per gallon and Mobil premium, which costs $1.399 per gallon. If Mobil regular gives you 26.4 miles per gallon (highway driving) and Mobil premium gives you 31.7 miles per gallon, which gas gives you the lower cost per mile? Hint: How much does it cost to drive one mile using both gases?
Solution:
regular: $1.199 = 4.54 cents/mile 26.4 premium: $1.399 = 4.41 cents/mile 31.7 Almost everyone who plays the lottery knows that they are overpaying for a ticket. They have only an infinitesimal chance of winning, but, as the tag line of an ad touting the New York State lottery says, "Hey, you never know." OK, let's assume a payoff of $15 million. If any $2-ticketholder's chance of winning were one in 20 million, how much is that ticket really worth?
N EXT S TEP
Okay, three sections down, one to go. Once again, let me ask you how things are going. If they're going well, then you're ready for the final section, which introduces percentages. If not, you know the drill. Go back over anything that needs going over. Just let your conscience be your guide.
143
SECTION
IV
P ERCENTAGES
P
ercentages are the mathematical equiva-
lent of fractions and decimals. For example, 1 = 0.5 = 50%. In 2 baseball, a 300 batter is someone who averages three hundred base hits every thousand times at bat, which is the same as thirty out of 30 3 a hundred ( 100 or 30%) or three out of ten ( 10 ). It means he gets a hit 30% of the time that he comes to bat. Let's take a close look at the relationship among decimals, fractions, 1 1 and percentages. We'll begin with the fraction, 100 . How much is 100 as a percent? It's 1%. And how much is the decimal, 0.01, as a percent? Also 1%. 1 10 That means, then, that 100 = 0.01 = 1%. How about 0.10 and 100 ? As a percent, they're both equal to 10%.
145
PERCENTAGES
Now I'm going to throw you a curve ball. How much is the number 1 as a decimal, a fraction, and as a percent? The number one may be written as 1.0, 1 (or 100 ), or as 100%. 1 100 It's easy to go from fractions and decimals to percents if you follow the procedures outlined in this section. It doesn't matter that much whether you can verbalize these procedures. In math the bottom line is always the same—coming up with the right answer. When you have completed this section, you will be able to find percentages, convert percentages into fractions and decimals, and find percentage changes, percentage distribution, and percentages of numbers. In short, you will have learned everything you will ever need to know about percentages.
146
LESSON 16
In this lesson, you'll learn how to convert decimals into percents and percents into decimals. You'll find out how and when to move the decimal point for each type of conversion. This easy conversion process will lead you to the more difficult process of converting between fractions and percents in the next lesson.
C ONVERTING D ECIMALS INTO P ERCENTS AND P ERCENTS INTO D ECIMALS
D
ecimals can be converted into percents
by moving their decimal points two places to the right and adding a percent sign. Conversely, percents can be converted into decimals by removing the percent sign and moving their decimal points two places to the left.
CONVERTING DECIMALS TO PERCENTS
You know that the same number can be expressed as a fraction, as a decimal, or as a percent. For example, 1 = 0.25. Now what percent is 1 and 4 4 0.25 equal to? The answer is 25%. Just think of these numbers as money: one quarter equals 25 cents, or $0.25, or 25% of a dollar.
147
PERCENTAGES
Here's how to figure it out. Start with the decimal, 0.25. Let's convert it into a percent. What you do is move the decimal point two places to the right and add a percent sign: .25 = 25.% = 25% When we have a whole number like 25, we don't bother with the decimal point. If we wanted to, we could, of course, write 25% like this: 25.0%.
( (
P ROBLEM S ET
I'd like you to convert a few decimals into percents.
1. 2. 3.
0.32 = 0.835 = 1.29 =
4. 0.03 = 5. 0.41 =
Solutions
1. 2. 3.
0.32 = 32% 0.835 = 83.5% 1.29 = 129%
4. 0.03 = 3% 5. 0.41 = 41%
Now we'll add a wrinkle. Convert the number 1.2 into a percent. Go ahead. I'll wait right here. What did you get? Was it 120%? What you do is add a zero to 1.2 and make it 1.20, and then move the decimal two places to the right and add the percent sign. What gives you the right to add a zero? Well, it's okay to do this as long as it doesn't change the value of the number, 1.2. Since 1.2 = 1.20, you can do this. Can you add a zero to the number 30 without changing its value? Try it. Did you get 300? Does 30 = 300? If you think it does, then I'd like to trade my $30 for your $300. Ready for another problem set? All right, then, here it comes.
Did you get them right? Good! Then you're ready for another wrinkle. Please convert the number 5 into a percent. What did you get? 500%? Here's how we did it. We started with 5, added a decimal point and a couple of zeros: 5 = 5.00. Then we converted 5.00 into a percent by moving the decimal point two places to the right and adding a percent sign: 5.00 = 500.% = 500%. Here's another group of problems for you.
( (
CONVERTING PERCENTS TO DECIMALS
Let's shift gears and convert some percentages into decimals. Problem: What is the decimal equivalent of 35 percent? Solution: 35% = 35.0% = .350% = 0.35 Problem: What is the decimal equivalent of 150 percent? Solution: 150% = 150.0% = 1.500% = 1.5 Let's talk about what we've been doing. To convert a percent into a decimal form, drop the percent sign and move the decimal point two places to the left. In other words, do the opposite of what you did to convert a decimal into a percent. Now I'd like you to do this problem set.
( ( ( (
You may have noticed that in problem 18, I expressed the answer as 4.0 or 4. By convention, when we express a whole number, we don't use the decimal point. Similarly, in problem 20, we can drop the decimal from 1.0 and express the answer as 1. Problem: What is the decimal equivalent of 0.3%? Solution: 0.3% = .003% = .003 Again, all you need to do is drop the percent sign and move the decimal point two places to the left. Do this problem set, so you can move on to even more exciting things.
( (
N EXT S TEP
So far, we've been converting decimals into percents and percents into decimals. Remember that every percent and every decimal has a fractional equivalent. So next, let's convert fractions into percents and percents into fractions.
152
LESSON 17
In this lesson, you'll learn how to convert fractions into percents and percents into fractions. You'll discover some handy shortcuts and other math tricks to get you to the right answer every time.
C ONVERTING F RACTIONS INTO P ERCENTS AND P ERCENTS INTO F RACTIONS
I
n the previous lesson, I said that a number
could be expressed as a fraction, as a decimal, or as a percent. I said that 1 = 0.25 = 25%. Read on to find out how this works—how frac4 tions can be converted to percents and percents to fractions. You'll even learn more than one way to do these conversions.
CONVERTING FRACTIONS INTO PERCENTS
You may remember that in Lesson 13, you had a great time converting fractions into decimals. So 1 is converted into 0.25 by dividing 4 into 1: 4 .25 4 1.00
153
2
PERCENTAGES
Now let's try another way of getting from 1 to 25%. We're going to use 4 an old trick that I mentioned previously; it's actually a law of arithmetic. The law says that whatever you do to the bottom of a fraction, you must also do to the top. In other words, if you multiply the denominator by a certain number, you must multiply the numerator by that same number. Let's start with the fraction 1 : 4
1 × 25 4 × 25
=
25 100
What did we do? We multiplied the numerator and the denominator by 25. Why 25? Because we wanted to get the denominator equal to 100. Having 100 on the bottom of a fraction makes it very easy to convert that fraction into a percent. 25 All right, we have 100 , which comes out to 25%. How did we do that? We removed the 100, or mathematically, we multiplied the fraction by 100, then added a percent sign. In other words,
25 100
×
100 1
25 = 1 100 ×
100 1
1
= 25%
Incidentally, percent means per hundred. Fifty-seven percent, then, means 57 per hundred. And 39 percent means 39 per hundred. This is exactly the same process as converting a decimal into a percent. The decimal 0.25 becomes 25% when we move the decimal point two places to the right and add a percent sign. Moving a decimal two places to the right is the same as multiplying by 100. Similarly, when we changed 25 the fraction 100 into a percent, we also multiplied by 100 and added a percent sign. Now you do this one. Problem: Write Solution: 34% So what you did was multiply would you convert you to try it.
