6 Chapter Rational Numbers and Proportional Reasoning Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

6-2 Addition, Subtraction, and Estimation with Rational Numbers Mixed Numbers Properties of Addition for Rational Numbers Subtraction of Rational Numbers Estimation with Rational Numbers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Addition of Rational Numbers Area model Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Addition of Rational Numbers Number-line model Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Addition of Rational Numbers with Like Denominators If are rational numbers, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Addition of Rational Numbers with Unlike Denominators If are rational numbers, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-4 Find each of the following sums: a. b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-4 (continued) c. d. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-4 (continued) e. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Mixed Numbers Numbers that are made up of an integer and a fractional part of an integer. A mixed number is a rational number, and therefore, it can always be written in the form Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-5 Change each of the following mixed numbers to the form where a and b are integers. a. b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-6 Change to a mixed number. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Addition for Rational Numbers Additive Inverse Property For any rational number , there exists a unique rational number , called the additive inverse of , such that Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Addition for Rational Numbers Properties of the additive inverse for rational numbers are analogous to those of the additive inverse for integers. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Addition for Rational Numbers Addition Property of Equality If are any rational numbers such that , and if is any rational number, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Subtraction of Rational Numbers If are any rational numbers, then is the unique rational number such that Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Subtraction of Rational Numbers If are any rational numbers, then If are any rational numbers, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-8 Find each difference. a. b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-9 Add or subtract. Write answers in simplest form. a. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 6-9 (continued) b. First, write each fraction in simplest form: Now add the simplified fractions: Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-9 (continued) c. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-9 (continued) d. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Estimation with Rational Numbers Many of the estimation and mental math techniques that are used with whole numbers also work with rational numbers. Estimation plays an important role in judging the reasonableness of computations. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-10 A sixth-grade class is collecting cans to take to the recycling center. Becky’s group brought the following amounts (in pounds). About how many pounds does her group have all together? Use front-end estimation and then adjust by using 0, and 1 as reference points. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-10 (continued) The front-end estimate is 1 + 3 + 5 = 9. The adjustment is The adjusted estimate is 9 + 2 = 11. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-11 Estimate each of the following. a. b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.