Focusing on Fractions
C. David Pilmer
Nova Scotia School for Adult Learning
Adult Education
2007
This resource is the intellectual property of the Adult Education Division of the Nova Scotia
Department of Labour and Advanced Education.
The following are permitted to use and reproduce this resource for classroom purposes.
• Nova Scotia instructors delivering the Nova Scotia Adult Learning Program
• Canadian public school teachers delivering public school curriculum
• Canadian nonprofit tuition-free adult basic education programs
The following are not permitted to use or reproduce this resource without the written
authorization of the Adult Education Division of the Nova Scotia Department of Labour and
Advanced Education.
• Upgrading programs at post-secondary institutions
• Core programs at post-secondary institutions
• Public or private schools outside of Canada
• Basic adult education programs outside of Canada
Individuals, not including teachers or instructors, are permitted to use this resource for their own
learning. They are not permitted to make multiple copies of the resource for distribution. Nor
are they permitted to use this resource under the direction of a teacher or instructor at a learning
institution.
Table of Contents
Preface………………………………………………………………………………………. ii
What Do You Already Know About Fractions?……………………………………………. iv
Tracking Your Progress…………………………………………………………………….. ix
Level II Materials
Area Models for Fractions…………………………………………………………………..
Improper Fractions and Mixed Numbers……………………………………………………
More Fractions………………………………………………………………………………
Equivalent Fractions………………………………………………………………………...
Measuring in Inches…………………………………………………………………………
Comparing Fractions………………………………………………………………………..
Concept Chart for Fractions…………………………………………………………………
1
12
15
22
31
33
39
Bridging Materials from Level II to Level III
Estimating the Addition and Subtraction of Fractions………………………………………
Adding Fractions…………………………………………….................................................
Subtracting Fractions………………………………………………………………………..
Multiplying Fractions………………………………………………………………………..
Dividing Fractions…………………………………………………………………………..
Charting Your Own Course…………………………………………………………………
40
44
55
61
70
78
Learning Logs…………………………………………………………………………….....
Glossary….………………………………………………………………………………….
79
83
Additional Items (Intervention Materials)
Area Model (Proper Fractions)……………………………………………………………...
Area Models (Improper Fractions & Mixed Numbers)……………………………………..
Money and Fractions………………………………………………………………………...
Number Magnitude Questions Using Area Models…………………………………………
Questions Involving Benchmark Numbers……………………………………....................
Number Magnitude Questions………………………………………………………………
Simplest Form Questions……………………………………………………………………
Another Approach to Addition and Subtraction……………………………………………
84
85
86
88
89
90
91
92
List of Websites……………………………………………………………………………..
93
Answers……………………………………………………………………………………... 95
NSSAL
i
Focus on Fractions
Preface
This resource was developed for the Level II Math program within the Nova Scotia School for
Adult Learning (NSSAL). The resource was designed to give adult learners a conceptual
understanding of fractions and their operations prior to the introduction of algorithms. Although
explanations have been provided at the beginning of each section, instructors are expected to
provide contextual introductions, elaborations, and additional examples to their learners. The
instructors should also be prepared to intercede and provide additional resources, if learners are
becoming confused or frustrated with a concept or concepts. Several sections of this resource
close with open-ended questions. These questions have more than one acceptable answer, can be
addressed using different strategies, and may demonstrate different levels of sophistication
among learners. It is important that the instructor take some time to initiate discussions
regarding the variety of learner solutions to these open-ended questions.
One of the problems with traditional adult textbooks in mathematics is that they tend to focus on
the mechanics of math. Little time is spent asking learners to make sense of the math and
therefore some learners are left with the misconception that mathematics is about memorizing
procedures. Although many middle school math textbooks require learners to investigate
concepts through activities, the texts are full of images and contexts that are inappropriate for
adult learners. Additionally, the extensive use of the investigative approach found in these
middle school textbooks is time intensive; an issue that can be problematic for many adult
learners. There are also concerns that relying too heavily on the investigative approach can be
detrimental to low-achieving math students. Several research studies have concluded that more
scaffolded and/or direct lessons can better serve the learning needs of these low-achieving
students. This resource attempts to blend many of the positive attributes of middle school math
textbooks with those found in traditional adult texts.
Each section finishes with a wrap up statement and a questionnaire called “Reflect Upon Your
Learning.” The wrap up statement is comprised of one or two sentences that focuses on the main
idea or ideas in lesson. The questionnaire asks the learner to respond to a series of statements
regarding their level of understanding of the concepts covered in the lesson. Learners should
complete this questionnaire as they complete a lesson. Their responses can be useful to the
instructor when he/she is trying to determine if more time should be spent on the concept. The
learners have another opportunity to demonstrate their level of understanding when they
complete the learning logs found at the end of this resource. It is recommended that the learners
complete each part of the log as they complete each corresponding section of this resource. As
an incentive, instructors could allow the adult learners to use these completed logs when writing
tests or quizzes.
There are a variety of ways of representing fractions visually.
(1)
(2)
(3)
(4)
Traditional Area Models Using Squares or Circles
Area Model Manipulatives (Pattern Blocks, Fraction Pieces, Geoboards)
Length Models (Fraction Strips)
Set Models
Since time is a factor for many adult learners, it was decided that this resource would focus on
traditional area models using squares or circle. This is not to say that the other representations
should not be considered in your classroom. Explorations involving pattern blocks can be
beneficial to some learners and therefore should not be dismissed. The decision to focus on
traditional area model representations for fractions was also based on the availability of
NSSAL
ii
Focus on Fractions
interactive online math activities (applets). Many of the applets work solely with the traditional
representation. You will notice that applicable applets have been identified throughout this
resource.
Having learners work with all four visual representations of fractions is not recommended.
Research states that when too many representational systems are used, some learners, especially
those with learning disabilities, may not notice similarities and may not make the necessary
connections. In a Level II Math course, it is reasonable to assume that some of the learners will
have undiagnosed learning disabilities. Focusing on only one or two visual representations is
imperative.
The first six sections of this resource focus primarily on the magnitude of fractions. These
sections correspond with outcomes in the Level II Math curriculum. The last five sections focus
on operations (addition, subtraction, multiplication, and division) with fractions and are
classified as “bridging materials” between the Level II and Level III curricula. It is
recommended that Level II learners should be exposed to these bridging materials if they plan on
enrolling in Level III Math, and if time permits. You will notice that the operations sections only
involve proper and improper fractions. Work with mixed numbers will be left for the Level III
course.
It should be stressed that this resource is one of many resources available to Level II Math
instructors. There are portions of this document that may not serve the needs of your learner.
Exercise your professional judgment. If you have materials that better meet your learners’ needs
and still meet the outcomes, use them.
This resource, or sections of this resource, may also be used by Level III Math instructors since
fractions are revisited in the Level III program. Most Level III courses are delivered by the Nova
Scotia Community College using individualized student instruction (ISI). Under this form of
instruction students are working at different speeds and on different concepts within the same
classroom. With these learners it is important to determine their level of mathematical
knowledge and understanding prior to assigning work. Having adults work on materials that
they already know can be problematic for the learner and instructor. To address this issue, this
resource includes a section titled “What Do You Already Know About Fractions?” Level III
learners can attempt the questions in this section of the resource. The instructor can check their
answers and based on their performance, decide what sections of this resource can be skipped by
the learner.
Some of you may be troubled with the definition for improper fraction found in the third section
of this resource.
An improper fraction is a fraction whose numerator is greater than or equal to its
denominator. Improper fractions are equal to or greater than 1.
Some resources state that improper fractions are fractions whose numerators are greater than the
denominator. They will then state that the unit fraction occurs when the numerator and the
denominator are equal. Use the definition that you are comfortable with.
Special thanks to the reviewers of this document.
Dr. Derek Berg (MSVU)
Meredith Hutchings (NSSAL)
Dianne Gray (ALACBC)
Sharon McCready (English Program Services)
NSSAL
iii
Focus on Fractions
What Do You Already Know About Fractions?
In this package of materials, there are 12 different sections on fractions. Based on what you
already know about fractions, you may not have to do all twelve sections. Try to answer the
questions listed in the blocks below. Do not seek assistance. Do not use a calculator. This is
not a test. It’s only used to determine if you have enough prior knowledge of fractions to skip
some of the sections in this package of material. If you can correctly answer all the questions in
a particular block , then you can skip the corresponding sections in this package.
Block of Questions
Block 1
(a) What fraction would represent the
shaded portion of the following area
model?
Sections You Can Skip
Area Models for Fractions
(pages 1 to 11)
(b) For each of these fractions, determine if it is closest to 0,
1
, or 1.
2
5
(i) The fraction
is closest to _____.
6
(ii) The fraction
2
is closest to _____.
9
(iii) The fraction
7
is closest to _____.
15
(c) Put these fractions in order from smallest to largest.
2 7 1 5
Answer: ____, ____, ____, ____
, , ,
7 7 7 7
(d) Put these fractions in order from smallest to largest.
8 9 2 5
Answer: ____, ____, ____, ____
,
,
,
8 10 13 9
Block 2
(a) Put these fractions in order from smallest to largest.
7 1 9 4
Answer: ____, ____, ____, ____
, , ,
5 5 5 5
Improper Fractions and Mixed
Numbers
(pages 12 to 14)
(b) Put these fractions in order from smallest to largest.
6 1 9 7
Answer: ____, ____, ____, ____
, , ,
7 3 9 5
NSSAL
iv
Focus on Fractions
Block 3
(a) Change each of these improper fractions to a mixed
number.
11
13
(i)
(ii)
= _______
= _______
6
4
More Fractions
(pages 15 to 21)
(b) Change each of these mixed numbers to an improper
fraction.
4
1
(ii) 2 = _______
(i) 1 = _______
7
9
(c) Put these fractions in order from smallest to largest.
15 10 2 6
Answer: ____, ____, ____, ____
,
, ,
7 3 9 5
Block 4
(a) Change each of these fractions to their simplest form.
12
7
(i)
(ii)
= _______
= _______
15
35
Equivalent Fractions
(pages 22 to 30)
(b) State whether the following pairs of fractions are
equivalent or not.
9
6
___________________________
(i)
and
9
12
(ii)
6
15
and
8
20
___________________________
Block 5
(a) Using a ruler, measure the following line segment in
inches.
(i)
Measuring in Inches
(pages 31 to 32)
(ii)
(iii)
NSSAL
v
Focus on Fractions
Block 6
(a) Determine which of these fractions is larger by changing
them to a common denominator.
4 3
,
7 5
Comparing Fractions
(pages 33 to 38)
(b) Put these fractions in order from smallest to largest.
9 2 11 1 13 4
, , , , ,
8 3 11 12 8 7
Block 7
Estimate each of these sums or difference.
1
1
(a) 2 + 3
25
30
(b) 1
(c)
Estimating the Addition and
Subtraction of Fractions
(pages 40 to 43)
12
14
+6
13
15
13 5
−
12 9
Block 8
(a) Figure out each of these sums.
2 4
5 3
(i)
(ii) +
+
3 5
6 4
Adding Fractions
(pages 44 to 54)
1
3
pounds of finishing nails, pounds of
2
4
5
common nails, and pounds of galvanized nails. How
8
many pounds of nails do you have in total?
(b) You have
NSSAL
vi
Focus on Fractions
Block 9
(a) Figure out each of these differences.
4 2
7 1
(i)
(ii)
−
−
7 5
8 6
(b) You have a piece of lumber that is
Subtracting Fractions
(pages 55 to 60)
7
of an inch thick. If
8
3
of an inch off the thickness of the lumber,
16
then how thick is the newly planed piece of lumber?
you plane
Block 10
(a) Multiply the following fractions. In some cases you will
have to put the answer in its simplest form.
6 4
5 2
(i) ×
(ii) ×
7 3
7 3
(iii) 4 ×
5
8
(b) A recipe calls for
making
NSSAL
Multiplying Fractions
(pages 61 to 69)
3
of a cup of flour. If you are only
4
1
of the recipe, how much flour is needed?
2
vii
Focus on Fractions
Block 11
(a) Solve each of the following..
2
7 2
(i)
(ii)
÷4
÷
11
9 3
Dividing Fractions
(pages 70 to 77)
(b) You have 10 pounds of flour in a bag. You are removing
2
it from the bag using a container that can hold
pounds
5
of flour. Assuming that you are filling the container each
time, how many times will you use the container to
completely empty the bag?
NSSAL
viii
Focus on Fractions
Tracking Your Progress
This page allows you to keep track of your progress though this material on fractions.
Level II Materials
Area Models for Fractions…………………………………….
Improper Fractions and Mixed Numbers……………………..
More Fractions………………………………………………..
Equivalent Fractions…………………………………………..
Measuring in Inches…………………………………………..
Comparing Fractions………………………………………….
Concept Chart for Fractions…………………………………..
Date
Started
Date
Completed
Date
Started
Date
Completed
1
12
15
22
31
33
39
Bridging Materials from Level II to Level III
Estimating the Addition and Subtraction of Fractions………..
Adding Fractions……………………………………………...
Subtracting Fractions………………………………………….
Multiplying Fractions…………………………………………
Dividing Fractions……………………………………………
Charting Your Own Course…………………………………..
40
44
55
61
70
78
Learning Logs…………………………………………………………………………………79
Glossary……………………………………………………………………………………… 83
NSSAL
ix
Focus on Fractions
Area Models for Fractions
Introduction
You may not realize it, but you’ve been working with fractions for years. Every time you handle
money, you are dealing with fractions.
•
Consider a quarter. It’s worth 25 cents and takes 4 quarters to make 1 dollar. As a
1
fraction, you would say that one 25 cent coin is
of a dollar.
4
•
Consider a 50 cent coin. This is a fairly rare coin that is only made to commemorate
special events. It takes 2 fifty cent coins to make 1 dollar. As a fraction, you would say
1
that one 50 cent coin is
of a dollar
2
Those of you who have worked in trades (carpentry, electrical, plumbing…) have also been
exposed to fractions. In the trades, most measurements are made using imperial measure
(inches, feet, pounds, ounces, cups, gallons…). Fractions are frequently used in imperial
measure. A carpenter would not ask for a piece of wood to be cut to a length of 6.75 inches,
3
rather the carpenter would state the measurement as 6 inches.
4
If you need to learn about fractions, the first place to start is with area models. Area models are
pictures that represent fractions.
Explanation:
If you use a square for your area model, then the fraction
3
can be drawn in a variety ways. All
4
of these are acceptable representations of this fraction.
Each of the above figures is divided into 4 equal parts and 3 of those parts are shaded in. That is
3
why each area model represents the fraction . The 3 is called the numerator and the 4 is the
4
denominator.
2
because the square
5
is divided into 5 equal parts and 2 of those parts are shaded in.
The area model to the right represents the fraction
NSSAL
1
Focus on Fractions
Area models can be useful when you are trying to compare the magnitude (size) of different
fractions.
5
2 5 1
Example: Order the fractions , , , and from smallest to largest.
3 12 9
6
2
5
5
1
3
6
12
9
1
must be the smallest number since the smallest portion of the square is shaded.
9
5
must be the largest number since the greatest portion of the square is shaded.
6
Sometimes it’s useful to plot the fractions on a number line and compare them to familiar
1
numbers like 0, , and 1 (benchmark numbers) when trying to understand the size of a
2
fraction.
5
2
5
1
6
3
12
9
0
1
1
2
The order from smallest to largest is
1 5 2 5
, , , .
9 12 3 6
Questions:
1. For each area model, figure out the fraction that the model represents.
(a)
(b)
(c)
(d)
Answer: ____
NSSAL
Answer: ____
Answer: ____
2
Answer: ____
Focus on Fractions
2. In question 1, which one is the smallest fraction? Which one is the largest fraction?
Smallest Fraction: _______
Largest Fraction: _______
3. Do all of the following area models represent the fraction
4. Does the following area model represent the fraction
5
? Why or why not?
6
1
? Why or why not?
