# MathTR-Gr7-Chap03

```MathTR-Gr7-Chap03 7/22/04 5:25 PM Page 82
Fraction Operations
Key Words
equivalent fractions
common denominator
multiple
proper fraction
improper fraction
mixed number
numerator
denominator
Curriculum Expectations
MAJOR EXPECTATIONS
Number Sense and Numeration
7m1, 7m11, 7m18, 7m19, 7m23, 7m26
CONTRIBUTING EXPECTATIONS
Number Sense and Numeration
7m5, 7m6, 7m7, 7m16, 7m24, 7m25
Geometry and Spatial Sense
7m47
Chapter Problem
A Chapter Problem is introduced in the Chapter Opener. Having students
discuss their understanding of how to answer the Chapter Problem will
You may wish to have students complete the Chapter Problem Revisits that
occur throughout the chapter. These Mini-Chapter problems are particularly
useful for special education students and those working at Level 1 or 2 because
these revisits will assist students in doing the Chapter Problem Wrap-Up on
page 111.
Alternatively, you may wish to assign only the Chapter Problem Wrap-Up
when students have completed Chapter 3. The Chapter Problem Wrap-Up
is a summative assessment.
82
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Planning Chart
Section
Suggested Timing
Core Questions and
Typical Assignment
Teacher’s Resource
Blackline Masters
Chapter Opener
• 10 min (optional)
Assessment
Tools
Interventions
Materials and
Technology Tools
Formative
Assessment:
Chapter Problem
• 40 min
tool on BLM GR3A to
assess which questions
individual students need
to complete
BLM GR3A Letter to Parents
BLM GR3B Create a Fraction
Mind Map
BLM GR3C Chapter 3
Diagnostic Checklist
BLM GR3D Investigate Fractions
Diagnostic
Assessment:
BLM GR3A
BLM GR3B
BLM 3A Writing
Fractions
BLM 3B Comparing
and Ordering
Fractions
BLM 3C Multiples
BLM 3D Equivalent
Fractions
•
•
•
•
•
•
Using Manipulatives
• 40 min
Core Questions: 1, 11, 15, 16
Typical Assignment: 1–3, 5,
7, 9–11, 13, 14, 16 [17–19
Levels 3 and 4]
Master 5 Tangram
Master 8 Grid Paper
BLM 3.1A Pattern Block
Worksheet
BLM 3.1B Add Fractions Try This!
Checkbric
BLM 3.1C Extra Practice
Formative
Assessment:
Chapter Problem,
#10, Try This!, #16
BLM 3.1B
BLM 3E Chapter
Problem Revisit
BLM 3F Section
3.1 Try This!
• pattern blocks
• pencil crayons
3.2 Subtract Fractions
Using Manipulatives
• 40 min
Core Questions: 1–10, 11,
12, 15
Typical Assignment: 1–3, 5,
7, 8, 10, 12–15 [16, 17 for
Level 4]
Master 5 Tangram
BLM 3.1A Pattern Block
Worksheet
BLM 3.2A Subtract Fractions
Try This! Checkbric
BLM 3.2B Extra Practice
Formative
Assessment:
Chapter Problem,
#14, Try This!, #15
BLM 3.2A
BLM 3G Chapter
Problem Revisit
BLM 3H Section
3.2 Try This!
• pattern blocks
• pencil crayons
3.3 Find Common
Denominators
• 40 min
Core Questions for Levels
1 and 2: 1–9, 12, 14
Typical Assignment: 1–3,
5, 8–11, 13, 14 [15 for
Level 4]
BLM 3.3A Common Denominators
Try This! Checkbric
BLM 3.3B Extra Practice
Formative
Assessment:
Try This!, #14
BLM 3.3A
BLM 3I Section 3.3
3.3 Try This!
• paper
• pencil crayons
Fractions Try This! Checkbric
BLM 3.4B Extra Practice
Formative
Assessment:
Chapter Problem,
#12, Try This!, #20
BLM 3.4A
BLM 3.5A More Fraction
Problems Try This! Checkbric
BLM 3.5B Extra Practice
Formative
Assessment:
Try This!, #18
BLM 3.5A
Fractions Using a
Common Denominator
• 40 min
3.5 More Fraction
Problems
• 40 min
Core Questions: 1–9, 11,
13, 14
Typical Assignment: 1–11,
13–17 (12, 18, 19 for
Level 4)
Chapter 3 Review
• 40 min
pattern blocks
fraction circles
base ten blocks
tangrams
fraction strips
cuisenaire rods
• pattern blocks
BLM 3J Section 3.5
Try This!
BLM 3K Chapter 3
Wordsearch
Chapter 3
Practice Test
• 40 min
BLM 3.6A Chapter 3 Test
BLM 3.6B Chapter 3 Test
Assessment
Summative
Assessment:
BLM 3.6A
BLM 3.6B
Chapter Problem
BLM 3.6C Chapter 3
Problem Wrap-Up Rubric
BLM 3.6D Chapter 3 Mark
Summary
Summative
Assessment:
BLM 3.6C
BLM 3L Chapter
Problem Wrap-Up
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Suggested Timing
40 min
Materials
pattern blocks
fraction circles
base ten blocks
tangrams
fraction strips
cuisenaire rods
Related Resources
BLM GR3A Letter to Parents
BLM GR3B Create a Fraction
Mind Map
BLM GR3C Chapter 3
Diagnostic Checklist
BLM GR3D Investigate Fractions
• Have students add the terms proper fraction, improper
fraction, mixed number, numerator, denominator, multiples,
and equivalent fractions, with definitions and examples, to
their personal math dictionaries.
Diagnostic Assessment
Prior to starting Chapter 3, explain that the next topic is about
fractions. Allow students to do a think-pair-share about fractions.
Discuss with students when they have used fractions in
everyday life and what they know about fractions. You may
wish to brainstorm and develop a mind map that includes
references to fractions. Encourage students to talk about
what they know. Try to elicit ideas from all class members.
Method 1: Challenge students to show how much they know
about fractions. Hand out BLM GR3B Create a Fraction Mind
Map. Ask students to complete the statement and then use
the boxes to create a mind map about the fraction 134. While
students are working, circulate and identify which skills students
already have and which they need to review. Students can
colour the areas in which they are comfortable. The colour will
initiating conversations to confirm possible assessments.
Use BLM GR3C Chapter 3 Diagnostic Checklist to assess
whether students are ready to start the work on fractions.
