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Teacher Guide
Unit 3: Geometry
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UNIT
3
“Geometry and spatial sense are
fundamental components of
mathematics learning. They offer
ways to interpret and reflect on
our physical environment and
can serve as tools for the study
of other topics in mathematics
and science.”
– National Council of Teachers of Mathematics
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Geometry
Mathematics Background
What Are the Big Ideas?
• Geometric figures can be identified, described, compared, and
classified in different ways.
• Polygons can be constructed, given angle and side measures.
• Figures can be viewed from different perspectives. The ability to
perceive and identify a figure builds understanding of relationships
among figures and objects.
How Will the Concepts Develop?
Students estimate and measure acute, obtuse, right, straight, and reflex
angles. They use a 360° protractor to measure and construct these angles.
Students are introduced to concave and convex polygons. They name and
sort polygons according to side and angle properties.
Students use a protractor, a ruler, and a compass to construct polygons,
given side and/or angle measures.
FOCUS STRAND
Geometry and Spatial Sense
SUPPORTING STRAND
Measurement
They use a computer to draw polygons and to measure sides and angles.
Students construct and sketch nets of objects, and identify the objects
associated with different nets.
Students use an isometric drawing or views of an object to build the
object with linking cubes. They also do the reverse — sketch an object,
given an isometric drawing or views of the object.
Why Are These Concepts Important?
Exploring geometric concepts and figures helps students develop
relationships between and among different figures. As students explore
geometry, they develop spatial sense. This allows them to visualize figures
and objects and to “see” a 3-D object from a 2-D drawing of it. As students
sort and classify geometric figures, they develop their logic and reasoning
skills. These skills can be applied in other branches of mathematics and
in other disciplines.
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Unit 3: Geometry
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Curriculum Overview
The codes refer to the 2005 Revised Curriculum.
Launch
Cluster 1 — Investigating, Classifying,
and Constructing Figures
Angle Hunt
Overall Expectations
• Classify and construct polygons
and angles. (6m43)
Specific Expectations
• Sort and classify quadrilaterals
by geometric properties related to
symmetry, angles, and sides, through
investigation, using a variety of
tools and strategies. (6m46)
• Measure and construct angles up to
180° using a protractor, and classify
them as acute, right, obtuse, or
straight angles. (6m48)
• Construct polygons using a variety
of tools, given angle and side
measurements. (6m49)
Lesson 1:
Investigating Angles
Lesson 2:
Classifying Figures
Lesson 3:
Strategies Toolkit
Lesson 4:
Constructing Figures
Technology:
Using The Geometer’s
Sketchpad to Draw and
Measure Polygons
Cluster 2 — Illustrating and Constructing
Objects
Overall Expectations
• Sketch three-dimensional figures,
and construct three-dimensional
figures from drawings. (6m44)
Specific Expectations
• Build three-dimensional models
using connecting cubes, given
isometric sketches or different
views of the structure. (6m50)
• Sketch, using a variety of tools,
isometric perspectives and different
views of three-dimensional figures
built with interlocking cubes. (6m51)
Lesson 5:
Nets of Objects
Lesson 6:
Illustrating Objects
Show What You Know
Unit Problem
Angle Hunt
Unit 3: Geometry
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Curriculum Across the Grades
Grade 5
Grade 6
Grade 7
Students distinguish
among polygons, regular
polygons, and other
2-dimensional figures,
prisms, right prisms,
pyramids, and other
3-dimensional objects.
Students sort and classify
quadrilaterals by side
and angle properties
and by symmetry.
Students use a variety of
tools to construct related
lines, angle bisectors, and
perpendicular bisectors.
Students use a protractor
to measure and construct
angles up to 180°. They
classify angles as acute,
right, obtuse, or straight.
Students sort and
classify triangles and
quadrilaterals by side
and angle properties
and by symmetry.
Given side and angle
measures, students use
a variety of tools to
construct polygons.
Students investigate to
identify the minimum side
and angle information
needed to describe a
unique triangle.
Students use a protractor
to measure and construct
angles to 90°. They
name angles as acute,
right, obtuse, or straight.
Students identify and
classify triangles by side
and angle properties.
They use a variety of
tools to construct
triangles, given side
and angle measures.
Students identify prisms
and pyramids from their
nets. They use a variety
of tools to construct nets
of prisms and pyramids.
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Unit 3: Geometry
Students use linking
cubes to build models
of objects shown in
isometric drawings or
in top, front, and side
view drawings.
Students use a variety of
tools to make isometric
drawings and draw top,
front, and side views
of objects.
Students investigate to
discover relationships
among area, perimeter,
corresponding sides, and
corresponding angles of
congruent figures.
Students use concrete
materials to investigate
the angles between the
faces of a prism. They
identify right prisms.
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Additional Activities
Angle Tic-Tac-Toe
Sorting Quadrilaterals
For Extra Practice (Appropriate for use after Lesson 1)
Materials: Angle Tic-Tac-Toe (Master 3.11),
protractors
For Extra Practice (Appropriate for use after Lesson 2)
Materials: Sorting Quadrilaterals (Master 3.12),
triangular dot paper, square dot paper, scissors
The work students do: Students play with a
partner. Students use a circular grid. They say the
coordinates of a point, then mark the point on the grid.
The first coordinate tells the distance from the centre.
The second coordinate tells the angle measure. One
player uses Xs, the other uses Os. The first player to
get 3 points in a row along a line or around the
circle wins the game.
The work students do: Students work alone. They
draw one example of each quadrilateral: square,
rectangle, rhombus, trapezoid, kite, parallelogram,
irregular quadrilateral. Students then choose 2
attributes, sort the quadrilaterals, and record the
sorting on a Venn diagram.
Take It Further: Students choose 3 attributes, then
sort the quadrilaterals.
Take It Further: Students develop their own game
using a circular grid.
Visual/Spatial
Individual Activity
Kinesthetic, Social,
Logical/Mathematical
Partner Activity
String Polygons
Build It
For Extra Practice (Appropriate for use after Lesson 4)
Materials: String Experiments (Master 3.13), 2 m
of string or yarn tied into a loop
For Extra Practice (Appropriate for use after Lesson 6)
Materials: Build It (Master 3.14), linking cubes,
triangular dot paper, grid paper
The work students do: Students work in groups
of 4. One student decides the attributes of a polygon.
The other members of the group put their hands
inside the loop of string and pull back to create the
polygon. Each hand represents a vertex. Once the
group has correctly made the polygon, students
switch roles. Students continue playing until each
group member has had at least one turn describing
a polygon.
The work students do: Students work with a
partner. Away from view, each person builds an
object from linking cubes, then draws views of the
object. Students trade views and try to build their
partner’s object. Students compare each object to
the original.
Take It Further: A student describes the attributes
one at a time, and the other students attempt to
make the polygon after each attribute.
Take It Further: Students draw each object on
triangular dot paper.
Kinesthetic/
Visual/Spatial
Partner Activity
Kinesthetic/Social
Partner Activity
Unit 3: Geometry
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Planning for Unit 3
Planning for Instruction
Lesson
vi
Time
Suggested Unit time: 2 weeks
Materials
Program Support
The right to reproduce or modify this page is restricted to purchasing schools.
This page may have been modified from its original. Copyright © 2006 Pearson Education Canada Inc.
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Planning for Assessment
Purpose
Tools and Process
Recording and Reporting
Unit 3: Geometry
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L A U N C H
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Angle Hunt
LESSON ORGANIZER
15–20 min
Curriculum Focus: Activate prior knowledge about
two-dimensional figures and three-dimensional solids,
their names, and angles.
ASSUMED PRIOR KNOWLEDGE
✓
✓
✓
Students can identify and name a variety of figures.
Students can describe the attributes of two-dimensional
figures.
Students can describe different types of angles.
ACTIVATE PRIOR LEARNING
Invite students to examine the various figures
on pages 80 and 81 of the Student Book.
Discuss the first question in the Student Book.
Record students’ answers on chart paper.
(I see squares, rectangles, rhombuses, trapezoids,
triangles, pentagons, hexagons, and quadrilaterals.)
Discuss the second question in the Student Book.
(I see figures that have 4 sides and 4 vertices. These
figures are quadrilaterals. I see congruent trapezoids
as the backs of chairs. They have matching angles
and matching sides. Each trapezoid has one pair of
parallel sides. I see parallelograms on the carpet.
Each parallelogram has 2 pairs of parallel sides.
The loudspeaker is surrounded by a square that
has 4 equal sides and 4 equal angles.)
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Unit 3 • Launch • Student page 80
Discuss the third question in the Student Book.
(In each trapezoid, there are 2 acute angles and 2
obtuse angles. On the board, there is a right triangle,
which has an angle of 90°. On the other board, there
is a hexagon with an angle that is greater than 180°.)
Discuss the fourth question in the Student Book.
(An acute angle has a measure less than 90°. An obtuse
angle has a measure between 90° and 180°. All angles
have two arms.)
Invite students to look around the classroom
to identify figures.
Have students play a game of “What Figure
Am I?” One student silently identifies a figure
in the room. Students take turns to ask a
yes/no question about the figure, until the
figure is identified.