3
5. You are going to use an interactive activity found on the internet. Using Google, search the
following key words: Shodor Interactive Activities. Once you are on the site, scroll down
and find the activity Fraction Sorter. It’s a game where you are asked to create the area
models (using circles or squares) for two fractions and then figure out which one is the larger
fraction. Play the game three times, making sure that you print out a copy of each round of
the game.
6. Create an area model for each of the following fractions.
5
1
3
(a)
(b)
(c)
6
5
8
NSSAL
3
(d)
4
5
Focus on Fractions
7. Look at the area models you created in question 6. Based on models, figure out whether the
1
fraction is closest to 0, , or 1. If it’s close to 0, then a very small portion of the whole
2
square is shaded. If it’s close to 1, then a very large portion of the whole square is shaded. If
1
it is close to , approximately half of the square is shaded.
2
5
1
(a) The fraction is closest to _____.
(b) The fraction is closest to _____.
6
5
3
4
(c) The fraction is closest to _____.
(d) The fraction
is closest to _____.
8
5
8. Take the fractions in question 6 and put them in order from smallest to largest.
Smallest
Largest
_______
_______
_______
_______
9. You are going to use an interactive activity found on the internet. Using Google, search the
following key words: NCTM Illuminations Activities. Once you are on this site, type in
fraction into the advanced options and press “search.” Different activities will show up on
the screen. Select Fraction Model I.
This tool allows you to examine area models for different fractions. There are two sliders of
the screen. The top slider allows you to change the numerator of the fraction. The bottom
slider allows you to change the denominator of the fraction. When this is done, an area
model is created, and the number is displayed as a fraction, decimal and percent. We’re not
concerned with decimal and percent answers at this time.
For each of the fractions in the table, use the internet tool to figure out if the fraction is
1
closest to 0, , or 1. The area model helps you figure this out. If it’s close to 0, then a very
2
small portion of the whole square is shaded. If it’s close to 1, then a very large portion of the
1
whole square is shaded. If it is close to , approximately half of the square is shaded.
2
Two sample questions have been done for you.
(a)
(c)
NSSAL
Fraction
4
5
2
5
5
5
Closest to:
1
(b)
(d)
4
Fraction
9
20
3
5
1
10
Closest to:
1
2
Focus on Fractions
(e)
(g)
(i)
(k)
Fraction
9
10
17
20
1
8
5
8
Closest to:
Fraction
11
20
3
20
7
8
(f)
(h)
(j)
10. Complete each of these statements with the numbers 0,
Closest to:
1
, or 1.
2
(a) When the number in the numerator of a fraction is close to the number in the denominator
7 9 15
of the fraction (examples: , , ), then the fraction is closest to the number ___.
8 11 18
(b) When the number in the numerator of a fraction is very small compared to the number in
1 2 3
the denominator of the fraction (examples: , , ), then the fraction is closest to the
8 11 18
number ___.
(c) When the numerator of the fraction is about half the size of the denominator of the
4 6 14
fraction (examples: , , ), then the fraction is closest to the number ___.
9 11 30
(d) When the numerator of the fraction is the same number as the denominator of the fraction
3 8 11
(example: , , ), then the fraction is equal to ___.
3 8 11
11. Without drawing an area model, figure out if each fraction is closest to 0,
Fraction
(a)
(d)
(g)
NSSAL
17
18
3
25
14
15
Closest to:
Fraction
(b)
(e)
(h)
Closest to:
Fraction
(c)
2
27
8
14
1
30
(f)
(i)
5
1
, or 1.
2
Closest to:
10
11
9
20
16
30
Focus on Fractions
Fraction
(j)
8
17
1
100
(m)
Closest to:
Fraction
(k)
Closest to:
(l)
2
19
7
12
(n)
Fraction
Closest to:
19
20
13
15
(o)
12. For each question, a pair of fractions has been provided. Circle the larger number. For some
questions, you may want to try to visualize the area model for each fraction. For other
1
questions, you may want to figure out if the fractions are close to 0,
, or 1.
2
3
7
4
1
(a)
(b)
10 10
5
5
(c)
5
8
1
8
(d)
1
5
1
3
(e)
1
7
1
10
(f)
3
8
3
5
(g)
3
7
5
7
(h)
7
8
2
3
(i)
1
9
4
5
(j)
4
9
11
12
(k)
6
7
1
4
(l)
2
11
5
9
(m)
9
10
5
11
(n)
5
6
2
9
(o)
1
30
6
7
(p)
11
12
7
16
13. A number line has been provided. There are four arrows that identify the approximate
8 3 1
5
locations of the fractions , , and . Figure out where each fraction would
8 8 8
8
approximately be on the number line. Be able to explain why.
0
NSSAL
1
2
1
6
Focus on Fractions
14. A number line has been provided. There are four arrows that identify the approximate
3 4 1
11
locations of the fractions , , and
. Figure out where each fraction would
3 7 10
12
approximately be on the number line. Be able to explain why.
0
1
2
1
15. In each case, take the fractions and place them in order from smallest to largest. If you are
struggling with questions (a) to (d), try visualizing or drawing the area models for each
1
fraction. For the rest of these questions, determining if a fraction is close to 0,
, or 1, or
2
equal to 1 would be useful when attempting to order these fractions.
Smallest
Largest
(a)
3 5 1 7
, , ,
7 7 7 7
____
____
____
____
(b)
5 1 7 2
, , ,
9 9 9 9
____
____
____
____
(c)
3 10 11 7
, , ,
11 11 11 11
____
____
____
____
(d)
3 1 1 4
, , ,
5 5 20 5
____
____
____
____
(e)
1 1 1 1
, , ,
7 2 3 9
____
____
____
____
(f)
2 2 2 2
, , ,
3 9 5 7
____
____
____
____
(g)
4 1 9 9
, , ,
7 12 9 11
____
____
____
____
(h)
4 11 3 1
, , ,
9 12 3 11
____
____
____
____
NSSAL
7
Focus on Fractions
Smallest
Largest
(i)
9 3 1 5
, , ,
10 8 12 8
____
____
____
____
(j)
1 1 6 11
, , ,
3 8 6 20
____
____
____
____
(k)
1 7 1 7
, , ,
21 9 3 13
____
____
____
____
(l)
2 9 1 7
, , ,
19 10 19 12
____
____
____
____
(m)
19 5 1 17
, , ,
20 12 5 20
____
____
____
____
16. A number line has been provided. There are five arrows that identify the approximate
9 1 3 1
5
locations of the fractions , , , and . Figure out where each fraction would
9 6 7 4
9
approximately be on the number line. Be able to explain why.
0
1
2
1
17. There are 21 adults in a night school course. The class is comprised of 8 males and 13
females.
(a) What fraction of the class is female?
Answer:
______
(b) What fraction of the class is male?
Answer:
______
18. Three friends chipped in to purchase a $10 lottery ticket. One friend contributed $2. Another
friend contributed $5. You contributed $3. If you had a winning ticket, what fraction of the
winnings should you get if you wanted to be fair to the friends?
Answer:
NSSAL
8
______
Focus on Fractions
19. Maggie has 26 employees working at her warehouse. On Monday, 3 of her employees were
absent due to illness.
(a) What fraction of Maggie’s employees was absent Monday?
Answer: ______
(b) What fraction of Maggie’s employees was present Monday?
Answer: ______
20. A baker offers to cut you a piece of cake but lets you choose among four different fractions
5 1 1
3
of the cake; , , , or .
7
8 10 12
(a) If you wanted the largest piece of cake, what fraction would you take?
Answer: ______
(b) If you wanted the smallest piece of cake, what fraction would you take? Answer: ______
21. A car dealer has agreed to drop the price of the car by
1
1
or
. Which one should you
20
15
choose to get the best deal and why?
22. State all the fractions between 0 and
1
that have a denominator of 7.
2
Open-ended Questions (There is more than one correct answer for each of these questions.)
23. Find two fractions that are between 0 and
24. Find two fractions that are between
1
.
2
1
and 1.
2
25. Find a fraction that is very close to, but slightly larger than
NSSAL
9
1
.
2
Focus on Fractions
26. Find a fraction that is very close to, but slightly smaller than
1
.
2
You’re the Instructor (With these types of the questions, you are going to be the instructor. You
will have to examine a student’s work and provide some sort of feedback
to him or her.)
1
3
or
, is larger. Her answer and
2
16
explanation is shown below. Her answer is incorrect. Explain to her how she should have
worked out the answer. You can include diagrams in your explanation.
27. Chantelle was asked to determine which fraction ,
Chantelle’s Answer and Explanation:
3
1
3
is larger than
The 3 in the numerator of the fraction
is larger than the 1
16
2
16
1
in the numerator of the other fraction, . The 16 in the
2
3
denominator of the fraction
is larger than the 2 in the
16
1
denominator of the other fraction, . Based on this, I figure
2
3
1
is larger than .
that
16
2
Your Explanation:
Wrap-Up Statement:
Being able to compare a fraction to more familiar numbers (benchmark numbers) like 0,
1
,
2
and 1 can be useful when you are trying to understand the size of the fraction.
NSSAL
10
Focus on Fractions
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 27. Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
(g)
NSSAL
I understand all of the concepts covered in the section, “Area
Models for Fractions.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can draw the area model for a fraction.
1
I can figure out if a fraction is closest to 0, or 1.
2
I can put fractions in order from smallest to largest.
I can do the word problems like the ones found in this section.
11
1
2
3
4
5
1
2
3
4
5
1
1
2
2
3
3
4
4
5
5
1
2
3
4
5
1
1
2
2
3
3
4
4
5
5
Focus on Fractions
Improper Fractions and Mixed Numbers
Explanation:
A proper fraction is a fraction whose numerator is less than the denominator (examples:
1 3
8
, , and ). Proper fractions are between 0 and 1.
2 16
9
2
Example: is a proper fraction.
3
An improper fraction is a fraction whose numerator is greater than or equal to its denominator.
Improper fractions are equal to or greater than 1.
4
Example: is an improper fraction.
3
All improper fractions can be written as mixed numbers. A mixed number contains a whole
4
number and a fraction. For example, the improper fraction
can also be written as the mixed
3
1
number 1 .
3
Questions:
1. Express each area model as an improper fraction and as a mixed fraction.
(a)
Improper Fraction:
Mixed Number:
(b)
Improper Fraction:
Mixed Number:
(c)
Improper Fraction:
Mixed Number:
NSSAL
12
Focus on Fractions
2. Create an area model to represent each of the following.
1 9
2 7
(a) 1 or
(b) 2 or
4 4
5 5
3. Circle the improper fractions in the following list.
8
2
3
9
6
6
1
5
7
2
10
7
5
9
13
10
4. In each case, put the numbers in order from smallest to largest.
Smallest
Largest
(a)
5 9 1 7
, , ,
8 8 8 8
_____ _____ _____ _____
(b)
10 1 11 13
, , ,
10 10 10 10
_____ _____ _____ _____
(c)
6 5 3 9
, , ,
5 5 5 5
_____ _____ _____ _____
(d)
8 1 1 10
, , ,
7 2 7 10
_____ _____ _____ _____
(e)
6 13 1 3
, , ,
11 13 9 2
_____ _____ _____ _____
(f)
4 2 6 1
, , ,
3 19 6 2
_____ _____ _____ _____
(g)
6 6 7 1
, , ,
5 11 5 8
_____ _____ _____ _____
5. Look at the following mixed and improper fractions.
1 6
1 9
2 7
3 10
1 =
2 =
1 =
1 =
5 5
5 5
7 7
4 4
2
3 11
=
4 4
2
2 8
=
3 3
(a) How can you change a mixed number to an improper fraction?
NSSAL
13
Focus on Fractions
(b) How can you change an improper fraction to a mixed number?
Wrap-Up Statement
Improper fractions and mixed numbers are equal to or greater than 1. If you didn’t figure out
question 5, don’t worry. This topic will be covered in the next lesson.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 5. Select your response to
each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
NSSAL
I understand all of the concepts covered in the section,
“Improper Fractions and Mixed Numbers.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can draw the area model for an improper fraction or mixed
number.
I can order fractions from smallest to largest even if an
improper fraction is included with the other fractions.
14
1
2
3
4
5
1
2
3
4
5
1
1
2
2
3
3
4
4
5
5
1
2
3
4
5
Focus on Fractions
More Fractions
Explanation:
Writing a Mixed Number as an Improper Fraction
Step 1: Multiply the whole number part by the denominator of the fraction.
Step 2: Add that number to the numerator.
Step 3: Then write the result as the numerator of the improper fraction. The denominator
will remain the same.
2
as an improper fraction.
5
2 (3 × 5) + 2
3 =
5
5
15 + 2
=
5
17
=
5
1
as an improper fraction.
4
1 (2 × 4 ) + 1
2 =
4
4
8 +1
=
4
9
=
4
Ex. Write 2
Ex. Write 3
Writing an Improper Fraction as a Mixed Number
Step 1: Divide the numerator by the denominator.
Step 2: Write down the whole number answer.
Step 3: Then write down any remainder above the denominator.
11
as a mixed number.
7
11
is 11 ÷ 7.
Another way to write
7
When you divide11 by 7, you get
Ex. Write
21
as a mixed number.
4
21
Another way to write is 21 ÷ 4.
4
When you divide 21 by 4 you get
Ex. Write
5 with a remainder of 1. Write
down the 5 and then write
down the remainder of 1 over
the denominator.
1
Answer : 5
4
1 with a remainder of 4. Write
down the 1 and then write
down the remainder of 4 over
the denominator.
4
Answer :1
7
Questions:
1. Change each of these mixed numbers to improper fractions.
2
1
(a) 1 =
(b) 1 =
5
2
NSSAL
15
Focus on Fractions
2
(c) 1 =
7
6
(e) 1 =
7
3
(g) 2 =
5
1
(i) 3 =
2
4
(d) 1 =
5
1
(f) 2 =
2
3
(h) 2 =
7
4
(j) 3 =
5
Answers to Question 1 (They are not in order.)
3
9
5
7
13
13
2
5
2
2
5
7
19
5
11
5
9
7
7
5
17
7
1
6
2
2. Change each of these improper fractions to mixed numbers.
5
5
(b) =
(a) =
4
3
11
9
(c)
(d) =
=
6
4
15
7
(e)
(f)
=
=
8
3
19
19
(g)
(h)
=
=
8
4
19
16
(i)
(j)
=
=
6
3
Answers to Question 2 (They are not in order.)
3
2
5
1
2
7
5
1
1
1
4
4
6
3
3
8
4
3
1
1
4
2
3
8
2
1
3
3
1
4
3. You have been given a list of fractions. For each, figure out if the fraction is between 0 and
1
1
, between
and 1, between 1 and 2, or greater than 2. Place a check mark in the
2
2
appropriate column.
Fraction
(a)
(b)
(c)
NSSAL
Between
1
0 and
2
Between
1
and 1
2
Between
1 and 2
Greater
than 2
9
8
6
7
11
4
16
Focus on Fractions
Fraction
(d)
Between
1
0 and
2
Between
1
and 1
2
Between
1 and 2
Greater
than 2
1
9
11
7
7
12
13
6
(e)
(f)
(g)
4. In each case, put the numbers in order from smallest to largest.
Smallest
Largest
(a)
8 3 7 1
, ,1 ,1
8 8 8 8
_____ _____ _____ _____
(b)
9
1
9 1
,1 ,1 ,
10 10 10 2
_____ _____ _____ _____
(c)
2 1 10 3
, , ,
11 15 9 2
_____ _____ _____ _____
(d)
5 4 13 1
, , ,
3 5 6 4
_____ _____ _____ _____
(e)
7 7 11 9
, , ,
4 2 5 8
_____ _____ _____ _____
5. Identify which number doesn’t belong and briefly explain why you made your selection.
There may be more than one correct answer and/or explanation.