9 50 34 100 34 100
as a percent.
by 100 and add a percent sign. How
into a percent? Don't wait for me to do it. I want
154
CONVERTING FRACTIONS AND PERCENTS
I hope you did it like this:
9×2 50 × 2
=
18 100
= 18%
Do you follow what I did? I multiplied the top (or numerator) by 2 and the bottom (or denominator) by 2. Am I allowed to do that? Yes! You are allowed to multiply the numerator and denominator of a fraction by the same number because it does not change that fraction's value. Why did I multiply the numerator and denominator by 2? Because I wanted to change the denominator into 100, so that I could easily convert this fraction into a percent. So whenever you get the chance, convert the denominator into 100. It can make your life a lot easier.
MORE DIFFICULT CONVERSIONS
So far you've been very lucky. Every fraction has been quite easy to convert into hundredths and then, the number written over 100 is read as 17 89 a percentage. For instance, 100 = 17% and 100 = 89%. But what if you have a fraction that cannot easily be converted into hundredths, like 3 ? 8 Problem: How do you change
3 8
into a percent?
Solution: You do it in two steps. First you change .375 8 3.00 0 Then you move the decimal point two places to the right and add a percent sign: .375 = 37.5% = 37.5%.
( (
3 8
into a decimal:
64
I did that one. Now you do this one. Problem: Change Solution: 42.5%
17 40
into a percent.
D OING S HORT D IVISION I NSTEAD
OF
L ONG D IVISION
Do you remember the trick I showed you in Lesson 14 when we did some fast division? Dividing 40 into 17 must be done by long division, which is what I'll bet you did. However, there is a shortcut you can take. Here's my trick:
17 40
=
1.7 4.0
What did I do? I divided the numerator, 17, by 10, and then I divided the denominator, 40, by 10. (You can easily divide a number by 10 by simply moving its decimal point one space to the left.) But why did I bother to divide 17 and 40 by 10? Why would I rather have 1.7 than 17 ? 4.0 40 Because then we can do short division instead of long division. Of course,
156
CONVERTING FRACTIONS AND PERCENTS
if you happen to be using a calculator, then there is no difference between long and short division. But you're not doing that here, are you? .425 4 1.700
12
= 42.5%
In general, when you need to divide the denominator of a fraction into the numerator, first reduce the fraction to the lowest possible terms, and then, if possible, divide the numerator and denominator by 10 or even 100 if that can get you from long division to short division. One thing I need to mention before you do the next problem set is how to treat a repeating decimal. You'll discover that for problems 9 and 10 you'll get to the point where the same numbers keep coming up. You can divide forever and the problem never comes out even. The thing to do in this case is to stop dividing and round when you get to the tenth of a percent. When you get to the solutions for problems 9 and 10, you'll see what I mean. Well, it's time for another problem set. Are you ready? All right, then, here it comes.
By convention, we usually round to one decimal place. So if you rounded to a whole number or to two or three decimal places, then your answers may have differed just a bit from mine. So how did you do? Did you get everything right? If you did, then you can pass GO, collect $200, and go directly to the next lesson. But if you didn't get all of these right, then please stay right here and work out the next set of problems. You've heard the saying "practice makes perfect." Now we'll prove it.
The answers to problems 16 and 17 were reduced to their lowest possible forms. We did that too with problem 19, but by convention, we express any number divided by itself as 1. In problem 20, we reduced the improper fraction 250 to the mixed number 2 1 . 100 2 Problem: Now convert 93.6% into a fraction. Solution: 93.6% =
93.6 100
=
936 1000
=
117 125
The first step should be familiar: Get rid of the percentage sign and place 93.6 over 100. To get rid of the decimal point, we multiply the numerator, 93.6, by 10, and we multiply the denominator, 100, by 10. 936 That gives us 1000 , which can be reduced to 117 . Sometimes we leave frac125
160
CONVERTING FRACTIONS AND PERCENTS
tions with denominators of 100 and 1,000 as they are, even though they 936 can be reduced. So if you leave this answer as 1000 , it's okay. Problem: Now change 1.04% into a fraction. Solution: 1.04% =
1.04 100
For problem 25, since we don't really want to leave our answer as an improper fraction, we should convert it into a mixed number. This situation rarely comes up, so you definitely should not lose any sleep over it.
N EXT S TEP
In this section, you've seen that every number has three equivalent forms—a decimal, a fraction, and a percentage. Now you can go on to finding percentage changes.
161
LESSON 18
This lesson will show you how to find and understand percentage changes. You can use this knowledge to figure out many practical percentage questions that arise in your daily life.
F INDING P ERCENTAGE C HANGES
I
f you went to any college graduation and
asked the first ten graduates you encountered to do the first problem in this lesson, chances are that no more than one or two of them would come up with the right answer. And yet percentage changes are constantly affecting us—pay increases, tax cuts, and changes in interest rates are all percentage changes. When you've completed this lesson, if someone should walk up to you and ask you to calculate a percentage change, you'll definitely be prepared.
CALCULATING PERCENTAGE CHANGE
Let's get right into it. Imagine that you were earning $500 and got a $20 raise. By what percentage did your salary go up? Try to figure it out.
163
PERCENTAGES
We have a nice formula to help us solve percentage change problems. Here's how it works: Your salary is $500, so that's the original number. You got a $20 raise; that's the change. The formula looks like this: percentage change =
change original number
Next, we substitute the numbers into the formula. And then we solve 20 1 it. Once we have 500 , we could reduce it all the way down to 25 and solve it using division: =
$20 $500
=
2 50
=
4 100
= 4%
.04 25 1.00 = 4% – 1.00 Try working out this next problem on your own. Problem: On New Year's Eve, you made a resolution to lose 30 pounds by the end of March. And sure enough, your weight dropped from 140 pounds to 110. By what percentage did your weight fall? Solution: percentage change =
change original number
What is the percentage change if Becky's weight goes from 150 to 180 pounds? What is the percentage change if Tom's weight goes from 130 to 200 pounds? If Jessica's real estate taxes rose from $6,000 to $8,500, by what percentage did they rise? Harriet's time for running a mile fell from 11 minutes to 8 minutes. By what percentage did her time fall?
2.
3.
4.
Solutions
1. 2.
Percentage change = Percentage change =
change original number change original number
= =
30 150 70 130
= =
1 5 7 13
=
20 100
= 20%
.538 = 53.8 13 7.000 – 6 5 XX 50_ – 39_ 110 – 104 6
3.
Percentage change =
change original number
=
$2,500 $6,000
=
25 60
=
5 12
.4166 = 41.7 12 5.0000 – 4 8 XXX 20__ – 12__ 80_ – 72_ 80 – 72
165
PERCENTAGES
4.
Percentage change =
change original number
=
3 11
.2727 = 27.3% 11 3.0000 – 2 2 XXX 80__ – 77__ 30_ – 22_ 80 – 77
PERCENTAGE INCREASES
Pick a number. Any number. Now triple it. By what percentage did this number increase? Take your time. Use the space in the margin to calculate the percentage. What did you get? Three hundred percent? Nice try, but I'm afraid that's not the right answer. I'm going to pick a number for you and then you triple it. I pick the number 100. Now I'd like you to use the percentage change formula to get the answer. (Incidentally, you may have gotten the right answer, so you may be wondering why I'm making such a big deal. But I know from sad experience that almost no one gets this right on the first try.) So where were we? The formula. Write it down in the space below, substitute numbers into it, and then solve it.
