western canadian teacher guide - sd67 (okanagan...

Teacher Guide

Western

Western Canadian

Unit 4: Multiplication andDivision

4UNIT

“Students exhibit computationalfluency when they demonstrateflexibility in the computationalmethods they choose,understand and can explainthese methods, and produceaccurate answers efficiently. Thecomputational methods that astudent uses should be based onmathematical ideas that thestudent understands well,including the structure of thebase-ten number system,properties of multiplication anddivision, and numberrelationships.”

Principles and Standards of School

Mathematics, NCTM, 2000

Mathematics Background

What Are the Big Ideas?

• Mathematical operations are related. For example, a multiplicationsentence can be rewritten using other operations: 7 � 5 = 6 � 5 + 5.

• Multiplication involves counting groups of equal size and finding howmany there are in all.

• Multiplication and division are related. You can use related facts tohelp you divide. For example, if you know 5 � 7 = 35, you also know35 � 7 = 5 and 35 � 5 = 7.

• Solutions to multiplication and division problems can be verified byusing estimation, a calculator, or the inverse operation.

How Will the Concepts Develop?

Students use skip counting, patterns, and number relationships to learn,organize, and memorize the basic multiplication facts.

Students use their knowledge of place value and the basic multiplicationfacts to explore the patterns involved in multiplying by 10, 100, 1000,and their multiples. Students learn strategies for multiplying 2-digitnumbers by 1-digit numbers, and for estimating the products of 2-digitnumbers and 1-digit numbers.

Students use arrays and other models to investigate the relationshipbetween multiplication and division, and to learn and organize the basicdivision facts. Students develop strategies for dividing 2-digit numbersby 1-digit numbers, with and without remainders.

Why Are These Concepts Important?

Fluency with computations involving whole numbers is essential in theworld. Numbers are all around us. A good understanding of number,and the meanings of and the relationships between the operations ofmultiplication and division is imperative. Students who understand thestructure of numbers and the relationships among them can work withthem flexibly and confidently.

FOCUS STRANDSNumber ConceptsNumber Operations

SUPPORTING STRANDPatterns and Relations

Multiplication and Division

ii Unit 4: Multiplication and Division

Unit 4: Multiplication and Division iii

Lesson 1:Skip CountingLesson 2:Multiplying by Numbers to 9Lesson 3:Other Strategies for MultiplyingLesson 4:Exploring Multiplication PatternsLesson 5:Estimating ProductsLesson 6:Strategies for MultiplicationLesson 7:Strategies Toolkit

Curriculum Overview

General Outcomes• Students demonstrate a number

sense for whole numbers 0 to 10 000.

• Students apply arithmetic operationson whole numbers, and illustratetheir use in creating and solvingproblems.

• Students use and justify anappropriate calculation strategy ortechnology to solve problems.

Specific Outcomes• Students use skip counting (forward

and backward) to support anunderstanding of patterns inmultiplication. (N2)

• Students demonstrate and describethe process of multiplication (3-digitby 1-digit), using manipulatives,diagrams, and symbols. (N13)

• Students recall multiplication factsto 81 (9 � 9 on a multiplicationgrid). (N15)

• Students verify solutions tomultiplication problems, usingestimation and calculators. (N16)

• Students verify solutions tomultiplication problems, using theinverse operation. (N17)

• Students justify the choice ofmethod for multiplication, usingestimation strategies, mentalmathematics strategies,manipulatives, algorithms, andcalculators. (N18)

General Outcomes• Students demonstrate a number

sense for whole numbers 0 to 10 000.

• Students apply arithmeticoperations on whole numbers, andillustrate their use in creating andsolving problems.

• Students use and justify anappropriate calculation strategy ortechnology to solve problems.