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LITERATURE CONNECTIONS FOR THE UNIT
Mummy Math: An Adventure in Geometry by Cindy
Neuschwander. Henry Holt, 2005.
ISBN: 0805075054
Stuck inside a pyramid with only each other, their dog Riley,
and geometric hieroglyphics to help them find their way,
Matt and Bibi must use their math knowledge to solve the
riddles on the walls and locate the burial chamber.
Fold Me a Poem by Kristine O’Connell. Harcourt Inc., 2005.
ISBN: 0152025014
Join a young boy as he creates a world filled with origami
creatures of all shapes and sizes out of brightly coloured paper.
Fold-Along Stories: Quick & Easy Origami Tales for Beginners
by Christine Petrell Kallevig. Storytime Ink International, 2001.
ISBN: 0962876992
Twelve short stories are illustrated by the progressive folding
steps of twelve origami models.
REACHING ALL LEARNERS
Some students may benefit from using
the virtual manipulatives on the e-Tools
CD-ROM. The e-Tools appropriate for this unit include
Geometry Shapes and Geometry Drawing.
DIAGNOSTIC ASSESSMENT
What to Look For
What to Do
✔ Students can identify
and name a variety
of figures.
Extra Support:
Post pictures of different figures, their names, and their attributes in the classroom,
so students can refer to them as they work.
✔ Students can
describe the
attributes of twodimensional figures.
Post examples of different types of angles, in different orientations, so students
can identify types of angles when no arm is horizontal or vertical.
✔ Students can
describe different
types of angles.
Unit 3 • Launch • Student page 81
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L E S S O N
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Investigating Angles
40–50 min
LESSON ORGANIZER
Curriculum Focus: Estimate, measure, and draw angles to
360°. (6m48)
Acute
Optional
쐍 Angles for Lesson 1
쐍 180° protractors
Explore (Master 3.6)
쐍 360° protractors
Vocabulary: acute angle, right angle, obtuse angle, straight
angle, reflex angle
Assessment: Master 3.2 Ongoing Observations: Geometry
Student Materials
Right
Obtuse
Key Math Learnings
1. Angles can be named and sorted according to their measures.
2. Angles can be measured and constructed using a protractor.
Numbers Every Day
Some students may wish to record the numbers in a
place-value chart.
• 113 321, 121 232, 123 231, 123 321
• 4 242 444, 4 344 342, 4 432 344, 4 432 413
Curriculum Focus
DURING
In this lesson, students measure and construct angles up
to 360°. The curriculum requires students to measure and
construct angles to 180°. You may wish to have students
focus on the angles that measure 180° or less.
Ongoing Assessment: Observe and Listen
BEFORE
Get Started
Use the pictures at the top of page 82 to review
the types of angles. To extend the review, ask
students to identify examples of each type of
angle in objects in the room.
Introduce Explore. Distribute copies of Master 3.6
to students. Have 360° or 180° protractors
available for student use.
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Unit 3 • Lesson 1 • Student page 82
Explore
Ask questions, such as:
• How did you estimate the measures of ⬔A
and ⬔B?
(I know the angle at the corner of a piece of paper
is 90°. If I fold one side onto the other, the fold line
divides the 90° angle into two 45° angles. Angle A is
about 45°. I know⬔B is greater than 180°, because
I compare it to a protractor that measures 180°. Two
180° protractors together make 360°. So, I estimate
⬔B is greater than 180° and less than 360°, but
closer to 360°.)
• How did you estimate the measures of ⬔C
and ⬔D?
(I used the right angle on a piece of paper to estimate
that ⬔C is greater than 90° and less than 180°. I
estimate it is halfway between 90° and 180°, which
is 135°. Angle D is greater than 180° and less than
360°, but closer to 180°.)
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REACHING ALL LEARNERS
Alternative Explore
Materials: protractors
Students work in pairs. One student draws an acute angle.
The other student estimates, then measures, the angle
formed. Students take turns to draw and measure acute,
obtuse, and reflex angles. Have students explain how the
measures of the angles in the first pair can be used to
estimate the measures of the angles in the second pair.
Early Finishers
Have students draw different angles on cards, then record
the angle measures on the backs of the cards. They trade
cards with a partner to measure and check.
Common Misconceptions
➤ Students do not know whether to use the inner or outer
scale when they measure an angle.
How to Help: Have students estimate the measure first and
identify the angle as acute, right, obtuse, straight, or reflex.
After they place the protractor correctly, they can read the
measure that is closest to their estimate.
ESL Strategies
ESL students benefit from hearing vocabulary and instructions
repeatedly with visual cues and demonstrations. Use key
words frequently in dialogue and point to the word and
illustrations from the text as often as possible.
• How are the angles in each pair the same?
How are they different?
(In each pair of angles, one angle is less than 180°
and the other angle is greater than 180°. It does not
matter if the angle with the lesser measure is acute,
right, or obtuse, the other angle in the pair is always
greater than 180°.)
• How can you use a 180° protractor to measure
the greater angle?
(I can extend one of the arms to make a straight line.
I measure the angle on the straight line; it is 180°.
Now I can measure the angle formed by the extension
I drew and the other arm of the original angle. I add
the measure of this angle to 180°.)
• What is the sum of the angles in each pair?
(The sum of the angles in each pair is 360°.)
AFTER
Connect
Invite students to share the strategies they used
to estimate and to measure the angles.
Ask:
• Why is it important to estimate when
measuring angles?
(There are two scales on a protractor. If I estimate
the measure of the angle, I am less likely to make
an error in measuring the angle.)
• How can you use what you know about one
angle to find the measure of the other angle?
(I can measure the smaller angle in each pair and
subtract its measure from 360°.)
Review the types of angles at the top of page 83.
Ask students to identify other examples of these
angles from objects in the room. Have students
sketch a variety of each type of angle.
Unit 3 • Lesson 1 • Student page 83
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Sample Answers
2. a)
205°
b)
200°
c)
Acute, 62°
Reflex, 270°
Reflex, 190°
270°
d)
107°
32°
300°
e)
192°
350°
331°
233°
Demonstrate the steps to measure and to draw
a reflex angle.
If time permits, have students draw different
angles and use protractors to measure them.
Remind students that the measure of an angle
does not change if the angle is rotated or moved
to a different location.
Practice
Questions 1 to 6 require protractors and rulers.
It may be helpful for students to trace the angles
and extend the arms to allow more accurate
measurements.
Encourage students to discuss their work and
to compare their results. This feedback helps
to correct errors as they occur while students
practise estimating, measuring, and constructing.
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Unit 3 • Lesson 1 • Student page 84
Assessment Focus: Question 5
Students will probably use a guess-and-check
strategy. They may draw a variety of reflex
angles, from 181° to 359°, and estimate or
measure the other angle. Students should
recognize that an acute angle and an obtuse angle
can be paired with a reflex angle, but a straight
angle is paired with another straight angle.
Students who need extra support to complete the
Assessment Focus questions may benefit from
using the Step-by-Step masters (Masters 3.15
to 3.19).
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5. a) Yes; an acute angle is less than 90° and a reflex angle is
90°
b)
45°
135°
0°, 360°
180°
c)
6. a)
225°
315°
270°
b)
greater than 180°; a complete turn is 360°; if one angle
is acute, the other angle will be between 270° and 360°.
Yes; an obtuse angle is between 90° and 180° and a
reflex angle is greater than 180°; a complete turn is 360°;
if one angle is obtuse, the other angle will be between 180°
and 270°.
No; a straight angle is 180°; the other angle will always
also measure 180°.
I would measure the related acute or obtuse angle and
subtract its measure from 360°.
I know 360° 245° 115°, so I would draw an angle
of 115°; the related angle is 245°.
REFLECT: The sum of the measures of a reflex angle and the
related acute or obtuse angle is 360°. Here is an acute angle
of 65° and a reflex angle of 295°. 65° 295° 360°
65°
295°
Here is an obtuse angle of 165° and a reflex angle of 195°.
165° 195° 360°
165°
195°
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Knowledge and Understanding
✔ Students can use a protractor to
measure and to draw angles less
than 360°.
Extra Support: Copy 360° protractors onto transparencies.
Cut the copies into wedges of 90°, 135°, 180°, and 270°.
Use the wedges to estimate the measures of angles.
Students can use Step-by-Step 1 (Master 3.15) to complete
question 5.
Communication
✔ Students can explain how to draw
or to measure an angle.
Extra Practice: Students can complete Extra Practice 1
(Master 3.22).
Students can do the Additional Activity, Angle Tic-Tac-Toe
(Master 3.11).
Extension: Challenge students to draw quadrilaterals with
different numbers of reflex, obtuse, right, and acute angles.
Which combinations of angles are possible?
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
Unit 3 • Lesson 1 • Student page 85
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Classifying Figures
40–50 min
LESSON ORGANIZER
Curriculum Focus: Sort and classify figures different ways.
(6m46)
Optional
쐍 Step-by-Step 2 (Master 3.16)
scissors
쐍 Extra Practice 1 (Master 3.22)
chart paper
square dot paper (PM 25)
Figures for Lesson 2 Explore (Master 3.7)
Figures for Lesson 2 Practice (Master 3.8)
Vocabulary: convex polygon, concave polygon
Assessment: Master 3.2 Ongoing Observations: Geometry
Student Materials
쐍
쐍
쐍
쐍
쐍
Key Math Learning
Figures can be identified, described, compared, and classified
in different ways.