If you are using Method 1, note the amount and quality of
information students provide about writing fractions, comparing
and ordering fractions, multiples, and equivalent fractions. Use
this assessment to decide which parts of the Get Ready section
each student needs to do.
Method 2: Have students discuss and then develop a journal
to explain what they know about the topic and how they use
84
fractions in everyday life. Use BLM GR3C Chapter 3 Diagnostic
Checklist to identify what you are looking for in each response.
Reinforce the Concepts: Students who have difficulty with D, K,
and L do not have a mind picture of what 134 looks like, nor
have they made a connection between fractions and their own
lives. Have these students work with pattern blocks and other
concrete materials to help them develop a mind picture of
fractions with simple denominators. BLM GR3D Investigate
Fractions may assist them in this area.
Teaching Suggestions
Introduction
Students need to have experience with the topics
Writing Fractions, Comparing and Ordering
Fractions, Multiples, and Writing Equivalent
Fractions in order to understand and complete
this chapter. For students having difficulty with
understanding fractions, use a concrete approach
and continually reinforce with visuals such as number
lines and diagrams. Encourage students to work in
pairs or to discuss difficulties as a class.
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Accommodations
Language
Colour-code math terms by strand (Number sense and
Numeration, Measurement, etc.), and post on a math word
wall (e.g., measurement terms might be colour-coded green).
Visual/Perceptual/Spatial/Motor
Students may find it helpful to use rectangular grids to compare
and order fractions or to find equivalent fractions and fractions
in lowest terms.
ESL
Explain to students that, in the English language, new words can
be created by adding a different suffix, or ending, to a root
word. For example, this chapter is called Fraction Operations.
Explain that to “operate” means to do a task. An “operator”
is the person who does the task (suffix -or), and an “operation”
Common Errors
• Students may have trouble identifying the denominator
from diagrams.
Rx Have students verbalize the number of parts in each whole,
and remind them that the total number of parts in each
whole is the denominator.
• Students may have trouble comparing fractions with different
denominators.
Rx Remind students that only equal items can be compared,
thus the total number of parts (denominators) must be equal.
Have students list several multiples of each denominator
and circle the ones that are common.
3
1
1. a) ; 1
2
2
7
3
b) ; 1
4
4
• Students may include the number as the first multiple.
8
2
c) ; 2
3
3
2. Diagrams may vary.
a)
b)
Rx Remind students that every multiple is the product of the
number and a natural number. When listing multiples,
the number 1.
c)
d)
• Students may forget to multiply or divide both the numerator
and denominator by the same number.
e)
Rx Encourage students to represent the fractions visually
by drawing.
f)
3. Diagrams may vary.
1
1
a) b) 4
3
1
1
1
–
–
–
8
4
4
Interventions
2
c) 3
5
–
8
1
–
3
BLM 3A Writing Fractions, BLM 3B Comparing and Ordering
Fractions, BLM 3C Multiples, and BLM 3D Equivalent Fractions
provide extra practice for students who need it.
2
–
3
4.
0
1
–
2
3
–
4
1
5. a) 2, 4, 6, 8, 10
b) 4, 8, 12, 16, 20
c) 5, 10, 15, 20, 25
3
–
2
3
1–
4
2
3
6. 15
4 1
7. a) ; 12 3
2 1
b) ; 8 4
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Using Manipulatives
3.1
Suggested Timing
40 min
Materials
pattern blocks
pencil crayons
Related Resources
Master 5 Tangram
Master 8 Grid Paper
BLM 3.1A Pattern Block Worksheet
BLM 3.1B Add Fractions Try This! Checkbric
BLM 3.1C Extra Practice
Specific Expectations
7m1, 7m7, 7m18, 7m24, 7m25, 7m26, 7m47
The Get Ready sections Writing Fractions,
Comparing and Ordering Fractions, and Equivalent
Fractions provide the needed skills for this section.
You may wish to have students complete these
sections before starting this section.
1. Draw a diagram to represent each fraction.
1
a) 2
2
b) 3
3
c) 8
2. Draw a diagram representing an equivalent
fraction for each fraction in question 1.
3. Calculate the area.
8 mm
4. Explain how these triangles are related.
X
86
Mental Math
6. Estimate.
7. Estimate.
34
46
53
62
78
39
242
177
746
10. 446 214
A
B Z
know it is isosceles.
8. Estimate.
14 mm
C
5. Draw an isosceles triangle. Show how you
Y
MHR • Mathematics 7: Making Connections Teacher’s Resource
9.
\$86.49
25.35
MathTR-Gr7-Chap03 7/22/04 5:26 PM Page 87
Near Compatible Estimation: When estimating,
have students consider what pairs of numbers
add to a multiple of 10. For example, estimate
76 44 19 26 53.
Either introduce this idea before students do
the Warm-Up or suggest it afterwards as a
possible way of estimating the answers for
questions 6–8.
Discover the Math Answers (pages 86–87)
Teaching Suggestions
Introduction
As a class, discuss the section opener. Provide
pairs of students with pattern blocks and have
them determine some of the relationships among
the pattern block pieces.
Mental Math—Do the following imaging exercise
with students. Read the clues. Allow sufficient time
for students to visualize each clue. Then, have students
draw a picture of each clue.
• Think of a circle.
• Think of one third of that circle.
• Think of another third.
• Draw what is left.
1
1
1
2. 2 trapezoids: 1; 3 rhombuses: 2
2
3
1
1
1
1
1
1
1; 6 triangles: 3
3
6
6
6
6
1
1
1
1
1; 1 trapezoid, 3 triangles: 6
6
2
6
1
1
1; 1 trapezoid, 1 rhombus, 1 triangle:
6
6
1
1
1
1
1; 1 rhombus, 4 triangles: 2
3
6
3
1
1
1
1
1; 2 rhombuses, 2 triangles:
6
6
6
6
2
1
1
1
3
6
6
3. Reflect Concrete materials and diagrams make
the answer visual. I can see what fractions make
a whole without always naming the fraction.
Pattern blocks represent part(s) of a whole.
ESL
Discuss the terms “manipulatives” and “concrete materials.”
Ask students to name some examples of these materials. How
would students refer to manipulatives or concrete materials in
their native language?
Have students name each shape and write the value of the
fractions that make it. For example, if a hexagon is 1, then 6
triangles or
Discover the Math
Have students work through the investigation. Discuss
any questions as a class.
