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Teacher Guide
Unit 3: Geometry
UNIT
3
“The study of geometry provides
students with an opportunity to
connect mathematics to the
world. Teachers should select
activities that involve the
recognition and classification of
shapes and figures and
operations on objects that are
familiar to the students.”
W. George Cathcart et al.
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Geometry
Mathematics Background
What Are the Big Ideas?
• Two-dimensional figures have attributes related to their sides
and angles.
• Lines can be classified as horizontal, vertical, parallel, intersecting,
or perpendicular.
• Quadrilaterals are figures with four sides that can be sorted according
to their side and angle attributes.
• Two figures are similar if they have the same shape, and congruent if
they have the same size and shape.
• Angles can be measured using concrete materials.
• Many objects in our world resemble three-dimensional solids.
FOCUS STRAND
Shape and Space: 3-D Objects
and 2-D Shapes
SUPPORTING STRANDS
Patterns and Relations
Number Concepts
Number Operations
• Pyramids and prisms have faces that are figures and are named for a
particular face called the base.
• A net is a cutout of connected figures that can be folded to make a
model of a solid.
• Models of solids can be built that show only the edges and vertices of
the solid.
How Will the Concepts Develop?
Students investigate the attributes of figures. They measure angles in
non-standard units using concrete materials.
Students use attributes of figures to sort quadrilaterals according to side
length, the number and position of parallel sides, and the number of
right angles.
Students investigate congruent and similar figures. They explore new
figures by making combinations of smaller, congruent figures.
Students explore the relationship among the faces of solids. They match
solids to objects in their environment. Students sort solids according to
attributes, such as the number of faces, vertices, and edges. Students
investigate the volumes obtained by creating larger solids from
smaller ones.
Students apply their knowledge to construct the skeleton of a castle
made of different solids. They design walls in the castle that show
different quadrilaterals.
Why Are These Concepts Important?
Active exploration of geometric properties leads students to reason
effectively about spatial concepts. Geometry tasks help students to
develop and build a foundation that will help them understand deeper
mathematical concepts they will encounter in later grades. As students
investigate geometric properties and relationships, their work is closely
connected with other mathematical strands, such as Measurement,
Number Concepts, and Patterns and Relations.
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Unit 3: Geometry
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Curriculum Overview
Launch
Cluster 1: Exploring Figures
Under Construction
General Outcomes
Specific Outcomes
Lesson 1:
• Students describe, classify,
construct, and relate 3-D objects
and 2-D shapes, using
mathematical vocabulary.
• Students identify and sort specific
quadrilaterals, including squares,
rectangles, parallelograms, and
trapezoids. (SS22)
• Students classify angles in a variety
of orientations according to whether
they are right angle, less than right
angle, or greater than right angle.
(SS21)
• Students recognize, from everyday
experience, and identify:
- point
- line
- parallel lines
- intersecting lines
- perpendicular lines
- vertical lines
- horizontal lines (SS20)
Congruent Figures
General Outcomes
Specific Outcomes
Lesson 8:
• Students describe, classify,
construct, and relate 3-D objects
and 2-D shapes, using
mathematical vocabulary.
• Students design and construct nets
for pyramids and prisms. (SS17)
• Students relate nets to 3-D objects.
(SS18)
• Students compare and contrast:
- pyramids
- prisms
- pyramids and prisms (SS19)
Faces of Solids
Lesson 2:
Exploring Angles
Lesson 3:
Measuring Angles
Lesson 4:
Exploring Sides in Quadrilaterals
Lesson 5:
Exploring Angles in Quadrilaterals
Lesson 6:
Attributes of Quadrilaterals
Lesson 7:
Similar Figures
Technology:
Using a Computer to Explore
Pentominoes
Cluster 2: Exploring Solids
Lesson 8A:
Exploring Nets of Solids
Lesson 9:
Solids in Our World
Lesson 10:
Designing Skeletons
Lesson 11:
Strategies Toolkit
Show What You Know
Unit Problem
Under Construction
Unit 3: Geometry
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Curriculum across the Grades
Grade 3
Grade 4
Grade 5
Students identify and
count faces, vertices, and
edges of 3-D objects.
Students identify and sort
specific quadrilaterals,
including squares,
rectangles,
parallelograms, and
trapezoids.
Students construct,
analyse, and classify
triangles according to the
side measures.
Students identify and
name faces of a 3-D
object with appropriate
2-D names.
Students describe and
name pyramids and
prisms by the shape of
the base.
Students demonstrate that
a rectangular solid has
more than one net.
Students compare and
contrast two 3-D objects.
Students recognize
congruent (identical) 3-D
objects and 2-D shapes.
Students explore,
concretely, the concepts
of perpendicular, parallel,
and intersecting lines on
3-D objects.
Students classify angles in
a variety of orientations
according to whether they
are right angle, less than
right angle, or greater
than right angle.
Students design and
construct nets for
pyramids and prisms.
Students relate nets to
3-D objects.
Students compare and
contrast:
- pyramids
- prisms
- pyramids and prisms
Students recognize, from
everyday experience, and
identify:
- point
- line
- parallel lines
- intersecting lines
- perpendicular lines
- vertical lines
- horizontal lines
Students build, represent,
and describe geometric
objects and shapes.
Students identify and
name polygons according
to the number of sides,
angles, and vertices
(3, 4, 5, 6, or 8).
Students cover a given
2-D shape with tangram
pieces.
Students complete the
drawing of a 3-D object,
on grid paper, given the
front face.
Students determine,
experimentally, the
minimum information
needed to draw a given
2-D shape.
Materials for This Unit
Tracing paper, wax paper, straws, old magazines, and Plasticine are used
in this unit. Identify or bring some objects to the classroom that have
shapes easily identified as the various solids.
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Unit 3: Geometry
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Additional Activities
Look Out for Angles
Congruent Figures
For Extra Practice (Appropriate for use after Lesson 2)
Materials: Look Out for Angles (Master 3.14), old
magazines, scissors, glue, heavy paper, a card with a
square corner
For Extra Practice (Appropriate for use after Lesson 1)
Materials: Congruent Figures (Master 3.15), scissors,
2-cm grid paper (PM 21), triangular grid paper (PM 24),
glue, heavy paper
The work students do: Students examine old
magazines for pictures that display a variety of angles.
The work students do: Each student draws 10 foursided figures on square or triangular grid paper. The
students initial each figure, and then cut them out.
They cut out the angles, compare them with the card,
and then sort them into 3 groups: angles less than,
equal to, or greater than a right angle.
Students create an angle collage that reflects the 3 types
of angles.
Take It Further: Students draw a picture that includes
items with angles that are less than, equal to, or greater
than a right angle.
Social
Partner Activity
Students work together to attempt to identify congruent
figures in both sets of figures.
Students glue the congruent figures onto heavy paper.
Students explain how they know figures are congruent.
Take It Further: Students repeat the activity by
drawing figures with other than four sides.
Spatial/Social
Partner Activity
Go Fish for Faces
Prisms and Pyramids
For Extra Practice (Appropriate for use after Lesson 8)
Materials: Go Fish for Faces (Master 3.16), Face-Off
game cards (Master 3.12), models of solids
For Extra Practice (Appropriate for use after Lesson 8A)
Materials: Prisms and Pyramids (Master 3.17), models
of various prisms and pyramids, 4-column charts (PM 18)
The work students do: Students play a variation of
the game, Go Fish.
The work students do: Students select models of
2 different prisms. They name the prisms, then work
together to count the faces, edges, and vertices of
each model.
Students use Face-Off game cards and take turns to
collect the faces of solids.
During each turn, a player asks the other player if she
has a face card that he needs to complete a solid. If the
other player has this card, she gives it to him. If not, she
tells him to “go fish,” and he takes a card from the deck.
Play continues until one player runs out of cards, or all
the cards have been used.
The player who has no cards left, or who has the fewer
cards left when all the cards have been used, wins.
Take It Further: Add more face cards, such as
pentagons and hexagons.
They record their findings in a table, then describe the
similarities and differences. Students repeat the activity
using models of 2 different pyramids.
Take It Further: Students choose 1 prism and
1 pyramid. They describe how the models are alike
and how they are different.
Kinesthetic/Linguistic
Partner Activity
Kinesthetic/Social
Partner Activity
Unit 3: Geometry
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Planning for Unit 3
Planning for Instruction
Lesson
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Unit 3: Geometry
Time
Suggested Unit time: 3–4 weeks
Materials
Program Support
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Program Support
Curriculum Focus
This unit has been tailored to provide a fit with your
curriculum. Lessons that address content not required by your
curriculum have been identified as optional, and can be
omitted. Lessons that contain some relevant and some
extraneous content have been annotated with suggestions for
modifications. In addition, some new material has been added
to this unit to ensure complete coverage of your curriculum.
Unit 3: Geometry
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Planning for Assessment
Purpose
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Unit 3: Geometry
Tools and Process
Recording and Reporting
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LAUNCH
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Under Construction
LESSON ORGANIZER
10–15 min
Curriculum Focus: Activate prior learning about
two-dimensional figures and three-dimensional solids.
Vocabulary: figure, solid
ASSUMED PRIOR KNOWLEDGE
✓ Students can name and describe different
✓
(two-dimensional) figures and (three-dimensional) solids
according to their attributes.
Students can describe what makes figures and solids alike
and different.
ACTIVATE PRIOR LEARNING
Discuss the first question in the Student Book.
Have students provide several examples of
figures in the picture of the castle.
(There are triangles and rectangles in the scaffolding
and trapezoids in the doorframe.)
Record student responses on chart paper or on
an overhead transparency. Keep them to display
at the end of the unit.
Have students look for figures in the classroom.
Discuss the second question. Model how to
describe figures using mathematical language.
For example, there are three rectangles in the
scaffolding. A rectangle has opposite
sides equal.
(Sample answers: Some figures form the faces of solids,
while others are outlined by a frame; some figures have
different side lengths; some figures have the same size
and shape.)
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Unit 3 • Launch • Student page 68
Draw a rectangle on the board, and then hold
up a book. Ask:
• How are this solid and this figure alike? How
are they different? (The rectangle on the board
has 2 dimensions: length and width. The book has 3
dimensions: length, width, and height. The front and
back covers of the book are rectangles.)
Discuss the third and fourth questions. Focus
attention on properties of solids, such as the
faces, edges, and vertices. (Some solids have
circular faces, while others have rectangular faces; some
solids have more vertices than others.)
Discuss the fifth question.
Tell students that they will use what they learn
about figures and solids at the end of the unit to
design a castle.
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LITERATURE CONNECTIONS FOR THE UNIT
Sir Cumference and the Sword in the Cone: A Math Adventure
by Cindy Neuschwander. Watertown, MA: Charlesbridge
Publications, 2003.
ISBN 1570916004
Sir Cumference, Radius, and Sir Vertex search for Edgecalibur,
the sword that King Arthur has hidden in a geometric solid.
Sir Cumference and the Great Knight of Angleland: A Math
Adventure by Cindy Neuschwander. Watertown,
MA: Charlesbridge Publications, 2002.
ISBN 157091169X
Radius sets out to rescue the King of Lell. He discovers acute,
obtuse, straight, and right angles, as well as parallel lines, all
using his precious ”medallion” (a protractor).
The Warlord’s Puzzle by Virginia Walton Pilegard. New York:
Pelican Books, 2000.
ISBN 1565544951
An ancient Chinese warlord is delighted when an artist brings him
a beautiful square tile. The artist drops the tile and it breaks into
seven pieces (tans). When the Warlord offers a reward to anyone
who can fix the tile, a simple peasant boy solves the puzzle.
REACHING ALL LEARNERS
Some students may benefit from using the virtual
manipulatives on the eTools CD-ROM.
The eTools appropriate for this unit include Geometry Shapes.
DIAGNOSTIC ASSESSMENT
What to Look For
What to Do
✔ Students understand
that figures and
solids can be
described by their
attributes.
Extra Support:
✔ Students can
describe what makes
figures and solids
alike and different.
✔ Students use the
correct mathematical
language to describe
geometric concepts
of figures and solids.
Students create reference charts that list attributes of different figures.
Work on this skill during Lesson 4.
Students may benefit from a review of the concepts of depth and height. Discuss
why it is possible to fill a cube with water, but not a square.
Work on this skill during Lesson 8.
Students may benefit from making and posting a chart that contains the names of
the figures and solids with which they are familiar. Refer students to the Glossary
for their definitions.
Continue to add to the chart throughout the unit.
Unit 3 • Launch • Student page 69
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LESSON 1
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Congruent Figures
40–50 min
LESSON ORGANIZER
Curriculum Focus: Identify and construct congruent figures.
Teacher Materials
two large, congruent triangles
tape
tracing paper
Student Materials
Optional
geoboards and geobands Step-by-Step 1 (Master 3.18)
square dot paper (PM 22) Extra Practice 1 (Master 3.31)
tracing paper
Figures 1 (Master 3.6)
Vocabulary: congruent figures
Assessment: Master 3.2 Ongoing Observations: Geometry
Key Math Learnings
1. Congruent figures have the same size and shape.
2. A figure may need to be flipped or turned to determine if it
is congruent to another figure.
3. Some figures can be divided into congruent parts.
Numbers Every Day
Questions such as this may help students who have difficulty with
the place value of large numbers. Some students may benefit
from modelling large numbers on place-value mats.
Curriculum Focus
In this lesson, students explore congruent figures. This
material is not directly required by your curriculum, but it
is recommended as a review.
• What can we say about the parallelograms?
(All of the parallelograms are identical. If we flipped
the parallelogram that slopes to the right, it would
match the parallelogram that slopes to the left.)
BEFORE
Remind students that figures with the same
shape and size are congruent figures.
Get Started
Tape two congruent triangles to the board.
Orient them differently.
Have students examine the pattern in the
Student Book.
Ask questions, such as:
• What are some figures you can see in the
pattern? (The large rectangle that surrounds the
pattern, a triangle, a parallelogram sloping to the
right, a parallelogram sloping to the left, a circle)
• What makes one figure different from
another? (They have different shapes and different
sizes; some are curved while others have
straight sides.)
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Unit 3 • Lesson 1 • Student page 70
Draw attention to the triangles on the board.
Ask:
• How could we find out if these triangles are
congruent?
Invite a volunteer to take one triangle from the
board and place it on top of the other triangle
to show that they are congruent.
Present Explore. Students can use a large book
or a large piece of stiff paper folded in half to
conceal their geoboards.
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REACHING ALL LEARNERS
Alternative Explore
Materials: 2-cm grid paper (PM 21), Colour Tiles or congruent
paper squares (make from PM 21)
Students work in pairs. One student creates a figure with Colour
Tiles or congruent paper squares. She describes her figure to her
partner, who then tries to draw the figure on 2-cm grid paper.
The students compare the figures to see if they are congruent.
One method would be to trace the original figure, and then try
to make it coincide with the new figure.
Common Misconceptions
➤Students may not recognize congruent figures that are
oriented differently. For example, when two congruent squares
have different orientations, a student may say the figures are a
square and a diamond, and not recognize they are congruent.
How to Help: Remind students that a figure may need to be
rotated or flipped to determine if it is congruent to another figure.
ESL Strategies
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4
When discussing a common figure, such as a square, ask an
English learner if she can say the figure’s name in her primary
language. Placing value on a student’s primary language can
help create a more comfortable learning environment.
Sample Answers
1. Tracings of the figures in each pair have the same size
Pairs of congruent figures: A and K; C and L; D and J; E and H.
DURING
and shape.
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• What attributes of your figure will give your
partner the best clues?
(Number of sides, side lengths, number of vertices)
• How do the pegs in the geoboard help you
describe your figure? (I can describe how many
pegs are included in each side of my figure.)
Listen to hear if students describe their figures
using geometric terms. If necessary, model how
to describe a figure by the number and length
of its sides, and the number of vertices.
AFTER
Connect
Invite a volunteer to explain how she
determined if her partner’s figure was
congruent to her figure. Have students suggest
different ways to check for congruency.
Elicit from students that tracing a figure allows
you to rotate or flip the figure to see if you can
make it coincide with another figure. This is
outlined in Connect.