LaunchAt the Garden Centre

Cluster 1: Multiplying Whole Numbers

Cluster 2: Dividing Whole Numbers

Specific Outcomes• Students use skip counting (forward

and backward) to support anunderstanding of patterns indivision. (N2)

• Students demonstrate and describethe process of division (2-digit by a1-digit), using manipulatives,diagrams, and symbols. (N14)

• Students recall division facts to 81(9 � 9 on a multiplication grid).(N15)

• Students verify solutions to divisionproblems, using estimation andcalculators. (N16)

• Students verify solutions to divisionproblems, using the inverseoperation. (N17)

• Students justify the choice of methodfor division, using estimationstrategies, mental mathematicsstrategies, manipulatives,algorithms, and calculators. (N18)

Show What You Know

Unit ProblemAt the Garden Centre

Lesson 8:Dividing by Numbers from 1 to 7Lesson 9:Dividing by Numbers from 1 to 9Lesson 10:Division with RemaindersLesson 11:Using Base Ten Blocks to DivideLesson 12:Another Strategy for Division

iv Unit 4: Multiplication and Division

Curriculum across the Grades

Grade 3

Students count by 2s, 5s,10s, and 100s to 1000,using random startingpoints.

Students recognize andexplain if a number isdivisible by 2, 5, or 10.

Students usemanipulatives, diagrams,and symbols withmaximum products anddividends to 50, todemonstrate and describethe processes ofmultiplication anddivision.

Students recallmultiplication facts to 49(7 � 7 on amultiplication grid).

Students calculateproducts and quotients,using estimationstrategies and mentalmathematics strategies.

Grade 4

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

Students demonstrate anddescribe the process ofmultiplication (3-digit by1-digit), usingmanipulatives, diagrams,and symbols.

Students demonstrate anddescribe the process ofdivision (2-digit by 1-digit), usingmanipulatives, diagrams,and symbols.

Students recallmultiplication anddivision facts to 81 (9 � 9 on amultiplication grid).

Students verify solutionsto multiplication anddivision problems, usingestimation andcalculators.

Students verify solutionsto multiplication anddivision problems usingthe inverse operation.

Students justify the choiceof method formultiplication anddivision, using:

• estimation strategies• mental mathematics

strategies• manipulatives• algorithms• calculators

Grade 5

Students estimate,mentally calculate,compute, or verify, theproduct (3-digit by 2-digit) and quotient (3-digit divided by 1-digit) of whole numbers.

Students multiply anddivide decimals tohundredths, concretely,pictorially, andsymbolically, using single-digit, whole numbermultipliers and divisors.

Students solve problemsinvolving multiple stepsand multiple operations,and accept that othermethods may be equallyvalid.

Multiplication BingoFor Extra Practice (Appropriate for use after Lesson 2)Materials: Multiplication Bingo (Master 4.11), BlankBingo Cards (Master 4.11b), Multiplication Chart(PM 29), counters

The work students do: Students work in groups.One student is the caller. Each player fills in the squaresof her blank Bingo card with different products selectedfrom the Multiplication Chart. The caller calls two factorsfrom a multiplication fact, for example, 6 � 7. The otherplayers cover the product on their cards, if they have theproduct on their cards. The first player to cover a row,column, or diagonal on her or his card is the winner.

Take It Further: Students play Multiplication Bingoagain. This time they write the factors (for example,5 � 2) in the squares of their Bingo cards and the callercalls out the products.

SocialGroup Activity

Additional Activities

Division Tic-Tac-ToeFor Extension (Appropriate for use after Lesson 9)Materials: Division Tic-Tac-Toe (Master 4.13), countersin two different colours

The work students do: Students work in pairs. Eachplayer uses one counter colour. Player 1 chooses anumber from List 1. Player 2 chooses a number from List2. Player 2 divides the number from List 1 by the numberfrom List 2. If the number divides evenly and the answeris on the Tic-Tac-Toe grid, he covers it with a counter.Players switch roles. The game continues until one playerwins by getting 3 of her counters in a row, column, ordiagonal, as in Tic-Tac-Toe.

Take It Further: Players create a new Tic-Tac-Toegame. They choose their own numbers for List 1, List 2,and the Tic-Tac-Toe grid.

Logical/Mathematical/SocialPartner Activity

Spinning ProductsFor Extra Support (Appropriate for use after Lesson 5)Materials: Spinning Products (Master 4.12), Spinners(Master 4.12b), calculators

The work students do: Students work in pairs.Players take turns to spin the pointers on two spinners.Both players record the numbers the pointers land onand estimate their product. They then use a calculator tofind the product of the numbers. The player whoseestimate is closer to the product scores a point. The firstplayer to score 10 points wins.

Take It Further: Players take turns to spin Spinner A.Each player chooses two factors whose product is closeto the number the pointer lands on. The player whoseproduct is closer to the number scores a point.

