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Teacher Guide

Western

Western Canadian

Unit 10: Patterns in Numberand Geometry

Cover Gr5_TG_WCP U10 02/03/2005 2:44 PM Page 3

UNIT

“Mathematicians continue tosearch and discover newpatterns and relationships.Applications of mathematicalpatterns have led to solutions ofreal-world problems that wereonce thought to be unsolvable.Patterns are powerful ideas.”

John A. Van de Walle, Elementary and

Middle School Mathematics Fifth Edition

Mathematics Background

What Are the Big Ideas?

• Tables and graphs can be used to make predictions.

• Patterns can be represented by concrete models, in tables, and by graphs.

• Patterns exist throughout mathematics and nature.

How Will the Concepts Develop?

Students investigate numerical and geometric patterns, and express themmathematically in words, pictures, and/or symbols.

Students are introduced to the idea that looking for patterns andrearranging factors can make multiplying easier.

Students analyze patterns and create models of them. They representpatterns with tables and extend patterns. Students graph patterns andanalyze the graphs.

Students work with Fibonacci numbers and discover the prevalence ofthe Fibonacci sequence.

Students model and extend tiling patterns with manipulatives and using a computer.

Why Are These Concepts Important?

Looking for patterns in fractions improves understanding. Recognizingdifferent representations of a pattern allows the student to makemathematical connections and strengthen algebraic reasoning.

Working with Fibonacci numbers and the Fibonacci sequence illustratespatterns found in nature.

Giving opportunities to discover patterns lays the foundation for theunderstanding of advanced geometric properties and theorems in thelater grades. The ability to recognize and apply patterns is a vital skill in many careers.

FOCUS STRANDPatterns and Relations: Patterns

SUPPORTING STRANDSNumber: Number Operations,Statistics and Probability: Data Analysis, Shape and Space:Transformations

Patterns in Number and Geometry

ii Unit 10: Patterns in Number and Geometry

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Unit 10: Patterns in Number and Geometry iii

Lesson 1:Patterns in Multiplication

Curriculum Overview

General Outcomes• Students construct, extend, and

summarize patterns, . . . usingrules, charts, mental mathematics,and calculators.

• Students apply arithmeticoperations on whole numbers . . .and illustrate their use in creatingand solving problems.

Specific Outcomes• Students develop charts to record

and reveal patterns. (PR1)• Students generate and extend

number patterns from a problem-solving context. (PR4)

• Students estimate, mentallycalculate, compute or verify, theproduct (3-digit by 2-digit) . . . of whole numbers. (N11)

LaunchSquares Everywhere!

Cluster 1: Investigating Number Patterns

Show What You Know

Unit ProblemSquares Everywhere!


summarize patterns, includingthose found in nature, using rules,charts, mental mathematics, andcalculators.

• Students develop and implement aplan for the collection, display, andinterpretation of data to answer aquestion.

Specific Outcomes• Students develop charts to record

and reveal patterns. (PR1)• Students describe how a pattern

grows, using everyday language inspoken and written form. (PR2)

• Students construct and expandpatterns in two and three dimensions,concretely and pictorially. (PR3)

• Students generate and extendnumber patterns from a problem-solving context. (PR4)

• Students predict and justify patternextensions. (PR5)

• Students evaluate the graphicpresentation of the data to ensureclear representation of the results.(SP4)

• Students display data by hand or bycomputer in a variety of ways,including . . . broken-line graphs.(SP6)

Cluster 2: Representing PatternsLesson 2:Graphing PatternsLesson 3:Another Number PatternLesson 4: Strategies Toolkit

Lesson 5: Tiling PatternsTechnology:Using a Computer to ExploreTiling Patterns


summarize patterns . . . usingrules, charts, mental mathematics,and calculators.

• Students describe motion in termsof a slide, a turn, or a flip.

Specific Outcomes• Students construct and expand

patterns in two and threedimensions, concretely andpictorially. (PR3)

• Students cover a surface, using oneor more tessellating shapes. (SS22)

• Students create tessellations, usingregular polygons. (SS23)

Cluster 3: Exploring Tiling

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iv Unit 10: Patterns in Number and Geometry

Curriculum across the Grades

Materials for This Unit

For Lesson 5, bring in a variety of examples of tiling patterns, such asquilts, wallpaper, flags, tiles, and rugs, for students to look at. You mayfind examples of such patterns in books or magazines.

Grade 4

Students identify andexplain mathematicalrelationships and patterns,using grids/tables/objects, Venn/Carroll/tree diagrams, graphs,objects or models, andtechnology.

Students make and justifypredictions, usingnumerical and non-numerical patterns.

Students use skipcounting (forward andbackward) to support anunderstanding of patternsin multiplication anddivision.

Grade 5

Students develop charts torecord and revealpatterns, and generateand extend numberpatterns from a problem-solving context.

Students estimate,mentally calculate,compute or verify, theproduct (3-digit by 2-digit) of wholenumbers.

Students describe how apattern grows, usingeveryday language inspoken and written form.They construct andexpand patterns in twoand three dimensions,concretely and pictorially,and predict and justifypattern extensions.

Students evaluate thegraphic presentation ofthe data to ensure clearrepresentation of theresults, and display databy hand or by computerin a variety of ways,including . . . broken-linegraphs.

Students cover a surface,using one or moretessellating shapes. Theycreate tessellations, usingregular polygons.

Grade 6

Students represent,visually, a pattern toclarify relationships andto verify predictions.

Students summarize arelationship, usingeveryday language inspoken or written form.

Students createexpressions and rules todescribe, complete, andextend patterns andrelationships.

Gr 5 Unit 10 FM WCP 2/16/05 11:45 PM Page iv

Additional Activities

Unit 10: Patterns in Number and Geometry v

Cover UpFor Extra Practice (Appropriate for use after Lesson 1)Materials: Cover Up (Master 10.7), hundred charts(PM 13), counters

The work students do: With a partner, students usea hundred chart to play a game that involves factorsand multiples. They look for patterns and applypatterning strategies to win the game. Many games willend when one player chooses a prime number greaterthan 50. This challenges students to improve theirmultiplication and division skills and to gain a betterunderstanding of factors and multiples.

Take It Further: Students extend the game board to a200 chart.

Logical/MathematicalPartner Activity

Win with LessFor Extension (Appropriate for use after Lesson 3)Materials: Win with Less (Master 10.9), Pattern Blocks(PM 28), paper bags, pencil and paper

The work students do: Students play a game tryingto create Pattern Block figures with a minimum perimeter.They place several Pattern Blocks in a paper bag and,without looking, draw 3 figures from the bag. The objectof the game is to use all three shapes to make a figurethat has the least perimeter. Students look for patterns in the figures and apply patterning strategies to win the game.

Take It Further: Challenge students to play the gamedrawing 4, 5, or 6 Pattern Blocks from the bag.

Kinesthetic/Social/Visual/SpatialPartner Activity

Graph ItFor Extension (Appropriate for use after Lesson 2)Materials: Graph It (Master 10.8), Colour Tiles, 1-cmgrid paper (PM 23), pencil crayons

The work students do: Students work with a partnerto extend a growing pattern. In a table, students recordthe frame number and the area of the figure. Theyextend the table of values and use the table to graph thepattern on grid paper. Students describe the pattern ruleand use it to predict larger areas that follow the pattern.

Take It Further: Students use congruent squares tomake and record a different area pattern.

Logical/Mathematical/KinestheticPartner Activity

Painted PatternsFor Extension (Appropriate for use after Lesson 5)Materials: Painted Patterns (Master 10.10), 4 coloursof Snap Cubes, paper and pencil

The work students do: Students work in pairs. Theybuild cube structures using Snap Cubes. They begin witha 2 by 2 by 2 cube structure. Students imagine that thecubes are dipped in paint and they have to figure outhow many Snap Cubes would have: 3 painted faces, 2 painted faces, 1 painted face, and no painted faces.Students look for patterns in a 3 by 3 by 3 cube, andthen a 4 by 4 by 4 cube. They may use different colours to represent the Snap Cubes with each numberof faces painted.Students record the data in a chart and look for patterns.

Take It Further: Have students extend the activity forsingle-layer prisms (for example, a 1 by 1 by 2rectangular prism).

Kinesthetic/Social/Visual/SpatialPartner Activity

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vi Unit 10: Patterns in Number and Geometry

Planning for Unit 10

Planning for Instruction

Lesson Time Materials Program Support

Suggested Unit time: About 2 weeks

Gr 5 Unit 10 FM WCP 2/16/05 11:45 PM Page vi

The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2005 Pearson Education Canada Inc.

Unit 10: Patterns in Number and Geometry vii

Purpose Tools and Process Recording and Reporting

Planning for Assessment

Gr 5 Unit 10 FM WCP 2/16/05 11:45 PM Page vii


2 Unit 10 • Launch • Student page 346

Squares Everywhere!

LESSON ORGANIZER

Curriculum Focus: Activate prior knowledge about patterns innumber and geometry.

