Download - Grade 7 Math Unit 5
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Grade 7 Mathematics Curriculum Outcomes 159Outcomes 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
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Grade 7 Mathematics Curriculum Outcomes 160Outcomes with Achievement Indicators
Unit 5
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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 patternblocks is an effective way to model the addition and subtraction of fractions. It creates aconcrete 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 animportant 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 cant see the problem.
G. K. Chesterton (1874 1936)
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 162
Outcomes with Achievement IndicatorsUnit 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 fractionsusing pattern blocks, clocks and fraction circles. It primarilydemonstrates like denominators, but includes some examples
in which one of the denominators is a simple multiple of theother. Teachers will need to model several more examples
using these manipulatives in order to ensure studentunderstanding. Students should also have the opportunity to
model using the manipulatives since they are hands-onexperiences.
Lesson 5.2 uses fraction strips and number lines to support the
same indicators. Students should be able to use the models tounderstand fractional equivalents and how they can be useful
when adding fractions and changing them to their simplestform.
Using the fractions strips and number line masters in the
ProGuide pp. 6467, students will combine both the fractionstrips and number lines to model sums and to illustrate the
concept of common denominators.
7N5.2 Determine the sumof two given positive
fractions with like
denominators.
7N5.1 Model addition of
positive fractions, usingconcrete representations,
and record symbolically.
7N5.3 Determine acommon denominator for
a given set of positivefractions.
7N5.4 Simplify a givenpositive fraction byidentifying the common
factor between thenumerator and
denominator.
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 163
Outcomes with Achievement IndicatorsUnit 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 totalvalue 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 inthe Pencil and Paperexercise above. Each student would receive a
card with either the addition sentence, or the pattern blockrepresentation. They mix-up and match-up within the class to findtheir partner. Each group must then explain to another group, or to
their class, why they belong together.
Resources/Notes
The national library of
virtual manipulativesprovides an interesting
activity on adding usingcommon denominators
with various models athttp://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 withFractions
TR: ProGuide, pp. 46 &pp. 711
Master 5.13, 5.18, 5.27Master 5.10, 5.11, 5.14,
5.15, 5.16, 5.17, 5.19,5.28
PM 28, PM 25CD-ROM Unit 5 Masters
ST: pp. 178180
ST: pp. 181185
Practice and HW Bookpp. 106108
pp. 109111
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 164
Outcomes with Achievement IndicatorsUnit 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]
(Contd)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
In the previous lessons, students used models to add using likedenominators. 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 multiplythe 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 to1
0, ,12 ) developed in
Unit 3, students will estimate the solution and use their
estimate to verify the reasonableness of the answer obtainedusing the algorithm.
(This elaboration is continued on the next two page spread)
7N5.5 Model addition of
positive fractions withunlike denominators,
using concreterepresentations, and
record symbolically.
7N5.6 Determine the sumof two given positive
fractions with unlikedenominators.
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 165
Outcomes with Achievement IndicatorsUnit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Pencil and Paper
1. Create three addition sentences that give the same sum as6 3
12 12+ . You cannot use like denominators in the sentences
you create.
2. Magic square. The sum of each row, column and diagonal inthis magic square must equal 1. Find the missing values.
Magic Square Solution
512
7
12
1
3
1
4
16
512
512
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 is 1
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 withFractions
TR: ProGuide, pp. 1215Master 5.14, 5.15, 5.16,
5.17, 5.20, 5.29PM 27
CD-ROM Unit 5 Masters
ST: pp. 186189
Practice and HW Book
pp. 112114
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 166
Outcomes with Achievement IndicatorsUnit 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]
(Contd)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Here is another example of adding fractions with unlikedenominators.
Find the sum of the fractions:
6
1
4
3+
Students should think4
3is a little bit more than a half and
6
1is
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, thecommon denominator that is chosen should be the lowest
common denominator. Simply multiplying the denominatorsof the fractions being adding or subtracted willnot guarantee a
lowest common denominator. The lowest common
denominator for6
1
4
3+ is 12, not 24.