Curriculum Focus
The curriculum requires students to sort and classify
quadrilaterals by geometric properties related to
symmetry. There is an Additional Activity in Unit 7
related to this concept.
BEFORE
Get Started
Have students look at the figures at the top
of page 86 in the Student Book. Ask:
• How are the figures alike? How are
they different?
(All figures have straight sides. All of them are
polygons. Two figures are triangles, two are
quadrilaterals, and one is a hexagon.)
• What is a regular figure?
(A regular figure has equal sides and equal angles.)
• What is an irregular figure?
(An irregular figure does not have all sides equal
and does not have all angles equal.)
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Unit 3 • Lesson 2 • Student page 86
• Which figures are regular? Which are
irregular? How do you know?
(Figures A, C, and D are regular; they have equal
sides and angles. Figures B and E are irregular.
Figure B has unequal sides; figure E has unequal
sides and angles.)
Introduce Explore. Distribute copies of Master 3.7,
chart paper, and scissors.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• How might you identify the attributes
of a figure?
(I could measure the side lengths and angles and
I could count the number of sides.)
• Which figures are regular? Irregular?
(Figures B and F are regular. All the other figures
are irregular.)
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ALL LEARNERS
Alternative Explore
Materials: Figures for Lesson 2 Explore (Master 3.7)
Students select one figure. They describe the figure using as
many attributes from page 86 as they can. They then choose
two different figures and use the listed attributes to describe
how they are alike and how they are different.
Early Finishers
List the attributes on page 86 in order, from those that include
the most figures to those that include the fewest. Order the
figures on page 86 from those that include the most attributes
from the list, to those that include the fewest.
Common Misconceptions
➤ Some students may find the number of attributes
overwhelming.
How to Help: Reduce the number of attributes in Explore
and gradually add to the list. Have students work with the
three attributes related to the type of angle or the four
attributes related to the type of polygon.
• How did you decide how to draw the loops?
(I chose two attributes so that it is possible for a
figure to have both attributes. Then I drew overlapping
loops. If it were not possible for one figure to have both
attributes, I would draw loops that do not overlap.)
AFTER
Connect
Invite students to share their sorting rules.
Discuss how students determined which figure
had the most attributes and which attribute
described the most figures.
What you might see:
Regular
Parallel sides
L
F
A
C
B
Reflex angle
D
E
I
L
J
E
G
Obtuse angle
A
K
J
D
C
I
H
F
B
You may wish to have students conduct a
“museum tour.” One student from each group
stays with its chart to answer questions. The
other members tour each of the other groups.
Ask:
• Which figure had the most attributes from
the list?
(Figure E: parallel sides, irregular, pairs of equal sides,
hexagon, reflex angle, acute angle)
The fewest?
(Figure F: regular, equal sides, triangle, acute angle)
• Which attribute describes the most figures?
(Irregular; 10 of the 12 figures are irregular.)
Present Connect. Ask:
• Why are trapezoids also quadrilaterals?
(Any figure with 4 sides is a quadrilateral.
A trapezoid has 4 sides.)
• Why are trapezoids not parallelograms?
(A trapezoid has only 1 pair of parallel sides.
A parallelogram has 2 pairs of parallel sides.)
Unit 3 • Lesson 2 • Student page 87
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Sample Answers
1. a) Convex irregular pentagon, 3 obtuse angles, 2 acute
angles, no equal sides or angles, no parallel sides
b) Concave irregular quadrilateral, no equal sides or angles,
one reflex angle, 3 acute angles, no parallel sides
c) Concave irregular hexagon, no equal sides or angles,
one reflex angle, 2 acute angles, 2 obtuse angles, one
right angle.
2. The pentagon and dodecagon have the attribute.
a) All the polygons in the first column have reflex angles.
All the polygons in the second column do not have
reflex angles.
b) Students’ figures will vary: any figure with a reflex angle
3. Answer will vary. Here is one example. The angles and sides
are not easy to measure, so I chose attributes I could identify
by looking at the figures.
Concave polygon
Quadrilateral
K
B
E
A
H
D
J
F
L
G
C
I
Reflex angle
• Where would a kite be placed on the Venn
diagram? Why?
(In the quadrilaterals region. A kite has 2 pairs of
adjacent sides equal, but no parallel sides, so it cannot
go inside the trapezoid loop.)
Point out the convex and concave polygons on
page 86 in the Student Book. Ask students to
find examples of convex and concave polygons
in the classroom.
Practice
Distribute copies of Master 3.8 for questions 3
to 5. Students can use either square or triangular
dot paper for question 6. Question 7 requires
square dot paper and rulers. Have toothpicks,
wooden stir sticks, geoboards, grid paper, and
other tools available for students. Encourage
students to show their thinking in labelled
pictures, in numbers, and in words.
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Unit 3 • Lesson 2 • Student page 88
Assessment Focus: Question 6
For parts a and b, students may list all the
figures with the first attribute, then identify
those that also have the second attribute. For
part c, have students explain why we use
“exactly” here and explain what changes if
we delete “exactly.”
Labelled examples for each question should be
explained in writing. Students should recognize
that there are many solutions for each question.
They should also recognize counterexamples;
that is, examples that do not meet the criteria.
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4. Answers will vary. Here is one example.
Pentagon
A
K
G
At least one
right angle
F
B
H C
I
D
L
J
E
Regular polygon
A, C, D,
F, G, I
6. Students’ answers should include the art described, drawn
on dot paper.
a) Rhombus, equilateral triangle, regular pentagon, concave
hexagon with all sides equal, regular hexagon, and so on
b) Any concave irregular quadrilateral, or chevron
c) Any trapezoid that is not a parallelogram, or figures
with more than 4 sides but only 2 parallel sides
7. Comparisons should include the number of sides, side
lengths, angle measures, parallel lines. Students’ answers
should include art.
The sum of the angles in
each quadrilateral is 360°.
REFLECT: Figures can be sorted according to the number of
sides, the lengths of the sides, the angle measures, the number
of equal sides and/or angles, the number of pairs of parallel
sides, and the number of lines of symmetry. Students’ answers
should include art.
About
About
About
About
23
110
60
700
Numbers Every Day
Students should be able to describe the estimation strategy and
why the estimate is reasonable.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Knowledge and Understanding
✔ Students understand that figures can
be sorted and classified by angle and
side properties.
Extra Support: Have students choose one attribute, then use
a geoboard and geobands to make as many different figures
as possible with that attribute. Students record their figures on
square dot paper.
Students can use Step-by-Step 2 (Master 3.16) to complete
question 6.
Communication
✔ Students can use appropriate
mathematical terms to describe,
compare, and classify geometric figures.
Application
✔ Students can identify, describe,
compare, and classify geometric
figures in different ways.
Extra Practice: Students can complete Extra Practice 1
(Master 3.22).
Students can do the Additional Activity, Sorting Quadrilaterals
(Master 3.12).
Extension: Students work in pairs. They try to make as many
figures as possible with 2 attributes from the list on page 86.
They record each figure on dot paper.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
Unit 3 • Lesson 2 • Student page 89
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Strategies Toolkit
40–50 min
LESSON ORGANIZER
Curriculum Focus: Check and reflect. (6m3)
Key Math Learning
Check and reflect is an important step in solving problems.
It helps to ensure the accuracy and reasonableness of solutions.
BEFORE
Get Started
Have students look back over their work from the
past few lessons. Ask them to find any inaccurate
answers. Explain that mistakes are a natural part
of problem solving in mathematics, but they
may be corrected by reflecting and checking.
Present Explore.
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Unit 3 • Lesson 3 • Student page 90
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• Does Paolo’s figure meet the criteria? Explain.
(No; the figure Paolo drew does not have any
parallel sides.)
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REACHING ALL LEARNERS
Early Finishers
Have students design their own polygon riddles similar
to those in Explore and Connect, or “Find the mystery
attribute,” from Practice.
Sample Answers
Practice
1. The attribute is parallel sides. The trapezoid and pentagon
in the third column of the chart have parallel sides.
2. a) No, the answer is not reasonable. I cannot divide 2046
by 13 and get a quotient that is about one-half of 2046.
I know that 2046 2 is 1023, so 2046 13 should be much
less than 1023. I used estimation to check: 2046 13 is
about 2000 10, which is 200.
b) No, since the answer is not reasonable, it cannot be correct.
When we divide 2046 by 13, we first think “2000 13
is about 100,” and write 1 above the first 0. There should
not be a 0 in the quotient. The correct quotient is 157 R5.
REFLECT: It is important to check a solution to ensure it is
accurate, that nothing was left out, and that the solution
answers the question.
• How could Paolo change the figure so his
solution is correct?
(Paolo must redraw one side so it is parallel to
another side. He should check that the new figure
is a pentagon with no lines of symmetry and exactly
one obtuse angle.)
• Is it possible to draw more than one
pentagon to solve this problem?
(Yes; all pentagons will be concave with one
reflex angle, but their side lengths and angles
may be different.)
AFTER
Connect
What you might see:
Invite students to share their strategies and
solutions for the problem from Explore. Ensure
students check the given criteria have been met.
