Chapter 6 Rational Numbers and Proportional Reasoning Copyright

Chapter 6 Rational Numbers and Proportional Reasoning Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

6 -1 The Set of Rational Numbers Students will be able to understand explain • Different representations for rational numbers. • Equal fractions, equivalent fractions, and the simplest form of fractions. • Ordering of rational numbers. • Denseness property of rational numbers. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 2

Definition Numerator, Denominator numerator denominator Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 3

Uses of Rational Numbers Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 4

Rational Number Models Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 5

Meaning of a Fraction To understand the meaning of any fraction, using the parts-to-whole model, we must consider each of the following: 1. The whole being considered. 2. The number b of equal-size parts into which the whole is divided. 3. The number a of parts of the whole that are selected. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 6

Definition Proper fraction A fraction where Examples are Improper fraction A fraction such that Examples are Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 7

Number Line Model What numbers are represented on the number line? Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 8

Equivalent or Equal Fractions Equivalent fractions are numbers that represent the same point on a number line. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 9

Fraction Strips Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 10

Fundamental Law of Fractions Let be any fraction and n a nonzero whole number, then Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 11

Example Find a value for x such that Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 12

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 © 2016, 2013, 2010 Pearson Education, Inc. Slide 13

Example Write each of the following fractions in simplest form if they are not already so: a. b. c. d. cannot be simplified Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 14

Example (continued) e. f. g. cannot be simplified Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 15

Equality of Fractions Show that Method 1: Simplify both fractions to the same simplest form. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 16

Equality of Fractions Show that Method 2: Rewrite both fractions with the same least common denominator. LCM(42, 35) = 210, so Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 17

Equality of Fractions Method 3: Rewrite both fractions with a common denominator. A common multiple of 42 and 35 is 42 · 35 = 1470. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 18

Equality of Fractions Two fractions and , d 0 are equal if and only if ad = bc. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 19

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 © 2016, 2013, 2010 Pearson Education, Inc. Slide 20

Denseness of Rational Numbers Given rational numbers there is another rational number between these two numbers. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 21

Example Find two fractions between Because is between Now find two fractions equal to respectively, but with greater denominators. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 22

Example (continued) are all between so they are between Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 23

Example Show that the sequence increasing. …, is Because the nth term of the sequence is the next term is We need to show that for all positive integers n, Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 24

Example (continued) The inequality is true, if, and only if, which is true for all n. Thus, the sequence is increasing. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 25

Denseness of Rational Numbers Let be any rational numbers with positive denominators, where Then Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 26