10–15 min

L A U N C H

ASSUMED PRIOR KNOWLEDGE

Students can recognize an area pattern and can describeit in informal language.Students can recognize a pattern in geometry and extendthe pattern.

�

�

ACTIVATE PRIOR LEARNING

Have students examine the patterns in theStudent Book.

Ask the first question in the Launch.(I see geometric patterns.)

Discuss the second and third questions posed inthe Student Book. Have students share theiranswers with a partner and the class. Recordthe answers on the board. (The patterns are the same because they all use figuresthat give the impression of squares; they are differentbecause one pattern uses shaded rhombuses to give theimpression of squares and another uses alternatingshaded squares.)

Ask:• What other figures do you see commonly

used in area patterns?(Cross, hexagon, L-shape, T-shape)

• Why do you think many area patterns usesquares? (Because squares leave no gaps in an area pattern)

• Are other quadrilaterals commonly used inarea patterns? Why do you think so?(Yes. Quadrilaterals form area patterns with nogaps.)

Tell students that, in this unit, they willinvestigate number patterns and patterns ingeometry. They will use number patterns to makepredictions and use Input/Output tables to lookfor relationships. Graphing a pattern enrichesstudents’ understanding of it. They will also learnthat number patterns can have geometricrepresentations. At the end of this unit, studentswill continue to look for patterns in squares andcreate a design using squares.

Gr 5 Unit 10 Launch WCP 02/03/2005 2:47 PM Page 2

LITERATURE CONNECTIONS FOR THE UNIT

Math Wizardry for Kids by Margaret Elizabeth Kenda, Tim Robinson (Illustrator), and Phyllis S. Williams. Barron’sEducational Series, Incorporated, 1995.ISBN 0812018095In this book, there are activities for finding math in nature, art,music, dance, poetry, gamesmanship, magic, and espionage.There is enough diversity here to ensure that at least someactivities will appeal to even the most die-hard math adversary.

Math-Terpieces: The Art of Problem-Solving by Greg Tang.Scholastic, Inc., 2003.ISBN 0439443881In Math-Terpieces, Greg Tang continues to challenge kids withhis innovative approach to math, and uses art history to expandhis vision for creative problem-solving.

Fibonacci Fun: Fascinating Activities with Intriguing Numbers byTrudi Hammel Garland. Dale Seymour Publications, 1997.ISBN 1572322659From “Raising Rabbits” to “Prickly Pinecones,” 24 easy-to-use,reproducible activities and projects help introduce students toFibonacci numbers.

DIAGNOSTIC ASSESSMENT

What to Look For

✔ Students canrecognize an areapattern and describeit in informallanguage.

✔ Students canrecognize a patternin geometry andextend the pattern.

What to Do

Extra Support:

Students who have difficulty finding patterns may benefit from more specificquestions, such as:• How is the chess game board the same as the other game boards?• How would you describe the pattern in terms of colour and shape?Work on this skill during Lessons 2, 3, 5, and Technology.

Students who cannot describe a pattern may benefit from modelling or preparingan illustration. Bring in a chess board or checker board to show to the class. Tofind the number of squares in a chess board or checker board, students could startby counting single squares, or 1 by 1 squares, then 2 by 2 squares, and so on.Work on this skill during Lessons 2, 3, 5, and Technology.

Unit 10 • Launch • Student page 347 3

Some students may benefit from using the virtualmanipulatives on the e-Tools CD-ROM. The

e-Tools appropriate for this unit include Geometry Shapes andPlace-Value Blocks.

REACHING ALL LEARNERS

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4 Unit 10 • Lesson 1 • Student page 348

Patterns inMultiplication

LESSON ORGANIZER

Curriculum Focus: Use patterns to solve multiplicationproblems. (PR1, PR4, N11)Student Materials Optional� calculators � Step-by-Step 1 (Master 10.11)

� Extra Practice 1 (Master 10.17)Vocabulary: product, factorAssessment: Master 10.2 Ongoing Observations: Patterns in Number and Geometry

40–50 min

L E S S O N 1

Key Math Learnings1. When multiplying numbers, the factors can be reordered or

regrouped to facilitate finding the product.2. Factors in a product can be further broken down to facilitate

the calculation.3. Multiplication and division are inverse operations.

That is, each undoes the other.

BEFORE Get S tar ted

Remind students of patterns in multiplicationthey have seen in earlier grades.

Display these products:439 � 10 439 � 100 439 � 1000

Invite a volunteer to provide the answers.(4390, 43 900, 439 000)

Ask:• Why is it easy to find these products without

using a calculator? (We use a base 10 number system. So, multiplyingby 10 moves the digits 1 place to the left, and we adda zero as a place-value holder. 100 = 10 � 10, so multiplying by 100 moves the digits 2 places tothe left, and we add 2 zeroes as place-value holders.1000 = 10 � 10 � 10, so multiplying by 1000moves the digits 3 places to the left, and we add 3 zeroes as place-value holders.)

• How can you find 439 � 20 without acalculator? (20 = 10 � 2. Use the previous result. 439 � 20 = 439 � 10 � 2 = 4390 � 2 = 8780)

• How can you use mental math to find 15 � 9?(I think of an array for 9 � 15. It would have 9 rowsof 10, and 9 rows of 5, which can be written as 9 � 10 + 9 � 5. This gives me 90 + 45, which is 135.)

Present Explore. Tell students they should writedown their prediction before using a calculator.Remind students that it will be easier to noticepatterns if their work is presented in an orderlymanner; for example, they could write theirpredictions and calculations in a 2-column chart.

360

360360360360360360

1050720400

1620

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Unit 10 • Lesson 1 • Student page 349 5

DURING Exp lore

Ongoing Assessment: Observe and Listen

As pairs of students work on the first part inExplore, ask:• How are the products similar?

(All the products are the same: 360)• How are the factors in the questions related?

(9, 18, and 36 are multiples of 9; 2, 4, 5, and 10 arefactors of 20.)

• Does the order in which you multiply thefactors affect the product? (No, the product is always the same, no matter inwhich order you multiply the factors.)

For the second part, ensure students find all thedifferent arrangements that can be made. Ask:• How do you know you have found all the

different possible arrangements? (I made a list, and found that I can arrange the

3 numbers in 6 different ways. I know I have found allthe different arrangements because I cannot reorder thenumbers without repeating a row in my list.)

• What is an example of an order that makesmultiplication easiest? (An order that produces a pair with a product thatis a multiple of 10)

As students work on the third part, ask:• Did you break any of the numbers apart to

make it easier to multiply mentally? Explain.(Yes, I broke down the multiples of 10 and tried toregroup. For example, 20 = 2 � 10, so 4 � 5 � 20= 4 � 5 � 2 � 10 = 4 � 10 � 10 = 400)

• How did you find the other products? (To multiply 5 � 21, I used 21 = 20 + 1, so 5 � 21 = 5 � 20 + 5 � 1 = 100 + 5 = 105. So, 5 � 21 � 10 = 105 � 10 = 1050)

Alternative ExploreMaterials: index cardsStudents work in pairs. They choose 3 different 1- or 2-digitnumbers and write these numbers on separate cards. Have themtrade cards with a partner. The partner arranges the cards in asmany ways as he or she can and describes which order makesmultiplication the easiest. Have students predict each product, asin Explore. Continue the rest of Explore as outlined in the StudentBook.

Early FinishersStudents can use patterns to create and solve problems with four1- or 2-digit factors.

Common Misconceptions➤Students may not know their basic multiplication facts and thus

have difficulty multiplying.How to Help: Have a 10 � 10 multiplication chart available forstudents to use. Encourage students to practise multiplication factsdaily by playing simple games, such as multiplication bingo.

ESL StrategiesModel the use of different multiplication words by displayingthem on a word wall. Include a multiplication sentence with thewords “factors” and “product” underneath the sentence.



Sample Answers1. Possible strategies: Use numbers that are easy to multiply.

Multiply by multiples of 10.2. a) 15 � 19 = 15 � 20 � 15

= 300 � 15= 285

15 � 21 = 15 � 20 + 15= 300 + 15= 315

b)

c) 14 � 20, 16 � 20; since the product of 15 � 20 is 300, 14 � 20 would be 20 less and 16 � 20 would be 20more.

6. 45 � = 2475; 558. 33 � 30 is the same as 33 � 10 � 3. I can multiply

33 � 10 first to get 330, then 330 � 3 to get 990. The area of the gym floor is 990 m2.

9. b) � 56 = 420010. First multiply 2 � 5 to get 10, and then multiply 7 � 8 to get

56. Then multiply 10 � 56, which makes 560.

300

20

15 285

19

15 315

21

15


As students try to find the missing factors inthe fourth part, they may decide to use guessand check or to divide. Ask them how theydecided on the initial value to check and howthey knew to refine the value by comparing theresulting product with the required value. Askstudents if they can think of an alternatestrategy that would produce the resultimmediately. If necessary, remind students ofrelated facts, such as 4 � 6 = 24, so 24 � 4 = 6,and 24 � 6 = 4. Remind them thatmultiplication and division undo each other.

