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Concept Book Concept Book A Mathematics Reference for Teachers and Students Authors: America’s Choice® developed and field-tested Ramp-Up to Algebra and Ramp-Up to Pre-Algebra over a ten-year period. Initial development started with a grant from the Office of Educational Research and Improvement (OERI) of the U.S. Department of Education, and was based on the most recent research in mathematics. During that ten-year period many authors, reviewers, and math consultants—both from the United States and internationally—contributed to this course. The materials underwent three major revisions as we analyzed field-test results and consulted math content experts. During the entire time, Phil Daro guided the development of the entire Ramp-Up to Algebra and Ramp-Up to Pre-Algebra curriculum. See the Getting Started Teacher Resource guide for a complete list of people involved in this effort. This work is protected by United States copyright laws and is provided solely for the use of teachers and administrators in teaching courses and assessing student learning in their classes and schools. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. Copyright © 2009 Pearson Education, Inc., or its affiliates. All Rights Reserved. Printed in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions, write to Pearson Curriculum Group Rights & Permissions, One Lake Street, Upper Saddle River, New Jersey 07458. ISBN 13: 978-1-60637-744-4 4 5 6 7 8 9 10 11 12 15 14 13 12 11 Table of Contents Chapter 1. Mathematical Reasoning 1 2. Properties and Notation 9 3. Using Algebra to Solve Problems 23 4. Place Value 45 5. The Number Line 61 6. Addition and Subtraction 69 7. Multiplication and Division 89 8. Factors and Multiples 117 9. Fractions and Decimals 139 10. Percents 173 11. Positive and Negative Numbers 185 12. Graphs 199 13. Geometric Figures 211 Concept Book | iii Table of Contents Chapter 14. Geometric Measure 231 15. Geometric Similarity 255 16. Units and Quantities 271 1 7. Ratio 291 18. Rates 305 19. Proportional Relationships 315 20. Graphing Relationships 327 2 1. Exponents and Roots 335 22. Equations 355 23. Expressions 373 24. Representing Data 385 Index 397 iv | Concept Book CHAPTER MATHEMATICAL REASONING 1 Particular Statements and General Statements — —————— Mathematical statements can be true or untrue, and they can be about particular things or about more general classes of things. Example The statement 2 + 2 = 2 • 2 is a true statement about some particular numbers. The statement 3 + 3 = 3 • 3 is another statement, but it is untrue. The statement x = x + 1 is a general statement but it is untrue, no matter what value is chosen for x. In words, this untrue statement says, “A number is equal to 1 more than itself.” This can never happen. The statement a + b = b + a is a general statement about adding two numbers a and b. It is a true statement for any numbers a and b. The most important statements in mathematics are of this kind, general and true. Concept Book | 1 Chapter 1 Always, Sometimes, Never — ——————————————————— General statements about numbers in mathematics are either true, false, or conditional. In your Student Book, this is referred to this as always true, sometime true or never true. Example The statement a + b = b + a is true. It is the commutative property of addition and holds for all numbers a and b. The statement a = a + 1 is false (never true). It holds for no number a. The statement a + a = a • a is conditional (sometimes true.) It is true for some values of a, and false for other values. It is true for 2, since 2 + 2 = 4, and 2 • 2 = 4. It is false for many numbers, such as 3, since 3 + 3 = 6, but 3 • 3 = 9. Examples and Counterexamples — ——————————————— To explain why a general statement is conditional (sometimes true), you need two things: An example showing a case when it is true, such as x = 2 in x + x = x • x A counterexample showing a case when it is untrue, such as x = 3 in x + x = x • x Example x + x = x • x is conditional (sometimes true). x = 2 makes the equation true. This is the example. x = 3 makes the equation false. This is the counterexample 2 | Concept Book MATHEMATICAL REASONING Example Any whole number can be expressed as the sum of consecutive numbers. You could experiment with examples: 11 = 5 + 6 22 = 4 + 5 + 6 + 7 5 = 3 + 2 37 = 18 + 19 After these examples, you might be tempted by induction to conclude that the statement is true. But, 8 is a counterexample, proving by deduction that the statement is not true. In fact, 2, 4, 8 and all powers of 2 can not be expressed as sums of consecutive whole numbers. True and Untrue for Different Types of Numbers ——————— Often the truth of a statement depends on the type of number you are discussing. Example When you write about factors and multiples, it is normally assumed that you are writing about whole numbers. So, even though 1 • 14 = 7, you would not say that 1 and 14 are 2 2 factors of 7, since 1 is not a whole number. 