Download 6.1. The Set of Fractions Problem (Page 216). A child has a set of 10

CHAPTER 6
Fractions
6.1. The Set of Fractions
Problem (Page 216). A child has a set of 10 cubical blocks. The lengths
of the edges are 1 cm, 2 cm, 3 cm, . . . , 10 cm. Using all the cubes, can the
child build two towers of the same height by stacking one cube upon another?
Why or why not?
Strategy 11 – Solve an Equivalent Problem.
This strategy may be appropriate when
• You can find an equivalent problem that is easier to solve.
• A problem is related to another problem you have solved previously.
• A problem can be represented in a more familiar setting.
• A geometric problem can be represented algebraically, or vice versa.
• Physical problems can be easily represented with numbers or symbols.
Solution.
This problem can be stated as an equivalent problem: Can the numbers 1
through 10 be put into two sets with equal sums?
There are two possibilities: both sets have the same odd sum or the same even
sum.
That means that the sum of the blocks is even.
But
1 + 2 + 3 + · · · + 10 = 55,
an odd number. Thus towers of equal height cannot be built.
89
⇤
90
6. FRACTIONS
Sometimes whole numbers are inadequate to describe a mathematical situation.
For example, in the picture below, how much of the pizza remains?
But using fractions: five-eighths of the pizza remains.
There are two dictinct uses of fractions in elementary mathematics:
(1) Part-to-whole model – the number of parts of a whole to be considered.
Above, the pizza was cut iunto 8 equivalent pieces, and 5 remain. We use
the fraction 58 to represent 5 out of 8 equivalent pieces.
In general, the fraction ab or a/b represents a of b equivalent parts. a is the
numerator, b is the denominator.
Question: Can b = 0 in this model?
Our definition, however, allows for a fraction like 72 . How do we interpret
or model this?
Four equivalent objects are divided into 2 equivalent pieces each, and 7 of
the eight pieces are chosen.
6.1. THE SET OF FRACTIONS
91
(2) Abstract meaning as number.
Recall that the numerals 58 and 72 used above are symbols that represent
the abstract idea of number, which represents a quantity.
Despite di↵erences in size and shape, the above figures share the common attribute that the same relative amount is shaded, 5 out of 8 parts. This attribute
is represented by the fraction 58 .
Definition (Fractions). A fraction is a number that can be represented by
a
an ordered pair of whole numbers (or a/b) where b 6= 0. The set of fractions
b
is
na
o
F =
a and b are whole numbers, b 6= 0 .
b
When considering a fraction as a number, the focus is on the relative amount.
Despite the di↵erences in the figures above, 14 represents the relative amount
shaded. We are focusing on the “number” aspect of a fraction.
But in terms of the “part-to-whole” relationship, focusing on fractions as numerals,
many numerals can be used. In the middle figure, besides 14 , we can use 28 and
3
many others. The same for the rightmost figure, where we also use 12
.
92
6. FRACTIONS
Equivalent fractions – two fractions that represent the same relative amount.
The figures below illustrate 3 fractions equivalent to 14 .
A clearer visualization, in that the unit strips are the same size, are the fraction
strips below.
How do we check for equivalent fractions without using pictures?
Example.
4
2·2
6
2·3
=
,
=
.
10 5 · 2 15 5 · 3
4
6
This means that 10
and 15
can be obtained from 25 by equally subdividing each
of the original 5 parts into 2 or 3 equal parts, respectively.
an
a
Simplified fraction – when
, n 6= 1, is replaced by .
bn
b
Determining when fractions are equal:
(1)
4
10
and
6
15
are equal since both simplify to 25 .
6.1. THE SET OF FRACTIONS
(2)
4
4 · 15
6
6 · 10
=
and
=
10 10 · 15
15 15 · 10
Since 4 · 15 = 6 · 10 and the denominators are the same,
93
4
10
=
6
15 .
Notice the cross products of these fractions. They are the equal.