KinestheticPartner Activity

Unit 4: Multiplication and Division v

Multiplication and DivisionPuzzle SquaresFor Extra Practice (Appropriate for use after Lesson 11)Materials: Multiplication and Division Puzzle Squares(Master 4.14), Puzzle Squares (Master 4.14b), scissors

The work students do: Students work alone.Students cut out the 16 puzzle squares. They multiply ordivide to find the answers to the questions on the puzzlesquares. Students fit the squares together, matchingquestions and/or answers that are equal. When this isdone correctly, a 4 by 4 square is again formed.

Take It Further: Students create their own puzzlesquares for others to solve.

Logical/MathematicalIndividual Activity

vi Unit 4: Multiplication and Division

Planning for Unit 4

Planning for Instruction

Lesson Time Materials Program Support

Suggested Unit time: 3–4 weeks

Unit 4: Multiplication and Division vii

Lesson Time Materials Program Support

Purpose Tools and Process Recording and Reporting

Planning for Assessment

2 Unit 4 • Launch • Student page 118

At the Garden Centre

LESSON ORGANIZER

Curriculum Focus: Activate prior learning about multiplyingand dividing whole numbers.

L A U N C H

ASSUMED PRIOR KNOWLEDGE

Students can count by 2s, 5s, and 10s.Students can recognize and use patterns in amultiplication chart.

✓✓

ACTIVATE PRIOR LEARNING

Invite students to examine the trays of seedlingson page 119 of the Student Book.

Ask questions, such as:• Look at the first three trays of seedlings at the

top of page 119. Are these trays the samesize? (Yes)

• How many rows does each of these trayshave? How many seedlings are in each row?(Each tray has 2 rows with 6 seedlings in each row.)

• Look at the large tray at the bottom of page119. How many rows does this tray have?How many seedlings will fit in each row?(Each tray has 2 rows. Three seedlings will fit ineach row.)

Discuss the first question in the Student Book.Record the answer on the board. (Sample answer: There are 36 seedlings.)

Discuss the second and third questions in theStudent Book. Record each way on the board.

Elicit from students that they can find thenumber of seedlings by counting, multiplying,or adding. (Sample answers: I noticed there are 6rows of 6 seedlings; 6 � 6 = 36. I multiplied2 � 6 = 12, and then added 12 + 12 + 12 = 36. Inoticed there are 18 rows of 2 seedlings; 18 + 18 = 36.)

Discuss the fourth question in the StudentBook. Ensure students recognize that they needto divide. (Sample answer: She needs 6 smaller boxes; 36 � 6 = 6.)

Tell students that, in this unit, they will learnhow to multiply and divide whole numbers. Atthe end of the unit, they will demonstrate whatthey have learned by solving problems relatedto the Garden Centre in the Unit Problem.

10–15 min

LITERATURE CONNECTIONS FOR THE UNIT

The Grapes of Math: Mind Stretching Riddles by Greg Tang.New York: Scholastic Press, 2001.ISBN 043921033XThrough patterns, grouping, and creative thinking, the problemsto be solved will have children adding, subtracting, andmultiplying. Throughout, Tang sneaks in useful visual strategiesthat can be used in solving other computation problems.

Each Orange Had 8 Slices by Paul Giganti. Harper Trophy;reprint edition, 1999.ISBN 068813985XEach orange has 8 slices and each slice has 2 seeds. Childrenhave fun working their way through math puzzles that relate tothe world outside the classroom.

The Doorbell Rang by Pat Hutchins. Harper Trophy, 1989.ISBN 0688092349Victoria and Sam learn about division when they must sharetheir cookies with more and more guests.

DIAGNOSTIC ASSESSMENT

What to Look For

✔ Students can countby 2s, 5s, and 10s.

✔ Students canrecognize and usepatterns in amultiplication chart.

What to Do

Extra Support:

Students who have difficulty counting by 2s, 5s, and 10s, may benefit from colouringeach pattern on a hundred chart. Have them use a different colour each time.Work on this skill during Lesson 1.

Students who have difficulty recognizing and using patterns in a multiplicationchart may benefit from examining single rows and columns, and from comparingcorresponding rows and columns.Work on this skill during Lesson 2.

Unit 4 • Launch • Student page 119 3

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

The e-Tools appropriate for this unit include Place-Value Blocksand Counters.