Let's go over this problem step by step. We picked a number, 100. Next, we tripled it. Which gives us 300. Right? Now we plug some numbers into the formula. Our original number is 100. And the change when we go from 100 to 300? It's 200. From there it's just arithmetic: 200 = 200%. 100 Percentage change =
change original number
=
200 100
= 200%
166
FINDING PERCENTAGE CHANGES
This really isn't that hard. In fact, you're going to get really good at just looking at a couple of numbers and figuring out percentage changes in your head. Whenever you go from 100 to a higher number, the percentage increase is the difference between 100 and the new number. Suppose you were to quadruple a number. What's the percentage increase? It's 300% (400 – 100). When you double a number, what's the percentage increase? It's 100% (200 – 100).
P ROBLEM S ET
Here's a set of problems, and I guarantee that you'll get them all right. What's the percentage increase from 100 to each of the following?
5. 6. 7.
150 320 275
8. 500 9. 425
Solutions
5. 6. 7.
50 100 220 100 175 100
= 50% = 220% = 175%
8. 9.
400 100 325 100
= 400% = 325%
The number 100 is very easy to work with. Sometimes you can use it as a substitute for another number. For example, what's the percentage increase if we go from 3 to 6? Isn't it the same as if you went from 100 to 200? It's a 100% increase. What's the percentage increase from 5 to 20? It's the same as the one from 100 to 400. It is a 300% increase. What we've been doing here is just playing around with numbers, seeing if we can get them to work for us. As you get more comfortable with numbers, you can try to manipulate them the way we just did.
167
PERCENTAGES
PERCENTAGE DECREASES
Remember the saying "whatever goes up must come down"? If Melanie Shapiro was earning $100 and her salary were cut to $93, by what percent was her salary cut? Solution: The answer is obviously 7%. More formally, we divided the change in salary, $7, by the original salary, $100: $7/$100 = 7%. What would be the percentage decrease from 100 to 10? Solution: 90/100 = 90%. Here's one last problem set, and, once again, I'll guarantee that you'll get them all right.
P ROBLEM S ET
What is the percentage decrease from 100 to each of the following numbers?
10.
150
11.
20
12.
92
13.
50
Solutions
10. 12.
39/100 = 39% 8/100 = 8%
11. 13.
80/100 = 80% 50/100 = 50%
Now I'm going to throw you another curve ball. If a number—any number—were to decline by 100%, what number would you be left with? I'd really like you to think about this one. What did you get? You should have gotten 0. That's right—no matter what number you started with, a 100% decline leaves you with 0.
168
FINDING PERCENTAGE CHANGES
N EXT S TEP
Being able to calculate percentage changes is one of the most useful of all arithmetic skills. If you feel you have mastered it, then go on to the next lesson. If not, you definitely want to go back to the beginning of this lesson and make sure you get it right the second time around.
169
LESSON 19
In this lesson, you'll learn how to calculate percentage distribution for several realworld scenarios. You'll find out that all percentage distributions add up to 100. You'll discover how you can check your answers after completing a problem and how to get the information you need when posed with a percentage distribution question.
P ERCENTAGE D ISTRIBUTION
P
ercentage distribution tells you the num-
ber per hundred that is represented by each group in a larger whole. For example, in Canada, 30% of the people live in cities, 45% live in suburbs, and 25% live out in the country. When you calculate percentage distributions, you'll find that they always add up to 100% (or a number very close to 100, depending on the exact decimals involved). If they don't, you'll know that you have to redo your calculations. A class had half girls and half boys. What percentage of the class was girls, and what percentage of the class was boys? The answers are obviously 50% and 50%. That's all there is to percentage distribution. Of course the problems do get a bit more complicated, but all percentage distribution problems start out with one simple fact: The distribution will always add up to 100%.
171
PERCENTAGES
Here's another one. One-quarter of the players on a baseball team are pitchers, one-quarter are outfielders, and the rest are infielders. What is the team's percentage distribution of pitchers, infielders, and outfielders? Pitchers are 1 , or 25%; outfielders are also 1 , or 25%. So infielders 4 4 must be the remaining 50%. Try doing the next percentage distribution on your own. Problem: If, over the course of a week, you obtained 250 grams of protein from red meat, 150 from fish, 100 from poultry, and 50 from other sources, what percentage of your protein intake came from red meat and what percentage came from each of the other sources? red meat fish poultry other 250 grams 150 grams 100 grams + 50 grams 550 grams
Try to work this out to the closest tenth of a percent. Hint: 550 grams = 100%. Solution: red meat = 11
656
250 550
=
25 55
=
5 11
= 45.5%
.4545 5.00 00
150 550
fish = 11
=
15 55
=
3 11
= 27.3%
.2727 3.0 00 0
100 550
838
poultry = 11
=
10 55
=
2 11
= 18.2%
.1818 2.0000
50 550
9 29
other = 11
172
=
5 55
=
1 11
= 9.1%
.0909 1.0000
1
PERCENTAGE DISTRIBUTION
Check:
45.5 27.3 18.2 + 9.1 100.1
31
When doing percentage distribution problems, it's always a good idea to check your work. If your percentages don't add up to 100 (or 99 or 101), then you've definitely made a mistake, so you'll need to go back over all your calculations. Because of rounding, my percentages added up to 100.1. Occasionally you'll end up with 100.1 or 99.9 when you check, which is fine. Are you getting the knack? I certainly hope so because there's another problem set straight ahead.
P ROBLEM S ET
Calculate to the closest tenth of a percent for these problems.
1.
Denver has 550,000 Caucasians, 150,000 Hispanics, 100,000 African-Americans, and 50,000 Asian-Americans. Calculate the percentage distribution of these groups living in Denver. Be sure to check your work. Eleni Zimiles has 8 red beads, 4 blue beads, 3 white beads, 2 yellow beads, and 1 green bead. What is the percentage distribution of Eleni's beads? Georgia-Pacific ships 5,000 freight containers a week. Fifteen hundred are sent by air, two thousand three hundred by rail, and the rest by truck. What percentage is sent by air, rail, and truck, respectively? In the mayor's election Ruggerio got 45 votes, Casey got 39 votes, Schultz got 36 votes, and Jones got 28 votes. What is the percentage distribution of the vote? In Middletown 65 families don't own a car; 100 families own one car; 108 families own two cars; 70 families own three cars; 40
173
2.
3.
4.
5.
PERCENTAGES
families own four cars; and 17 families own five or more cars. What is the percentage distribution of car ownership?
N EXT S TEP
Congratulations on learning how to calculate percentage distribution. The next lesson shows how to find percentages of numbers. Go for it!
178
LESSON 20
In this lesson, you'll learn how to find percentages of numbers. We'll start with the percentage of your pay that the Internal Revenue Service collects.
F INDING P ERCENTAGES N UMBERS
OF
T
he Internal Revenue Service charges dif-
ferent tax rates for different levels of income. For example, most middle-income families are taxed at a rate of 28 percent on some of their income. Suppose that one family had to pay 28 percent of $10,000. How much would that family pay?
Solution: $10,000 × .28 = $2,800
You'll notice that we converted 28 percent into the decimal .28 to carry out that calculation. We used fast multiplication, which we covered in Lesson 14.
179
Problem: How much is 73.5 percent of $12,416.58? Solution: $12,416.58 ___× .735 6208290 3724974_ 8691606__ $9,126.18630 = $9,126.19 Most people try to leave around a 15 percent tip in restaurants. In New York City, where the sales tax is 8.25 percent, customers often just double the tax. But there is actually another very fast and easy way to calculate that 15 percent tip. Let's say that your bill comes to $28.19. Round it off to $28, the nearest even dollar amount. Then find 10 percent of $28, which is $2.80. Now what's half of $2.80? It's $1.40. How much is $1.40 plus $2.80? It's $4.20. Let's try a much bigger check—$131.29. Round it off to the nearest even dollar amount—$132. What is 10 percent of $132? It's $13.20. And how much is half of $13.20? It's $6.60. Finally, add $13.20 and $6.60 together to get your $19.80 tip.
