Fractions and Rational Numbers

1
Chapter 6

• Fractions and Rational Numbers

2
DAY 1
3
Homework QuestionsChapter 5
4

• Counting Numbers 1, 2, 3, . . .
• Whole Numbers 0, 1, 2, 3, . . .
• Integers . . . , -2, -1, 0, 1, 2 . . .

5
Rational Numbers

• p/q p and q are integers and q ? 0
• Ratio of integers
• Any number that can be written as the ratio of
  integers

6

• Counting Numbers 1, 2, 3, . . .
• Whole Numbers 0, 1, 2, 3, . . .
• Integers . . . , -2, -1, 0, 1, 2 . . .
• Rational Numbers
• p/q p and q are integers and q ? 0

7

• Rationals p/q p and q are integers and q ?
  0
• Rational Numbers
• Closed with respect to addition?
• Closed with respect to multiplication?
• Closed with respect to subtraction?
• Closed with respect to division?

8

• Fractions were first introduced in measurement
  problems, to express a quantity that is less than
  a whole unit.

9
Figure 6.2Page 342

• What is this a picture of?
• 3/5?
• 5/3?
• 3/8?
• 5/8?
• Why is this not a good question?
• What would be a better question?
• What might a child be thinking who answers 3/5?

10
To interpret the meaning of any fraction a/b we
must

• Agree on the unit. What is the whole thing we
  are considering?
• Understand that the unit is subdivided into b
  parts of equal size. The denominator tells us
  how many pieces total.
• Understand that we are considering a of the parts
  of the unit. The numerator tells us how many of
  the pieces are being considered.

11

• Numerator
• Denominator

12
Models for Fractions

• Colored Regions Figure 6.3, Page 343
• Set Model Figure 6.4, Page 345
• Fraction Strips Figure 6.5, Page 345
• Numberline Figure 6.6, Page 346

13
Equivalent Fractions

• Fractions that express the same quantity are
  called equivalent fractions.
• Figure 6.7, Page 346
• We like to use the name that is the simplest
  because it is easier to think about what it means
  and easier to do calculation with.

14

• Identity element for multiplication is 1. That
  means we can multiply any rational number by 1
  and not change the identity.
• It is also true that we can divide any rational
  number by 1 and not change the identity.

15
We will make equivalent fractions by either
multiplying by 1or dividing by 1
16
We will make equivalent fractions by either
multiplying by 1or dividing by 1
17
Equivalent Fractions
18

• If two fractions are equivalent their cross
  products are equivalent.
• If the cross product of two fractions are
  equivalent, the fractions are equivalent.

19
Are the two fractions equal?
20
Find m if
21
Fractions in Simplest Form

• Reducing Fractions
• Simplifying Fractions

22
Simplify

• By finding the biggest thing that will divide
  into both the top and bottom. (WHAT IS THAT
  CALLED?)
• By finding the GCF.

23
Simplify

• By finding the prime factorization.

24
Simplify

• By dividing successively by common factors. (This
  is what you usually do.)

25
Simplify
26
Finding Common Denominator

• You can always find A common denominator by
  multiplying the two denominators together.
• 8 x 10 80, so

27

• It is sometimes worthwhile to find the common
  positive denominator that is as small as
  possible.
• The smallest number that both of the denominators
  divide into (LCM) will be the preferred common
  denominator.

28

• LCM(8,10)

29
Rename the fractions using the Least Common
Denominator
30
Find the Least Common Denominator
31
Find the Least Common Denominator
32
Find the Least Common Denominator
33
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6

34
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1
• 11/13

35
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1
• 11/13 1
• 1/15

36
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1
• 11/13 1
• 1/15 0
• 8/9

37
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1
• 11/13 1
• 1/15 0
• 8/9 1
• 2/13

38
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1
• 11/13 1
• 1/15 0
• 8/9 1
• 2/13 0
• 6/13

39
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1 2/51
• 11/13 1
• 1/15 0
• 8/9 1
• 2/13 0
• 6/13 ½

40
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1 2/51 0
• 11/13 1 33/35
• 1/15 0
• 8/9 1
• 2/13 0
• 6/13 ½

41
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1 2/51 0
• 11/13 1 33/35 1
• 1/15 0 4/9
• 8/9 1
• 2/13 0
• 6/13 ½

42
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1 2/51 0
• 11/13 1 33/35 1
• 1/15 0 4/9 ½
• 8/9 1 4/100
• 2/13 0
• 6/13 ½

43
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1 2/51 0
• 11/13 1 33/35 1
• 1/15 0 4/9 ½
• 8/9 1 4/100 0
• 2/13 0 7/12
• 6/13 ½

44
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1 2/51 0
• 11/13 1 33/35 1
• 1/15 0 4/9 ½
• 8/9 1 4/100 0
• 2/13 0 7/12 ½
• 6/13 ½ 6/7