1
6
1
1
6
6
1
6
1
6
1
6
1. Encourage students to
see how they can mix fractions. For example, if a hexagon is 1,
then 1 trapezoid and 3 triangles or 12 1
6
1
6
1
6
1. Also,
have them consider what might happen if a rhombus or
trapezoid equals 1.
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Accommodations
Visual/Perceptual/Spatial/Motor
Have students place pattern block pieces on top of a hexagon
to see the spacing to make one whole.
Ongoing Assessment
Communicate the Ideas
• Check that students understand the concept of equivalent
fractions. You may wish to have students explain the Key
Ideas in their own words.
• Ask students to explain why each fraction need not be a
part of equal size.
Common Errors
• Students may not represent the fractions as having the same
number of parts.
Rx Have students place the equal pattern block pieces on top
of the shape to identify equivalence.
Accommodations
Language/ Visual/ Perceptual/ Spatial/ Motor
• Display the Key Ideas on chart paper and discuss as a class.
• Provide Master 5 Tangram to help students with questions
13 and 14.
Interventions
• Have students use BLM 3.1A Pattern Block Worksheet to
visualize the addition for question 12.
• Students who need extra practice can use BLM 3.1E
Extra Practice and Mathematics 7 Workbook, pages 23–25.
• Use BLM 3E Chapter Problem Revisit for students who need
assistance with question 10.
• Use BLM 3F Section 3.1 Try This! for those students who are
having difficulty with question 16.
Method 1: Have students use pattern blocks to create
a hexagon. Using manipulatives allows students to
see the results in a concrete fashion.
Method 2: Have students use BLM 3.1A Pattern
Block Worksheet to identify the parts of each whole
and to determine which shapes make one whole.
Students can colour the shapes they are showing and
write the related addition statement below the shapes.
Method 3: Use BLM 3.1A Pattern Block Worksheet
as a transparency. Colour the identified sections in
the appropriate colour (green triangle; blue rhombus; red trapezoid). As a class, identify
which shapes would complete the cartoons. Write
the related addition statement below each shape.
The Example illustrates how to add fractions using
concrete materials and diagrams.
Have students try both methods to solve the
problem.
Key Ideas
Ongoing Assessment
Try This!
Use BLM 3.1B Add Fractions Try This! Checkbric for formative
assessment of the material covered in this section.
88
Have students read and review the Key Ideas.
Have students answer and discuss the Communicate
the Ideas questions.
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Communicate the Ideas Answers (page 87)
1
–
6
1
–
6
1
–
6
1
–
6
3
–
6
1
–
6
1
–
6
ⴝ
ⴝ
1
–
2
3. Answers will vary depending on method is used.
(pages 88–89)
1
1
4. a) b) 2
3
5
1
5. a) b) 6
2
4 1
1
1
1
4
6. a) ; 6 6
6
6
6
6
1 1
1
1
7. a) ; b)
2 3
6
2
2 1
1
2
c) ; d)
3 2
6
3
2
2
1
8. a) b) or c)
3
6
3
3
6
9. a) or 1
b) or 1
3
6
Communicate the Ideas
Suggest that students consider the use of concrete
materials in their answer to question 3. Use the
2 1
1
2
b) ; 3 3
3
3
5 1
1
5
; 6 2
3
6
1
1
1
1; 1
2
3
6
5
6
5
c) 6
2
3
1 (whole puzzle)
5
5
1
1
2
2
11. a) 1 1
b) 1 1
3
3
3
3
1
1
c) 1 1 2
2
2
2
1
2
1
2
1
12. a) b) c) d) 1
3
3
3
3
3
3
1
1
1
1
13. a) 1
b) 1
2
2
2
2
1
1
1
c) 1
2
4
4
1
1
1
14. 4
4
2
15. a) &amp; b) Answers may vary.
Question Planning Chart
Core questions for Levels 1 &amp; 2: 1–11, 14, 15, 16
Typical assignment: 3, 5, 7, 9–11, 13, 14, 16
(17–19 for Level 4 and some Level 3s)
17. Diagrams may vary.
1
–
3
1
–
3
For question 15, encourage students to use
links to a web site where they can build pattern
block diagrams on-screen.
1
–
6
1
1
1
5
– + – + – = –
3
3
6
6
5
b) 1
6
18. a) Diagrams may vary.
1
–
2
1
–
2
1
–
3
1
–
6
1
–
3
5
19. 1
6
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Subtract Fractions
Using Manipulatives
3.2
Suggested Timing
40 min
Materials
pattern blocks
pencil crayons
Related Resources
Master 5 Tangram
BLM 3.1A Pattern Block Worksheet
BLM 3.2A Subtract Fractions Try This! Checkbric
BLM 3.2B Extra Practice
Specific Expectations
7m5, 7m6, 7m7, 7m16, 7m18, 7m24, 7m25,
7m26, 7m47
1.
Write the fraction represented in this diagram.
2.
Draw a diagram to represent an equivalent
fraction for question 1.
3.
Add using concrete materials or diagrams.
1
1
3
6
a) 90
Mental Math
5.
Classify the triangle in two
ways. Give reasons for your
classifications.
2450
3252
743
8. Estimate.
9. \$875.86
10. 8.6 6.2
1
2
2
3
Draw a parallelogram. Explain
how to find the area.
7. Estimate.
193
346
208
892
310
240
b) 4.
6. Estimate.
A
60&deg;
60&deg;
B
60&deg;
C
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Discover the Math Answers (page 90)
1
1
1
1
1
1
1. 3 ways: 1 ; 1 ; 1 ;
2
6
6
6
6
3
fraction represented in the question. Physically
removing the blocks being subtracted shows
what is remaining.
Accommodations
Language/Visual/Perceptual/
Spatial/Motor
how to take away 12. Encourage
students to make as many fractions
greater than 12 but less than 1 as
they can using pattern blocks or
BLM 3.1A Pattern Block Worksheet.
Then, ask them to remove 12 by
circling or highlighting a trapezoid
or its equivalent.
ESL
Discuss the similarities and
differences among the terms
“illustration,” “diagram,”
“sketch,” and “drawing.”
Teaching Suggestions
Strategies
Introduction
In this section, students learn how to subtract fractions using manipulatives.
Discuss the section opener as a class. Provide pairs of students with pattern
blocks. Have students determine what fraction of the hexagon pictured is
and is not covered.
This investigation uses
modelling to have students
subtract. Students could
share and explain how they
are using this strategy.
Invite students to try each
other’s techniques.
