chapter 7 geometry of polygons and polyhedra (e)/files/blackline...occur throughout the chapter...

CHAPTER 7 Geometry of Polygons andPolyhedra

S p e c i f i c Cu r r i c u l u m O u t co m e s

M a j o r O u t c o m e s

E1 recognize, name, describe, and construct polygons

E2 predict and generate polygons that can be formed with a transformation or

composition of transformations of a given polygon

E3 make and apply generalizations about the properties of regular polygons

E4 make and apply generalizations about tessellations of polygons

E5 construct polyhedra using one type of regular polygonal face, and describe

and name the resulting Platonic Solids

E6 construct semi-regular polyhedra and describe and name the resulting

solids, and demonstrate an understanding about their relationships to the

Platonic Solids

E7 make and apply generalizations about angle relationships

C h a p t e r Pro b l e m

A chapter problem is introduced in the chapter opener. This chapter problem has

students designing a new ball sport that will use one of the polyhedra explored in the

chapter. You may wish to have students complete the chapter problem revisits that

occur throughout the chapter (section 7.1 question 15, section 7.2 question 9,

section 7.3 question 14, and section 7.4 question 9). These simpler versions provide

scaffolding for the chapter problem and offer struggling students some support. The

revisits will assist students in preparing their response for the Chapter Problem

Wrap-Up on page 329.

Alternatively, you may wish to assign only the Chapter Problem Wrap-Up

when students have completed Chapter 7. The Chapter Problem Wrap-Up is a sum-

mative assessment.

Key Wordspolygonregular polygonvertexline of symmetryorder of rotational symmetryconcave polygonconvex polygoncongruentsimilartessellationpolyhedronPlatonic solidvertex regularityArchimedean solidtruncate

Get Ready Wordsinterior anglescongruentsimilarline of symmetryorder of rotational symmetrytranslationreflectionrotation

220 MHR • Mathematics 7 : Focus on Understanding Teacher ’s Resource

Planning Chart

Section Suggested Timing

Teacher’s ResourceBlackline Masters Assessment Tools Adaptations

Materials andTechnology Tools

Chapter Opener• 15 min (optional)

Get Ready• 60 min

• BLM 7GR Parent Letter• BLM 7GR Extra Practice

7.1 Explore Polygonsand Their Properties• 180 min

• BLM 7.1 Extra Practice Formative Assessment:• BLM 7.1 AssessmentQuestion, #12

• BLM Venn Diagram • pattern blocks• transparent mirrors

7.2 Tessellations• 120 min

• BLM 7.2 Extra Practice Formative Assessment:Question #8

• The GeometryTemplate®

7.3 Regular Polyhedra• 180 min


• Polydron® pieces:equilateral triangles,squares, pentagons.• The GeometryTemplate®• square dot paper• isometric dot paper• bags

7.4 Semi-RegularPolyhedra• 120 min


• Polydron® pieces:equilateral triangles,squares, pentagons,hexagons• isometric dot paper

Chapter 7 Review• 90 min

• BLM 7R Extra Practice

Chapter 7 Practice Test• 90 min

Summative Assessment:• BLM 7PT Chapter 7Test

Chapter Problem Wrap-Up• 60 min

• BLM 7CP ChapterProblem Wrap-UpRubric

Chapter 7 • MHR 221

Get Ready

W A R M - U P

Compare using the proper symbol ( < , > , or = ).

1. � (<) 2. 2 � 1 (=)

Calculate mentally.

3. � 48 <12> 4. 21 � <12>

5. 20% of 110 <22> 6. 5% of 280 <14>

Evaluate.

7. (–9) + (–6) + (+4) <–11> 8. (–12) – (–17) <+5>9. (–6) � (–9) <+54> 10. (–42) ÷ (+7) <–6>11. (–2) � (–3)2 <–18> 12. –3(5 – 6) <+3>13. (–38) + (–64) <–102> 14. (–46) + (–72) + (–54) <–172>15. (+12) � (–35) <–420>

A S S E S S M E N T F O R L E A R N I N G

Before starting Chapter 7, explain that the topic is the geometry of polygons and

polyhedra. The chapter involves the study of polygons and their properties, tessella-

tions, regular polyhedra (Platonic solids), and semi-regular polyhedra (Archimedean

solids). Draw students’ attention to their spatial awareness by asking if they are good

at giving or receiving directions to an unknown place. Can they point to the north?

Describe a foreign object and have students draw a picture of what they think the

object might look like. Next, discuss what 2-D and 3-D objects look like. Challenge

students to visualize what 1-D, 4-D, and 5-D objects would look like. Can these

objects be drawn? Have students analyze their own spatial awareness. Some students

will have good spatial awareness, while others will have to work at it.

Discuss with students when they have used polygons, tessellations, and 3-D

solids before, and what they know about these geometric figures. You may wish to

brainstorm and develop a mind map for each topic or start the development of a

graphic organizer to be used throughout the chapter. After students have discussed

these geometric figures, have them complete the assessment suggestions below in

pairs or individually. This assessment is designed to provide you and your students

with information about their readiness for the chapter. After strengths and weaknesses

have been identified, students can work on appropriate sections of the Get Ready.

Method 1: Have students develop a journal entry to explain what they know about the

topics and how they use geometry in their everyday language or in their everyday lives.

Method 2: Challenge students to show how much they know about each topic.

Encourage them to use words, numbers, and diagrams to show what they know.

4

7

1

4

1

4

3

4

5

8

5

9

Related Resources• BLM 7GR Parent Letter• BLM 7GR Extra Practice

Suggested Timing60 min


R e i n fo rce t h e Co n ce p t s

Have those students who need more reinforcement of the prerequisite skills com-

plete BLM 7GR Extra Practice.

T E A C H I N G S U G G E S T I O N S

Before starting the Get Ready, you could use the information from the Discover the

Math, Part A, Step 1 in section 7.1 to make a chart of pictures and names of polyhe-

dra for students’ reference. Have students complete all of the Get Ready questions

before starting the chapter.

