Grade 7 Mathematics
Unit 5
Operations with Fractions
Estimated Time: 24 Hours
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
159
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
160
Unit 5: Operations with Fractions
Unit 5 Overview
Introduction
Students will focus on developing skills and understanding the addition and subtraction of fractions. The
big ideas in this unit are:
• Equivalent fractions represent the same quantities.
• The concept of equivalent fractions is very useful when comparing, ordering, simplifying, and
operating with fractions.
• The use of manipulatives such as fraction strips and fraction circles, number lines, and pattern
blocks is an effective way to model the addition and subtraction of fractions. It creates a
concrete base for a traditionally difficult concept.
• Addition and subtraction of fractions requires common denominators.
• Estimation strategies for these two operations are based on using benchmarks like 0,
1 1 3
, , etc.
4 2 4
Context
The students will model, using manipulatives, the addition and subtraction of fractions. They will be
encouraged to informally generalize rules for these operations that are based on their investigations.
Through the use of these investigations, and guidance from the teacher, the students will discover the need
to use common denominators when adding, subtracting, comparing and ordering fractions.
They will discover the algorithm for adding and subtracting fractions. Once again estimation will play an
important role in helping students to decide if their answers are “sensible.” The students will then apply
these algorithms to adding and subtracting mixed numbers.
Why are these concepts important?
Developing a good understanding of adding and subtracting fractions will permit students to:
• Understand real-life situations that require fractions such as;
The clock ("a quarter 'till").
Electricians (gauge/length of wires).
Plumbing (thickness of pipe, diameter of pipe, length of pipe).
Carpenters (thickness/length/width of wood).
Engineers (just math equations).
Metal fabrication (length/width/gauge of metal).
Taxes/budgeting (obvious math involved).
Cooking (measurements like HALF a cup...).
In your car (km PER hour, km PER liter).
Paying for things in general (1 penny is 1/100 of a dollar, writing out checks.)
• Be ready to learn and understand future topics in math such as algebra and proportions.
“It isn't that they can't see the solution. It is that they can’t see the problem.”
G. K. Chesterton (1874 – 1936)
Grade 7 Math Curriculum Guide
161
Strand: Number
General Outcome: Develop Number Sense
Specific Outcome
Elaborations: Suggested Learning and Teaching Strategies
It is expected that students will:
7N5. Demonstrate an
understanding of adding
and subtracting positive
fractions and mixed
numbers, with like and
unlike denominators,
concretely, pictorially and
symbolically (limited to
positive sums and
differences).
[C, CN, ME, PS, R, V]
Achievement Indicators
7N5.1 Model addition of
positive fractions, using
concrete representations,
and record symbolically.
7N5.2 Determine the sum
of two given positive
fractions with like
denominators.
7N5.3 Determine a
common denominator for
a given set of positive
fractions.
7N5.4 Simplify a given
positive fraction by
identifying the common
factor between the
numerator and
denominator.
Lesson 5.1 in the student text briefly models like fractions
using pattern blocks, clocks and fraction circles. It primarily
demonstrates like denominators, but includes some examples
in which one of the denominators is a simple multiple of the
other. Teachers will need to model several more examples
using these manipulatives in order to ensure student
understanding. Students should also have the opportunity to
model using the manipulatives since they are hands-on
experiences.
Lesson 5.2 uses fraction strips and number lines to support the
same indicators. Students should be able to use the models to
understand fractional equivalents and how they can be useful
when adding fractions and changing them to their simplest
form.
Using the fractions strips and number line masters in the
ProGuide pp. 64–67, students will combine both the fraction
strips and number lines to model sums and to illustrate the
concept of common denominators.
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
162
Strand: Number
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Pencil and Paper
Write an addition sentence to represent the total fraction of each
hexagon that is shaded. Use an addition sentence to find the total
value of the shaded hexagons in each case.
A.
Resources/Notes
The national library of
virtual manipulatives
provides an interesting
activity on adding using
common denominators
with various models at
http://nlvm.usu.edu/en/na
v/frames_asid_106_g_3_
t_1.html?from=category_
g_3_t_1.html
B.
C.
Informal Observation
An alternative, but similar activity would be to create cards with
addition sentences and their equivalents in pattern blocks as used in
the Pencil and Paper exercise above. Each student would receive a
card with either the addition sentence, or the pattern block
representation. They mix-up and match-up within the class to find
their partner. Each group must then explain to another group, or to
their class, why they belong together.