P ROBLEM S ET
1. 2. 3. 4. 180
How much is 13 percent of 150? Find 34.5 percent of $100. How much is 22.5 percent of $390? Find 78.2 percent of $1,745.
N EXT S TEP
You've already done some applications, so the next lesson will be easy. Let's find out.
181
LESSON 21
In this lesson, you'll be able to pull together all of the math skills you've mastered in this section and apply them to situations you may encounter in your daily life. You'll see how practical the knowledge that you've gained is and how often mathematical questions arise that you now know the answers to.
A PPLICATIONS
C
an you believe it? You're about to begin
the last lesson in this book. Many of the problems you'll be solving here are ones you may encounter at work, at home, or while driving or shopping. Before you get started, how about a few practice problems? First, a markup problem. Stores pay one price for an item, but they almost always charge a higher price to their customers. We call that process a markup. For instance, if a store owner pays $10 for a radio and sells it for $15, by what percentage did she mark it up? Percentage markup =
$5 $10
= .50 = 50%
183
PERCENTAGES
Markdown is another common commercial term. Suppose a store advertises that every item is marked down by 40%. If a CD was originally selling for $8, what would its marked-down price be during the sale? Sale price = $8 – ($8 × 0.40) = $8 – $3.20 = $4.80
Try this next one yourself. Problem: Imagine that you're earning $250 a week and receive a raise of 10 percent. How much is your new salary? Solution: New salary = $250 + ($250 × 0.10) = $250 + $25 = $275
Problem: Suppose you went on a big diet, and your weight fell by 20%. If you started out weighing 150 pounds, how much would you weigh after dieting? Solution: New weight = 150 – (150 × 0.20) = 150 – 30 = 120
Here's another type of problem. Problem: What percentage of 100 is 335? Solution:
335 100
= 3.35 = 335%
Congratulations! You've just gotten a $100 salary increase. How much of that $100 do you actually take home if you have to pay 15 percent in federal income tax, 7.65 percent in payroll tax, and 2.5 percent in state income tax?
P ROBLEM S ET
Let's apply what you've learned about percentages to some real-life situations. Check your work with the solutions at the end of the lesson.
1.
The Happy Day Nursing Home increased the number of beds from 47 to 56. By what percentage did they increase? If one-quarter of all Americans live in cities, what is the percentage of Americans who do not live in cities? Three people ran for state Senator. If Marks got one-third of the vote and Brown got one-fifth of the vote, what percentage of the vote did Swanson receive? If you had four pennies, two nickels, three dimes, and a quarter, what percentage of a dollar would you have? If 8.82 is the average score in a swim meet and you had a score of 9.07, by what percentage did your score exceed the average? A dress is marked up 65% from what it cost the store owner. If the store owner paid $20 for the dress, how much does she charge? A suit on sale is marked down 40% from its regular price. If its regular price is $170, how much is its sale price? Of 319 employees at the Smithtown Mall, 46 were out sick. What percentage of employees were at work that day?
2.
3.
4.
5.
6.
7.
8.
185
PERCENTAGES
9.
Henry Jones gets a hit 32.8% of the times he comes to bat. What is his batting average? (Hint: a batting average is a decimal expressed in thousandths.)
10. You're driving at 40 mph and increase your speed by 20%. How
fast are you now going?
11. You cut back on eating and your $50 weekly food bill falls by 30%.
What is your new food bill?
12. Jason Jones was getting 20 miles per gallon. But when he slowed
down to an average speed of 70 mph, his gas mileage rose by 40%. What is his new gas mileage?
13. If you were making $20,000 and got a 15% pay increase, how
much would you now be making?
14. A school that had 650 students had a 22% increase in enrollment.
How much is its new enrollment?
15. What would your percentage score on an exam be if you got 14
questions right out of a total of 19 questions?
16. What percentage of a dollar is $4.58? 17. If you needed $500 and had saved $175, what percentage of the
$500 had you saved?
18. The University of Wisconsin alumni association has 45,000
members. Four thousand five hundred are women 40 and under; seven thousand nine hundred are women over 40; twelve thousand eight hundred are men 40 and under; the remainder of members consists of men over 40. Find the percentage distribution of all four membership categories. Remember to check your work.
If a storewide sale sounds too good to be true, it probably is. Like this one: "All prices reduced by 50%. On all clothing, take off an additional 30%. And on item with red tags, take off an additional 25%." OK, doesn't that mean that on red-tagged clothing you take off 105%, which means that on a dress originally priced at $100, the store gives you $5 to take it off their hands? Evidently not. How much would you actually have to pay for that dress?
Solution First take off 50%: $100 × .50 = $50. Now take off another 30%: $50 × .70 = $35. Notice the shortcut we just took. Instead of multiplying $50 × .30, getting $15, and subtracting $15 from $50 to get $35, we saved ourselves a step by multiplying $50 by .70. Finally, we take off another 25%: $35 × .75 = $26.25. Notice that we take the same shortcut, instead of multiplying $35 by .25, and subtracting $8.75 from $35. To summarize, we take 50% off the original $100 price, then 30% off the new $50 price, and then 25% off the price of $35. A price reduction from $100 to $26.25 is not too shabby, but that's a far cry from a reduction of $105.
N EXT S TEP
Congratulations on completing the 21 lessons in this book! You deserve a break. Don't look now, but there is one more chance for you to exercise the skills you've learned thus far. After you've taken a well-deserved break, check out the final exam.
193
F INAL E XAM
Y
ou didn't think you'd actually be able to
get out of here without taking a final exam, did you? If you know this stuff cold, then this exam will be a piece of cake. And if you don't do so well, you'll see exactly where you need work, and you can go back to those specific lessons so that you can master those concepts.
L AST S TEP
You know the drill. I put the lesson numbers on each set of problems so you'd know what lessons to go back and review if you needed further help. If you're doing fine and are ready to go on to more complicated math, turn to the appendix called Additional Resources to see what to tackle next.
210
A DDITIONAL R ESOURCES
A
re you ready to tackle algebra, or would
you like to work your way through another book like this one to get more practice? Two very similar books are these: • Practical Math in 20 Minutes a Day by Judith Robinovitz (LearningExpress, order information at the back of this book) • Arithmetic the Easy Way by Edward Williams and Katie Prindle (Barron's)
Three other books, which cover much of the same material but also introduce very elementary algebra, as well as some business math applications, are these: • Business Mathematics the Easy Way by Calman Goozner (Barron's)
211
Algebra is traditionally taught in a three-year sequence. If you've mastered fractions, decimals, and percentages, then you're definitely ready to tackle elementary (or first-year) algebra. Unfortunately, many of the algebra books you'll run across assume a prior knowledge of elementary algebra, or rush through it much too quickly. Two of my own books, Practical Algebra and Quick Algebra Review, do just that. There are, however, two elementary algebra books that I do recommend: • Prealgebra by Alan Wise and Carol Wise (Harcourt Brace) • Let's Review Sequential Mathematics Course 1 by Lawrence S. Leff (Barron's) Whatever course you follow, just remember that doing math can be fun and exciting. So don't stop now. You'll be amazed at how much further you can go. | 677.169 | 1 |
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Radical Expressions and Equations
This chapter introduces the concept of radical expressions/equations at the Algebra 1 level. Students will first learn the properties of square roots and associated operations to include solving basic radical equations. Next the chapter looks at the application of radicals and how they help solve many problems in algebra. Also the chapter will cover the Pythagorean Theorem and the Distance and Mid-Point formula. | 677.169 | 1 |
Homework
Homework Help
You are encouraged to seek help through all the means available to you: instructors, resource center, internet, tutors, etc. The internet can do your homework for you, especially the computational parts. However, it is your responsibility to seek only those means of help through which you learn. Specifically, when you write your solutions, you must write them alone, in your own words, using your textbook and course notes if necessary, not copying from other notes, websites, friends, or any other source. This means you can work on problems with your friends, your tutor, or your dog, but you must not copy answers during the discussion; instead, you must write in your own words, afresh from your own newly improved brain, after the discussing and working has been done. This is course policy, but it is also common sense study habits. Failure to follow this policy may result in a grade of zero.