Present Connect. Model the thinking process
by drawing on an overhead projector or the
board as you work through Marg’s solution
to the problem.
Unit 3 • Lesson 3 • Student page 91
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Constructing Figures
40–50 min
LESSON ORGANIZER
Curriculum Focus: Use a protractor, a ruler, and a compass
to construct figures. (6m49)
Teacher Materials
Optional
쐍 demonstration compass
Student Materials
Optional
쐍 tangrams (PM 29)
쐍 Step-by-Step 4 (Master 3.17)
쐍 protractors
쐍 Extra Practice 2 (Master 3.23)
쐍 rulers
쐍 compasses
쐍 triangular dot paper (PM 26)
Vocabulary: quadrilateral, arc, segments, vertex, congruent,
convex, hexagon
Assessment: Master 3.2 Ongoing Observations: Geometry
Key Math Learnings
1. Figures can be combined to create other figures.
2. A figure can be constructed using a protractor and a ruler,
or a compass and a ruler, given the lengths of its sides and
the measures of its angles.
BEFORE
Get Started
DURING
Explore
Remind students how a figure can be
decomposed into smaller figures.
Ongoing Assessment: Observe and Listen
Draw a square on the board. Draw one
diagonal. Ask students which two figures
make up the square.
(Two congruent right isosceles triangles)
• How can you make a quadrilateral with two
tangram figures?
(Two small triangles can be joined to form a square
if I join the triangles along the longer side. If I join
them along the shorter side, I make a parallelogram.)
• Which quadrilaterals could have an angle that
measures 135º?
(A trapezoid and a parallelogram; the square and a
small triangle can be put together to make a trapezoid.
One angle measures 135°.)
• How do you know the measure of the angle
is 135°?
(I know a square has 90° angles. I used my
protractor to measure the equal angles in the right
triangle. Each equal angle is 45°. So, when I put the
triangle and square together, one angle where they
meet is 135°.)
Draw a pentagon on the board. Draw one
diagonal. Ask students which two figures
make up the pentagon.
(Answers will vary, depending on the shape of the
pentagon, a quadrilateral and a triangle.)
Introduce Explore. Explain that students will
be doing the opposite to your demonstration.
They will combine 2 or more figures to make
a larger figure.
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Unit 3 • Lesson 4 • Student page 92
Ask questions, such as:
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REACHING ALL LEARNERS
Alternative Explore
Materials: Pattern Blocks
Have students use 2 or more Pattern Blocks to make
polygons with two 120° angles.
Common Misconceptions
➤ For question 4, students have difficulty drawing a figure
when no side lengths are provided.
How to Help: Explain that this means students can choose
the length of the first line segment they draw. Have students
draw any line segment, then construct one of the given
angles at each end. Since they do not yet know that the sum
of the angles in a triangle is 180°, they will need to measure
the third angle to check that it is correct.
• How many different quadrilaterals with
two 135º angles were you able to make?
(Four: one is a parallelogram made from two
small triangles; another is a trapezoid made with
the parallelogram and one small triangle; a third
is made from the two small triangles and the
parallelogram; a fourth is a trapezoid made with
the medium-sized triangle and the parallelogram.)
AFTER
Connect
Invite students to share their quadrilaterals.
What you might see:
135°
135°
135°
135°
135°
135°
135°
Ask students to share how they made the
quadrilaterals. Ask:
• What strategies did you use to make a
quadrilateral with two angles that each
measure 135°?
(I used trial and error. I tried putting different tans
together. Very few arrangements made a quadrilateral.
Most resulted in a pentagon or a hexagon.)
• How did you decide which tans to use?
(The parallelogram has two 135º angles so I wanted
to use this figure. When it is placed next to a small
triangle, it makes a trapezoid.)
Explain and demonstrate the safe use of a
compass. If possible, obtain a large demonstration
compass. Begin by drawing a line segment on
the board. Set the compass to a width greater
than the length of the line segment. Place the
compass point at one end of the line segment.
135°
Unit 3 • Lesson 4 • Student page 93
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Sample Answers
2. a) For example: Two small triangles and one medium triangle
create a square.
b) The ruler and protractor construction should be identical
to the sketch created in part a.
3. a) Only one triangle is possible. The triangle is scalene.
The triangle is a right triangle.
b) Only one triangle is possible. The triangle is acute.
The triangle is equilateral.
c) Only one triangle is possible. The triangle is acute. The
triangle is isosceles. I checked with my classmates and,
in each case, the triangles we drew were congruent, so
I inferred that only one triangle can be drawn in each case.
4. The triangles may not be congruent, but they will be similar.
The triangles are not congruent because they do not coincide
with each other if one is placed on top of another.
6. a) Many possible concave hexagons have three or more sides
3 units long and angles of 60° and 240°. For example:
60°
60°
240°
240°
60°
60°
60° 240°
240°
60°
60°
Draw a circle. Keep the compass at the same
width. At the other end of the line segment,
draw a congruent circle.
The two circles overlap above and below the
line segment. Draw line segments from one
point of intersection to each endpoint of the
original line segment.
Ask students what figure has been created.
(An isosceles triangle)
Draw line segments from the other point of
intersection to the endpoints of the original
line segment.
Ask students what figure has now been created.
(A second isosceles triangle; the two triangles form
a rhombus.)
Review Connect. You may wish to demonstrate
how to draw a rough sketch first.
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Unit 3 • Lesson 4 • Student page 94
Practice
Rulers, protractors, and compasses are needed
for most questions. Tangrams are needed for
question 2. Triangular dot paper is required
for question 6.
Some students may benefit from first making
a labelled rough sketch of each figure.
Assessment Focus: Question 6
Most students will use trial and error to create
concave hexagons. Some students may choose
to draw a rough sketch first. Some will realize
that the dots on the triangular dot paper form
equilateral triangles with 60° angles. They can
apply this knowledge to create other angles that
are multiples of 60°. Students should realize
that some of the hexagons are rotations of
hexagons they have already drawn, and, so,
are not different.
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6. b) For example: I drew a line segment 3 units long. Then
I drew an angle of 240°. Then I drew another segment
3 units long, and drew an angle of 60°. Then I drew
another line segment 3 units long. Finally, I drew 3 more
segments to make the hexagon.
9. a) Pentagon 1: ⬔QRM 131°; ⬔RMN 98°; ⬔MNP 131°
Pentagon 2: ⬔QRM 50°; ⬔RMN 262°; ⬔MNP 49°
b) The second pentagon is concave. The first is convex.
⬔B 69°; ⬔C 81°; ⬔D 30°
REFLECT: A pentagon has five sides. For this concave
70°
Yes; the lengths of the sides
can be different.
6 cm
pentagon, three sides must be equal and one angle must
measure 240°. I would use a ruler and a protractor to construct
a 240° angle with equal arms; this is ⬔ABC. The 240° angle
is reflex. It makes the pentagon concave. Next I would draw
a line segment, AE, from one endpoint of one arm that has
the same length as the arms of the angle. I would draw the
remaining two sides (DE and CD) with a ruler and pencil to
make the pentagon.
C
A
The second pentagon is concave and
the first is convex.
B
240°
E
D
170
343
122
178
Numbers Every Day
Have students explain the strategies they used each time.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Communication
✔ Students can explain how to
construct a figure, given side and
angles measures.
Extra Support: Students who have difficulty using a 180°
protractor may find a 360° protractor easier to use.
Students can use Step-by-Step 4 (Master 3.17) to complete
question 6.
Application
✔ Students can use a variety of tools
to construct figures, given angle and
side measures.
Extra Practice: Students can do the Additional Activity,
String Polygons (Master 3.13).
Students can complete Extra Practice 2 (Master 3.23).
Extension: Students use a pencil, a ruler, and a protractor
to construct their name using capital letters only. Students will
have to draw letters that have curved parts, such as B and D,
with straight line segments to replace the curves.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
Unit 3 • Lesson 4 • Student page 95
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Using The Geometer’s
Sketchpad to Draw
and Measure Polygons
LESSON ORGANIZER
40–50 min
Curriculum Focus: Use a computer to draw polygons and to
measure sides and angles. (6m49)
Student Materials
쐍 computers with The Geometer’s Sketchpad or AppleWorks
Key Math Learning
A computer can be used to draw and measure polygons.
BEFORE
Tell students that they will be using The
Geometer’s Sketchpad to draw polygons and to
measure the lengths of the sides and the angles.
DURING
Ongoing Assessment: Observe and Listen
Watch to ensure students understand and
follow the procedures carefully.
Instructions for drawing polygons with
AppleWorks:
1. Open a new drawing document
in AppleWorks.
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Unit 3 • Technology • Student page 96
2. To check the Ruler settings:
Click: Format
Select: Rulers, then click: Ruler Settings…
In the pop-up window, select Centimeters
and set Divisions to 10. Click: OK
3. To draw a regular polygon:
Click the Regular Polygon Tool.
Click: Edit, then click: Polygon Sides
Type in the number of sides you want.
Click: OK
Click and hold down the mouse button.
Drag the cursor until the polygon is the
size and shape you want. Release the
mouse button.