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
Math Makes Sense 7
Lesson 5.1
Lesson 5.2
Unit 5: Operations with
Fractions
TR: ProGuide, pp. 4–6 &
pp. 7–11
Master 5.13, 5.18, 5.27
Master 5.10, 5.11, 5.14,
5.15, 5.16, 5.17, 5.19,
5.28
PM 28, PM 25
CD-ROM Unit 5 Masters
ST: pp. 178–180
ST: pp. 181–185
Practice and HW Book
pp. 106–108
pp. 109–111
163
Strand: Number
General Outcome: Develop Number Sense
Specific Outcome
Elaborations: Suggested Learning and Teaching Strategies
It is expected that students will:
In the previous lessons, students used models to add using like
denominators. They also modelled unlike denominators when
one denominator was a multiple of the other.
7N5. Demonstrate an
understanding of adding
and subtracting positive
fractions and mixed
numbers, with like and
unlike denominators,
concretely, pictorially and
symbolically (limited to
positive sums and
differences).
[C, CN, ME, PS, R, V]
(Cont’d)
Achievement Indicators
7N5.5 Model addition of
positive fractions with
unlike denominators,
using concrete
representations, and
record symbolically.
Lesson 5.3 develops the addition algorithm for fractions.
The addition of fractions with unlike denominators that are not
simple multiples of each other will require students to multiply
the numerator and denominator of each fraction by the same
number. Example:
7N5.6 Determine the sum
of two given positive
fractions with unlike
denominators.
Ideally, students should use the Least Common Multiple
(LCM) of the unlike denominators.
1
Through the use of benchmarks (close to 0, ,1 ) developed in
2
Unit 3, students will estimate the solution and use their
estimate to verify the reasonableness of the answer obtained
using the algorithm.
(This elaboration is continued on the next two page spread…)
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
164
Strand: Number
General Outcome: Develop Number Sense
Resources/Notes
Suggested Assessment Strategies
Pencil and Paper
1. Create three addition sentences that give the same sum as
6
3
+ . You cannot use like denominators in the sentences
12 12
you create.
2. Magic square. The sum of each row, column and diagonal in
this magic square must equal 1. Find the missing values.
Magic Square
5
12
7
12
1
4
1
3
Solution
1
5
6
12
7
1
12
3
1
1
4
4
5
12
1
12
1
2
3. A tangram is a square puzzle that is divided into seven shapes.
1
A. Suppose piece A is . What are the values of pieces B, C,
4
D, E, F and G?
B. What is the sum of A and B?
C. If you subtract D from the whole puzzle, what value
remains?
D. Which two tangram pieces add up to the value of C?
E. Invent a problem on your own and solve it.
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
Math Makes Sense 7
Lesson 5.3
Unit 5: Operations with
Fractions
TR: ProGuide, pp. 12–15
Master 5.14, 5.15, 5.16,
5.17, 5.20, 5.29
PM 27
CD-ROM Unit 5 Masters
ST: pp. 186–189
Practice and HW Book
pp. 112–114
165
Strand: Number
General Outcome: Develop Number Sense
Specific Outcome
Elaborations: Suggested Learning and Teaching Strategies
It is expected that students will:
Here is another example of adding fractions with unlike
denominators.
7N5. Demonstrate an
understanding of adding
and subtracting positive
fractions and mixed
numbers, with like and
unlike denominators,
concretely, pictorially and
symbolically (limited to
positive sums and
differences).
[C, CN, ME, PS, R, V]
(Cont’d)
Achievement Indicators
7N5.5 Model addition of
positive fractions with
unlike denominators,
using concrete
representations, and
record symbolically.
(continued)
7N5.6 Determine the sum
of two given positive
fractions with unlike
denominators.
(continued)
Find the sum of the fractions:
3 1
+
4 6
3
1
is a little bit more than a half and
is
4
6
less than a half so the answer should be close to 1. Then they
can use the previous algorithm to calculate:
Students should think
3 1
+
4 6
3 3 1 2
= × + ×
4 3 6 2
9
2
= +
12 12
11
=
12
Finally, they should look at their answer and ask themselves if
11
is reasonable based on their estimate of 1.
12
Note: When a common denominator must be found, the
common denominator that is chosen should be the lowest
common denominator. Simply multiplying the denominators
of the fractions being adding or subtracted will not guarantee a
lowest common denominator. The lowest common
3 1
denominator for + is 12, not 24.
4 6
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
166
Strand: Number
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Performance
Use pattern blocks to create a design on triangular grid paper
(Program Master 27). Then use fraction addition to name the
design. Consider the flower design illustrated in Appendix 5-A. It is
possible to use several different addition sentences to name the
same design.
