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

NCTM Standard: Rational Numbers as Fractions K–2: Children should understand and represent commonly used fractions. 3–5: Children should develop an understanding of fractions to include them as parts of unit wholes, parts of a collection, as locations on number lines, and as divisions of whole numbers. 6–7: Children work flexibly with fractions to solve problems, compare and order fractions, and find their locations on a number line. They understand and use ratios and proportions to represent quantitative relationships. (p.33) Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

6-1 The Set of Rational Numbers Equivalent or Equal Fractions Simplifying Fractions Equality of Fractions Ordering Rational Numbers Denseness of Rational Numbers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Numerator, Denominator Definition Rational numbers The set of numbers Q such that Numerator, Denominator numerator denominator Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Uses of Rational Numbers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Rational Number Models Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Meaning of a Fraction To understand the meaning of any fraction, using the parts-to-whole model, we must consider each of the following: The whole being considered. The number b of equal-size parts into which the whole is divided. The number a of parts of the whole that we are selecting. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition Proper fraction A fraction where Examples are Improper fraction A fraction such that Examples are Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Number Line Model What numbers are represented on the number line? Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Equivalent or Equal Fractions Equivalent fractions are numbers that represent the same point on a number line. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Fraction Strips Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Fundamental Law of Fractions Let be any fraction and n a nonzero whole number, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-1 Find a value for x such that Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Simplifying Fractions A rational number is in simplest form if b > 0 and GCD(a,b) = 1; that is, if a and b have no common factor greater than 1. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-2 Write each of the following fractions in simplest form if they are not already so: a. b. c. d. cannot be simplified Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-2 (continued) e. f. g. cannot be simplified Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Equality of Fractions Show that Method 1: Simplify both fractions to the same simplest form. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Equality of Fractions Method 2: Rewrite both fractions with the same least common denominator. LCM(42, 35) = 210, so Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Equality of Fractions Method 3: Rewrite both fractions with a common denominator. A common multiple of 42 and 35 is 42 · 35 = 1470. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Equality of Fractions Two fractions and , d  0 are equal if and only if ad = bc. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Ordering Rational Numbers If a, b, and c are integers and b > 0, then if and only if a > c. If a, b, c, and d are integers and b > 0, d > 0, then if and only if ad > bc. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Denseness of Rational Numbers Given rational numbers there is another rational number between these two numbers. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-3a Find two fractions between Because is between Now find two fractions equal to respectively, but with greater denominators. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 6-3a (continued) are all between so they are between Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 6-3b Show that the sequence is increasing. Because the nth term of the sequence is the next term is We need to show that for all positive integers n, Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 6-3b (continued) The inequality is true, if, and only if, which is true for all n. Thus, the sequence is increasing. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Denseness of Rational Numbers Let be any rational numbers with positive denominators, where Then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.