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REACHING ALL LEARNERS
Early Finishers
Have students use The Geometer’s Sketchpad to measure the
perimeter and area of their polygons. They can observe the
effect on the area and the perimeter of changing the size
and shape of the polygon.
REFLECT: When I use a computer to draw polygons, it is easy
to measure the side lengths and the angles. I can also
change the size and shape of the polygon by clicking and
dragging a side or vertex. It would take me much longer to
do this by hand.
4. To draw an irregular polygon:
Click the Polygon Tool.
Click and drag to make each side of
the polygon.
Double-click when you have finished.
5. Use the centimetre grid to measure the
lengths of the sides of the polygon. Where
the sides do not follow the grid lines, the
length will need to be estimated.
6. To save your polygons:
Click: File, then click: Save As…
Give your file a name. Click: Save
7. To print your polygons:
Click: File, then click: Print
Click: OK
AFTER
Invite volunteers to share their polygons with
the class. Discuss the methods they used to
estimate the lengths of sides that did not follow
the grid lines.
Unit 3 • Technology • Student page 97
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Nets of Objects
optional
LESSON ORGANIZER
Lesson Focus: Identify, design, and sketch nets of objects.
Teacher Materials
Optional
Student Materials
쐍
쐍
쐍
쐍
쐍
쐍
쐍 cereal box
Optional
쐍 Step-by-Step 5 (Master 3.18)
쐍 Extra Practice 2 (Master 3.23)
scissors
tape
polyhedrons
triangular dot paper (PM 26)
1-cm grid paper (PM 23)
Diagrams for Lesson 5 Explore (Master 3.9)
Vocabulary: tetrahedron
Assessment: Master 3.2 Ongoing Observations: Geometry
Key Math Learning
A net shows the faces of an object. The net can be cut out and
folded to make the object.
Numbers Every Day
• 4 682 000; 4 700 000
• 803 092 000; 803 100 000
• 9 990 000; 10 000 000
Curriculum Focus
In this lesson, students identify and draw nets of objects.
This material is not required by the curriculum. This is
a review of concepts covered in Grade 5.
BEFORE
Get Started
Hold up a cereal box, or some other package
shaped like a rectangular prism. Engage
students in a discussion of the solid. Focus
on the properties of the prism.
Ask questions, such as:
• What is the name of this object?
(It is a rectangular prism.)
How do you know?
(It has 6 rectangular faces arranged so that opposite
faces are congruent.)
• If I were to cut along enough edges to be able
to flatten the cardboard, what would you see?
(Six rectangles joined together to make a net of
the prism.)
Open the box flat to show its net. Point out to
students that there are tabs and some sections
that overlap. Nets of prisms and pyramids may
not have tabs.
Present Explore. Distribute copies of Master 3.9.
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REACHING ALL LEARNERS
Early Finishers
Have students explore the different arrangements of the square
and four isosceles triangles that are the nets of a square
pyramid. Have them find as many different arrangements as
they can. Students could use Polydrons if they are available.
Common Misconceptions
➤ In question 2, students cannot draw the net for the
octagonal pyramid.
How to Help: Provide students with a model of the pyramid.
Have them trace the base, then flip the pyramid so the base
of one triangular face aligns with one side of the octagonal
face. Students trace this face, and repeat the tracing until
there are 8 congruent isosceles triangles attached to the base.
Sample Answers
3. a) Rectangular pyramid
b) Triangular prism
c) Not a net; if the diagram were cut out and folded, the
adjacent sides of the triangular faces would not match;
they have different lengths.
d) Not a net; there are only five square faces; six square
faces are needed to make a cube.
Cube
Square pyramid
Triangular prism
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• Which diagrams are the nets of objects?
(Diagram A is a net of a pentagonal prism. Diagram B
is not a net of an octagonal pyramid; two faces would
overlap. Diagram C is a net of a triangular prism.
Diagram D is a net of a square pyramid. Diagram E
is a net of a triangular prism.)
• How could you change diagram B to make it
a net for an octagonal prism?
(I would move the right triangle where there are
three triangles together, and put it on the left side of
the single triangle. I would have to make sure I placed
the triangle so it would not overlap any other triangle
when I folded the net.)
• How do you identify the object from the net?
(I look at the number of faces and the shapes of the
faces. A prism has two congruent faces for its bases.
The other faces are rectangles. The number of
rectangular faces is determined by the base. For
example, a triangular prism has 3 rectangular faces
and a pentagonal prism has 5 rectangular faces. A
pyramid has one base and triangular faces. The
number of triangular faces is determined by the base.)
• What must be true for a diagram to be a net?
(There must be the correct number of faces. For
example, a rectangular prism has 6 faces and a
triangular pyramid has 4 faces. The faces must be
arranged so that no faces overlap when the net is
folded. Also, the lengths of sides that join to make
edges must be equal.)
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4. a)
The base is an octagon. There
must be 8 congruent isosceles
triangular faces.
b)
c)
d)
5. a)
b)
AFTER
Connect
Invite students to share the ways in which they
check if a diagram is a net. Then ask questions,
such as:
• There are nets for two triangular prisms.
How are these nets different?
(The triangular faces are different; one has
equilateral triangles, the other has right isosceles
triangles. The arrangement of the faces is different, too.
One net has the triangles on either side of a rectangle.
The other net has the rectangles arranged around
one triangle.)
Review Connect. Ask students how they could
change the diagram of faces for a rectangular
prism so it is a net.
(One of the smaller faces could move to the opposite end
of the rectangle that has the other smaller face attached
to it.)
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Practice
Students will need polyhedrons, if available,
for most questions. Grid paper is required for
question 7.
Assessment Focus: Question 6
A pyramid is defined by its base. Most students
will begin by drawing a triangular base with a
triangle on each side, then use trial and error to
rearrange the faces so they still represent a net.
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c)
d)
e)
Square pyramid
Rectangular prism
Rectangular pyramid
Pentagonal pyramid
Triangular pyramid
If each diagram was cut out and folded, it would make an
object. The two sides that join to form each edge are equal.
6. a) There are two possible nets
for a triangular pyramid, or
regular tetrahedron.
b)
1
2
4
5
3
6
1
1
2
4
5
5
6
3
6
3
2
4
c) For example: Any triangle in part a could be the base for
the tetrahedron, but in part b there is only one triangle
that is the base: the equilateral triangle. There are 4 nets
for part b but only 2 for part a.
REFLECT: To be sure a diagram is a net of an object, I could
trace it and cut it out. Then fold it to see if it makes a solid.
For a prism, there are two congruent bases and the same
number of rectangles as there are number of edges on a
base. A pyramid has one base. The other faces are triangles.
The number of triangles is equal to the number of edges on
the base.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Knowledge and Understanding
✔ Students can identify objects from
their nets.
Extra Support: Have students use Polydrons or Frameworks
to construct models of objects and decompose them to form nets.
Students can use Step-by-Step 5 to complete question 6.
Application
✔ Students can sketch and construct
nets of objects.
Extra Practice: For each diagram in question 3 that is not a
net, have students trace the diagram and change it so it is a net.
Students can complete Extra Practice 2 (Master 3.23).
Communication
✔ Students can explain how to draw a
net for an object using appropriate
mathematical language.
Extension: There are eleven different nets for a cube.
Challenge students to find all the different nets.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
Unit 3 • Lesson 5 • Student page 101
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Illustrating Objects
40–50 min
LESSON ORGANIZER
Curriculum Focus: Build objects, then sketch them.
(6m59, 6m51)
Student Materials
Optional
쐍 linking cubes
쐍 Step-by-Step 6 (Master 3.19)
쐍 triangular dot paper
쐍 Extra Practice 3 (Master 3.24)
(PM 26)
쐍 1-cm grid paper (PM 23)
Vocabulary: front view, top view, side view
Assessment: Master 3.2 Ongoing Observations: Geometry
Key Math Learning
Objects can be represented as isometric drawings and as
top/front/side views.
Numbers Every Day
For example: 18, 23, 28, 33; any whole number with 3 or 8
in the units place has a remainder of 3 when divided by 5.
Numbers with 0 or 5 in the units place are multiples of 5, so
do not have remainders when divided by 5. Adding 3 to these
numbers will produce the required numbers.
BEFORE
Get Started
Hold up a rectangular prism.
Ask questions, such as:
• How many faces does this prism have?
(It has 6 faces.)
• How would you describe the faces?
(Each face is a rectangle. There are pairs of
congruent faces.)
• Suppose you wanted to draw this prism as
you see it. How many faces would you draw?
(I would draw 2 or 3 faces. That is the maximum
number of faces I can see at one time.)
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Unit 3 • Lesson 6 • Student page 102
Rotate the prism so that three faces are visible.
Ask:
• How many vertices can you see?
(I see 7 vertices; 4 on the top face and 3 around the
bottom edges.)
Present Explore.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• How do you know the object you built is
a rectangular prism?
(My object has 6 faces, 8 vertices, and 12 edges.
These are the attributes of rectangular prisms.)
• What are the dimensions of your
rectangular prism?
(My prism has dimensions 5 by 4 by 3.)
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REACHING ALL LEARNERS
Common Misconceptions
➤ Students have difficulty constructing isometric drawings.