REACHING ALL LEARNERS

4 Unit 4 • Lesson 1 • Student page 120

Skip Counting

L E S S O N 1

LESSON ORGANIZER

Curriculum Focus: Count on using numbers up to 10. (N2) (PR1)Teacher Materials� hundred chart transparency (PM 13)� multiplication chart transparency (Master 4.6)Student Materials Optional� hundred charts (PM 13) � Step-by-Step 1 (Master 4.12)� multiplication charts � Extra Practice 1 (Master 4.25)

(Master 4.6) � metre sticks� 10 by 10 multiplication

charts (PM 15)� number cubes labelled

4, 5, 6, 7, 8, and 9� countersVocabulary: multipleAssessment: Master 4.2 Ongoing Observations:Multiplication and Division

40–50 min

Key Math Learnings1. The multiples of a number are found by starting with the

number and skip counting by the same number.2. Patterns on a hundred chart and patterns on a multiplication

chart are related.

TEACHING TIP

Preparing MaterialsIn Practice question 4, students neednumber cubes labelled 4, 5, 6, 7, 8,and 9. If you do not have these, youcan cover the faces of a standard diewith stickers and write the numbers onthe stickers. The sums of the numberson opposite faces of the number cube

must be equal.

BEFORE Get S tar ted

Use the hundred chart transparency. Start at 2and count on by 2s. Have a student record thenumbers you circle on the board. Ask:• What patterns do you see?

(All the numbers are even. They start at 2 and go upby 2 each time. The ones digits form a repeatingpattern with core: 0, 2, 4, 6, 8.)

Use the overhead transparency of the partiallycompleted multiplication chart. Ask:• Where do you see this number pattern on the

multiplication chart? (The pattern 2, 4, 6, 8, 10,... is in the row of the multiplication chart that hasnumbers with 2 as a factor. It is also in the columnthat has numbers with 2 as a factor.)

Present Explore. Hand out copies of the hundredchart and the multiplication chart.

DURING Exp lore

Ongoing Assessment: Observe and Listen

Ask questions, such as:• What numbers do you write when you start

at 3 and count on by 3s? (3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...)

Sample Answers1. a) Student answer should include a hundred chart with all

multiples of 3, 6, and 9 shaded. 3, 6, 9, 12, ... 99b) 6, 12, 18, 24, ... 96; Multiples of 6 are also multiples of 3.c) 9, 18, 27, ... 99; Multiples of 9 are also multiples of 3.

2. c) The numbers 8, 16, 24, 32, and 40 are in both lists. Everyother multiple of 4 is a multiple of 8.

3. c) The numbers 10, 20, 30, 40, and 50 are in both lists.Every other multiple of 5 is a multiple of 10.

Unit 4 • Lesson 1 • Student page 121 5

• What numbers do you write when you startat 5 and count on by 5s? (5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ...)

• What numbers do you write when you startat 7 and count on by 7s? (7, 14, 21, 28, 35, 42, 49, 56, 63, 70, ...)

Listen for students to make the connectionbetween skip counting by a number and therow for that number on the multiplication chart.

AFTER Connec t

Invite students to share the patterns they found.Have students make a list of the patterns to helpthem as they continue to master the basic facts.

Use Connect to introduce the term multiple. Ask:• How can you tell if a number is a multiple of

2? 5? 10? (A number is a multiple of 2 if it has 0, 2,4, 6, or 8 in the ones place. A number is a multiple of

5 if it has 0 or 5 in the ones place. A number is amultiple of 10 if it has 0 in the ones place.)

• What do you notice about the multiples of 1?(The multiples of 1 are the counting numbers.)

Prac t i ce

Questions 1, 4, and 7 require a hundred chart.Questions 2 and 3 require a multiplicationchart. Question 4 requires counters and anumber cube labelled 4, 5, 6, 7, 8, and 9.

Assessment Focus: Question 5

Students should recognize that since 4 is amultiple of 2, multiples of 4 are also multiplesof 2. They should understand that the reverse isnot true.

Students who need support to completeAssessment Focus questions may benefit fromthe Step-by-Step masters (Masters 4.12 to 4.22).