AFTER Connec t

Invite various students to share their answersand strategies. Connect demonstrates thatfinding factors that are easy to multiply andregrouping factors can help with mental math.

Prac t i ce

Have calculators available for all questions.

Assessment Focus: Question 8

Students should demonstrate they are able tobreak numbers apart into factors that are easierto multiply. Some students might need to drawan array, and then try different ways ofbreaking the array into 2, until they are able tofind a way to break the numbers apart andregroup the factors.

Students who need extra support to completeAssessment Focus questions may benefit fromthe Step-by-Step Masters (Masters 10.11–10.14).

450 960

250

3400

250 1250160

285 315

11 98 12

8 711 9


REFLECT: I can use patterns to multiply by looking for numbersthat make multiplication easy, such as 2, 5, or 10. For example, in 2 � 17 � 5, I would first multiply 2 � 5 to get 10, and then multiply 10 � 17 to get 170.


ASSESSMENT FOR LEARNING

What to Look For

Reasoning; Applying concepts ✔ Students can use more than one

strategy to solve multiplicationproblems.

✔ Students look for multiplicationpatterns to predict products.

Accuracy of procedures✔ Students use mental math to multiply.

✔ Students use multiplication facts todetermine the missing factors.

What to Do

Extra Support: Have students model Practice questions withBase Ten Blocks. Students can use Step-by-Step 1 (Master 10.11)to complete question 8.

Extra Practice: Students can complete the Additional Activity,Cover Up (Master 10.7).Students can complete Extra Practice 1 (Master 10.17).

Extension: Have students use a set of digit cards from 1 to 10,or a deck of cards with the face cards removed. The Ace will beused as 1. Challenge students to create and solve multiplicationsentences using 5 cards as the factors.

Recording and ReportingMaster 10.2 Ongoing Observations:Patterns in Number and Geometry

Numbers Every DayPattern rules: Start at 3.75, add 3.75 each time.Start at 2.4, multiply by 3 each time.Start at 2.25, add 3.5 each time.

5.75 16.25, 19.75

194.4, 583.2, 1749.615, 18.75, 22.5

999999999

75 km

25 1524 42



Graphing Patterns

Key Math Learnings1. Models, tables, and graphs can be used to represent

patterns.2. Models, tables, and graphs can be used to extend and

predict patterns.3. A graph provides a picture of a table of values.

LESSON ORGANIZER

Curriculum Focus: Use models, tables, and graphs torepresent patterns. (PR1, PR2, PR3, PR4, PR5, SP4, SP6)Teacher Materials� overhead Colour Tiles (optional)� overhead grid transparency (PM 23)Student Materials Optional� Colour Tiles � 2-column charts (PM 17)� 1-cm grid paper (PM 23) � Step-by-Step 2 (Master 10.12)� Perimeter of a Square � Extra Practice 1 (Master 10.17)

(Master 10.6)� pencil crayons� rulersVocabulary: unit length, perimeter, horizontal axis, verticalaxis, broken-line graphAssessment: Master 10.2 Ongoing Observations: Patterns inNumber and Geometry

80–100 min

L E S S O N 2


Draw a 2-cm by 2-cm square on an overheadgrid transparency. Ask:• How can you find the perimeter of this

square? (Count the units along the outside of thefigure.)

• What is another way of finding the perimeterof the square? (Multiply the side length by 4.)

On the overhead grid, draw the vertical andhorizontal axes for a broken-line graph. Ask:• What is the line that goes straight up and

down called? (The vertical axis)• What is the line that goes straight across

called? (The horizontal axis)

Write a scale on the horizontal and verticalaxes. Ask:• What is the scale on my horizontal axis?

(1 square represents 2 units.)

• What is the scale on my vertical axis?(1 square represents 4 units.)

• What else do I need on my graph?(A title and labels for the vertical and horizontal axes)

Mark a point on the graph. Ask:• How can you identify where the point is on

the graph?(It is 4 units along the horizontal axis and 8 unitsupward in the direction of the vertical axis.)

Mark another point, higher up and to the righton the graph, and join the two points. Ask:• What happens when line segments go up to

the right? (The graph is increasing.)

Students may use grid paper or Colour Tiles forExplore. Explain the concept of “1 unit long,”referring to the side length of a congruent square.You may want to model the first two frames ofthe pattern on an overhead grid transparency.


Sample Answers1. a) Pattern rule: multiply hours worked by 6 to get the

amount earned.


Present Explore. Remind students to record theside length and the perimeter of each square ina table.

DURING Exp lore


Ask questions, such as:• As you increase the side length of the square,

what happens to the perimeter?(The perimeter increases.) How do you know? (I counted the units alongthe outside of the square. In my table, I also see thatthe perimeter increases as the side length increases.)

• What does the broken-line graph tell youabout the pattern? (As the side length increases,the perimeter also increases.) How do you know? (The line segments on theline graph go up to the right.)

• How are the broken-line graph and table ofvalues related? (Each row in the table representsone point on the broken-line graph. The number inthe first column refers to the distance along thehorizontal axis. The number in the second columnrefers to the distance along the vertical axis.)

AFTER Connec t

Have students share their work and theirfindings. Ask:• What patterns do you see in the perimeter of

the square compared to the side length of thesquare? (The perimeter is 4 times the side length.)How do you know? (I multiply the number inthe column labelled Side Length by 4, and get thenumber in the column labelled Perimeter.)

Discuss the table in Connect. Ask:• What is the pattern in the row labelled

Length? (Start at 24. Subtract 4 each time.)

Alternative ExploreMaterials: square dot paper (PM 25)Have students draw squares on dot paper, as in Explore. Askthem to find the area and the side length of each square, recordthe data in a table, and create a broken-line graph.

Early FinishersStudents can investigate patterns in side lengths and perimetersof equilateral triangles by making a table and a graph.

Common Misconceptions➤Some students build rectangles that are not larger squares.How to Help: Help students define the pattern correctly bymodelling the first three frames.➤Some students cannot make the connection between the table

representation and the graphic representation.How to Help: Have students find each point on the graph, andread aloud the corresponding numbers from the horizontal axisand the vertical axis. Have students find the row in the table thatcontains these numbers.


Numbers Every DayStudents may round one or both factors to the nearest 10.

About 800About 1500About 2700About 200


b) c) On the graph, 3 h is midway between2 h and 4 h, so the correspondingearning is midway between $12 and$24. Juanita earned $18. In the table,I would follow the pattern rule andmultiply 3 by 6 to get 18.

2. b) The numbers in the Perimeter c)

column follow the pattern: Start at 6; add 4 each time.

3. b) d)

4. a) Input: Start at 2. Add 2 each time.Output: Start at 7. Add 2 each time.

b) c) 7 is halfway between 6 and 8. Thecorresponding output is halfwaybetween 11 and 13. The outputnumber is 12.

d) 14 is halfway between 13 and 15.The corresponding input is halfwaybetween 8 and 10. The inputnumber is 9.

Input/Output

15

13

11

9

7

Ou

tpu

t

0 106 84

Input

2

Areas

15

12

9

6

3

Are

a (

sq

ua

re u

nit

s)

0 53 42

Frame

1

Frame Area (square units)

1 32 63 94 125 15

Perimeters

22

18

14

10

6

Pe

rim

ete

r (u

nit

s)

0 53 42

Frame

1

Juanita's Earnings

48

36

24

12

Am

ou

nt

Ea

rne

d (

$)

0 8642

Hours

• What is the pattern in the row labelledWidth? (Start at 2. Add 4 each time.)

Direct students to follow the steps in theStudent Book. Ensure they label and title theirbroken-line graphs. Model the graph on anoverhead grid transparency. Make theconnection between the table data and thegraphic representation (the broken-line graph).Ask:• What is the width of a rectangle with

perimeter 52 cm and length 14 cm? (12 cm)• What is the length of a rectangle with

perimeter 52 cm and width 8 cm? (18 cm)• What is happening on the broken-line graph?

(The length is decreasing.) How do you know? (The line segments are going down to the right.)

Prac t i ce

Students will need grid paper for all questions.


Students should correctly choose 5 inputnumbers that are multiples of one number.Students use this information and choose anoperation to create an Input/Output machinewith the correct output numbers. They shouldexplain that a broken-line graph is arepresentation of the Input/Output machine.Students correctly draw a broken-line graphbased on their Input/Output table.


Frame 4 Frame 5

26

24

2 103 144 185 22


5. a) The output numbers will be greater, but they will notdouble. The graph will still look the same, but the outputnumbers on the vertical axis will be different.

b) c)

The graph does not change. Only the input and outputnumbers have changed.

6.

a) b)

As the input numbers increase, so do the output numbers.The broken-line graph is a representation of theInput/Output machine.

REFLECT: I prefer to display a pattern in a table if the numbersincrease rapidly, because it is difficult to graph largenumbers.