2 Concept Book | 3 Chapter 1 Example Think about the statement a • b > a. If you think about whole numbers and multiplications you might think that this statement is always true for all numbers a and b. But looking at a larger set of numbers, you need to break down the possible values of a and b into all their separate cases. The following table sorts out when the statement a • b > a is true and when it is false. b > 1 b = 1 b < 1 a > 0 True False, a • b = a False, a • b < a a = 0 False, a • b = a False, a • b = a False, a • b = a a < 0 False, a • b < a False, a • b = a True Using this analysis, the sometimes true statement a • b > a can be replaced with these two statements: a • b > a is true for a > 0 and b > 1, and also for a < 0 and b < 1. a • b ≤ a is true for a > 0 and b ≤ 1, and also for a < 0 and b ≥ 1, and for a = 0. Always true statements are more useful to mathematicians than sometimes true statements, because always true statements help them examine groups of values. 4 | Concept Book MATHEMATICAL REASONING Equations ——————————————————————————————— The truth of an equation is often undecided until the variable is replaced with a chosen value. Example x + x = x • x Replacing x with 2 gives 2 + 2 = 2 • 2, which is a true statement. Replacing x with 0 gives 0 + 0 = 0 • 0, which is a true statement. Replacing x with 3 gives 3 + 3 = 3 • 3, which is a false statement. The values of the variable that make the equation a true statement are called the solutions to the equation. You can use algebra to show that x = 0 and x = 2 are the only solutions to the equation x + x = x • x. The equation x2 + 2x = x(x + 2) follows from the distributive property. It is true for all values of x. An equation true for all values of a variable is called an identity. The equation x = x + 1 has no solutions. Finding Out and Saying Why ——————————————————— In mathematics, you can find interesting number patterns or statements that appear to be true in a large number of cases. Justifying or explaining why these number patterns or statements are true in all cases is also part of mathematics. This mathematical process is called a justification. Concept Book | 5 Chapter 1 Example Finding the Pattern Look for patterns in the calculations below. Note that many digits are reversed, and that many of the equations involve subtracting or dividing by 9. 73 − 37 = 36 36 ÷ 9 = 4 987 − 789 = 198 198 ÷ 9 = 22 82 − 28 = 54 54 ÷ 9 = 6 786 − 687 = 99 108 ÷ 9 = 12 75 − 57 = 18 18 ÷ 9 = 2 150 − 051 = 99 135 ÷ 9 = 15 98 − 89 = 9 9 ÷ 9 = 1 9753 − 3579 = 6174 6174 ÷ 9 = 686 Claim: If the difference is calculated between a number and the number with digits reversed, this difference is a number that is divisible by 9. Saying Why—The Justification A mathematical justification must go beyond only writing examples. To justify the statement for two-digit numbers, you could write a two-digit number as 10a + b, where a is the tens digit and b is the ones digit: for example, 73 = 10 • 7 + 3, where a = 7 and b = 3. With the digits reversed, the number is 10b + a: 37 = 10 • 3 + 7. Given (10a + b) the number (10b + a) the digits reversed (10a + b) − (10b + a) the difference Then = 10a + b − 10b − a distributive property = 10a − a − 10b + b rearranging terms = 9a − 9b collecting like terms = 9(a − b) distributive property 9(a − b) is a multiple of 9 6 | Concept Book MATHEMATICAL REASONING Tools for Mathematical Reasoning — —————————————— To write a mathematical justification, you can use these tools: • Definitions—for example, the definition of an even number • Properties—for example, the distributive property • Previously known results—for example, the sum of angles in a triangle • Given information—a fact included in the problem statement Justification for • Diagrams and words—to explain your reasoning • Letters—to represent the variables Tools • Examples and counterexamples—for sometimes true statements Concept Book | 7 Chapter 1 A Proof Using Number Properties ———————————————— An arithmetic sequence is a list of numbers that are an equal distance apart on the number line, such as 92, 96, 100, 104, 108.
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