Practice
Distribute copies of Master 3.6 for Question 1.
Questions 1 and 3 require tracing paper.
Question 4 requires a geoboard and geobands,
or square dot paper (PM 22).
Assessment Focus: Question 4
Students understand that dividing a larger
figure into smaller, congruent figures is the
same as dividing the larger figure into equal
parts. However, equal areas do not necessarily
mean congruent parts. Students understand
that the figure in part c can be divided into
4 congruent figures in more than 1 way.
Unit 3 • Lesson 1 • Student page 71
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2. a) No; both the figures are triangles, but they have different
sizes.
b) Yes; the figures have the same shape and the same size.
One figure is turned.
c) No; both the figures are parallelograms, but they have
different sizes.
3. Students show their original figure, and a tracing of the same
figure. The tracing is congruent to the original figure.
4. In part c, we can divide the figure into 4 congruent rectangles
in 3 different ways, and into 4 congruent triangles. Students
may identify different orientations of a divided rectangle as
different congruent figures. So, they may say 6 ways. There is
only 1 way to divide the figure in part a into 3 congruent
triangles. It is possible to divide the figure in other ways, but
the triangles do not have the same shape and size, and are
not congruent. Similarly, there is only 1 way to divide the
figure in part b into 3 congruent rectangles.
REFLECT: I would trace the figure. My tracing is congruent to the
original figure because it has the same size and shape.
Student should show the figure they created by tracing
another figure.
part c
Making Connections
Math Link: Have students look for patterns of congruent
figures at home. They could write a description of these patterns
using mathematical terms. For example, the ceiling is made of
congruent square tiles.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that congruent
figures have the same size and shape.
Extra Support: Have students cut matching figures from folded
paper. For example, fold a piece of paper in half, and cut a figure
across the fold. Unfold the figure, and then cut along the fold to
create two congruent figures.
Students who need help can use Step-by-Step 1 (Master 3.18) to
complete question 4.
Applying procedures
✔ Students can create congruent figures.
Communicating
✔ Students can describe congruent
figures using the correct mathematical
language.
Extra Practice: Have students complete the Additional Activity,
Congruent Figures (Master 3.15).
Students can complete Extra Practice 1 (Master 3.31).
Extension: Challenge students to divide Pattern Blocks into
congruent figures. Students may wish to trace the Pattern Blocks
on triangular grid paper (PM 24).
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
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Unit 3 • Lesson 1 • Student page 72
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ESSON 2
Exploring Angles
LESSON ORGANIZER
40–50 min
Curriculum Focus: Use non-standard units to measure
angles. (SS21)
Teacher Materials
Pattern Blocks for the overhead projector, or Pattern Blocks
transparency (PM 25)
non-permanent markers
6-division protractor transparency (Master 3.7)
Student Materials
Optional
Pattern Blocks (PM 25)
Step-by-Step 2 (Master 3.19)
tracing paper
Extra Practice 1 (Master 3.31)
6-division protractor
(Master 3.7)
rulers
Vocabulary: angle, protractor, right angle, vertex, baseline
Assessment: Master 3.2 Ongoing Observations: Geometry
Key Math Learnings
1. Two sides of a figure or 2 lines meet at a vertex to form
an angle.
2. An angle that forms a square corner is a right angle.
3. Concrete, non-standard units can be used to measure,
compare, and sort angles.
4. Protractors marked with non-standard units can be used to
measure, compare, and sort angles.
Curriculum Focus
This lesson introduces the concept of angles and
comparing a given angle to a right angle (SS21),
as well as measuring angles using non-standard units
(not required). To modify this lesson, have students
complete just the first part of Explore, question 4 of
Practice, and Reflect. Assign Extra Practice 1b
(Master 3.31b).
Display an orange square next to the green
triangle. Trace the square. Draw an arc to
indicate the angle between two sides of the
square. Tell students that two lines that meet in
a square corner make a right angle.
Introduce the convention of indicating a right
angle with a small square instead of an arc.
Display a tan rhombus Pattern Block.
BEFORE
Get Started
Display a green triangle Pattern Block on the
overhead projector. Trace the triangle. Draw an
arc to indicate the angle between two sides of
the triangle. Explain that the 2 sides meet at a
vertex forming an angle. Explain the use of an
arc to show an angle.
Ask:
• How can you describe the angles?
(The 2 smaller angles are equal and are less than a
right angle. The 2 larger angles are equal and are
greater than a right angle.)
Assign the first part of Explore. If assigning the
second part (optional), remind students that the
small angle in the tan rhombus is one unit.
They must record their angle measures with this
unit. Students may name this unit “small tan.”
Unit 3 • Lesson 2 • Student page 73
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REACHING ALL LEARNERS
Alternative Explore
Materials: heavy paper, rulers, scissors
Students mark, then cut out, 2 angles of different size from
heavy paper. They use these angle measurers to measure angles
in the classroom. They should measure each angle with both
measurers, and then compare the results. Challenge students to
construct an angle measurer for right angles.
Early Finishers
Have students use the smaller angle in the tan rhombus Pattern
Block to measure angles in the classroom. They re-measure each
angle using a different Pattern Block, and then compare
the results.
Common Misconceptions
➤Some students see equal angles with different arm lengths
and/or orientations as being unequal.
How to Help: Draw several equal angles that have different
arm lengths and orientations. Have students compare the angles
by tracing each angle and placing them one on top of the other.
➤Students read the wrong set of numbers on the protractor.
How to Help: Have students practise placing the centre of the
protractor on the angle’s vertex, and rotating the protractor so
that its baseline aligns with the angle’s lower arm. As the
students measure, have them count the units from 0 as they move
along the protractor towards the angle’s upper arm.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• How can you tell if an angle is greater than
or less than a right angle? (I can compare it
with the corner of the orange square Pattern Block.)
• How many small tan units fit in the small
angle of the blue rhombus Pattern Block? (2)
How many fit in the large angle? (4)
Invite a volunteer to demonstrate on the
overhead projector how he measured an angle
in the yellow hexagon, using the smaller angle
in the tan rhombus. Ask:
• How many small tan rhombuses fit in each
angle in the yellow hexagon? (4)
• What is the measure of each angle in the
yellow hexagon? (4 small tans)
• How could we make measuring a large angle
easier? (We could tape some tan rhombuses together.)
Watch to ensure that students measure with the
correct angle in the tan rhombus.
Elicit from students that they can make an
angle measurer by tracing tan rhombus blocks
on tracing paper or on a transparency.
AFTER
Explain that a device used to measure angles is
a protractor.
Connect
Invite volunteers to explain how they
determined which Pattern Blocks have angles
larger than right angles, and which have angles
less than right angles. (The yellow hexagon, blue
rhombus, and red trapezoid have at least 1 angle
greater than a right angle.)
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Unit 3 • Lesson 2 • Student page 74
Invite a volunteer to read the instructions in
Connect that explain how to measure an angle
with a protractor. As she reads, model the steps
on the overhead projector.
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Sample Answers
2. For example:
1 green triangle angle and 2 small tan rhombus angles fit in
1 yellow hexagon angle.
1 green triangle angle and 2 small tan rhombus angles fit in
1 blue rhombus large angle.
2 red trapezoid small angles fit in 1 yellow hexagon angle.
3. For example:
1 orange square has 4 right angles.
3 tan rhombus small angles form 1 orange square angle.
1 tan rhombus small angle and 1 green triangle angle form
1 orange square angle.
1 blue rhombus large angle is greater than 1 orange
square angle.
Each angle: 2 units
Small angle: 1 unit; large angle: 2 units
Small angle: 1 unit; large angle: 2 units
Making Connections
Math Link: Discuss how early surveyors created maps by
carrying instruments, such as theodolites and telescopes, great
distances. Now, airplanes and satellites take photographs of the
land below to help surveyors with their work.
= 52
= 124
17
+ 35
52
71
+ 53
124
Distribute 6-division protractors (Master 3.7)
and have students use them to measure the
angle in Connect.
Emphasize that students should place the
baseline of the protractor along an arm of the
angle, and its centre on the vertex of the angle.
Ask:
• Why do you think the protractor has two sets
of numbers? (To allow you to measure from either
arm of the angle)
Model how to measure angles with different
orientations. Remind students to make sure
they read the correct set of numbers on
the protractor.
Numbers Every Day
Students should use each number once. They should think about
the value of each number. For example, each number is worth
more as a tens digit than as a ones digit. Challenge students to
find the smallest sum and the largest sum if the numbers can be
used more than once.
(11 + 11 = 22, 77 + 77 = 154)
Practice
Questions 1, 2, and 3 require Pattern Blocks.
Questions 5 and 6 require a 6-division
protractor (Master 3.7). Students could use an
orange square Pattern Block for question 4.
Assessment Focus: Question 6
The student constructs an angle by drawing
two lines that meet. He may create an angle
less than, greater than, or equal to a right angle.
The student measures and records the angle
measure with a number and a unit. He
communicates clearly how he used a protractor
to measure the angle.
Make sure students understand that the
measure of an angle is independent of the
lengths of the arms of the angle. An angle does
not necessarily have arms of equal length.
Unit 3 • Lesson 2 • Student page 75
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5. a) 3 small tans
b) 1 small tan
c) 3 small tans
d) About 4 small tans
e) About 31/2 small tans f) 1 small tan
6. I drew a line and marked a point at one end. From this point,
I drew another line to make an angle. I placed my protractor
with the baseline along one arm of the angle, and the centre
at the angle’s vertex. I used the arc that was numbered
counterclockwise to measure the angle. It is between
2 and 3 units.
REFLECT: I can compare the angle to another right angle, such
as the corner of an orange square Pattern Block, or the corner
of a piece of paper.
right
angle
right angle
less than a
right angle
greater than a
right angle
greater than a right
angle
less than a
right angle
Curriculum Focus
Your curriculum requires that students recognize and identify:
point, line, parallel, intersecting, perpendicular, vertical, and
horizontal lines (SS20).
The Curriculum Focus Activity, Exploring Lines (Master 3.38) is
provided to cover this outcome. Have students complete this
activity after this lesson.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that an angle is
formed when two lines meet.
Extra Support: Have students use 3 paper angles, one smaller
than a right angle, one larger than a right angle, and one equal
to a right angle to measure the angles of classroom objects.
Students who need help can use Step-by-Step 2 (Master 3.19) to
complete question 6.
✔ Students recognize that an angle may
be greater than, equal to, or less than
a right angle.
Applying procedures
✔ Students can measure angles using
concrete angles in non-standard units.
✔ Students can measure angles in
non-standard units using a protractor.
Extra Practice: Students can complete the Additional Activity,
Look Out for Angles (Master 3.14).
Students can complete Extra Practice 1 (Master 3.31).
Extension: Students investigate which combinations of
connected Pattern Block angles make one complete turn.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
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Unit 3 • Lesson 2 • Student page 76
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ESSON 3
Measuring Angles
optional
LESSON ORGANIZER
Curriculum Focus: Measure angles using a protractor.
Teacher Materials
protractor
Optional
tracing paper or wax paper Step-by-Step 3 (Master 3.20)
protractors
Extra Practice 2 (Master 3.32)
rulers
scissors
Vocabulary: degree
Assessment: Master 3.2 Ongoing Observations: Geometry
Student Materials
Key Math Learnings
1. Angles are measured with a standard semicircular protractor
divided into 180 equal slices.
2. Each division on a protractor represents 1 degree.
Between 5 and 6 slices
Between 3 and 4 slices
Curriculum Focus
DURING
This lesson discusses measuring angles with a standard
protractor. This is not required by your curriculum. If you
choose to do this lesson, allow 40–50 minutes.
Ongoing Assessment: Observe and Listen
BEFORE
Get Started
Have students examine the protractor they
made in Lesson 2. Ask:
• How could you make the protractor more
accurate? (Make the congruent slices smaller.)
Explore
Ask questions, such as:
• What makes this protractor different from the
one you used in Lesson 2?
(This protractor has 8 congruent slices instead of 6.)
• How does the number of congruent slices on
a protractor affect the measurement of
angles? (More slices means each slice is smaller,
and we can measure with greater precision.)
Present Explore. Ensure students understand
how to fold the paper. First fold the length in
half. The second and third folds are in half
diagonally.
Suggest students run the edge of a ruler along
each fold to make a sharp crease.
Unit 3 • Lesson 3 • Student page 77
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REACHING ALL LEARNERS
Early Finishers
Have students construct a given angle using a ruler and a
protractor.
Common Misconceptions
➤Students cannot read a standard protractor because the units
are too small.
How to Help: Photocopy a standard protractor. White out the
1º markings, leaving only the 10º markings. Copy the modified
protractor onto a transparency for students to use. After they
have practised measuring angles with the 10º markings,
introduce the 1º markings. Alternatively, enlarge the copy of the
protractor. This will make it easier to read, but will not affect
angle measurements.
➤Students think that an angle becomes larger when measured
with a smaller unit.
How to Help: Have students construct, then measure angles.
Students should gain a better understanding of what affects an
angle’s measure by constructing their own angles.
=
=
=
=
32
62
92
122
All ones digits are 2, and the other
digit (or digits) is a multiple of 3.
AFTER
Connect
Invite volunteers to share their angle
measurements. Record them on the board.
Ask:
• How could we make our protractor even
more precise?
(Divide it into smaller congruent slices.)
Students can roughly divide each congruent
slice on their protractor in half, using a pencil
and ruler to create 16 congruent slices.
Have students use their improved protractors to
re-measure the angles in Explore. Invite
volunteers to record the new measures on the
board next to the original measures.
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Unit 3 • Lesson 3 • Student page 78
Ask:
• Have the sizes of the angles changed? (No)
• What has changed? (More slices fit in the angle,
so the number of slices we need to measure the angle
increases; the units have changed.)
• Suppose we divided each of the 16 slices in
half. What would happen to each angle? The
angle measurements?
(Each angle would be the same size, but we would
need twice the number of units to measure it.)
• Which of the tools you have used to measure
angles is the most precise? (The protractor with
16 slices gives the most precise measure. The angle is
more likely to be near one of the divisions; it is
easier to see which division is closest to the angle.)
Elicit from students that the more divisions
there are in the protractor, the more precisely it
can measure angles.
Home
60º
Less than 90º
90º
Equal to 90º
140º
Greater than 90º
C,B,A
Show students the standard protractor in
Connect. Point out that it has 18 large divisions,
each of which is divided into 10 smaller,
congruent slices, for a total of 180 slices. Tell
students that each of these slices is 1 degree.
Model the steps in Connect for measuring an
angle with a standard protractor. Model how to
record the measure using the standard
degree notation, º.
Remind students to count from the baseline
that aligns with one arm of the angle.
Remind students how to indicate a right angle
using a small square in the angle:
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Numbers Every Day
One strategy is to take 1 from the second number and add it to
the first number to make 10, or a multiple of 10. In the numbers
that are being added, the first number starts at 9 and we add 10
each time. The second number starts at 23 and we add 20 each
time. In the sums, we start at 32 and add 30 each time. All ones
digits in the sums are 2, and the other digit (or digits) is a
multiple of 3.
Practice
All questions require a standard protractor.
Assessment Focus: Question 4
Students create three angles, one that is less
than, one that is greater than, and one that is
equal to a right angle. They should measure
their angles with a standard protractor. Some
students may use a protractor to help construct
the angles.
Students who need extra support to complete
Assessment Focus questions may benefit from
the Step-by-Step masters (Masters 3.18 to 3.28).
Unit 3 • Lesson 3 • Student page 79
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Sample Answers
2. c) ELF
d) MAN
4. a)
b)
E, F, H, L, T (I may also be included.)
K, X, Y
A, K, M, N, V, W, X, Y, Z
c)
I can check my angles by measuring them with a protractor.
The angle in part a measures 45º. The angle in part b
measures 90º. The angle in part c measures 150º.
90º
120º
REFLECT: I put the baseline on the arm of the angle. I look along
the curve of the protractor until I reach the other arm of the
angle. I measure the angle to the nearest 10º, then count the
smaller marks for a more precise measure to the nearest 1º.
