Section 3.1
PRE-ACTIVITY
PREPARATION
Introduction to Fractions
Fraction notation is used as the numeric description of choice in a variety of
common contexts. Consider the following examples in various contexts.
Sewing:
You purchase 7/8 of a yard of fabric to make place mats.
Cooking:
A cookie recipe calls for 3/4 of a teaspoon of cinnamon.
Market research:
A survey reveals that twelve out of every thirty people,
or 12/30, prefer the convenience of take-out to home
cooking.
Quality control:
For every two thousand light bulbs coming off the assembly
line, two of them (2/2000) are defective.
Construction:
Forty-two acres of land are split evenly into sixty-three lots for a new subdivision, yielding
42/63 of an acre per lot.
Home decorating: A length of leftover carpeting seven yards long and three feet wide can be cut evenly to
make four scatter rugs, each 7/4 yards long.
Demographics:
In a college psychology class comprised of thirteen men and seventeen women, 13/17
(“thirteen to seventeen”) represents the comparison of the number of male students to the
number of female students.
Conversions:
To convert sixty milligrams to centigrams, you multiply the sixty milligrams by 1/10 (that
is, “1 centigram to 10 milligrams”).
LEARNING OBJECTIVES
•
Learn the various relationships fractions can represent.
•
Set up fractions to represent given information.
•
Interpret fractions using appropriate mathematical language and/or diagrams.
•
Convert between improper fractions and mixed numbers.
223
Chapter 3 — Fractions
224
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
quotient
TO
LEARN
conversion
improper fraction
convert
mixed number
comparison
numerator
denominator
proper fraction
fraction
ratio
fraction bar
simple fraction
fraction notation
BUILDING MATHEMATICAL LANGUAGE
A fraction is, in its broadest definition, the quotient of two quantities. It has three components—a
top number called the numerator, a bottom number called the denominator, and a fraction bar
separating them to indicate their relationship.
Example:
numerator
denominator
5
8
fraction bar
This chapter will only consider what are sometimes referred to as simple fractions, those fractions
whose numerators and denominators are whole numbers.
5
3.5
For example,
is a simple fraction but
is not.
8
10
It is always important to examine the context in which a fraction is being used. By its components, a
fraction represents the information of a relationship, and the sort of relationship determines into which of
four broad categories the fraction fits:
Category 1 — Part of a Whole Unit
Category 2 — Part of a Group of Objects
Category 3 — Division into Equal Portions
Category 4 — Comparison of Two Quantities
Section 3.1 — Introduction to Fractions
225
Categories 1 and 2 — Parts of a Whole or Parts of a Group
The fractions with which you are most familiar may be those that represent specific parts:
— of a whole unit which can be divided into equal parts (a sheet cake, a pizza, a cup)
(Category 1)
OR
— of a group of objects (a classroom of students, a box of assorted chocolates,
a bouquet of flowers) (Category 2)
5
In both categories, the fraction
is read as “five eighths.”
8
In Category 1, the fraction implies
five out of eight equal parts of a whole.
In Category 2, “five eighths” implies
five out of a group of eight.
Refer back to the first four examples in the Introduction to this section—Sewing, Cooking, Market research, and
Quality control—in which the numerators denote the number of parts being considered, and the denominators
denote either the total number of parts in the specified whole unit or in the specified group.
Complete the following tables for these Category 1 and 2 fractions using the information found on
page 207. An additional example for each has been filled in for you.
Category 1
Additional Example: Carpentry: You trim
Context
Example:
Carpentry
Sewing
Cooking
5
of an inch from a piece of molding to make it fit.
16
What is the
whole unit?
The whole unit is
broken up into how
many equal parts?
How many
parts are being
considered?
What is the
fractional
representation?
1 inch
16
5
5
16
Chapter 3 — Fractions
226
Category 2
Additional Example: Merchandising: For every seventeen toasters on display at the chain store, six of
⎛6⎞
them are of a particular brand name ⎜⎜ ⎟⎟.
⎜⎝17 ⎟⎠
Context
Example:
Merchandising
Identify the
group.
The whole unit is
broken up into how
many equal parts?
How many
parts are being
considered?
What is the
fractional
representation?
display of
toasters
17
6
6
17
Market research
Quality Control
Category 3 — Division into Equal Parts
A fraction in this category represents a quantity split apart evenly.
In this category,
5
implies 5 units broken up into 8 equal portions (“five divided by eight”).
8
Note: This division results in
5
, or “five eighths” of a unit per portion.
8
The Construction and Home decorating examples in the Introduction fit into Category 3.
With that in mind, use the information found on page 207 to complete the following table.
Category 3
Additional Example: Culinary arts: Six pizzas split evenly among thirteen people allows each
6
person
of a pizza.