Definition (Fraction Equality).
a
c
a c
Let and be any fractions. Then = if an only if ad = bc.
b
d
b d
Theorem.
a
Let be any fraction and n a nonzero whole number. Then
b
a an na
=
=
.
b
bn
nb
Example.
15 25
=
since 15 · 40 = 25 · 24 = 600.
24 40
15 21
6=
since 15 · 32 = 480 and 24 · 21 = 504.
24 32
Note.
(1) Every fraction (a number) has an infinite number of representations (numerals), and any one of these numerals may be used to name it.
(2) A fraction is in its simplest form or lowest terms when its numerator and
denominator have no common prime factors.
(3) Equals refers to fractions as numbers, while equivalent refers to fractions as
numerals.
(4) Every whole number is a fraction.
94
6. FRACTIONS
Example.
5=
5 10 15
0 0 0
=
=
= · · · and 0 = = = = · · · .
1
2
3
1 2 3
A fraction is improper if the numerator is larger than the denominator.
Example.
9
4
is an improper fraction.
A mixed number is a combination of a whole number with a fraction.
Example. 2 14 is a mixed number
Ordering Fractions
Fraction strips can be used to order fractions.
One can also use the fraction number line to order fractions.
3 5
3
5
3 16
< since is to the left of on the fraction number line. Similarly, < .
7 7
7
7
7
3
6.1. THE SET OF FRACTIONS
95
Definition (Less than for Fractions).
a
b
a b
Let and be any fractions. Then < if and only if a < b.
c
c
c c
Note. A corresponding statement holds for “greater than,” “less than or
equal to,” and “greater than or equal to.”
Example.
5
7

since 5  7.
13 13
Theorem (Cross-Multiplication of Fraction Inequality).
a
c
a c
Let and be any fractions. Then < if an only if ad < bc.
b
d
b d
Why?
Example. Let
a
9
c 7
=
and = .
b 11
d 8
a
9
9·8
72 ad
=
=
=
=
b 11 11 · 8 88 bd
c 7 7 · 11 77 bc
= =
=
=
d 8 8 · 11 88 bd
9
7
Thus
< since 9 · 8 < 11 · 7 (i.e., we compare the numerators when the
11
8
denominators are equal).
Density Property of Fractions
There are gaps between the whole numbers on the whole-number line.
96
6. FRACTIONS
With the introduction of fractions, many new fractions appear in the gaps in
the fraction number line.
Theorem (Density Property of Fractions).
a
c
a c
Let and be any fractions where < . Then
b
d
b d
a a+c c
<
< .
b
b+d d
Proof.
a c
Since < , ad < bc. Then
b d
ad + ab < bc + ab =) a(b + d) < b(a + c) =)
a a+c
<
.
b
b+d
The other half is similar. Try it!
⇤
Note.
a+c
a c
(1)
is an incorrect answer students sometimes get for + .
b+d
b d
(2) This theorem means that between any two fractions there is another fraction
(the density property) and gives us a method for finding such a fraction.
(The average of two fractions is also between them.)
(3) This means there are infinitely many fractions between any two fractions.
(4) Even after all the fractions have been added to the number line, the closest
number that gives the per cent of the number line filled is 0%. The rest is
filled by irrational numbers.
6.2. FRACTIONS: ADDITION AND SUBTRACTION
97
6.2. Fractions: Addition and Subtraction
Addition and Its Properties
Definition (Addition of Fractions with Common Denominators).
a
c
Let and be any fractions. Then
b
b
a c a+c
+ =
.
b b
b
This definition is simply an extension of whole-number addition (in the numerator). It can be illustrated by the region and number-line models shown
below.
Example.
3 4 13
+ =
9 9
9
3
9
12 6
+
=
=
14 14 14 7
2 1
+ ?
3 4
This can be illustrated using fraction strips, a blend of the two models above.
But what about
98
6. FRACTIONS
2 1 2·4 1·3
8
3
11
+ =
+
=
+
=
3 4 3 · 4 4 · 3 12 12 12
Definition (Addition of Fractions with Unlike Denominators).
a
c
Let and be any fractions. Then
b
d
a c ad + bc
+ =
.
b d
bd
To add fractions with di↵erent denominators, we first find equivalent fractions
with common denominators, and then add the numerators, placing the sum
over the common denominator.