Making Connections
At Home: Street signs with angles less than 90º include a
yield sign.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that the size of
an angle remains the same, no matter
which unit is used to measure it.
Extra Support: Have students trace, then measure, angles in
figures in the classroom.
Students who need help can use Step-by-Step 3 (Master 3.20) to
complete question 4.
✔ Students understand that a smaller unit
gives a more precise measurement.
Extra Practice: Have students find, then trace, angles from any
part of the Student Book. They then use a protractor to measure
angles in degrees.
Students can complete Extra Practice 2 (Master 3.32).
Applying procedures
✔ Students can measure angles with a
standard protractor.
Communicating
✔ Students can communicate and record
the measurement of angles accurately,
using the correct terminology
and symbols.
Extension: Have students use a ruler to draw a triangle. They
measure the 3 angles of the triangle in degrees, then add the
measures. Repeat the activity with 3 different triangles. Have them
record any pattern they see in the sums.
(The sum of the angles inside a triangle is always 180º.)
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
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Unit 3 • Lesson 3 • Student page 80
150º
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ESSON 4
Exploring Sides in
Quadrilaterals
40–50 min
LESSON ORGANIZER
Curriculum Focus: Discover attributes of quadrilaterals
related to side lengths. (SS22)
Teacher Materials
chart paper
markers
Optional
Quadrilaterals 1 (Master 3.8) Step-by-Step 4 (Master 3.21)
rulers
Extra Practice 2 (Master 3.32)
3-column charts (PM 18)
geoboards and geobands
square dot paper (PM 22)
Venn diagrams (PM 28)
Vocabulary: quadrilateral, diagonal, hatch mark, kite
Assessment: Master 3.2 Ongoing Observations: Geometry
Student Materials
Key Math Learnings
1. A quadrilateral is a 4-sided figure.
2. A quadrilateral can be classified according to its side
lengths, diagonal lengths, and the number of parallel sides.
BEFORE
Get Started
Have students find examples of different
4-sided figures in the classroom. Explain that
we call a figure with four sides a quadrilateral.
Remind students of the quadrilaterals they
know, such as squares, rectangles,
parallelograms, rhombuses, and trapezoids.
Present Explore. Distribute copies of
Quadrilaterals 1 (Master 3.8) for students to use.
Ensure students understand that a diagonal
joins two opposite vertices in a quadrilateral.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• Which figures do you get when you draw a
diagonal in a quadrilateral? (Two triangles)
• Which quadrilaterals have all four sides the
same length? (Quadrilaterals A, C, G, and F;
squares, and rhombuses)
• Which quadrilateral has all sides of different
lengths? (Quadrilateral I, a trapezoid)
• How many diagonals can you draw in
each quadrilateral? (2)
• How do the two diagonals in a square
compare in length? (They have the same length.)
Listen to hear if students use mathematical
terminology. They should refer to the figures as
quadrilaterals, and try to use the correct name
for each.
Unit 3 • Lesson 4 • Student page 81
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REACHING ALL LEARNERS
Alternative Explore
Materials: 60-cm loops of string, square dot paper (PM 22)
Students work in pairs. Each student hooks the string with one
finger on each hand to form 2 vertices. Students create
quadrilaterals by stretching the string taut. They then change the
position of their fingers to create a new quadrilateral. They
should record each quadrilateral on dot paper, then compare its
side lengths and diagonal lengths.
Early Finishers
One student describes a quadrilateral in the classroom to
another student. She describes as many attributes as necessary
for the other student to identify the quadrilateral.
Common Misconceptions
➤Students do not recognize that some quadrilaterals may be
described using different names. For example, a square is a
rectangle, but a rectangle is not necessarily a square.
How to Help: Illustrate this concept using attributes of students.
A student can be a girl and in grade 4. A boy in grade 4 shares
one of these attributes, but not both. Similarly, a square has 4
sides equal and diagonals that are equal. A rectangle shares
one of these attributes, but not both.
➤Students think that “2 parallel sides”, and “1 pair of parallel
sides” are different attributes.
How to Help: Point out that these phrases mean the same thing.
If 1 side is parallel to another side, the 2 sides form “1 pair of
parallel sides.” Ensure students understand that, for 2 sides to be
parallel, they do not have to be the same length.
AFTER
Connect
Invite volunteers to share their findings about
the identity, side lengths, and diagonal lengths
of each quadrilateral. Discuss how we sort
quadrilaterals according to the different
attributes.
Elicit from students that there are many ways
to sort quadrilaterals.
Ask:
• Can you think of another way to classify
quadrilaterals using their sides?
(Sort according to the number of parallel sides.)
Create a 3-column chart on chart paper. Title
the chart “Quadrilaterals.” Label the first
column “Name,” and the second column “Sides
and Diagonals.”
Record students’ findings from Explore in the
first and second columns of the Quadrilaterals
chart. Reserve the third column for Lesson 5.
16
Unit 3 • Lesson 4 • Student page 82
For example,
Quadrilaterals
Name
Sides and Diagonals
Square
4 equal sides
Opposite sides parallel
Equal diagonals
Rhombus 4 equal sides
Opposite sides parallel
Use the diagrams in Connect to check the class
chart. Point out that some quadrilaterals belong
to more than one type. For example, a rhombus
with equal diagonals is a square.
Model how to indicate equal sides with hatch
marks, and parallel sides with small arrows.
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Sample Answers
1. On a 5 by 5 geoboard, the rectangles may have any of these
dimensions: 1 by 1, 1 by 2, 1 by 3, 1 by 4, 1 by 5, 2 by 2,
2 by 3, 2 by 4, 2 by 5, 3 by 3, 3 by 4, 3 by 5, 4 by 4,
4 by 5, and 5 by 5.
The lengths, widths, and areas of the rectangles are different.
2. a) Squares: 1 by 1, 2 by 2, 3 by 3,… and rhombuses
The number of each depends on the size of the geoboard.
b) All squares, rhombuses, rectangles, and parallelograms
have 2 pairs of parallel sides. Possible figures depend on
the geoboard. (See answer to 1 and 2a)
c) All trapezoids have 2 parallel sides. Some can have no
equal sides. Possible trapezoids depend on the geoboard.
3. A kite has 2 pairs of equal, adjacent sides.
= 121
= 165
= 132
Numbers Every Day
The ones and tens digits in each number are the same. All the
sums have a tens digit one more than the ones digit.
Ask questions, such as:
• If a quadrilateral does not have 4 equal
sides, what might it be?
(A rectangle, a parallelogram, or a trapezoid)
• If we started with a rhombus, then made one
pair of its parallel sides longer than the other
pair, which quadrilateral would we have?
(A parallelogram)
• What would we call a parallelogram that has
equal diagonals? (A rectangle or a square)
Practice
Questions 1, 2, 5, and 6 require square dot
paper. Questions 1 and 2 require geoboards
and geobands.
Students can use a Venn diagram for question 4.
Question 3 introduces a new quadrilateral,
the kite. See the Math Note on page 19 for
more information.
Assessment Focus: Question 6
Students know that a parallelogram has two
pairs of opposite sides that are equal. Students
use these properties to draw a parallelogram on
dot paper. Some students might label their
parallelogram with hatch marks and small
arrows to indicate equal sides and parallel
sides. Students then describe something that is
not true about a parallelogram, something that
is sometimes true about a parallelogram, and
something that is always true about a
parallelogram. These descriptions should
indicate the depth of the student’s
understanding of parallelograms in particular,
and quadrilaterals in general.
Unit 3 • Lesson 4 • Student page 83
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4. a) Loop 1 — Diagonals of different lengths: C, D, E, F, G
Loop 2 — 2 pairs of equal sides: A, B, D, F, G
Loop 1
Loop 2
C
E
D
F
G
A
B
b) Loop 1 — Diagonals of equal length: A, B
Loop 2 — 2 equal sides: A, B, D, F, G
Loop 2
C
Loop 1
A
B
D
F
G
E
5. b) This cannot be done. If we join all the dots, there are 15
squares. 15 is not divisible by 4.
It is possible to make 4 rectangles, but they are not
congruent.
6. a) A parallelogram never has sides of all different lengths.
b) A parallelogram sometimes has two diagonals the same
length.
c) A parallelogram always has two pairs of opposite sides equal.
REFLECT: If all sides of a quadrilateral have equal length, it
All
All
Some
Some
could be a square or a rhombus. If two pairs of sides have
equal length, it could be a parallelogram, a rectangle, or a
kite. If the equal sides are adjacent, the quadrilateral is a kite.
If the equal sides are opposite, the quadrilateral is a
parallelogram or a rectangle.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students recognize the attributes of
different quadrilaterals.
Extra Support: Have students select and record one or two
attributes that a specific quadrilateral does not have. For example,
a rectangle does not have all sides of different lengths.
Students can use Step-by-Step 4 (Master 3.21) to complete question 6.
Applying procedures
✔ Students can sort and classify
quadrilaterals according to their side
lengths, their diagonal lengths, and
the number of parallel sides.
Communicating
✔ Students use the correct terms to
describe the attributes of quadrilaterals.
Extra Practice: Have students cut out squares, rectangles,
parallelograms, rhombuses, and trapezoids they find in old
magazines. They should sort each quadrilateral according to side
lengths and number of parallel sides.
Students can complete Extra Practice 2 (Master 3.32).
Extension: Have students draw quadrilaterals on heavy paper, cut
out the quadrilaterals, and then join them to make new
quadrilaterals. Students explore the attributes of the new
quadrilaterals.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
18
Unit 3 • Lesson 4 • Student page 84
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ESSON 5
Exploring Angles in
Quadrilaterals
LESSON ORGANIZER
70º
Kite
95º
90º
110º70º
60º
120º
90º
Square
95º
90º
110º
70º
Parallelogram
70º110º
Rectangle
90º
90º
90º
90º Rectangle
120º
Trapezoid
90º
90º
35º
110º
Rhombus
110º 70º
Kite
60º
125º
70º
90º
90º
110º
110º
Rhombus
70º
90º
60º
125º
75º
110º
90º
90º 90º
Square
110º
Parallelogram
70º
90º
150º
40–50 min
Curriculum Focus: Discover attributes of quadrilaterals related
to angle measures. (SS21, SS22)
Student Materials
Optional
protractors, 6-Division
Step-by-Step 5 (Master 3.22)
Protractors (Master 3.7),
Extra Practice 3 (Master 3.33)
or tracing paper and rulers
Quadrilaterals 2 (Master 3.9)
geoboards and geobands
square dot paper (PM 22)
rulers
Venn diagrams (PM 28)
Assessment: Master 3.2 Ongoing Observations: Geometry
90º
90º Trapezoid
30º
Key Math Learning
A quadrilateral can be classified according to its angles.
Math Note
Trapezoids, Kites, and Parallelograms
In this book, a trapezoid is defined as a quadrilateral
with at least 1 pair of parallel sides. This is an inclusive
definition, by which a square, a rectangle, a rhombus,
and a parallelogram are trapezoids.
A kite is defined as having 2 pairs of equal, adjacent
sides. This definition is inclusive of squares and
rhombuses. Some people define the kite as a convex
quadrilateral, using chevron or deltoid to name the
concave figure.
Similarly, a rectangle, a square, and a rhombus
are parallelograms.
Students learn these definitions in Lesson 6.
BEFORE
Get Started
Draw some quadrilaterals on the board or use
the Quadrilaterals class chart from Lesson 4 to
review the attributes of different quadrilaterals.
Ask:
• Which attributes do a rhombus and a square
share? (They have 4 equal sides.)
• What makes a square different? (A square has
2 equal diagonals, while a rhombus may not.)
Note: students will later learn that a square is a
special case of a rhombus.
Elicit from students that another attribute of a
square is each of its 4 angles is a right angle.
Present Explore. Distribute copies of
Quadrilaterals 2 (Master 3.9).
Curriculum Focus
If students have not measured angles with a protractor,
they can trace angles on tracing paper, and then use the
tracing to check for congruent angles.
Students can use an orange square Pattern Block to check
for right angles.
Unit 3 • Lesson 5 • Student page 85
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REACHING ALL LEARNERS
Alternative Explore
Materials: tangram (PM 26), 6-division protractor (Master 3.7),
square dot paper (PM 22)
Have students work in pairs to make different quadrilaterals with
the tans of a tangram. Students should draw each quadrilateral
on square dot paper, measure its angles, and record what they
notice about the angles in each quadrilateral.
Early Finishers
Have students use a ruler to draw a quadrilateral. They measure
the 4 angles in the quadrilateral, and then add the measures.
Students should repeat the activity with 3 different quadrilaterals,
and then record any pattern they see in the sums. (The sum is
always the same: 12 small tans using the 6-division protractor
(Master 3.7).)
Common Misconceptions
➤Some students have difficulty measuring angles in
quadrilaterals.
How to Help: Have students use a ruler to extend the arms of
the angle before they measure it. Ensure students align the
baseline of the protractor with one arm of the angle, and the
vertex of the angle with the centre of the baseline.
609, 670, 683, 694
2536, 2635, 3256, 6253
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• How do you know if one angle is equal to
another? (I use a tracing of an angle to compare it
with another angle.)
Watch for students who have difficulty
measuring angles within the same
quadrilateral.
Suggest students label each angle with its
measure to keep track of what they have done.
AFTER
Connect
Invite volunteers to identify each quadrilateral
and share their angle measurements with the
class. As students present their findings, ask
questions, such as:
• What kind of quadrilateral is figure B?
(Parallelogram)
• How do you know?
(It has two pairs of opposite angles that are equal.)
• How could we change a rhombus to make it
square? (Make all of its angles right angles.)
Discuss the angles in each type of quadrilateral.
Include these attributes in the third column of
the Quadrilaterals class chart started in Lesson 4.
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Unit 3 • Lesson 5 • Student page 86
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Sample Answers
1. a)
b)
c) It is not possible to make a quadrilateral with only 3 right
angles. If there are 3 right angles, the 4th angle must be a
right angle or the sides will not meet. For example,
square, rhombus,
rectangle,
parallelogram,
trapezoid, kite
rectangle,
parallelogram, trapezoid
parallelogram,
trapezoid
3. a) All sides equal: A, D
rhombus,
parallelogram,
trapezoid,
kite
Angle greater than a right angle: C, D, E, F, G, H
trapezoid
Loop 1
Loop 2
A
D
CEFG
H
kite
trapezoid
quadrilateral
B
Numbers Every Day
Students can use place value. If the numbers in the highest place
are the same, compare the numbers in the second-highest place,
and so on. For example, in the first set of numbers, the hundreds
digit is 6. Compare the tens digits. The number 609 is the only
number with a 0 as the tens digit, so it is the least number.
Practice
For example:
Quadrilateral
Sides and Diagonals
Angles
Square
4 equal sides
Opposite sides
parallel
Equal diagonals
4 right angles
Rhombus
4 equal sides
Opposite sides
parallel
Diagonals may not
be equal
Opposite
angles equal
Ensure students understand the term “opposite
angles.” Have them point to opposite angles in
different quadrilaterals.
Check the class chart against the illustrations
in Connect.
Question 1 requires a geoboard, geobands, and
square dot paper. Have Protractors (Master 3.7)
available. Question 3 requires a Venn diagram.
Assessment Focus: Question 4
Students use their understanding of the angle
and side properties of quadrilaterals to explain
why a given quadrilateral does not belong to a
particular group. Some students may also use
the diagonal properties of quadrilaterals.
Students who need extra practice can complete
the Extra Practice masters (Masters 3.31 to 3.36)
on the CD-ROM.
If students have measured accurately, they may
notice that the sum of the angles in a
quadrilateral is 360º. This knowledge is not a
curriculum expectation in grade 4, but accept it
as an attribute if students suggest it.
Unit 3 • Lesson 5 • Student page 87
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b) Loop 1 has 1 pair of opposite angles equal: A, B, C, D, G
Loop 2 has no right angles: C, D, E, F, G, H
Loop 1
Loop 2
AB
C
D
G
E
F
H
4. a) A square has 4 equal sides.
b) A rectangle has 4 right angles.
c) A rhombus has 4 equal sides.
d) A kite has 2 pairs of equal, adjacent sides.