Journal
1. If a problem required you to add fourths and thirds, is it
possible for the sum to be sixths? Why or why not? You may
use an example or a diagram to help you explain your answer.
2. If a problem required you to add fourths and thirds, is it
possible for the sum to be sevenths? Why or why not? You may
use an example or a diagram to help you explain your answer.
Interview
A classmate missed yesterday’s class. When solving a problem
5 5 10
today he suggested that + = . How would you convince him
6 8 14
that this is not a reasonable solution?
Math Makes Sense 7
Lesson 5.3
(continued)
Game/Activity
Refer to Appendix 5-B for the Connect Three game.
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
167
Strand: Number
General Outcome: Develop Number Sense
Specific Outcome
Elaborations: Suggested Learning and Teaching Strategies
It is expected that students will:
Lesson 5.4 of the student text begins with subtraction
involving unlike denominators using pattern blocks. Students
will learn that addition and subtraction of fractions with unlike
denominators uses the same algorithm. Teachers may wish to
model several examples using fraction circles or fraction
strips.
4 1
For example: −
5 5
In this case, students must understand that they are simply
removing one part of a set of equivalent quantities. This can be
4
using fraction strips or fraction
demonstrated by modelling
5
1
circles and removing one portion representing . The answer
5
3
is the remaining portion of .
5
7N5. Demonstrate an
understanding of adding
and subtracting positive
fractions and mixed
numbers, with like and
unlike denominators,
concretely, pictorially and
symbolically (limited to
positive sums and
differences).
[C, CN, ME, PS, R, V]
(Cont’d)
Achievement Indicators
7N5.7 Model subtraction
of positive fractions, using
concrete representations,
and record symbolically.
7N5.8 Determine the
difference of two given
positive fractions with like
denominators.
7N5.9 Determine the
difference of two given
positive fractions with
unlike denominators.
The subtraction of fractions with unlike denominators that are
not simple multiples of each other will require students to
multiply the numerator and denominator of each fraction by
the same number. This is identical to the algorithm used for
addition.
Ideally, students should use the Least Common Multiple
(LCM) of the unlike denominators.
1
Through the use of benchmarks (close to 0, ,1 ) developed in
2
Unit 3, students will estimate the solution and use their
estimate to verify the reasonableness of the answer obtained
using the algorithm.
(This elaboration is continued on the next two page spread…)
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
168
Strand: Number
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Observation
Ask students to use concrete materials or diagrams to show why the
following is an incorrect procedure.
3 1 3 −1 2 1
− =
= =
8 4 8−4 4 2
Informal Observation
Students can play the game Tic-Tac-Toe Fractions. A really useful
game for adding and subtracting fractions. See ProGuide (Page V)
and Master 5.8a, 5.8b and 5.8c.
Math Makes Sense 7
Lesson 5.4
Lesson 5.5
Unit 5: Operations with
Fractions
TR: ProGuide, pp. 17–20
& pp. 21–24
Master 5.12, 5.14, 5.15,
5.16, 5.17, 5.21, 5.30
Master 5.14, 5.15, 5.16,
5.17, 5.22, 5.31
CD-ROM Unit 5 Masters
ST: pp. 191–194
ST: pp. 195–198
Practice and HW Book
pp. 115–117
pp. 118–120
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
169
Strand: Number
General Outcome: Develop Number Sense
Specific Outcome
Elaborations: Suggested Learning and Teaching Strategies
It is expected that students will:
Find the difference of the fractions:
7N5. Demonstrate an
understanding of adding
and subtracting positive
fractions and mixed
numbers, with like and
unlike denominators,
concretely, pictorially and
symbolically (limited to
positive sums and
differences).
[C, CN, ME, PS, R, V]
(Cont’d)
Achievement Indicators
7N5.7 Model subtraction
of positive fractions, using
concrete representations,
and record symbolically.
(continued)
7N5.8 Determine the
difference of two given
positive fractions with like
denominators.
(continued)
4 1
−
9 3
4
1
is a little bit less than a half and is
9
3
a little less than a half. The difference between them should
therefore be almost 0 or just a little bit more than 0.
Students should think
4 1
−
9 3
4 1 3
= − ×
9 3 3
4 3
= −
9 9
4−3
=
9
1
=
9
Finally, they should look at their answer and ask themselves if
1
is reasonable based on their estimate of something a little
9
bit more than 0.
7N5.9 Determine the
difference of two given
positive fractions with
unlike denominators.