(a)
(b)
NSSAL
7
8
27
29
1
6
12
13
3
4
5
6
12
11
1
1
17
1
5
Focus on Fractions
(c)
1
6
5
6
2
6
1
7
(d)
11
10
8
8
3
3
1
1
3
12
5
(f)
7
15
9
17
5
9
9
10
(g)
1
2
1
6
1
7
1
10
(h)
1
2
7
8
0.5
50%
(e)
2
2
3
4
2
4
7
6. You are going to use an interactive activity found on the internet. Using Google, search the
following key words: NCTM Illuminations Activities. Once you are on this site, type in
fraction into the advanced options and press “search.” Different activities will show up on
the screen. Select Fraction Model I.
This tool allows you to examine area models for different fractions. There are two sliders of
the screen. The top slider allows you to change the numerator of the fraction. The bottom
slider allows you to change the denominator of the fraction. When this is done, an area
model is created, and the number is displayed as a fraction, decimal and percent.
NSSAL
18
Focus on Fractions
For each of the fractions in the two tables, use the internet tool to:
- express the fraction as a decimal,
- express the fraction as a percent.
Notice that there are two charts; one for proper fractions and the other for improper fractions.
You’ll need to compare these two charts to answer question 7.
Two sample questions have been done for you.
(a)
(b)
(c)
(d)
(e)
Proper
Fraction
4
5
3
8
19
20
2
5
7
8
6
10
Decimal
Percent
0.8
80%
Improper
Fraction
6
5
(a)
9
8
(b)
11
8
(c)
20
20
(d)
17
10
(e)
8
5
Decimal
Percent
1.2
120%
Note: At this time you are not expected to be able to convert a fraction to a decimal or percent
by hand. Many of you would only have limited experience with decimals and percents
and should not feel that you need to learn these concepts before continuing with the
Focus on Fraction materials.
7. (a) Based on your answers to question 6, you can say that any improper fraction will
generate a decimal number greater than or equal to ____.
(b) Based on your answers to question 6, you can say that any improper fraction will
generate a percentage greater than or equal to _______%.
8. A number line has been provided. There are six arrows that identify the approximate
5 11 13 5
1
locations of the fractions
. Figure out where each fraction would
, , , and
12 6 12 5
24
approximately be on the number line. Be able to explain why.
0
NSSAL
1
2
1
19
1
1
2
2
Focus on Fractions
Open-ended Questions (There is more than one correct answer for each of these questions.)
9. Find two fractions that are greater than 1 but less than 2.
10. Find a fraction that is very close to, but slightly larger than 1.
11. Find a fraction that is very close to, but slightly smaller than 1.
12. Find a fraction that is very close to, but slightly smaller than 2.
13. Find a fraction that is very close to, but slightly larger than 2.
You’re the Instructor (With these types of the questions, you are going to be the instructor. You
will have to examine a student’s work and provide some sort of feedback
to him or her.)
14. Andrea was asked to put four fractions in order from smallest to largest. Her answer is
shown below. Did she do it correctly? If this question was worth 4 points on a quiz, what
would you give her as a mark out of 4? Correct any mistakes that she may have made.
1 6 8 5
, , ,
9 13 7 6
3
to an improper fraction. His answer and explanation are
4
shown below. Explain to him what he did wrong and give him the correct answer.
15. Ethan was asked to change 2
Ethan’s Answer and Explanation
I multiplied the 3 by the 2, and then added 4. That means that I now have
10
10 fourths.
4
Your Explanation and Answer:
NSSAL
20
Focus on Fractions
Wrap-Up Statement
Although we know that all improper fractions are equal to or greater than 1, it is important to
be able to change them to mixed fractions so that you can compare the sizes of different
improper fractions.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 15. Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
NSSAL
I understand all of the concepts covered in the section, “More
Fractions.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can change a mixed number to an improper fraction.
I can change an improper fraction to a mixed number.
21
1
2
3
4
5
1
2
3
4
5
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
Focus on Fractions
Equivalent Fractions
Explanation:
Consider the following area models.
1
2
2
4
3
6
9
18
In all four cases, the same part of the whole has been shaded. When this happens you can say
that the fractions are equivalent fractions.
1 2 3 9
= = =
2 4 6 18
Questions:
1. The area model for
2
is shown below.
3
(a) Circle the area models below that show a fraction equivalent to
2
?
3
(b) For each of the area models that you circled in question 1(a), state the corresponding
fraction.
2. The area model for
NSSAL
3
is shown below.
4
22
Focus on Fractions
Create three area models that are equivalent to the area model for
3
. Also state the
4
corresponding fraction for each of your new models.
3. You are going to use an interactive activity found on the internet. Using Google, search the
following key words: National Library Virtual Manipulative Utah State. Once you are
on the site, select Numbers & Operations. A list of lessons will appear in the main body of
the page. Select the item called Fractions-Equivalent. In this game, you are supplied with
an area model for a particular fraction. You divide the model into more pieces in an attempt
to find and enter an equivalent fraction. Play at least four successful rounds of the game and
print off your results for each round.
Explanation:
You may be asking yourself the following questions.
How do I create equivalent fractions without using an area model?
How can I tell if fractions are equivalent without using an area model?
Creating Equivalent Fractions
To create an equivalent fraction, multiply or divide the numerator and denominator of a fraction
by the same number.
Example: Create three equivalent fractions for
6 × 3 18
=
10 × 3 30
6 × 5 30
=
10 × 5 50
6
.
10
6÷2 3
=
10 ÷ 2 5
Determining if Fractions are Equivalent
There are two methods that can be used to figure out if fractions are equivalent.
Method 1
This method involves changing fractions to their simplest form and comparing them. A fraction
is written in its simplest form when the numerator and denominator have no common factors
other than one.
3
is written in its simplest form because the only number that
4
divides into both 3 and 4 without leaving a remainder is 1 (i.e. 1 is the only common factor).
For example, the fraction
NSSAL
23
Focus on Fractions
6
is not written in its simplest form because the number 3 divides into the
15
numerator, 6, and the denominator, 15 without leaving a remainder (This means 3 is a
2
.
common factor of 6 and 15). When the fraction is simplified, the equivalent fraction is
5
This is accomplished by dividing the numerator and denominator by 3.
The fraction
Example: Figure out if these fractions are equivalent.
4
10
(a) and
7
35
4
is in its simplest form because the only number that divides into 4 and 7 without
7
leaving a remainder is 1.
10
needs to be put in its simplest form. The number 5 divides into both the numerator
35
and denominator without leaving a remainder (This means 5 is a common factor of 10
and 35).
10 10 ÷ 5
=
35 35 ÷ 5
2
=
7
Since
(b)
4
2
4
10
is not equal to , then and
are not equivalent fractions.
7
35
7
7
21
12
and
35
20
21
needs to be put in its simplest form. The number 7 is a common factor of 21 and 35.
35
21 21 ÷ 7
=
35 35 ÷ 7
3
=
5
12
needs to be put in its simplest form. The number 4 is a common factor of 12 and 20.
20
12 12 ÷ 4
=
20 20 ÷ 4
3
=
5
Since
NSSAL
3 3
21
12
are equivalent.
= , then the fractions
and
5 5
35
20
24
Focus on Fractions
Method 2
Any two fractions are equivalent if their cross products are equal.
Example: Figure out if the fractions are equivalent.
(a)
6
8
and
21
28
6
21
8
28
6 × 28 = 168
21 × 8 = 168
Since the cross products are equal (both equal 168), then
(b)
6
8
.
=
21 28
15
6
and
33
14
15
33
6
14
15 × 14 = 210
33 × 6 = 198
Since the cross products are not equal, then
15
6
are not equivalent.
and
33
14
Questions:
4. Create at least three equivalent fractions for each of the fractions supplied below.
5
5
(a)
(b)
7
6
(c)
2
10
(d)
4
3
(e)
12
8
(f)
4
12
5. Change each of these fractions to their simplest form.
6
8
(a)
(b)
12
10
(c)
NSSAL
9
21
(d)
25
30
50
Focus on Fractions
(e)
12
18
(f)
35
40
(g)
28
20
(h)
35
63
(i)
3
30
(j)
36
24
(k)
21
49
(l)
30
27
Answers to Question 5 (They are not in order.)
1
7
3
3
1
7
10
10
5
7
2
2
8
9
3
5
5
9
10
7
3
7
2
3
4
5
6. You are going to use an interactive activity found on the internet. Using Google, search the
following key words: 321Know AAA Math. Once you are on the site, select Fractions
from the menu on the left side of the page. A list of lessons will appear in the main body of
the page. Select the item called Equivalent Fractions. If you scroll down the page, you will
find a section called Practice. It’s a game where you are asked to put identify the equivalent
fractions. Play at least ten rounds of the game and print your final results.
7. State whether the following pairs of fractions are equivalent or not.
(a)
6
9
and
12
18
(b)
1
3
and
5
15
(c)
2
6
and
14
36
(d)
6
9
and
14
21
(e)
6
24
and
5
20
(f)
6
8
and
27
36
NSSAL
26
Focus on Fractions
(g)
21
16
and
12
10
(h)
8
3
and
20
5
8. You are going to use an interactive activity found on the internet. Using Google, search the
following key words: Shodor Interactive Activities. Once you are on this site, scroll down
and find the activity called Equivalent Fraction Pointer. With this activity you create
equivalent fractions and their corresponding area models. Complete three rounds of this
activity, printing off the results from each round.
9. With each of these questions, four fractions have been supplied. Three of the fractions are
equivalent fractions. Circle the one fraction that isn’t equivalent to the other three.
(a)
6
8
1
5
3
4
(c)
5
6
15
18
4
5
9
12
10
12
(b)
6
12
1
2
4
8
(d)
7
10
3
5
15
25
2
3
6
10
10. Place the appropriate symbol ( <, >, or = ) between each pair of fractions. In some cases you
may wish to simplify one or both of the fractions.
Note:
< means less than
> means greater than
(a)
2
5
4
10
(b)
2
6
4
5
(c)
7
6
6
7
(d)
2
8
1
4
(e)
3
4
15
20
(f)
3
8
8
9
(g)
1
5
1
6
(h)
2
6
7
21
(i)
5
6
9
8
(j)
4
10
6
15
(k)
8
6
12
9
(l)
7
9
6
11
NSSAL
27
= means equal to
Focus on Fractions
(m)
6
9
8
20
(n)
9
12
16
20
11. A work shift at a fast food restaurant is 8 hours. What fraction of an employee’s work shift is
represented by 4 hours?
Answer: ____
12. There are 12 inches in a foot. What fraction of a foot is represented by 8 inches?
Answer:
____
13. There are 100 centimetres in a metre. What fraction of a metre is 20 cm?
Answer:
____
14. Concrete is made by mixing 1 part cement to 2 parts water to 9 parts gravel.
(a) What fraction of the mixture is cement?
Answer:
____
(b) What fraction of the mixture is water?
Answer:
____
(c) What fraction of the mixture is gravel?
Answer:
____
(d) What fraction of the mixture is not gravel?
Answer:
____
Answers to Questions 11 to 14 (They are not in order.)
1
2
1
1
1
12
3
5
6
4
5
6
3
4
1
2
15. A number line has been provided. There are six arrows that identify the approximate
5 7 3 12 1 23
7
locations of the fractions
. Figure out where each fraction
, , ,
,
,
, and
10 6 3 8 12 12
12
would approximately be on the number line. Be able to explain why.
0
NSSAL
1
2
1
28
1
1
2
2
Focus on Fractions
16. If you can change an improper fraction to its simplest form, can it ever be equivalent to a
proper fraction? Explain.
You’re the Instructor (With these types of the questions, you are going to be the instructor. You
will have to examine a student’s work and provide some sort of feedback
to him or her.)
5
. His answer and explanation are shown
6
below. His answer is incorrect. Explain to him how he should have done the question.
17. Eric was asked to make an equivalent fraction for
Eric’s Answer and Explanation:
I just added 2 to both the numerator and denominator of the
5
7
is equivalent to .
fraction.
6
8
Your Explanation:
Wrap Up Statement
Changing fractions to their simplest form can be useful when you are trying to compare the
size of different fractions.
NSSAL
29
Focus on Fractions
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 17. Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
I understand all of the concepts covered in the section,
“Equivalent Fractions.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can figure out if two fractions are equivalent.
I can change a fraction to its simplest form.
I can do word problems similar to the ones found in this
section.
1
2
3
4
5
1
2
3
4
5
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
For Your Information (FYI)
Did you know that a multiplication table can be used to generate equivalent fractions? Consider
the two rows that have been highlighted in the table below.
×
1
2
3
4
5
1
1
2
3
4
5
2
2
4
6
8
10
3
3
6
9
12
15
4
4
8
12
16
20
5
5
10
15
20
25
Now look at the equivalent fractions that can be made using the values from these two rows.
2 4
6
8 10
=
=
=
=
5 10 15 20 25
NSSAL
30
Focus on Fractions
Measuring in Inches
With the introduction of the metric system, fewer computations require the use of fractions. The
focus became the use of decimals. Many jobs however still rely heavily on imperial measure
(pounds, miles, feet, inches, pints,…) which in turn rely heavily on fractions.
Consider a carpenter’s ruler or tape measurer that is marked in centimeters and inches.
Carpenters typically work only in inches even though the metric measures are also present on the
ruler or tape measurer. On these measuring devices, one inch is typically divided into sixteenths.
That means that every inch on the ruler is broken down into 16 smaller and equal sections.
8
1
or
16 2
1
16
15
16
0
1
6
3
or 1
16
8
2
1
4
1
or 2
16
4
2
Questions:
1. Measure the length of each line segment in inches.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Answers to Question 1 (They are not in order. You don’t need both answers for each.)
14
7
or
16
8
14
7
1
or 1
16
8
NSSAL
8
1
or 1
16
2
2
1
2
or 2
16
8
1
1
6
3
or 1
16
8
10
5
or
16
8
31
2
1
or
16
8
4
1
1
or 1
16
4
4
1
or
16
4
16
or 1
16
12
3
or
16
4
2
1
1
or 1
16
8
Focus on Fractions
2. Bring in five scrap pieces of lumber. You want five pieces that
are rectangular prisms (see diagram). Using your ruler, measure
the length, width, and height of the lumber in inches, and record
that information in the table.
Scrap of Lumber
A
Length
Width
Height
B
C
D
E
Wrap-Up Statement:
You have to understand fractions if you plan to measure in inches.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 4. Select your response to
each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
NSSAL
I understand all of the concepts covered in the section,
“Measuring in Inches.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can measure in inches to the nearest sixteenth of an inch.
32
1
2
3
4
5
1
2
3
4
5
1
1
2
2
3
3
4
4
5
5
Focus on Fractions
Comparing Fractions
Explanation:
You’ve spent a lot of time dealing with the magnitude (size) of fractions. Presently, you know
two methods for handling these types of questions.
1. The first method involves using benchmark numbers like 0,
Example 1
Order from smallest to largest.
5 13 2
, ,
11 14 25
Answer:
2 5 13
, ,
25 11 14
1
, and 1.
2
Explanation:
5
1
is closest to
because 5 is almost half
11
2
13
2
of 11. The fraction
is closest to 1. The fraction
14
25
is closest to 0.
The fraction
2. The second method works only if the fractions you’re looking at have the same denominator.
Example 2
Order from smallest to largest.
3 1 2
, ,
4 4 4
Answer:
1 2 3
, ,
4 4 4
Explanation:
1 fourth is the smallest and 3 fourths is the largest. The
area models support this.
2
4
1
4
3
4
What happens when neither of these methods works? Consider the following question.
Example 3
Which is larger
5
2
or ?
7
3
1
and 1, so the benchmark
2
method doesn’t work. These fractions don’t have the same denominator, so the
second method doesn’t work.
Both of these fractions are somewhere between
To do this question you’ll have to change the two fractions to a common denominator.
To find the common denominator, first list all the multiples of the two denominators; 7
and 3.
Multiples of 7: 7, 14, 21, 28, 35, 42, …
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …
NSSAL
33
Focus on Fractions
Look for the least common multiple for 7 and 3. It’s 21. That means that each fraction
5
will be changed to its equivalent fraction where the denominator is 21. In the case of ,
7
the denominator of 21 is achieved by multiplying the numerator and denominator by 3.
