Download Fractions and Rational Numbers

Section 6.1 – Basic Concepts of Fractions and Rational Numbers
What is a fraction?
2
.
3
Two-thirds of the set are shaded circles.
Several models are used for fractions. Example: model the fraction
Discrete model:
Geometric models:
number line
2
3
0
1
3
The vector is two-thirds of a unit.
2
3
1
Cuisenaire rods are a manipulative number line model.
orange
blue
brown
black
dark green
yellow
10
9
8
7
6
5
purple
light green
red
white
4
3
2
1
How many ways may the Cuisenaire rods be used to represent
2
?
3
Two whites are two-thirds of a light green.
A red is two-thirds of a light green.
Two reds are two-thirds of a dark green.
A purple is two-thirds of a dark green.
Two light greens are two-thirds of a blue.
A dark green is two-thirds of a blue.
Area
rectangle (or fraction strips)
regular polygons
Two-thirds of the rectangle is shaded.
Two-thirds of the triangle is shaded.
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circle Shade two-thirds of the following circle.
Volume
Two-thirds of the cylinder is shaded.
Dienes' Multibase blocks may also be used for volume illustrations.
Base Three Dienes’ Multibase Blocks
Block
Flat
Long
Unit
1
1
1
1
3
9
27
Base Three Dienes’ Multibase Blocks
Two units are two-thirds of a long.
Two longs are two-thirds of a flat.
Two flats are two-thirds of a block.
Definition. Let a and b be integers such that b  0. A common fraction is an ordered pair written as
a
or a/b. The integer
b
a is called the numerator and the nonzero integer b is called the denominator.
a
. The slanted format, a/b, is for writing
b
fractions in a typed sentence. Many students who learn to write fractions only in the slanted form have
problems interpreting mixed numbers and working with rational expressions in algebra.
For these reasons, all fractions in this course must be written in the vertical form.
Note. The most proper way to write a fraction is in the vertical format,
Note. For students beginning to understand fractions, the denominator represents the number of equivalent parts that
some entity is divided into, and the numerator represents the number of equivalent parts under consideration;
i.e., the numerator is the number of equivalent parts out of the whole.
Definition. A rational number is a number that is equivalent to a common fraction, that is, it may be written as a common
fraction. The set of rational numbers, denoted by Q, may be written in set-builder notation as
a


Q   x : x  where a and b are integers with b  0.
b


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Equivalent Fractions
Definition. Fractions that represent the same quantity are said to be equivalent fractions.
Example. Show that
2 4 6 8 10
, , , , ,
3 6 9 12 15
are equivalent fractions by using a fraction strip illustration.
We use an equal sign to denote equivalent fractions. By the above illustration, we have
2 4 6 8 10
  


3 6 9 12 15
2
Based on this pattern write the next five fractions that are equivalent to .
3
Note. The numerators are multiples of 2 and the denominators are multiples of 3.
2 2  2 3 2 4  2 5 2





3 2  3 3 3 4  3 5 3
We have modeled the following property:
The Fundamental Law of Fractions Let
equivalent to any fraction of the form
Definition. A fraction
a
a an
a
be a fraction. Then 
, for any nonzero integer n, that is,
is
b
b bn
b
an
.
bn
a
is said to be in simplest form or lowest terms provided |a| and |b| are relatively prime and b is a
b
positive integer.
The procedure for determining the simplest form for a fraction is to find the greatest common factor for the
numerator and denominator, then apply the Fundamental Law of Fractions.
Examples. Simplify each of the following:
12 2  6 2
42 3  14 3




18 3  6 3
70 5  14 5
120
126 42  3 42


183 61 3 61
120 = 23 · 3 · 5 and 180 = 22 · 32 · 5 so the GCD(120, 180) = 22 · 3 · 5 = 60.
180
120
180

2  60
3  60
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
2
3
Note. Do not use terms such as reduce or cancel; they lead many students to misunderstand the concepts involved. The
x
concept of factoring is important in simplifying. Consider when a student is asked to simplify 2
in algebra.
x x
Wrong
x
x
0