When working through Angle Properties, check that students understand that

the sum of the interior angles of a triangle is 180°. Students have been comparing

angles to right angles since Grade 1. When working through Congruent and SimilarFigures, check that students understand the difference between the two terms. Use

tracing paper or overhead transparencies to help identify the figures.

When working through Polygon Symmetry, check that students understand

that a line of reflective symmetry divides a figure into two congruent figures that are

reflective images of each other. Different shaped boxes (especially ones that are not

square or rectangular) are a good tool to review rotational symmetry. Mark one

corner of the lid and see how many times the lid can be placed on the box. For more

activities to review rotational symmetry, see the Nova Scotia Department of

Education’s Mathematics Grade 7: A Teaching Resource. When working through

Transformations, check that students understand that reflections, rotations, and

translations produce congruent figures.

Co m m o n E r ro r s

• Some students may think that the pentagons in question 2, part c)are congruent.

Rx Remind students that two figures with the same shape but different sizes are

similar not congruent.

• Some students may have difficulty distinguishing between the types of

transformations, particularly reflections and rotations.

Rx Remind students of the definitions of translations, rotations, and reflections

and provide extra practice questions similar to question 4.

L i t e ra c y Co n n e c t i o n s

Venn diagrams are used frequently in this chapter to classify polygons in different

ways. Venn diagrams are useful organizers for comparing and contrasting concrete

relationships, and help students remember the characteristics for concepts that have

similarities and differences. It is important to compare only two or three properties

at a time, as more than this may make it confusing for students.


Wh i p - Aro u n d G a m e

At the end of the unit, students could use the language learned in the chapter to

create a card game similar to the Loops game for other students to play. Students

might be assigned to small groups to create this game.

To Create: Have students write a list of vocabulary words from the chapter and

give their definitions. You may allow all students to choose the words they wish to

use or students might be assigned a category within the chapter. Once the students

have their rough copy, they can use index cards to organize their good copy. The

good copy should be edited to make sure that the spelling and punctuation are

correct. Once the good copy is complete, choose 10 to 15 definitions and write a

question and answer for each definition similar to the way Loops cards are written.

(Example: Question: Who has a three-sided figure with equal side lengths? Answer:

I have an equilateral triangle.) Make the cards like Loops cards, staggering the

questions and answers so the final card has the question for the answer on the first card.

To Play: Each person takes a card. The first person reads out the answer then

the question, The person with the card that answers the question reads their answer

then their question. The game continues until all the cards have been read.

G e t R e a d y An s we r s

1. a) 50° b) 70° c) 75°

2. a) congruent b) neither c) similar d) neither

3. a) 3; 3 b) 4; 4 c) 1; 1

4. a) translation of 2 units to the right and 1 unit up b) reflection along a vertical

mirror line c) rotation of 90° clockwise about the turn centre C and translation

of 2 units to the right


7.1 Explore Polygons and Their Properties

W A R M - U P

Multiply using the Compatible Factors strategy.

1. 2 � 13 � 5 <130> 2. 20 � 7 � 5 <700>3. 11 � 4 � 2 � 25 <2200> 4. 5 � 7 � 7 � 2 <490>5. 5 � 5 � 6 � 2 � 2 <600> 6. 9 � 8 � 50 � 2 <7200>7. 25 � 5 � 8 � 4 <4000> 8. 4 � 7 � 3 � 250 <21 000>9. (–4) � (–6) � (+25) <+600>10. (+2) � (+29) � (–5) <–290>11. (–7) � (+15) � (–2) <+210>12. (–11) � (–5) � (+2) � (–6) <–660>13. (–15) � (+4) � (–5) � (+5) <+1500>14. (–5) � (–5) � (+7) � (–2) � (+2) <–700>15. (–4) � (+62) � (+250) <–62 000>

Co m p at i b l e Fa c to r s St rat e g y

Sometimes a multiplication problem can be made easier to calculate mentally by

changing the order and looking for “nice” numbers.

For example, the problem 25 � 5 � 9 � 2 � 4 can be very difficult to calculate

mentally if you do the problem in the order in which it appears. However, by using

the Associative Property to change the order and looking for compatible factors, you

can make some easy calculations to get the answer.

If you multiply 25 � 4 to get 100, and 5 � 2 to get 10 you will now be multiplying

100 � 9 � 10 which is a much simpler problem. The answer is 9000.

Another example:


In this section, students continue to learn about the properties of polygons. Students

coming into Grade 7 can recognize many polygons but are not necessarily able to

give accurate descriptions. In chapter 10, students will be defining what is explicitly

required to name one kind of polygon and not others that could be similar in some

respects. Chapter 7 deals with exploring all of the properties of polygons. Reviewing

the Van Hiele model in VandeWalle’s Elementary and Middle School Mathematics will

(+7) � (–6) � (–50) � (–2)

= (–42) � (+100)= –4200

100

25 � 5 � 9 � 2 � 4

10

Materials• pattern blocks• transparent mirrors

Related Resources• BLM Venn Diagram• BLM 7.1 Extra Practice

Specific CurriculumOutcomesE1 recognize, name,

describe and constructpolygons

E3 make and applygeneralizations aboutthe properties of regularpolygons


Link to Get ReadyStudents should havedemonstratedunderstanding of all sectionsof the Get Ready prior tobeginning this section.


give you more insight into what is expected of students in Level 2 of the Van Hiele

levels of geometric thought.

Discuss regular and not regular polygons. Emphasize the term polygon and try

to avoid using shape or figure when describing polygons. You could add a circle or a

‘pizza slice’ to the not regular polygons to show that a closed figure is not always a

polygon. Start with the properties that all polygons have in common. Ask students to

define a regular polygon based on their observations. Try to use not regular polygon

rather than irregular polygon for polygons that are not regular.

D i s cove r t h e M at h

Before starting Part A, review the term congruent so that students know how to join

congruent sides of the pattern blocks. Demonstrate an example and a non-example

of joining congruent sides using pattern blocks on the overhead. Have students

predict what different polygons could be made before joining the pattern blocks.