Written Homework Assignments
Due Thursday, September 6th: 7.124, 70, 72, 86 (note: these are some harder substitutions; do the easier ones on webwork first if you are having trouble); 8.112, 27 (for this problem, please slice the water into horizontal slices and do an integral; then compare to a simpler geometric method to compute the volume!); 8.27, 29, 36. Extra problem: Compute the volume of a 5-dimensional sphere, using the formula for the volume of a 4-dimensional sphere. Solution here. SOLUTIONS: click on each number above.
Due Thursday, September 13th: 7.21, 22, 23, 26 (note: these are some harder integration-by-parts problems; do the easier ones on webwork first if you are having trouble); 8.214. 8.312, 15, 16, 25, 31, 34, 41, 42. SOLUTIONS: click on each number above.
Due Thursday, September 20th: 7.78, 12, 16, 36, 43, 50; Extra Problem 1: Find the antiderivative of xnex, where n is a positive integer. That is, figure out what the answer looks like as a formula depending on n. Hint: Do a few small cases to figure out the general pattern. Explain what you do. Extra Problem 2: Suppose that every undergraduate at CU takes a True-False test by guessing. How many questions does the test need to have so that the expected number of perfect scores is closest to 1? What about if it is a multiple choice test where each problem has 5 options? For the multiple choice test, do you still expect the scores on the test to be distributed as a bell curve (hint: play with the quincunx)? What is the average grade? SOLUTIONS: click on each number above (solutions to extra problems here).
Due WEDNESDAY, October 3rd: Note the due date one day early! NOTE: This is a large assignment, and it will count double: once for a Written Homework and also a Webwork (it replaces the Webwork due Monday the 1st). 8.43, 8, 11, 28. Chapter 7 Review26, 42, 66, 92, 126, 127, 153, 169, 174 (all these can be done by hand with the methods we've learned). Chapter 8 Review5, 19, 28, 41, 47, 49, 50, 51, 52, 53. October 11th: 9.142, 43, 45, 52, 55 9.23, 7, 12 Extra Problem: If a sequence is given by recursive formula Sn = 2Sn-1 + 3Sn-2, and S3 = 9 and S4=27, then a) what are S1 and S2; and b) what is a closed form for Sn? Check that your closed formula works for the first four terms. Finally, think back to the tutorial, where we used a recursive formula to verify a closed formula by checking that if the closed formula is true for term Sn, then it is true for term Sn+1. Use the same method here to prove your closed formula is correct. This method is called induction. Solution to extra problem here.
Due Thursday, November 1st: 10.229, 30, 32 10.34, 19 10.47, 21 Extra Problem: Find all power series which are equal to their own derivative (i.e. write down a general power series and determine what has to be true about its coefficients; using this, explain what is the full set of solutions).
Due WEDNESDAY, November 7th: Note the due date one day early! NOTE: This is a large assignment, and it will count double: once for a Written Homework and also a Webwork (it replaces the Webwork due Monday the 5th). Chapter 9 Review: 1, 3, 5, 6, 30, 31, 34, 36, 37, 41, 43, 45, 47, 54, 58, 71, 72, 76-80 Chapter 9 Check Your Understanding: 2, 7, 8, 12, 20, 28, 44 (for `Check Your Understanding' questions, provide a justification for why the answer is True/False) Chapter 10 Review: 3, 12, 17, 21, 28, 30, 32, 34 (note: think before finding a series the 'direct' way; it may be easier to use one of the methods we've seen) Chapter 10 Check Your Understanding: 20, 22-24 November 15th: 11.21,2,4,6,10,11,12; for some of these you will need to draw on the slope fields: you can find printable versions of these fields here (please make sure you use the right one for the right problem! compare with your text). Problem 1: A color, A black-white (note! solutions are in the back of the book; draw *different* curves (there are infinitely many to choose from) for full credit, Problem 4: B color, B black-white, Problem 11: C color, C black-white (if you use the black-white versions, please draw with a color pen); 11.46,12,20. | 677.169 | 1 |
Year 10 maths coursework
Affordable 5 classes home tuition in Maths English Science. Ditional Maths Year 10 e Learning Videos; Revision Workbook for IGCSE Add Maths Year 10 by topics; Additional Maths Year. IGCSE Add Maths Exam Solutions. Eds Huddersfield Bradford tutors Key Stage 1 4 Primary GCSE SATs 11+ exams Education First. Ditional Maths Year 10 e Learning Videos; Revision Workbook for IGCSE Add Maths Year 10 by topics; Additional Maths Year. The General Certificate of Education (GCE) Advanced Level, or A Level, is a secondary school leaving qualification in the United Kingdom, offered as a main. The Mathematical Tripos is the taught mathematics course in the Faculty of Mathematics at the University of Cambridge. IGCSE Add Maths Exam Solutions. This time of year I've usually been carefully considering for months the. Eds Huddersfield Bradford tutors Key Stage 1 4 Primary GCSE SATs 11+ exams Education First. Is the oldest Tripos examined in? International Foundation Certificate (IFC) in Legal Studies. R International Foundation Certificate (IFC) in Legal Studies is the ideal preparation for progressing. Affordable 5 classes home tuition in Maths English Science. For! Follow us on twitter for access to Google drive and first downloads on resources and lessons weteachmaths Visit weteachmaths. I always have an idea of the ones that are next on my list it's carefully planned out. The General Certificate of Education (GCE) Advanced Level, or A Level, is a secondary school leaving qualification in the United Kingdom, offered as a main.
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In certain classes, all it's going to take to move an exam is take note getting, memorization, and remember. However, exceeding in a very math class requires a different form of energy. You can't basically exhibit up for a lecture and enjoy your teacher "talk" about algebra and . You understand it by accomplishing: paying attention in school, actively learning, and solving math complications – even when your instructor hasn't assigned you any. For those who end up battling to carry out properly inside your math course, then check out finest site for resolving math issues to determine the way you could become an improved math college student.
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Math programs comply with a purely natural development – every one builds on the understanding you have received and mastered from your past class. When you are getting it challenging to comply with new ideas in school, pull out your outdated math notes and assessment preceding substance to refresh you. Be sure that you meet up with the prerequisites before signing up for just a class.
Overview Notes The Evening Right before Class
Detest any time a instructor phone calls on you and you've forgotten ways to address a selected dilemma? Stay clear of this instant by reviewing your math notes. This will assist you figure out which concepts or thoughts you'd prefer to go more than at school another working day.
The considered doing homework every evening could appear frustrating, but if you would like to succeed in , it's important that you continuously exercise and master the problem-solving approaches. Make use of your textbook or on the net guides to work via top math challenges on a weekly foundation – even though you've got no research assigned.
Use the Nutritional supplements That come with Your Textbook
Textbook publishers have enriched fashionable publications with further materials (which include CD-ROMs or on the net modules) that can be utilized to support college students achieve further apply in . Some of these supplies can also include a solution or explanation guideline, which often can allow you to with operating via math complications all by yourself.
Go through In advance To stay In advance
If you would like to lessen your in-class workload or maybe the time you shell out on homework, make use of your spare time immediately after college or within the weekends to browse ahead for the chapters and ideas that should be covered the subsequent time you might be in school.