REFLECT: I knew that you could sort quadrilaterals by side
length and by the number of parallel sides. I learned that you
can also sort quadrilaterals by the size of their angles. For
example, a parallelogram, a rhombus, a square, and a
rectangle all have opposite angles equal. A trapezoid can
have 0, 2, or 4 right angles. A square and a rectangle must
have 4 right angles.
Making Connections
Math Link: Have students look for things in their neighbourhood
that are parallel. For example, opposite sides of the street are
parallel, and fence pickets are parallel. Ask students what might
happen if train tracks were not parallel.
Art: Have students draw scenes with components that are
parallel. Discuss how we make parallel lines come together in
the distance when we draw pictures.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students recognize the attributes of
different quadrilaterals.
Extra Support: Have students select any two figures, except the
squares (Figures C and J), from the quadrilaterals in Explore. They
use the attributes of quadrilaterals to explain why the two figures
are not squares.
Students can use Step-by-Step 5 (Master 3.22) to complete question 4.
Applying procedures
✔ Students can measure angles in
quadrilaterals.
✔ Students can classify quadrilaterals
according to angle size.
Communicating
✔ Students can communicate
quadrilateral attributes and use
geometric terms to explain the
relationships among quadrilaterals.
Extra Practice: Students create a design on grid paper using
quadrilaterals. Before they begin, students assign colours to the
quadrilaterals according to angle attributes. For example, all
figures with four right angles are red, all figures with only 1 right
angle are blue, and so on.
Students can complete Extra Practice 3 (Master 3.33).
Extension: Have students choose a quadrilateral from Explore.
Students use a ruler and protractor to draw the quadrilateral.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
22
Unit 3 • Lesson 5 • Student page 88
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ESSON 6
Attributes of
Quadrilaterals
40–50 min
LESSON ORGANIZER
Curriculum Focus: Relate attributes to quadrilaterals. (SS22)
Teacher Materials
5 long lengths of string
5 file cards labelled: baseball and soccer, soccer,
baseball, ball sports, all sports
6 file cards labelled: square, parallelogram, rhombus,
rectangle, trapezoid, kite
Quadrilaterals Venn diagram transparency (Master 3.10)
(optional)
Student Materials
Optional
Quadrilaterals Venn
Step-by-Step 6 (Master 3.23)
diagrams (Master 3.10)
Extra Practice 3 (Master 3.33)
geoboards and geobands
square dot paper (PM 22)
triangular dot paper (PM 23)
rulers
tangrams (PM 26)
Vocabulary: adjacent
Assessment: Master 3.2 Ongoing Observations: Geometry
Key Math Learnings
1. Quadrilaterals have certain attributes.
2. Quadrilaterals can be sorted into sets and subsets.
BEFORE
Get Started
Use string or skipping ropes to create a Venn
diagram on the classroom floor similar to the one
shown in the Student Book.
It must be large enough to accommodate at
least 5 students.
Invite 5 volunteers to be sorted. Give a sport
file card to each volunteer.
Ask:
• Why is “ball sports” placed on the secondlargest loop? (Ball sports are sports, so they
belong inside the “all sports” loop.)
• Why is “baseball” inside one of the inner
loops? (Baseball is a ball sport, so it belongs inside
the “ball sports” loop.)
all sports
ball sports
baseball
Ask the class to arrange the volunteers in the
Venn diagram. Have each volunteer stand in
the appropriate loop and hold up his card.
Alternatively, draw the Venn diagram on the
board, or use a Quadrilateral Venn diagram
transparency on an overhead projector. Have
students label the diagram correctly.
soccer
baseball and soccer
Leave the Venn diagram and its labels in place.
Post the Quadrilaterals class chart from
Lesson 5. Students may also refer to the Connect
section of this lesson.
Unit 3 • Lesson 6 • Student page 89
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REACHING ALL LEARNERS
Early Finishers
Have each student draw a quadrilateral on square dot paper
(PM 22), and then write instructions explaining how to make the
quadrilateral. They should then give the instructions to another
student, who will try to draw a congruent figure without looking
at the original drawing.
Common Misconceptions
➤Students may have difficulty explaining why one type of
quadrilateral is not another type of quadrilateral.
How to Help: Have students sketch each quadrilateral and then
label parallel sides, equal sides, and equal angles. They then
look for attributes that are the same and attributes that are
different in each quadrilateral.
Making Connections
Art: Have students create designs that use combinations of
different quadrilaterals.
Present Explore. Remind students that a
trapezoid is a quadrilateral with 2 parallel
sides, or 1 pair of parallel sides, and that many
other quadrilaterals are classified as trapezoids.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask:
• Which quadrilateral belongs in the middle
loop? (The square; it has the attributes of all the
other quadrilaterals in the diagram.)
• Which quadrilateral has the fewest
attributes? (The trapezoid has only 2 parallel
sides, or 1 pair of parallel sides.)
• What attributes does a kite have?
(A kite has 1 pair of opposite angles equal, and
2 equal, adjacent sides.)
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Unit 3 • Lesson 6 • Student page 90
AFTER
Connect
Invite volunteers to discuss their Venn
diagrams. Place the quadrilateral file cards into
the Venn diagram on the floor according to
students’ descriptions.
Alternatively, draw the correct diagram on the
board, or label the transparency on the
overhead projector.
For example,
trapezoid
parallelogram
rhombus
rectangle
square
kite
Ask:
• What is true about all quadrilaterals in the
diagram?
(They have at least 1 pair of parallel sides.)
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Sample Answers
Same: Figures A and B have 2
pairs of parallel sides, all
sides equal, and opposite
angles equal.
Different: Figure A has 4 right
angles; Figure B has no
right angles.
2. a) A parallelogram has at least one pair of parallel sides.
b) A trapezoid does not have two pairs of parallel sides.
c) A rectangle has at least one pair of parallel sides.
d) A trapezoid may not have all right angles.
3. You can move the geoband twice to create 4 right angles and
all sides of equal length.
4. d) I have only one pair of parallel sides. What am I? Answer:
A trapezoid
5. a)
b)
No, squares and rhombuses must be divided into 4 smaller
figures to get figures with equal side lengths.
Rhombus
Rectangle or square
Trapezoid, square, or rectangle
Numbers Every Day
3, 23, 43, 63, 83, 103, 123, 143, 163
• Why is a square placed in the area where the
two loops overlap? (A square is a trapezoid, a
parallelogram, a rectangle, and a rhombus.)
• Where would the kite fit on this diagram?
(A kite goes outside the loops. It is a quadrilateral,
but not a trapezoid, because it has no parallel sides.)
Discuss the chart in Connect. If necessary, update
the Quadrilaterals class chart and then post it
for reference.
These instructions are for a TI-108 calculator. Use the instructions
in Unit 1, Lesson 3 for a TI-10 calculator. Consult the owner’s
manual for other calculators. Remind students to look for a pattern,
and to predict the next 4 terms, before checking with a calculator.
Pattern rule: start at 3 and add 20 each time.
Assessment Focus: Question 4
Students use their understanding of
quadrilateral attributes to solve the riddles.
Students should list all of the quadrilaterals
that satisfy each riddle, but in some cases, there
will only be one. Students create a new riddle
using these attributes. Some students may
include a riddle using diagonal length.
Practice
Question 1 requires a ruler. Questions 3 and 5
require a geoboard, geobands, and square dot
paper. Question 6 requires a tangram. Reflect
requires square or triangular dot paper.
Unit 3 • Lesson 6 • Student page 91
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6. b) Combine C and E, or F and G, to make a square.
Combine C and B, or E and D, to make a trapezoid.
Combine E, D, and C to make a trapezoid.
Students’ answers should include sketches.
7. Alternative answer: each figure in the first set has at least 1
right angle. No figure in the second set has 1 right angle. The
attribute is 1 right angle. Figures B and C share this attribute.
parallelogram
square
REFLECT:
1 pair of parallel sides,
2 right angles
This parallelogram has 2 pairs of opposite sides equal. It has
2 pairs of opposite angles equal. It has no right angles. This
kite has 2 pairs of equal, adjacent sides. It has 1 pair of
opposite angles equal, and no right angles.
no right angles
2 right angles
2 right angles
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that different
quadrilaterals may share
some attributes.
Extra Support: Have students compare three quadrilaterals at a
time. They can use a simpler Venn diagram, such as PM 28.
Students can use Step-by-Step 6
(Master 3.23) to complete question 4.
Applying procedures
✔ Students use the attributes of
quadrilaterals to sort and
classify them.
Extra Practice: Have students look for
quadrilaterals in old magazines, cut them
out, and then sort them on the
Quadrilaterals Venn Diagram (Master 3.10).
Students can complete Extra Practice 3 (Master 3.33).
Extension: Have students sort Pattern Blocks according to side
length and angle attributes. They can create a new Venn diagram
for the Pattern Blocks.
Recording and Reporting
Maser 3.2 Ongoing Observations:
Geometry
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Unit 3 • Lesson 6 • Student page 92
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ESSON 7
Similar Figures
optional
LESSON ORGANIZER
Curriculum Focus: Identify similar figures.
Teacher Materials
chart paper
4 by 5 photo and 8 by 10 photo, or similar drawings
scaled on chart paper
Student Materials
Optional
rulers
Step-by-Step 7 (Master 3.24)
Figures 2 (Master 3.11)
Extra Practice 4 (Master 3.34)
Vocabulary: similar
Assessment: Master 3.2 Ongoing Observations: Geometry
Key Math Learnings
1. Similar figures have equal corresponding angles; side lengths
may differ.
K is similar to N.
M is similar to T and Q.
A is similar to F.
C is similar to D.
E is similar to G.
2. The side lengths in the larger of 2 similar figures are
multiples of the corresponding side lengths in the smaller of
the figures.
Curriculum Focus
DURING
In this lesson, students explore similar figures. This
material is not required by your curriculum. If you choose
to complete this lesson, allow 40–50 minutes.
Ongoing Assessment: Observe and Listen
BEFORE
Get Started
Discuss the photographs in the Student Book.
Show students a 4 by 5 photograph and an
8 by 10 photograph.
Ask:
• Are these 2 photographs congruent? (No, they
have the same shape, but they are not the same size.)
Tell students that we call two figures with the
same shape, but with different sizes, similar
figures.
Present Explore. Distribute copies of Masters
3.11 and 3.11b. These are enlargements of the
figures in Explore.
Explore
Ask questions, such as:
• How do you know that Figures A and F are
similar? Are they also congruent?
(A and F are squares, with 4 equal sides and 4
right angles. They have the same shape. A has side
lengths of 3 units. F has side lengths of 1 unit. These
figures are similar, but not congruent.)
• How can you check if two figures are
similar? (I count the number of squares on each
side. Each side length in the larger figure could be
twice, three times, four times, and so on, the side
length in the smaller figure. The matching angles in
2 similar figures must be equal.)
Watch for students who have difficulty finding
similar triangles because they cannot count the
number of squares in a triangle’s sides. Suggest
they measure the angles in the triangles to see
if the matching angles are equal.
Unit 3 • Lesson 7 • Student page 93
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REACHING ALL LEARNERS
Alternative Explore
Materials: tangram (PM 26)
Students work in pairs to find congruent figures in a set of tans.
They then look for another figure that is similar to the
congruent figures.
=
=
=
=
Early Finishers
Students work in pairs to describe figures that are similar to
those in Explore. One student chooses a figure. The other
describes a similar figure that is not congruent to any of the
other figures illustrated in Explore. They should then switch roles.
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Common Misconceptions
➤Students may not recognize the relationships between the side
lengths of similar figures.
How to Help: Encourage students to use patterning. They can
measure the corresponding sides of two similar figures, and then
try to find a pattern rule to relate these side lengths. For
example, what do I multiply each side length of the smaller
figure by to get the side lengths of the larger figure?
ESL Strategies
Have students use the word similar in a sentence that does not
describe two geometric figures. Discuss the difference between
the mathematical term similar, and the everyday word similar.
Numbers Every Day
Students can add or subtract 1 or 2 from a known fact. For
example, 15 + 14 = 15 + 15 – 1 = 29.
AFTER
Connect
Invite volunteers to share the similar figures
they found and how they determined the
figures were similar.
Ask:
• How can you use patterns to help check if
two figures are similar? (Compare the sides in
the smaller figure to those in the larger figure. The
number you need to multiply by to create a larger,
similar figure must be the same for each side.)
Have students verify that the figures in Connect
are similar by counting squares and comparing
side lengths.
Ask:
• What could you do to create Figure B from
Figure A? (Multiply each side length in Figure A
by 2 and keep the angles the same.)
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Unit 3 • Lesson 7 • Student page 94
Practice
Have grid paper available for Questions 1, 3, 4,
and Reflect.
Assessment Focus: Question 3
Students use the properties of quadrilaterals to
answer parts a and b. They understand that all
squares have the same shape and are similar,
but that rectangles have different shapes, and
are therefore not always similar. Students
understand that triangles also have different
shapes, and are not all similar. Some students
may show that if you use diagonals to divide
two differently shaped rectangles, differently
shaped triangles will result. Students should
include drawings of the figures to support
their answers.
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Sample Answers
1. a) Similar; the triangles have matching angles equal, and
each side length of the large triangle is 2 times the side
length of the small triangle.
b) Not similar; the first figure is a triangle and the second
figure is a parallelogram.
c) Not similar; the first figure is a rhombus and the second
figure is a parallelogram.
d) Similar; the rhombuses have matching angles equal, and
each side length of the larger rhombus is 2 times the
matching side length of the smaller rhombus.
2. Rectangle B’s side lengths are 2 times as long as Rectangle A’s.
Rectangle C’s side lengths are 3 times as long as Rectangle A’s.
Rectangle D is 5 times as long as Rectangle A, but only 4
times as wide. It is not similar to Rectangle A, B, or C.
3. a) Yes. All squares have 4 equal sides and 4 equal angles, so
different-sized squares are similar. For example:
Rectange A is similar to Rectangle B and Rectangle C.
b) No. Rectangles may have different shapes.
c) No. Triangles may have different shapes.
4. a) No; all rectangles have 4 right angles and opposite sides
equal, but two rectangles may have different shapes. For
example, one rectangle may measure 1 cm by 2 cm, and
another 1 cm by 3 cm.
b) No; 2 rectangles may have different shapes.
REFLECT: If two figures have matching angles equal, and the
side lengths of the smaller figure are multiplied by the same
number to get the side lengths of the larger figure, then the
figures are similar.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that similar
figures have the same shape.
Extra Support: Provide students with several rectangles drawn
on square grid paper. Some rectangles should be similar, and some
should not. Have students find the side lengths of each rectangle,
and then look for patterns in the side lengths of pairs of rectangles.
Students can use Step-by-Step 7 (Master 3.24) to complete
question 3.
Applying procedures
✔ Students can recognize similar figures
by comparing angles and side lengths
of figures.
Extra Practice: Students can do the Additional Activity, Similar
and Congruent Quadrilaterals (Master 3.15).
Students can complete Extra Practice 4 (Master 3.34).
Extension: Have students find and measure a figure in the
classroom. They should list the measurements that a congruent
figure would have, and those that a similar figure could have.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
Unit 3 • Lesson 7 • Student page 95
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TECHNOLOGY
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Using a Computer to
Explore Pentominoes
LESSON ORGANIZER
optional
Curriculum Focus: Use a computer to look at patterns and
puzzles.
Teacher Materials
dominoes (optional)
Student Materials
AppleWorks or Microsoft Word
Vocabulary: pentominoes
Key Math Learning
Computers can be used to explore composite figures and
geometric patterns.
Curriculum Focus
Pentominoes are composite figures made from
5 congruent squares.
In this lesson, students use a computer to create and find
all possible pentominoes. This activity could also be done
with congruent paper squares or Colour Tiles.
Extend the lesson by having students make another set of
composite figures using 5 green triangle Pattern Blocks
(PM 25). Students can record their findings on triangular
grid paper (PM 24).
BEFORE
Discuss how the computer is used in many
different occupations to make jobs faster and
more efficient.
Distribute some dominoes for students to
inspect. Have students describe a domino using
geometric terms.