The Example explores concave and convex polygons. Students have been using the

terms concave and convex since Grade 3. Another way to test convexity is to make a

polygon on a geoboard with a rubber band. Put a piece of string around the perime-

ter of the polygon and pull it tight. If the string touches all of the vertices of the poly-

gon then the polygon is convex. If it does not touch all of the vertices then the

polygon is concave. This could be done with the class to promote understanding.


For Part B, give students BLM Venn Diagram from Mathematics Blackline Masters

Grades P to 9 to sort the pattern blocks in questions 1. For question 2, you could

print the polygons on card stock so students could place the polygons in the Venn

diagram, rather than just writing the letter for each polygon. All angles that appear

to be right angles but are not marked as such should be checked by comparing with

the square from the pattern block set. Note: Triangle F is not isosceles.

For question 3, have students trace the polygons on a transparency or tracing

paper to explore rotational symmetry and angle congruency. Mark one vertex with a

dot in order to test rotational symmetry. Give students a list of polygon properties

and have them look at angles, diagonals, sides, reflective symmetry, and rotational

symmetry. Have transparent mirrors available for diagonals and reflective symmetry.

Devote a great deal of time to question 3. Note: for question 5, the yellow circle label

should be Has Only One Line of Symmetry.

D i s cove r t h e M at h An s we r s

P a r t A

1. a), b), c), d), e), f) Drawings may vary. 2. a), b), c) Answers may vary.

3. a) Answers may vary. c) Congruent not regular polygons give more than one type

of polygon when joined but congruent regular polygons give only one polygon.

4. Answers may vary. a) use 3 rhombi b) use 3 triangles and 1 trapezoid c) use 2

rhombi and 1 triangle d) use 2 rhombi and 1 trapezoid

5. No. Examples may vary. Two congruent squares can be joined to form a

rectangle which has 4 sides like each square.

6. Answers may vary.


P a r t B

1. a) red circle; rhombus, trapezoid; blue circle: triangle, hexagon; inner circle:

square b) red circle: triangle, square; blue circle: trapezoid; inner circle:

rhombus, hexagon c) red circle: hexagon; blue circle; triangle; inner circle:

square, rhombus, trapezoid

2. Polygons that have both properties listed in the Venn diagram.

3. Descriptions may vary. A parallelogram, B rectangle, C regular pentagon, D

square, E equilateral triangle, F right triangle, G concave quadrilateral, H

regular hexagon, I rhombus, J pentagon, K isosceles triangle, L concave

pentagon, M concave hexagon, N kite, O scalene triangle, P quadrilateral Q

right triangle R quadrilateral

4. a) red circle: B, C, D, E, H, I; blue circle: A, B, C, D, E, F, G, H, I, J, L, M, N; inner

circle: B, C, D, E, H, I b) red circle: G, L, M; blue circle: A, B, C, D, E, F, H, I, J, K,

N, O, P, Q, R c) red circle: A, C, H, I, J, M, N, P, R; blue circle: C, H, J, L, M; inner

circle: C, H, J, M d) red circle: A, B, D, H, I, J, L, M; blue circle: A, B, C, D, E, H, I;

inner circle: A, B, D, H, I

5. yellow circle: F, G, J, K, L, M, N; red, blue, and yellow inner circle: A, B, C, D, E,

H, I; outside diagram: O, P, Q, R

6. a) red circle: At Least One Pair of Congruent Angles; blue circle: At Least One

Right Angle b) red circle: Irregular Polygons; blue circle: Convex Polygons

7. a), b) Answers may vary. 8. Answers may vary.

Journal

Students could use these prompts for question 6.• Examples of regular polygons I see in everyday life are …

• Some polygons that are not regular that I see in everyday life are …

Co m m u n i c ate t h e Key I d e a s

Have students work in groups to answer and discuss questions 1 and 2. For question3, assign a different polygon to each group and have them present their ideas. Then,

answer part b) as a class. Use this opportunity to assess student readiness for the

Check Your Understanding questions.

Co m m u n i c ate t h e Key I d e a s An s we r s

1. a) Azar. A rhombus does not have all angles equal. b) Answers may vary.

2. Answers may vary. 3. a), b), c), d) Answers may vary

4. If all the diagonals of a polygon are contained inside the polygon, then it is

convex. Otherwise, it is concave. Examples may vary.

O n g o i n g A s s e s s m e nt

• Can students demonstrate a working knowledge of the vocabulary

presented in this section and in previous grades?


C h e c k Yo u r U n d e r s t a n d i n g

Q u e s t i o n P l a n n i n g C h a r t

For question 3, part d), use The Geometry Template® or the polygon sheets in

Mathematics Blackline Masters Grades P to 9, pages 80 to 83. If you are assigning

questions for homework, provide students with pattern blocks or Power Polygons®

to take home. Store the manipulatives in baggies to keep them from getting lost.

For question 12, you should only assume that all line segments are congruent.


• Students think that any closed figure is a polygon.

Rx Review the definition of a polygon. Explain what is and is not a polygon

before defining a regular polygon and not regular polygon.

I nt e r ve nt i o n

• Students who struggle with the vocabulary terms may benefit from cue

cards that can be used for drill and practice or for reference when complet-

ing questions.

A S S E S S M E N T

Q u e s t i o n 1 2 , p a g e 3 0 6 , An s we r s

a) Dodecagon or 12-gon. It has 12 sides. b) No. Not all the angles are equal.

c) Concave. It has 3 reflex angles.

d) 3

e) Yes. Order of rotational symmetry is 3.

A D A P T A T I O N S

BLM 7.1 Assessment Question provides scaffolding for question 12.

BLM 7.1 Extra Practice provides additional reinforcement for those who need it.

Level 1 Knowledge andUnderstanding

Level 2 Comprehension of

Concepts and Procedures

Level 3 Application and Problem Solving

1, 3a), 5b) –e). 6b), c), 9a),10a)–d), 11a)–c), 12, 13, 15

2, 3b)–d), 4, 5a), 6a), 7, 8,9b)–e), 10e), 11d)

14, 16


E x t e n s i o n

Assign question 16. You may wish to reduce the number of Check Your Understanding

questions to provide students with extra time to work on the Extend question.