Evaluate Aged Exams and Classroom Illustrations
The do the job you are doing at school, for homework, and on quizzes can present clues to what your midterm or last exam will appear like. Make use of your aged assessments and classwork to produce a own review guidebook for your personal impending test. Appear in the way your teacher frames inquiries – this really is almost certainly how they'll seem on your own check.
Learn to Do the job Through the Clock
This can be a preferred study idea for men and women using timed exams; specifically standardized checks. When you have only 40 minutes for just a 100-point exam, then you can optimally expend 4 minutes on each and every 10-point concern. Get details regarding how very long the take a look at will be and which types of thoughts will be on it. Then approach to assault the easier inquiries initial, leaving by yourself ample time and energy to expend about the far more difficult types.
Maximize your Assets for getting math homework assist
If you're getting a tough time comprehending ideas in school, then be sure to get support beyond course. Question your pals to create a analyze team and go to your instructor's business several hours to go around tricky problems one-on-one. Attend study and critique classes whenever your teacher announces them, or seek the services of a non-public tutor if you need a person.
Talk To Yourself
Whenever you are reviewing complications for an examination, try out to explain out loud what technique and strategies you accustomed to get the solutions. These verbal declarations will arrive in handy during a take a look at if you should remember the steps you ought to choose to locate a remedy. Get further apply by making an attempt this tactic that has a friend.
Use Research Guides For Excess Exercise
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Arithmetic of Complexity and Dynamical structures is an authoritative connection with the elemental instruments and ideas of complexity, platforms concept, and dynamical platforms from the point of view of natural and utilized arithmetic. advanced structures are platforms that contain many interacting components having the ability to generate a brand new caliber of collective habit via self-organization, e.
Each year scholars pay up to $1000 to check prep businesses to organize for the GMAT. you can now get an identical guidance in a ebook. GMAT Prep path offers the similar of a two-month, 50-hour path. even supposing the GMAT is a tough try, it's a very learnable try out. GMAT Prep path provides a radical research of the GMAT and introduces various analytic thoughts to help you immensely, not just at the GMAT yet in company tuition to boot.
This publication includes refereed papers that have been provided on the thirty fourth Workshop of the foreign tuition of arithmetic "G. Stampacchia," the foreign Workshop on Optimization and keep watch over with functions. The e-book includes 28 papers which are grouped in response to 4 large subject matters: duality and optimality stipulations, optimization algorithms, optimum keep an eye on, and variational inequality and equilibrium difficulties.
In those essays, David Harvey searches for enough conceptualizations of house and of asymmetric geographical improvement that would support to appreciate the recent ancient geography of worldwide capitalism. the speculation of asymmetric geographical improvement wishes extra exam: the intense volatility in modern political monetary fortunes throughout and among areas of the area financial system cries out for larger historical-geographical research and theoretical interpretation.
If the equation solves to x = any number, then the graph is a vertical line. - If the equation solves to y = any number, then the graph is a horizontal line. - When graphing a linear inequality, the line will be dotted if the inequality sign is < or > . If the inequality signs are either ≥ or ≤ , the line on the graph will be a solid line. Shade above the line when the inequality sign is ≥ or > . Shade below the line when the inequality sign is < or ≤ . Inequalities of the form x >, x ≤, x <, or x ≥ number, draw a vertical line (solid or dotted).
STEP 4. Combine like terms on each side of the equation or inequality. STEP 5. If there are variables on both sides of the equation, add or subtract one of those variable terms to move it to the other side. Combine like terms. STEP 6. If there are constants on both sides, add or subtract one of those constants to move it to the other side. Combine like terms. STEP 7. If there is a coefficient in front of the variable, divide both sides by this number. This is the answer to an equation. However, remember: MATHEMATICS/SCIENCE MS 34 TEACHER CERTIFICATION STUDY GUIDE Dividing or multiplying an inequality by a negative number will reverse the direction of the inequality sign. | 677.169 | 1 |
Mind Power Math - High School and College Math | 1.46 GB Genre: eLearning
This set of six CDs offers high school students basic and useful practice in algebra 1, algebra 2, statistics, geometry, trigonometry and calculus. In the Algebra 1 CD, for example, students choose from 12 chapters and pick a sub topic. After a written (and pictorial) demonstration of the concept (e.g. ratio and proportion), students answer 10 questions to show comprehension. Hint and Solution icons lead to useful and straightforward help, and general progress is described after each section. All things considered, although dry in format and somewhat lacking in depth, this handy set of CDs offers understandable explanations of difficult subjects.
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A Walk Through Combinatorics: An Introduction to Enumeration by Miklos Bona
It is a textbook for an introductory combinatorics direction that could take in one or semesters. an intensive record of difficulties, starting from regimen workouts to analyze questions, is incorporated. In each one part, there also are routines that include fabric no longer explicitly mentioned within the previous textual content, on the way to supply teachers with additional offerings in the event that they are looking to shift the emphasis in their direction. simply as with the 1st variation, the recent version walks the reader during the vintage elements of combinatorial enumeration and graph thought, whereas additionally discussing a few fresh growth within the region: at the one hand, offering fabric that may aid scholars research the fundamental strategies, and nonetheless, displaying that a few questions on the vanguard of study are understandable and available for the gifted and hard-working undergraduate.The simple themes mentioned are: the twelvefold method, cycles in variations, the formulation of inclusion and exclusion, the suggestion of graphs and bushes, matchings and Eulerian and Hamiltonian cycles. the chosen complex themes are: Ramsey concept, development avoidance, the probabilistic process, partly ordered units, and algorithms and complexity. because the aim of the publication is to motivate scholars to benefit extra combinatorics, each attempt has been made to supply them with a not just worthwhile, but additionally relaxing and interesting examining.
The basic principles touching on computation and recursion evidently locate their position on the interface among good judgment and theoretical computing device technology. The contributions during this ebook offer an image of present rules and strategies within the ongoing investigations into the constitution of the computable and noncomputable universe.
A Polish house (group) is a separable, thoroughly metrizable topological house (group). This publication is set activities of Polish teams, in connection with--or from the perspective of--the topic of descriptive set idea. Descriptive set idea is the examine of definable units and capabilities in Polish areas.
This can be an advent to brooding about basic arithmetic from a categorial perspective. The objective is to discover the implications of a brand new and primary perception concerning the nature of arithmetic. Foreword; word to the reader; Preview; half I. the class of units: 1. units, maps, composition; half II.
Graph concept is a space in discrete arithmetic which reports configurations (called graphs) regarding a collection of vertices interconnected through edges. This e-book is meant as a normal creation to graph conception and, specifically, as a source booklet for junior students and lecturers examining and educating the topic at H3 point within the new Singapore arithmetic curriculum for junior university.
Additional info for A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory (2nd Edition)
Sample text
Let us split our group of players into One Step at a Time. The Method of Mathematical Induction 31 two groups of size 2k each. Have both groups play a round-robin tournament. By the induction hypothesis, that is possible in 2k — 1 rounds. Then denote the players in the two groups 01,02, •• • ,02* and 61,62, • • • , &2fc- Have them play 2k rounds as follows. In the first round, a; plays 6*. In the second round en plays 6j+i, modulo 2k, that is, aik plays 61. Continue this way, in round j , a,i will play bi+j.
We have to tell how many of these arrangements differ only because of these labels. The five red flowers could be given five different labels in 5! different ways. The three yellow flowers could be given three different labels in 3! different ways. The two white flowers could be given two different labels in 2! different ways. Moreover, the labeling of flowers of different colors can be done independently of each other. Therefore, the labeling of all ten flowers can be done in 5! -3! -2! different ways once the flowers are planted in any of A different ways.