2
In the case of , the denominator of 21 is achieved by multiplying the numerator and
3
denominator by 7.
5 5×3
=
7 7×3
15
=
21
2 2×7
=
3 3× 7
14
=
21
Therefore
5
2
is larger than .
7
3
Example 4
Put the numbers
5 3 9 1
5
, , , , and in order from smallest to largest.
3 20 11 20
6
Answer:
Start by grouping the fractions based on the benchmark numbers 0,
1
3
are close to 0.
and
20
20
1
and 1.
2
5
9
and are close to 1.
11
6
5
is an improper fraction therefore it is greater than 1.
3
If you have any fractions with the same denominator, then you can figure out the
larger and smaller fractions.
1
3
is smaller than
.
20
20
On your last step, you might have to change fractions to a common denominator.
5 5 × 11
9
9×6
=
=
6 6 × 11
11 11 × 6
9
5
is smaller than .
55
54
11
6
=
=
66
66
For this question, we had to use all three methods to figure out the size of the
fractions
1 3 9 5 5
The proper order is
, , , , .
20 20 11 6 3
NSSAL
34
Focus on Fractions
Questions:
1. For each of these questions, two fractions have been supplied. Figure out which of the two
fractions is larger by changing the fractions to a common denominator.
1 2
4 3
(b) ,
(a) ,
3 7
7 5
(c)
3 7
,
4 9
(d)
2 3
,
3 5
(e)
4 7
,
5 9
(f)
8 5
,
10 6
(g)
3 1
,
8 3
(h)
1 2
,
4 7
(i)
3 5
,
4 8
(j)
7 2
,
9 3
NSSAL
35
Focus on Fractions
2. For each of the following pairs of fractions, circle the larger fraction and then indicate the
method that you used to figure out this. The choices for the methods are as follows.
1
Method A - compare fractions to the benchmark numbers 0,
, and 1
2
Method B - compare fractions that already have a common denominator
Method C - change fractions to a common denominator, then compare
Circle the larger
fraction.
3
5
7
7
(a)
Method
(A, B, or C)
(b)
Circle the larger
fraction.
7
1
15
19
(c)
5
8
3
5
(d)
1
10
19
21
(e)
2
5
4
9
(f)
12
13
10
13
(g)
5
11
4
9
(h)
2
7
3
10
(i)
7
12
5
12
(j)
1
8
7
9
Method
(A, B, or C)
3. Put the numbers in order from smallest to largest. Remember that you have three different
strategies to choose from. Use the easier strategies first. Changing fractions to a common
denominator takes a lot more time so this strategy should be used last.
5 3 1 5
(a) , , ,
4 8 20 8
(b)
1 13 9 5
, , ,
40 9 10 11
(c)
13 3 11 2
, , ,
10 7 12 5
NSSAL
36
Focus on Fractions
(d)
19 1 21 3
, , ,
20 3 20 8
(e)
2 7 8 1 5
, , , ,
11 9 7 11 6
(f)
11 1 5 8 2
, , , ,
6 10 7 8 3
4. You are going to use an interactive activity found on the internet. Using Google, search the
following key words: National Library Virtual Manipulative Utah State. Once you are
on the site, select Numbers & Operations. A list of lessons will appear in the main body of
the page. Select the item called Fractions-Comparing. In this game, you are supplied with
two area models for two fractions. You divide one or both area models into more pieces in
an attempt to find and enter fractions with common denominators. Play at least four
successful rounds of the game and print off your results for each round.
5. Find the next four terms in each sequence. Make sure that the terms are in their simplest
form.
1 1 3 1
(a) , , , , _____, _____, _____, _____
8 4 8 2
(b)
1 1 1 1 5 1
, , , , , , _____, _____, _____, _____
12 6 4 3 12 2
Open-ended Questions (There is more than one correct answer for these questions.)
6. If
?
1
is greater than , but less than 1 , then what might the missing number be?
7
2
NSSAL
37
Focus on Fractions
7. If
4
1
is less than , then what might the missing number be?
?
2
8. If
?
is greater than 1, but less than 2, then what might the missing number be?
9
9. If
2
?
is less than , then what might the missing number be?
5
3
10. A fellow classmate put the fractions
1
3 2 1 7
into two groups. What might the
, , , , and
10
4 5 3 10
groups be?
Wrap Up Statement
You now have three strategies that you can use when attempting to understand the magnitude
of fractions.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 10. Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
NSSAL
I understand all of the concepts covered in the section,
“Comparing Fractions.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can order fractions from smallest to largest if they all have the
same denominator.
I can order fractions from smallest to largest by comparing the
1
fractions to the benchmark numbers 0, , and 1.
2
I can order fractions from smallest to largest by changing
fractions to a common denominator.
38
1
2
3
4
5
1
2
3
4
5
1
1
2
2
3
3
4
4
5
5
1
2
3
4
5
1
2
3
4
5
Focus on Fractions
Concept Chart for Fractions
Definitions
Characteristics
Fractions
Examples
NSSAL
Diagrams
39
Focus on Fractions
Estimating the Addition and Subtraction of Fractions
Explanation:
One of the biggest problems many learners encounter learning how to add and subtract fractions
is that they get wrapped up in the rules and don’t take the time to see if their final answer is
reasonable. That’s why you’re going to spend some time working on your estimation skills
rather than focusing on the rules. Estimation is the process of finding a reasonable
approximation for the final answer, without doing the formal calculation. On previous
worksheets, you spent a significant amount of time comparing fractions to familiar numbers like
1
0, , and 1. This skill can be very useful when attempting to estimate the sum or difference of
2
two fractions.
Example:
Estimate each of these sums.
14
7
is close to 1. The proper fraction
is also
15
6
7 14
close to 1. To estimate the + , think 1 + 1. This means that the
6 15
sum will be close to 2.
(a)
7 14
+
6 15
The improper fraction
(b)
1
2
+
30 45
Both of these fractions are very small numbers that are close to 0. To
estimate
1
2
, think 0 + 0. That means that the sum will be close
+
30 45
to 0.
(c)
1 5
+
16 9
(d)
10 8
+
9 17
NSSAL
1
5
1
is close to 0. The fraction is close to . That
16
9
2
means that when you add these two fractions, the answer should be
1
close to .
2
The fraction
10
8
is close to 1. The proper fraction
is
9
17
1
1
1
close to . Since 1 + = 1 , then you can say that the sum of the
2
2
2
1
two original fractions is close to 1 .
2
The improper fraction
40
Focus on Fractions
(f) 1
1
7
1
7
is close to , then the mixed number 3
is close to 3 .
2
15
2
15
3
3
is close to 0, then the mixed number 2
is close to 2.
Since
50
50
1
1
Since 3 + 2 = 5 , then you can say that the sum of the two original
2
2
1
mixed numbers is close to 5 .
2
7
3
+2
15
50
Since
5
8
+2
12
15
The mixed number 1
(e) 3
5
1
8
is close to 1 . The mixed number 2
is
12
2
15
1
1
1
close to 2 . Since 1 + 2 = 4 , then you can say that the sum of the
2
2
2
two original mixed numbers is close to 4.
Example:
Estimate each of these differences.
2
12
12 2
The fraction
is close to 1. The fraction
is close to 0. To
(a)
−
37
13
13 37
12 2
estimate
, think 1 - 0. Since 1 − 0 = 1 , then you know that the
−
13 37
difference between the two original fractions is close to 1.
(b)
13 5
−
12 9
(c) 3
1
1
−1
15
20
(d) 4
2
11
−1
31
20
NSSAL
5
13
1
is close to 1. The fraction is close to . To
9
12
2
1
13 5
1 1
estimate
− , think 1 − . Since 1 − = , then you know that
12 9
2 2
2
1
the difference between the two original fractions is close to .
2
The fraction
1
1
is close to 3. The mixed number 1
is
15
20
close to 1. Since 3 − 1 = 2 , then you know that the difference between
these two mixed fractions is close to 2.
The mixed number 3
2
11
is close to 4. The mixed number 1
is
31
20
1
1
1
close to 1 . Since 4 − 1 = 2 , then you know that the difference
2
2
2
1
between these two original mixed fractions is close to 2 .
2
The mixed number 4
41
Focus on Fractions
Questions:
1. For each sum, check the column that is the best estimate.
Close to 0
(a)
Close to
1
2
Close to 1
Close of 1
1
2
Close to 2
7 15
+
8 14
1 4
+
20 9
1
1
+
12 15
21 7
+
20 13
9
5
+
17 11
8 11
+
9 10
2
1
+
31 25
12 1
+
13 32
8
9
+
17 10
8
1
+
15 40
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2. Estimate each of these sums or differences.
1
12
5
1
(b) 3 + 4
(a) 1 + 4
20
13
12
32
5
7
(c) 2 + 3
9
8
(e)
13 7
−
12 13
(g) 8
NSSAL
8
6
−4
17
14
(d)
21 2
−
22 53
(f) 6
14
1
−2
15
20
(h) 8
7
2
−3
15
45
42
Focus on Fractions
(i) 9
3
7
−6
50
13
(j) 7
Answers to Question 2 (They are not in order.)
1
1
1
1
8
5
6
7
3
2
2
2
2
2
14
10
−4
15
21
4
1
5
1
2
1
2
6
3. You are asked to check a fellow student’s work. This student was asked to add two fractions
together. He decided that all he had to do was add the numbers in the numerator, and add the
numbers in the denominator. He wrote down the following.
6 12 18
+
=
13 25 38
Demonstrate using your estimation skills, why this student’s answer has to be wrong.
Wrap Up Statement
Although you don’t know how to add or subtract fractions at this time, you can estimate the
answer.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 3. Select your response to
each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
NSSAL
I understand all of the concepts covered in the section,
“Estimating the Addition and Subtraction of Fractions.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can estimate the sum of two fractions.
I can estimate the difference of two fractions.
43
1
2
3
4
5
1
2
3
4
5
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
Focus on Fractions
Adding Fractions
Explanation:
Suppose you were given the following problem.
You walked 6 yards, and then walked 24 inches. How far did you walk?
The answer is not 30. The units of measure are not the same; one is measured in yards and the
other in inches. You need to change them to the same units of measure first. You have two
choices.
First Choice: Change Yards to Inches
If 1 yard equals 36 inches, then 6 yards equals 216 inches (note: 6 × 36 = 216 ).
Total distance = 216 inches + 24 inches
= 240 inches
Second Choice: Change the Yards and Inches to Feet
If 1 yard equals 3 feet, then 6 yards equals 18 feet (note: 3 × 6 = 18 ).
24
If 12 inches equals 1 foot, then 24 inches equals 2 feet (note:
= 2 ).
12
Total Distance = 18 feet + 2 feet
= 20 feet
So how does this apply to the addition and subtraction of fractions? Consider the next two
examples.
Example 1.
3 2
+
7 7
5
=
7
Example 2.
2 3
+
5 4
8 15
=
+
20 20
23
=
20
In this example, the two fractions have a common denominator. In this
case, the denominator is 7.
You have 3 sevenths plus 2 sevenths. The answer will be 5 sevenths. It’s
as if you were dealing with the same unit of measure; sevenths. That’s
why it’s easy to add these two fractions.
In this example, the two fractions have different denominators. This
makes the question more challenging because at this point we don’t know
how to add fifths and fourths. It’s like trying to add different units of
measure.
You need to create a common denominator. It’s similar to creating the
same units of measure. In this case, the common denominator is 20. The
2
8
3
15
fraction
is equivalent to
. The fraction
is equivalent to
.
5
20
4
20
Now you have 8 twentieths plus 15 twentieths. The answer will be 23
twentieths.
The Big Question:
NSSAL
How do you figure out a common denominator for two fractions with
different denominators?
44
Focus on Fractions
Let’s try solving the following problem two different ways. Both methods require that you make
a common denominator.
Example 3
Evaluate
1 1
+
3 2
Method 1: Using Area Models
1
1
and .
3
2
Notice that the first model is divided into columns, and
the second model is divided into rows. This is
deliberate. We can’t add these fractions in their present
form.
You start by drawing the area models for
+
+
Answer:
5
6
In the next step, we put two rows on the first model,
and put three columns on the second model. By doing
this, both models are now dealing with sixths, rather
2
than thirds and halves. The first model is , which is
6
1
3
equivalent to . The second model is
, which is
6
3
1
equivalent to .
2
You have 2 sixths plus 3 sixths. The answer is 5 sixths.
Method 2: The Mathematical Procedure
The denominators (3 and 2) are different. List the
1 1
+
multiples of 3 and 2 separately.
3 2
1× 2 1× 3
=
+
Multiples of Three: 3, 6, 9, 12, 15, …
3× 2 2× 3
2 3
Multiples of Two: 2, 4, 6, 8, 10, 12, 14, …
= +
6 6
Notice that the least common multiple of 3 and 2 is 6.
5
=
This means that the common denominator will be 6.
6
Now you write each fraction as an equivalent fraction
whose denominator is 6.
You have 2 sixths plus 3 sixths. The answer is 5 sixths.
NSSAL
45
Focus on Fractions
Example 4
Evaluate
1 2
+
6 3
Method 1: Using Area Models
1
2
and .
6
3
Notice that the first model is divided into columns, and
the second model is divided into rows. This is
deliberate. We can’t add these fractions in their present
form.
You start by drawing the area models for
+
In the next step, we put three rows on the first model,
and put six columns on the second model. By doing
this, both models are now dealing with eighteenths,
3
,
rather than sixths and thirds. The first model is
18
12
1
which is equivalent to . The second model is
,
18
6
2
which is equivalent to .
3
+
Answer:
5
6
You have 3 eighteenths plus 12 eighteenths. The
15
answer is 15 eighteenths. The answer
can be
18
5
changed to its simplest form, .
6
Method 2: The Mathematical Procedure
The denominators (6 and 3) are different. List the
1 2
+
multiples of 6 and 3 separately.
6 3
1 2× 2
= +
Multiples of Six: 6, 12, 18, 24, 30, …
6 3× 2
1 4
Multiples of Three: 3, 6, 9, 12, 15, 18, …
= +
6 6
Notice that the least common multiple of 6 and 3 is 6.
5
=
This means that the common denominator will be 6.
6
Now you write each fraction as an equivalent fraction
whose denominator is 6.
You have 1 sixths plus 4 sixths. The answer is 5 sixths.
Important Note: The area model technique that was shown for this example didn’t create the
least common denominator, although it ultimately gave the correct answer.
Questions:
NSSAL
46
Focus on Fractions
1. Solve each of the following. Note that all of these sums already involve common
denominators. At the end of your answer, you may have to put the fraction in its simplest
form and/or change it to a mixed number.
3
4
2 6
(a)
(b)
+
=
+ =
10 10
11 11
(c)
5
6
+
=
14 14
(d)
6 1
+ =
7 7
(e)
5
3
+
=
16 16
(f)
3 1
+ =
6 6
(g)
7
8
+
=
20 20
(h)
6 4
+ =
9 9
(i)
8 9
+ =
11 11
(j)
5 7
+ =
8 8
(k)
23 5
+
=
10 10
(l)
7 9
+ =
6 6
Answers to Question 1 (They are not in order.)
1
2
7
1
2
6
1
1
2
1
3
3
10
9
3
11
6
3
4
1
2
1
1
2
8
11
2
4
5
11
14
2. Two questions involving the addition of fractions have been solved using area models. For
each of the partial solutions,
- state the original question,
- state the second step where fractions with a common denominator have been created,
and
- state the final answer.
(a)
Original Question:
+
Second Step:
+
Final Answer:
NSSAL
47
Focus on Fractions
(b)
Original Question:
+
Second Step:
+
Final Answer:
3. Use area models to evaluate
1 2
+ .
4 3
+
+
Final Answer:
4. (a) List the multiples of 6.
_____, _____, _____, _____, _____, . . .
(b) List the multiples of 4.
_____, _____, _____, _____, _____, . . .
(c) What is the least common multiple of 4 and 6?