C h e c k Yo u r U n d e r s t a n d i n g An s we r s

1. a) regular b) not regular c) regular d) not regular


3. a) triangle: 3; 3. square: 4; 4. pentagon: 5; 5. hexagon: 6; 6. b) The number of lines

of symmetry equal order of rotational symmetry. c) regular heptagon: 7; 7.

regular octagon: 8; 8.

4. triangle: 60°; square: 90°; hexagon: 120°; rhombus: 60° and 120°; trapezoid: 60°

and 120°.

5. a) hexagon c) hexagon d) 6 e) Concave. It has 1 reflex angle.

6. a) 6 rhombi. rhombus; concave hexagon; concave heptagon; concave decagon;

concave 12-gon; concave 12-gon. 12 sides. c) Concave. It has 6 reflex angles.

7. a) Answers may vary. b) No. The interior angles in a regular heptagon cannot be

made using pattern blocks.

8. a), b), c), d) Answers may vary.

9. a) triangle: 0. pentagon: 2; 5. hexagon; 3; 9. b) The number of diagonals from one

vertex is increasing by 1. The total number of diagonals is increasing according

to the pattern 2, 3, 4, … . c) regular octagon: 5; regular decagon: 7 d) regular

heptagon: 14; regular decagon: 35

10. a) hendecagon or 11-gon b) Yes. All sides and angles are congruent. c) 8 d) 11 e) 11

11. a) decagon or 10-gon b) Concave. It has 5 reflex angles. c) 5 d) 5

13. circle a): E; circle b): B, F, Q; circle c): A, G, M, O, P, R; a) and b) inner circle: D;

a) and c) inner circle: C, H, I; b) and c) inner circle: J, L, N; outside of diagram: K

14. Neither. A polygon is regular if all sides and all angles are congruent.

15. a), b), c) Answers may vary.



7.2 Tessellations

W A R M - U P

Choose the best estimate.

1. + <1 or >1 (<1) 2. + 1 <4 or >4 (>4)

Calculate mentally.

3. � 72 <12> 4. � 45 <35>

5. 4% of 650 <26> 6. 75% of 360 <270>

Calculate using compatible numbers or near compatible numbers.

7. (–25) + (–87) + (–75) <–187>8. (–166) + (+243) + (–241) <–164>9. (–149) + (+246) + (–100) <–3>

Calculate mentally using the halve/double strategy.

10. (–45) � (+6) <–270> 11. (+22) � (–15) <–330>12. (–25) � (–42) <1050>

Evaluate using compatible factors.

13. (–5) � (–23) � (+2) <230>14. (–2) � (+9) � (–5) � (–8) <–720>15. (+4) � (+9) � (–5) � (+25) <–4500>


In this section, students continue to learn about tessellating polygons on a plane.

Read the Discover the Math activity completely before teaching the lesson. Try some

of the activities to familiarize yourself with what students will be doing.


In Part A, it is very important to do question 2 with other pattern blocks before

moving on to the next questions so that students see that the sum of interior angles

must be 360° in order to tessellate the plane. Before beginning a discussion on

Schlafli notation, discuss the paragraph about the honeycomb diagram. If students

do not see how the numbers correspond to the diagram, refer to the first diagram in

question 2 and describe the tessellation as {3,3,3,3,3,3}. Spend time on vertex regu-

larity and what it means, as this is key to continuing the activity.

7

9

1

6

3

4

22

3

2

5

3

7

Materials• The Geometry Template®

Related Resources• BLM 7.2 Extra Practice

Specific CurriculumOutcomesE2 predict and generate

polygons that can beformed with atransformation orcomposition oftransformations of agiven polygon

E4 make and applygeneralizations abouttessellations of polygons

E7 make and applygeneralizations aboutangle relationships


Link to Get ReadyStudents should havedemonstrated understandingof all sections of the GetReady prior to beginning thissection.


The following table gives the measures of interior angles of regular polygons

for your reference. Chapter 10 will deal with the sum of interior angles in polygons,

so do not be tempted to explain this ahead of time.

For Part B, have students read the Communicate Mathematically box to see how

to write Schlafli notation for semi-regular tessellations. A vertex and the complete tes-

sellation can be described by the polygons meeting at a vertex. Some students might

ask about tessellations that contain two different semi-regular or regular tessellations

and therefore two types of vertices to name. These are called demi-regular tessella-

tions and exceed the course outcomes. For question 1d), some students may build tes-

sellations that are not semi-regluar because they do not have vertex regularity.


P a r t A

1. a) triangle, square, and hexagon 2. 360°

3. The interior angles of regular polygons that do not tessellate do not divide

evenly into 360°.

4. Regular polygons with interior angles that divide evenly into 360° will tessellate.

5. a) The sum of the angles meeting at any vertex must equal 360°. b) No. Regular

polygons with more than 8 sides have interior angles that do not divide evenly

into 360°.


P a r t B

1. a), b) Answers may vary. c) No. If they have the same Schlafli notation then they

cannot be different tessellations. d) Because they use more than one kind of

regular polygon.

2. a) Yes. The interior angles around the point add to 360° (135° + 135° + 90°).

b) Because more than one kind of regular polygon is used. c) {4,8,8}

3. a), b) Answers may vary. c) yes d) Answers may vary.

4. Answer may vary. 5. a), b) Answers may vary. 6. a), b) Answers may vary.

7. a) The sum of the angles meeting at any vertex must be 360°.b) The polygons

must be regular and the sum of the angles meeting at any vertex must be 360°.


P a r t C

1. a) All of the quadrilaterals will tessellate the plane. b) Strategies may vary.

c) All quadrilaterals will tessellate the plane.

2. c) Strategies may vary. d) All triangles will tessellate the plane.