13
Context
Example:
Culinary arts
Construction
Home decorating
Identify the quantity
to be distributed
evenly.
What is the
number of equal
portions?
six pizzas
13
What is the fractional
representation?
6
of a pizza per person
13
Section 3.1 — Introduction to Fractions
227
Category 4 — Comparison of Two Quantities
A fraction in this category represents a comparsion of two quantities (a ratio).
In this category,
5
is read as “five to eight.”
8
The final two examples in the Introduction (Demographics and Conversions) fit into this category.
With that in mind, use the information found on page 207 to complete the following table.
Category 4
Additional Example: Gardening: The directions for mixing up the liquid fertilizer call for two parts
fertilizer concentrate to eight parts water; that is, mix in the ratio of 2 .
8
Context
Example:
Gardening
Which is the initial
quantity?
What is the
quantity it is
compared to?
What is the fractional
representation of the
comparison?
2 parts fertilizer
8 parts water
2
8
Demographics
Conversions
Additional Descriptions for Fractions
A proper fraction has a numerator less than its denominator.
3
“three fourths”
4
THINK
3 out of 4 parts of a whole unit that
has been divided into 4 equal parts
make up less than one whole unit.
Chapter 3 — Fractions
228
An improper fraction has a numerator which is the same as or greater than its denominator.
7
“seven sixths”
6
7 pieces, when there are six
pieces in a whole, make up
more than one whole unit.
THINK
8 of 8 items in a group make one entire group
8
“eight eighths”
8
THINK
Or,
8 of 8 equal parts in a whole make up a whole
unit.
A mixed number combines a whole number and a fractional part of a whole. The attached fractional
part must always be a proper fraction.
THINK
2
2 whole units and 3 out 5 pieces of another whole unit.
3
“two and three fifths”
5
⎛
It implies the addition of the whole number and the fraction ⎜⎜2 +
⎜⎝
the addition sign is routinely omitted.
3 ⎞⎟
⎟ ; and except for calculator entries,
5 ⎟⎠
To convert a number is write it in another form. Improper fractions can be converted to mixed
numbers and mixed numbers to improper fractions.
Converting Between Mixed Numbers and Improper Fractions
As you will learn later in this chapter, it will be necessary to write a mixed number in its improper fraction
notation for multiplication and division. Equally as important, you will write an improper fraction as a mixed
number when presenting your final answer to a computation with fractions.
Following are the methodologies for converting between the two forms. As you follow the steps, note that
an understanding of what the denominator represents is a particularly key piece of information for each
conversion.
Section 3.1 — Introduction to Fractions
229
METHODOLOGIES
Converting an Improper Fraction to a Mixed Number
►
►
17
Example 1: Convert 5 to a mixed number.
23
Example 2: Convert 3 to a mixed number.
Try It!
Steps in the Methodology
Step 1
Identify the
denominator.
Step 2
Determine
whole
number part.
Step 3
Determine
fraction part.
Identify the number of parts
in a whole (the denominator).
Divide the total number of
parts (the numerator) by the
number of parts in a whole
(the denominator) to identify
how many whole units or
groups there are.
Determine from the
remainder the fractional part
of a whole.
Special No remainder
Case: (see page 231, Models)
Step 4
Present the
answer.
Step 5
Validate your
answer.
Example 1
17 implies 5 parts in a
5 whole.
3
5 17
3 whole units
)
−1 5
3
5 17
2 of 5 parts in a
whole remain.
)
2
5
−1 5
2
3 whole units +
2/5 of another
VISUALIZE
Present the whole number
connected to the fractional
part as your mixed number
answer.
Validate by changing the
mixed number to an improper
fraction.
3
2
5
(parts in a whole)
3×5 = 15
(parts in 3 whole units)
15+2 = 17 (total parts)
17
9
5
Example 2
Chapter 3 — Fractions
230
Converting a Mixed Number to an Improper Fraction
►
►
2
to an improper fraction.
9
7
Example 2: Convert 3 to an improper fraction.
8
Example 1: Convert 8
Try It!
Steps in the Methodology
Step 1
Identify the
fraction part.
Step 2
Identify the
denominator.
Step 3
Determine the
parts from
the whole
number.
Step 4
Add the parts
from the
fraction.
Example 1
Identify the fractional portion
of the mixed number.
2
is the fractional portion
9
2
of 8
9
Identify the number of parts in
each whole unit or group (the
denominator).
2
implies 9 parts in a
9 whole.
Multiply the number of whole
units by the number of parts in
each whole (the denominator).
8×9
8 whole units × 9 parts
per whole = 72 parts
Add the number of fractional
parts (the numerator of the
fractional part).