Alternative ExploreMaterials: metre sticks, multiplication charts (Master 4.6)Students use metre sticks as number lines from 1 to 100. Theyskip count by 3s, 4s, 5s, up to 10s on the number line, andrecord the numbers. Students use these numbers to complete themultiplication chart and discuss any patterns they notice.

Early FinishersHave students use their list of multiples of 3 to 50 from Practicequestion 6. Students list the multiples of 4 to 50. Ask students touse the numbers in these two lists to list the multiples of12 to 50.


12, 15, 1850, 60, 7035, 42, 4932, 37, 42

4, 8, 12, 16, 20, 24, 28, 32, 36, 40

8, 16, 24, 32, 40, 48, 56, 64, 72, 80

10, 20, 30, 40, 50, 60, 70, 80, 90, 1005, 10, 15, 20, 25, 30, 35, 40, 45, 50

Numbers Every DayEnsure students understand they can skip count starting at anynumber, but skip counting only gives the multiples of a numberwhen you start at that number and count on by thesame number.

5. a) 4 is a multiple of 2, so every multiple of 4 is also a multipleof 2. For example, 8 is a multiple of 4 and 2.

b) 6 is a multiple of 2, but 6 is not a multiple of 4.c) The even numbers are multiples of 2, so multiples of even

numbers are also multiples of 2.d) 3 is an odd number. 6 is an even number. 6 is a multiple

of 3.6. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32,

34, 36, 38, 40, 42, 44, 46, 48, 503, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 486, 12, 18, 24, 30, 36, 42, 48The numbers that appear in both lists are multiples of 6.

7. a) 11, 22, 33, 44, 55, 66, 77, 88, 99b) I coloured the multiples of 11. I say these numbers when I

start at 11 and count on by 11.8. I can start at 3 and count on by 3s. If I say 36, then it is a

multiple of 3.

REFLECT: I can find multiples of a number by starting at anumber and counting on by the same number. For example, tofind the multiples of 7, I start at 7 and count on by 7s. I canuse a hundred chart or a number line to help me.


ASSESSMENT FOR LEARNING

What to Look For

Applying procedures✔ Students can skip count by 2s, 3s, 4s,

5s, 6s, 7s, 8s, 9s, and 10s to 100.

✔ Students can list the multiples ofa number.

Understanding concepts✔ Students can use patterns in the ones

digits to tell if one number is amultiple of another number.

What to Do

Extra Support: Have students colour the multiples of a numberby skip counting on a hundred chart.Students can use Step-by-Step 1 (Master 4.12) to complete question 5.

Extra Practice: In small groups, students play the game, Buzz.Players take turns. One player chooses a target number from 1 to10. She counts by 1s to 100, saying “Buzz” in place of the targetnumber and all multiples of the target number. For example, if thetarget number is 3, the player would say: 1, 2, Buzz, 4, 5, Buzz, ...Students can complete Extra Practice 1 (Master 4.25).

Extension: Students extend the patterns in the multiplication chartto find multiples of numbers greater than 10.

Recording and ReportingMaster 4.2 Ongoing Observations:Multiplication and Division

TrueFalse

TrueFalse


L E S S O N 2

Multiplying byNumbers to 9

Key Math Learnings1. Patterns can be used to remember basic multiplication facts.2. Changing the order of the factors in a multiplication question

does not change the product.

LESSON ORGANIZER

Curriculum Focus: Use patterns to multiply. (N2, N15) (PR1)Teacher Materials� multiplication chart transparency (PM 15)Student Materials Optional� calculators (for question 11) � Step-by-Step 2 (Master 4.13)

� Extra Practice 1 (Master 4.25)Vocabulary: multiplication fact, factor, productAssessment: Master 4.2 Ongoing Observations:Multiplication and Division

40–50 min


Use the multiplication chart transparency. Ask:• On a multiplication chart, where are the

numbers that are multiplied? (The numbers that are multiplied are in the firstrow and the first column.)

• On a multiplication chart, where do you findthe answers? (All the numbers that are not in the first row or thefirst column are the answers.)

• How do you find the answer to amultiplication question, such as 5 � 6? (I find the number 5 in the first column and thenumber 6 in the first row. I find the number that isin the same row as 5 and in the same column as 6.This number, 30, is the answer.)

Present Explore. Encourage students to multiplyby different numbers.