Example.
7 2 7 · 3 2 · 8 21 16 37
+ =
+
=
+
=
8 3 8 · 3 3 · 8 24 24 24
13
7
13 · 12 + 15 · 7 156 + 105 261 29
+
=
=
=
=
15 12
15 · 12
180
180 20
But we can also change the fractions to the least common denominator (LCD)
by taking the LCM of the two denominators.
6.2. FRACTIONS: ADDITION AND SUBTRACTION
99
Example (redone).
7 2 21 16 37
+ =
+
=
8 3 24 24 24
13
7
52 35 87 29
+
=
+
=
=
15 12 60 60 60 20
ha c c ai
Addition of fractions has the closure, commutative + = + , associative
⇣
h a 0b d0 ad ba i
a c⌘ e a ⇣c e⌘
+ + = + +
, and identity + = + =
properties,
b d
f
b
d f
b d d b
b
and these are used to simplify computations.
Example.
3 ⇣9
5⌘
+
+
=
8
14 8
3 ⇣5
9⌘
+
+
=
8
8 14
⇣3 5⌘ 9
+
+
=
8 8
14
9
9
23
1+
= 1 , a mixed number, or .
14
14
14
Example (mixed number to improper fraction).
3
3 20 3 23
5 =5+ =
+ = , or
4
4
4
4
4
(shortcut) 5
3 5 · 4 + 3 23
=
= .
4
4
4
Example (improper fraction to mixed number).
53 49 4
4
4
=
+ =7+ =7 .
7
7
7
7
7
100
6. FRACTIONS
Problem (Page 232 #4c).
1
1
+
=
54 ⇥ 75 ⇥ 132 32 ⇥ 5 ⇥ 133
32 ⇥ 13
53 ⇥ 75
+
=
32 ⇥ 54 ⇥ 75 ⇥ 133 32 ⇥ 54 ⇥ 75 ⇥ 133
(32 ⇥ 13) + (53 ⇥ 75)
2100992
=
=
32 ⇥ 54 ⇥ 75 ⇥ 133
32 ⇥ 54 ⇥ 75 ⇥ 133
28 ⇥ 29 ⇥ 283
.
32 ⇥ 54 ⇥ 75 ⇥ 133
Problem (Page 232 #6a).
5
2
7 + 13 =
8
3
⇣5
2⌘
(7 + 13) +
+
=
8 3
⇣ 15
16 ⌘
31
20 +
+
= 20 +
24 24
24
⇣
7⌘
7
7
20 + 1 +
= (20 + 1) +
= 21 .
24
24
24
6.2. FRACTIONS: ADDITION AND SUBTRACTION
Problem (Page 232 #15).
(a)
(b)
(c)
⇣2
5⌘ 3
+
+ =
5 8
5
⇣5
2⌘ 3 5 ⇣2 3⌘ 5
5 13
+
+ = +
+
= +1=1 = .
8 5
5 8
5 5
8
8
8
4 ⇣2
5⌘
+
+2
=
9
15
9
4 ⇣ 5
2 ⌘ ⇣4
5⌘ 2
2
2
47
+ 2 +
=
+2
+
=3+
=3 = .
9
9 15
9
9
15
15
15 15
⇣ 3
5⌘ ⇣ 8
1⌘
1 +3
+ 1 +2
=
4
11
11
4
⇣ 3
⇣
⇣
1⌘ ⇣ 5
8⌘
13 ⌘
2⌘
1 +2
+ 3 +1
= (3 + 1) + 4 +
=4+ 5+
=
4
4
11
11
11
11
9
2
101
=
.
11
11
101
102
6. FRACTIONS
8
3
Problem (Page 232 #17a). Estimate 5 + 6 .
9
13
(1) Range estimation:
8
3
5 + 6  5 + 6  6 + 7 or
9
13
8
3
11  5 + 6  13.