(continued)
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
170
Strand: Number
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Math Makes Sense 7
Lesson 5.4
Lesson 5.5
(continued)
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
171
Strand: Number
General Outcome: Develop Number Sense
Specific Outcome
Elaborations: Suggested Learning and Teaching Strategies
It is expected that students will:
Now that the models for addition and subtraction have been
studied separately by the students, the same models and skills
can now be used in the study of mixed fractions.
7N5. Demonstrate an
understanding of adding
and subtracting positive
fractions and mixed
numbers, with like and
unlike denominators,
concretely, pictorially and
symbolically (limited to
positive sums and
differences).
[C, CN, ME, PS, R, V]
(Cont’d)
Lessons 5.6 and 5.7 explore the subtraction of mixed numbers
using fraction circles, number lines and fraction strips. Lesson
5.7 also introduces Cuisenaire rods as a model for subtracting
mixed fractions. Teachers may consult the link for use of this
model in the resource section of this guide.
Achievement Indicators
7N5.10 Model addition
and subtraction of mixed
numbers with like
denominators, using
concrete representations,
and record symbolically.
7N5.11 Determine the
sum or difference of two
mixed numbers with like
denominators.
7N5.12 Model addition
and subtraction of mixed
numbers with unlike
denominators, using
concrete representations,
and record symbolically.
7N5.13 Determine the
sum and difference of two
mixed numbers with
unlike denominators.
When adding and subtracting mixed fractions students may
approach the problem in different ways. They may choose to
keep the mixed fraction form or, they may change the mixed
fractions to improper fractions.
For addition:
Mixed Fraction Form
2
5
1 +1
9
6
2 2
5 3
= 1 × +1 ×
9 2
6 3
4
15
=1 + 1
18 18
19
=2
18
1
= 2 and 1
18
1
=3
18
Improper Fraction Form
2
5
1 +1
9
6
11 11
= +
9 6
11 2 11 3
= × + ×
9 2 6 3
22 33
=
+
18 18
55
=
18, 36, 54...
18
1
=3
18
(This elaboration is continued on the next two page spread…)
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
172
Strand: Number
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Interview
3
3
and 1 −
4
10
Without calculating, explain how you could determine which
answer would be greater.
Consider the following two problems: 1 −
Journal
An introduction to
Cuisenaire rods and their
use in the study of
fractions can be found at
http://teachertech.rice.ed
u/Participants/silha/Lesso
ns/cuisen2.html
1
5
Describe at least two ways you can calculate 4 − 2 .
2
6
Math Makes Sense 7
Lesson 5.6
Lesson 5.7
Unit 5: Operations with
Fractions
TR: ProGuide, pp. 25–29
& pp. 30–34
Master 5.13, 5.14, 5.15,
5.16, 5.17, 5.23, 5.32
Master 5.13, 5.14, 5.15,
5.16, 5.17, 5.24, 5.33
PM 28
CD-ROM Unit 5 Masters
ST: pp. 199–203
ST: pp. 204–208
Practice and HW Book
pp. 121–122
pp. 123–124
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
173
Strand: Number
General Outcome: Develop Number Sense
Specific Outcome
Elaborations: Suggested Learning and Teaching Strategies
It is expected that students will:
For subtraction:
7N5. Demonstrate an
understanding of adding
and subtracting positive
fractions and mixed
numbers, with like and
unlike denominators,
concretely, pictorially and
symbolically (limited to
positive sums and
differences).
[C, CN, ME, PS, R, V]
(Cont’d)
Achievement Indicators
7N5.10 Model addition
and subtraction of mixed
numbers with like
denominators, using
concrete representations,
and record symbolically.
7N5.11 Determine the
sum or difference of two
mixed numbers with like
denominators.
Mixed Fraction Form
4
2
2 −1
7
3
4 3 2 7
= 2 × −1 ×
7 3 3 7
12 14
= 2 −1
21 21
Students will be challenged
by 12–14 and therefore must
think about regrouping.
Students should think:
21
12 14
1 and
and
−1
21
21 21
which will allow them to
calculate:
33 14
1 −1
21 21
19
=
21
Improper Fraction Form
4
2
2 −1
7
3
18 5
= −
7 3
18 3 5 7
= × − ×
7 3 3 7
54 35
=
−
21 21
19
=
21
7N5.12 Model addition
and subtraction of mixed
numbers with unlike
denominators, using
concrete representations,
and record symbolically.
7N5.13 Determine the
sum and difference of two
mixed numbers with
unlike denominators.
