fractions and rational numbers 6.1 the basic concepts of fractions and rational numbers 6.2 addition...
TRANSCRIPT
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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
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6.1
Slide 6-2
The Basic Concepts of Fractions and Rational Numbers
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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
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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
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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)
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MODELS FOR FRACTIONS:COLORED REGIONS
A shape is chosen to represent the unit and is then subdivided into subregions of equal size.
14
412
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MODELS FOR FRACTIONS:THE SET MODEL
Each subset A of U corresponds to the fraction
310
.( )( )n An U
of the apples have worms.
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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.
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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.
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THE FUNDAMENTAL LAW OF FRACTIONS
Let be a fraction. Then ab
, for any integer 0. nanbn
ab
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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
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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
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28
48
7 4
12 4
7
12
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Example 6.3 Finding Common Denominators
Find equivalent fractions to with a common denominator of 12.
1 and
4
56
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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
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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
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4 3 39Since and , there are
10 1 132 7
three different rational numbers; , 3, and .5 4
25
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6.2
Slide 6-16
Addition and Subtraction of Fractions
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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
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MODELING ADDITION OF FRACTIONSWITH COLORED REGIONS
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MODELING ADDITION OF FRACTIONSWITH THE NUMBER-LINE
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MODELING ADDITION OF FRACTIONSWITH UNLIKE DENOMINATORS
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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
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MIXED NUMBERS
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Example 6.7 Working with Mixed Numbers
a. Give an improper fraction for 3 .17120
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3 173
1 120
17120
3 120 1 17
120
360 17
120
377
120
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Example 6.7 Working with Mixed Numbers
b. Give a mixed number for
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.355133
355133
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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
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MODELING SUBTRACTION OF FRACTIONS WITH FRACTION STRIPS
1
4
7
12
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5
6
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6.3
Slide 6-27
Multiplication and Division of Fractions
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DEFINITION:MULTIPLICATION OF FRACTIONS
Let be fractions.
Then their product is given by
and a c
b d
. a c ac
b d bd
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Example 6.10 Calculating Products of Fractions
5 2
8 3
ab
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Example 6.12 Multiplying Fractions on the Number Line
Illustrate why with a number-line diagram.
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4 8
5 15
23
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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
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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
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6.4
Slide 6-33
The Rational Number System
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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
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Example 6.18 Subtracting Rational Nmbers
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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.