(2 congruent squares placed so that the sides join)
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Unit 3 • Technology • Student page 96
Tell students that a pentomino is made of 5
squares placed so that the sides join. Elicit from
students that a domino can have only one
shape, but pentominoes can have several
shapes.
Explain to students that they will create and
explore pentominoes with a computer. Ensure
students understand that pentominoes can be
joined together to create a pattern. Several
pentominoes may be joined to make a
rectangle, a square, or another figure. In this
way, students can make pentomino puzzles.
Note: AppleWorks and many other computer
programs are developed in the United States.
Point out to students that in the United States,
the word centimetre is spelled centimeter, and
will appear this way in most computer
programs.
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REACHING ALL LEARNERS
Alternative Explore
Materials: 2-cm grid paper (PM 21), Colour Tiles or congruent
paper squares (PM 21)
Have students use 5 congruent paper squares to create
pentominoes. They should find as many pentominoes as possible,
and then sketch each one on 2-cm grid paper.
Early Finishers
Have students create composite figures using Pattern Blocks. For
example, students make composite figures with green triangle
Pattern Blocks.
Common Misconceptions
➤Students may see two congruent pentominoes with a different
orientation as two different pentominoes.
How to Help: Have students print their pentominoes, and then
use tracing paper to trace the pentominoes and check for pairs
that are congruent.
DURING
Ongoing Assessment: Observe and Listen
Watch to ensure students understand they
should first find the 12 different pentominoes,
and then use them to make a puzzle.
Instructions for creating pentominoes using
Microsoft Word:
1. Open a new document in Microsoft Word.
2. Set the measurement units to centimetres.
Click Tools. Click Options.
Click the tab labelled General.
Look for Measurement units. Click
Click Centimeters.
Click OK.
3. Display the Drawing toolbar.
Click Tools. Click Customize.
Click the tab labelled Toolbars.
The box next to ‘Drawing’ should have a
check mark in it. If not, click on the box.
Click OK.
4. Select grid settings.
Click Draw. Click Grid.
The box next to Snap objects to grid should
have a check mark in it. If not, click on the
box. Click OK.
Grid settings should be:
Horizontal Spacing: 1 cm
Vertical Spacing: 1 cm
If they are not, click in each box and
enter 1 cm.
The box next to Display gridlines on screen
should have a check mark in it. If not, click
on the box.
Click OK.
Unit 3 • Technology • Student page 97
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The 12 pentominoes
REFLECT: I started with 5 squares in a row. Then I started with 4
squares in a row, and moved 2 squares to create 2 new
pentominoes. Then I started with 3 squares in a row and
moved 2 squares to create 8 new pentominoes. Then I made
1 more pentomino by connecting 2 rows of 2 squares, with
one square on the side, to form a “W.”
5. To draw a square,
click the Rectangle Tool
on the
Drawing toolbar.
The cursor will look like this: +
Hold down the Shift key while you click
and hold down the mouse button.
Drag the cursor. Release the mouse button.
6. To change the size of the square, click twice
on the square to select it.
Click Size.
Enter 2 cm for the width and 2 cm for
height.
Click OK.
7. To colour a square, click twice on the square
to select it.
Click Colours and Lines.
Click
next to Fill.
Select a colour.
Click OK.
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Unit 3 • Lesson Technology • Student page 98
Steps 8–12 are the same as in the Student
Book.
AFTER
Students should share their patterns and
puzzles with classmates. They should explain
how they created their puzzles and patterns.
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ESSON 8
Faces of Solids
LESSON ORGANIZER
40–50 min
Curriculum Focus: Identify and sketch the faces of
solids. (SS19)
Teacher Materials
Face-Off game cards or transparencies (Master 3.12)
(enough to make the faces of a square-based pyramid)
models of solids
Student Materials
Optional
models of solids
Step-by-Step 8 (Master 3.25)
Face-Off game cards
Extra Practice 4 (Master 3.34)
(3 copies of Master 3.12)
scissors (if game cards are not yet cut out)
Vocabulary: solid, face, base
Assessment: Master 3.2 Ongoing Observations: Geometry
Key Math Learnings
1. A face is a flat surface of a solid.
2. A solid can be identified by looking at its faces.
3. A solid is named for a particular face, called the base.
Math Note
Bases
The definition of base is the figure that forms the cross
section of a solid. For example, if you cut horizontally
through a square-based pyramid, the cut surface is a
square. The base of a square pyramid is a square. In
this book, “-based” is left off the names of solids. For
example, a square-based pyramid is a square pyramid,
and a triangular-based prism is a triangular prism. A
square prism is another name for a rectangular prism
with 2 faces that are congruent squares.
BEFORE
Get Started
Show students a square Face-Off game card on
the overhead projector. Tell students to
imagine that the square is the base of a solid.
Explain that the figure in the base of a solid
names the solid.
Ask:
• Which solids could have this square for a
base? (Cube, square pyramid, rectangular prism)
• Which solids could not have this square for
a base?
(A triangular pyramid, a cylinder, and so on)
Display 4 triangle Face-Off game cards next to
the square.
Ask:
• Which solid could have the square as a base
and 4 triangles as its other faces?
(Square pyramid)
Present Explore. Discuss the rules of the FaceOff game.
If the Face-Off game cards have not been cut
out, have students do this before they play.
Ensure that each group has enough Face-Off
game cards. They will need to cut apart 3
copies of Master 3.12.
Unit 3 • Lesson 8 • Student page 99
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REACHING ALL LEARNERS
Alternative Explore
Materials: models of solids
Students work in pairs. Each student selects one model, and then
traces some, but not all of its faces. The students exchange their
tracings and use them to identify their partner’s solid. They
record which figure is the base, and which figures are the faces.
Early Finishers
Introduce a “wild card” to the Face-Off game. This card can be
placed over a face card to change the identity of the solid being
created. For example, a student can switch the solid from a
triangular pyramid to a square pyramid by covering a triangle
face card with a square face card.
Common Misconceptions
➤Students may have difficulty identifying the bases of a solid.
How to Help: Remind students that a pyramid has one base,
and a prism has two bases. Suggest they start by identifying the
figure that appears least often in the solid.
As students play, suggest they refer to the
illustrations of solids in the Student Book, or to
models of solids.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• What solid could have a square as one face?
(Cube, any prism with square faces, a square
pyramid)
• What solid could not have a square as one
face? (A triangular pyramid, a cylinder, a sphere)
• What solid are you trying to show with your
game cards? (A rectangular prism)
• How many cards will you need to complete
this solid? (I have 1 square. A rectangular prism
has 6 faces, so I need 5 more cards.)
• What other faces will complete this solid?
(I need 1 square, and 4 rectangles.)
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Unit 3 • Lesson 8 • Student page 100
Watch to see if students continue to refer to the
illustrations in the Student Book as the game
proceeds. Listen to hear if students use the
words face and base, and the correct name for
each solid.
AFTER
Connect
Invite a group to share how they played the
game. They can demonstrate using transparent
Face-Off game cards on the overhead projector.
Have students describe the solid they made
with their cards. For example, 2 congruent
triangles and 3 congruent rectangles could form
a triangular prism.
If necessary, model how to use the appropriate
mathematical terms when describing figures
and solids.
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Making Connections
Science: Crystals often have the shape of familiar solids or
combinations of solids.
Have students investigate the faces of different crystals. For
example, a halite or salt crystal is a cube. A diamond crystal has
the shape of 2 square pyramids with bases that touch, also
called an octahedron. All faces of a diamond crystal are
triangles. A quartz crystal is a hexagonal prism with hexagonal
pyramid ends. Quartz crystals have rectangular and triangular
faces.
Refer students to the table in Connect. Explain
that the table shows the same figures as the
Face-Off game cards, but the faces are grouped
together beside the solid they form.
Practice
Have models of solids available for all
questions.
Encourage students to handle the solids and to
look at them from various aspects.
Assessment Focus: Question 4
Students use the properties of solids. For part a,
students understand that all pyramids have 1
base and triangular faces. They know that a
pyramid is named for the shape of its base. For
part b, students understand that prisms have 2
congruent bases, and that a prism is named for
the shape of its bases.
Students who need extra support to complete
Assessment Focus questions may benefit from
the Step-by-Step masters (Masters 3.18 to 3.28).
Unit 3 • Lesson 8 • Student page 101
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Sample Answers
2. a) 3 pairs of congruent rectangles
b) Circle
c) 2 congruent pentagons, 3 congruent rectangles, and
2 other congruent rectangles
Rectangular prism; rectangle
Cone; circle
Pentagonal prism; pentagon
3. Triangular prism (2 triangles and 3 rectangles); rectangular
pyramid (1 rectangle and 4 triangles); cube (6 squares)
Student answers should include sketches.
4. a) Square pyramid: square base and 4 congruent triangles
b) Pentagonal prism: 2 congruent pentagons and 5 congruent
rectangles
The base is the figure that names the solid. A pyramid has
1 base and triangular faces. Prisms have 2 bases and
rectangular faces.
REFLECT: A pyramid has one base that determines its name. The
other faces are triangles. A prism has 2 bases that determine
its name. The other faces are rectangles. The faces that are
not the base of a pyramid meet at 1 point. Students may draw
on personal experiences to describe other similarities and
differences.
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3
9
1
Numbers Every Day
29
Remind students that they can trade 1 ten for 10 ones to
represent the number in different ways.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students can identify a solid by
looking at its faces.
Extra Support: Students trace the faces of solids. On each
tracing, they write the name of the figure that forms the face.
Students use these tracings to explain the differences among the
different solids.
Students use Step-by-Step 8 (Master 3.25) to complete question 4.
✔ Students can sketch the faces of
a solid.
Communicating
✔ Students use appropriate
mathematical terms to describe the
faces of a solid.
Extra Practice: Students can do the Additional Activity, Go
Fish for Faces (Master 3.16).
Students can complete Extra Practice 4 (Master 3.34).
Extension: Students can explore the faces of solids other than
pyramids and prisms. For example, they can explore a solid
with 12 faces.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
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Unit 3 • Lesson 8 • Student page 102
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SSON 8A
Sample Answers
1. a) No
b) Yes
c) Yes
2. There are several possible nets. For example:
3. a)
Exploring Nets
of Solids
b)
LESSON ORGANIZER
Curriculum Focus: Draw nets for prisms and pyramids.
(SS17, SS18, SS19)
Student Materials
Optional
Lesson 8A (Master 3.13)
Step-by-Step 8A (Master 3.29)
cereal boxes
Extra Practice 5 (Master 3.35)
Toblerone box
scissors
tape
2-cm grid paper (PM 21)
Vocabulary: net, rectangular prism, triangular prism
Assessment: Master 3.2 Ongoing Observations: Geometry
c)
4. a) This solid must be a cube.
b) For example,
5. For example,
The square at the end of the rectangle can
move to any of the 3 rectangles.
REFLECT: Students should draw the net of a
rectangular prism. The net should reflect the
typical shape of a chocolate box, with 2 large,
congruent rectangles (or squares), and 2 pairs
of long, thin rectangles.
BEFORE
40–50 min
Key Math Learnings
1. Recognize and construct nets for rectangular prisms and cubes.
2. Investigate different nets for rectangular prisms and cubes.
Get Started
Show students a cardboard box, such as a cereal box. Ask:
• What solid does this box resemble? (Rectangular prism)
• Which figures can you see in this solid? (Rectangles)
Discuss what the box might look like if it were flattened.
Present Explore. Remind students to cut along the edges only until they can flatten the box. They
should not cut all of the pieces apart. They should sketch the fold lines of the box on their tracing.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• How did you decide which edges to cut? (We started with a long side, and then tried to flatten the box. We
looked at which sides were keeping us from flattening the box, and then cut along those edges.)
• How would you describe the figures you are forming by tracing over the fold lines? (There is 1 pair
of congruent rectangles, and another 4 congruent rectangles in the rectangular prism. There is 1 pair of congruent
triangles, and 3 other congruent rectangles in the triangular prism.)
Watch students to see if they understand they should be able to fold their box back together.
Unit 3 • Lesson 8A
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AFTER
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Connect
Invite volunteers to share their tracings. Ask:
• What is the same about all of the tracings? (Rectangular prisms all contain rectangles: 2 pairs of congruent
rectangles, and 1 more pair of congruent rectangles. The triangular prisms contain 1 pair of congruent triangles
and 3 congruent rectangles.)
• What is different? (The base and side rectangles have different sizes. The triangular prism contains both
triangles and rectangles.)
Introduce the term net as an arrangement of connected figures that can be folded to make a solid.
Discuss the steps in Connect that describe how to create a net for a rectangular prism.
Elicit from students that some pieces of a net can be moved to create a new net, but that any net
must fold to create a solid.
Practice
Questions 2, 3, 4, and Reflect require 2-cm grid paper.
Assessment Focus: Question 5
Students understand that the net must fold into a rectangular prism. They know a rectangular prism
has 6 faces, 2 of which are bases. Students see that one face of the prism is a square, and 2 faces are
congruent rectangles. They add 1 more congruent square, and 2 more congruent rectangles to
complete the net. Students should suggest where faces can be moved to create new nets.
REACHING ALL LEARNERS
Common Misconceptions
➤Students have difficulty visualising a net being folded, and cannot arrange the faces in the net correctly.
How to Help: Students should work with concrete materials while investigating nets. Have students practise folding a
net, cutting it apart, re-arranging the faces, and then folding it again. They can then move on to the more abstract
concept of drawing a net on grid paper.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that a net is a
cutout arrangement of figures that can
be folded to make a model of a solid.
Extra Support: Students can tape congruent paper squares
and rectangles together to create nets. They then fold their nets
along the taped edges to see if they fold into solids.
Students can use Step-by-Step 8A (Master 3.26) to complete
question 5.
Applying procedures
✔ Students can recognize and draw nets
for prisms and pyramids.
Extra Practice: Have students complete the Additional Activity,
Prisms and Pyramids (Master 3.17).
Students can do Extra Practice 5 (Master 3.35).
Extension: Students can explore nets of other solids such as a
hexagonal prism and a pentagonal pyramid.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
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Unit 3 • Lesson 8A
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ESSON 9
Solids in Our World
LESSON ORGANIZER
40–50 min
Curriculum Focus: Recognize and sort solids. (SS19)
Teacher Materials
models of solids
Optional
cards with solid sorting
methods
Step-by-Step 9 (Master 3.27)
Extra Practice 5 (Master 3.35)
Assessment: Master 3.2 Ongoing Observations: Geometry
Student Materials
Venn diagram (PM 28)
Plasticine
Key Math Learnings
1. Many new-world objects resemble solids.
2. Solids can be sorted according to their attributes.
Math Note
Modified Solids
Objects that resemble solids may have shapes that are
slightly different from those of their geometric
counterparts.
For example, a waste paper basket is roughly the shape
of a cylinder, but it is wider at one end.
The cylindrical shape has been modified to make the
object more useful.
You may wish to have students speculate why a particular
object’s shape has been modified from that of a true
geometric solid.
BEFORE
Get Started
Have students examine the photograph of the
farm in the Student Book. Ask:
• What makes the objects in this photograph
different from one another?
(The objects have different sizes and shapes; some
are rounded, while others have straight sides.)
• What solids can you see in this photograph?
(The tall silo is a cylinder with a half-sphere on top;
there are barns shaped like prisms. The tower in the
front looks like a prism with a square pyramid roof.)
Display a model of a cylinder and compare it to
the classroom garbage can.
Ask:
• How is the garbage can like a cylinder?
(It has a curved surface and a circular base.)
• How is the garbage can different from the
model of the cylinder? (One end of the garbage
can is wider than the other end. It has only 1 base.)
Present Explore. Students can record the results
of their sorting in a Venn diagram. Encourage
students to find a variety of ways to sort.
Unit 3 • Lesson 9 • Student page 103
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REACHING ALL LEARNERS
Alternative Explore
Students match objects illustrated in the photograph of the farm
to different solids. For example, the silo is the shape of a
cylinder, and the barn is a pentagonal prism. They sort the solids
in the farm scene according to different geometric attributes,
such as shapes of faces, numbers of faces, and so on.
