fractions and rational numbers 6.1 the basic concepts of fractions and rational numbers 6.2 addition...
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Fractions and Rational Numbers
6.1 The Basic Concepts of Fractions and Rational Numbers
6.2 Addition and Subtraction of Fractions6.3 Multiplication and Division of Fractions6.4 The Rational Number System
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
6.1
Slide 6-2
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.
ab
Slide 6-3Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
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.
ab
Slide 6-4Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
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)
Slide 6-5Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
MODELS FOR FRACTIONS:COLORED REGIONS
A shape is chosen to represent the unit and is then subdivided into subregions of equal size.
14
412
Slide 6-6Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
MODELS FOR FRACTIONS:THE SET MODEL
Each subset A of U corresponds to the fraction
310
.( )( )n An U
of the apples have worms.
Slide 6-7Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
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.
Slide 6-8Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
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.
Slide 6-9Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
THE FUNDAMENTAL LAW OF FRACTIONS
Let be a fraction. Then ab
, for any integer 0. nanbn
ab
Slide 6-10Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
THE CROSS-PRODUCT PROPERTY OF EQUIVALENT FRACTIONS
The fractions are equivalent if,
and only if, ad = bc. That is,
and c
d
ab
, if, and only if, . c
ad bcd
ab
Slide 6-11Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
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.
ab
Slide 6-12Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
28
48
7 4
12 4
7
12
Example 6.3 Finding Common Denominators
Find equivalent fractions to with a common denominator of 12.
1 and
4
56
Slide 6-13Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
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.
ab
Slide 6-14Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Example 6.4 Representing Rational Numbers
How many different rational numbers are given in this list of five fractions?
4 39 7, 3, , , and
10 13 4
25
Slide 6-15Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
4 3 39Since and , there are
10 1 132 7
three different rational numbers; , 3, and .5 4
25
6.2
Slide 6-16
Addition and Subtraction of Fractions
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
DEFINITION:ADDITION OF FRACTIONS
Let two fractions have a
common denominator. Then their sum is the fraction given by
and a c
b b
.
a c a c
b b b
Slide 6-17Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
MODELING ADDITION OF FRACTIONSWITH COLORED REGIONS
Slide 6-18Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
MODELING ADDITION OF FRACTIONSWITH THE NUMBER-LINE
Slide 6-19Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
MODELING ADDITION OF FRACTIONSWITH UNLIKE DENOMINATORS
Slide 6-20Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
MIXED NUMBERS
A mixed number can always be rewritten in the standard form
bAc
Ac b
c c
Ac b
c
23
53 5 2
5
17
5
Slide 6-21Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
MIXED NUMBERS
Slide 6-22Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Example 6.7 Working with Mixed Numbers
a. Give an improper fraction for 3 .17120
Slide 6-23Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
3 173
1 120
17120
3 120 1 17
120
360 17
120
377
120
Example 6.7 Working with Mixed Numbers
b. Give a mixed number for
Slide 6-24Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
.355133
355133
DEFINITION:SUBTRACTION OF FRACTIONS
Let be fractions.
Then
if, and only if,
and a c
b d
a c e
b d f
. a c e
b d f
Slide 6-25Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
MODELING SUBTRACTION OF FRACTIONS WITH FRACTION STRIPS
1
4
7
12
Slide 6-26Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
5
6
6.3
Slide 6-27
Multiplication and Division of Fractions
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
DEFINITION:MULTIPLICATION OF FRACTIONS
Let be fractions.
Then their product is given by
and a c
b d
. a c ac
b d bd
Slide 6-28Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Example 6.10 Calculating Products of Fractions
5 2
8 3
ab
Slide 6-29Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Example 6.12 Multiplying Fractions on the Number Line
Illustrate why with a number-line diagram.
Slide 6-30Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
4 8
5 15
23
THE INVERT-AND-MULTIPLY ALGORITHM FOR DIVISION OF FRACTIONS
, a c a d ad
b d b c bc
Note that this is a process for dividing fractions, not a definition of division.
where 0.c
d
Slide 6-31Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Example 6.15 Dividing Fractions
Compute.
a.
b.
Slide 6-32Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
1
8
34
1
68
68
1
or 8
34
8 24
61 4
34
14 2
3
16
7 25 3
3 6 7
256
25 3
6 7
25 1
2 7
25
14
6.4
Slide 6-33
The Rational Number System
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
DEFINITION:NEGATIVE OR ADDITIVE INVERSE
Let be a rational number.
Its negative, or additive inverse,
written is the rational number
a
b
,a
b.
ab
Slide 6-34Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Example 6.18 Subtracting Rational Nmbers
Slide 6-35Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
7
6
34
22 4
3
14
4 7
4 6
3 64 6
18 2824
10
24
1 22 4
4 3
3 86
12 12 11
612
Compute.
a.
b.