Discover the Math
The purpose of this investigation is to provide students with practice in
subtracting fractions using manipulatives. Prior to doing this activity, allow
students time to freely explore the pattern blocks and familiarize themselves
with the fraction relationships.
1
Method 1: Have students use pattern blocks to subtract . Ideally, each
2
student should have a set of manipulatives. Remind students that a trapezoid
represents 12, and to take it away, the first fraction needs to be greater than
1 trapezoid. Students may find it useful to use BLM 3.1A Pattern Block
Worksheet for recording purposes.
Common Errors
• Students may not represent
the fractions with parts of
equal size.
Rx Encourage students to use
manipulatives to demonstrate
the amount that is being
removed. This will help
them see that they can only
subtract pattern blocks that
are available.
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Journal
Ask students to reflect on their preferred method for
subtracting fractions (e.g., using manipulatives, using diagrams,
or using subtraction sentences). Which method do they like
best, and why?
Ongoing Assessment
• Are students able to use pattern blocks to represent
subtracting fractions?
• Do students use parts of equal size—and use them correctly?
• Do they remove the blocks represented by the fraction that
is being subtracted?
• Do they correctly identify the fraction that remains?
Communicate the Ideas
Listen as students model verbal communication. Can they
explain a variety of ways to represent a fraction? Can they
show visually how to subtract fractions?
Accommodations
Language/Visual/Perceptual/Spatial/Motor
For question 12 in Check Your Understanding, allow students
to give an oral explanation or to use visuals. Encourage
students to use tangrams to manipulate the values. Provide
Master 5 Tangram.
Interventions
Students who have difficulty with subtraction may wish to visit
the web site listed with question 15 on page 89. Have them
“virtually” explore subtraction.
Ongoing Assessment
Try This!
BLM 3.2A Subtract Fractions Try This! Checkbric provides
formative assessment for student work in this section.
Interventions
• Use BLM 3G Chapter Problem Revisit to assist students who
are having difficulty with question 14.
• You may wish to use BLM 3H Section 3.2 Try This! for those
students who need extra scaffolding for question 15.
• BLM 3.2B Extra Practice and Mathematics 7 Workbook,
pages 26–28, provides extra basic practice.
Questions 16 and 17 ask students to think of two hexagons as
one whole. After students complete the questions, ask them
to describe the strategies they used in solving the problems.
Method 3: Use BLM 3.1A Pattern Block Worksheet
as a transparency. Colour the identified sections in
the appropriate colour (green triangles; blue rhombuses; red trapezoid). As a class, identify
when 12 could be taken away. Record the responses
below each visual as a subtraction sentence. Have
students use BLM 3.1A Pattern Block Worksheet
to record the findings.
The Example illustrates two methods for subtracting
fractions: using concrete materials and using diagrams.
Discuss with students what is happening in each
subtraction problem and how the method shown
divides the shape into sixths. Many students will
find it useful to show 12 as sixths in order to subtract.
Discuss how to get that equivalent fraction.
Key Ideas
Method 2: Students can use BLM 3.1A Pattern
Block Worksheet to identify parts that are greater
than 12. Have students highlight the 12, as represented
by a trapezoid, that is being removed. Then, have
them use a subtraction sentence to record what
they have done.
Have students read and review the Key Ideas section.
Remind students that fractions must be represented
by equal-sized parts.
Communicate the Ideas
Have students answer and discuss the
Communicate the Ideas questions.
92
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7. Diagrams may vary.
a)
b)
c)
1
1
8. a) 2
3
d)
2
1
b) 3
6
9. Diagrams may vary.
a)
b)
10. Diagrams may vary.
a)
b)
c)
d)
11. Diagrams may vary.
a)
b)
Question Planning Chart
Core questions for Levels 1 and 2: 1–10, 11, 12, 15
Typical assignment: 1–3, 5, 7, 10, 12–15 (16, 17
for Level 4 and some Level 3s)
Communicate the Ideas Answers (page 91)
1.
–2
3
–2
3
–2
3
2.
–2 ⴚ –1
3
6
1
1
12. 2
4
1
13. a) 1 2
1
b) 1 4
3
14. Answers may vary. Season puzzle: 1 (number
4
5
of pieces left); Colour puzzle:1 8
1
16. a) 4
(pages 92–93)
17. a) Diagrams may vary.
1
3. a) 1 3
1
1
b) 3
6
c)
1
1
5. a) 2
6
1
1
c) 2
3
6. Diagrams may vary.
a)
b)
1
3
c) 1 4
4
5
2
c) 6
6
4. Diagrams may vary.
a)
b)
2
1
b) 3
6
1
b) 1 4
c)
b) In this example, 1 is represented by a
2
whole hexagon and 13 is represented by
two blue rhombi.
c) The numerical answer is represented by pattern
blocks twice the size as those that would
represent the answer if 1 hexagon 1 whole.
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3.3
Find Common Denominators
Suggested Timing
40 min
Materials
paper
pencil crayons
Related Resources
BLM 3.3A Common Denominators Try This! Checkbric
BLM 3.3B Extra Practice
Specific Expectations
7m1, 7m7, 7m11, 7m16, 7m24, 7m25
The Get Ready section Multiples provides the needed
skills for this section. You may wish to have students
review this material before starting this section.
Balancing Strategy: This is similar to the near
compatible estimation strategy discussed in section
3.1. It involves changing the numbers to make the
addition easier to compute. For example, you might
to the other. The final sum is maintained. Once
students have practised this strategy, encourage
them to look for places to use it.
For example, have them add 68 57.
Think: 68 2 70
70 50 120
94
Now, take that 2 away from the 7.
120 5 125
When discussing these strategies with students,
have them consider what other mental strategy
they might use. For example, another way to add
68 57 is to use the front-end strategy discussed
in section 1.3.
60 50 110
8 7 15
110 15 125
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1. Add using concrete materials or diagrams.
1
3
1
2
1
6
a) 2
3
b) 2. Subtract using concrete materials
or diagrams.
1
2
2
3
a) 1 1
6
b) 3. Find a common denominator for:
1
2
1
3
1
2
a) and 1
4
b) and 4. What is the perimeter?
4.5 cm
P
Q
S
R
2 cm
Mental Math
Teaching Suggestions
6. 49 55
7. 68 37
9. Estimate.
10. Estimate.
\$ 245.60
53.22
198.19
8429.6
3231.8
8. 159 33
Introduction
As a class, discuss the section opener visual. Have students discuss in pairs
how to determine the fraction of the rectangle that is covered.
