Fractions and Rational Numbers 6.1 The Basic Concepts of Fractions and Rational Numbers 6.2 Addition and Subtraction of Fractions 6.3 Multiplication and Division of Fractions 6.4 The Rational Number System Copyright © 2012, 2009, and 2006, Pearson Education, Inc.

The Basic Concepts of Fractions and Rational Numbers 6.1 The Basic Concepts of Fractions and Rational Numbers Copyright © 2012, 2009, and 2006, Pearson Education, Inc.

THE MEANING OF A FRACTION To interpret the meaning of any fraction we must: • agree on the unit; understand that the unit is subdivided into b parts of equal size; understand that we are considering a of the parts of the unit. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-3

• The integer a is called the numerator of the fraction. DEFINITION: FRACTION A fraction is an ordered pair of integers a and b, b ≠ 0, written or a/b. • The integer a is called the numerator of the fraction. The integer b is called the denominator of the fraction. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-4

• What is the unit? (the whole) Into how many equal parts has the unit MODELS FOR FRACTIONS A physical or pictorial representation of a fraction must clearly answer the following questions: • What is the unit? (the whole) Into how many equal parts has the unit been subdivided? (the denominator) How many of these parts are under consideration? (the numerator) Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-5

MODELS FOR FRACTIONS: COLORED REGIONS A shape is chosen to represent the unit and is then subdivided into subregions of equal size. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-6

MODELS FOR FRACTIONS: THE SET MODEL Each subset A of U corresponds to the fraction of the apples have worms. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-7

MODELS FOR FRACTIONS: FRACTION STRIPS The unit is defined by a rectangular strip of cardstock. A set of fraction strips typically contains strips for the denominators 1, 2, 3, 4, 6, 8, and 12. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-8

MODELS FOR FRACTIONS: THE NUMBER-LINE Fractions can be modeled by subdividing the unit interval into equal parts determined by the denominator and then counting off the number of those parts determined by the numerator. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-9

THE FUNDAMENTAL LAW OF FRACTIONS Let be a fraction. Then Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-10

THE CROSS-PRODUCT PROPERTY OF EQUIVALENT FRACTIONS The fractions are equivalent if, and only if, ad = bc. That is, Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-11

FRACTIONS IN SIMPLIEST FORM A fraction is in simplest form if a and b have no common divisor larger than 1 and b is positive. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-12

Example 6.3 Finding Common Denominators Find equivalent fractions to with a common denominator of 12. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-13

DEFINITION: RATIONAL NUMBERS A rational number is a number that can be represented by a fraction , where a and b are integers, b ≠ 0. Two rational numbers are equal if, and only if, they can be represented by equivalent fractions. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-14

Example 6.4 Representing Rational Numbers How many different rational numbers are given in this list of five fractions? Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-15 15

Addition and Subtraction of Fractions 6.2 Addition and Subtraction of Fractions Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-16

DEFINITION: ADDITION OF FRACTIONS Let two fractions have a common denominator. Then their sum is the fraction given by Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-17

MODELING ADDITION OF FRACTIONS WITH COLORED REGIONS Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-18

MODELING ADDITION OF FRACTIONS WITH THE NUMBER-LINE Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-19

MODELING ADDITION OF FRACTIONS WITH UNLIKE DENOMINATORS Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-20

A mixed number can always be rewritten in the standard form MIXED NUMBERS A mixed number can always be rewritten in the standard form Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-21

MIXED NUMBERS Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-22

Example 6.7 Working with Mixed Numbers a. Give an improper fraction for Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-23 23

Example 6.7 Working with Mixed Numbers b. Give a mixed number for Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-24 24

DEFINITION: SUBTRACTION OF FRACTIONS Let be fractions. Then if, and only if, Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-25

MODELING SUBTRACTION OF FRACTIONS WITH FRACTION STRIPS Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-26

Multiplication and Division of Fractions 6.3 Multiplication and Division of Fractions Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-27

DEFINITION: MULTIPLICATION OF FRACTIONS Let be fractions. Then their product is given by Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-28

Example 6.10 Calculating Products of Fractions Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-29

Example 6.12 Multiplying Fractions on the Number Line Illustrate why with a number-line diagram. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-30 30

THE INVERT-AND-MULTIPLY ALGORITHM FOR DIVISION OF FRACTIONS where Note that this is a process for dividing fractions, not a definition of division. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-31

Example 6.15 Dividing Fractions Compute. a. b. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-32 32

The Rational Number System 6.4 The Rational Number System Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-33

DEFINITION: NEGATIVE OR ADDITIVE INVERSE Let be a rational number. Its negative, or additive inverse, written is the rational number Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-34

Example 6.18 Subtracting Rational Nmbers Compute. a. b. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 6-35 35