Input/Output

20

16

12

8

4

Output

0 106 84

Input

2

Input Output2 44 86 128 16

10 20

Input Outputx 2

Input/Output

25

21

17

13

9

Output

0 2012 168

Input

4

Input Output4 98 13

12 1716 2120 25



What to Look For

Reasoning; Applying concepts ✔ Students understand that a table of

values can be represented by a graph.

✔ Students understand that a graph canbe used to predict patterns.

Accuracy of procedures✔ Students create a table of values.

✔ Students graph a table of values.

What to Do

Extra Support: Have pairs of students work through thequestions.Students can use Step-by-Step 2 (Master 10.12) to completequestion 6.

Extra Practice: Students can complete Extra Practice 1 (Master 10.17).

Extension: Students can do the Additional Activity, Graph It(Master 10.8).


79

111315



Another NumberPattern

Key Math Learnings1. Number patterns and geometric patterns exist throughout

nature.2. The Fibonacci sequence is a special number pattern, and can

be found in nature.3. Many patterns can be found in the Fibonacci sequence.

LESSON ORGANIZER

Curriculum Focus: Explore number patterns in context. (PR1, PR2, PR5)Student Materials Optional� dominoes � Step-by-Step 3 (Master 10.13)� calculators � Extra Practice 2 (Master 10.18)Vocabulary: Fibonacci sequence, Fibonacci number,consecutiveAssessment: Master 10.2 Ongoing Observations: Patterns inNumber and Geometry

40–50 min

L E S S O N 3


Review some of the patterns students havealready seen in previous grades. Write thesenumber patterns on the board:

2, 4, 6, 8, 10, ... 1, 3, 5, 7, 9, ...

Ask:• What patterns do you see?

(First pattern: Start at 2. Add 2 each time.Second pattern: Start at 1. Add 2 each time.)

• How are the two sets of numbers the same?How are they different?(They both increase the same way, by adding 2 eachtime. They are different because they start at adifferent number, and the patterns contain differentnumbers.)

Write this number pattern on the board:2, 5, 3, 6, 4, 7, 5, ...

Ask:• What pattern do you see in these numbers?

(Start at 2. Add 3 to get the next number, thensubtract 2 to get the number after that. Keep adding3, then subtracting 2.)

• How else could you describe the pattern?(For the 1st, 3rd, 5th, 7th term, and so on: Start at 2.Add 1 each time. For the 2nd, 4th, 6th term, and soon: Start at 5. Add 1 each time.)

• How would you describe the differencebetween the first two number patterns, andthe third one?(In the first two number patterns, we add the samenumber each time. In the third number pattern, weadd, then subtract.)

Point out that a series of numbers can followmany different kinds of patterns, and thepattern can be described in different ways.Present Explore. Write the three number patternson the board.

42, 6865, 10563, 102

Making ConnectionsNature: Write the first 6 terms of the Fibonacci sequence onthe board. Have students count out the number of petals oneach flower.


Sample Answers2. a) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,

987b) I chose 3, 5, and 8; 3 � 8 = 24; 5 � 5 = 25. The product

of the middle number times itself is one greater than theproduct of the outside numbers.

c) I chose 1, 2, and 3; 1 � 3 = 3; 2 � 2 = 4. The middleproduct is one greater than the outside product.

3. b) 2, 5, and 21 are all Fibonacci numbers.


DURING Exp lore


Ask questions, such as:• How does the first pattern grow?

(The next number is found by adding the twonumbers before it.)

• How are the 3 patterns the same? (They follow the same pattern rule: the first 2 termsare the same, and each term after is the sum of theprevious 2 terms.)

• How are the patterns different?(They all start with a different number.)

AFTER Connec t

Invite students to share their work with therest of the class. Ask:• How are the number patterns in this lesson

different from the ones we saw at thebeginning of the unit?

(In the previous patterns, we weren’t adding any ofthe terms to get the next term.)

Review Connect. Introduce the term Fibonaccisequence. Explain that Leonardo Fibonaccidiscovered this number sequence around theyear 1200. Many mathematicians are stilldiscovering new patterns in this sequence.Highlight Math Link, and count the number ofpetals on each flower to demonstrate that thenumber is a Fibonacci number.

Prac t i ce

Students will need dominoes for question 6.


Students see that the emerging pattern in theresults is the Fibonacci sequence. Students whorealize this in part a can use the Fibonaccisequence to make their prediction for part b.Other students will use the dominoes to buildthe rectangles and check.

Alternative ExploreMaterials: counters or Cuisenaire RodsStudents can model and develop the Fibonacci sequence usingcounters or Cuisenaire Rods, and display the results in a table.Students could also model the first pattern in Explore with counters.

Early FinishersStudents can start with 1, 1, 1, ... to create a pattern where,after the first three terms, each term is the sum of the threeprevious numbers. (1, 1, 1, 3, 5, 9, 17, ...)

ESL StrategiesHave students conduct a three-step interview based on thisstatement: Describe the Fibonacci sequence. The first two studentsconduct the interview while the third student takes notes. Thestudents switch roles until each student has had a turn taking notes.


9; 64; 55

13 or 34

233609

4 2521


4. 3, 8, and 55 are all Fibonacci numbers. This is always true.Check it for one more pair of Fibonacci numbers.

5. These numbers are to the left and to the right of 21. Thenumber of spirals are consecutive Fibonacci numbers.

6. a) 1 domino: 1 rectangle; 2 dominoes: 1 rectangle; 3 dominoes: 2 rectangles; 4 dominoes: 3 rectangles

b) 8 rectangles. I could check my prediction by building therectangles. Since the number of rectangles are theFibonacci numbers, I could also look up the 6th number inthe sequence to see that it is 8.

c) My results are the Fibonacci sequence.

REFLECT: Start at 1. Repeat the number 1. Add the first twonumbers together to make the third number: 1 + 1 = 2. Addthe second and third number to make the fourth number: 1 + 2 = 3. Add the third and fourth number to make the fifthnumber, and so on: 2 + 3 = 5.



What to Look For

Reasoning; Applying concepts ✔ Students understand that all

Fibonacci-like patterns follow thesame rule, but can have different startnumbers.

✔ Students recognize Fibonacci patternsin nature.

Accuracy of procedures✔ Students can identify the Fibonacci

pattern.

✔ Students can extend the Fibonaccipattern.

What to Do

Extra Support: Help students recognize the pattern by havingthem choose pairs of consecutive terms in the Fibonacci sequenceand find their sum. Students should compare the sum to the sumof other pairs. In this way, they will see that the pairs of numbersand their sum are all part of the Fibonacci sequence. Students can use Step-by-Step 3 (Master 10.13) to completequestion 6.

Extra Practice: Students can complete Extra Practice 2 (Master 10.18).

Extension: Students can research Fibonacci and the Fibonaccisequence on the Internet.Students can do the Additional Activity, Win with Less(Master 10.9).


16

Numbers Every DayFor 1, 2, and 4, reorder to find compatible numbers that areeasy to multiply with.4 � 3 � 5 = (4 � 5) � 35 � 7 � 2 = (5 � 2) � 76 � 3 � 5 = (6 � 5) � 3

6070

90


Choreographers are designers of dance andmovement. They create and arrange the dancesteps in musical productions. Their work isvisible in many different forms, from the danceroutines performed by boy bands and theirbacking groups to the graceful movements ofthe principal dancer in a classical ballet.

Choreographers usually specialize in a form ofdance or movement, such as:

• classical ballet• contemporary dance• musical theatre productions• ice dance shows• coordinating the movements of models in

fashion shows.

Choreographers can also specialize in featurefilms, commercials, music videos, and TVshows. The opening and closing events for theOlympics are designed by a choreographer.

The role of a choreographer is to plan andrehearse the way dancers will perform. Theymay invent new routines. They could choosenew music or interpret existing works. Theyhave to plan each movement of the dancers.They may have to use a form of dance notationto record the movements they design.

Choreographers work with producers, artisticdirectors, musicians, and costume designers,and discuss the requirements of theperformance. They often assist in choosingspecific dancers for certain roles.

Choreographers start by developing ideas forcreating dances, keeping notes and sketches torecord influences and trying out ideasinformally with colleagues. Choreographersfrequently absorb artistic influences from otherart forms, such as theatre, the visual arts, andarchitecture.

Unit 10 • World of Work • Student page 359 15

W O R L D O F W O R K

Choreographer



Strategies Toolkit

Key Math Learnings1. Interpret a problem involving patterns.2. Solve a problem by drawing a diagram.

Sample Answers1. a) The pattern is the Fibonacci sequence, starting with the

number 2.2. A

I

BE

D C

J H

F G

LESSON ORGANIZER

Curriculum Focus: Interpret a problem and select anappropriate strategy. (PR1, PR3, PR4, PR5)Student Materials Optional

� countersAssessment: PM 1 Inquiry Process Check List, PM 3 Self-Assessment: Problem Solving

40–50 min

L E S S O N 4


Present Explore. Review the key steps inproblem solving: understand the problem, planhow to solve the problem, solve it, then lookback to check the solution.