Mental Math—Have students order the following fractions from least to
greatest using benchmarks and number sense, not common denominators:
4 3 2
1
, , . Students can use zero, , and 1 as benchmarks.
5 6 8
2
Discover the Math
This investigation introduces students to a concrete method of finding a
common denominator.
Method 1: Have students do the investigation as shown in the text.
Encourage them to see how the two ways of paper folding result in a
common denominator.
Method 2: Have students work through the investigation by drawing
diagrams representing the fractions. Students then can model the fractions
by shading in the appropriate sections. Have them practise the technique
with some other simple fractions.
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Discover the Math Answers (page 94)
3. a) 12
4
6. 12
3
b) 3; 12
7. Reflect Folding the paper to represent the
denominators of each fraction will show a multiple
that both denominators have in common.
Literacy Connections
Have students practice using
“…” in their lists of multiples.
ESL
In their own words students
write an explanation for finding
a common denominator.
Ongoing Assessment
• Do students understand
what is meant by a common
denominator?
• Can they find a common
denominator using one or
more methods?
Strategies
what pattern they see in
the following two series of
numbers: 2, 4, 6, 8 and 3, 6,
9, 12. Have them predict the
next common multiple for
each series.
Method 3: Have students identify the first five multiples of the numbers 4
and 3. Have students circle the common multiples.
The Example illustrates three methods for finding a common denominator.
Reinforce that all of the methods work and are useful.
Key Ideas
Have students read and review the Key Ideas section. Ensure that students
understand how to use each solution strategy.
Communicate the Ideas
Have students answer and discuss the Communicate the Ideas questions.
Question Planning Chart
Ongoing Assessment
Core questions for Levels 1 and 2: 1–9, 12, 14
Typical assignment: 1–3, 5, 8–11, 13, 14 (15 for Level 4 and some Level 3s)
Communicate the Ideas
Check that students realize they
can find a common denominator
in several ways, including paper
folding, using a diagram, and
listing multiples.
96
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Ongoing Assessment
Try This!
• BLM 3.3A Common Denominators Try This! Checkbric provides
formative assessment for student work in this section.
• Level 4 performance can be demonstrated by showing
multiple ways to determine which fraction is greater.
Journal
Have students complete the following statement in
their journals: The best way to find a common denominator
is... because...
Accommodations
Language/Visual/Perceptual/Spatial/Motor
For questions 9 and 12, allow students to demonstrate
their understanding through paper folding or diagrams. For
question 12, allow students to explain their understanding
and process orally.
Interventions
• Use BLM 3I Section 3.3 Try This! for those students who are
having difficulty with question 14.
• Students having difficulty finding more than one common
denominator may benefit from listing the multiples of the
denominators until they identify two common multiples.
• BLM 3.3B Extra Practice and Mathematics 7 Workbook,
Encourage students to use paper folding and/or diagrams for at least questions
4, 7, and 8, even if they are able to find a common denominator mentally.
Questions 11 and 12 are good application questions. Question 14 allows
students to investigate fractions through comparing and to use their skills
at finding common denominators.
Communicate the Ideas Answers (page 96)
1. No. She could draw a diagram or use multiples
to find a common denominator of 40.
2. 24 and 42 are both multiples of 3 and 6. Other
possible common denominators include: 6, 12,
18, 30, and 36.
3. Any multiple of 6 could be used; 6 is best because
it is the smallest common denominator.
4. Answers may vary; lowest common denominators
b) 8
c) 12
d) 10
given a) 12
5. Answers may vary; lowest common denominators
b) 21
c) 20
d) 24
given a) 15
4
2 3
1
a) ; 6
3 6
2
5
1 6
2
c) ; 15
3 15
5
18
3 20
5
b) ; 24
4 24
6
8
2 6
1
d) ; 12
3 12
2
a) 6
b) 24
c) 15
d) 12
a) 4
b) 10
a) 15, 30
b) 12, 24
a) 6, 12, 18
b) 8, 16, 24
11. 12, 24, 36
12. a) 12, 18
b) Answers may vary. For example, you could
find common multiples of 2 and 6.
6 5
5 2
a) 10; , b) 8; , 10 10
8 8
14. Methods may vary.
5
2
a) b) 6
5
15. Answers may vary. For example, you could find
common multiples of 2, 3, and 4.
a) 12 or 24
b) 60
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Using a Common Denominator
3.4
Suggested Timing
40 min
Materials
pattern blocks
Related Resources
BLM 3.4A Add and Subtract Fractions Try This! Checkbric
BLM 3.4B Extra Practice
Specific Expectations
7m1, 7m6, 7m7, 7m11, 7m16, 7m18, 7m19,
7m24, 7m25, 7m26
Strategies
Discuss how the strategy of model making helps students
visualize a problem and arrive at a solution.
Interventions
Allow students to use fraction circles or diagrams to model
questions concretely.
1. State two common denominators for each
pair of fractions.
1
4
1
6
a) and 1
2
6. 159 33
7
8
b) and 7. \$2.49 \$0.24
2. Write 3 equivalent fractions for each
fraction.
1
3
a) Mental Math
8. Estimate.
2.4 6.3 7.8 9.2
3
4
b) 9. 100 1
3. Use diagrams to subtract: 1 6
10. 200 145
4. Identify the numerator and denominator
3
8
of .
1
4
1
2
5. Use diagrams to add: 98
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Literacy Connections
• Adding and Subtracting Fractions: Have students
explain why it is not necessary to subtract the
denominator.
• Remembering Common Denominators: Have students
brainstorm other ways that can help them to remember
to use common denominators.
Accommodations
Language/Visual/Perceptual/Spatial/Motor
Have students make a diagram of the section-opener problem
and write a subtraction sentence about it.
Teaching Suggestions
Introduction
As a class, discuss the section opener. Draw students’ attention to the photograph. Discuss how large the pizza was, what fraction was eaten, and what
is left.
Discover the Math
The purpose of this investigation is to provide students with different
methods for adding and subtracting fractions.
• Example 1 illustrates the procedure for adding fractions using pattern
blocks. This method reinforces the concrete understanding of adding
fractions. The Example also offers an algorithm strategy of solving by
using a common denominator. The solution is semiconcrete because the
common denominator is determined using paper folding.
L
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Strategies
Encourage students to
continue making models to
until they are confident
Ongoing Assessment
Communicate the Ideas
with different denominators?