How to Help: Have students take one linking cube and orient
it on their desk so three faces are visible. Have students state
the number of vertices that are visible. Point out that 3 vertices
on the top face and 3 on the bottom face form a hexagon.
Have students draw this hexagon, then mark a dot in the
middle for the 7th vertex. Students then join this vertex to 3
other vertices to complete the cube. Shading the top face of
the drawing helps with orientation and suggests depth.
Sample Answers
1. a)
b)
2. a)
Front Side
c)
b)
Top
Front
Side
Top
• How did you draw your prism on triangular
dot paper?
(I drew a vertical line 5 units long. Then, I drew a
line 4 units long that went up to the left. Next, I
drew a line 3 units long that went up to the right.
Finally, I drew the rest of the vertical lines and
diagonal lines to complete the prism.)
• How did you draw the views of the prism?
(I placed the prism on the desk. I looked at it from
the front. I saw a 5 by 4 rectangle, so I drew that on
grid paper as the front view. Then I looked at the prism
from the side. I saw a 5 by 3 rectangle, so I drew that
on grid paper as the side view. Finally, I looked down
on the prism. I saw a 4 by 3 rectangle, so I drew that
as the top view.)
AFTER
Connect
Invite students to share their prisms, their
drawings on triangular dot paper, and their
views. Discuss the methods students used to
draw their prisms on triangular dot paper.
Ask students to share any problems and
challenges they had in making their drawings
and what they did to overcome them.
Unit 3 • Lesson 6 • Student page 103
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c)
Top
Front
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Side
3. a)
b)
4. b)
Top
5. a)
Top
Front
Front
Side
Side
b)
Have volunteers describe any problems they
had constructing a prism from the drawing or
views prepared by a classmate.
Review Connect. Have the prism at the top of
page 103 on display as you review. If students
have had limited or no opportunity to work with
triangular dot paper, use the same prism on
page 103, but rotate it so the height is 4 units.
Use a transparency of triangular dot paper on
the overhead projector to illustrate how to draw
the prism in this new position.
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Unit 3 • Lesson 6 • Student page 104
Practice
Have linking cubes available. Students also
require triangular dot paper (PM 26) and 1-cm
grid paper (PM 23).
Assessment Focus: Question 5
Some students may build relatively complex
objects. If they have difficulty drawing their
objects, suggest they build a simpler object.
Students should position the object in front
of them as they wish to draw it.
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6. a)
Front
Side
b)
Front
Side
7. b)
REFLECT: Using isometric dot paper helps with the drawing. I
first draw all the vertical edges. Then, I draw the edges that go
up to the right. Finally, I draw the edges that go up to the left.
Making Connections
Math Link: Engineers, draftspeople, and industrial designers
draw and interpret isometric drawings and views of objects.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Knowledge and Understanding
✔ Students can identify objects from
views or drawings on triangular
dot paper.
Extra Support: Students who have difficulty remembering
which face they have drawn should number the top, front, and
side views as 1, 2, and 3, then draw them in order.
Application
✔ Students can build an object with
linking cubes and draw the object.
Extra Practice: Have students arrange 5 cubes differently
from the arrangements in question 6, then draw the views and
the isometric drawing for each arrangement. Students can
complete Extra Practice 3 (Master 3.24).
Students can do the Additional Activity, Build It (Master 3.14).
Extension: Challenge students to build different objects that
have the same set of top, front, and side views.
Recording and Reporting
Master 3.2 Ongoing Observations:
Geometry
Unit 3 • Lesson 6 • Student page 105
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S H O W W H A T Y O U Home
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40–50 min
LESSON ORGANIZER
Student Materials
쐍
쐍
쐍
쐍
쐍
rulers
protractors
compasses
linking cubes
triangular dot paper (PM 26)
Assessment
About 100°
Obtuse, 115°
Master 3.1 Unit Rubric: Geometry
Master 3.4 Unit Summary: Geometry
Sample Answers
2. a)
About 45°
Acute, 53°
4. K
About 320°
Reflex, 325°
L
175°
4 cm
b)
N
210°
c)
8 cm
5. a)
110°
d)
5 cm
350°
70°
9 cm
50°
3. The common attribute is all sides equal.
Figures P, Q, and R are quadrilaterals.
Each has all sides equal.
Figures F, G, and H are also quadrilaterals.
Each has at least 2 equal sides,
so that is not the required attribute.
Each of figures A and B has all sides equal,
while figure C has 2 pairs of equal sides,
and figure D has no equal sides.
28
Unit 3 • Show What You Know • Student page 106
M
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7. a)
70° and 110°; 9 cm and 5 cm
Net; triangular pyramid
Not a net
Net; pentagonal prism
b)
8. For example: a cube made from 9 linking cubes; the drawing
will show 7 vertices and all appropriate lines that represent
the individual cubes.
ASSESSMENT FOR LEARNING
What to Look For
Knowledge
✔ Question
✔ Question
✔ Question
and Understanding
1: Students are able to estimate, classify, and measure angles.
2: Students are able to construct angles using a ruler and a protractor.
4: Students can construct polygons, given side and angle measures.
Thinking
✔ Question 3: Students can solve problems related to attributes of figures.
Communication
✔ Question 8: Students can draw views and an isometric drawing of an object.
Application
✔ Question 7: Students are able to sketch nets for a given solid.
Recording and Reporting
Master 3.1 Unit Rubric: Geometry
Master 3.4 Unit Summary: Geometry
Unit 3 • Show What You Know • Student page 107
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U N I T
P R O B LHome
E M
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Angle Hunt
LESSON ORGANIZER
80–100 min
Student Groupings: 2 to 4
Student Materials
쐍
쐍
쐍
쐍
Angle Hunt Game Cards (Master 3.10)
protractors
rulers
blank game cards
Assessment
Master 3.3 Performance Assessment Rubric: Angle Hunt
Master 3.4 Unit Summary: Geometry
Teaching Notes for the Cross Strand Investigation Ziggurats are
in the Additional Assessment Support module.
Display the answers recorded in the Unit Launch
and review the questions and answers. Refer
students to the list of Key Words and the
Learning Goals in the Unit Launch to clarify
the purpose for the Unit Problem. Refer to the
Check List on page 109 to focus on expectations
about student work.
Invite a volunteer to read Part 1 aloud. Have
students play the game in Part 1. Encourage
students to use different objects. You may wish
to suggest that once an object has been used,
it cannot be used again during the game. Use
the information from the Check List and the
Performance Assessment Rubric: Angle Hunt to
clarify what is expected as the students play.
30
Unit 3 • Unit Problem • Student page 108
Introduce Part 2 and have students play the
game again, but with the new rules.
Introduce Part 3. Discuss what different kinds
of cards could be used. Discuss how the new
game rules might change the game. Have students
create their games.
Listen for how students use vocabulary.
Observe how they estimate, measure, and
sketch angles and figures. Do they estimate
before measuring and constructing, and reflect
on the reasonableness of their answers? Observe
how well students recognize figures by side
and angle measures.
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Reflect on the Unit
I learned that there are these types of angles: an acute angle is
less than 90°; an obtuse angle is between 90° and 180°; a straight
angle is 180°; and a reflex angle is between 180° and 360°.
I learned that a convex polygon has all angles less than 180°, and
a concave polygon has at least one angle greater than 180°.
I also learned that an object can be represented on paper by
its views (top, front, and side) and on triangular dot paper, when
its three dimensions are seen.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Thinking
✔ Students can select an appropriate
strategy and use it to solve a problem.
Extra Support: Make the game accessible.
Have students play the game using only the Sketch cards or the
Find cards. Add the others after students are comfortable. For
Part 3, limit variables. Direct students to vary either the cards
or the rules but not both. Help them to focus on one type of
change, for example, angle measures.
Communication
✔ Students use correct geometric
language to explain their answers.
✔ Students give clear explanations of
how the game is played.
Application
✔ Students can sketch different angles,
nets, and figures with given attributes.
Recording and Reporting
Master 3.3 Performance Assessment Rubric: Angle Hunt
Master 3.4 Unit Summary: Geometry
Unit 3 • Unit Problem • Student page 109
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Evaluating Student Learning: Preparing to Report:
Unit 3 Geometry
This unit provides an opportunity to report on the Geometry and Spatial Sense 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:
Most Consistent Level of Achievement*
Strand: Geometry and
Spatial Sense
Knowledge and
Understanding
Thinking
Communication
Application
Overall
Ongoing Observations
3
2
2
3
2/3
Work samples or
portfolios; conferences
3
3
2
3
3
Show What You Know
3
3
3
3
3
Unit Test
3
2
2
3
2/3
Unit Problem
Angle Hunt
3
3
3
2
3
Achievement Level for reporting
3
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 mostly heavily weighted.
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 later in the unit may be more heavily weighted.
Show What You Know
Teachers may choose to assign some or all of these questions. Master 3.1 Unit Rubric:
Geometry may be helpful in determining levels of achievement.
#1, 2, 4, and 6 provide evidence of Knowledge and Understanding; #3 provides evidence of
Thinking; #5, 7, and 8 provide evidence of Application; 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 Application; Part B provides evidence of Knowledge and Understanding;
Part C provides evidence of Thinking; all parts provide evidence of Communication.