DURING Exp lore


Ask questions, such as:• Which strategy will you use to solve the

problem? (I will draw a diagram.)• Explain what is meant by the phrase,

“… after 8 rounds of division.”(A round is like a frame in a growing pattern, soafter 8 rounds would be the same as after 8 framesof the pattern.)

Have students share their solutions with theclass. They should explain any strategies used.

AFTER Connec t

Present Connect. Students should recognize thatthe pattern follows a Fibonacci sequence. Theycopy and continue the diagram to check theirprediction.

Ask:• How many cows are there after 5 years? (8)• What pattern do you see in the number of

cows? (The Fibonacci sequence: 1, 1, 2, 3, 5, 8, ...)

Prac t i ce

Encourage students to refer to the Strategies listto assist in selecting an appropriate strategy.

256

8


2. (continued)A total of 35 isosceles triangles can be found.Draw additional lines so that each vertex is connected withevery other vertex. Label each vertex and each point ofintersection with a letter. Then make an organized list of thetriangles.Singles (10): ABG, GBH, BHC, HCI, CID, IDJ, DJE, JEF, EFA,FAGDoubles (10): AHB, GBC, BIC, DHC, CJD, DIE, EDF, AEJ,EGA, AFBTriples on the outside (5): DCE, EAD, AEB, ACB, BDCTall in the middle (5): DAC, EBD, AEC, ADB, BECInternal corners (5): DBF, EBI, AHD, CGE, ACJ

REFLECT: When I solve a problem with predictable changes I canuse a diagram to solve the problem. A diagram lets me see thepattern clearly. For example, in question 2, when I drew thepentagon and all its diagonals, and then labelled all possiblevertices, I could count the number of triangles inside easily.



What to Look For

Problem solving✔ Students can select an appropriate

strategy for solving a problem.

✔ Students can justify their solutions.

Communicating✔ Students can describe their strategy

clearly, using appropriate language.

What to Do

Extra Support: For question 1, have students use triangulargrid paper to model the diagram. For question 2, have themdraw the diagonals of a square first and then label all possiblevertices of triangles to find a strategy. Then ask them to draw thediagonals and label all possible vertices of triangles of a regularpentagon to solve the problem.

Extra Practice: Have students repeat question 2 using aregular hexagon.

Extension: Challenge students to create their own mazeproblem that is similar to question 1.

Recording and ReportingPM 1 Inquiry Process Check ListPM 3 Self-Assessment: Problem Solving

Common Misconceptions➤Students have difficulty using the strategy draw a diagram.How to Help: Students can use counters to model the patternfirst, then draw a diagram of the model.


22, 4

42



Tiling Patterns

Key Math Learnings1. A tiling pattern is a geometric pattern that extends in

different directions in a predictable way.2. A tiling pattern has no gaps or overlaps.

LESSON ORGANIZER

Curriculum Focus: Model and extend tiling patterns. (PR3, SS22, SS23)Teacher Materials� overhead Pattern BlocksStudent Materials Optional� Pattern Blocks (PM 28) � Step-by-Step 5 (Master 10.14)� triangular dot paper � Extra Practice 2 (Master 10.18)

(PM 26)� 1-cm grid paper (PM 23)� square dot paper (PM 25)Vocabulary: tilingAssessment: Master 10.2 Ongoing Observations: Patterns inNumber and Geometry

40–50 min

L E S S O N 5

Numbers Every DayStudents should recognize that they need to convert the fractions todecimal equivalents, or vice versa, and then order them from leastto greatest.

7�14

�, 7.5, 8.5, 8�34

�; 6.2, 6�170�, 7�

120�, 7.5;

2.25, 2�34

�, 3.25, 3�12

�; �180�, 1.5, �

1108�, 2.5


Have a variety of examples of tiling patternssuch as quilts, wallpaper, flags, tiles, and rugsavailable for students to look at. Ask:• What are some of the tiling patterns you see

in the classroom?(There is a tiling pattern in the floor. It is made upof small rectangles that make up a square. Thesquare repeats. The rug has a pattern of trianglesand squares.)

Present Explore. Explain to students that therecannot be any gaps or overlaps in a tilingpattern; it has to cover an entire region. Allowtime for students to experiment with PatternBlocks. Explain that they will make a tilingpattern on triangular dot paper that coversabout one-half of a page.

DURING Exp lore


Ask questions, such as:• Name a combination of 2 or 3 Pattern Blocks

that cover the surface with no gaps oroverlaps. (Square and triangle)

• Name a combination of 2 or 3 Pattern Blocksthat cannot make a tiling pattern. (Tan rhombus and hexagon)

AFTER Connec t

Have volunteers use overhead Pattern Blockson an overhead projector to model theirpatterns. Invite the class to describe the pattern.

Review the example in Connect. Ask:• How do the pentagons form a rectangle?

(The red pentagon and the orange pentagon fittogether to form a rectangle.)


Sample Answers1. a) T-shaped and U-shaped polygons

b)

2. a) Irregular pentagons; a shape formed by joining 4 pentagonsrepeats. To extend the pattern, I would keep drawing repeatingshapes along each row, and keep adding new rows.

b) Squares and parallelograms; the purple square with 4parallelograms repeats. To extend the pattern, I would keepadding a purple square with 4 parallelograms.

• How do you extend the pattern?(Continue repeating the rectangle formed from apair of pentagons until the grid is covered.)

• Will the pattern leave any gaps or overlaps?Explain. (No, a tiling pattern with rectangles doesnot leave any gaps or overlaps.)

• Could you extend the pattern using only oneirregular pentagon? Explain.(No, if you only use one pentagon in the tilingpattern, there will be gaps.)

Highlight that the rectangle in the StudentBook is 2 units wide and 3 units long. Ask:• How many rows of rectangles are there in

the grid? (4) How many columns? (6)• What is the area of the grid? (6 � 4 = 24)• What does the number 24 mean?

(24 rectangles cover the surface.)• How many pentagons of each colour are in

one row? (6) In one column? (4)• How many pentagons of each colour cover

the grid? (6 � 4 = 24)

Encourage students to use terminology such asgrid, area, tile, overlap, repeat, and expandwhen describing a tiling pattern.

Students could use the information in Connect toextend a pattern and predict the number offigures needed to tile a surface area.

Prac t i ce

Students will need grid paper for question 1,and square dot paper for question 3.


Students should make sure the pattern coversthe grid with no gaps or overlaps. Somestudents may model the pattern first usingPattern Blocks, and then record the pattern onsquare dot paper. Others will create and recordthe pattern on square dot paper.


c) 48 of each. I know thisbecause the T-shape andU-shape make a 3 by 3square, and 48 squaresfit on a surface that is 18 units by 24 units.

Early FinishersHave students use 4 types of Pattern Blocks to create tilingpatterns on their desks. Challenge students to complete otherstudents’ patterns.

ESL StrategiesModel the idea of tiling by using specific visual materials forthe student to copy. For example, have a picture of a wallpaperpattern that the student could copy using Pattern Blocks.





What to Look For

Reasoning; Applying concepts✔ Students can create more than one

tiling pattern using the samecombination of figures.

Accuracy of procedures✔ Students can identify and extend tiling

patterns.

What to Do

Extra Support: Draw a pattern on square or triangular dotpaper. Have students use two different colours to illustrate thepattern.Students can use Step-by-Step 5 (Master 10.14) to completequestion 3.

Extra Practice: Have students design and colour a tilingpattern on triangular dot paper.Students can complete Extra Practice 2 (Master 10.18).

Extension: Students can do the Additional Activity, PaintedPatterns (Master 10.10).


3.

The pattern is made up of right triangles, isosceles triangles,and 2 kinds of irregular pentagons. The part of the patternthat repeats is a rectangle formed by an isosceles triangle with2 right triangles on either side, plus 2 irregular pentagons.

REFLECT: I could use two figures that fit together to form asquare or a rectangle. For example, an isosceles triangle andan irregular pentagon.

I know that squares form a tilingpattern with no gaps or overlaps. Iwould repeat the square to extend mytiling pattern.


Unit 10 • Technology • Student page 365 21

T E C H N O L O G Y

Using a Computer toExplore Tiling Patterns

Key Math LearningComputers can be used to create patterns.

LESSON ORGANIZER

Curriculum Focus: Use a computer to explore tiling patterns.(PR3, SS22, SS23)Student Materials� computers with AppleWorks or The Geometer’s Sketchpad

40–50 min

BEFORE

Explain to students that they will use acomputer to explore tiling patterns.Ask:• What is a tiling pattern?

(It is a pattern that has no gaps or overlaps.)

Students with stronger computer skills can bepaired with students who are not as familiarwith computers.

DURING


Watch to ensure that students read and followthe instructions carefully.

Instructions for creating a growing patternusing The Geometer’s Sketchpad:

1. Open a new sketch in The Geometer’s Sketchpad. Click File.Then click New Sketch.

2. Select the Point Tool.Click to draw a point.Do this 4 times.

3. Select the Selection Arrow Tool.Click to select all four points.

4. Click Construct. Then click QuadrilateralInterior. This connects the points with line segments to form a quadrilateral.