72
+2
74 total parts
VISUALIZE
Step 5
Present the
answer.
Step 6
Validate your
answer.
Present your fraction answer:
74
9
total number of parts
number of parts in a whole
Validate by changing the
improper fraction to a mixed
number.
8
9 74
−72
2
)
8
2
9
9
Example 2
Section 3.1 — Introduction to Fractions
231
MODELS
Special Case: No Remainder
Convert each improper fraction:
A
►
63
7
Step 1:
Steps 2-4:
7 parts in a whole.
9
7 63
Answer: 9
−63
)
If there is no remainder—that is,
if the denominator divides into the
numerator evenly—there will be no
fractional part. The improper fraction
will convert to a whole number.
0
Step 5:
B
►
8
8
Validate: 9 × 7 (parts in a whole) = 63 (total parts)
THINK
8 divided by 8 equals 1, or
8 parts out of 8 total parts in a whole
63
9
7
Answer: 1
Writing Whole Numbers as Fractions
When computing with a combination of whole numbers and fractions, it may be necessary to write a whole
number, including the number one (1), in fraction notation. The following techniques demonstrate how you
can easily do so. Each uses a Special Property of Division Involving One (1) as its starting point.
TECHNIQUES
First recall the Special Property of Division that states that any number divided by itself equals one (1). Therefore, in order to write the number one (1) as a fraction, the numerator and the denominator must be the same.
Writing One (1) as a Fraction
Technique
To write one (1) as a fraction, you can write it as
Examples: 1 =
1=
any number
.
that same number
4
4
THINK
One whole is 4 parts out of 4 parts in a whole (as in Category 1).
25
25
THINK
One entire group is 25 out of a group of 25 (as in Category 2).
Chapter 3 — Fractions
232
Next, recall the Special Property of Division that states that any number divided by one (1) equals that same
number.
any number
= that same number
1
Therefore, the simplest way to write any whole number as a fraction is to use the whole number as the
numerator and one (1) as the denominator.
Writing Any Whole Number as a Fraction
Technique
To write a given whole number as a fraction,
you can simply write it as the given whole number .
1
9
15
38
15 =
38 =
Examples: 9 =
1
1
1
Additional Note:
Keeping in mind that a fraction can also represent a division, you might also write the number 9, for example,
as 18 , or 27 , or 36 , and so on; and the number 15 as 30 , or 45 , or 60 , and so on.
2
3
4
2
3
4
ADDRESSING COMMON ERRORS
Incorrect
Process
Issue
Incorrectly
determining the
number of parts
in a whole when
converting from
a mixed number
to an improper
fraction
5
1 5+1 6
=
=
3
3
3
Correct
Process
Resolution
Identify the denominator
of the fraction component
of the mixed number.
It indicates the number
of parts in a whole for
conversion.
1 implies 3 parts in a whole
3
5 × 3 = 15
15 + 1 =16
Answer: 16
3
5
)
5
Validate: 3 16
−15
5
1
9
3
1
Incorrectly
determining the
total number of
components in a
group
In a class of 8
women and 7 men,
what fractional
part of the class is
comprised of men?
7
8
The total number in a
group is determined by
adding all components of
the group.
This total becomes
the denominator when
representing fractional
parts of the group.
8 women
+7 men
15 total in group (class)
7
Men comprise
of the
15
class.
Section 3.1 — Introduction to Fractions
Issue
Incorrectly
identifying
the number of
portions when
subdividing a
quantity
233
Incorrect
Process
Mom has three children who will share
four candy bars. How
many bars will each
child get?
3c
chi
children
hildren
ars
4 bars
Correct
Process
Resolution
When subdividing a
quantity, what is being
divided becomes the
numerator, and the
number of portions
becomes the denominator.
In this case, four candy bars
are to be shared (subdivided)
among three children.
4 candy bars
3 children (portions)
3
of a bar each
4
Not comparing
equal parts in a
whole
=
Whereas the parts of a
group need not be of equal
size (for example, a group
of students), a fraction
of one whole item truly
describes that fractional
part of the whole only if
the pieces are all the same
size.
This rectangle has
been
into three
n cut
c int
parts.
Therefore
s The
e
the sha
shaded
part
ed p
represents
1/3 of
t 1/
the rectangle.
4
1
or 1 bars per child
3
3
In the diagram at the
left, none of the pieces
represents 1/3 of the
rectangle because the pieces
are not all the same size.
One third would be
represented by the shaded
section in the rectangle
below.