Early Finishers
Students work in pairs. One student chooses a rule to sort models
of solids and/or classroom objects that resemble solids. He shows
the sorted solids to his partner. She tries to guess the sorting rule.
ESL Strategies
Provide visual references for the names of figures and solids.
Display labelled drawings showing faces, edges, and vertices.
Provide additional time to discuss attributes and sorting rules.
If students sort by 2 attributes, they will need a
diagram with 2 overlapping loops. If students
sort by more than 2 attributes, have them add
one loop to their diagram for each
additional attribute.
DURING
Explore
For example, solids with rectangular faces and
solids with triangular faces could be sorted into
2 overlapping sets. A rectangular prism belongs
to the rectangular face set, a triangular pyramid
belongs to the triangular face set, and a
triangular prism belongs to the overlapping set.
A cylinder belongs to neither set, and would be
placed outside of the loops.
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• What object did you find that resembles a
rectangular prism? (A tissue box, a new eraser)
• What attribute(s) are you using to sort the
solids? (I am sorting by the number of faces.)
• How can you sort your objects in a different
way? (I can sort by the number of bases, or by the
shapes of the bases.)
Watch to see if students sort into two sets, more
than two sets, or into overlapping sets.
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Unit 3 • Lesson 9 • Student page 104
AFTER
Connect
Invite volunteers to share their sorting methods
and attributes. Elicit from students that
different sorting methods are correct, as long as
they follow consistent rules or criteria.
Discuss the different attributes by which the
solids in Connect are sorted. Have students
re-sort their solids according to these attributes:
number of faces, number of edges, number of
vertices, and shapes of faces.
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Sample Answers
1. The ice cream cone has a vertex, a circular base, and a
cube
cone
sphere
rectangular
prism
curved surface. It is longer and thinner than the model cone,
and has an extra band around the open end.
The tissue box has 6 square faces. Unlike the model cube, it
has an opening in one face.
The cereal box has 6 rectangular faces, but the faces are a
different size than the faces in the model of the
rectangular prism.
The marble is smaller than the model of the sphere.
3. a) Has circular faces: cone, cylinder
Has more than 6 faces: hexagonal prism,
hexagonal pyramid
Has circular faces
Cone
Cylinder
Has more than 6 faces
Hexagonal
prism
Hexagonal
pyramid
square pyramid
rectangular prism
Cube, Triangular pyramid
= 134
= 179
= 252
pentagonal prism
Numbers Every Day
Successful students will combine several strategies, choosing the
most appropriate for each situation. For example, in the first
subtraction, they could count on from 322:
322 (+100) = 422 (+20) = 442 (+10) = 452 (+4) = 456
They would count on by: 100 + 20 + 10 + 4 = 134
Practice
Question 3 requires a Venn diagram.
Assessment Focus: Question 4
Students focus on the attributes of prisms. They
might use models of solids, or objects in the
classroom that resemble prisms, to help explain
their answers. Students should recognize that a
cube is a special type of rectangular prism, just
as a square is a special type of rectangle.
Unit 3 • Lesson 9 • Student page 105
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b) No circular faces: triangular pyramid, cube, hexagonal
prism, hexagonal pyramid
Less than 6 faces: triangular pyramid, cylinder, cone
Has no circular faces
Cube
Hexagonal Triangular
prism
pyramid
Hexagonal
pyramid
Has less than 6 faces
Cylinder
Cone
4. a) The number of vertices in a rectangular prism is always 8.
This does not change if the prism changes shape, so no
rectangular prism has 6 vertices.
b) Cubes have 6 square faces. Squares are special types of
rectangles, so a cube is also a rectangular prism.
c) Rectangular prisms do not always have square faces, so
only some rectangular prisms are cubes.
d) Triangular prisms have 5 faces, but the bases are triangles
and the other faces are rectangles. The faces of a
triangular prism are not all congruent.
5. A sharpened pencil is best represented as a cone on a
hexagonal prism, or a cone on a cylinder.
Student answer should show a sketch of the 2 solids joined to
make the shape of a sharpened pencil.
No
All
Some
No
REFLECT: I picture solids that I could put together to make the
object. If the object has a round part, it might have a cone,
cylinder, or sphere in it. If it is more like a box, it might have
a cube or rectangular prism in it.
Making Connections
At Home: Students can identify solids in their neighbourhood.
For example, many houses are pentagonal prisms.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that a solid in the
environment might not look exactly
like a model of the solid.
Extra Support: Give students a set of cards. Each card
describes a different way to sort solids. Have the students choose
a card, and then sort the models of solids according to the
criteria on the card. They then draw another card and sort the
solids again.
Students can use Step-by-Step 9 (Master 3.27) to complete
question 4.
Applying procedures
✔ Students recognize solids in their
own environment.
✔ Students can sort solids in a variety
of ways.
Extra Practice: Have students sort solids according to the
number of faces and bases, and then replace each model with a
classroom object that resembles it.
Students can complete Extra Practice 5 (Master 3.35).
Extension: Have students discuss how and why some solids are
modified to create functional objects.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
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Unit 3 • Lesson 9 • Student page 106
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Home
SSON 10
Designing Skeletons
optional
LESSON ORGANIZER
Curriculum Focus: Design and make skeletons of solids.
Teacher Materials
models of solids
Optional
scissors
Step-by-Step 10 (Master 3.28)
straws
Extra Practice 6 (Master 3.36)
Plasticine
3-column chart (PM 18)
Vocabulary: skeleton, edge, vertex
Assessment: Master 3.2 Ongoing Observations: Geometry
Student Materials
Key Math Learning
Skeletons show the edges and vertices of solids.
Curriculum Focus
The content of this lesson is not specifically required by the
Grade 4 curriculum. However, it is a review of work from
previous years. If you choose to do this lesson, allow
40–50 minutes.
BEFORE
Get Started
Discuss the photo of the building under
construction in the Student Book.
Ask questions, such as:
• What things make up the skeleton of this
building? (Wooden walls, a floor)
• What figures can you see in this skeleton?
(Rectangles, triangles, and squares)
• What might fill in the skeleton?
(Walls and windows)
Draw attention to the models of the solids.
Ask:
• How would constructing a skeleton of a
solid be similar to constructing a skeleton for
a building? (I build the frame of the solid. The
frame of the solid is like the frame of the building.)
• How would it be different?
(A building has extra supports, while the skeleton of
a solid only has edges and vertices.)
Present Explore. Remind students to use
mathematical words, such as edge and vertex,
when talking with their partners.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as,
• What materials do you need?
(I need a straw for each edge, and a ball of
Plasticine for each vertex.)
• What was your first step?
(I counted the edges and vertices in the solid. I
counted out 1 straw for each edge, and made 1 ball
of Plasticine for each vertex.)
• How do the different edges compare? How
did you give this information in your
instructions? (Some edges are twice as long as
others. I said I had to cut some straws in half.)
Unit 3 • Lesson 10 • Student page 107
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REACHING ALL LEARNERS
Alternative Explore
Materials: pretzels and marshmallows
Instead of using straws and Plasticine, students can construct
skeletons using pretzels and marshmallows. They can later eat
their skeletons.
Early Finishers
Students play Name That Skeleton with a partner. One student
states either the number of edges or the number of vertices in a
skeleton. The other student tries to name the skeleton. Students
switch roles and repeat.
Common Misconceptions
➤Students use toothpicks or straws in places that are not edges.
For example, they might use a toothpick or a straw across the
face of a cube.
How to Help: Draw students’ attention to the edges on the
models. Remind them that a straw or a toothpick in the skeleton
represents an edge of the solid.
Edges
Vertices
4
square or rectangular
pyramid
triangular prism
12
8
• How did you decide the order of your
instructions? (I gave the instructions in the order
that I constructed the skeleton.)
Review the information in the table in Connect.
You could have students make a similar chart
for their skeletons.
Watch to see if a student can follow his
partner’s instructions. Listen for comments
regarding omitted details or vague wording.
Model how to show equal edges with
hatch marks.
Practice
AFTER
Connect
Invite a volunteer to show a skeleton she
constructed. Discuss with the class, ways to
describe the skeleton.
• How many vertices does this skeleton
have?(8)
• How many edges does it have? (12)
• How do the lengths of the edges in this
skeleton compare? (They are all the same length.)
• Which solid does this skeleton represent?
(A cube)
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Unit 3 • Lesson 10 • Student page 108
Question 1 requires a 3-column chart. Question
2 requires Snap Cubes or congruent cubes.
Students can use models of solids.
Assessment Focus: Question 3
Students recognize that, although there are a lot
of possible edges, there are only 6
marshmallows, so it is only possible to make
skeletons with 6 vertices or less. They
determine which solids have six vertices or
less. They rule out solids for which skeletons
cannot be made with straight edges, such as
cones and cylinders.
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Sample Answers
2. a) You would need 8 marshmallows.
b) 16 marshmallows and 28 toothpicks
c) 20 marshmallows and 34 toothpicks
3. Triangular pyramid
Square pyramid
Rectangular pyramid
Triangular prism
Pentagonal pyramid
4. Skeletons cannot be made for cones, cylinders, and spheres
because these solids do not have edges.
REFLECT: Pyramids and prisms have vertices and edges, which
make it easy to create skeletons. Skeletons for solids with all
edges equal are the easiest to construct.
1,2,3,4,1,2,3,4; core 1,2,3,4
1,2,3,4,5,6,7,8; start at 1 and add 1.
1,2,4,7,11,16,22,29; start at 1. Add 1, then
increase the number you add by 1 each time.
1,2,1,2,1,2,1,2; core 1,2
Numbers Every Day
Remind students that number patterns can grow or repeat.
Students can make patterns that grow by adding or multiplying
by the same number each time, or by adding increasing
numbers each time. They can use calculators if necessary.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that solids are
built around frames of vertices
and edges.
Extra Support: Have students use a non-permanent marker to
mark edges on a model of a solid. They then use a different
colour to mark the vertices on the model. Students use these
markings to help build a skeleton of the solid.
Students can use Step-by-Step 10 (Master 3.28) to complete
question 3.
Applying procedures
✔ Students can make skeletons of solids.
Problem solving
✔ Students choose appropriate
strategies to solve problems involving
geometric models.
Extra Practice: Students can complete Extra Practice 6
(Master 3.36).
Extension: Have students create large-scale skeletons by using
cardboard rolls from paper towels or toilet paper, and masking
tape. Alternatively, they could use tightly rolled newspapers.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
Unit 3 • Lesson 10 • Student page 109
45
LESSON 11
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Strategies Toolkit
40–50 min
LESSON ORGANIZER
Curriculum Focus: Interpret a problem and select an
appropriate strategy.
Teacher Materials
models of solids
Student Materials
Snap Cubes or congruent cubes
2-column charts (PM 17)
Vocabulary: volume
Assessment: PM 1 Inquiry Process Check List,
PM 3 Self-Assessment: Problem Solving
There are 8 possible prisms. The dimensions
are: 1 by 1 by 36; 1 by 2 by 18;
1 by 3 by 12; 1 by 4 by 9; 1 by 6 by 6; 2 by 2 by 9; 2 by 3 by 6; 3 by 3 by 4.
Key Math Learning
Using a model can help solve problems involving the building of
rectangular prisms.
BEFORE
Get Started
Show students a variety of models of
rectangular prisms. Elicit from students that
each rectangular prism takes up a certain
amount of space. Tell students that we call this
space the solid’s volume.
Present Explore. Remind students that the solids
must be rectangular prisms, and all of them
must have a volume of 36 cubes. Suggest
students record their work in a 2-column chart.
Note: Use a smaller volume if you do not have
enough cubes to give 36 to each group.
Use any amount but 24.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• How are you going to solve the problem?
(I will construct different rectangular prisms with
my cubes.)
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Unit 3 • Lesson 11 • Student page 110
• How do cubes help you?
(I can use them to see what I am doing. By
arranging cubes in an organized way, I can keep
track of which prisms I have made.)
• How do you know this prism has a volume
of 36 cubes? (I used 36 cubes to build it, so its
volume is 36 cubes.)
Watch how the students approach the problem.
Observe how they make use of available
materials. Do they work systematically in order
to find all of the different possible prisms?
AFTER
Connect
Invite volunteers to share their strategies for
solving the problem. Display or describe the
different rectangular prisms that have a volume
of 36 Snap Cubes.
Work through the problem in Connect as a class.
Discuss methods of checking if all possible
prisms have been found.
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REACHING ALL LEARNERS
Common Misconceptions
➤Students may see two congruent prisms with different
orientations as two different prisms.
How to Help: Have students count the number of cubes along
the length, the width, and the height of the prism. Any prisms for
which these 3 measures are the same are congruent prisms.
Alternatively, have students trace around each different face.
Any prisms that have the same set of faces are congruent prisms.
Sample Answers
1. Cubes have square faces, so each edge must have the same
There are 6 possible prisms. The dimensions are: 1 by 1 by 24;
1 by 2 by 12; 1 by 3 by 8; 1 by 4 by 6; 2 by 2 by 6; 2 by 3 by 4.
length. Possible cubes are: 1 by 1 by 1; 2 by 2 by 2;
3 by 3 by 3; and 4 by 4 by 4.
2. There are 2 rectangular prisms with a volume of 9
Snap Cubes: 1 by 1 by 9; and 1 by 3 by 3.
There are 4 rectangular prisms with a volume of 18
Snap Cubes: 1 by 1 by 18; 1 by 2 by 9; 1 by 3 by 6; and
2 by 3 by 3.
REFLECT: Building a model gives you a way to show your
thinking that is easy for others to understand. I built a model
with a length of 8 units, a width of 1 unit, and a height of
3 units. My friend built a model with a length of 3 units, a
width of 1 unit, and a height of 8 units. We could see that we
had built the same model. If we did not have models to look
at, we might have thought these rectangular prisms were
different prisms.
Ask:
• What other strategies might we use to solve
this problem?
(An organized list would help keep track of the
prisms, and make it easier to look for any patterns.)
Practice
Have Snap Cubes or congruent cubes available.
Students could use a 2-column chart to record
their answers.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Problem solving
✔ Students solve problems using
geometric models.
Extra Support: Give each of 4 students 12 Snap Cubes or
congruent cubes. Have them work together to find the
4 possible prisms.
✔ Students work systematically to find
all solutions to a geometric problem.
Extra Practice: Repeat Explore with 16 cubes.
Extension: Have students investigate volumes that can only be
modelled with one rectangular prism. For example, 7 cubes and
13 cubes
Recording and Reporting
PM 1 Inquiry Process Check List
PM 3 Self-Assessment: Problem Solving
Unit 3 • Lesson 11 • Student page 111
47
S H O W W H A T Y O U K NHome
OW
LESSON ORGANIZER
Quit
40–50 min
Student Materials
protractors
straws
scissors
Plasticine
Snap Cubes
Assessment: Masters 3.1 Unit Rubric: Geometry,
3.4 Unit Summary: Geometry
less than a
right angle
greater
than a right
angle
less than a
right angle
Curriculum Focus
Do not assign question 5 if students did not complete the lesson
on similar figures. Question 1 can be modified by having students
tell if each angle is less than or greater than a right angle.
Square
Rectangle
Trapezoid
Sample Answers
2. For example:
The figure has 2 parallel sides: A, B, C, E
The figure has 1 right angle: A, C, E
Student answer should include a Venn diagram.
3. A is a square. A square has 4 sides equal, and
4 right angles.
B is a trapezoid. A trapezoid has 2 parallel sides.
E is a rectangle. A rectangle has opposite sides equal, and
4 right angles.
4. a) A rectangle is not a square, because a square has 4 sides
equal. A rectangle only has opposite sides equal.
b) A square is a rectangle because a rectangle has
opposite sides equal. A square has 4 sides equal, so
its opposite sides are equal.
c) A rhombus is not always a square because a square
has 4 right angles. A rhombus does not always have
4 right angles.
d) A rhombus is a parallelogram because a
parallelogram has opposite sides equal. A rhombus
has 4 sides equal, so its opposite sides are equal.