Selection Arrow Tool

Point Tool

StraightedgeTool


5. Click outside the quadrilateral to deselectall the points.To change the size of the quadrilateral:Click on a point on the figure. Hold down the mouse button. Drag the point until it is where you want it.The attached line segments will stretch.Release the mouse button.Continue to select and move points untilyour quadrilateral is the shape you want.

6. To draw a polygon, select the Point Tool.Click to draw a point.This will be one vertex of your polygon.Continue to draw points until you have as many vertices as you wish.

7. Select the Selection Arrow Tool.Click to select the points in order.Click Construct.Then click Polygon Interior.This forms a polygon with the points asvertices.

8. To change the size of a polygon, repeatStep 5.

9. To colour a figure, click to select the figurewith the Selection Arrow Tool.Click Display. Then click Color.Select red, or any other colour that you wish.

10.To copy a figure, select the SelectionArrow Tool.Click on the figure to select it.Click Edit. Then click Copy.Click Edit. Then click Paste.

11.To translate a figure, follow Step 10 tocopy the figure.Click the copy.Hold down the mouse button.Drag the copy to where you want it.

12.To reflect a figure, select the StraightedgeTool. Click and drag the mouse to create aline. This will be your mirror line.Select the Selection Arrow Tool.

22 Unit 10 • Technology • Student page 366

Early FinishersHave students create a tiling pattern with a core of four or moredifferent figures. Ask them to write about how they extended thepattern. Did they use translations, rotations, reflections, or acombination of the three?

ESL StrategiesIf possible, pair students with the same native language.



REFLECT: A designer would use a computer to create tilingpatterns because it is faster to extend the tiling pattern with acomputer. The designer can correct or make changes to thepattern quickly, and print out many copies of the pattern.

Unit 10 • Technology • Student page 367 23

Click on the line you just created to select it.Click Transform. Then click Mark Mirror.With the Selection Arrow Tool, click onthe figure you want to reflect.Click Transform. Then click Reflect.

13. To rotate a figure, select the SelectionArrow Tool.Click on a point on the figure around whichyou want to rotate the figure.Click Transform. Then click Mark Centre.Click on the figure to select it.Click Transform. Then click Rotate.Change the number of degrees to 90 for a �14

� turn, to 180 for a �12

� turn or to 270 for a �34

�

turn.

14.Use Steps 2 to 13 to create a tiling pattern.

15.To save your pattern:Click File.Then click Save As.Give your file a name.Click Save.

16.To print your pattern:Click File.Then click Print.Click OK.

AFTER

Invite volunteers to share their tiling patternswith the class, and discuss how they createdthem. Elicit from students that using the copyand paste function makes it easy to extend atiling pattern.

In Step 9 in the Student Book, to rotate a �14

� turn, the programrequires the measure of the angle in degrees. You may need toexplain the reference “90 degrees” in the student text.

Math Note


Sample Answers1. I rearranged the factors to make multiplication easier.

a) First, I multiplied 2 � 5 to get 10. Then I multiplied 10 � 14 to get 140.

b) First, I multiplied 2 � 5 to get 10. Then I multiplied 23 � 10 to get 230.

c) First, I multiplied 2 � 5 to get 10. Then I multiplied150 � 10 to get 1500.

d) First, I multiplied 2 � 44 to get 88. Then I multiplied88 � 10 to get 880.

e) First, I multiplied 3 � 25 to get 75. Then I multiplied75 � 10 to get 750.

f) First, I multiplied 5 � 20 to get 100. Then I multiplied100 � 17 to get 1700.

3. a)

Frame 4 Frame 5b) Frame Number Number of Tiles

1 32 103 214 365 55

c) I noticed the number of tiles in each layer starts at 3,and goes up by 2 each time. The number of layersstarts at 1 and goes up by one each time; this is alsoequal to the Frame Number. I need to multiply thenumber of tiles in each layer by the number of layers(or frame number). By the 6th frame, there are 13tiles in each layer; I multiplied 13 � 6 to get 78.

d)

4. My pattern starts with 5 tiles in the shape of a cross (four arms). Each frame has one more layer than the frame before; so, the number of layers equals the framenumber. Each frame has one more tile added to eacharm than the frame before.

80

70

60

50

40

30

20

10

Nu

mb

er

of

Til

es

0 53 42

Frame Number

1 6

Colour Tiles

24 Unit 10 • Show What You Know • Student page 368

LESSON ORGANIZER

Student Materials� Pattern Blocks (PM 28)� calculatorsAssessment: Masters 10.1 Unit Rubric: Patterns in Numberand Geometry, 10.4 Unit Summary: Patterns in Number andGeometry

40–50 min

S H O W W H AT Y O U K N O W

140 230 1500880 750 1700

150

1112

98

78

Frame Number Number of Tiles1 52 183 394 685 105


5. a) Time (min) Number of Swings1 602 1203 1804 2405 300

b) Start at 60. Add 60 each time.c)

6. I know because the 3 numbers are consecutive Fibonaccinumbers, ending in 13. The numbers are 5, 8, and 13.

7. No, he cannot combine all three to make a tiling patternbecause there will be gaps.For example:

Number of Swings

300

240

180

120

60Nu

mb

er

of

Sw

ing

s

0 53 42

Time (min)

1

Unit 10 • Show What You Know • Student page 369 25

5, 8


What to Look For

Reasoning; Applying concepts✔ Questions 3 and 4: Student can construct and expand patterns.

✔ Question 6: Student recognizes Fibonacci patterns in nature.

Accuracy of procedures✔ Questions 1 and 2: Student uses mental math to multiply.

✔ Questions 3 and 4: Student can predict and justify pattern extensions.

✔ Question 5: Student can create and graph a table of values.

Problem solving✔ Question 7: Student can solve problems by applying a patterning strategy.

Communication✔ Questions 4: Student can describe how a pattern grows.

Recording and ReportingMaster 10.1 Unit Rubric: Patterns in Number and GeometryMaster 10.4 Unit Summary: Patterns in Number and Geometry


Remind students of the Learning Goals in theUnit Launch. Students will use these skills andconcepts as they complete the Unit Problem.

Review the patterns in the games in the UnitLaunch. Remind students that patterns can bedescribed and extended in many ways.

Students work individually on the Unit Problem.Have 1-cm grid paper, coloured pencils, and 2-column charts available. Invite volunteers toread aloud the instructions to the three parts ofthe Unit Problem. Ensure students understandwhat they are expected to do. In Part 1, theyshould draw a broken-line graph of the datathey plotted in a table, and properly label theaxes and title the graph. Remind students thatthey must show their work.

Have a volunteer read aloud the Check List.Remind students that they should use the CheckList to assess their work before handing it in.

You may also wish to provide students withcopies of Master 10.3 Performance AssessmentRubic: Squares Everywhere!, so they willknow how their work will be assessed.

26 Unit 10 • Unit Problem • Student page 370

Squares Everywhere!LESSON ORGANIZER

Student Grouping: IndividualStudent Materials� 1-cm grid paper (PM 23)� coloured pencils� 2-column charts (PM 17)Assessment: Masters 10.3 Performance Assessment Rubric:Squares Everywhere!, 10.4 Unit Summary: Patterns inNumber and Geometry

40–50 min

U N I T P R O B L E M

Sample ResponsePart 1Student work should include:• An accurate sketch of the squares on grid paper• A table of values as shown:• A statement that the side

length multiplied by itself givesthe area.

• A prediction of 144 cm for aside length of 12 cm

• A broken-line graph that plots area versus side length,without connected points. Thehorizontal axis is labelled SideLength (cm), and the verticalaxis is labelled Area (cm2).

Side Area (cm2)Length (cm)

1 12 43 94 165 256 367 498 64


Part 2Student work should include:• An accurate sketch of 4 squares drawn• A table of values as shown:• Recognition that Part 1 and

Part 2 show the same pattern.To get the area, multiply theside length by itself. In Part 1,we increased the side length,while in Part 2, we decreasedthe side length.

Part 3Student work should include:• A colourful square pattern that extends far enough so that the

pattern is clearly visible

Reflect on the Unit:Patterns follow rules. Once I know a pattern rule, I can use it toextend and predict the pattern. I can represent a pattern by atable or a graph. When I look at a table or a graph, I can seethe patterns in the numbers and come up with the rule. Forexample, the pattern rule: Multiply the input number by itself,can be written in this table or shown on this graph.

20

16

12

8

4

Ou

tpu

t

0 3 42

Input

1

Input/OutputInput Output1 12 43 94 16

Unit 10 • Unit Problem • Student page 371 27


What to Look For

Reasoning; Applying concepts ✔ Students apply and explain patterning

rules appropriately.

Accuracy of procedures✔ Students follow patterning rules

accurately to extend and creategeometric and/or numeric patterns.

Problem solving✔ Students choose appropriate

patterning strategies to solveproblems and create an area pattern.

What to Do

Extra Support: Break the task into manageable parts.Some students may have difficulty recognizing the pattern. Havethem model the pattern with congruent squares first, and thenhave them record the pattern on grid paper. You may want torecord the table of values for Part 1 and for Part 2 on the board.As a class, have students complete the initial entries, and thenextend the pattern on their own.