3. a) 360° b) Not regular polygons will tessellate if the sum of the angles meeting at

any vertex is 360° and the polygons have vertex regularity.

equilateral triangle: 60° regular octagon: 135°

square: 90° regular nonagon: 140°

regular pentagon: 108° regular decagon: 144°

regular hexagon: 120° regular hendecagon: 147.27° (147°)

regular heptagon: 128.57° (129°) regular dodecagon: 150°


Journal

Students could use these journal prompts for Part A, question 6.

• Examples of tessellations outside my classroom are …

• People who make these designs need to understand geometry because …

Students could use this journal prompt for Part B, question 8.

• I prefer [regular / semi-regular] tessellations because …


Have students work in groups to answer and discuss all of the Communicate the Key

Ideas questions. Use this opportunity to assess student readiness for the Check Your

Understanding questions.


1. a) A tiling pattern that covers a plane with no overlaps or gaps. b) Examples may

vary.

2. A regular tessellation is made using only one type of congruent polygon.

A semi-regular tessellation is made using 2 or more types of congruent polygons.

3. a) The number of sides in each polygon around a vertex in a tessellation.

b) Answers may vary.

4. No. It must also have vertex regularity.



Note: For questions 1a) and 2, the second tessellation is neither regular or semi-

regular. This question presents a good opportunity for students to demonstrate their

understanding of the differences between the two tessellations. You may wish to pres-

ent the question such that “neither” is another choice. For question 8, the Schlafli

notation should be {3,3,3,4,4} not {3,3,4,4}.


• Allow students who struggle with tessellations to work with a partner so

they can see the transformation(s) required to create the tessellations.





1, 2, 3, 4b), 5a), 8a), b), 9 4a), 5b), 6, 7a), 8c) 5c), 7b), 10, 11


A S S E S S M E N T

Q u e s t i o n 8 , p a g e 3 1 3 , An s we r s

a) 5 b) 3 triangles and 2 squares c)



E x t e n s i o n

Assign questions 10 and 11. You may wish to reduce the number of Check Your

Understanding questions to provide students with extra time to work on the Extend

questions. For question 10, you may want provide students with the interior angle

measures for the given polygons. Refer to the interior angle measure chart supplied

above.


1. a) regular; neither; semi-regular b) 4 squares; 6 triangles; 1 triangle, 2 squares, 1

hexagon c) {4,4,4,4}; not applicable; {3,4,6,4}

2. Neither. It does not have vertex regularity.

3. a) triangles and hexagons b) 2 triangles and 2 hexagons

c)

4. a), b) Answers may vary

5. a) translation b) yes c) Yes, Use a translation.

6. a) translation b) rotation and translation

7. a) neither b) No. The hands are not congruent.

9. a), b) Answers may vary.

10. a) The interior angles do not divide evenly into 360°. b) A polygon with greater

than 12 sides does not have interior angles that divide evenly into 360°.

11. Rotate one bird at its wingtip by 60° six times and then translate this figure.


7.3 Regular Polyhedra

W A R M - U P

Each dividend is broken up into parts. Determine the missing part.

1. 232 ÷ 4 = (200 ÷ 4) + (� ÷ 4) <32>2. 434 ÷ 7 = (420 ÷ 7) + (� ÷ 7) <14>3. 828 ÷ 9 = (� ÷ 9) + (18 ÷ 9) <810>

Divide using the Break Dividend Into Parts strategy.

4. 224 ÷ 7 <32> 5. 459 ÷ 9 <51>6. 496 ÷ 8 <62> 7. 1760 ÷ 4 <440>8. 4320 ÷ 6 <720> 9. 1545 ÷ 5 <309>


10. (+384) ÷ (–6) <–64> 11. (–581) ÷ (–7) <83>12. (–344) ÷ (+8) <–43> 13. (+2545) ÷ (–5) <–509>14. (–7360) ÷ (–8) <920> 15. (–2070) ÷ (+9) <–230>

B re a k D i v i d e n d I nt o Pa r t s St rat e g y

This strategy for division works well when the dividend can be broken into two parts

that are both easily divided by the given divisor.

Example:

In 4250 ÷ 5, 4250 can be broken into two parts, 4000 and 250. Both of these

numbers can be divided easily.

4250 ÷ 5 = (4000 ÷ 5) + (250 ÷ 5)

= 800 + 50

= 850

Another example:

In 372 ÷ 6, try to find a multiple of 6 that will come close to (but less than) 372.

A close multiple would be 360. Then 12 remains to be divided, which is also a

multiple of 6.

372 ÷ 6 = (360 ÷ 6) + (12 ÷ 6)

= 60 + 2

= 62


In this section, students learn about polyhedra. Review the language that shows how

3-D polyhedra are similar to 2-D polygons. Students will not have used polyhedra

language before. For polygons, students counted the number of sides but for poly-

hedra, students will count the number of faces. The counting leads to the names in

both cases. A polygon with six sides is called a hexagon and a polyhedron with six

faces is called a hexahedron.

Materials• Polydron® pieces:

equilateral triangles,squares, pentagons

• The Geometry Template®• square dot paper• isometric dot paper• bags


Specific CurriculumOutcomesE5 construct polyhedra

using one type ofregular polygonal face,and describe and namethe resulting PlatonicSolids


Link to Get ReadyStudents should havedemonstratedunderstanding of AngleProperties and Congruentand Similar Figures in theGet Ready prior to beginningthis section.


It is very important to read the Discover the Math completely and do theactivities yourself before teaching this section. Have students work in groups of

four to make the activities manageable. Make sure that students fully understand ver-

tex regularity from section 7.2 before starting this section. Students need several

opportunities to build the Platonic solids. Building each solid three times seems to

be appropriate for students to become aware of the properties of polygons.

Demonstrate how the hinge works in the Polydron® set. Remind students that

the logo must be up on both pieces or down on both pieces to make a hinge

connection. Demonstrate by making a pyramid or a prism, which are familiar from

elementary school. Students can stick numbered sticky notes to the faces of the solids

as they count so they do not lose track.