45
Estimating Fractions

• Classify as close to 1, ½, or 0
• 5/6 1 2/51 0
• 11/13 1 33/35 1
• 1/15 0 4/9 ½
• 8/9 1 4/100 0
• 2/13 0 7/12 ½
• 6/13 ½ 6/7 1

46
Fill in the numerator or denominator to make the
fraction close but less than 1.

• __
• 27

47
Fill in the numerator or denominator to make the
fraction close but less than 1.

• 26 __
• 27 12

48
Fill in the numerator or denominator to make the
fraction close but less than 1.

• 26 11 ___
• 27 12 75

49
Fill in the numerator or denominator to make the
fraction close but less than 1.

• 26 11 74
• 27 12 75
• ___
• 8

50
Fill in the numerator or denominator to make the
fraction close but less than 1.

• 26 11 74
• 27 12 75
• 7 9
• 8

51
Fill in the numerator or denominator to make the
fraction close but less than 1.

• 26 11 74
• 27 12 75
• 7 9 3
• 8 10

52
Fill in the numerator or denominator to make the
fraction close but less than 1.

• 26 11 74
• 27 12 75
• 7 9 3
• 8 10 4

53
Fill in the numerator or denominator to make the
fraction close but less than ½ .

• ___
• 100

54
Fill in the numerator or denominator to make the
fraction close but less than ½ .

• 49 __
• 100 25

55
Fill in the numerator or denominator to make the
fraction close but less than ½ .

• 49 12 _
• 100 25 9

56
Fill in the numerator or denominator to make the
fraction close but less than ½ .

• 49 12 4
• 100 25 9
• 7

57
Fill in the numerator or denominator to make the
fraction close but less than ½ .

• 49 12 4
• 100 25 9
• 7 11
• 15

58
Fill in the numerator or denominator to make the
fraction close but less than ½ .

• 49 12 4
• 100 25 9
• 7 11 8
• 15 23

59
Fill in the numerator or denominator to make the
fraction close but less than ½ .

• 49 12 4
• 100 25 9
• 7 11 8
• 15 23 17

60
Ordering Rational NumbersTHINK!

• Fill in the blank with gt, lt or

61
Ordering Rational NumbersTHINK!

• Fill in the blank with gt, lt or

62
Ordering Rational NumbersTHINK!

• Fill in the blank with gt, lt or

63
Ordering Rational NumbersTHINK!

• Fill in the blank with gt, lt or

64
Ordering Rational NumbersCommon Denominator

• Fill in the blank with gt, lt or

65
Ordering Rational NumbersCommon Denominator

• Fill in the blank with gt, lt or

66
Ordering Rational NumbersCross Product?
67
Ordering Rational NumbersCross Product?

• 19 207 180 501
  95 104
• 9 140 80 50 45
  40

68
Ordering Rational Numbers

• Think! Visualize
• Get a common denominator
• Find the cross product

69
Fill in the proper relation
70
Fill in the proper relation
71
Fill in the proper relation
72
Fractions with Cuisenaire Rods
73
Day 2
74
Homework QuestionsPage 354
75

• 3 apples 2 apples 5 apples
• 3x 2x 5x
• 3 eighths 2 eighths 5 eighths
• 3 2 5
• 8 8 8

76

• 3 eighths 2 eighths 5 eighths
• Figure 6.9, Page 360

77
How do you find the sum of two fractions with a
common denominator?
78
How do you find the difference of two fractions
with a common denominator?
79

• Make it a problem that you know how to solve.
• Find a common denominator.

80
Show the steps to the answer as if you were
showing a fifth grader.
81
Show the steps to the answer as if you were
showing a fifth grader.
82
Show the steps to the answer as if you were
showing a fifth grader.
83
Show the steps to the answer as if you were
showing a fifth grader.
84
Proper Fractions and Mixed Numbers

• The sum of a counting number and a fraction is
  referred to as a mixed number.
• 2 ¾ is read Two and three-fourths

85
Proper Fractions and Mixed Numbers

• 2 ¾

86
Proper Fractions and Mixed Numbers

• 2 ¾
• 11/4

87

• 11/4
• How many times can you get 4 4ths?
• How many 4ths are left over?

88
Proper Fraction

• A fraction a/b such that 0 a lt b is called a
  proper fraction.
• 11/4 is an improper fraction because
• 11 gt 4

89

• Write the 439 as a mixed number.
• 19

90

• In mixed number form we know that the number is
  slightly larger than 23. That was not evident in
  the improper fraction form.
• The improper form is more convenient for
  arithmetic and algebra.
• The mixed number form is easiest to understand
  for practical applications.