Students who have difficulty creating a square pattern in Part 3may benefit from seeing models of geometric patterns.Magazines, wallpaper, and the Internet are excellent sources ofthese types of patterns.

Recording and ReportingMaster 10.3 Performance Assessment Rubric: Squares Everywhere!Master 10.4 Unit Summary: Patterns in Number and Geometry

Length (cm) Area (cm2)16 2568 644 162 4


28

Evaluating Student Learning: Preparing to Report: Unit 10 Patterns in Number and Geometry This unit provides an opportunity to report on the Patterns and Relations: Patterns strand. Master 10.4: Unit Summary: Patterns in Number and Geometry provides a comprehensive format for recording and summarizing evidence collected.

Here is an example of a completed summary chart for this Unit: Key: 1 = Not Yet Adequate 2 = Adequate 3 = Proficient 4 = Excellent

Strand: Patterns and Relations: Patterns

Reasoning; Applying concepts

Accuracy of procedures

Problem solving

Communication Overall

Ongoing Observations 2 2 3 3 2/3

Strategies Toolkit (Lesson 4)

3 3

Work samples or portfolios; conferences

3 2 3 3 3

Show What You Know 3 3 3 3 3 Unit Test 2 2 3 3 2/3 Unit Problem Squares Everywhere!

3 3 3 2 3

Achievement Level for reporting 3

Recording How to Report Ongoing Observations

Use Master 10.2 Ongoing Observations: Patterns in Number and Geometry to determine the most consistent level achieved in each category. Enter it in the chart. Choose to summarize by achievement category, or simply to enter an overall level. Observations from late in the unit should be most heavily weighted.

Strategies Toolkit (problem solving)

Use PM 1: Inquiry Process Check List with the Strategies Toolkit (Lesson 4). Transfer results to the summary form. Teachers may choose to enter a level in the Problem solving column and/or Communication.

Portfolios or collections of work samples; conferences or interviews

Use Master 10.1 Unit Rubric: Patterns in Number and Geometry to guide evaluation of collections of work and information gathered in conferences. Teachers may choose to focus particular attention on the Assessment Focus questions. Work from late in the unit should be most heavily weighted.

Show What You Know Teachers may choose to assign some or all of these questions. Master 10.1 Unit Rubric: Patterns in Number and Geometry may be helpful in determining levels of achievement. #3, 4, and 6 provide evidence of Reasoning; Applying concepts; #1–5 provide evidence of Accuracy of procedures; #7 provides evidence of Problem solving; all questions provide evidence of Communication.

Unit Test Master 10.1 Unit Rubric: Patterns in Number and Geometry may be helpful in determining levels of achievement. Part A provides evidence of Application of mathematical procedures; Part B provides evidence of Understanding of concepts; Part C provides evidence of Problem solving; all parts provide evidence of Communication.

Unit performance task Use Master 10.3 Performance Assessment Rubric: Squares Everywhere! The Unit Problem offers a snapshot of students’ achievement. In particular, it shows their ability to synthesize and apply what they have learned.

Student Self-Assessment Note students’ perceptions of their own progress. This may take the form of an oral or written comment, or a self-rating.

Comments Analyze the pattern of achievement to identify strengths and needs. In some cases, specific actions may be planned to support the learner.

Learning Skills

PM 4: Learning Skills Check List Use to record and report throughout a reporting period, rather than for each unit and/or strand.

Ongoing Records

PM 10: Summary Class Records: Strands PM 11: Summary Class Records: Achievement Categories PM 12: Summary Record: Individual Use to record and report evaluations of student achievement over several clusters, a reporting period, or a school year. These can also be used in place of the Unit Summary.


29

Name Date

Unit Rubric: Patterns in Number and Geometry

Not Yet Adequate Adequate Proficient Excellent


• shows understanding by appropriately demonstrating and explaining: – patterns in

multiplication – patterns in tables

and on graphs – tiling patterns

may be unable to demonstrate or explain: – patterns in



partially able to demonstrate and explain: – patterns in



demonstrates and explains: – patterns in

multiplication – patterns in tables and

on graphs – tiling patterns

in various contexts, demonstrates and explains: – patterns in

multiplication – patterns in tables and

on graphs – tiling patterns


• accurately: – uses mental math

to multiply – identifies a pattern

rule and predicts the next element

– identifies and extends tiling patterns

limited accuracy; major errors in: – using mental math

to multiply – identifying a pattern

rule and predicting the next element

– identifying and extending tiling patterns

somewhat accurate; makes several minor errors in: – using mental math to

multiply – identifying a pattern



generally accurate; makes few minor errors in: – using mental math to




accurate; makes very few or no errors in: – using mental math to




Problem-solving strategies

• chooses and carries out a range of patterning strategies to solve and create problems in number and geometry (e.g., using calculators and computers, making models, creating charts and tables)

with assistance, chooses and carries out a limited range of appropriate patterning strategies, rarely resulting in an accurate solution

with limited assistance, chooses and carries out appropriate patterning strategies, frequently resulting in an accurate solution

chooses and carries out appropriate patterning strategies to solve problems, usually resulting in an accurate solution

chooses and carries out appropriate patterning strategies to solve problems accurately; may offer an innovative approach

Communication

• describes patterns and explains reasoning clearly and precisely

needs assistance to describe patterns and explain reasoning; often unclear and imprecise

describes patterns and explains reasoning with some clarity and precision

describes patterns and explains reasoning clearly and precisely

describes patterns and explains reasoning clearly, precisely, and confidently

• presents work clearly presents work with little clarity

presents work with some clarity

presents most work clearly and precisely

presents work clearly and precisely

Master 10.1


30

Name Date

Ongoing Observations: Patterns in Number and Geometry

The behaviours described under each heading are examples; they are not intended to be an exhaustive list of all that might be observed. More detailed descriptions are provided in each lesson under Assessment for Learning.

STUDENT ACHIEVEMENT: Patterns in Number and Geometry * Student Reasoning; Applying

concepts Accuracy of procedures

Problem solving Communication

Demonstrates and explains: – patterns in tables


Identifies and extends patterns Uses mental math

to multiply

Uses appropriate strategies to solve and create problems involving patterns

Describes patterns and explains reasoning clearly Presents work clearly

* Use locally or provincially approved levels, symbols, or numeric ratings.

Master 10.2


31

Name Date

Performance Assessment Rubric: Squares Everywhere!

Not Yet

Adequate Adequate Proficient Excellent


• shows understanding of patterns by explaining patterns and predictions involving side lengths and area of squares

shows very limited understanding by giving inappropriate explanations

shows limited understanding giving appropriate but incomplete explanations

shows understanding by giving appropriate explanations

shows thorough understanding by giving appropriate and complete explanations


• accurately: – records

information in tables

– constructs a graph using data in a table

– identifies patterns – uses patterns to

predict area from given side length

completes table and constructs graph with major errors or omissions limited accuracy; major errors or omissions in identifying patterns and making predictions

completes table and constructs graph with several minor errors or omissions somewhat accurate; several minor errors in identifying patterns and making predictions

completes table and constructs graph with few minor errors or omissions generally accurate; few errors or omissions in identifying patterns and making predictions

completes table and constructs graph with no minor errors or omissions accurate; very few or no errors in identifying patterns and making predictions

Problem-solving strategies

• chooses and carries out appropriate patterning strategies to solve problems and create a tiling pattern

chooses and carries out a limited range of appropriate patterning strategies, with incomplete or seriously flawed results

chooses and carries out some appropriate patterning strategies, with partially successful results

chooses and carries out appropriate patterning strategies, with generally successful results

chooses and carries out innovative, appropriate, and effective patterning strategies, with consistently successful results

Communication • describes patterns

and explains reasoning clearly and precisely

needs assistance to describe patterns and explain reasoning; often unclear and imprecise

describes patterns and explains reasoning with some clarity and precision

describes patterns and explains reasoning clearly and precisely

describes patterns and explains reasoning clearly, precisely, and confidently

• presents work clearly, including labels

presents work with little clarity

presents work with some clarity



Master 10.3


32

Name Date

Unit Summary: Patterns in Number and 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: Patterns and Relations: Patterns



Problem solving

Communication Overall

Ongoing Observations

Strategies Toolkit (Lesson 4)

Work samples or portfolios; conferences

Show What You Know

Unit Test

Unit Problem Squares Everywhere!

Achievement Level for reporting

*Use locally or provincially approved levels, symbols, or numeric ratings. Self-Assessment:

Comments: (Strengths, Needs, Next Steps)

Master 10.4


33

Name Date

To Parents and Adults at Home … Your child’s class is starting a mathematics unit on patterns in number and geometry. Patterns help us make sense of the world we live in and allow us to pose and solve problems. Patterns occur regularly in mathematics and can be represented by numbers and in the discipline of geometry. They exist in tiled floors, quilts, game boards, logos, and the flags of the world. As children learn to identify, extend, and create patterns, they develop powerful connections between number relations and geometric properties. In this unit, your child will:

• Use patterns to solve multiplication problems. • Model patterns in tables and on graphs. • Analyze and create tiling patterns. • Use a computer to create a tiling pattern.