Common Errors
• Students may try to add and
subtract fractions that have
different denominators.
• Example 2 shows the comparable procedure for subtracting fractions.
Two methods are depicted: Method 1 uses manipulatives; Method 2 uses
multiples. Review the thought balloons with students.
• Example 3 illustrates two strategies for adding the same fraction: using
manipulatives and multiplying by a whole number. Review the thought
balloon with students.
Key Ideas
Have students read and review the Key Ideas section. Ask them to explain
in their own words why a common denominator is needed to add or
subtract fractions.
Communicate the Ideas
Have students answer and discuss their solutions to the Communicate
the Ideas questions.
Rx Use pictures to explain why
fractions that have different
denominators. Encourage
students to draw a diagram
or fold paper to find a
common denominator.
100
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Question Planning Chart
Core questions for Levels 1 and 2: 1–4, 5–10,11–13, 20
Typical assignment: 1–4, 5–6, 8–10,12, 14, 15, 17, 18, 20 (21 for
Level 3 and some Level 2s; 19, 22, 23 for Level 4 and some Level 3s)
Communicate the Ideas Answers (page 101)
1. Diagrams may vary.
Ongoing Assessment
Try This!
• BLM 3.4A Add and Subtract
Fractions Try This! Checkbric
provides formative assessment
for material covered in
this section.
• Level 4 performance will be
evident by the clarity and
completeness of the explanation.
Interventions
• Review with students how to
find common denominators.
• BLM 3.4B Extra Practice and
Mathematics 7 Workbook,
reinforcement.
2. Use 6 as a common denominator because it is the smallest common multiple
3
2
1
of both 2 and 3. 6
6
6
5
3
8
3. Answers may vary. For example, use 6 as a common denominator: 6
6
6
1
1
1
3 3
1
4. ; 6
6
6
6 6
2
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Interventions
Encourage students to identify
a pattern in Question 11 on
page 102. Have them visualize
two halves. They might find it
easier to say, “2 of one half”,
“3 of one third”, etc.
• Questions 19 and 23 will
challenge students to identify
common multiples from three
numbers.
• Question 22 will encourage
students to consider order of
operations with fractions. They
may wish to do the Making
Connections activity on page 103
in connection with this question.
2
3
5
1
2
3
5. a) b) 6
6
6
4
4
4
2
3
5
1
3
4
c) d) 8
8
8
6
6
6
1
1
1
3
6. a) b) c) d) 4
8
3
8
3
4
1
4
7. a) b) or 1
c) 4
3
3
5
4
1
3
3
2
1
a) b) 6
6
6
4
4
4
6
5
1
7
2
5
c) d) 10
10
10
8
8
8
2
5
7
a) 10
10
10
6
1
10
16
c) or 1
15
15
15
15
1
3
1
10. a) 3 or 1
2
2
2
2
11. a) 1 b) They all equal one.
2
3
1
10
13
b) or 1
12
12
12
12
1
3
4
2
d) or 6
6
6
3
1
5
2
b) 5 or 1
3
3
3
c) 1
1
8
12. Answers may vary. Total number of pieces: 8 = or 1;
8
8
1
3
Number of pieces that did not fall out: 3 8
8
3
15
1
4
12
13. a) 5 or 7
b) 3 9 or 4
2
2
2
3
3
102
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Making Connections (page 103)
Review order of operations with students. Have them
challenge each other with fraction questions that include
5
12
5
6
a) b) Journal
Have students answer the following questions in
section? What is the most challenging?
(pages 101–103)
20. Diagrams may vary.
a)
ⴙ
9
16
25
25
14. ; Since is greater than 1, the
24
24
24
24
addition shows that together they shovelled
more than 1 whole driveway. One or both of
their statements are incorrect.
1
–
4
1
–
2
b)
are not the same size.
7
16. a) 12
ⴙ
1
–
6
5
–
6
9 10
9
2
1
5
10 1
1
3
c) or , or . 12 12
12
3
6
6
12 4
2
4
3
21. Diagrams may vary. 1
4
ⴙ
9
2
1
2
1
17. is greater. ,
10
5
2
5
2
2
1
5 9
5
, 3
6
6 10
6
5
2
1
5
3 3
1
18. 1 is greater. 1 , 1 , 8
3
3
8
8 8
3
13
1
19
19. a) or 1
b) 12
12
20
ⴝ
2
–
3
ⴝ
5
b) 12
3
–
4
ⴙ
b)
ⴝ
1
ⴙ
1
–
2
ⴝ
1
–
4
3
1–
4
1
3
25
1
22. a) 1 or 2
3
4
12
2
2
1
11
1
b) 2 or 1
5
2
10
10
2
1
3
19
19
23. ; Since is less than 1, the
5
4
10
20
20
addition shows that together they did not clean
all the windows. They should not be paid the
full amount.
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3.5
More Fraction Problems
Suggested Timing
40 min
Related Resources
BLM 3.5A More Fraction Problems Try This! Checkbric
BLM 3.5B Extra Practice
Specific Expectations
7m1, 7m5, 7m6, 7m7, 7m11, 7m16, 7m18, 7m19,
7m23, 7m24, 7m25, 7m26
Compatible Number Estimation: Students
can use compatible numbers to help them
subtract. For example, when subtracting
750
1014 766, think of the pairing
250
to make 1000.
Think: 766 is 16 more than 750, so the
But, 1014 is 14 greater than 1000, so the
answer will be only 2 different.
1. List four multiples of 5 and 10.
5. Draw this triangle:
• one side measures 5 cm
• one side joins a 25&deg; angle and a 40&deg; angle
2. Rewrite each expression with a common
denominator, and solve.
1
3
1
2
a) 3
8
1
3
b) 3. Write each repeated addition as a
multiplication, and evaluate.
1
1
1
1
4
4
4
4
1
1
1
1
b) 3
3
3
3
1
4
a) Mental Math
6. Estimate. 106 77
8. Estimate. 508 250 9. 5.9 3.6
10.
78.9
45.2
4. Draw a composite shape. Show how to
split it to find the area.
104
7. Estimate. 1000 498
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Literacy Connections
Have students apply the steps outlined in the Literacy
Connections to the question in the Introduction.
Strategies
• Make a Picture or Diagram: Emphasize that a rough
sketch will often work.
• Look for a Pattern: Ask students how looking for a
pattern might help them to solve Example 2.
Common Errors
• Students do not know where to begin in solving a problem.