(All Cont’d)
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
174
Strand: Number
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
An introduction to
Cuisenaire rods and their
use in the study of
fractions can be found at
http://teachertech.rice.ed
u/Participants/silha/Lesso
ns/cuisen2.html
Math Makes Sense 7
Lesson 5.6
Lesson 5.7
(continued)
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
175
Strand: Number
General Outcome: Develop Number Sense
Specific Outcome
Elaborations: Suggested Learning and Teaching Strategies
It is expected that students will:
Throughout the sections on adding and subtracting fractions, it
is necessary for students to simplify their answers. Simplified
answers may be proper fractions, improper fractions or mixed
numbers in simplest form depending on the context of the
problem.
1
Example: Kyra is making cookies. She has 2 bags of
4
2
chocolate chips. She adds 1 of these bags to her cookie
3
dough.
a) What fraction of the total amount of chocolate chips is left?
11
b) Kyra then decides to add
bags of butterscotch chips to
12
the dough as well. How many bags of chips does Kyra use in
total to bake her cookies?
1
For part a), students should think “ 2 bags” is a little more
4
2
than two bags. Kyra then uses 1 bags which is a little less
3
than two bags. Therefore she has two little bits or about half a
bag left over.
Then they calculate:
1
2
2 −1
Students must reflect
4
3
upon their answer to
9 5
= −
determine if it is
4 3
reasonable.
9 3 5 4
= × − ×
4 3 3 4
In this case, seven
twelfths is very close to a
27 20
=
−
half.
12 12
7N5. Demonstrate an
understanding of adding
and subtracting positive
fractions and mixed
numbers, with like and
unlike denominators,
concretely, pictorially and
symbolically (limited to
positive sums and
differences).
[C, CN, ME, PS, R, V]
(Cont’d)
Achievement Indicators
7N5.14 Simplify the
solution to a given
problem involving the
sum or difference of two
positive fractions or
mixed numbers.
7N5.15 Solve a given
problem involving the
addition or subtraction of
positive fractions or
mixed numbers, and
determine if the solution is
reasonable.
=
7
12
7
of a bag of
12
chocolate chips left.
(This elaboration is continued on the next two page spread…)
Kyra has
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
176
Strand: Number
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Journal
Is it possible to find two mixed numbers which add together to form
a whole number?
Explain your answer and, if possible, give an example to support
your explanation.
Pencil and Paper
1. Andrew plays guitar in a rock band. For a song that is 36
1
3
measures long he plays for 4 measures, rests for 8
2
8
1
measures, plays for another 16 measures, rests for 2 measures
4
and plays for the last section. How many measures are in the
last section?
2. This week, Mark practised piano for 3
6
1
h, played soccer for
2
Math Makes Sense 7
Lesson 5.6
Lesson 5.7
(continued)
1
1
h, and talked on the phone for 4 h.
4
3
A. How many hours did Mark spend practising piano and
playing soccer?
B. Hour many more hours did Mark spend playing soccer
than talking on the phone?
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
177
Strand: Number
General Outcome: Develop Number Sense
Specific Outcome
Elaborations: Suggested Learning and Teaching Strategies
It is expected that students will:
7N5. Demonstrate an
understanding of adding
and subtracting positive
fractions and mixed
numbers, with like and
unlike denominators,
concretely, pictorially and
symbolically (limited to
positive sums and
differences).
[C, CN, ME, PS, R, V]
(Cont’d)
Achievement Indicators
7N5.14 Simplify the
solution to a given
problem involving the
sum or difference of two
positive fractions or
mixed numbers.
(continued)
7N5.15 Solve a given
problem involving the
addition or subtraction of
positive fractions or
mixed numbers, and
determine if the solution is
reasonable.
(continued)
For part b) students should think 2
1
bags is a little more than
4
11
is almost one full bag, but not quite.
12
Therefore Kyra uses a little more than 3 bags of chips in total.
two bags and
Then they calculate:
1 11
2 +
4 12
1 3 11
=2 × +
4 3 12
3 11
=2 +
12 12
14
=2
12
14 2
=2 ÷
12 2
7
=2
6
1
= 2 and 1
6
1
=3
6
Kyra used 3
Note that the final answer
must be simplified.
Students must reflect upon
their answer to determine if
it is reasonable.
In this case, the answer is
very close to the estimate.
1
bags of chips
6
in total.
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
178
Strand: Number
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Math Makes Sense 7
Lesson 5.6
Lesson 5.7
(continued)
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
179
Strand: Number
Grade 7 Mathematics Curriculum Outcomes
Outcomes with Achievement Indicators
Unit 5
180