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Grade 7 Mathematics Curriculum Outcomes 159
Outcomes with Achievement Indicators
Unit 5
Grade 7 Mathematics
Unit 5
Operations with Fractions
Estimated Time: 24 Hours
[C] Communication [PS] Problem Solving [CN] Connections [R] Reasoning
[ME] Mental Mathematics [T] Technology
and Estimation [V] Visualization
Grade 7 Mathematics Curriculum Outcomes 160
Outcomes with Achievement Indicators
Unit 5
Unit 5: Operations with Fractions
Grade 7 Math Curriculum Guide 161
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,
4
3,
2
1,
4
1etc.
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)
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 162
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Specific Outcome
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
Elaborations: Suggested Learning and Teaching Strategies
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.
7N5.2 Determine the sum
of two given positive
fractions with like
denominators.
7N5.1 Model addition of
positive fractions, using
concrete representations,
and record symbolically.
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.
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 163
Outcomes with Achievement Indicators
Unit 5
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.
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.
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
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
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 164
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Specific Outcome
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
Elaborations: Suggested Learning and Teaching Strategies
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.
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:
Ideally, students should use the Least Common Multiple
(LCM) of the unlike denominators.
Through the use of benchmarks (close to 1
0, ,12
) developed in
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…)
7N5.5 Model addition of
positive fractions with
unlike denominators,
using concrete
representations, and
record symbolically.
7N5.6 Determine the sum
of two given positive
fractions with unlike
denominators.
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 165
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Pencil and Paper
1. Create three addition sentences that give the same sum as
6 3
12 12+ . You cannot use like denominators in the sentences
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 Solution
5
12
7
12
1
3
1
4
1
6
5
12
5
12
7
12
1
3
1
12
1
4
1
4
1
2
3. A tangram is a square puzzle that is divided into seven shapes.
A. Suppose piece A is1
4. What are the values of pieces B, C,
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.
Resources/Notes
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
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 166
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Specific Outcome
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
Elaborations: Suggested Learning and Teaching Strategies
Here is another example of adding fractions with unlike
denominators.
Find the sum of the fractions:
6
1
4
3+
Students should think 4
3is a little bit more than a half and
6
1 is
less than a half so the answer should be close to 1. Then they
can use the previous algorithm to calculate:
12
11
12
2
12
9
2
2
6
1
3
3
4
3
6
1
4
3
=
+=
×+×=
+
Finally, they should look at their answer and ask themselves if
12
11is reasonable based on their estimate of 1.
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
denominator for 6
1
4
3+ is 12, not 24.
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)
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 167
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies
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
today he suggested that5 5 10
6 8 14+ = . How would you convince him
that this is not a reasonable solution?
Game/Activity
Refer to Appendix 5-B for the Connect Three game.
Resources/Notes
Math Makes Sense 7
Lesson 5.3
(continued)
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 168
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Specific Outcome
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
Elaborations: Suggested Learning and Teaching Strategies
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.
For example: 4 1
5 5−
In this case, students must understand that they are simply
removing one part of a set of equivalent quantities. This can be
demonstrated by modelling 4
5 using fraction strips or fraction
circles and removing one portion representing1
5. The answer
is the remaining portion of3
5.
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.
Through the use of benchmarks (close to 1
0, ,12
) developed in
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…)
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.
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 169
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies
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.
Resources/Notes
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
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 170
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Specific Outcome
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
Elaborations: Suggested Learning and Teaching Strategies
Find the difference of the fractions:
3
1
9
4−
Students should think 9
4is a little bit less than a half and
3
1 is
a little less than a half. The difference between them should
therefore be almost 0 or just a little bit more than 0.
9
1
9
34
9
3
9
4
3
3
3
1
9
4
3
1
9
4
=
−=
−=
×−=
−
Finally, they should look at their answer and ask themselves if
9
1 is reasonable based on their estimate of something a little
bit more than 0.
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)
7N5.9 Determine the
difference of two given
positive fractions with
unlike denominators.
(continued)
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 171
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Math Makes Sense 7
Lesson 5.4
Lesson 5.5
(continued)
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 172
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Specific Outcome
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
Elaborations: Suggested Learning and Teaching Strategies
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.