Unit performance task
Use Master 3.3 Performance Assessment Rubric: Angle Hunt. 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’ perception 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
PM 10: Summary Class Records: Strands
PM 11: Summary Class Records: Achievement Categories
PM 12: Summary Record: Individual
Use to record and report throughout a reporting period, rather
than for each unit and/or strand.
Use to record and report evaluations of student achievement over
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
Master 3.1
Categories/Criteria
Level 1
Level 2
Level 3
Level 4
demonstrates limited
understanding of
geometric concepts in:
– classifications and
comparisons
– attributes and
relationships
– explorations with
computer programs
demonstrates some
understanding of
geometric concepts in:
– classifications and
comparisons
– attributes and
relationships
– explorations with
computer programs
demonstrates
considerable
understanding of
geometric concepts in:
– classifications and
comparisons
– attributes and
relationships
– explorations with
computer programs
demonstrates thorough
understanding of
geometric concepts in:
– classifications and
comparisons
– attributes and
relationships
– explorations with
computer programs
uses appropriate
strategies to solve
problems that involve
constructing figures, or
building and drawing
objects, with limited
effectiveness
uses appropriate
strategies to solve
problems that involve
constructing figures, or
building and drawing
objects, with some
effectiveness
uses appropriate
strategies to solve
problems that involve
constructing figures,
or building and
drawing objects,
with considerable
effectiveness
uses appropriate
strategies to solve
problems that involve
constructing figures,
or building and
drawing objects, with
a high degree of
effectiveness
limited effectiveness;
unable to explain
reasoning and
procedures clearly;
rarely uses appropriate
terms and symbols
some effectiveness;
explains reasoning and
procedures with some
clarity; sometimes
uses appropriate terms
and symbols
considerable
effectiveness; explains
reasoning and
procedures clearly,
using appropriate
terms and symbols
high degree of
effectiveness; explains
reasoning and
procedures clearly and
precisely, using the
most appropriate terms
and symbols
presents diagrams and
drawings with limited
clarity and limited use
of appropriate
conventions
presents diagrams
and drawings with
some clarity and some
use of appropriate
conventions
presents diagrams
and drawings with
considerable clarity;
uses appropriate
geometric conventions
presents diagrams and
drawings with a high
degree of clarity; uses
appropriate geometric
conventions
limited effectiveness;
makes major errors
or omissions in:
– identifying,
classifying, and
constructing figures
– estimating,
measuring, naming,
and constructing
angles
– building and
illustrating objects
some effectiveness;
somewhat accurate,
with several minor
errors or omissions in:
– identifying,
classifying, and
constructing figures
– estimating,
measuring, naming,
and constructing
angles
– building and
illustrating objects
considerable
effectiveness;
generally accurate,
with few minor errors
or omissions in:
– identifying,
classifying, and
constructing figures
– estimating,
measuring, naming,
and constructing
angles
– building and
illustrating objects
high degree of
effectiveness; accurate
and precise, with very
few or no errors in:
– identifying,
classifying, and
constructing figures
– estimating,
measuring, naming,
and constructing
angles
– building and
illustrating objects
Knowledge and Understanding
• shows understanding by
describing and explaining:
– classification and
comparison of geometric
figures
– attributes and
relationships
– explorations of geometric
concepts with computer
programs
Thinking
• plans and carries out
appropriate strategies
to solve problems that
involve constructing
figures, or building
and drawing objects
Communication
• explains reasoning and
procedures clearly, using
appropriate terminology
and symbols
• presents diagrams and
drawings clearly, using
appropriate geometric
conventions
Application
• applies geometric skills
appropriately to:
– identify, classify, and
construct figures
– estimate, measure, name,
and construct angles
– build and illustrate
objects
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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 under Assessment for Learning.
STUDENT ACHIEVEMENT: Geometry
Student
Knowledge and
Understanding
Thinking
Communication
Application
• Demonstrates and
explains attributes
and relationships
• Uses appropriate
strategies to pose and
solve problems that
involve constructing
figures or building and
drawing objects
• Explains reasoning
and procedures
clearly, using
appropriate terms
• Presents diagrams
and drawings clearly
• Names, classifies,
and constructs figures
and angles
• Builds and draws
objects
Level 1 – very limited; Level 2 – somewhat or limited; Level 3 – satisfactory; Level 4 thorough
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Performance Assessment Rubric:
Angle Hunt
Master 3.3
Categories/Criteria
Level 1
Level 2
Level 3
Level 4
Knowledge and Understanding
• explanations and
instructions show
understanding of attributes
and relationships of
angles and figures
explanations and
instructions show
limited understanding
of attributes and
relationships
explanations and
instructions show
some understanding
of attributes and
relationships
explanations and
instructions show
considerable
understanding of
attributes and
relationships
explanations and
instructions show
thorough
understanding of
attributes and
relationships
uses a few simple
strategies with limited
success to:
– solve problems
related to the
game cards
– create a new
game, including
game cards
uses some appropriate
strategies with some
success to:
– solve problems
related to the
game cards
– create a new
game, including
game cards
uses appropriate
strategies with
considerable
success to:
– solve problems
related to the
game cards
– create a new
game, including
game cards
uses appropriate, often
innovative, strategies
with a high degree of
success to:
– solve problems
related to the
game cards
– create a new
ame, including
game cards
limited effectiveness;
unable to communicate
responses, game
cards, and instructions
clearly
some effectiveness;
communicates
responses, game
cards, and instructions
with some clarity
considerable
effectiveness;
communicates
responses, game
cards, and instructions
clearly
a high degree of
effectiveness;
communicates
responses, game
cards, and instructions
clearly and precisely
limited effectiveness;
makes major errors
or omissions in:
– recognizing
geometric attributes
in everyday objects
– estimating and
measuring angles
– constructing and
drawing angles
and figures
some effectiveness;
somewhat accurate in:
– recognizing
geometric attributes
in everyday objects
– estimating and
measuring angles
– constructing and
drawing angles
and figures
considerable
effectiveness;
generally accurate in
– recognizing
geometric attributes
in everyday objects
– estimating and
measuring angles
– constructing and
drawing angles
and figures
high degree of
effectiveness; accurate
and precise in:
– recognizing
geometric attributes
in everyday objects
– estimating and
measuring angles
– constructing and
drawing angles
and figures
Thinking
• uses appropriate
strategies to:
– solve problems related to
the game cards
– create a new game,
including game cards
Communication
• communicates clearly,
using appropriate
geometric language
and conventions in
responses, game cards,
and instructions
Application
• accurately applies
geometric skills to:
– recognize geometric
attributes in everyday
objects
– estimate and measure
angles
– construct and draw
angles and figures
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Master 3.4
Date
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: Geometry and
Spatial Sense
Knowledge and
Understanding
Thinking
Communication
Application
Ongoing Observations
Work samples or
portfolios; conferences
Show What You Know
Unit Test
Unit Problem:
Angle Hunt
Achievement Level for reporting
*Use Ontario Achievement Levels 1, 2, 3, 4
Self-Assessment:
Comments: (Strengths, Needs, Next Steps)
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Master 3.5
Date
To Parents and Adults at Home …
Your child’s class is beginning a mathematics unit on geometry. Through
daily activities, students will explore a variety of two-dimensional figures and
three-dimensional objects to develop a deeper understanding of their attributes.
Students will investigate angle measures and draw figures using a ruler, a protractor,
and a compass.
In this unit, your child will:
• Estimate, measure, and construct angles to 360º.
• Classify figures by side and angle properties.
• Construct figures.
• Identify, sketch, and draw nets of solids.
• Build and draw objects.
Geometry and spatial awareness are important elements in understanding
mathematics. Geometry provides students with a strong link between mathematics
and the world around them.
Here are some suggestions for activities to do at home.
• Look for figures and objects, and estimate the sizes of the angles that you
see. Use 90°, 180°, and 360° as referents to make more refined estimates.
• Go on an angle hunt. Look for objects with angles that are less than 90°, 90°,
between 90° and 180°, 180°, and between 180° and 360°.
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Master 3.6
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Angles for Lesson 1 Explore
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Master 3.7
Date
Figures for Lesson 2 Explore
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Master 3.8
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Figures for Lesson 2 Practice
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Master 3.9a
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Diagrams for Lesson 5 Explore
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Master 3.9b
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Diagrams for Lesson 5 Explore (continued)
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Diagrams for Lesson 5 Explore (continued)
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Master 3.10
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Angle Hunt Game Cards
Find a regular polygon.
Find an irregular polygon.
Find a figure with
more than one line
of symmetry.
Find an obtuse angle.
Find an acute angle.
Find a right angle.
Find a polygon with a
reflex angle.
Find a figure with exactly
one line of symmetry.
Find a reflex angle.
Find a scalene triangle.
Find an isosceles triangle.
Find an equilateral
triangle.
Find a figure with
parallel lines.
Sketch a 30° angle.
Sketch a 60° angle.
Sketch a 135° angle.
Sketch a 225° angle.
Sketch a 315° angle.
Sketch a 270° angle.
Sketch a 45° angle.
Find a 330° angle.
Find a 45° angle.
Find a 60° angle.
Find a 240° angle.
Find a 120° angle.
Find a figure with more
than one acute angle.