Patterns occur in many forms. Ask your child to describe some of the patterns learned in class. Often adults use geometric patterns to solve problems. Masons, artists, and graphic designers all routinely use patterns in their line of work. Discuss with your child that many different careers use patterns in geometry. Making some connections with your child fosters a deeper understanding of the mathematics that is taught in school. Talk about how you use patterns in your daily life. The possibilities are endless! Here are some suggestions for activities you can do at home:

• Search the Internet to discover the Fibonacci sequence. • Look for number and geometry patterns in tiles, quilts, game boards, etc. in

your home.

Master 10.5


34

Name Date Perimeter of a Square

Master 10.6


35

Name Date

Additional Activity 1: Cover Up Work with a partner. You will need a hundred chart and counters of 2 colours. The game is played on a hundred chart.

Player 1 chooses any even number and puts a counter on that number.

Player 2 chooses any number that is either a multiple or a factor of Player 1’s number, and places a different-coloured counter on that number.

Players continue to take turns choosing numbers. A player can choose any

uncovered number as long as it is a multiple or a factor of the previous chosen number.

The game continues until one player cannot cover a number. When this

happens, the other player is the winner. Take it Further: Extend the game board, using a 200 chart.

Master 10.7


36

Name Date

Additional Activity 2: Graph It Work with a partner. You will need Colour Tiles and 1-cm grid paper.

The shapes below are made with Colour Tiles. Look for patterns in the area of each shape.

Frame 1 Frame 2 Frame 3 Frame 4

Determine the area of the shapes in Frame 5 and Frame 6. Create a table to organize your findings.

Draw a graph to display your data.

Explain how you could determine the area of the shape in any frame using

either a table of values or a graph. Take it Further: Create your own growing area pattern. Complete a matching table. Have a classmate graph your pattern on 1-cm grid paper.

Master 10.8


37

Name Date

Additional Activity 3: Win with Less Work with a partner. You will need Pattern Blocks, a paper bag, and pencil and paper.

Place an assortment of Pattern Blocks in a paper bag.

Without looking, each player chooses 3 Pattern Blocks and places them together to form a new figure.

The object of the game is to make a figure with the least perimeter.

Count the number of sides on each of the new figures.

The player with the least perimeter gets a point.

The first person to score 10 points wins the game. Take it Further: Repeat the activity with each player choosing 4, 5, or 6 Pattern Blocks from the bag.

Master 10.9


38

Name Date

Additional Activity 4: Painted Patterns Work with a partner. You will need different coloured Snap Cubes, a pencil, and a sheet of paper. Use Snap Cubes to model a growing cube pattern.

Begin with a 2 by 2 by 2 cube structure. Imagine that the cube has been dipped in paint. Try to figure out how many Snap Cubes would have 3 painted faces, 2 painted faces, 1 painted face, or no painted face.

Use a table. Record the frame number, the number of cubes with only

3 painted faces, the number of cubes with only 2 painted faces, the number of cubes with only 1 face painted, and the number of cubes with no face painted.

Continue to build cube structures.

You may want to use one colour to represent the Snap Cubes with 1 face

painted, another colour to represent the Snap Cubes with 2 faces painted, and a third colour to represent Snap Cubes with 3 faces painted.

Continue to record your data in the table.

Look for patterns in the table.

Describe any patterns that you see.

Take it Further: Try the activity for rectangular prisms.

Master 10.10


39

Name Date

Step-by-Step 1 Lesson 1, Question 8

You will need 0.5-cm grid paper. Step 1 On grid paper, outline a 33 by 30 array. Step 2 Draw a horizontal or a vertical line to break the array into 2 parts. Step 3 How did you decide where to draw the line?

________________________________________________________ Step 4 Write a multiplication sentence for each of the 2 smaller arrays.

________________________________________________________ Are these products easy to find? Explain.

________________________________________________________ If not, draw a different line in Step 2, and break the array into

2 different parts. Repeat Steps 3 and 4.

________________________________________________________ ________________________________________________________

Step 5 Find the product for each of the 2 smaller arrays you created.

________________________________________________________ Step 6 Find the sum of the products in Step 5: ________________________ What is the area of the gym floor?

________________________________________________________

Master 10.11


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Name Date


Step 1 Place an operation (× , ÷ , +, – ) and a number in the box. Input Output

Step 2 Complete the table.

Use the operation and the number from Step 1 to find the Output. Input Output

1 3 5 7 9

Step 3 Graph the Input/Output numbers.

Create your own intervals for the Output axis.

Input/Output

1 3 5 7 9 Input

Step 4 Look at your graph. Describe it.

________________________________________________________ ________________________________________________________

Master 10.12

Output


41

Name Date


Step 1 The length of a domino is 2 units. Its width is 1 unit. With dominoes, make a rectangle with 1 side 2 units long. How many different rectangles can you make with 1 domino? ___________ How many different rectangles can you make with 2 dominoes? _________ How many different rectangles can you make with 3 dominoes? _________ How many different rectangles can you make with 4 dominoes? _________

Step 2 Predict the number of different rectangles you could make with 6 dominoes. _________ Step 3 How can you check that your prediction is right?

________________________________________________________

________________________________________________________ Step 4 What are the first 6 Fibonacci numbers?

________________________________________________________ Step 5 How do the numbers of rectangles you found relate to the Fibonacci

numbers? ________________________________________________________

Master 10.13


42

Name Date


Step 1 Draw 2 different figures on square dot paper.

Step 2 Can these figures be put together without any gaps or overlaps? _____

If they can, then go to the next step. If not, change one or more of the figures.

Step 3 Use your figures to make a tiling pattern.

Step 4 Describe your pattern.

_________________________________________________________

_________________________________________________________

_________________________________________________________

_________________________________________________________

Master 10.14


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Name Date

Unit Test: Unit 10 Patterns in Number and Geometry Part A 1. Multiply. Use mental math. a) 2 × 13 × 5 = ______ b) 27 × 5 × 2 = ______ c) 2 × 186 × 5 = ______ d) 2 × 43 × 10 = ______

e) 3 × 13 × 10 = ______ f) 4 × 50 × 2 = ______ 2. Multiply.

13 × 30 = ______ Use this multiplication fact to find the missing factors. a) × 30 = 360 b) × 30 = 420 c) 330 = × 30 d) 450 = × 30 3. Find each missing factor. a) × 50 = 1000 b) × 100 = 1300 c) 1620 = × 45 d) 870 = × 30

e) 55 × = 2475 f) 1020 = 20 ×

Master 10.15a


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Name Date

Unit Test continued Part B 4. Evan is paid $8 per hour to mow lawns. The table shows his earnings.

Hours Amount Earned ($) 1 8 3 24 5 40 7 56

a) Write a pattern rule for the amount earned.

__________________________________________________________

b) Draw a broken-line graph to display the data.

c) Suppose Evan works 6 h. Use the graph to find how much he will earn.

__________________________________________________________

Master 10.15b


45

Name Date

Unit Test continued 5. Suppose you followed the Fibonacci pattern, but started at 7.

7, 7, 14, … Show the first 10 numbers in this pattern.

__________________________________________________________ Part C 6. a) Create a tiling pattern using more than two different figures.

b) Describe your pattern.

__________________________________________________________

__________________________________________________________

Master 10.15c


46

Name Date

Sample Answers Unit Test – Master 10.15 Part A 1. a) 130 b) 270 c) 1860 d) 860 e) 390 f) 400 2. 390 a) 12 b) 14 c) 11 d) 15 3. a) 20 b) 13

c) 36 d) 29 e) 45 f) 51

Part B 4. a) Multiply the number of hours worked by $8.

b)

c) $48 5. 7, 7, 14, 21, 35, 56, 91, 147, 238, 385 Part C 6. a)

b) My tiling pattern uses 2 T-shaped figures, 2 kinds of isosceles right triangles (4 of each), and 2 squares.

Master 10.16


47

Extra Practice Masters 10.17–10.19 Go to the CD-ROM to access editable versions of these Extra Practice Masters.


Program Authors

Peggy Morrow

Ralph Connelly

Ray Appel

Daryl M.J. Chichak

Cynthia Pratt Nicolson

Jason Johnston

Bryn Keyes

Don Jones

Michael Davis

Steve Thomas

Jeananne Thomas

Angela D’Alessandro

Maggie Martin Connell

Sharon Jeroski

Jim Mennie

Trevor Brown

Nora Alexander

Copyright © 2005 Pearson Education Canada Inc.

All Rights Reserved. This publication is protected by copyright,and permission should be obtained from the publisher prior toany prohibited reproduction, storage in a retrieval system, ortransmission in any form or by any means, electronic, mechanical,photocopying, recording, or likewise. For information regardingpermission, write to the Permissions Department.

Printed and bound in Canada

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