Chapter Integers 5 5 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

NCTM Standard: Integers  explore numbers less than 0 by extending the number line and through familiar applications. (p. 148)  develop meaning for integers and represent and compare quantities with them. (p. 214) Principles and Standards recommends that students in grades 3−5 be able to Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

NCTM Standard: Integers and Number Theory Middle-grades students should also work with integers. In lower grades, students may have connected negative integers in appropriate ways to informal knowledge derived from everyday experiences, such as below-zero winter temperatures or lost yards on football plays. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

NCTM Standard: Integers and Number Theory In the middle grades, students should extend these initial understandings of integers. Positive and negative integers should be seen as useful for noting relative changes or values. Students can also appreciate the utility of negative integers when they work with equations whose solution requires them, such as 2x + 1 = 7 (pp. 217–218) Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

5-1Integers and the Operations of Addition and Subtraction Representations of Integers Integer Addition Number-Line Model Absolute Value Properties of Integer Addition Integer Subtraction Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

The set of integers is denoted by I: The negative integers are opposites of the positive integers. – 4 is the opposite of positive 4 3 is the opposite of – 3 Representations of Integers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 5-1 For each of the following, find the opposite of x. a.x = 3 − x = − 3 b.x = − 5 − x = 5 c.x = 0 − x = 0 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Integer Addition Chip Model Black chips represent positive integers and red chips represent negative integers. Each pair of black/red chips neutralize each other. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Integer Addition Charged-Field Model Similar to the chip model. Positive integers are represented by +’s and negative integers by –’s. Positive charges neutralize negative charges. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Integer Addition Number-Line Model Positive integers are represented by moving forward (right) on the number line; negative integers are represented by moving backward (left). Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 5-2 The temperature was − 4°C. In an hour, it rose 10°C. What is the new temperature? − = 6 The new temperature is 6°C. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Integer Addition Pattern Model Beginning with whole number facts, a table of computations is created by following a pattern = = = = 4 Basic facts 4 + − 1 = − 2 = − 3 = − 4 = − 5 = − − 6 = − 2 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Absolute Value The absolute value of a number a, written |a|, is the distance on the number line from 0 to a. Absolute value is always positive or zero. |4| = 4 and | − 4| = 4 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Definition Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 5-3 Evaluate each of the following. a.|20||20| = 20 b.| − 5|| − 5| = 5 c.|0||0| = 0 d. − | − 3 | − | − 3| = − 3 e.|2 + − 5||2 + − 5| = | − 3| = 3 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Integer Addition Integer addition has all the properties of whole- number addition. Given integers a, b, and c. Closure property of addition of integers a + b is a unique integer. Commutative property of addition of integers a + b = b + a. Associative property of addition of integers (a + b) + c = a + (b + c). Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Integer Addition Identity element of addition of integers 0 is the unique integer such that, for all integers a, 0 + a = a = a + 0. Uniqueness of the additive inverse For every integer a, there exists a unique integer − a, the additive inverse of a, such that a + − a = 0 = − a + a. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of the Additive Inverse By definition, the additive inverse, − a, is the solution of the equation x + a = 0. For any integers a and b, the equation x + a = b has a unique solution, b + − a. For any integers a and b − ( − a) = a and − a + − b = − (a + b). Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 5-4 Find the additive inverse of each of the following. a. − (3 + x)3 + x b.a + − 4 − (a + − 4) = − a + 4 c. − 3 + − x − ( − 3 + − x) = 3 + x Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Integer Subtraction Chip Model for Subtraction To find 3 − − 2, add 0 in the form 2 + − 2 (two black chips and two red chips) to the three black chips, then take away the two red chips. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Integer Subtraction Charged-Field Model for Subtraction To find − 3 − − 5, represent − 3 so that at least five negative charges are present. Then take away the five negative charges. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Integer Subtraction Number-Line Model While integer addition is modeled by maintaining the same direction and moving forward or backward depending on whether a positive or negative integer is added, subtraction is modeled by turning around. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Integer Subtraction Pattern Model for Subtraction Using inductive reasoning and starting with known subtraction facts, find the difference of two integers by following a pattern. 3 − 2 = 1 3 − 3 = 0 3 − 4 = 3 − 5 = 3 − 2 = 1 3 − 1 = 2 3 − 0 = 3 3 − − 1 = −1−1 −2−24 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Integer Subtraction We compute 3 – 7 as follows: 3 – 7 = n if and only if 3 = 7 + n. Because 7 + – 4 = 3, then n = – 4. Subtraction Using the Missing Addend Approach Subtraction of integers, like subtraction of whole numbers, can be defined in terms of addition. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Definition Subtraction For integers a and b, a − b is the unique integer n such that a = b + n. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 5-5 Use the definition of subtraction to compute the following: a.3 − 10 Let 3 − 10 = n. Then 10 + n = 3, so n = − 7. b. − 2 − 10 Let − 2 − 10 = n. Then 10 + n = − 2, so n = − 12. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Integer Subtraction Subtraction Using Adding the Opposite Approach Subtracting an integer is the same as adding its opposite. For all integers a and b a − b = a + − b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 5-6 Using the fact that a − b = a + − b, compute each of the following: a.2 − 82 − 8 = 2 + − 8 = − 6 c. − 12 − − 5 − 12 − − 5 = − 12 + − ( − 5) = − = − 7 b.2 − − 8 2 − − 8 = 2 + − ( − 8) = = 10 d. − 12 − 5 − 12 − 5 = − 12 + − 5 = − 17 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 5-7 Write expressions equal to each of the following without parentheses. a. − (b − c) − (b − c) = − (b + − c) = − b + − ( − c) = − b + c b. a − (b + c) a − (b + c) = a + − (b + c) = a + ( − b + − c) = (a + − b) + − c = a + − b + − c Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 5-8 Simplify each of the following. a. 2 − (5 − x) 2 − (5 − x) = 2 + − (5 + − x) = 2 + − 5 + − ( − x) = 2 + − 5 + x = − 3 + x or x − 3 b. 5 − (x − 3) 5 − (x − 3) = 5 + − (x + − 3) = 5 + − x + − ( − 3) = 5 + − x + 3 = 8 + − x = 8 − x Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 5-8 (continued) Simplify each of the following. c. − (x − y) − y − (x − y) − y = − (x + − y) + − y = [ − x + − ( − y)] + − y = ( − x + y) + − y = − x + (y + − y) = − x Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Order of Operations Recall that subtraction is neither commutative nor associative. An expression such as 3 − 15 − 8 is ambiguous unless we know in which order to perform the subtractions. Mathematicians agree that 3 − 15 − 8 means (3 − 15) − 8. Subtractions are performed in order from left to right. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 5-9 Compute each of the following. a.2 − 5 − 52 − 5 − 5 = − 3 − 5 = − 8 b.3 − − = − = − 1 c.3 − (7 − 3)3 − (7 − 3) = 3 − 4 = − 1 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.