NSSAL
48
Answer: _____
Focus on Fractions
5. For each question, two numbers have been supplied. In the first column, list at least the first
five multiples of the first number. In the second column, list the first five multiples of the
second number. In the last column, state the least common multiple (LCM).
Multiples of the First Number
Multiples of the Second Number
LCM
(a) 5, 2
(b) 4, 10
(c) 2, 6
(d) 5, 3
(e) 6, 8
6. Fill in the blanks.
(a)
3 1
+
5 4
3 × 4 1× 5
=
+
5× 4 4× 5
=
=
20
+
20
20
The multiples of 5 are 5, 10, 15, 20, 25, 30, …
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The LCM of 5 and 4 is ____.
We have to multiply the numerator and denominator of
3
by 4 to make our common denominator.
5
We have to multiply the numerator and denominator of
1
by ____ to make our common denominator.
4
(b)
5 1
+
6 4
5×
1×
=
+
6×
4×
10 3
=
+
12 12
=
12
1
=1
12
NSSAL
The multiples of 6 are ____, ____, ____, ____, …
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The LCM of 6 and 4 is ____.
We have to multiply the numerator and denominator of
5
by ____ to make our common denominator.
6
We have to multiply the numerator and denominator of
1
by ____ to make our common denominator.
4
49
Focus on Fractions
(c)
2 3
+
3 5
2×
=
3×
=
=
3×
5×
8
=1
8
The LCM of 3 and 5 is ____.
We have to multiply the numerator and denominator of
3
by ____ to make our common denominator.
5
15
5
= +
8 8
The multiples of 5 are 5, 10, 15, 20, 25, 30, …
We have to multiply the numerator and denominator of
2
by ____ to make our common denominator.
3
15
3 5
+
4 8
3×
=
4×
=
+
+
15 15
=1
(d)
The multiples of 3 are ____, ____, ____, ____, ____, …
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
+
5
8
The multiples of 8 are ____, ____, ____, ____, …
The LCM of 4 and 8 is ____.
We have to multiply the numerator and denominator of
3
by ____ to make our common denominator.
4
5
because it already has
8
the denominator that we need.
We don’t have to change the
7. Figure out each of these sums. Show all your work.
1 1
1 2
(a) + =
(b) + =
2 5
4 5
(c)
NSSAL
3 1
+ =
5 6
(d)
50
2 3
+ =
3 4
Focus on Fractions
(e)
4 3
+ =
5 4
(f)
2 3
+ =
3 5
(g)
1 3
+ =
6 4
(h)
3 1
+ =
8 6
(i)
4 5
+ =
9 6
(j)
1 5
+
=
6 12
(k)
2 1
+
=
3 12
(l)
4 8
+
=
5 15
(m)
7 3
+ =
8 4
(n)
1 3 5
+ + =
2 4 8
Answers to Question 7 (They are not in order.)
7
4
13
13
5
23
11
7
1
1
1
15
20
24
8
30
12
12
8
NSSAL
51
1
5
12
1
1
3
1
5
18
1
11
20
1
3
8
3
4
7
10
Focus on Fractions
8. A freight truck is carrying computer components and office furniture. There is
computer components and
1
ton of
2
5
ton of office furniture. What is the total weight of the load?
8
3
1
inch plywood onto
inch tongue and groove
4
8
boards. How thick is the flooring at this stage?
9. For the flooring in your kitchen, you nail
1
3
1
cup of sour cream with cup of salsa sauce and
cup of cooked ground
3
4
2
hamburger meat. How many cups of the mixture do you now have?
10. You mix
11. Angela is making U-shaped brackets by bending a straight piece of metal in two places.
How long is the piece of metal given the following measurements? All the measurements are
in inches.
7
8
3
4
7
8
12. A sales clerk at a candy shop mixes
of chocolate-covered raisins and
3
3
pound of chocolate-covered almonds with
pound
8
4
1
pound of chocolate-covered cherries. How much does
2
the mixture weigh?
Answers to Questions 8 to 12 (They are not in order.)
3
1
1
7
2
1
1
1
2
8
12
8
1
5
8
7
8
Open-ended Questions (There is more than one correct answer for each of these questions.)
13. Two fractions add to
NSSAL
1
. What might the fractions be?
2
52
Focus on Fractions
14. Create a word problem using
1 1
+ .
2 3
15. Three fractions add to 1. What might the fractions be?
? 1
1
+ must be less than , then what might the missing number be? Verify your answer
2
11 9
by working out the sum.
16. If
You’re the Instructor (With these types of the questions, you are going to be the instructor. You
will have to examine a student’s work and provide some sort of feedback
to him or her.)
2 4
+ . Her answer and explanation are shown below. Her
5 7
answer is incorrect. Explain to her how she should have done the question.
17. Janice was asked to work out
Janice’s Answer and Explanation:
To add fractions, you just have to add the numbers in the
2 4 6
+ =
numerators of the original fractions to get the number in the
5 7 12
numerator of the new fraction. You then add the numbers in
1
=
the denominators of the original fractions to get the number
2
in the denominator of the new fraction. I also had to change
my answer to its simplest form by dividing the numerator and
denominator by 6.
Your Explanation:
NSSAL
53
Focus on Fractions
18. Three students were given the same question but they all arrived at different answers. Which
student did the question correctly? Circle the first mistake made by each of the other two
students.
Student #1
Student #2
Student #3
4 1 4 + 3 1+ 5
4 1 4 × 3 1× 5
4 1 4 × 3 1× 5
+ =
+
+ =
+
+ =
+
5 3 5+3 3+5
5 3 5× 3 3× 5
5 3 5× 3 3× 5
7 6
12 5
12 5
= +
=
+
=
+
8 8
15 15
15 15
13
17
17
=
=
=
8
15
30
2
5
=1
=1
15
8
Wrap Up Statement
Adding fractions is easiest when the fractions already have a common denominator. If they
don’t have a common denominator, you need to first figure out the least common multiple for
the denominators. Then you will write each fraction as an equivalent fraction whose
denominator is the least common multiple. Now that you have created the common
denominator, you can add the fractions. Remember that you might have to put the answer in
simplest form and/or express it as a mixed number.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 18. Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
(g)
NSSAL
I understand all of the concepts covered in the section, “Adding
Fractions.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can add fractions using area models.
I can add fractions that already have common denominators.
I can add fractions by making common denominators.
I can do word problems similar to the ones in this section.
54
1
2
3
4
5
1
2
3
4
5
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
Focus on Fractions
Subtracting Fractions
Explanation:
Subtracting fractions is easy if the fractions already have common denominators.
Example 1
5 2
−
7 7
3
=
7
In this example, the two fractions have a common
denominator. In this case, the denominator is 7.
You have 5 sevenths minus 2 sevenths. The answer will be
3 sevenths.
If the original fractions don’t have a common denominator, you have to figure out an equivalent
fraction for each one such that a common denominator is created.
Example 2
1 1
−
2 3
1× 3 1× 2
=
−
2 × 3 3× 2
3 2
= −
6 6
1
=
6
The denominators (2 and 3) are different. List the multiples of 2
and 3 separately.
Multiples of Two: 2, 4, 6, 8, 10, 12, 14, …
Multiples of Three: 3, 6, 9, 12, 15, …
Notice that the least common multiple of 2 and 3 is 6. This
means that the common denominator will be 6.
Now you write each fraction as an equivalent fraction whose
denominator is 6.
You have 3 sixths minus 2 sixths. The answer is 1 sixth.
Example 3
2 1
−
3 6
2× 2 1
=
−
3× 2 6
4 1
= −
6 6
3
=
6
1
=
2
The denominators (3 and 6) are different. List the multiples of 3
and 6 separately.
Multiples of Three: 3, 6, 9, 12, 15, 18, …
Multiples of Six: 6, 12, 18, 24, 30, …
Notice that the least common multiple of 3 and 6 is 6. This
means that the common denominator will be 6.
Now you write each fraction as an equivalent fraction whose
denominator is 6.
You have 4 sixths minus 1 sixth. The answer is 3 sixths which
can be simplified to 1 half.
NSSAL
55
Focus on Fractions
Questions:
1. Solve each of the following. Note that all of these differences already involve common
denominators. At the end of your answer, you may have to put the fraction in its simplest
form.
5 4
7 2
(a) − =
(b) − =
6 6
9 9
(c)
7 3
− =
8 8
(d)
7
3
− =
12 12
(e)
10 1
− =
12 12
(f)
13 3
−
=
25 25
(g)
7
5
− =
16 16
(h)
19 3
−
=
20 20
Answers to Question 1 (They are not in order.)
7
1
4
5
2
3
5
9
5
10
2. Fill in the missing numbers.
(a)
6 3
−
7 4
6 × 4 3× 7
=
−
7× 4 4×7
=
=
28
−
28
28
1
2
3
4
1
6
1
8
The multiples of 7 are ____, ____, ____, ____, ____, …
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The LCM of 7 and 4 is ____.
We have to multiply the numerator and denominator of
6
7
by 4 to make our common denominator.
We have to multiply the numerator and denominator of
3
4
by 7 to make our common denominator.
(b)
5 3
−
6 8
5×
3×
=
−
6×
8×
20 9
=
−
24 24
=
24
The multiples of 6 are ____, ____, ____, ____, ____, …
The multiples of 8 are ____, ____, ____, ____, ____, …
The LCM of 6 and 8 is ____.
We have to multiply the numerator and denominator of
5
6
by ____ to make our common denominator.
We have to multiply the numerator and denominator of
3
8
by ____ to make our common denominator.
NSSAL
56
Focus on Fractions
3. Figure out each of these differences. Show all your work.
3 2
4 1
(a) − =
(b) − =
4 5
5 3
(c)
3 2
− =
4 3
(d)
4 1
− =
5 2
(e)
5 1
− =
7 2
(f)
7 1
− =
16 4
(g)
5 2
− =
6 3
(h)
5 1
− =
6 4
(i)
1 1
− =
6 9
(j)
5 1
− =
8 6
NSSAL
57
Focus on Fractions
(k)
5 1
− =
6 2
(l)
Answers to Question 3. (They are not in order.)
2
7
11
1
7
3
3
5
20
24
18
15
10
14
9 1
− =
10 2
1
6
3
5
7
12
1
3
1
12
3
16
4. You are going to use an interactive activity found on the internet. Using Google, search the
following key words: Shodor Interactive Activities. Once you are on this site, scroll down
and find the activity called Equivalent Bounded Fraction Finder. With this activity you
will:
- provide two proper fractions (smaller fraction is the left bound, larger fraction is the
right bound)
- create area models for these two fractions
- figure out the difference of these two fractions and express the answer as an area
model.
Complete three rounds of this activity, printing off the results from each round.
Some of the remaining word problems require you to subtract fractions, while others require you
to add fractions. Make sure you think about this before working out the solution.
3
of an inch thick. When the wood is run repeatedly through a planer,
4
3
the planer shaves
of an inch off. Figure out the thickness of the planed lumber.
16
5. A piece of wood is
6. Figure out the length of the following shaft.
3
4
NSSAL
58
"
7
8
"
1
2
"
Focus on Fractions
7
1
of a gallon of gas in your lawn tractor. You ran the tractor and now only of a
8
4
gallon of gas remains. How much gas did you use?
7. You had
8. Figure out the length of side A.
13
16
7
8
"
5
8
"
"
1
3
8
"
Side A
9. A small piece of metal measuring
15
15
inch by
inch has to have a hole drilled
16
16
1
of an inch of metal between the hole and the
4
side of metal, what is the diameter of the hole?
in the center. If there must be
7
inch shaft be
8
11
reduced so that the diameter will be an
inch
16
shaft?
10. How much must the diameter of a
Answers to Questions 5 to 10 (They are not in order.)
7
5
7
9
8
16
16
8
NSSAL
59
7
16
1
3
16
2
1
8
Focus on Fractions
Open-ended Questions (There is more than one correct answer for each of these questions.)
11. When you subtract two fractions you get
1
. What might the fractions be?
2
12. Create a word problem that would be solved using
1 1
− .
2 3
? 1
1
but less than 1, then what might the missing number be?
− must be greater than
2
7 9
Verify your answer by working out the difference.
13. If
Wrap Up Statement
When you subtract fractions, you must have a common denominator. The process is very
similar to the one you use when adding fractions.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 13. Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
NSSAL
I understand all of the concepts covered in the section,
“Subtracting Fractions.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can subtract fractions that already have common
denominators.
I can subtract fractions by making common denominators.
I can do word problems similar to the ones in this section.
60
1
2
3
4
5
1
2
3
4
5
1
1
2
2
3
3
4
4
5
5
1
1
2
2
3
3
4
4
5
5
Focus on Fractions
Multiplying Fractions
You may not realize it, but most of you have been multiplying fractions for years. Consider the
following real world problems.
Half of the earnings are mine. If we earned $50, how much do I get?
The three of us collected recyclables. Therefore one third of the profit belongs to me. If
the profit was $90, how much should be coming to me?
Three quarters of a mixture is flour. If there are 8 cups of mixture, how many cups of
flour were used?
Many of you can already answer these questions. You probably know that half of 50 is 25. You
probably know that one third of 90 is 30. You might know that three quarters of 8 is 6. All of
these can be expressed as multiplication questions.
1
× 50 = 25
2
1
× 90 = 30
3
3
×8 = 6
4
If 1 quarter of 8 is 2, then
3 quarters of 8 must be 6.
In all of these examples, you’re multiplying a fraction by a whole number. What happens when
you’re multiplying a fraction by another fraction? Let’s look at area models to help us out with
this.
Example 1
1 1
×
3 2
The question could be presented in this manner.
1
1
What is of ?
3
2
Based on this, you would start with the area model for
In the next step you would divide the
1
.
2
1
into thirds.
2
In the next step, you would shade only one of those thirds.
In the last step, you look at that newly shaded region and ask
yourself what it represents in terms of the whole square. In this
case, the shaped region represents one sixth of the whole square.
Therefore
NSSAL
1 1 1
× =
3 2 6
61
Focus on Fractions
Example 2
2 1
×
3 3
The question could be presented in this manner.
2
1
What is of ?
3
3
Based on this, you would start with the area model for
In the next step you would divide the
1
.
3
1
into thirds.
3
In the next step, you shade two of the thirds.
In the last step, look at that newly shaded region and ask yourself
what it represents in terms of the whole square. In this case, the
shaped region represents two ninths of the whole square.
Therefore
2 1 2
× =
3 3 9
Questions:
1
of 8?
2
_____
(b) What is
1
of 9?
3
_____
(c) What is
1
of 20?
4
_____
(d) What is
1
of 60?
10
_____
(e) What is
2
of 12?
3
_____
(f) What is
3
of 100?
10
_____
1. (a) What is
2. There are partially completed solutions below. Complete the solutions and state the final
answer.
(a)
1 3
×
2 4
The question could be presented in this manner.
What is
1
3
of ?
2
4
Final
Answer:
NSSAL
62
Focus on Fractions
(b)
3 1
×
4 4
The question could be presented in this manner.
What is
3
1
of ?
4
4
Final
Answer:
(c)
1 1
×
3 4
The question could be presented in this manner.
What is
1
1
of ?
3
4
Final
Answer:
(d)
1 4
×
3 5
The question could be presented in this manner.
What is
1
4
of ?
3
5
Final
Answer:
(e)
3 3
×
5 4
The question could be presented in this manner.
What is __________?
Final
Answer:
Answers to Questions 1 and 2 (They are not in order.)
3
4
9
3
5
8
3
16
15
20
8
NSSAL
63
6
1
12
7
10
30
4
Focus on Fractions
3. You going to take a few minutes to examine the answers you obtained using area models.
In the two examples, we obtained the following.
1 1 1
2 1 2
× =
× =
3 2 6
3 3 9
In question 2, you obtained the following. (Fill in your final answers.)
1 3
3 1
1 1
1 4
3 3
× =
× =
× =
× =
× =
2 4
4 4
3 4
3 5
5 4
Based on these answers, explain how you would multiply fractions without using area
models.