Rx Encourage students to reference their solution strategy list.
Using a diagram helps make the problem more concrete.
This can point toward a solution strategy.
• Students may use the incorrect mathematical operation.
Rx Encourage students to list words that refer to addition (e.g.,
sum, altogether, total) and subtraction (e.g., difference,
leftover, remaining) to help them identify the necessary
operation.
Ongoing Assessment
Are students able to identify and apply appropriate strategies?
Teaching Suggestions
Introduction
As a class, discuss the section opener. Discuss the photograph of sandwiches
that are remaining. Consider how each sandwich was cut, the number of
remaining pieces, and how to combine them.
Discover the Math
In this investigation, students learn about different strategies that they can
use for solving problems containing fractions.
Method 1: Have students work in pairs to develop a strategy for solving
the sandwich question. Allow students to work for 10 minutes, and then
have them present their solution strategies. Develop a list of the different
strategies used. Discuss the content of the thought balloon.
Method 2: Have students complete each example using manipulatives.
• The concrete method used in Example 1 will reinforce student
understanding of adding fractions in a problem-solving format.
• Example 2 needs a patterning strategy because different items are added.
Discuss the content of the thought balloon, and when, in real life, students
might need to use this technique.
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Communicate the Ideas Answers (page 106)
1. Answers may vary. You could draw a diagram.
1
8
1
2. Strategies may vary. 8 or 1
6
6
3
4. Each whole sandwich is 4 quarters, and there are
1
6
1
6 quarters in total. So, 6 or 1. There
4
4
2
1
are 1 sandwiches.
2
Ongoing Assessment
Communicate the Ideas
• Can students identify the
strategies they use to solve
problems?
• Are students able to incorporate
new strategies as they solve
problems?
Journal
one journal entry from this chapter
that they would like to share. If
you wish, give them time to write
a good copy. Collect and assess
the journal using Assessment
Master 11 Journal Assessment
Rubric. You may wish to respond
personally to student thoughts and
ideas as reflected in the journal.
Key Ideas
Have students read and review the Key Ideas section. Encourage students
to detail their solution to the sandwich problem. Discuss why a different
strategy should be used to check answers.
Communicate the Ideas
Have students answer and discuss their solutions to the Communicate the Ideas
questions. Discuss why it is useful to know a number of different strategies.
Question Planning Chart
Core questions for Levels 1and 2: 1–4, 5–9, 11, 15, 18
Typical assignment: 1–4, 5, 7, 9, 11, 13, 15, 16, 18 (19, 20 for Level 4
and some Level 3s)
Accommodations
Language/Visual/Perceptual/
Spatial/Motor
Question 11 is a good question to
diagram. Once students understand
the visual, they will be able to
approach the problem.
106
Questions 13 and 14 are good communication questions. Questions 15 and
17 are good application questions.
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Ongoing Assessment
Try This!
• BLM 3.5A More Fraction Problems Checkbric provides formative
assessment for student work in this section.
• Level 4 students should be encouraged to solve the same
problem in two ways, to compare their strategies, and
identify the more efficient solution.
Question 19 presents a good challenge. Stronger students will
comment that the diagonals also add to 1. After doing this
question, students may wish to develop their own magic
squares using fractions.
Interventions
• BLM 3J Section 3.5 Try This! provides scaffolding for
question 17.
• BLM 3.5B Extra Practice and Mathematics 7 Workbook,
pages 35–36, provides extra practice for those who need it.
(pages 106–107)
17
1
5. a) or 4
4
4
5
6. a) 12
5
1
7. or 1
4
4
13
5
8. or 1
8
8
7
1
9. or 3
2
2
17
5
b) or 2
6
6
3
b) 8
27
3
c) or 3
8
8
2
c) 5
1
15. a) 5
1
1
b) or 3
7
16. a)
5m
15 m
b) 75 m2
2
17. a) 3
14
1
12
1
10. sandwiches: or 3; oranges: or 1
4
2
8
2
3
1 8
2
1
11. 1, 1 or 1
2
2 6
6
3
c) 40 m
b) 7
c) 15
5
3
1
18. a) top row: ; middle row: or ;
12
12
4
1
bottom row: 12
19. 6
12. 18
5
13. a) blue: , purple:
16
6
3
green: or 16
8
11
b) c)
16
1
14. 5
3
2
1
, white: or ,
16
16
8
13
16
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Suggested Timing
40 min
Related Resources
BLM 3K Wordsearch
Accommodations
Visual/Perceptual/Spatial/Motor
• Encourage students to use diagrams wherever possible.
Pattern blocks or fraction circles will reinforce the concrete
meaning of fractions.
Language/Memory
• Allow students to refer to personal math dictionaries,
index card files, or notes.
Interventions
• Have students complete BLM 3K Wordsearch to review
Key Words.
• Have students identify problems, and then suggest they
review those sections in the chapter.
• Mathematics 7 Workbook, pages 37–38, provides
1
4
1
4
1
2
1. Show why 1.
7
8
Teaching Suggestions
2. Draw a diagram to represent 1 .
1
6
2
3
3. Find a common denominator for and .
1
3
1
3
1
3
1
3
4. Evaluate in two ways. 5. Your lunch has two sandwiches cut into
quarters. After lunch, two pieces are left.
How many sandwiches is this?
Mental Math
6. Estimate. 1024 250
Using the Practice Review
Provide an opportunity for the students to discuss
any questions, consider alternative strategies, and
difficult. After they complete the Chapter Review,
encourage students to make a list of questions that
caused them difficulty and include the related
sections and teaching examples. They can use
this to focus their studying for the final test on
the chapter’s content.
7. Estimate: 1008 650
8. Estimate: 10 089 7486
9.
108
78 655
4 432
10. 1000 282
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Ongoing Assessment
Chapter Review
This is an opportunity for students to assess themselves by
completing selected questions and checking their answers. They
can then revisit any questions with which they had difficulty.
15. Diagrams may vary. Make sure that the fractions
are the same as those shown in question 14.
2
1
16. a) 6
6
4
1
b) 6
2
17. Diagrams may vary. Make sure that the fractions
are the same as those in question 16.