The left-hand side is a sum that is not an arithmetic series or a geometric series, so we could not use the known formulae for those series. Moreover, the right-hand side look slightly counter-intuitive; for example, it is not clear how the number 6 will show up in the denominator. The method of mathematical induction, however, solves this problem effortlessly as we will see below. Solution. (1) The Initial Step. If m = 1, then the left-hand side is 1, and so is the right-hand side, so the statement is true. | 677.169 | 1 |
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Application-oriented introduction relates the subject as closely as possible to science. In-depth explorations of the derivative, the differentiation and integration of the powers of x, and theorems on differentiation and antidifferentiation ... | 677.169 | 1 |
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Description: The preparation necessary for the profitable study of the following course of Mathematics is a knowledge of common Arithmetic, and some acquaintance with the principles of Geometry, as taught in Euclid's Elements. A student ignorant of these initiatory, but most important departments of elementary science, would scarcely seek his first lessons therein from a book such as this. We shall commence the volume now in the hands of the reader, with a treatise on Algebra.
Fundamentals of Practical Mathematics by George Wentworth - Ginn and Co. This work reviews four fundamental operations with integers and fractions, the practical use of percentage, the applications of proportion, the elements of mensuration, the use of the formula and the equation, the finding of roots, and trigonometry. (3058 views)
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We will have a short quiz on Chapter 6 tomorrow, based on the reading, homework, and what we've discussed in class. Tonight, please finish reading chapter 6 (though skip the section on straightening an association for now; the quiz will not include that section).
Your second quiz of Unit 1 is tomorrow! This quiz will cover word problems and adding/subtracting/multiplying polynomials. You got the Quiz 2 Review in class today, so finish that for homework (and see the answers here).
You may also want to work on some IXL practice problems for additional practice. I've linked the list below. | 677.169 | 1 |
graphDS is a program for the Nintendo DS that can evaluate mathematical expressions and graph Cartesian, parametric, and polar equations.
Release notes:
In v0.9.1, I focused on fixing bugs with graphing. Graphing will no longer result in random pixels and lines being plotted and I've also refined my algorithms so that weird things don't happen near the edges of the screen. Please be aware that I also resized the graphscreen, so there will be a white space 4 pixels wide around the entire graphing area | 677.169 | 1 |
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This e-book bargains with numerous subject matters in algebra valuable for laptop technology functions and the symbolic therapy of algebraic difficulties, declaring and discussing their algorithmic nature. the subjects lined variety from classical effects resembling the Euclidean set of rules, the chinese language the rest theorem, and polynomial interpolation, to p-adic expansions of rational and algebraic numbers and rational services, to arrive the matter of the polynomial factorisation, particularly through Berlekamp's procedure, and the discrete Fourier rework. easy algebra suggestions are revised in a sort suited to implementation on a working laptop or computer algebra system.
Hans-Paul Schwefel explains and demonstrates numerical optimization equipment and algorithms as utilized to laptop calculations--which might be rather beneficial for hugely parallel desktops. The disk includes all algorithms offered within the ebook.
This quantity offers a concise advent to the method of nonstandard finite distinction (NSFD) schemes building and indicates how they are often utilized to the numerical integration of differential equations happening within the common, biomedical, and engineering sciences. those tools had their genesis within the paintings of Mickens within the 1990's and are actually starting to be generally studied and utilized through different researchers.
This ebook covers using Mathematica as programming language. a variety of programming paradigms are defined in a uniform demeanour, with totally labored out examples which are precious instruments of their personal correct. The floppy disk comprises various Mathematica notebooks and programs, priceless instruments for using all the tools mentioned
Moreover, before the point we have 0. It is easy to see that this is a general fact. 3 3. Let us see now an example with p = 10. Expand − 11 . We have 102 ≡ 2 1 mod 11, so the period is d = 2. Moreover, 10 − 1 = 11 · 9, so m = at = 3 · 9 = 27. 2. Remark. The usual decimal expression of a proper fraction ab with a denominator coprime with 10 can be obtained as above with p = 10. More precisely, having found the least d such that 10d ≡ 1 mod b, that is, the period, let 10d − 1 = bt. Then, as above, at m a = d = d .
Proof. First of all, note that if ∂a < ∂g, then necessarily ∂b < ∂f , otherwise ∂h = ∂(af + bg) = ∂bg ≥ ∂f + ∂g, against the hypothesis. Suppose now that ∂a ≥ ∂g; dividing we get a = gq + r with ∂r < ∂g; so, set a1 = a − gq and b1 = b + f q and we get a1 f + b1 g = h; since ∂a1 < ∂g we find ∂b1 < ∂f , by the above argument. 5. Let f and g be two relatively prime polynomials. Then there exist two polynomials a and b with ∂a < ∂g and ∂b < ∂f , and such that af + bg = 1. Moreover, a and b are uniquely determined.
4 Series expansion of rational functions A polynomial f (x) with coefficients in a field (which in what follows will be the field Q of rationals) is a linear combination of the monomials 1, x, x2 , . , with coefficients equal to zero from some point on: f (x) = a0 + a1 x + · · · + an xn + 0 · xn+1 + · · · . In this form a polynomial is a (finite) series of powers of x, or in base x. If p(x) is an arbitrary polynomial of first degree, we have analogously the expansion: f (x) = c0 + c1 p(x) + c2 p(x)2 + · · · + cn p(x)n + 0 · p(x)n+1 + · · · in base p(x). | 677.169 | 1 |
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The discovery of new algorithms for dealing with polynomial equations, and their implementation on fast, inexpensive computers, has revolutionized algebraic geometry and led to exciting new applications in the field. This book details many uses of algebraic geometry and highlights recent applications of Grobner bases and resultants. This edition contains two new sections, a new chapter, updated references and many minor improvements throughout.
"The book has been very successful. It succeeded in establishing a bridge between modern computer science and classical algebraic geometry." (Gerhard Pfister, Zentralblatt MATH, Vol. 1079, 2006)
"The authors a ] do not expect much from their reader in terms of mathematical prerequisites. a ] This reviewer thinks that the part of the intended audience consisting of graduate students interested in research in computational algebraic geometry will find this to be a very useful book. The many exercises throughout the text, and the mere fact that this is a very good and detailed overview of the subject will definitely make it desirable for this group." (Gizem Karaali, MathDL, September, 2005 | 677.169 | 1 |
Math Competition Links
American Mathematics Contest 8 (Middle School) The AMC 8 is a 25 question, 40 minute multiple choice examination in junior high school (middle school) mathematics designed to promote the development and enhancement of problem solving skills. The examination provides an opportunity to apply the concepts taught at the junior high level to problems that not only range from easy to difficult but also cover a wide range of applications.
American Mathematics Contest 10 (Secondary Grades) The AMC 10 is a 25-question, 75-minute multiple choice examination in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts. Calculators are allowed. The main purpose of the AMC 10 is to spur interest in mathematics and to develop talent through the excitement of solving challenging problems in a timed multiple-choice format. The problems range from the very easy to the extremely difficult.
American Mathematics Contest 12 (Secondary Grades) The AMC 12 is a 25-question, 75-minute multiple choice examination in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts. Calculators are allowed. The main purpose of the AMC 12 is to spur interest in mathematics and to develop talent through solving challenging problems in a timed multiple-choice format. Because the AMC 12 covers such a broad spectrum of knowledge and ability there is a wide range of scores. The National Honor Roll cutoff score, 100 out of 150 possible points, is typically attained or surpassed by fewer than 3% of all participants. The AMC 12 is one in a series of examinations (followed in the United States by the American Invitational Examination and the USA Mathematical Olympiad) that culminate in participation in the International Mathematical Olympiad, the most prestigious and difficult secondary mathematics examination in the world.
The Mandelbrot Competition (Secondary Grades) In those ten years the contest has grown to two divisions encompassing students from across the United States as well as from several foreign countries. Nearly half of the competitors in the USA Math Olympiad in the last couple of years have been Mandelbrot competitors. The Mandelbrot Competition is split into two divisions: Division A for more advanced problem solvers and Division B for less experienced students.