4. Multiply the following fractions. In some cases, you will have to put the fraction in its
simplest form.
1 2
1 1
(a) × =
(b) × =
3 5
4 5
(c)
2 3
× =
5 7
(d)
3 5
× =
4 7
(e)
2 1
× =
5 4
(f)
3 1
× =
4 6
(g)
2 3
× =
3 5
(h)
1 2
× =
4 3
(i)
3 5
× =
5 6
(j)
4 3
× =
5 8
(k)
3 5
× =
10 8
(l)
4 3
× =
9 8
Answers to Question 4 (They are not in order.)
5
1
6
2
1
3
1
8
35
2
15
16
20
18
NSSAL
64
3
10
1
10
2
5
1
6
15
28
1
6
Focus on Fractions
5. In question 4, you were multiplying two proper fractions together. Take a look at the
products you got when you attempt to answer the following multiple choice question.
The product of any two proper fractions is always:
(a) less than one.
(b) equal to one.
(c) greater than one.
(d) none of the above
2
× 7 ? Can you easily visualize two thirds of seven? It
3
doesn’t work out to be a whole number. So how else can we do it?
6. How do you multiply something like
Answer:
7
. Once you do this, you simply follow
1
the rules for multiplying two fractions.
Remember that 7 can be expressed as
2
2 7
×7 = ×
3
3 1
14
=
3
2
=4
3
Important Note:
2
, you would
3
handle it in the same manner. The question
7 2
could be expressed as × .
1 3
If the question had been 7 ×
Keep this example in mind when you answer the following questions.
3
2
(b)
(a) × 2 =
×3 =
7
11
(c) 2 ×
3
=
10
(d)
1
×4 =
6
(e)
3
×4 =
5
(f)
5
×5 =
6
(g)
2
×9 =
3
(h)
4
×7 =
5
(i)
5
×2 =
6
(j)
4
×5 =
15
(k) 4 ×
7
=
8
(l) 6 ×
Answers to Question 6 (They are not in order.)
6
2
1
6
1
3
1
1
4
5
3
11
3
3
7
6
5
3
NSSAL
65
5
=
9
2
2
5
1
2
3
2
5
9
3
1
2
3
5
6
Focus on Fractions
7. In question 6, you were multiplying a proper fraction by a whole number. Take a look at the
products you got when you attempt to answer the following multiple choice question.
The product of any proper fraction with any whole number is always:
(a) less than one.
(b) equal to one.
(c) greater than one.
(d) smaller than the whole number in the original question.
8. What do you think would happen if you multiplied an improper fraction by a whole number?
Provide a couple of examples to confirm your thinking.
Some of the remaining word problems require you to multiply fractions, while others require you
to add or subtract fractions. Make sure you think about this before working out the solution.
3
cup of vinegar. If you wanted to
4
triple the recipe, how much vinegar would you need?
9. A recipe for a homemade cleaning solution requires
10. If the diameter of a hole is
7
inch, what is the radius of the
8
hole?
radius
diameter
11. Three quarters of a pizza is left over from last night’s party. You want half of the remaining
pizza. How much of the original pizza does your piece represent?
NSSAL
66
Focus on Fractions
12. The cross-sectional view of a pipe is provided.
Based on the information in the diagram, figure out
the outer diameter of the pipe.
13. A dog groomer uses a 36 gallon container to wash dogs. If she only fills it
1
4
"
7
8
"
1
4
"
3
full of water,
4
how many gallons of water does she use?
14. A recipe calls for
1
1
cup of sugar. How much sugar should be used if only of the recipe is
2
3
being made?
15. A drawn line is
15
1
inch long. If you erase
inch from the end of the line, then how long is
4
16
the line now?
1
1
cup of sugar. At another stage, you add cup of sugar.
3
4
How much sugar has been added in total to the recipe?
16. At one stage in a recipe, you add
NSSAL
67
Focus on Fractions
5
inch plywood. If he stacks 10 sheets of plywood on top of each other,
8
how high is the stack?
17. Brian is stacking
18. For each hour that an oil burner runs, it uses
for
3
gallon of fuel. If the burner is only running
4
2
hour, then how much fuel is used?
3
Answers to Questions 8 to 18 (They are not in order.)
5
3
1
1
1
7
6
2
8
4
6
4
12
12
1
3
8
1
2
7
16
27
11
16
Open-ended Questions (There is more than one correct answer for each of these questions.)
19. The product of two fractions is
5
. What might the fractions be?
6
20. Create a word problem that would be solved using
21. The product of three fractions is
NSSAL
1 1
× .
2 3
7
. What might the fractions be?
24
68
Focus on Fractions
? 6
1
but less than 1, then what might the missing number be?
× must be greater than
2
5 7
Verify your answer by working out the product.
22. If
23. One-quarter of the staff have digital cable. How many people might be on staff and how
many have digital cable?
Wrap Up Statement
You don’t need common denominators to multiply fractions. You multiply the numbers in
the numerators of your original fractions to get the number in the numerator of the new
fraction. You also multiply the numbers in the denominators of your original fractions to get
the number in the denominator of the new fraction. Once you have this product, you should
check to see if it needs to be changed to its simplest form.
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 23. Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
(f)
NSSAL
I understand all of the concepts covered in the section,
“Multiplying Fractions.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can multiply fractions using area models.
I can multiply fractions and change the answer to its simplest
form.
I can do word problems similar to the ones in this section.
69
1
2
3
4
5
1
2
3
4
5
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
1
2
3
4
5
Focus on Fractions
Dividing Fractions
Explanation:
We’ll start with a question. How many one-quarters are in three? The first time you encounter
this type of question, you’ll find it easiest to answer using area models.
In the first step, you draw the area model for three and the area model for one-quarter.
In the next and final step, we want to fit as many of the area models for one-quarter into
the area model for three. You can fit 12 one-quarter area models into the area model for
three. Therefore the answer is 12.
Answer each of the following.
How many halves are in one?
How many halves are in two?
How many halves are in three?
_____
_____
_____
How many one-thirds are in one?
How many one-thirds are in two?
How many one-thirds are in three?
_____
_____
_____
Many of you can now answer these types of questions with little assistance. What you may not
know is that these are really division questions involving fractions.
Consider the question 6 ÷ 2 . A past instructor probably told you that this question is asking you
to figure out how many 2’s are in 6. You would have answered 3 because there are three 2’s in
6. Based on this, when you are asked “How many one-thirds are in two?”, you now know that it
1
can be expressed as 2 ÷ .
3
If you reconsider the questions above, those questions and their answers can be expressed this
way.
1
1
1
2÷ =4
1÷ = 2
3÷ = 6
2
2
2
1÷
1
=3
3
2÷
1
=6
3
3÷
1
=9
3
You might see a pattern here. (If not, don’t worry about it.)
NSSAL
70
Focus on Fractions
2 1
÷ . This is
3 6
actually asking, “How many one-sixths are in two-thirds?” We’ll look at area models to solve
this.
First Step Draw the two area models.
What about questions where we divide a fraction by another fraction. Consider
Second Step - Figure out how many area models for one-sixth can be fit into the
area model for two-thirds.
There are 4 one-sixths in two-thirds.
2 1
Therefore: ÷ = 4
3 6
How do we do these types of questions without using area models? In other words, what are the
rules for dividing fractions?
Dividing Fractions
To divide two fractions, multiply the first fraction by the reciprocal of the
second fraction.
reciprocals
Therefore:
a c a d
÷ = ×
b d b c
a×d
=
b×c
Note:
b, c, and d are not equal to 0.
Let’s see if this works for two questions that we previously addressed using the area models.
2 1 2 6
÷ = ×
3 6 3 1
12
=
3
=4
It works!
3÷
1 3 4
= ×
4 1 1
12
=
1
= 12
It works!
Let’s do a few more examples where the answers don’t work out to be whole numbers.
NSSAL
71
Focus on Fractions
2 3 2 4
÷ = ×
3 4 3 3
8
=
9
4 2 4 3
÷ = ×
7 3 7 2
12
=
14
6
=
7
2
2 3
÷3= ÷
5
5 1
2 1
= ×
5 3
2
=
15
8 2 8 5
÷ = ×
3 5 3 2
40
=
6
20
=
3
2
=6
3
Questions:
1. (a) How many one-fifths are in one?
(b) How many one-fifths are in two?
(c) How many one-sevenths are in four?
_____
_____
_____
2. In a brief sentence, how would you explain what the question 4 ÷
1
means?
3
3 1
÷ .
4 8
Draw the two area models.
3. Use area models to help solve the question
First Step -
Second Step - Figure out how many area models for one-eighths can be fit into
the area model for three-fourths.
There are _____ one-eighths in three-fourths.
3 1
Therefore ÷ =
4 8
4. Solve each of the following.
3 1
(a) ÷ =
7 2
NSSAL
(b)
72
3 7
÷ =
5 2
Focus on Fractions
(c)
1 2
÷ =
5 3
(d)
1 1
÷ =
4 16
(e)
2 6
÷ =
5 5
(f)
4 2
÷ =
9 3
(g)
3 2
÷ =
4 3
(h)
3 2
÷ =
5 7
(i)
9 2
÷ =
10 5
(j)
5 5
÷ =
6 8
(k)
2
÷3=
5
(l)
2
÷4=
7
Answers to Question 4 (They are not in order.)
5
2
1
2
1
6
1
2
2
3
15
3
7
10
4
14
4
1
1
3
3
10
1
14
1
1
8
6
35
Look at the answers you got in question 4 when answering the next two multiple choice
questions.
NSSAL
73
Focus on Fractions
5. When a smaller fraction is divided by a larger fraction, then the quotient is always:
(a) less than 0.
(b) less than 1
(c) greater than 1.
(d) greater than 2.
6. When a larger fraction is divided by a smaller fraction, then the quotient is always:
(a) less than 0.
(b) less than 1
(c) greater than 1.
(d) greater than 2.
7. Solve the following. Please note that different questions involve different operations
(addition, subtraction, multiplication or division).
3 1
2 3
(a) + =
(b) × =
7 2
7 5
(c)
4 1
− =
5 3
(d)
5 3
÷ =
7 2
(e)
8 2
+ =
9 3
(f)
4 5
× =
3 2
(g)
5 1
− =
6 4
(h)
4
÷6 =
3
Answers to Question 7 (They are not in order.)
6
2
5
7
7
1
35
9
9
12
15
NSSAL
74
3
1
3
10
21
2
2
3
13
14
Focus on Fractions
Some of the remaining word problems require you to divide fractions, while others require you
to add, subtract, or multiply fractions. Make sure you think about this before working out the
solution.
8. You have seven-eighths of a pound of hamburger meat that you are making into small
meatballs. If each meatball is supposed to weigh one-sixteenth of a pound, how many
meatballs can you make?
9. Three-fourths of a pizza are divided equally among five people. How much of the original
pizza does each of the five people get?
3
inch counter top is comprised of a
4
1
particle board core with
inch laminate glued
16
on top. How thick is the particle board core?
10. A particular
11. Figure out the number of sheets of
3
4
"
core
5
inch plywood in a stack 25 inches high?
8
7
of an inch. If they need to cut 40 of
8
these pieces, what should be the minimum length of the original strip of sheet metal?
12. A strip of sheet metal is to be cut into pieces every
NSSAL
75
Focus on Fractions
13. A crank case on a motor has
1
2
quart of oil. If you add quart of oil, how much oil is now
4
3
in the crank case?
14. You have 12 pounds of flour in a bag. You are removing it from the bag using a container
2
pound of flour. Assuming that you are filling the container each time, how
that can hold
3
many times will you use the container to completely empty the bag?
Answers to Questions 8 to 14 (They are not in order.)
4
11
3
14
18
12
20
5
40
11
16
35
Open-ended Questions (There is more than one correct answer for each of these questions.)
15. The quotient of two fractions is
1
. What might the fractions be?
2
1
16. Create a word problem that would be solved using 4 ÷ .
3
Wrap Up Statement
You don’t need a common denominator to divide fractions. To divide two fractions, multiply
the first fraction by the reciprocal of the second fraction.
NSSAL
76
Focus on Fractions
Reflect Upon Your Learning
Fill out this questionnaire after you have completed questions 1 to 16. Select your response
to each statement.
1 - strongly disagree
2 - disagree
3 - neutral
4 - agree
5 - strongly agree
(a)
(b)
(c)
(d)
(e)
NSSAL
I understand all of the concepts covered in the section,
“Dividing Fractions.”
I do not need any further assistance from the instructor on the
material covered in this section.
I do not need any more practice questions.
I can divide fractions and change the answer to its simplest
form.
I can do word problems similar to the ones in this section.
77
1
2
3
4
5
1
2
3
4
5
1
1
2
2
3
3
4
4
5
5
1
2
3
4
5
Focus on Fractions
Charting Your Own Course
In this section, you are required to find or create eight word problems involving fractions.
Hopefully you will choose questions that are interesting to you. For example, if your future
plans include becoming a carpenter, then you may wish to choose math questions that only focus
on issues encountered in carpentry.
Finding these types of questions on the internet is surprising difficult. Most math sites don’t
include practical math problems that would be relevant to adults. You may want to use the
following classroom resources.
Title
Practical Problems in Mathematics for Heating &
Cooling Technicians
Practical Problems in Mathematics for Electricians
Practical Problems in Mathematics for Carpenters
Practical Problems in Mathematics for Welders
Practical Problems in Mathematics for Industrial
Technology
Mathematics for the Trades: A Guided Approach
Author(s)
Devore
Publisher
Thomson
Herman
Huth & Huth
Schell
Boatwright
Thomson
Thomson
Thomson
Thomson
Carman & Saunders
Pearson
You are required to write down each of the questions and then solve them. Show all your work.
NSSAL
78
Focus on Fractions
Learning Logs
This is your opportunity to summarize what you learned in each of the sections of this unit.
Space has also been provided to include sample questions and their solutions.
Summary
Area Models for Fractions (p. 1 to 11)
Example(s)
Improper Fractions and Mixed Numbers (p.12 to 14)
More Fractions (p. 15 to 21)
NSSAL
79
Focus on Fractions
Summary
Equivalent Fractions (p. 22 to 30)
Example(s)
Measuring in Inches (p. 31 and 32)
Comparing Fractions (p. 33 to 38)
NSSAL
80
Focus on Fractions
Summary
Estimating the Addition and Subtraction of Fractions
(p. 40 to 43)
Example(s)
Adding Fractions (p.44 to 54)
Subtracting Fractions (p.55 to 60)
NSSAL
81
Focus on Fractions
Summary
Multiplying Fractions (p. 61 to 69)
Example(s)
Dividing Fractions (p. 70 to 77)
NSSAL
82
Focus on Fractions
Glossary
denominator
a
, the number below the line is called the denominator.
b
The denominator represents the number of equal parts the whole has been divided into.
For a fraction written in the form
equivalent fractions
Equivalent fractions are fractions that when changed to their simplest forms, represent the
4 7 1
3
same number. For example, the fractions , , and are all equivalent fractions
8 14 2
6
1
because in their simplest forms they can all be expressed as .
2
fraction
A fraction is a number that expresses part of a whole, or part of a set. Fractions can also
be expressed as decimals or percents.
improper fraction
An improper fraction is a fraction whose numerator is greater than or equal to its
9
7 8
denominator (examples: , , and ). Improper fractions are equal to or greater than 1.
6 3
9
least common multiple
The least common multiple is the smallest number (greater than zero) that is a multiple of
a set of two or more numbers. For example, the least common multiple of 4 and 6 is 12.
mixed number
A mixed number is a number that is expressed as the sum of a whole number and a
2
2
proper fraction. For example 3 + can be written as the mixed number 3 . Any
5
5
improper fraction can be expressed as a mixed number. For example, the improper
9
2
fraction
can be expressed as the mixed number 1 .
7
9
numerator
For a fraction written in the form
a
, the number above the line is called the numerator.
b
proper fraction
A proper fraction is a fraction whose numerator is less than the denominator (examples:
1 3
8
, , and ). Proper fractions are between 0 and 1.