5. A and C are similar. A and C both have the shape of
squares with one corner removed, but C is larger than
A. B is a rectangle with one corner removed. It is not the
same shape as A or C, so it is not similar to A or C.
6. a) Square pyramid: 1 square base, 4 congruent
triangular faces, for a total of 5 faces
b) Rectangular prism: 2 congruent rectangular bases,
2 pairs of congruent rectangular faces, for a total of
6 faces
(Student answer should include a sketch of the faces.)
7. a) Cube; 6 congruent squares
b) Cylinder; 2 congruent circles
c) Rectangular prism; 4 congruent rectangles, and
2 larger congruent rectangles
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Unit 3 • Show What You Know • Student page 112
Yes
No
No
Yes
(Note: Some students may say that the sandwich is also a
rectangular prism, with faces similar to those of the
container.) (Student answers should include a sketch of
the faces.)
8. The marshmallows are the solid’s vertices. A student
would model them with Plasticine. The toothpicks are the
solid’s edges. A student would model these with straws.
A student can make, and then sketch, one of the
following:
Solid
Edges
Vertices
triangular pyramid
square pyramid
triangular prism
pentagonal pyramid
6
8
9
10
4
5
6
6
9. 1 by 1 by 20; 1 by 2 by 10; 1 by 4 by 5; 2 by 2 by 5
Total: 4
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SHOW YOUR BEST
Explain
Students are often confused about what to do when asked to
explain their answers. Share this tip with your students: Suggest
they think about, and write down, the strategies they used to
arrive at their answer.
Square pyramid
Rectangular prism
Encourage students to use the correct mathematical terminology
in their explanation. Model how to do this. For example, explain
how you know a square is a rhombus by identifying the
attributes of squares and rhombuses. (I know a square is a
rhombus because rhombuses have 4 equal sides and 4 opposite
angles equal. A square has 4 equal sides and 4 right angles, so
a square is a rhombus.)
4
ASSESSMENT FOR LEARNING
What to Look For
Reasoning; Applying concepts
✔ Question 2: Student demonstrates understanding by choosing appropriate attributes.
✔ Question 8: Student demonstrates understanding by identifying objects that model parts of the skeleton of
a solid.
Accuracy of procedures
✔ Question 1: Student describes angles relative to a right angle.
✔ Question 6: Student identifies and sketches faces of a solid.
Problem solving
✔ Question 9: Student chooses an appropriate strategy to find rectangular prisms with a specific volume.
Recording and Reporting
Master 3.1 Unit Rubric: Geometry
Master 3.4 Unit Summary: Geometry
Unit 3 • Show What You Know • Student page 113
49
UNIT PROBLEM
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Under Construction
LESSON ORGANIZER
80–100 min
Student Grouping: 4
Teacher Materials
models of solids
Student Materials
straws
scissors
Plasticine
1-cm grid paper (PM 20)
rulers
protractors
Assessment: Masters 3.1 Unit Rubric: Geometry,
3.3 Performance Assessment Rubric: Under Construction
Curriculum Focus
This Unit Problem contains content that is not required
by your curriculum.
Modify the problem as follows:
Part 1:
Part 2:
Have students design and construct nets for
various solids. The nets can be folded and
taped to form parts of the castle.
Rather than measuring the angles of the
figures, have students identify each angle as a
right angle, less than a right angle, or greater
than a right angle.
Have students use terms such as horizontal, vertical,
intersecting, parallel, and perpendicular to describe
some of the lines in their design.
Invite a volunteer to read Parts 1 and 2 aloud.
Students should work in groups of 3 or 4 to
complete the tasks.
Use information from the Check List and the
Performance Assessment Rubric: Under
Construction (Master 3.3) to help clarify what
is expected from the students’ work.
Have illustrations of castles available to
spark ideas.
Observe how students use the materials to
represent edges and vertices in their skeletons.
Listen to hear if students are using geometric
terms correctly.
Remind students of the castle in the Launch.
Refer to the ideas discussed during the Launch.
Observe how students organize the task to
ensure they include all parts of the problem.
Do they refer to the Student Book?
If responses to the Launch questions were
recorded, display them. Tell students that the
Unit Problem will show what they have learned
about angles, figures, and solids.
Students should have the opportunity to share
and display their skeletons, sketches, and
diagrams with the rest of the class.
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Sample Response
Part 1
The skeleton should include many different solids. The sketch
should be neat and accurate, with the various solids labelled.
Students may include a description of why they chose particular
shapes for parts of their castle. For example, a turret is in the
shape of a cylinder so that someone inside it can see in
all directions.
Part 2
The wall design should include different figures that students
have encountered in this unit. Students may explain why they
made certain figures congruent. Angles should be measured
accurately and labelled.
Reflect on the Unit
Students should sketch the main figures they used. They could
include the faces of solids from their skeleton, as well as the
figures from their wall design. The sketches should include
information about the angles and side-lengths of the figures.
Teaching notes for the Cross Strand Investigation, The Icing on
the Cake, are in the Additional Assessment Support module.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that a solid is
related to the figures that make its faces.
Extra Support: Make the problem accessible.
Applying procedures
✔ Students can correctly identify solids
and figures.
✔ Students can describe angles.
Communicating
✔ Students use geometric terms and
symbols correctly.
Some students may require more direction to complete both Parts
1 and 2. Suggest students use models of solids to create their
castle. They can then make, and join, the models of each solid.
Alternatively, have the students create a model for part of the
castle illustrated in the Student Book. For example, they could
make a model of a rectangular prism for the castle’s tower. They
could then add more elements to their model.
For Part 2, students could create and extend a design for the wall
of the castle tower.
Recording and Reporting
Master 3.3 Performance Assessment Rubric: Under Construction,
Master 3.4 Unit Summary: Geometry
Unit 3 • Unit Problem • Student page 115
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Evaluating Student Learning: Preparing to Report:
Unit 3 Geometry
This unit provides an opportunity to report on the Shape and Space: Geometry strand.
Master 3.4 Unit Summary: Geometry provides a comprehensive format for recording and summarizing
evidence collected.
Here is an example of a completed summary chart for this Unit:
Key:
1 = Not Yet Adequate
2 = Adequate
3 = Proficient
4 = Excellent
Strand: Shape and
Space: Geometry
Reasoning;
Applying
concepts
Accuracy of
procedures
Problem
solving
Communication
Overall
Ongoing Observations
2
2
1
2
2
Work samples or
portfolios; conferences
2
2
1
2
2
Show What You Know
2
2
2
2
2
Unit Test
2
3
2
Unit Problem
Under Construction
2
3
2
Strategies Toolkit
1
Achievement Level for reporting
1
2
2
2
2
Recording
How to Report
Ongoing Observations
Use Master 3.2 Ongoing Observations: Geometry to determine the most consistent level
achieved in each category. Enter it in the chart. Choose to summarize by achievement
category, or simply to enter an overall level.
Observations from late in the unit should be most heavily weighted.
Strategies Toolkit
(problem solving)
Use PM 1: Inquiry Process Check List with the Strategies Toolkit (Lesson 11). Transfer
results to the summary form. Teachers may choose to enter a level in the Problem solving
column and/or Communication.
Portfolios or collections of
work samples; conferences,
or interviews
Use Master 3.1 Unit Rubric: Geometry to guide evaluation of collections of work and
information gathered in conferences. Teachers may choose to focus particular attention on
the Assessment Focus questions.
Work from late in the unit should be most heavily weighted.
Show What You Know
Master 3.1 Unit Rubric: Geometry may be helpful in determining levels of achievement.
#2 and 8 provide evidence of Reasoning; Applying concepts; #1 & 6 provide evidence of
Accuracy of procedures; #9 provides evidence of Problem solving; all provide evidence of
Communication.
Unit Test
Master 3.1 Unit Rubric: Geometry may be helpful in determining levels of achievement.
Part A provides evidence of Accuracy of procedures; Part B provides evidence of
Reasoning; Applying concepts; Part C provides evidence of Problem solving; all parts
provide evidence of Communication.
Unit performance task
Use Master 3.3 Performance Assessment Rubric: Under Construction. The Unit Problem
offers a snapshot of students’ achievement. In particular, it shows their ability to synthesize
and apply what they have learned.
Student Self-Assessment
Note students’ perceptions of their own progress. This may take the form of an oral or
written comment, or a self-rating.
Comments
Analyse the pattern of achievement to identify strengths and needs. In some cases, specific
actions may be planned to support the learner.
Learning Skills
Ongoing Records
PM 4: Learning Skills Check List
Use to record and report throughout a reporting period, rather
than for each unit and/or strand.
PM 10: Summary Class Records: Strands
PM 11: Summary Class Records: Achievement Categories
PM 12: Summary Record: Individual
Use to record and report evaluations of student achievement over
several clusters, a reporting period, or a school year.
These can also be used in place of the Unit Summary.
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Unit Rubric: Geometry
Not Yet Adequate
Adequate
Proficient
shows some
understanding (may be
vague or incomplete);
partially able to:
– describe properties
– compare and sort
figures and solids
– explain or
demonstrate
relationships
shows understanding;
able to clearly and
appropriately:
Excellent
Reasoning;
Applying concepts
• shows understanding of shows little
understanding; may be
figures and solids by:
unable to:
– describing and
– describe properties
making
generalizations
– compare and sort
figures and solids
– comparing and
sorting
– explain or demonstrate
relationships
– explaining or
demonstrating
– describe geometric
relationships
properties in everyday
experiences
– describing examples
in everyday
experiences
– describe geometric
properties in
everyday experiences
– describe properties
– compare and sort
figures and solids
– explain or
demonstrate
relationships
– describe geometric
properties in
everyday experiences
shows thorough
understanding; in
various contexts, able
to precisely and
effectively:
– describe properties
– compare and sort
figures and solids
– explain or
demonstrate
relationships
– describe geometric
properties in
everyday
experiences
Accuracy of
procedures
• identifies and classifies
lines, angles, figures,
and solids according to
their attributes
• constructs and relates
nets to 3-D solids
makes major errors in:
– identifying and
classifying lines,
angles, figures, and
solids
– constructing and
relating nets
makes frequent minor
errors in:
– identifying and
classifying lines,
angles, figures, and
solids
– constructing and
relating nets
makes few errors in:
– identifying and
classifying lines,
angles, figures, and
solids
– constructing and
relating nets
makes no errors in:
– identifying and
classifying lines,
angles, figures, and
solids
– constructing and
relating nets
Problem-solving
strategies
• uses a range of
appropriate strategies
to investigate and
create geometric
problems
may be unable to use
appropriate strategies to
investigate and create
geometric problems
with limited help, uses
some appropriate
strategies to
investigate and create
geometric problems;
partially successful
uses appropriate
strategies to
investigate and create
geometric problems
successfully
uses appropriate, often
innovative strategies to
investigate and create
geometric problems
successfully
• explains reasoning and
procedures clearly
unable to explain
reasoning and
procedures clearly
partially explains
reasoning and
procedures
explains reasoning and
procedures clearly
explains reasoning and
procedures clearly,
precisely, and
confidently
• uses appropriate
geometric terms and
symbols (e.g., names
of lines, figures, and
solids)
uses few appropriate
mathematical terms or
symbols appropriately
uses some appropriate
mathematical terms
and symbols
uses appropriate
mathematical terms
and symbols
uses a range of
appropriate
mathematical terms
and symbols with
precision
Communication
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Master 3.2
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Ongoing Observations: Geometry
The behaviours described under each heading are examples; they are not intended to be an exhaustive list of all
that might be observed. More detailed descriptions are provided in each lesson under Assessment for Learning.
STUDENT ACHIEVEMENT: Geometry
Student
Reasoning;
Applying concepts
Describes
properties
Explains
relationships
Offers reasoned
predictions, and
generalizations
Accuracy of
procedures
Identifies and
classifies lines,
angles, figures, and
solids
Constructs and
relates nets to
solids
Problem solving
Solves/creates
problems involving
figures and solids
(including
constructions)
Use locally or provincially approved levels, symbols, or numeric ratings.
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Communication
Uses mathematcial
language and
symbols
(e.g., attributes)
Explains
procedures and
solutions
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Performance Assessment Rubric:
Under Construction
Not Yet
Adequate
Adequate
Proficient
Excellent
unable to explain or
apply:
– attributes of figures,
including angles
– congruence
– relationships between
figures and solids
partially explains, and
applies:
– attributes of figures,
including angles
– congruence
– relationships between
figures and solids
explains and applies:
– attributes of
figures, including
angles
– congruence
– relationships
between figures
and solids
thoroughly and
effectively explains and
applies:
– attributes of figures,
including angles
– congruence
– relationships between
figures and solids
makes major errors in:
– naming objects and
figures
– sketching figures
– describing angles and
lines
– constructing nets
makes frequent minor
errors in:
– naming objects and
figures
– sketching figures
– describing angles and
lines
– constructing nets
makes few errors in:
– identifying objects
and figures
– sketching figures
– describing angles
and lines
– constructing nets
rarely makes errors in:
– identifying objects and
figures
– sketching figures
– describing angles and
lines
– constructing nets
uses few effective
strategies to:
– design the castle and
build its model; may be
unworkable
– incorporate the
required figures into
window design
uses some appropriate
strategies, with partial
success, to:
– design the castle and
build its model; may
have major flaws
– incorporate the
required figures into
window design
uses appropriate
and successful
strategies to:
– design the castle
and build its model;
may have some
flaws
– incorporate the
required figures
into window design
uses innovative and
effective strategies to:
– design the castle and
build its model; may
have minor flaws
– incorporate the
required figures into
window design
• explains design clearly
unable to explain design
clearly
partially explains
design
explains design
clearly
explains design clearly,
precisely, and
confidently
• uses appropriate terms
and symbols related to
geometric properties
and relationships
(e.g., names of figures
and solids, congruent,
degrees)
uses few appropriate
mathematical terms or
symbols
uses some appropriate
mathematical terms
and symbols
uses appropriate
mathematical terms
and symbols
uses a range of
appropriate
mathematical terms and
symbols with precision
Reasoning; Applying
concepts
• shows understanding by
demonstrating,
explaining and applying
concepts in geometry,
including:
– attributes of figures,
including angles
– congruence
– relationships between
figures and solids (e.g.,
castle, wall, and sketch)
Accuracy of
procedures
• accurately:
– identifies objects and
figures
– sketches a variety of
figures (windows),
including congruent
figures, on graph paper
– describes angles
– constructs nets
Problem-solving
strategies
• uses appropriate
strategies to design:
– a castle model that can
be built from materials
– windows that include
congruent figures, and
examples of the figures
studied
Communication
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Master 3.4
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Unit Summary: Geometry
Review assessment records to determine the most consistent achievement levels for the assessments conducted.
Some cells may be blank. Overall achievement levels may be recorded in each row, rather than identifying
levels for each achievement category.
Most Consistent Level of Achievement*
Strand: Shape and Space:
Geometry
Reasoning;
Applying
concepts
Accuracy of
procedures
Problem
solving
Communication
Ongoing Observations
Strategies Toolkit
(Lesson 11)
Work samples or portfolios;
conferences
Show What You Know
Unit Test
Unit Problem:
Under Construction
Achievement Level for reporting
*Use locally or provincially approved levels, symbols, or numeric ratings.
Self-Assessment:
Comments: (Strengths, Needs, Next Steps)
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OVERALL
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Master 3.5
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To Parents and Adults at Home …
During the next three weeks, your child’s class will be exploring geometry.
Through daily activities, your child will explore the relationship between flat,
two-dimensional figures and solid, three-dimensional objects in the world
around them.
In this unit, your child will:
• Construct congruent figures.
• Explore angles.
• Recognize and identify horizontal, vertical, perpendicular, intersecting,
and parallel lines.
• Sort and classify figures.
• Explore solids.
• Build nets.
Geometry is an important part of a student’s mathematical experience.
Geometry provides students with a strong link between the mathematics they
learn in the classroom and the real world.
Here are some suggestions for activities to do at home.
Look around the kitchen for different objects that have the same shape as a
solid. For example, a can of soup is a cylinder, a cereal box is a rectangular
prism, and an orange is a sphere.
Find objects that have the same shape, but have different sizes. For example,
drinking glasses often have the same shape, but come in different sizes.