0
2
x  x x( x  1) x  1
Wrong
x
1
 2
2
x  x x 1
Correct
x
x 1
1


2
x  x x( x  1) x  1
Improper Fractions and Mixed Numbers.
9
1
1
1
Use fraction strips to illustrate the fraction
is equivalent to 2 . (Note 2 is defined to be 2  .)
4
4
4
4
=
Another model that may be used for the above problem is to use quarters and dollars.
9
4
1
2
4
Definition. An improper fraction is a common fraction in which the absolute value of the numerator is greater than the
absolute value of the denominator. A mixed number is a representation of a sum of an integer and a common fraction.
Examples. Use Dienes’ multibase blocks to show the equivalence of the improper fraction
8
and the mixed number
3
8 6 2
2
2
2
   2  2 .
, i.e. show
3 3 3
3
3
3
Assume a block represents one whole and a flat is one-third of a block.
2
8
3
2
2
3
Note. There is nothing "improper" about improper fractions. In some applications, it is often more appropriate to use an
improper fraction than it is to use a mixed number. Examples would be problems involving rates of change, ratios, or
odds.
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Comparing Fractions (Ordering Rational Numbers)
2
3
Compare
and
by the use of the area model with rectangles.
3
4
2
3
3
4
2 24 8


3 3  4 12
3 33 9


4 4  3 12
8
9

12 12
2 3

3 4
Compare each of the following
3 4
,
5 7
3 3  7 21


5 5  7 35
4 4  5 20


7 7  5 35
21 20

35 35
3 4

5 7
pairs of fractions.
7 8
,
12 15
7
7  5 35


12 12  5 60
8
8  4 32


15 15  4 60
35 32

60 60
7
8

12 15
23 15
,
60 36
60 = 22∙3∙5, 36 = 22∙32
LCM(60, 36) = 22∙32∙5 = 180
23 23  3 69


60 60  3 180
15 15  5 75


36 36  5 180
69
75

180 180
23 15

60 36
For each of the above problems, we did not need to find the lowest common denominator. We could have compared the
fractions by using the product of the denominators as the common denominator as follows:
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3 4
,
5 7
7 8
,
12 15
23 15
,
60 36
3 3  7 21


5 5  7 35
4 4  5 20


7 7  5 35
21 20

35 35
3 4

5 7
7
7 15 105


12 12 15 180
8
8 12
96


15 15 12 180
105 96

180 180
7
8

12 15
23 23  36 828


60 60  36 2160
15 15  60 900


36 36  60 2160
828
900

2160 2160
23 15

60 36
The above three problems motivate a short-cut for comparing two fractions. Since we need only compare the numerators
when we have a common denominator, we would not need to actually compute the denominators.
Cross-Product Rule.
a c
 if and only if ad < bc.
b d
Examples. We redo the above three problems with this short-cut.
3 4
,
5 7
7 8
,
12 15
23 15
,
60 36
3∙7 ? 4∙5
7 ∙ 15 ? 8 ∙ 12
23 ∙ 36 ? 15 ∙ 60
21 > 20
105 > 96
828 < 900
3 4

5 7
7
8

12 15
23 15

60 36
Caution. This short-cut causes problems for many students.
1. Students, who do not understand that the method is a shortcut for comparing the numerators, do not
remember where to place each product. So, they get the inequality going the wrong direction.
2. In algebra, they confuse the rule with solving equations and inequalities.
Therefore, my philosophy is that the method should not be taught to all students. Instead, allow the students who
discover the method for themselves use it provided they can justify why the method works.
Density Property of Fractions. Given any two distinct fractions, there is another fraction between them.
Examples. Find at least one fraction between each of the following pairs.
2
3
5
7
and
and
3
4
6
8
2 24 8
8  2 16




3 3  4 12 12  2 24
3 33 9
9  2 18




4 4  3 12 12  2 24
2 17 3


3 24 4
5 5  8 40


6 6  8 48
7 7  6 42


8 8  6 48
5 41 7


6 48 8
19
61
and
42
15
19 19  42 798


15 15  42 630
61 61 15 915


42 42 15 630
19 799 800 801




15 630 630 630
19 799 80 10 89  9




15 630 63 10 70  9
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914 61

630 42
457  2 61


315  2 42

19 799 80 89




15 630 63 70

457 61

315 42
Note. When simplifying fractions, we often use the greatest common factor, GCD. When comparing fractions, we often
use the least common multiple, LCM, for the denominator.
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