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.
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 Improper Fraction Form
18
13
18
112
18
192
18
151
18
41
3
3
6
51
2
2
9
21
6
51
9
21
=
=
=
+=
×+×=
+
and
18
13
...54,36,1818
55
18
33
18
22
3
3
6
11
2
2
9
11
6
11
9
11
6
51
9
21
=
=
+=
×+×=
+=
+
(This elaboration is continued on the next two page spread…)
7N5.11 Determine the
sum or difference of two
mixed numbers with like
denominators.
7N5.10 Model addition
and subtraction of mixed
numbers with like
denominators, using
concrete representations,
and record symbolically.
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.
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 173
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Interview
Consider the following two problems: 3
14
− and 3
110
−
Without calculating, explain how you could determine which
answer would be greater.
Journal
Describe at least two ways you can calculate1 5
4 22 6
− .
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
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
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 174
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Specific Outcome
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
(All Cont’d)
Elaborations: Suggested Learning and Teaching Strategies
For subtraction:
Mixed Fraction Form Improper Fraction Form
21
141
21
122
7
7
3
21
3
3
7
42
3
21
7
42
−=
×−×=
−
Students will be challenged
by 12–14 and therefore must
think about regrouping.
Students should think:
21
141
21
12
21
211 −andand
which will allow them to
calculate:
21
19
21
141
21
331
=
−
21
19
21
35
21
54
7
7
3
5
3
3
7
18
3
5
7
18
3
21
7
42
=
−=
×−×=
−=
−
7N5.11 Determine the
sum or difference of two
mixed numbers with like
denominators.
7N5.10 Model addition
and subtraction of mixed
numbers with like
denominators, using
concrete representations,
and record symbolically.
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.
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 175
Outcomes with Achievement Indicators
Unit 5
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)
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 176
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Specific Outcome
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
Elaborations: Suggested Learning and Teaching Strategies
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.
Example: Kyra is making cookies. She has 1
24
bags of
chocolate chips. She adds 2
13
of these bags to her cookie
dough.
a) What fraction of the total amount of chocolate chips is left?
b) Kyra then decides to add 11
12 bags of butterscotch chips to
the dough as well. How many bags of chips does Kyra use in
total to bake her cookies?
For part a), students should think “1
24
bags” is a little more
than two bags. Kyra then uses 2
13
bags which is a little less
than two bags. Therefore she has two little bits or about half a
bag left over.
Then they calculate:
12
7
12
20
12
27
4
4
3
5
3
3
4
9
3
5
4
9
3
21
4
12
=
−=
×−×=
−=
−
Kyra has 7
12of a bag of
chocolate chips left.
Students must reflect
upon their answer to
determine if it is
reasonable.
In this case, seven
twelfths is very close to a
half.
(This elaboration is continued on the next two page spread…)
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.
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 177
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies
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
measures long he plays for 1
42
measures, rests for 3
88
measures, plays for another 16 measures, rests for 1
24
measures
and plays for the last section. How many measures are in the
last section?
2. This week, Mark practised piano for 1
32
h, played soccer for
16
4 h, and talked on the phone for
14
3 h.
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?
Resources/Notes
Math Makes Sense 7
Lesson 5.6
Lesson 5.7
(continued)
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 178
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Specific Outcome
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
Elaborations: Suggested Learning and Teaching Strategies
For part b) students should think 1
24
bags is a little more than
two bags and 11
12is almost one full bag, but not quite.
Therefore Kyra uses a little more than 3 bags of chips in total.
Then they calculate:
6
13
6
112
6
72
2
2
12
142
12
142
12
11
12
32
12
11
3
3
4
12
12
11
4
12
=
=
=
÷=
=
+=
+×=
+
and
Kyra used 1
36
bags of chips
in total.
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.
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)
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 179
Outcomes with Achievement Indicators
Unit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Math Makes Sense 7
Lesson 5.6
Lesson 5.7
(continued)
Strand: Number
Grade 7 Mathematics Curriculum Outcomes 180
Outcomes with Achievement Indicators
Unit 5