Do Part A and Part B over two days. Materials (per group of four): one bag of 24

equilateral triangles, one bag of 12 squares, The Geometry Template® or isometric

dot paper and square dot paper. For each group, two students build a tetrahedron

and an octahedron and then unfold them to make many nets as possible. The other

two students build two cubes and then unfold them to make as many nets as possi-

ble. Both pairs trace their nets and compare the nets they have made with their part-

ners. On day two, the pairs of students in each group exchange tasks.

Do Part C and Part D over two days. Materials (per group of four): one bag of

24 equilateral triangles, one bag of 24 pentagons. One pair of students builds a tetra-

hedron and an octahedron and the other pair builds a dodecahedron On day two,

students exchange bags and build the other solids.

Do Part E over one day. Materials (per group of four): two bags of 20 equilat-

eral triangles. Have pairs of students construct an icosahedron. Note: Question 2c)should say icosahedron not dodecahedron. For Part F, students should rebuild all of

the five Platonic solids. For sample nets of the Platonic solids see Check Your

Understanding question 1, on page 318: v) tetrahedron, i) and vi) cube, iv) octahe-

dron, ii) dodecahedron, iii) icosahedron.


P a r t A

1. c) square-based pyramid: 5 faces, triangle-based pyramid: 4 faces d) Yes, triangle-

based pyramid. All the faces are regular polygons and it has vertex regularity.

2. c) a six-sided polyhedron made with triangular faces; an eight-sided polyhedron

made with triangular faces. d) The eight-sided polyhedron is regular; the six-

sided polyhedron is not. e) The eight-sided polyhedron has vertex regularity; the

six-sided polyhedron does not. f) {3,3,3,3}

3. a) 10 faces b) No. It does not have vertex regularity.

4. They both use only one type of regular polygon and have vertex regularity.

P a r t B

1. b) yes c) yes d) {4,4,4} 2. Answers may vary.

3. Answers may vary. There are 11 possible nets. 4. Answers may vary.


P a r t C

2. Answers may vary. 3. Answers may vary. 4. Answers may vary.


6. They both use only one type of regular polygon and have vertex regularity.

P a r t D

1. b) 12 faces c) yes d) yes e) {5,5,5} 2. Answers may vary.

P a r t E

2. a) 20 faces b) yes c) yes d) {3,3,3,3,3} 3. Answers may vary.

P a r t F

1. a) regular tetrahedron: 4; 4; 6; 4 + 4 = 8. octahedron: 8; 6; 12; 8 + 6 = 14.

dodecahedron: 12; 20; 30; 12 + 20 = 32. icosahedron: 20; 12; 30; 20 + 12 = 32.

b) The number of faces plus the number of vertices is two more than the

number of edges. c) F + V = E + 2

2. The number of faces and the number of vertices is equal to the number of

edges plus two.


Journal

Students could use these prompts for Part F, question 3.

• One place I saw a Platonic solid was …

• The solid was a …

• Another place I saw a Platonic solid was …

• The solid was a…


Students may wish to create a table to organize their answers for question 1, part a).

For question 2, have students refer to the definition of vertex regularity in section

7.2, page 309, if necessary. Use this opportunity to assess student readiness for the

Check Your Understanding questions.


1. a) regular tetrahedron: 4 equilateral triangle faces; cube: 6 square faces;

octahedron: 8 equilateral triangle faces; dodecahedron: 12 pentagonal faces;

icosahedron: 20 equilateral triangle faces b) Has vertex regularity and all faces

are congruent regular polygons

2. a) Each vertex is formed by the same number of congruent polygons.

b), c) Answers may vary.


• Can students describe the difference between a polyhedron and a polygon?




Have Polydron® pieces available for all questions, but encourage students to

visualize their answers before using manipulatives.


• Students may confuse the names of the polyhedra.

Rx Display labelled diagrams or models of the polyhedra. Have students copy

this information into their notebooks for reference.


• For some students, you may need to review how to construct polyhedra.

Have students work with a partner.

A S S E S S M E N T


a) No. An octahedron has 4 triangles around each vertex but the net has 5

triangles in a row.




V i s u a l / Pe rce p t u a l / S p at i a l / M o t o r

• Students with poor motor skills may have difficulty building the polyhedra.

Be sure to pair them with students who can help them during group work.

E x t e n s i o n

Assign questions 15 to 18. You may wish to reduce the number of Check Your

Understanding questions to provide students with extra time to work on the

Extend questions. Suggest that students build the five Platonic solids again to

answer question 15.





1, 2, 3c), 5c), 9, 10, 12a)–c) 3a), b), 4, 5a), b), 6, 7, 13, 14 8, 11, 12d), 15–18



1. A iii); B i), vi); C iv), vii); D ii); E v)

2. the first and third arrangements

3. a) Need four squares in a row but the net only has three. b) Move the blue square

in the top left corner to the end of one of the yellow or red squares.

4. 2 squares

6. 2 triangles

7. a) tetrahedron b) cube c) icosahedron

8. a) yes c) Answers may vary.

9. c) Answers may vary.

10. d) Drawings may vary.

11. Yes. Pentagonal dipyramid has 10 faces, 7 vertices, 15 edges: 10 + 7 = 15 + 2.

12. d) rotation

13. a) b) or c) No. + <


15. Cubes or a combination of tetrahedrons and octahedrons will fill space without

leaving gaps. Reports may vary.

16. triangular prism

17. hexagonal prism

18. a) No. The internal angles of a pentagon do not divide evenly into 360°.

b) The angles around a vertex can be less than 360° because the object is now

three-dimensional. f) dodecahedron

1

2

3

12

1

12

1

4

3

12

1

12


7.4 Semi-Regular Polyhedra

W A R M - U P

Multiply.

1. � 48 <8> 2. 56 � <35>

Evaluate.

3. 8% of 900 <72> 4. 70% of 120 <84>

5. Give your best estimate for 27% of 795. <214.65>

Multiply using the Compatible Factors strategy.