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the terminology and notation associated with fractions
how to determine the category in which a specific fraction fits by the context in which it appears
the meaning of the numerator in each of four broad categories of fractions
the meaning of the denominator in each of four broad categories of fractions
the process of converting between mixed numbers and improper fractions
the meaning of a mixed number
how to write any whole number as a fraction
Section 3.1
ACTIVITY
Introduction to Fractions
PERFORMANCE CRITERIA
• Interpreting a given fraction correctly
– correct meaning of its numerator for the context
– correct meaning of its denominator for the context
• Setting up a fraction correctly for a given context
– correct identification and placement of the numerator
– correct identification and placement of the denominator
• Converting between improper fractions and mixed numbers
– accurate presentation of the answer
– validation of the conversion
CRITICAL THINKING QUESTIONS
1. What are the three components of a fraction?
•
•
•
2. What are the four categories of fractions identified in the Pre-Activity?
•
•
•
•
3. In general, how can the number one (1) be written as a fraction? Give two examples.
4. In general, how can you represent any whole number other than one (1) as a fraction?
234
Section 3.1 — Introduction to Fractions
235
5. What is the meaning of a zero in the numerator for each of the four categories of fractions?
•
•
•
•
6. How do you determine how many parts are in a whole for a mixed number? for an improper fraction?
7. What are the relationships between the processes of converting between improper fractions and mixed
numbers and their validation techniques?
8. What happens to the size of the fraction when the numerator stays the same but the denominator increases?
Chapter 3 — Fractions
236
TIPS
FOR
SUCCESS
• When setting up a fraction for a given context, first determine the total number of parts in the whole (or
group). This number becomes the denominator.
• Use diagrams to visualize fractions.
DEMONSTRATE YOUR UNDERSTANDING
1. Draw a diagram to represent the following fractions and mixed numbers.
a) 2 as a Category 1 fraction
b) 3 as a Category 2 fraction
3
7
c)
3
1
as a Category 1 fraction
8
2. By using an example other than those already presented, describe the relationship and meaning represented
7
by
for each of the four categories of fractions discussed in this section.
9
Category 1
Category 2
Category 3
Category 4
Section 3.1 — Introduction to Fractions
237
3. There are 18 smokers and 45 nonsmokers in a group.
a) What fraction of the group are nonsmokers?
b) What fraction of the group are smokers?
c) What is the ratio (Category 4) of smokers to non-smokers?
4. Three diskettes are used from a box of ten.
a) What fraction of the diskettes is left?
b) What is the ratio (Category 4) of used diskettes to non-used diskettes?
5. Fill in the chart below.
Improper Fraction
a)
37
8
b)
c)
5
5
6
29
7
d)
e)
Mixed Number
10
52
13
11
12
Number of Parts
in a Whole
Validation of the
Conversion
Chapter 3 — Fractions
238
TEAM EXERCISES
1
tablespoons of salt.
2
Complete the following table for this context.
1. A soup recipe calls for 2
What is the
whole unit?
The whole unit is
broken up into how
many equal parts?
How many
parts are being
considered?
What is the fractional
representation? (2 forms)
2. Write the number five (5) as a fraction in ten different ways.
IDENTIFY
AND
CORRECT
THE
ERRORS
Identify the error(s), if any, in the following worked solutions. If the worked solution is incorrect, solve the
problem correctly in the third column. If the worked solution is correct, write “Correct” in the second column.
Worked Solution
What is Wrong Here?
1) What mixed number
17
is equivalent to
?
4
Identify the Errors
The division is incorrect.
The quotient should be 4
(not 3) with a remainder
of one. Four goes into
14 four times with a
remainder of one.
While 3
5
is correct, it is
4
not fully reduced.
Correct Process
4
4 17
−16
1
)
Answer: 4
4
1
4
1
4
Section 3.1 — Introduction to Fractions
Worked Solution
What is Wrong Here?
2) Write 2 1 as an improper
2
fraction.
3) Three cases of twenty-four
cans each and seven more
cans is equal to
cases.
4) 20 sub sandwiches to be
shared equally among
36 people will allow each
person
of a sub.
5) The fractional part of
this group of circles that
is shaded is
6) Write the whole number
three as a fraction.
239
Identify the Errors
Correct Process
240
Chapter 3 — Fractions
ADDITIONAL EXERCISES
1. Draw a diagram to represent the following fractions:
7
a)
as a Category 1 fraction
9
8
b)
as a Category 2 fraction
11
3
c) 5 for Category 1
4
2. There are seven yellow daisies in a “Happy Birthday Bouquet” of twenty daisies. The rest of the daisies
in the bouquet are white.
a) What fraction of the daisies are yellow?
b) What fraction of the daisies are white?
c) What is the ratio (Category 4) of yellow daisies to white daisies?
3. Write the each of the following as a mixed number.
32
a)
3
72
b)
8
c) 65
4
4. Write each of the following as an improper fraction.
5
a) 4
9
2
b) 12
3
c) 14