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Master 3.6
Figures 1
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6-Division Protractor
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Master 3.8
Quadrilaterals 1
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Quadrilaterals 2
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Master 3.10
Quadrilaterals Venn Diagram
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Master 3.11
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Figures 2
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Master 3.11b
Figures 2
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Master 3.12
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Face-Off Game Cards
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Master 3.13a
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LESSON 8A: Exploring Nets of Solids
EXPLORE
You will need a cereal box or a Toblerone box and a pair of scissors.
Cut along the edges of the box until you can lay it flat.
Place the flattened box on a large piece of paper.
Trace the box and cut out the tracing.
Use a ruler to draw the fold lines on the tracing.
Write about the figures you see.
Fold the tracing along the fold lines.
Show and Share
Share your tracing with another pair of students.
How are your tracings the same?
How are they different?
CONNECT
A cutout that we can fold to form a model of a solid is called a net.
We can make a net for a solid from its faces.
The faces must be arranged so that they can be folded to make the solid.
There are different ways to arrange the faces to make a net.
This rectangular prism has 2 congruent square faces and 4 congruent
rectangular faces.
Here are the steps to make a net for this prism.
Trace around a
square face
2 times.
Lesson Focus: Draw nets for prisms and pyramids.
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Master 3.13b
Trace around a
rectangular face
4 times.
Place the
rectangles as
shown.
Tape the longer
sides together.
Tape a square to
each end of one
rectangle.
To check that this
is a net, fold it to
make a
rectangular prism.
Here is another net for the same
rectangular prism. One of the
congruent squares is in a different
position.
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Master 3.13c
PRACTICE
1. Which of these pictures are nets of a cube? How do you know?
a)
b)
c)
2. How many different nets can you make for a cube? Draw each net on grid
paper. How do you know all of them are different?
3. Design and draw a net for:
a) a square pyramid
b) a triangular pyramid
c) a triangular prism
4. The net for a solid has 3 pairs of congruent rectangles.
a) What kind of solid is it? How do you know?
b) Draw a net for the solid.
5. This is part of a net for a rectangular prism. Copy this figure on grid
paper. Draw the other faces to complete the net. How many different
ways can you do this? Show your work.
Reflect
Draw a net that you could use to make a box to hold chocolates. What kind of
solid will your net make? Explain how you made your net.
Assessment Focus: Question 5
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Master 3.14
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Additional Activity 1:
Look Out for Angles
Work with a partner.
You will need old magazines, scissors, glue, a card with a square corner,
and heavy paper.
Look for angles in the magazines.
Cut out each angle.
Use the card to measure the angles as less than, equal to,
or greater than a right angle.
Sort the angles by these attributes:
•
Has all angles less than a right angle.
•
Has all right angles.
•
Has all angles greater than a right angle.
Glue the angles on heavy paper to make an angle collage.
Take It Further: Draw a picture. Include items with right angles, angles less
than a right angle, and angles greater than a right angle.
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Master 3.15
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Additional Activity 2:
Congruent Figures
Work with a partner.
You will need a ruler, triangular or square grid paper, scissors, glue,
and heavy paper.
Draw 10 four-sided figures each.
Write your initials on each figure.
Cut out each figure.
Place your figures and your partner’s figures on a table.
Look for congruent figures.
If you find no congruent figures, choose one figure and draw a figure
congruent to it on grid paper.
Glue each pair of congruent figures on heavy paper.
Write how you know the figures in each pair are congruent.
Take It Further: Repeat the activity. Draw figures that are not
four-sided figures.
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Additional Activity 3:
Go Fish for Faces
Play with a partner.
You will need 36 Face-Off game cards (Master 3.12) and models of solids.
Each card shows the face of a solid.
The goal is to use all your cards to make solids.
How to play:
1. Decide who will be the dealer.
The dealer deals 6 cards to each player.
Players do not show their cards.
The deck of remaining cards is placed face down.
2. Players take turns. Player A looks at his cards.
If the cards show the faces of a solid, he places the cards face up and
says the name of the solid.
3. If Player A cannot make a solid with his cards,
he asks Player B for a card he needs to complete a solid.
If Player B has this card, she gives it to Player A.
If Player B does not have this card, she tells Player A to “go fish.”
Player A takes a card from the deck.
4. Player B has a turn.
5. Play continues until one player has no cards left or until all the cards
have been used.
The first player to use all his cards, or the player with the fewer cards left
when all the cards have been used, is the winner.
Take It Further: Play the game again. Add cards that show different faces,
such as hexagons and pentagons.
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Additional Activity 4:
Prisms and Pyramids
Work with a partner.
You will need models of various prisms and pyramids, and 4-column charts.
Select 2 different prisms.
Name them.
Work together. Look at one of the prisms.
Count the number of faces, edges, and vertices.
Record your findings in a table.
Count the number of faces, edges, and vertices on the other prism.
Record your findings.
Tell how the prisms are similar.
Tell how the prisms are different.
Select 2 different pyramids.
Name them.
Look at one of the pyramids.
Count the number of faces, edges, and vertices.
Record your findings in a chart.
Count the number of faces, edges, and vertices on the other pyramid.
Record your findings.
Tell how the pyramids are similar.
Tell how the pyramids are different.
Take It Further: Choose 1 prism and 1 pyramid. Tell how the models are alike
and how they are different.
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Master 3.18
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Step-by-Step 1
Lesson 1, Question 4
Use a geoboard or square dot paper.
Make each figure.
Join the dots to divide each figure.
Check that you understand the meaning of “congruent.”
Step 1 Divide this figure into 3 congruent triangles.
Hint: Make each triangle 2 units long at the bottom.
Step 2 Divide this figure into 3 congruent rectangles.
Hint: Make 1 side of each rectangle 2 units long.
Step 3 Divide this figure into 4 congruent shapes.
Hint: Make 4 rectangles.
Which figure can you divide in different ways?
_______________________________________________________
Why can you not divide the other figures in different ways?
_______________________________________________________
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Master 3.19
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Step-by-Step 2
Lesson 2, Question 6
Step 1 Use a ruler and draw a line. Mark one end of the line with a dot.
Step 2 Use a ruler to draw another line that starts at the dot.
Step 3 Use a 6-division protractor transparency to measure your angle.
Place the baseline of the protractor on one line.
Place the centre mark of the protractor on the dot.
Count from 0 along the protractor until you reach the other line.
Read and record the angle’s measure.
_______________________________________________________
Step 4 Use the words baseline, arm, vertex, and degrees to explain what
you did.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
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Master 3.20
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Step-by-Step 3
Lesson 3, Question 4
Step 1 Look at the 90º mark on a protractor.
What kind of angle measures 90º?
_______________________________________________________
Step 2 Use a ruler to draw an angle you think is less than 90º.
Step 3 Use a ruler to draw an angle you think measures 90º.
Step 4 Use a ruler to draw an angle you think is greater than 90º.
Step 5 Use a protractor to check that each angle is the correct size.
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Step-by-Step 4
Lesson 4, Question 6
Step 1 List 3 attributes of parallelograms.
_______________________________________________________
_______________________________________________________
_______________________________________________________
Step 2 Use a ruler and draw a
parallelogram on the dots.
Step 3 Write something about a parallelogram that is never true.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Step 4 Write something about a parallelogram that is sometimes true.
_______________________________________________________
_______________________________________________________
_______________________________________________________
Step 5 Write something about a parallelogram that is always true.
_______________________________________________________
_______________________________________________________
_______________________________________________________
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Master 3.22
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Step-by-Step 5
Lesson 5, Question 4
Step 1 List some attributes of a square. Hint: Think about angles and sides.
_______________________________________________________
_______________________________________________________
Why is this quadrilateral not a square?
Step 2 List some attributes of a rectangle. Hint: Think about angles and sides.
_______________________________________________________
_______________________________________________________
Why is this quadrilateral not a rectangle?
Step 3 List some attributes of a rhombus.
_______________________________________________________
_______________________________________________________
Why is this quadrilateral not a rhombus?
Step 4 List some attributes of a kite.
_______________________________________________________
_______________________________________________________
Why is this quadrilateral not a kite?
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Master 3.23
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Step-by-Step 6
Lesson 6, Question 4
Use the “Attributes of Quadrilaterals” chart in your book to solve these
riddles.
All the figures are quadrilaterals.
Write down all the different figures you find for each riddle.
a) I do not have any right angles.
All my sides are the same length.
What am I?
_____________________________________________________
b) All 4 of my angles are right angles.
I have 2 pairs of equal sides.
What am I?
_____________________________________________________
c) I have 2 parallel sides.
I have 2 right angles.
What am I?
_____________________________________________________
d) Make up your own riddle by filling in two or more of these phrases:
I have _____ parallel sides.
I have _____ right angles.
I have _____ opposite sides equal.
I have _____ adjacent sides equal.
Trade riddles with a classmate.
Solve your classmate’s riddle.
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Master 3.24
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Step-by-Step 7
Lesson 7, Question 3
Step 1 What makes 2 figures similar?
Hint: Think about the lengths of sides and the sizes of angles.
_______________________________________________________
_______________________________________________________
Use words and pictures to show your answer for each of these questions.
Step 2 Are all squares similar?
________________________________
________________________________
________________________________
Step 3 Are all rectangles similar?
________________________________
________________________________
________________________________
Step 4 Are all triangles similar?
________________________________
________________________________
________________________________
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Master 3.25
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Step-by-Step 8
Lesson 8, Question 4
Step 1 Use words and pictures. Explain the difference between
a pyramid and a prism.
_________________________________
_________________________________
_________________________________
Step 2
Are these the faces of a pyramid or a prism? ______________________
What is the name of the solid? ___________________________________
How do you know? ____________________________________________
_______________________________________________________
_______________________________________________________
Step 3
Are these the faces of a pyramid or a prism? ______________________
What is the name of the solid? ___________________________________
How do you know? ____________________________________________
_______________________________________________________
_______________________________________________________
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Step-by-Step 8A
Lesson 8A, Question 5
This is part of a net for a rectangular prism.
Step 1 How many faces make up a rectangular prism? _________________
How many faces do you need to add to this figure to make a
rectangular prism? ________________________________________
Step 2 Copy the figure on grid paper.
Use the same paper and sketch the faces you need to add.
Step 3 Cut out the figure and the faces.
Place the cutouts together to make a net for a rectangular prism.
Use tape to join the cutouts.
Can you fold your creation to make a rectangular prism?
______________________________________________________________
Step 4 Sketch the net you made.
Cut apart your net, and re-arrange the pieces to make another net.
Sketch this net.
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Step-by-Step 9
Lesson 9, Question 4
Think about how to sort solids using faces, edges, and vertices.
Think about how to sort solids using the shapes of their bases.
Complete each sentence. Use “all,” “some,” or “no” to make each sentence true.
Explain how you know the sentence is true.
Step 1 _________________ rectangular prisms have 6 vertices.
This is true because _______________________________________
_______________________________________________________
Step 2 _________________ cubes are rectangular prisms.
This is true because _______________________________________
_______________________________________________________
Step 3 _________________ rectangular prisms are cubes.
This is true because _______________________________________
_______________________________________________________
Step 4 _________________ triangular prisms have 5 congruent faces.
This is true because _______________________________________
_______________________________________________________
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Master 3.28
Date
Step-by-Step 10
Lesson 10, Question 3
Step 1 Make a list of the solids you know.
Solid
Edges
Vertices
Step 2 Record the number of edges and the number of vertices in each solid.
Step 3 Use Plasticine and drinking straws to make skeletons for some of
these solids. Look for patterns.
Step 4 Underline the solids in your list that have skeletons with 20 or
fewer edges, and 6 or fewer vertices.
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Master 3.29
Date
Unit Test: Unit 3 Geometry
Part A
Use one tan Pattern Block.
1. Measure the side lengths of each figure.
Label each angle as a right angle (R), less than a right angle (L), or greater
than a right angle (G).
Figure
Side lengths
A
B
C
2. Which figures in Question 1 are congruent?
Explain your answer.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
3. Name the figure in Question 1.
What are the attributes of this figure?
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
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Master 3.29b
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Date
Unit Test continued
Part B
4. This hexagon is one face of a solid.
a) Sketch the other faces if this solid was a hexagonal prism.
b) Sketch the other faces if this solid was a hexagonal pyramid.
c) Look at the figures you sketched in parts a and b.
Which figures are congruent? How do you know?
__________________________________________________________
__________________________________________________________
__________________________________________________________
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Master 3.29c
Date
Unit Test continued
Part C
5. Use 1-cm grid paper.
a) Draw a rectangle.
b) Name all the solids you know that have a rectangular face.
__________________________________________________________
__________________________________________________________
__________________________________________________________
c) Draw the faces of each solid you named.
d) Give an example of an object that matches each solid you named in part b.
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
6. Use triangular dot paper.
a) Draw a net for a triangular pyramid and a net for a triangular prism.
b) Describe how your nets are the same and how they are different.
__________________________________________________________
__________________________________________________________
__________________________________________________________
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Master 3.30
Sample Answers
Unit Test – Master 3.29
Part A
1.
Figure
Side lengths
A
1 cm by 1 cm by 1 cm by 2 cm
B
2 cm by 2 cm by 2 cm by 4 cm
C
1 cm by 1 cm by 1 cm by 2 cm
2. Figures A and C are congruent. They have the
same size and shape.
3. All of the figures are trapezoids. A trapezoid
has one pair of parallel sides.
Part B
4. a)
Date
b)
Part C
5. a) Student should draw a rectangle on 1-cm
grid paper.
b) Triangular prism, rectangular prism,
rectangular pyramid
c) Student should draw the appropriate
number of faces needed to form solids
named in part b.
(See page 101 in Student Edition.)
d) Toblerone bar, cereal box, tent
6. a) Students should draw a net consisting of
4 congruent triangles that will fold into a
triangular pyramid, and a net consisting of
3 congruent rectangles and 2 congruent
triangles arranged so that it will fold into a
triangular prism.
b) The nets are the same because they both
have triangular bases. They are different
because the pyramid has 1 triangular base
and the prism has 2. The pyramid has
4 faces, 6 edges, and 4 vertices. The prism
has 5 faces, 9 edges, and 6 vertices.
c) All of the rectangles are congruent; all of
the triangles are congruent. The hexagon
is regular.
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Master 3.38
Date
Curriculum Focus Activity:
Exploring Lines
A horizontal line goes left and right.
A vertical line goes up and down.
Two lines that cross at a point are intersecting lines.
Two lines that intersect at right angles are perpendicular lines.
Two lines that never meet are parallel lines.
PRACTICE
1. Draw:
a) a pair of parallel lines that are vertical
b) a pair of intersecting lines that are not perpendicular
2. Look at these letters: A B D F H K L M N T V W X Y Z
Which letters have:
a) 2 pairs of parallel lines?
b) just 1 pair of perpendicular lines?
c) 1 pair of parallel lines?
d) just 1 horizontal line?
e) just 1 vertical line?
f) 1 pair of intersecting lines?
3. Use dot paper. Draw a figure with:
a) 2 pairs of parallel sides
b) 1 pair of perpendicular sides
4. Find a black and white picture in a magazine or newspaper.
a) Colour a horizontal line red.
b) Colour a vertical line orange.
c) Colour 2 other lines that are perpendicular blue.
d) Colour 2 different lines that are intersecting green.
e) Colour 2 different lines that are parallel yellow.
Activity Focus: Recognize and identify lines and points.
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Extra Practice Masters 3.31–3.37
Go to the CD-ROM to access editable versions of these Extra Practice Masters.
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Program Authors
Peggy Morrow
Ralph Connelly
Bryn Keyes
Jason Johnston
Steve Thomas
Jeananne Thomas
Angela D’Alessandro
Maggie Martin Connell
Don Jones
Michael Davis
Linden Gray
Sharon Jeroski
Trevor Brown
Linda Edwards
Susan Gordon
Copyright © 2004 Pearson Education Canada Inc., Toronto, Ontario
All Rights Reserved. This publication is protected by copyright,
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This book contains recycled product and is acid free.
Printed and bound in Canada
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