6. (–5) � (+19) � (–2) <190>7. (+20) � (+9) � (–5) <–900>8. (–7) � (–25) � (–4) <–700>9. (–2) � (+7) � (+250) <–3500>10. (+2) � (+5) � (–2) � (+26) � (+5) <–2600>


11. (–552) ÷ (–6) <92> 12. (+648) ÷ (+9) <72>13. (+4520) ÷ (–5) <–904> 14. (–3080) ÷ (+7) <–440>15. (–2640) ÷ (–8) <330>


In this section, students learn about the Archimedean solids. You may wish to

demonstrate the term truncate by building a cube out of clay and using thread or

floss to slice off, or truncate, one of the corners. Students are not expected to build

all the Archimedean solids but they should have opportunity to build some of them.

Many students play role-playing games with dice that are in the form of

Archimedean and Platonic solids. It might be useful to show some of these dice as an

introduction. If available, show students a regular and a semi-regular polyhedron.

Ask students to describe the similarities and differences.

It is very important to read the Discover the Math completely and do theactivities yourself before teaching this section.


Do Part A and B as a group activity and give students carefully guided instructions.

5

8

1

6

Materials• Polydron® pieces:

equilateral triangles,squares, pentagons,hexagons

• isometric dot paper


Specific Curriculum OutcomesE6 Construct semi-regular

polyhedra and describeand name the resultingsolids, and demonstratean understanding abouttheir relationships to thePlatonic Solids


Link to Get ReadyStudents should havedemonstratedunderstanding of Congruentand Similar Figures in theGet Ready prior to beginningthis section.



P a r t A

3. tetrahedron

4. a) The truncated tetrahedron can be formed by cutting off the corners of the

tetrahedron. b) 8 faces c) 4 regular hexagons and 4 equilateral triangles

5. a) 3 polygons b) yes c) {3,6,6}

6. b), d) Answers may vary.

7. It has vertex regularity like a Platonic solid, but is made of two kinds of regular

polygons instead of one.

P a r t B

3. The truncated octahedron can be can be formed by cutting off the corners of

the octahedron. An octahedron.

4. a) 3 polygons b) yes c) {4,6,6}

5. b), d) Answers may vary.

6. The polyhedron was formed by slicing off or truncating the corners or an

octahedron. It has vertex regularity like a Platonic solid, but is made of two

kinds of regular polygons instead of one.

The Example shows how to build a rhombicuboctahedron. You could set up an

activity centre with materials so that all students can build this solid sometime

during studying this section.


Have students work in groups to answer and discuss all of the Communicate the Key

Ideas questions. Use this opportunity to assess student readiness for the Check Your

Understanding questions.


1. a) They both have vertex regularity and are made of regular polygons.

b) Regular polyhedra are made of only one kind of regular polygon and

semi-regular polyhedra are made of more than one kind.

2. To slice off part of an object.

3. Slice off the corners of some Platonic solids while maintaining vertex regularity

to produce the Archimedean solids.


• Can students describe the difference between semi-regular polyhedra,

regular polyhedra, polygons, and regular polygons?




Have Polydron® pieces available for questions 2, 3, 7, 10, and 11. For question 2,

some students may benefit from using modelling clay and floss to actually cut the

corners off a cube.

A S S E S S M E N T


b) No. It does not have vertex regularity.



E x t e n s i o n

Assign questions 10 and 11. You may wish to reduce the number of Check Your

Understanding questions to provide students with extra time to work on the Extend

questions. For question 10, have students work with one or two other students to

build the snub dodecahedron. It will require a large number of Polydron® pieces.

Have students build some of the Archimedean solids to answer question 11.


1. 12 pentagons and 20 hexagons

2. a), c) Answers may vary. d) cuboctahedron


4. a) Answers may vary. b) Euler’s Formula applies to all Archimedean solids.

c) All Archimedean solids have vertex regularity. d) Answers may vary.

5. a) square b) pentagon, triangle, hexagon, decagon c) {3,4,5,4}; {4,6,10}

d) Polyhedron 1 (a small rhombicosidodecahedron) can be truncated to produce

polyhedron 2 (a great rhombicosidodecahedron).

6. snub cube

8. a) Answers may vary.

9. A truncated isocahedron. It is made of hexagons and pentagons.


10. a) triangles and pentagons b) {3,3,3,3,5} d) 60 vertices, 92 faces, 150 vertices. Yes.

60 + 92 = 150 + 2 e) Drawings may vary.






1, 2b), d), 3a), b), 4a), 5b),10a)

2a), c), 3c), 4b)–d), 5a), c),6–9

5d), 10b) –e), 11


Chapter 7 Review

W A R M - U P

Subtract by adding the opposite.

1. (–12) – (–17) <5> 2. (+73) – (–27) <100>3. (–56) – (+14) <–70>

Evaluate using compatible numbers or near compatible numbers.

4. (–18) + (–82) <–100> 5. (–76) + (+74) <–2>6. (+165) + (–347) + (+135) <–47>

Multiply using the halve/double strategy.

7. (–12) � (–55) <660> 8. (+22) � (–250) <–5500>9. (–15) � (+26) <–390>

Multiply using compatible factors.

10. (–36) � (+2) � (–5) <360>11. (–15) � (–6) � (+2) � (–2) <–360>12. (–5) � (+8) � (–5) � (+4) <800>


13. (–372) ÷ 4 <–93> 14. 567 ÷ (–9) <–63>15. (–4960) ÷ (–8) <620>


Us i n g t h e C h a p t e r R ev i ew

The students might work independently to complete the Chapter Review, and then

compare solutions in pairs. Alternatively, the Chapter Review could be assigned for

reinforcing skills and concepts in preparation for the Practice Test. Provide an

opportunity for the students to discuss any questions, consider alternative strategies,

and ask about questions they find difficult.

Note: question 4D has no answer since the tessellation is neither regular nor

semi-regular. You could ask students which notation is not matched and for the

remaining notation, provide a drawing of a tessellation that would match the

remaining notation. Example: tessellation of squares. For question 6 and 13,

students could use Polydron® pieces to construct the polyhedron. Have pattern

blocks available for questions 7, 8, and 14.

After students complete the Chapter Review, encourage them to make a list of

questions they found difficult, and to include the related sections. They can use this

list as a guide on what to concentrate their efforts on when preparing for the final

chapter test.