Mathcounts (Grades 7-8) Each year, more than 500,000 students participate in MATHCOUNTS at the school level. Those who do tell us that their experience as a "mathlete" is often one of the most memorable and fun experiences of their middle school years.
Math Problems of the Week (Grades K-12) The Problem of the Week is an educational web site that originates at the University of Mississippi. All the prizes are generously donated by CASIO electronics. All contest winners are chosen randomly from the pool of contestants that successfully solve that week's problem | 677.169 | 1 |
Holt pre-algebra workbook ePub Telecharger Gratuit
Please use this form if you would like to have this math solver on your website, free of charge. holt mcdougal algebra 1, geometry and algebra 2 student editions are now streamlined to focus holt pre-algebra workbook on a deeper understanding of math strategies and concepts. if you need support with algebra and in particular with rearranging formulas calculator or precalculus come visit us at algebra-equation.com. holt pre-algebra workbook holt mcdougal mathematics offers new student editions that focus on deeper understanding of math concepts. holt mcdougal algebra 1, geometry and algebra 2 student editions are now streamlined to holt pre-algebra workbook focus on a deeper understanding of math strategies and concepts pearson prentice hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum. we carry a large amount. shop now at houghton mifflin harcourt classzone book finder. holt mcdougal mathematics offers new student editions that focus on deeper understanding of math concepts. follow these simple steps to holt pre-algebra workbook find online resources for your book great selection of school curriculum books and resources from best selling publishers including popular brands like saxon math, houghton mifflin, wordly wise, modern. shop now at houghton mifflin harcourt holt pre-algebra workbook classzone book finder. classzone book finder. name: 10 science test-7th grade if you need support with algebra and in particular with rearranging formulas holt pre-algebra workbook calculator or precalculus come visit us at algebra-equation.com. follow these simple steps to find online resources for your book. holt mcdougal mathematics offers new student editions that focus on deeper understanding of math holt pre-algebra workbook concepts. follow these simple steps to find online resources for your book great selection of school curriculum books and resources from best selling publishers including popular brands like saxon math, houghton mifflin, wordly wise, modern. a new era in the middle school math curriculum and a new approach for reaching all students! in keeping with the common core standards, holt mcdougal mathematics. shop now at houghton mifflin harcourt classzone book finder. please use this form if you would like to have this math solver on your website, free of charge. name:.
Holt pre-algebra workbook ePub Gratuit
Please use this form if you would like to have this math solver on your website, free of charge. holt mcdougal mathematics offers new student editions that focus on deeper understanding of math concepts. we carry a large amount. holt mcdougal algebra 1, geometry and algebra 2 student editions are now streamlined to focus holt pre-algebra workbook on a deeper understanding of math strategies and concepts pearson prentice hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum. classzone book finder. name:. name:. we carry a large amount. holt mcdougal mathematics holt pre-algebra workbook offers new student editions that focus on deeper understanding of math concepts. in keeping with the common core standards, holt mcdougal mathematics. please use holt pre-algebra workbook this form if you would like to have this math solver on your website, free holt pre-algebra workbook of charge. shop now at houghton mifflin harcourt classzone book holt pre-algebra workbook finder. shop now at houghton mifflin harcourt classzone book finder. holt mcdougal algebra 1, geometry and algebra 2 student editions are now streamlined to focus on a deeper understanding of math strategies and concepts. if you need support with algebra and in particular with rearranging formulas calculator or precalculus come visit us at algebra-equation.com holt pre-algebra workbook 10 science test-7th grade if you need support with algebra and in particular with rearranging formulas calculator or precalculus come visit us at algebra-equation.com. holt mcdougal mathematics offers new student editions that focus on deeper understanding of math concepts. a new era in the middle school math curriculum and a new approach for reaching all students! follow these simple steps to find online resources for your book great selection of school curriculum books and resources from best selling publishers including popular brands like saxon math, houghton mifflin, wordly wise, modern.
Holt pre-algebra workbook eBook Herunterladen
Holt mcdougal algebra 1, geometry and algebra 2 student editions are now streamlined to focus on a deeper understanding of math strategies and concepts pearson prentice hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum. holt pre-algebra workbook name:. classzone book finder. name:. please use this form if you would like to have this math solver on your website, free of charge. holt mcdougal mathematics offers new student editions that focus on deeper understanding of math concepts. if you need support with algebra and in particular with rearranging formulas calculator or precalculus come visit us at algebra-equation.com. shop now at houghton mifflin harcourt classzone book finder. please use this form if you would like to have this math solver on your website, free of charge. follow these simple steps to find online resources for your book great selection of school curriculum books and resources from best selling publishers including popular brands like saxon math, houghton mifflin, wordly wise, modern. pearson prentice hall and our other respected imprints provide educational materials, technologies, assessments holt pre-algebra workbook and related services across the secondary curriculum quizlet provides test 7th grade holt science chapter 9 activities, flashcards and chapter 9, 7th grade holt life science chapter 9 & 10 holt pre-algebra workbook science test-7th grade if holt pre-algebra workbook you need support with algebra and in particular with rearranging formulas calculator or precalculus come visit us at algebra-equation.com. shop now at houghton mifflin harcourt classzone book finder. holt mcdougal mathematics offers new student editions that focus on deeper understanding of math concepts. holt pre-algebra workbook a new era in the middle school math curriculum and a new approach for reaching all students! shop now at houghton mifflin harcourt classzone book finder. holt mcdougal mathematics offers new student editions that focus on deeper understanding of math concepts. | 677.169 | 1 |
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About the Book: Numerical Methods This book is a complete, unified, informative, lucid, and adequate for learning and using of numerical methods for science students at undergraduate and postgraduate level. The book will also cater needs
About the Book: Diffrential Calculus Differential Calculus is written with an objective to present student-friendly content. It explores fundamental concepts in a step by step, easy and understandable way with numerous solved graded examp
It has been the authors experience that the overwhelming majority of students in MBA derivatives courses go on to careers where a deep conceptual, rather than solely mathematical, understanding of products and models is required. The first edition of D
About the Book: Calculus: 4th Edition This book offers students and instructors a mathematically sound text, robust exercise sets and elegant presentation of calculus concepts. The new edition has been updated with a reorganization of th
About the Book: Oxford New Enjoying Mathematics Class 7 : CceNew Enjoying Mathematics revised edition series it places emphasis on developing thinking and reasoning skills among students, by connecting the mathematics curriculum with real
About Author: Jose Paul is Ex- Director, Educational Planning Group at St. Xaviers School, Delhi. His main contribution to the field of Education is the numerous workshops he conducts country-wide on methodology of t
Fundamentals of Error Correcting Codes is an in-depth introduction to coding theory from both an engineering and mathematical viewpoint. It reviews classical topics, and gives much coverage of recent techniques which could previously only be found in spec
About the Book: Fundamentals of Complex Analysis With Applications to Engineering, Science, and Mathematics: 3rd Edition This is the best seller in this market. It provides a comprehensive introduction to complex variable theory and i
About the Book: Engineering Mathematics for Semester I and II The textbook on Engineering Mathematics has been created to provide an exposition of essential tools of engineering mathematics which forms the core of all branches of engineer
About the Book: NCERT Handbook Term 2 Mathematics Class 9 (OTBA + Value Questions)
The book "NCERT Handbook Term 2 Mathematics Class 9" is exclusively written for CBSE students of class 9.
OTBA, Open Text Based Assessme
About the Book: NCERT Handbook Term 2 Mathematics Class 10 (Value Questions)
The book NCERT Handbook Term 2 Mathematics Class 10 is exclusively written for CBSE students of class 10.
The book provides Quick Revision/ No | 677.169 | 1 |
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