9
13
(2) Front end with adjustment estimation:
8
3
5 + 6 ⇡ 11 + 1 = 12.
9
13
8
4
Problem (Page 232 #18a). Estimate 5 + 6 by using “rounding to the
9
7
1
nearest whole number or 2 estimation.”
8
4
1
1
5 + 6 ⇡ 6 + 6 = 12 .
9
7
2
2
Subtraction
Subtraction can be viewed as (1) take-away or (2) missing addend.
Definition (Subtraction of Fractions with Common Denominators).
a
c
Let and be any fractions with a c. Then
b
b
a c a c
=
.
b b
b
Example.
4 1 3
=
(take-away) or
7 7 7
4
7
1 3
3 1 4
= since + =
(missing addend).
7 7
7 7 7
6.2. FRACTIONS: ADDITION AND SUBTRACTION
This subtraction is illustracted by the following fraction strips:
Theorem (Subtraction of Fractions with Unlike Denominators).
a
c
a
c
Let and be any fractions where
. Then
b
d
b
d
a c ad bc
=
.
b d
bd
Problem (Page 232 # 10).
7 ⇣2 1⌘
=
8
3 6
7 ⇣ 12 3 ⌘ 7
9
7 1 7 4 3
=
=
=
= .
8
18
8 18 8 2 8 8 8
But
⇣7 2⌘ 1
=
8 3
6
⇣ 21 16 ⌘ 1
5
1
5
4
1
=
=
= .
24 24
6 24 6 24 24 24
Thus there is no associative property for subtraction.
103
104
6. FRACTIONS
Problem (Page 232 # 11b).
2
3
13
⇣
2⌘
13 +
3
⇣
5⌘
7+
= (13
8
7
5
=
8
7) +
⇣2
3
5⌘
16 15
=6+
=
8
24
1
1
145
=6 =
.
24
24
24
Problem (Page 242 # 11b).
6+
5
5
8
11
3
11
2
3=2
6
=
11
8
30
= .
11 11
6.3. Fractions: Multiplication and Division
Multiplication
Case 1 Whole number times a fraction.
Example.
4⇥
1
=
5
(can use repeated addition)
1 1 1 1 4
+ + + =
5 5 5 5 5
6.3. FRACTIONS: MULTIPLICATION AND DIVISION
Case 2 Fraction times a whole number.
Example.
1
⇥3=
4
(1st option – use commutativity)
3⇥
1 1 1 1 3
= + + =
4 4 4 4 4
1
3
1
⇥3=
(2nd option – think of 3)
4
4
4
Case 3 Fraction times (of) a fraction.
Example.
1 5
1 5
⇥ means of .
3 7
3 7
1 5
5
⇥ =
3 7 21
105
106
6. FRACTIONS
Definition (Multiplication of Fractions).
a
c
Let and be any fractions. Then
b
d
a c ac
· = .
b d bd
Example.
(1)
(2)
(3)
3
9
27
⇥
= .
5 14 70
4 18 72 24 · 3 24
⇥
=
=
= .
3 13 39 13 · 3 13
2
1 8 7 56 28 · 2 28
1
2 ⇥3 = ⇥ =
=
=
=9
3
2 3 2
6
3·2
3
3
An erroneous approach:
⇣2 1⌘
2
1
1
1
1
2 ⇥ 3 = (2 · 3) +
·
= 6 + = 6 6= 9
3
2
3 2
3
3
3
An area representation:
6.3. FRACTIONS: MULTIPLICATION AND DIVISION
107
(4) Simplify, then compute:
6 28 6 · 28 6 · 18 6 · 18
6 28 1 4 4
⇥
=
=
=
=
·
= · = .
7 18 7 · 18 28 · 7 28 · 7 18 7
3 1 3
(5) Cancelling:
a c a·c a·d a c
· =
=
= · .
b d b·d
b·c
d b
ha c c ai
Fraction multiplication has the closure, commutative · = · , associative
b d d b
h⇣ a c ⌘ e a ⇣ c e ⌘i
ha
i
a a
m
· · = · ·
, and identity ·1 = 1· = with 1 = , m 6= 0
b d f
b d f
b
b
b
m
properties.