Related Resources• BLM 7R Extra Practice



A S S E S S M E N T

Chapter Review

This is an opportunity for the students to assess themselves by completing selected

questions and checking the answers. They can then revisit any questions that they

found difficult.

Upon completing the Chapter Review, students can also answer questions such

as the following:

• Did you work by yourself or with others?

• What questions did you find easy? difficult? Why?

• How often did you have to ask a classmate to help you with a question? For

which questions?


Have students use BLM 7R Extra Practice for more practice.

R ev i ew An s we r s

1. Regular convex triangle: 3; 3. Not regular concave quadrilateral: 1; 1.

Regular convex pentagon: 5; 5. Not regular concave dodecahedron: 4; 4.

2. a) cube, regular tetrahedron, octahedron, dodecahedron, icosahedron

b) Has vertex regularity and all faces are of congruent regular polygons.

3. a) all three triangles b) Drawings may vary.

4. A iv), B i), C ii), D none, it is neither regular nor semi-regular

5. a) translation; reflection then translation b) yes

6. c) neither d) It does not have vertex regularity. e) 7 vertices, 10 faces, 15 edges. Yes.

7 + 10 = 15 + 2

7. a), b), c), d) Answers may vary. 8. Answers may vary.

9. a) dodecagon b) a square and a triangle c) Answers may vary.

10. a), b) Answers may vary. 11. a), b), c), d) Answers may vary.

12. The sum of the interior angles of a quadrilateral is 360°, so any quadrilateral

can be arranged so that the sum of the angles around any vertex on the plane

is 360°.

13. c) Platonic solid d) The solid has vertex regularity and is made of equilateral

triangles. e) 4 faces, 4 vertices, 6 edges. Yes. 4 + 4 = 6 + 2.

14. a) b) c) d)

15. 62 faces, 120 vertices, 180 edges. Yes. 62 + 120 = 180 + 2


Chapter 7 Practice Test


Us i n g t h e Pra c t i ce Te s t

This Practice Test can be assigned as an in-class or take-home assignment. If it is used

as an assessment, use the following guidelines to help you evaluate the students.

• Can students construct polygons and polyhedra?

• Can students describe a given polyhedron?

• Can students describe tessellations and polyhedra using Schlafli notation?

St u d y G u i d e

Use the following study guide to direct stu-

dents who have difficulty with specific ques-

tions to appropriate areas to review.

A S S E S S M E N T

After students complete the Practice Test, you may wish to use BLM 7PT Chapter 7Test as a summative assessment.

V i s u a l / Pe rce p t u a l / S p at i a l / M o t o r

• Allow the use of calculators.

• Let students give their answers verbally, either in an interview setting or recorded.

L a n g u a g e / M e m o r y

• Allow students to refer to personal math dictionaries, journals, index card

files, or notes.

Pra c t i ce Te s t An s we r s

1. D 2. A 3. C 4. C 5. D 6. B 7. D

8. a) semi-regular b) 3 triangles and 2 squares c) {3,3,4,3,4}

9. b) Alike: both made of equilateral triangles. Different: tetrahedron has 4 faces,

4 vertices, and 6 edges; octahedron has 8 faces, 6 vertices, and 12 edges.

c) Yes. Tetrahedron: 4 + 4 = 6 + 2; octahedron: 8 + 6 = 12 + 2.


11. The internal angles of a regular pentagon do not divide evenly into 360° so

regular pentagons cannot completely surround any point on the plane and

cannot tile a plane.

12. a) truncated tetrahedron b) 4 triangles and 4 hexagons d) Archimedean solid

e) {3,6,6}

13. a) 3 polygons b) square, hexagon, and dodecagon c)

Questions Refer to Section

1, 6, 7, 10 7.1

2, 8, 11, 13 7.2

4, 9 7.3

3, 5, 12 7.4

Related Resources• BLM 7PT Chapter 7 Test



Chapter 7 Chapter Problem Wrap-Up

1. Introduce the problem.

2. Clarify the assessment criteria by reviewing BLM 7CP Chapter Problem Wrap-Up Rubric with students.

3. Remind individual students that they have worked on the chapter problem

during chapter problem revisits throughout the chapter and that these will help

them. Students can also be directed to section 7.1 question 15, section 7.2

question 9, section 7.3 question 14, and section 7.4 question 9 at this point.

4. Brainstorm with students about their favourite ball sports. Discuss the rules of

the different sports and the shapes of the balls used. For a multicultural

connection, have students research sports from different countries. Ask students

to think about how they could invent a new sport that uses a new kind of ball.

5. Allow students time to work on the problem, either individually or in a group.

Students should prepare separate reports.

6. Keep copies of your own students’ work to show in future years.

O ve r v i ew o f t h e Pro b l e m

In the chapter problem revisits, students designed a team logo using polygons,

created a tessellation using the team logo, and selected polyhedra that were and were

not suitable to use as a soccer ball. In the chapter problem, students choose a ball for

a sport they invent and describe it using the vocabulary they learned.

A S S E S S M E N T

Use BLM 7CP Chapter Problem Wrap-Up Rubric to assess student achievement.

C r i t e r i a fo r a H i g h S co r i n g R e s p o n s e

• Student clearly describes the game and its rules.

• Student chooses an appropriate polyhedron for the game.

• Student clearly explains why the polyhedron is appropriate.

• Student includes all required information about the polyhedron.

• Student creates a game that works and would be interesting to play.

Wh at D i s t i n g u i s h e s Lowe r S co r i n g R e s p o n s e s

• Student may exclude information about the game or describe it poorly.

• Student may choose an inappropriate polyhedron or not explain their

choice of polyhedron.

• Student may not successfully describe the polyhedron.

• Student basically understands the problem and can make an attempt at

solving it – just cannot finish.

C h a p te r Pro b l e m Wra p - Up, p a g e 3 2 9 , An s we r

Answers may vary.


chapter 7 geometry of polygons and polyhedra (e)/files/blackline...occur throughout the chapter...

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