Find a figure with more
than one obtuse angle.
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Master 3.11
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Additional Activity 1:
Angle Tic-Tac-Toe
Play with a partner.
You will need protractors.
Use this circular grid. The object of the game is to get 3 points in a row along a line or
around a circle.
¾Player A chooses a distance from the centre, in units, and an angle measure.
She measures the angle from the horizontal line segment on the grid.
Player A marks a point on the circle to represent the angle measure
and distance.
She labels the point X.
¾Player B repeats the process.
He labels his point O.
¾
Play continues until one player has 3 points in a line or along a circle.
Take It Further
Students develop their own game using the circular grid.
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Master 3.12
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Additional Activity 2:
Sorting Quadrilaterals
Work on your own.
You will need triangular dot paper, square dot paper, and scissors.
Draw one example of each figure:
¾square
¾rhombus
¾parallelogram
¾rectangle
¾trapezoid
¾kite
¾irregular quadrilateral
Cut out the figures. Choose 2 attributes. Sort your figures.
Use a Venn diagram to record your sorting.
Take It Further
Choose 3 attributes. Sort your figures. Record your sorting.
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Master 3.13
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Additional Activity 3:
String Polygons
Work in a group of 4.
You will need 2 m of string or yarn tied into a loop.
¾ Take turns.
One student chooses a polygon.
She does not say its name.
She describes its attributes.
¾ The other members of the group put their hands inside
the loop of string and pull back to create the polygon.
Each hand represents a vertex.
Once the group has correctly made and named the polygon, switch roles.
¾ Continue playing until each group member has had
at least one turn describing a polygon.
Take It Further
Repeat the activity. Describe one attribute at a time.
The winner is the student who has to describe
the most attributes before the correct polygon is made.
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Master 3.14
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Additional Activity 4:
Build It
Work with a partner.
You will need linking cubes, grid paper, and triangular dot paper.
¾ Without your partner seeing,
build an object with linking cubes.
Do not build a rectangular prism.
¾ Draw as many views of your object as
your partner needs to build the object.
¾ Trade views with your partner.
Build your partner’s object.
¾ Compare objects and views.
If the object your partner built from your views
is different from your object, try to find the error.
Take It Further
Draw each object on triangular dot paper.
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Master 3.15
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Step-by-Step 1
Lesson 1, Question 5
Step 1
Draw an acute angle. Measure the angle.
What is the measure of the other angle formed
by the arms of the acute angle?
Name the other angle as obtuse, straight, or reflex.
Step 2
Draw an obtuse angle. Measure the angle.
What is the measure of the other angle formed
by the arms of the obtuse angle?
Name the other angle as obtuse, straight, or reflex.
Step 3
Draw a straight angle. Measure the angle.
What is the measure of the other angle formed
by the arms of the straight angle?
Name the other angle as obtuse, straight, or reflex.
Step 4
Is it possible to draw a reflex angle so the other angle
formed by the arms is acute? Obtuse? Straight? Explain.
___________________________________________________________
___________________________________________________________
___________________________________________________________
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Master 3.16
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Step-by-Step 2
Lesson 2, Question 6
You will need triangular dot paper and square dot paper.
Step 1
Use triangular dot paper.
Draw a triangle with all sides equal.
Draw a hexagon with all sides equal.
Use square dot paper.
Draw a quadrilateral with all sides equal.
Step 2
Use triangular dot paper.
Draw a quadrilateral with one reflex angle.
Use square dot paper.
Draw a quadrilateral with one reflex angle.
Use either dot paper.
Draw a different quadrilateral with one reflex angle.
Step 3
Use square dot paper.
Draw two parallel line segments with different lengths.
Join the segments to form a quadrilateral.
Draw a pentagon with exactly 2 parallel sides.
Draw a hexagon with exactly 2 parallel sides.
Step 4
Label each figure you drew with its name.
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Master 3.17
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Step-by-Step 4
Lesson 4, Question 6
Use triangular dot paper and a protractor.
Construct a concave hexagon.
The hexagon must have at least one angle with each measure:
60° and 240°. Three or more sides must be 3 units long.
Step 1
Any 2 adjacent dots are 1 unit apart.
Draw a line segment 3 units long.
Step 2
From the one endpoint of the line segment from Step 1,
draw another line segment 3 units long.
Measure and record the angle formed by the two line segments.
Step 3
From the endpoint of one of the line segments from Step 2,
draw another line segment 3 units long.
Measure and record the angle formed by the two line segments.
Step 4
Continue to draw line segments until you have drawn a hexagon.
Check that at least one angle measures 60°, at least one angle
measures 240°, and at least 3 sides are 3 units long.
Step 5
Repeat Steps 1 to 4 to draw a different concave hexagon.
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Master 3.18
Date
Step-by-Step 5
Lesson 5, Question 6
You will need a regular tetrahedron, a triangular pyramid, and triangular dot paper.
Step 1
Use the regular tetrahedron.
Trace one face.
Trace the same face on each side of the first triangle you drew.
Copy the diagram on dot paper. This is one net.
Arrange the faces to make a second net.
Step 2
Use the triangular pyramid that is not a regular tetrahedron.
Trace the face that is different from the other three faces.
Trace one of the three congruent faces on each side of the first triangle
you drew. Copy the diagram on dot paper.
Step 3
Use the triangular pyramid that is not a regular tetrahedron.
Trace the three triangular faces, so that one meets
another along one of the two equal sides.
Place the base of the pyramid so it touches one of the non-equal sides
of one of the triangles you drew.
Trace the base. Copy the diagram on dot paper. This is another net.
Step 4
Look at the two nets in Steps 2 and 3.
Try to arrange the faces a different way to make a different net.
Step 5
How are the nets in Step 1 like the nets in Steps 2, 3, and 4?
How are the nets different?
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Master 3.19
Date
Step-by-Step 6
Lesson 6, Question 5
You will need linking cubes, 1-cm grid paper, and triangular dot paper.
Step 1
Use 3 linking cubes.
Build an object that is not a rectangular prism.
Step 2
Place the object on the desk.
Look down on the object.
Draw its top view.
Look at the object from the front.
Draw its front view.
Look at the object from the side.
Draw its side view.
Step 3
To draw the object on triangular dot paper:
Draw a line segment for each vertical edge.
Draw a line segment for each edge that goes up to the right.
Draw a line segment for each edge that goes up to the left.
Step 4
Describe what you did in Steps 2 and 3.
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Master 3.20a
Date
Unit Test: Unit 3 Geometry
Part A
1. Use a ruler and a protractor. Draw an angle with each measure.
a) 140°
b) 240°
c) 340°
d) 40°
2. Name this polygon
List as many of its attributes as you can.
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Part B
3. Use triangular dot paper or square dot paper.
Draw a quadrilateral with each attribute:
a) 1 right angle
b) 1 reflex angle
c) 2 parallel sides
d) 2 equal sides that are not parallel
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Master 3.20b
Date
Unit Test: Unit 3 Geometry continued
4. Use these figures:
a) Choose 2 attributes. Sort the figures.
Record your sorting in a Venn diagram.
b) Choose 3 attributes. Sort the figures again.
Record your sorting in a Venn diagram.
5. Use a ruler and protractor.
a) Construct a trapezoid with one right angle and one 120° angle.
b) What are the measures of the other angles?
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Master 3.20c
Date
Unit Test: Unit 3 Geometry continued
6. Use a ruler and a compass.
Draw UMNP with sides 5 cm long.
What are the measures of the angles in UMNP?
7. a) Use linking cubes.
Build a rectangular prism.
Draw the prism on triangular dot paper.
b) Use the same linking cubes.
Build an object that is not a rectangular prism.
On grid paper, draw its front, top, and side views.
Part C
8. Use a ruler and a protractor.
Draw quadrilateral BCDE with these measures:
BC = 6 cm, DE = 9 cm, ∠BED = 50°, and ∠EDC = 60°
How many different quadrilaterals can you draw? Explain.
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Master 3.21
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Unit Test Sample Answers
Unit Test – Master 3.20
5. a) For example:
Part A
1.
b) 90º and 60º
2. Hexagon, 6 sides, concave, 1 reflex angle
(270°), 3 right angles, 1 acute angle,
1 obtuse angle, 3 parallel sides, irregular,
no lines of symmetry
6. a) All angles are 60°.
Part B
3. a)
b)
7. Answers will vary.
c)
d)
Part C
8. I can only draw one quadrilateral with these
measures. The angles are measured from
the ends of the longest side, DE. The length
of the side opposite DE is given, so only one
measure is possible for each of sides BE
and CD.
4. a)
b)
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Extra Practice Masters 3.22–3.25
Go to the CD-ROM to access editable versions of these Extra Practice Masters.
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Pearson-Math6TR-Un03-Cover 11/9/05 12:01 PM Page 2
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Program Authors
Peggy Morrow
Ralph Connelly
Jason Johnston
Bryn Keyes
Don Jones
Michael Davis
Steve Thomas
Jeananne Thomas
Nora Alexander
Linda Edwards
Ray Appel
Cynthia Pratt Nicolson
Carole Saundry
Ken Harper
Jennifer Paziuk
Maggie Martin Connell
Sharon Jeroski
Trevor Brown
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