Division
(1) As an extension of whole number division:
6 ÷ 2 = 3 since there are three groups of 2 in 6.
6 2
2 6
÷ = 3 since there are three groups of in .
7 7
7 7
108
6. FRACTIONS
Example.
12
3
3
12
÷
= 4 since there are four ’s in .
13 13
13
13
15
3
3
15
÷
= 5 since there are five ’s in .
25 25
25
25
Definition (Division of Fractions with Common Denominators).
a
c
Let and be any fractions with c 6= 0. Then
b
b
a c a
÷ = .
b b c
(2) Extension to Di↵erent Denominators:
4
7
4 · 11
7·9
44 63 44
÷
=
÷
=
÷
= .
9 11 9 · 11 11 · 9 99 99 63
a c ad bc ad
÷ =
÷
= .
b d bd bd bc
4 11 44
·
= .
9 7
63
Definition (Division of Fractions with Unlike Denominators — Invert the
Divisor and Multiply). ]
a
c
Let and be any fractions with c 6= 0. Then
b
d
a c a d
÷ = ⇥ .
b d b c
6.3. FRACTIONS: MULTIPLICATION AND DIVISION
Example.
3 4 3 7 21
÷ = · = .
5 7 5 4 20
Viewing invert and multiply as a measurement division problem.
Example. How many groups of size
1
are in 3?
2
1
Since two ’s fit in each whole, there are 3 ⇥ 2 = 6.
2
1
3 ÷ = 3 · 2 = 6.
2
2
Example. How many groups of size are in 4?
3
Since 1
1
2
groups of size fit in each whole, there are 4 ⇥ 1 12 = 4 ⇥ 32 = 6.
2
3
2
3
4 ÷ = 4 · = 6.
3
2
109
110
6. FRACTIONS
Divide the numerators and denominators approach – numerator divides numerator and denominator divides denominator.
Example.
Also note that
24 8 24 ÷ 8 3
÷ =
=
35 7 35 ÷ 7 5
Dividing whole numbers without remainders:
Example.
13 ÷ 5 =
13 5 13 1 13
3
÷ =
⇥ =
=2
1
1
1
5
5
5
Theorem.
For all whole numbers a and b, b 6= 0,
a
a÷b= .
b
Example.
A quotient of 2 with a remainder of 3 translates to 2
3 13
= .
5
5
6.3. FRACTIONS: MULTIPLICATION AND DIVISION
Note.
a
even when a and b are not whole numbers.
b
Example.
a÷b=
(1)
2 3
÷ =
3 4
2
3
3
4
=
(a complex fraction)
2
3
3
4
(2)
· 12 8
= .
· 12 9
p
5÷⇡ =
(3)
3 4
÷ =
5 7
3
5
4
7
=
3
5
4
7
p
5
.
⇡
21
· 74
21
20
=
=
1
20
· 74
Problem (Page 245 # 21). Calculate mentally:
(a)
52 ·
⇣7
52
8
(b)
7
8
52 ·
3
=
8
3⌘
4
1
= 52 · = 52 · = 26.
8
8
2
⇣2
5⌘ 3
+
+ =
5 8
5
⇣5
2⌘ 3 5 ⇣2 3⌘ 5
5
5
+
+ = +
+
= +1=1+ =1 .
8 5
5 8
5 5
8
8
8
111
112
6. FRACTIONS
(c)
36 ⇥ 2
(d)
3
=
4
⇣
3⌘
3
36 2 +
= 36 · 2 + 36 · = 72 + 27 = 99.
4
4
23 ·
3
4
+ 7 + 23 · =
7
7
⇣3 4⌘
3
4
23 · + 23 · + 7 = 23 +
+ 7 = 23 · 1 + 7 = 23 + 7 = 30.
7
7
7 7