2 16
9
reciprocal
To find the reciprocal of a fraction, simply "flip it over.” For example the reciprocal of
2
3
is .
3
2
NSSAL
83
Focus on Fractions
Area Models (Proper Fractions)
Express each of the following area models as a fraction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Answers (They are not in order.)
5
7
1
5
8
18
6
12
NSSAL
2
3
9
16
1
4
84
7
8
7
12
1
8
4
9
3
8
Focus on Fractions
Area Models (Improper Fractions & Mixed Numbers)
Express each of the following as a mixed number and as an improper fraction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Mixed Number Answers (They are not in order.)
3
1
2
3
5
1
1
1
1
2
1
1
4
2
9
4
6
3
1
1
6
Improper Fraction Answers (They are not in order.)
7
4
7
5
16
5
11
6
3
4
3
9
4
4
NSSAL
85
2
1
2
3
2
2
3
2
5
2
11
9
1
5
6
1
1
4
17
6
1
7
9
11
6
Focus on Fractions
Money and Fractions
You may not realize it, but you’ve been working with fractions for years. Every time you handle
money, you are dealing with fractions.
Consider a penny. It takes 100 pennies to make 1 dollar. As a decimal, you would say
that 1 penny or 1 cent is equal to $0.01. As a fraction, you would say that 1 penny or 1
1
of a dollar.
cent is equal to
100
Consider a quarter. It takes 4 quarters to make 1 dollar. As a decimal, you would say
that 1 quarter or 25 cents is equal to $0.25. As a fraction, you would say that 1 quarter or
25
1
25 cents is equal to
or of a dollar.
100
4
Consider a 50 cent coin. This is a fairly rare coin that is only made to commemorate
special events. It takes 2 fifty cent coins to make 1 dollar. As a decimal, you would say
that one 50 cent coin is equal to $0.50. As a fraction, you would say that one 50 cent
50
1
or
of a dollar.
coin is equal to
100
2
Questions:
1. Fill in the blanks.
(a) Consider a dime. It takes ______ dimes to make 1 dollar. As a decimal, you would say
that 1 dime or 10 cents is equal to $______. As a fraction, you would say that 1 dime or
10 cents is equal to _________ of a dollar.
(b) Consider a nickel. It takes ______ nickels to make 1 dollar. As a decimal, you would
say that 1 nickel or 5 cents is equal to $______. As a fraction, you would say that 1
nickel or 5 cents is equal to _________ of a dollar.
2. Complete the following table. In each case, you are given a specific number of coins. Figure
out the value of these coins in cents, dollars, and as a fraction of a dollar. Two sample
questions have been completed.
Coins
Ex. 1 dime, 1 nickel
15¢
Dollars
(decimal)
$0.15
Ex. 4 nickels, 2 pennies
22¢
$0.22
NSSAL
Cents
86
Fraction of a
Dollar
15  3 
 or 
100  20 
22  11 
 or 
100  50 
Focus on Fractions
Coins
(a)
1 nickel, 3 pennies
(b)
2 dimes, 3 nickels
(c)
1 quarter, 1 nickel
(d)
2 quarters, 2 dimes
(e)
3 dimes, 4 nickels
(f)
3 quarters, 1 dime
(g)
2 quarters, 4 pennies
(h)
7 dimes, 2 pennies
(i)
4 dimes, 6 pennies
(j)
5 nickels, 3 pennies
(k)
1 quarter, 3 dimes
(l)
3 quarters, 1 nickel
Cents
Dollars
(decimal)
Fraction of a
Dollar
(m) 9 dimes, 2 pennies
(n)
6 dimes, 1 nickel
Fraction Answers for Questions 1 and 2 (They are not in order. You don’t need both answers.)
50  1 
35  7 
46  23 
40  2 
5  1 
28  7 
 or 
 or 
 or 
 or 
 or 
 or 
100  2 
100  20 
100  50 
100  5 
100  20  100  25 
70  7 
30  3 
10  1 
84  21 
72  18 
8  2 
 or 
 or 
 or 
 or 
 or 
 or 
100  10  100  10  100  25  100  10 
100  25  100  25 
92  23 
54  27 
65  13 
55  11 
85  17 
80  4 
 or 
 or 
 or 
 or 
 or 
 or 
100  25  100  50  100  20  100  20  100  20 
100  5 
NSSAL
87
Focus on Fractions
Number Magnitude Questions Using Area Models
1. Write the corresponding fraction below each area model and then order the fractions from
smallest to largest.
(a)
Smallest to Largest
____, ____, ____
(b)
Smallest to Largest
____, ____, ____
(c)
Smallest to Largest
____, ____, ____
2. Draw the corresponding area model for each fraction and then order the fractions from
smallest to largest.
(a)
2
3
1
4
1
10
Smallest to Largest
____, ____, ____
(b)
1
5
8
9
3
8
Smallest to Largest
____, ____, ____
NSSAL
88
Focus on Fractions
Questions Involving Benchmark Numbers
The four rules presented here will help you do the question below.
•
•
•
•
When the number in the numerator of a fraction is very small compared to the number in
1 2 3
the denominator of the fraction (examples: , , ), then the fraction is closest to the
8 11 18
number 0.
When the numerator of the fraction is about half the size of the denominator of the
1
4 6 14
fraction (examples: , , ), then the fraction is closest to the number .
9 11 30
2
When the numerator of the fraction is the same number as the denominator of the fraction
3 8 11
(example: , , ), then the fraction is equal to 1.
3 8 11
When the number in the numerator of a fraction is close to the number in the denominator
7 9 15
of the fraction (examples: , , ), then the fraction is closest to the number 1.
8 11 18
Question:
Examine the fraction that has been provided and check off the column that best describes that
fraction.
Fraction
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
NSSAL
Closest to 0
Closest to
1
2
Closest to 1
Equal to 1
1
12
11
11
6
14
9
10
7
8
1
7
2
17
7
15
20
20
11
20
89
Focus on Fractions
Number Magnitude Questions
Put the numbers in order from smallest to largest.
3 7 4
4 6 2
(a) , ,
(b) , ,
9 9 9
5 5 5
(c)
8 5 3 11
, , ,
8 8 8 8
(d)
9 1 5
, ,
10 6 9
(e)
6 7 1
, ,
11 6 10
(f)
9 5 1
, ,
4 5 9
(g)
7 8 11
, ,
7 9 6
(h)
3 1 6 6
, , ,
3 12 5 10
(i)
5 10 1 1
, , ,
4 11 14 2
(j)
4 6 1 8 9
, , , ,
9 6 8 7 10
(k)
1 1 1
, ,
2 7 4
(l)
1 1 1
, ,
10 3 5
(m)
1 8 1
, ,
5 9 4
(n)
7 1 2 1
, , ,
5 3 3 4
(o)
5 11 4 13 1
, , , ,
9 8 4 8 6
(p)
1 8 4 9 1 3
, , , , ,
9 10 4 10 4 2
NSSAL
90
Focus on Fractions
Simplest Form Questions
A fraction is written in its simplest form when the numerator and denominator have no common
factors other than one.
3 5 1
9
are all in their simplest form.
Examples: , , , and
4 6 3
7
When a fraction is not in its simplest form, you must divide the numerator and denominator of
the fraction by the greatest common factor of the numbers in the numerator and denominator.
6 6÷2 3
35 35 ÷ 5 7
Examples:
=
=
=
=
8 8÷2 4
25 25 ÷ 5 5
Change each of the fractions to their simplest form.
2
10
(a) =
(b)
=
8
15
(c)
6
=
15
(d)
2
=
14
(e)
25
=
30
(f)
10
=
30
(g)
12
=
20
(h)
9
=
12
(i)
6
=
27
(j)
7
=
14
(k)
4
=
20
(l)
9
=
9
(m)
16
=
28
(n)
18
=
12
(o)
3
=
24
(p)
10
=
14
(q)
8
=
6
(r)
15
=
40
(s)
10
=
8
(t)
10
=
5
(u)
40
=
90
(v)
32
=
20
(w)
14
=
8
(x)
90
=
50
(y)
30
=
25
(z)
48
=
30
Answers (They are not in order.)
1
5
1
4
2
3
4
2
7
2
2
4
1
8
5
3
5
9
3
NSSAL
1
4
1
7
4
1
7
91
3
5
9
5
5
7
3
4
3
8
6
5
2
5
1
8
8
5
5
6
3
2
4
9
Focus on Fractions
Another Approach to Addition and Subtraction
Some learners find the addition and subtraction of fractions difficult because they struggle with
the process of finding the least common multiple (LCM) of the numbers in the denominator.
There is another approach that doesn’t require you to list the multiples but still allows you to
create a common denominator and ultimately find the correct sum or difference. The only
problem with this alternate approach is that it does not always create smallest common
denominator. That means that you will often have to do an extra step where you put your answer
in simplest form.
This alternate approach involves using the following formulas.
a c ad + bc
a c ad − bc
+ =
− =
b d
bd
b d
bd
Examples:
3 2
+
4 5
(3 × 5) + (4 × 2)
=
4×5
15 + 8
=
20
23
3
or 1
=
20
20
Questions:
3 1
(a) +
5 6
(d)
7 2
−
8 5
6 2
−
7 3
(6 × 3) − (7 × 2)
=
7×3
18 − 14
=
21
4
=
21
1 5
+
4 6
(1 × 6) + (4 × 5)
=
4×6
6 + 20
=
24
26
=
24
13
1
or 1
=
12
12
5 3
−
6 8
(5 × 8) + (6 × 3)
=
6×8
40 + 18
=
48
58
=
48
29
5
or 1
=
24
24
(b)
2 3
+
3 4
(c)
1 5
+
3 6
(e)
5 1
−
9 4
(f)
7 1
−
8 2
Answers (They are not in order.)
1
5
19
1
6
16
40
23
30
NSSAL
92
3
8
1
5
12
11
36
Focus on Fractions
List of Websites
Google Search
Directions and Description
Shodor Interactive Activities
Once you are on the site, scroll down and
find the activity Fraction Sorter. It’s a
game where you are asked to create the area
models (using circles or squares) for two
fractions and then figure out which one is
the larger fraction.
NCTM Illuminations Activities
Once you are on this site, type in fraction
into the advanced options and press
“search.” Different activities will show up
on the screen. Select Fraction Model I.
This tool allows you to examine area
models for different fractions. There are
two sliders of the screen. The top slider
allows you to change the numerator of the
fraction. The bottom slider allows you to
change the denominator of the fraction.
When this is done, an area model is created,
and the number is displayed as a fraction,
decimal and percent.
Page 4
Question 9
National Library Virtual
Manipulative Utah State
Once you are on the site, select Numbers
& Operations. A list of lessons will
appear in the main body of the page. Select
the item called Fractions-Equivalent. In
this game, you are supplied with an area
model for a particular fraction. You divide
the model into more pieces in an attempt to
find and enter an equivalent fraction.
Page 23
Question 3
321Know AAA Math
Once you are on the site, select Fractions
from the menu on the left side of the page.
A list of lessons will appear in the main
body of the page. Select the item called
Equivalent Fractions. If you scroll down
the page, you will find a section called
Practice. It’s a game where you are asked
to put identify the equivalent fractions.
Page 26
Question 6
Shodor Interactive Activities
Once you are on this site, scroll down and
find the activity called Equivalent
Fraction Pointer. With this activity you
create equivalent fractions and their
corresponding area models.
Page 27
Question 8
NSSAL
93
Page Number
and Question
Page 3
Question 5
Page 18
Question 6
Focus on Fractions
Google Search
Directions and Description
National Library Virtual
Manipulative Utah State
Once you are on the site, select Numbers
& Operations. A list of lessons will
appear in the main body of the page. Select
the item called Fractions-Comparing. In
this game, you are supplied with two area
models for two fractions. You divide one
or both area models into more pieces in an
attempt to find and enter fractions with
common denominators.
Shodor Interactive Activities
Once you are on this site, scroll down and
Page 58
find the activity called Equivalent
Question 4
Bounded Fraction Finder. With this
activity you will first provide two proper
fractions (smaller fraction is the left bound,
larger fraction is the right bound) and create
area models for these two fractions. You
will then figure out the difference of these
two fractions and express the answer as an
area model.
NSSAL
94
Page Number
and Question
Page 37
Question 4
Focus on Fractions
Answers
(h)
Please note that answers to open-ended
questions, and answers to questions
requiring written explanations or
drawings of area models are not supplied.
(i)
(j)
Pages 1 to 11
1
6
1
2
(c)
or
3
6
1. (a)
1
2. smallest:
6
(k)
7
9
7
(d)
18
(b)
7
largest:
9
3. Yes
4. No
7. (a) 1
1
(c)
2
(b) 0
(d) 1
Fraction Closest
to:
0
3
20
0
1
8
1
7
8
5
1
8
2
10. (a) 1
1
(c)
2
(b) 0
11. (a) 1
(c) 1
1
(e)
2
(g) 1
1
(i)
2
(k) 0
(b) 0
(d) 0
1
(f)
2
(h) 0
1
(j)
2
(l) 1
1
(n)
2
(m) 0
1 3 4 5
8.
, , ,
5 8 5 6
(o) 1
7
10
5
(c)
8
1
(e)
7
5
(g)
7
4
(i)
5
6
(k)
7
9
(m)
10
6
(o)
7
12. (a)
9.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Fraction Closest
to:
1
2
2
5
3
1
5
2
1
5
5
0
1
10
1
9
10
11
1
20
2
1
17
20
13.
NSSAL
(d) 1
95
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
4
5
1
3
3
5
7
8
11
12
5
9
5
6
11
12
1 3 5 8
, , ,
8 8 8 8
Focus on Fractions
14.
1 4 11 3
, , ,
10 7 12 3
1 3 5 7
, , ,
7 7 7 7
1 2 5 7
(b) , , ,
9 9 9 9
3 7 10 11
(c)
, , ,
11 11 11 11
1 1 3 4
(d)
, , ,
20 5 5 5
1 1 1 1
(e) , , ,
9 7 3 2
2 2 2 2
(f) , , ,
9 7 5 3
1 4 9 9
(g)
, , ,
12 7 11 9
1 4 11 3
(h)
, , ,
11 9 12 3
1 3 5 9
(i)
, , ,
12 8 8 10
1 1 11 6
(j) , , ,
8 3 20 6
1 1 7 7
(k)
, , ,
21 3 13 9
1 2 7 9
(l)
, , ,
19 19 12 10
1 5 17 19
(m) , , ,
5 12 20 20
15. (a)
21.
1
15
22.
1 2 3
, ,
7 7 7
Pages 12 to 14
1
5
, 1
4
4
11
5
(b)
, 1
6
6
8
2
(c) , 2
9
3
1. (a)
3.
8 3 6 13
, , ,
5 2 5 10
4. (a)
(b)
(c)
(d)
(e)
(f)
16.
1 1 3 5 9
, , , ,
6 4 7 9 9
17. (a)
18.
13
21
(g)
(b)
8
21
Pages 15 to 21
3
10
7
5
9
(c)
7
13
(e)
7
13
(g)
5
1. (a)
19. (a)
3
26
(b)
23
26
20. (a)
5
8
(b)
1
12
NSSAL
1 5 7 9
, , ,
8 8 8 8
1 10 11 13
, , ,
10 10 10 10
3 5 6 9
, , ,
5 5 5 5
1 1 10 8
, , ,
7 2 10 7
1 6 13 3
, , ,
9 11 13 2
2 1 6 4
, , ,
19 2 6 3
1 6 6 7
, , ,
8 11 5 5
96
3
2
9
(d)
5
5
(f)
2
17
(h)
7
(b)
Focus on Fractions
7
2
(i)
1
4
5
1
6
7
1
8
3
2
8
1
3
6
(e)
(g)
(i)
3. (a)
(b)
(c)
(d)
(e)
(f)
(g)
4. (a)
(b)
(c)
(d)
(e)
5. (a)
9
8
6
7
11
4
1
9
11
7
7
12
13
6
(b)
(c)
2
3
1
2
4
1
2
3
3
4
4
1
5
3
(b) 1
2. (a) 1
(c)
19
5
(j)
(d)
(f)
(h)
(j)
(d)
Between 1 and 2
Between
1
and 1
2
(e)
Greater than 2
Between 0 and
1
2
(f)
Between 1 and 2
Between
1
and 1
2
(g)
Greater than 2
(h)
3 8 1 7
, ,1 ,1
8 8 8 8
1 9
1
9
, ,1 ,1
2 10 10 10
1 2 10 3
, , ,
15 11 9 2
1 4 5 13
, , ,
4 5 3 6
9 7 11 7
, , ,
8 4 5 2
6.