1
18. Diagrams may vary. a) 3
2
1
b) or 6
3
a) 12
b) 20
7. G
a) 18
b) 24
1
1
1
1
1
b) 6
6
6
6
6
1
1
1
1
d) 6
6
6
6
a) 12
b) 10
1. D
5. A
2. F
6. C
1
1
8. a) 3
3
1
1
1
c) 2
6
6
3. H
4. B
9. Diagrams may vary.
5
5
4
2
a) b) c) or 6
6
6
3
1
1
1
23. a) 2
3
6
1
1
1
4
2
b) or 3
6
6
6
3
represent those shown in the question.
correct fractions from question 9.
1
1
1
b) 1
2
3
6
2
12. Diagrams may vary. a) 3
1
1
13. a) 2 b) 5 3
6
1
5
1
14. a) 1 b) 2
6
6
22. 12, 18
24. Diagrams may vary. Make sure that the fractions
10. Diagrams may vary. Check that they show the
11. a)
1
c) 6
5
b) 6
7
25. a) 30
7
1
26. or 2
3
3
1
13
b) or 1
12
12
6
4
1
1
27. a) red: or , blue: or ,
24
24
4
6
9
5
3
white: or , grey: 24
24
8
5
10
b) or 12
24
15
5
c) or 24
8
5
1
11
5
28. or 2 muffins; or 1 apples
2
2
6
6
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Suggested Timing
40 min
Related Resources
BLM 3.6A Chapter 3 Test
BLM 3.6B Chapter 3
Test Assessment
BLM 3.6C Chapter
Problem Wrap-Up Rubric
BLM 3.6D Chapter 3
Mark Summary
Summative Assessment
After students complete the Practice Test, you
may wish to use BLM 3.6A Chapter 3 Test as
a summative assessment. BLM 3.6B Chapter 3
student achievement.
Interventions
BLM 3L Chapter Problem
Wrap-Up provides scaffolding for
the Chapter Problem Wrap-Up
on page 111.
Accommodations
Visual/Perceptual/
Spatial/Motor
• Students can use BLM 3.1A
Pattern Block Worksheet to
complete the Chapter
Problem Wrap-Up.
Language/Memory
• Allow students to refer to
personal math dictionaries,
index card files, or notes.
110
Teaching Suggestions
Using the Practice Test
This practice test can be assigned as an in-class or take-home assignment.
If it is used as an assessment, use the following guidelines to help you
evaluate the students:
• Can students add fractions using manipulatives?
• Can students use pattern blocks to represent subtracting fractions?
• Are students able to find a common denominator?
• Are students able to add and subtract fractions using a common
denominator?
• Do students use different strategies to solve problems containing fractions?
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Chapter 3 Practice Test
1. B
2. D
3. B
4. C
5. A
6. a) Diagrams may vary.
2
1
1
b) 3
2
6
1
4
7. a) 4 5
5
2
10
3
b) 5 or 1
7
7
7
a) 4
b) 15
9. 12, 24
1
1
25
10. a) or 1
b) 24
30
24
3
11. a) 1 10
b) Strategies may vary. For example, you could
3
2
19
14
12. or 1. Eric is correct.
5
3
15
15
13. 16
14. a) Both are correct. Both 12 and 24 are multiples
of 2, 3, and 4.
23
11
b) or 1
12
12
Study Guide
Use the following study guide to direct students
who have difficulty with specific questions to
appropriate examples to review.
Question
1, 2
3
4, 5
6
7, 8, 9
10, 11, 12
13
14
Section(s)
3.1
3.2
3.1, 3.2
3.2
3.3
3.4
3.5
3.4
Refer to
Example
Example
Example, Example
Example
Example
Examples 1 and 2
Examples 1 and 2
Example
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This problem should be accessible to all students.
Less confident students will use materials that are
familiar, such as pattern blocks, whereas more
able students will create their own unique pattern
pieces to fit the conditions given. Some may try to
incorporate interlocking tabs, as are found with
jigsaw puzzles.
1. Introduce the problem.
2. Clarify the assessment criteria by reviewing
BLM 3.6C Chapter 3 Problem Wrap-Up
Rubric with students.
3. Remind individual students that they have
worked on the chapter problem during Chapter
Problem revisits throughout the chapter and
that these will help them.
4. Possibly share with all students the example
from the TR.
5. Allow students time to work on the problem,
either individually or in a group. They should
do separate puzzles and reports.
Summative Assessment
• Use BLM 3.6C Chapter 3 Problem Wrap-Up
Rubric to assess student achievement.
• Use BLM 3.6D Chapter 3 Mark Summary to
summarize student work in this chapter.
Level 3 Sample Response
1. a) • A rectangle divided into 6 equal sections
with 1 section red, 1 section blue, 2 sections
green, and the remaining 2 sections yellow.
• Using pattern blocks, there is.
– 1 yellow hexagon covered by
2 green triangles,
– one triangle labelled red
– one triangle labelled blue,
– 1 blue rhombus beside the triangles
is labelled green.
• Using centimetre cubes or coloured tiles,
there should be 6 pieces: 1 red, 1 blue,
2 green, and 2 yellow.
110a
b) • I know that 13 of the puzzle is green
because 13 is equal to 26, and 2 out of 6
c) •
1
3
of the puzzle is yellow because there
were 2 sections left over. 2 sections out
of 6 total sections is 26 or 13.
2. • 3 sections of the puzzle are red, 3 sections
are blue, 4 sections are green, and the
remaining 2 sections are yellow.
• Since 12 of the puzzle is either red or blue, this
means that 6 of the puzzle pieces are red
or blue.
• The fraction of green sections is double the
fraction of yellow sections. The 2 yellow
sections make up 16 of the puzzle. The 4
green pieces make up 13 of the puzzle. 16 is
half of 13.
• The sum of red and blue sections equals half
the puzzle, since 14 14 12.
• The sum of green and yellow sections equals
half the puzzle, since 13 16 12.
• The total number of blue and red pieces is
equal to the total number of green and
yellow pieces.
Level 3 Notes
• Students create appropriate puzzle pieces.
• Explanations show good understanding of
fractions.
• May have some minor errors.
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What Distinguishes Level 2
At this response level, look for the following:
– Puzzles are designed with slight errors or in the
incorrect proportions
– Few fraction relations provided
– Puzzles are inaccurate
– Explanations are simplistic or vague
What Distinguishes Level 4
At this response level, look for the following:
– Puzzle may represent an equivalence of 6.
For example, a rectangle divided into 12
equal sections with 2 sections red, 2 sections
blue, 4 sections green and the remaining 4
sections yellow.
– A puzzle in equivalent ratios
– Greater depth in explanation
– Creating a puzzle using fractions of their
own choosing
– A variety of fraction relations
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