91
Adding and SubtractingMixed Numbers
92
Adding and SubtractingMixed Numbers
93
Into the ClassroomPage 367
94

• ½ of 6

95

• ½ of 6 3
• ½ of 4/5

96

• ½ of 6 3
• ½ of 4/5 2/5
• ½ of ¾

97

• ½ of 6 3
• ½ of 4/5 2/5
• ½ of ¾ 3/8

98
Multiplication of Fractions

• ½ of 6 ½ x 6 3
• ½ of 4/5 ½ x 4/5 2/5
• ½ of ¾ ½ x ¾ 3/8

99
Multiply. Look for ways to shorten your work.
Simplify.
100
Multiply. Look for ways to shorten your work.
Simplify.
101
Multiply. Simplify.Make it a problem you know
how to solve.
102

• 10 2 ? How many 2s are in 10?

103

• 10 2 ? How many 2s are in 10? 5
• 10 ½ ?

104

• 10 2 ? How many 2s are in 10? 5
• 10 ½ ? How many ½s are in 10?

105

• 10 2 ? How many 2s are in 10? 5
• 10 ½ ? How many ½s are in 10? 20
• 12 ¼ ?

106

• 10 2 ? How many 2s are in 10? 5
• 10 ½ ? How many ½s are in 10? 20
• 12 ¼ ? How many ¼ s are in 12?

107

• 10 2 ? How many 2s are in 10? 5
• 10 ½ ? How many ½s are in 10? 20
• 12 ¼ ? How many ¼ s are in 12? 48
• 12 ¾ ?

108

• 10 2 ? How many 2s are in 10? 5
• 10 ½ ? How many ½s are in 10? 20
• 12 ¼ ? How many ¼ s are in 12? 48
• 12 ¾ ? How many ¾s are in 12?

109

• 10 2 ? How many 2s are in 10? 5
• 10 ½ ? How many ½s are in 10? 20
• 12 ¼ ? How many ¼ s are in 12? 48
• 12 ¾ ? How many ¾s are in 12? 16

110

• 3/2 ¼ ?

111

• 3/2 ¼ ? How many ¼ s are in 3/2?

112

• 3/2 ¼ ? How many ¼ s are in 3/2? 6

113
Get a common denominator
114

• How many ¼ s are in 6/4s?

115
(No Transcript)
116
(No Transcript)
117
Get a common denominator
118
(No Transcript)
119
(No Transcript)
120
(No Transcript)
121
(No Transcript)
122
(No Transcript)
123
(No Transcript)
124
(No Transcript)
125
How do you divide fractions?
126
Page 370

• Division of fractions rule emphasizes that
  division is the inverse of multiplication.
• HOWEVER, conceptual models such as repeated
  subtraction or sharing, are sometimes the best
  representation along with manipulatives or
  pictures to convey real understanding and foster
  the ability to problem solve.

127
Example 6.11Page 370

• 2 ½ acre grass playfield
• Grass seed bags _at_ ¾ acres each
• How many bags are needed?
• Will there be any extra for reseeding worn places?

128
Example 6.11Page 370

• 2 ½ acre grass playfield
• Grass seed bags _at_ ¾ acres each

129
Example 6.11Page 370

• 2 ½ acre grass playfield
• Grass seed bags _at_ ¾ acres each
• How many bags are needed?
• Will there be any extra?

130
Example 6.12Page 371

• 3 girls
• 7 ½ pounds total to carry
• Share the load equally

131
Example 6.12Page 371

• 3 girls
• 7 ½ pounds total to carry
• Share the load equally

132
(No Transcript)
133
(No Transcript)
134
(No Transcript)
135
(No Transcript)
136
Reciprocal?
137
Show, as if you are working with a fifth grader,
all the steps needed to compute the answers.
Simplify.
138
Show, as if you are working with a fifth grader,
all the steps needed to compute the answers.
Simplify.
139
Show, as if you are working with a fifth grader,
all the steps needed to compute the answers.
Simplify.
140
Show, as if you are working with a fifth grader,
all the steps needed to compute the answers.
Simplify.
141
Show, as if you are working with a fifth grader,
all the steps needed to compute the answers.
Simplify.
142
Show, as if you are working with a fifth grader,
all the steps needed to compute the answers.
Simplify.
143
Pattern Block Lab
144
Day 3
145
Homework QuestionsPage 375
146
PropertiesPage 381 and 383

• Closure Addition and Multiplication
• Commutative Addition and Multiplication
• Associative Addition and Multiplication
• Identity Element Addition and Multiplication
• Inverse Addition and Multiplication
• Distributive for Multiplication over Addition
• Multiplication by Zero

147
Density Property of Rational Numbers

• Between any two rational numbers there is some
  other rational number.

148
Find THE rational number halfway between the two
numbers.
149
Find THE rational number halfway between the two
numbers.
150
Find THE rational number halfway between the two
numbers.
151
Perform the computation mentally
152
Perform the computation mentally
153
Perform the computation mentally
154
Perform the computation mentally
155
Perform the computation mentally
156
Perform the computation mentally
157
Test Number Theory, Integers, Rationals
158
Fraction War!