7N5.5 Model addition of
positive fractions withunlike denominators,
using concreterepresentations, and
record symbolically.
(continued)
7N5.6 Determine the sumof two given positive
fractions with unlikedenominators.
(continued)
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 167
Outcomes with Achievement IndicatorsUnit 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 thedesign. Consider the flower design illustrated in Appendix 5-A. It is
possible to use several different addition sentences to name thesame design.
Journal
1. If a problem required you to add fourths and thirds, is itpossible 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 itpossible 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 yesterdays 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)
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 168
Outcomes with Achievement IndicatorsUnit 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]
(Contd)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Lesson 5.4 of the student text begins with subtractioninvolving unlike denominators using pattern blocks. Students
will learn that addition and subtraction of fractions with unlikedenominators uses the same algorithm. Teachers may wish to
model several examples using fraction circles or fractionstrips.
For example:4 1
5 5
In this case, students must understand that they are simplyremoving one part of a set of equivalent quantities. This can be
demonstrated by modelling 45
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 arenot simple multiples of each other will require students to
multiply the numerator and denominator of each fraction bythe 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 10, ,12
) developed in
Unit 3, students will estimate the solution and use theirestimate 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, usingconcrete representations,
and record symbolically.
7N5.8 Determine thedifference of two given
positive fractions with like
denominators.
7N5.9 Determine the
difference of two given
positive fractions withunlike denominators.
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 169
Outcomes with Achievement IndicatorsUnit 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 withFractions
TR: ProGuide, pp. 1720& pp. 2124
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. 191194
ST: pp. 195198
Practice and HW Book
pp. 115117pp. 118120
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 170
Outcomes with Achievement IndicatorsUnit 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]
(Contd)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Find the difference of the fractions:
3
1
9
4
Students should think9
4is a little bit less than a half and
3
1is
a little less than a half. The difference between them shouldtherefore 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
1is reasonable based on their estimate of something a little
bit more than 0.
7N5.7 Model subtraction
of positive fractions, usingconcrete representations,
and record symbolically.
(continued)
7N5.8 Determine thedifference of two given
positive fractions with likedenominators.
(continued)
7N5.9 Determine the
difference of two given
positive fractions withunlike denominators.
(continued)
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 171
Outcomes with Achievement IndicatorsUnit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies Resources/Notes
Math Makes Sense 7
Lesson 5.4
Lesson 5.5
(continued)
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 172
Outcomes with Achievement IndicatorsUnit 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](Contd)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Now that the models for addition and subtraction have beenstudied 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 numbersusing fraction circles, number lines and fraction strips. Lesson
5.7 also introduces Cuisenaire rods as a model for subtractingmixed 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 mayapproach the problem in different ways. They may choose to
keep the mixed fraction form or, they may change the mixedfractions to improper fractions.
For addition:
Mixed Fraction Form Improper Fraction Form
18
13
18112
18
192
18
151
18
41
3
3
6
51
2
2
9
21
6
51
9
21
=
=
=
+=
+=
+
and
18
13
...54,36,181855
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 twomixed numbers with like
denominators.
7N5.10 Model additionand subtraction of mixed
numbers with likedenominators, using
concrete representations,and record symbolically.
7N5.12 Model additionand subtraction of mixed
numbers with unlikedenominators, using
concrete representations,
and record symbolically.
7N5.13 Determine the
sum and difference of twomixed numbers with
unlike denominators.
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 173
Outcomes with Achievement IndicatorsUnit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Interview
Consider the following two problems:3
14
and3
110
Without calculating, explain how you could determine which
answer would be greater.
Journal
Describe at least two ways you can calculate1 5
22 6 .