(a)
(b)
(c)
(d)
1
; it’s the only fraction that isn’t
6
closest to 1
NSSAL
(e)
97
5
; it’s the only number that isn’t
6
greater than 1.
two possible responses:
1
; it’s the only one that doesn’t
7
have 6 as it’s denominator.
Or
5
; it’s the only fraction that isn’t
6
closest to 0
two possible responses:
11
; it’s the only number that isn’t
10
equal to one
Or
1; it’s the only number not expressed
as a fraction
12
; it’s the only number not
5
expressed as a mixed number
9
; it’s the only fraction that isn’t
10
1
closest to .
2
1
; it’s the only fraction that isn’t
2
closest to 0.
7
1
; all the rest are equal to
8
2
Proper
Fraction
3
8
19
20
2
5
7
8
6
10
Decimal
Percent
0.375
37.5%
0.95
95%
0.4
40%
0.875
87.5%
0.6
60%
Focus on Fractions
(a)
(b)
(c)
(d)
(f)
Improper
Fraction
9
8
11
8
20
20
17
10
8
5
Decimal
Percent
1.125
112.5%
1.375
137.5%
1
100%
1.7
170%
1.6
160%
(f)
5. (a)
(c)
(e)
(g)
(i)
(k)
7. (a) 1
(b) 100%
7. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
1 5 5 13 11
8.
, , , ,
24 12 5 12 6
14. Only the last two fractions are in the
wrong order. The correct answer is
1 6 5 8
, , , .
9 13 6 7
1
2
3
7
2
3
7
5
1
10
3
7
1. (a) Area Models 2, 4, and 5
8 6 12
(b)
, ,
12 9 18
4. Three possible answers have been
supplied for each. These are not the
only correct answers.
10 15 20
(a)
, ,
14 21 28
10 15 20
(b)
, ,
12 18 24
1 4 6
(c) , ,
5 20 130
8 12 16
(d) , ,
6 9 12
3 6 24
(e) , ,
2 4 16
98
(b)
(d)
(f)
(h)
(j)
(l)
2
3
7
(d)
10
9. (a)
(b)
10. (a) =
(c) >
(e) =
(g) >
(i) <
(k) =
(m) >
(b)
(d)
(f)
(h)
(j)
(l)
(n)
11.
1
2
12.
2
3
13.
1
5
4
5
3
5
7
8
5
9
3
2
10
9
equivalent
equivalent
not equivalent
equivalent
equivalent
equivalent
not equivalent
not equivalent
1
5
4
(c)
5
Pages 22 to 30
NSSAL
1 2 8
, ,
3 6 24
<
=
<
=
=
>
>
Focus on Fractions
1
12
3
(c)
4
14. (a)
15.
1
6
1
(d)
4
3
8
3
(i)
4
(g)
(b)
1 5 7 3 7 12 23
,
,
, , ,
,
12 10 12 3 6 8 12
2. (a)
(b)
Pages 31 to 32
10
5
1. (a)
or
16
8
4
1
(b)
or
16
4
14
7
(c)
or
16
8
2
1
(d) 1
or 1
16
8
3
6
(e) 1
or 1
16
8
14
7
(f) 1
or 1
16
8
16
(g)
or 1
16
12
3
(h)
or
16
4
8
1
(i) 1
or 1
16
2
2
1
(j)
or
16
8
4
1
(k) 1
or 1
16
4
2
1
(l) 2
or 2
16
8
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
3. (a)
(b)
(c)
(d)
(e)
Pages 33 to 38
(f)
1
1. (a)
3
7
(c)
9
4
(e)
5
3
(b)
5
2
(d)
3
5
(f)
6
2
7
7
(j)
9
(h)
5
, Method B
7
7
, Method A
15
5
, Method C
8
19
, Method A
21
4
, Method C
9
12
, Method B
13
5
, Method C
11
3
, Method C
10
7
, Method B
12
1
, Method A
8
1 3 5 5
, , ,
20 8 8 4
1 5 9 13
, , ,
40 11 10 9
2 3 11 13
, , ,
5 7 12 10
1 3 19 21
, , ,
3 8 20 20
1 2 7 5 8
, , , ,
11 11 9 6 7
1 2 5 8 11
, , , ,
10 3 9 8 6
5 3 7
5 6 7 8
, , , 1 or
, , ,
8 4 8
8 8 8 8
7 2 3 5
7 8 9 10
(b)
, , , or
, , ,
12 3 4 6
12 12 12 12
5. (a)
6. 4, 5, or 6
NSSAL
99
Focus on Fractions
7. any number equal to or greater than 9.
2.
8. any number equal to or greater than 10
and equal to or less than 17.
(a)
(b)
9. 1, 2, or 3
(c)
10. Possible Answers:
(d)
numbers with common
denominators/numbers not having
common denominators
(e)
numbers greater than one half/numbers
less than one half
(f)
(g)
numbers easily understood using
benchmark numbers/ numbers not easily
understood using benchmarks
(h)
(i)
Pages 40 to 43
1.
Sum
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
NSSAL
Sum or
Difference
1
12
1 +4
20
13
5
1
3 +4
12
32
5
7
2 +3
9
8
21 2
−
22 53
13 7
−
12 13
14
1
6 −2
15
20
8
6
8 −4
17
14
7
2
8 −3
15
45
3
7
9 −6
50
13
14
10
7 −4
15
21
7 15
+
8 14
1 4
+
20 9
1
1
+
12 15
21 7
+
20 13
9
5
+
17 11
8 11
+
9 10
2
1
+
31 25
12 1
+
13 32
8
9
+
17 10
8
1
+
15 40
(j)
Closest
to:
2
3.
1
2
0
Estimate
6
1
2
1
6
2
1
7
1
2
5
4
1
2
1
2
2
1
3
2
5
1 1
1
+ equals 1, not
2 2
2
Pages 44 to 54
1
2
1
1
1. (a)
(b)
2
(c)
0
(d)
(e)
1
(f)
1
1
2
1
2
(g)
(h)
100
7
10
8
11
11
14
1
1
2
2
3
3
4
1
1
9
Focus on Fractions
(e) 6, 12, 18, 24, 30, …
8, 16, 24, 32, 40, …
LCM: 24
6
11
1
(j) 1
2
4
(k) 2
5
2
(l) 2
3
(i) 1
1 2
+
2 3
3 4
+
6 6
7
1
or 1
6
6
2. (a)
(b)
3.
6. (a) The LCM of 5 and 4 is 20.
We have to multiply the numerator
1
and denominator of
by 5 to
4
make our common denominator.
(b) The multiples of 6 are 6, 12, 18, 24,
30,…
The LCM of 6 and 4 is 12.
We have to multiply the numerator
5
and denominator of
by 2 to
6
make our common denominator.
5 1
+
6 2
10 6
+
12 12
4
16
or 1
12
12
4
1
or 1
3
3
We have to multiply the numerator
1
and denominator of
by 3 to
4
make our common denominator
(c) The multiples of 3 are 3, 6, 9, 12, 15,
…
The LCM of 3 and 5 is 15.
We have to multiply the numerator
2
and denominator of
by 5 to
3
make our common denominator.
11
12
4. (a) 6, 12, 18, 24, 30, …
(b) 4, 8, 12, 16, 20, …
(c) 12
We have to multiply the numerator
3
and denominator of
by 3 to make
5
our common denominator.
4
Final Answer: 1
15
5. (a) 5, 10, 15, 20, 25,…
2, 4, 6, 8, 10, …
LCM: 10
(b) 4, 8, 12, 16, 20, …
10, 20, 30, 40, 50, …
LCM: 20
(d) The multiples of 8 are 8, 16, 24, 32,
40, …
(c) 2, 4, 6, 8, 10, …
6, 12, 18, 24, 30, …
LCM: 6
The LCM of 4 and 8 is 8.
We have to multiply the numerator
3
and denominator of
by 2 to
4
make our common denominator.
(d) 5, 10, 15, 20, 25, …
3, 6, 9, 12, 15, …
LCM: 15
NSSAL
101
Focus on Fractions
Final Answer: 1
3
8
12. 1
7
10
13
(b)
20
23
(c)
30
5
(d) 1
12
11
(e) 1
20
4
(f) 1
15
11
(g)
12
13
(h)
24
5
(i) 1
18
7
(j)
12
3
(k)
4
1
(l) 1
3
5
(m) 1
8
7
(n) 1
8
7. (a)
8. 1
9.
Pages 55 to 60
1. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(b) The multiples of 6 are 6, 12, 18, 24,
30, …
The multiples of 8 are 8, 16, 24, 32,
40, …
The LCM of 6 and 8 is 24.
We have to multiply the numerator
5
and denominator of
by 4 to
6
make our common denominator.
7
8
We have to multiply the numerator
3
and denominator of
by 3 to make
8
our common denominator.
11
Final Answer:
24
7
10. 1
12
1
2
NSSAL
5
9
1
6
1
2
1
3
3
4
2
5
1
8
4
5
2. (a) The multiples of 7 are 7, 14, 21, 28,
35, …
The LCM of 7 and 4 is 28.
3
Final Answer:
28
1
8
11. 2
5
8
102
Focus on Fractions
3. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
5.
(b)
(c)
(d)
(e)
(f)
7
15
7
20
1
12
3
10
3
14
3
16
1
6
7
12
1
18
11
24
1
3
2
5
2. (a)
(b)
(c)
(d)
(e)
4. (a)
(b)
(c)
(d)
(e)
9
16
(f)
6. 2
7.
1
8
(g)
5
8
(h)
(i)
7
8. 1
16
9.
(j)
7
16
(k)
(l)
3
10.
16
3
5
6
8
30
3
8
3
16
1
12
4
15
9
20
2
15
1
20
6
35
15
28
1
10
1
8
2
5
1
6
1
2
3
10
3
16
1
6
5. (a) less than one
Pages 61 to 69
1. (a) 4
NSSAL
6. (a)
103
6
7
Focus on Fractions
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
6
11
3
5
2
3
2
2
5
1
4
6
6
3
5
5
2
1
3
1
1
3
1
3
2
1
3
3
15.
11
16
16.
7
12
17. 6
18.
1
4
1
2
Pages 70 to 77
1. (a) 5
(b) 10
(c) 28
2. How many one-thirds are in four?
3. 6
4. (a)
7. (d) smaller than the whole number in the
original question
(b)
8. The product is greater than or equal to
the whole number in the original
question.
9. 2
(c)
(d)
(e)
1
4
(f)
10.
7
16
11.
3
8
(g)
(h)
(i)
3
12. 1
8
(j)
13. 27
(k)
1
6
(l)
14.
NSSAL
104
6
7
6
35
3
10
4
1
3
2
3
1
1
8
1
2
10
1
2
4
1
1
3
2
15
1
14
Focus on Fractions
5. (b) less than 1
(c)
6. (c) greater than 1
(d)
7. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
13
14
6
35
7
15
10
21
5
1
9
1
3
3
7
12
2
9
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
4
9
5
8
7
12
2
3
5
12
3
8
1
4
9
16
7
18
7
8
8. 14
Area Models (Improper Fractions &
Mixed Numbers)
Page 85
1 3
(a) 1 ,
2 2
2 5
(b) 1 ,
3 3
1 5
(c) 1 ,
4 4
5 11
(d) 1 ,
6 6
2 11
(e) 1 ,
9 9
1 7
(f) 1 ,
6 6
3 7
(g) 1 ,
4 4
1 5
(h) 2 ,
2 2
1 4
(i) 1 ,
3 3
3 11
(j) 2 ,
4 4
3
9.
20
10.
11
16
11. 40
12. 35
13.
11
12
14. 18
Additional Items
Area Models (Proper Fractions)
Page 84
1
(a)
8
1
(b)
6
NSSAL
105
Focus on Fractions
Number Magnitude Questions Using
Area Models
Page 88
1 1 5
1. (a) , ,
9 3 6
1 1 3
(b) , ,
6 2 4
2 5 13
(c) , ,
9 12 18
7 16
(k) 1 ,
9 9
5 17
(l) 2 ,
6 6
Money and Fractions
Pages 86 and 87
10  1 
1. (a) 10, 0.10,
 or 
100  10 
5  1 
(b) 20, 0.05,
 or 
100  20 
2.
(a)
¢
8
$
0.08
(b)
35
0.35
(c)
30
0.30
(d)
70
0.70
(e)
(f)
50
85
0.50
0.85
(g)
54
0.54
(h)
72
0.72
(i)
46
0.46
(j)
28
0.28
(k)
55
0.55
(l)
80
0.80
(m)
92
0.92
(n)
65
0.65
NSSAL
1 1 2
, ,
10 4 3
1 3 8
(b) , ,
5 8 9
2. (a)
Fraction
8  2 
 or 
100  25 
35  7 
 or 
100  20 
30  3 
 or 
100  10 
70  7 
 or 
100  10 
50  1 
 or 
100  2 
85  17 
 or 
100  20 
54  27 
 or 
100  50 
72  18 
 or 
100  25 
46  23 
 or 
100  50 
28  7 
 or 
100  25 
55  11 
 or 
100  20 
80  4 
 or 
100  5 
92  23 
 or 
100  25 
65  13 
 or 
100  20 
Questions Involving Benchmark Numbers
Page 89
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
1
12
11
11
6
14
9
10
7
8
1
7
2
17
7
15
20
20
11
20
Closest to 0
Equal to 1
1
2
Closest to 1
Closest to
Closest to 1
Closest to 0
Closest to 0
1
2
Equal to 1
Closest to
Closest to
1
2
Number Magnitude Questions
Page 90
2 4 6
(a) , ,
9 9 9
106
Focus on Fractions
3 4 7
, ,
5 5 5
3 5 8 11
(c) , , ,
8 8 8 8
1 5 9
(d) , ,
6 9 10
1 6 7
(e)
, ,
10 11 6
1 5 9
(f) , ,
9 5 4
8 7 11
(g) , ,
9 7 6
1 6 3 6
(h)
, , ,
12 10 3 5
1 1 10 5
(i)
, , ,
14 2 11 4
1 4 9 6 8
(j) , , , ,
8 9 10 6 7
1 1 1
(k) , ,
7 4 2
1 1 1
(l)
, ,
10 5 3
1 1 8
(m) , ,
5 4 9
1 1 2 7
(n) , , ,
4 3 3 5
1 5 4 11 13
(o) , , , ,
6 9 4 8 8
1 1 8 9 4 3
(p) , , , , ,
9 4 10 10 4 2
1
3
3
(g)
5
3
(h)
4
2
(i)
9
1
(j)
2
1
(k)
5
(l) 1
4
(m)
7
3
(n)
2
1
(o)
8
5
(p)
7
4
(q)
3
3
(r)
8
5
(s)
4
(t) 2
4
(u)
9
8
(v)
5
7
(w)
4
9
(x)
5
6
(y)
5
8
(z)
5
(b)
(f)
Simplest Form Questions
Page 91
1
(a)
4
2
(b)
3
2
(c)
5
1
(d)
7
5
(e)
6
NSSAL
107
Focus on Fractions
Another Approach to Addition and
Subtraction
Page 92
23
(a)
30
5
(b) 1
12
1
(c) 1
6
19
(d)
40
11
(e)
36
3
(f)
8
NSSAL
108
Focus on Fractions