Resources/Notes
An introduction to
Cuisenaire rods and theiruse in the study of
fractions can be found athttp://teachertech.rice.ed
u/Participants/silha/Lessons/cuisen2.html
Math Makes Sense 7
Lesson 5.6
Lesson 5.7
Unit 5: Operations withFractions
TR: ProGuide, pp. 2529& pp. 3034
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 28CD-ROM Unit 5 Masters
ST: pp. 199203
ST: pp. 204208
Practice and HW Bookpp. 121122
pp. 123124
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 174
Outcomes with Achievement IndicatorsUnit 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](Contd)
Achievement Indicators
(All Contd)
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 challengedby 1214 and therefore mustthink 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
2135
2154
7
7
3
5
3
3
7
18
3
5
7
18
3
21
7
42
=
=
=
=
7N5.11 Determine thesum or difference of twomixed numbers with like
denominators.
7N5.10 Model addition
and subtraction of mixednumbers with like
denominators, usingconcrete representations,
and record symbolically.
7N5.12 Model additionand subtraction of mixed
numbers with unlikedenominators, using
concrete representations,and record symbolically.
7N5.13 Determine thesum and difference of two
mixed numbers withunlike denominators.
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 175
Outcomes with Achievement IndicatorsUnit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies Resources/Notes
An introduction to
Cuisenaire rods and theiruse in the study of
fractions can be found athttp://teachertech.rice.ed
u/Participants/silha/Lessons/cuisen2.html
Math Makes Sense 7
Lesson 5.6
Lesson 5.7
(continued)
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 176
Outcomes with Achievement IndicatorsUnit 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]
(Contd)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Throughout the sections on adding and subtracting fractions, itis necessary for students to simplify their answers. Simplified
answers may be proper fractions, improper fractions or mixednumbers in simplest form depending on the context of the
problem.
Example: Kyra is making cookies. She has1
24
bags of
chocolate chips. She adds2
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 add11
12bags of butterscotch chips to
the dough as well. How many bags of chips does Kyra use intotal to bake her cookies?
For part a), students should think 1
24
bags is a little more
than two bags. Kyra then uses2
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:
127
12
20
12
27
4
4
3
5
3
3
4
9
3
5
4
9
3
21
4
12
=
=
=
=
Kyra has7
12of a bag of
chocolate chips left.
Students must reflectupon their answer to
determine if it isreasonable.
In this case, seven
twelfths is very close to ahalf.
(This elaboration is continued on the next two page spread)
7N5.14 Simplify the
solution to a givenproblem involving the
sum or difference of twopositive fractions or
mixed numbers.
7N5.15 Solve a given
problem involving theaddition or subtraction of
positive fractions ormixed numbers, and
determine if the solution isreasonable.
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 177
Outcomes with Achievement IndicatorsUnit 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 36measures long he plays for
14
2measures, rests for
38
8
measures, plays for another 16 measures, rests for 124
measures
and plays for the last section. How many measures are in thelast section?
2. This week, Mark practised piano for 132
h, played soccer for
16
4h, and talked on the phone for
14
3h.
A. How many hours did Mark spend practising piano andplaying soccer?
B. Hour many more hours did Mark spend playing soccerthan talking on the phone?
Resources/Notes
Math Makes Sense 7
Lesson 5.6
Lesson 5.7
(continued)
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 178
Outcomes with Achievement IndicatorsUnit 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]
(Contd)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
For part b) students should think1
24
bags is a little more than
two bags and11
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
6112
6
72
2
2
12
142
12
142
12
11
12
32
12
11
3
3
4
12
12
11
4
12
=
=
=
=
=
+=
+=
+
and
Kyra used1
36
bags of chips
in total.
Note that the final answermust be simplified.
Students must reflect upon
their answer to determine ifit is reasonable.
In this case, the answer is
very close to the estimate.
7N5.14 Simplify the
solution to a givenproblem involving the
sum or difference of twopositive fractions or
mixed numbers.
(continued)
7N5.15 Solve a givenproblem involving the
addition or subtraction ofpositive fractions or
mixed numbers, anddetermine if the solution is
reasonable.
(continued)
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 179
Outcomes with Achievement IndicatorsUnit 5
General Outcome: Develop Number Sense
Suggested Assessment Strategies Resources/Notes
Math Makes Sense 7
Lesson 5.6
Lesson 5.7
(continued)
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Strand: Number
Grade 7 Mathematics Curriculum Outcomes 180
Outcomes with Achievement Indicators