DURING Exp lore


Ask questions, such as:• How is the multiplication chart in Explore the

same as the chart in the introduction? Howis it different? (The charts are the same in thatthe numbers being multiplied are in the first rowsand the first columns and the answers are in therest of the charts. The charts are different in that thechart in Explore shows answers for questions from0 � 0 to 9 � 9, while the other chart showsanswers for questions from 1 � 1 to 7 � 7.)

• What are some multiplication facts you canwrite using this chart? (Any multiplication fact with factors from 0 to 9)


• Which multiplication facts can you write forthe answer 12? How did you find them?(2 � 6 = 12; 3 � 4 = 12; 4 � 3 = 12; 6 � 2 = 12;I looked for the number 12 in the multiplicationchart. I found it in 4 different places. I found thenumbers that were in the same row and column aseach answer.)

AFTER Connec t

Invite students to share some of themultiplication facts they wrote. Record factstogether whenever possible, such as 6 � 3 = 18and 3 � 6 = 18. Then ask:• What is a quick way to multiply by 0? How

do you know? (When I multiply by 0, the answeris always 0. For example, if I have 0 groups ofapples, I don’t have any apples. Also, if I have 5groups of nothing, I have nothing.)

• What is a quick way to multiply by 1? Howdo you know? (When I multiply by 1, the answeris always the other number that was multiplied; forexample, 1 � 6 = 6 and 9 � 1 = 9.)

Review the meanings of the terms factor andproduct. Discuss the strategies forremembering multiplication facts that arepresented in Connect.

Each square (that is, the product of a numberand itself) lies along the leading diagonal ofthe multiplication chart. For all the facts thatare not squares, learning one fact gives you twofacts. You might present this to students as“Learn one fact, and get one free.”

The strategy for multiplying by 9 in Connectmay cause confusion. Two separate rules areinvolved. Students may confuse the two rulesor try to apply the idea to other facts. Makesure students realize this strategy only worksfor multiplying by 9.

Early FinishersStudents can explore the patterns within a multiplication chart.Students choose any 2 by 2 square of products. They find thesum of each diagonal and compare the sums. Students repeatthis for other squares, and then look for patterns in the sums.

Common Misconceptions➤Students often confuse some of the multiplication facts with 8s

and 9s; for example, 7 � 8 = 56 and 6 � 9 = 54

How to Help: Have students use the patterns in themultiplication facts with 9. Alternatively, have students use factsthat are easier to remember. For example, 6 � 10 is 60, so 6 � 9 is 6 less than 60, or 54.


Sample Answers2. Some strategies are: use patterns to multiply by 2s; use the

multiplication chart.3. Some strategies are: use patterns to multiply by 5s; use the

multiplication chart.4. For parts a to d, I used the patterns for multiplying by 9. The

tens digit in the answer is 1 less than the number multiplied by9. The digits in the product add to 9. For part e, I doubled 8to get 16.

6. Students may explain how they found the point where a rowand a column on a multiplication chart meet.

Another strategy for multiplying by 9 is tomultiply the number by 10, then subtractthe number.

For example: 6 � 9 is 6 � 10 – 6 = 60 – 6 = 547 � 9 is 7 � 10 – 7 = 70 – 7 = 63

Students must have mastery of basicmultiplication facts. “Mastery” means a studentcan give the correct response in about 3 seconds. Students in grade 4 should beworking on mastery. Drill of multiplicationfacts is appropriate for students who have astrategy they understand.

Have a large multiplication chart on thebulletin board or multiplication tables thatstudents can refer to.

Prac t i ce

Question 11 requires a calculator.


Some students will multiply 2 by 7 to find thenumber of hours Emi walks her dog in oneweek. They will then either multiply by 5 oradd the result 5 times. If they choose tomultiply, they will be finding the product of a2-digit number and a 1-digit number. You maywish to allow students to use a calculator, oryou can encourage students to think addition.Other students will multiply 5 by 7 to find thenumber of days in 5 weeks. They will thendouble the result to find the answer.


7 0 2 0

2 4 6 810 12 14 16

525

1030

1535

2040

63 72 81 54 16

3 � 8

40 48 54 56 18

32 45 56 18 8

9. a) 2 � 6 = 12; 6 � 2 = 12; 3 � 4 = 12; 4 � 3 = 12b) 2 � 8 = 16; 8 � 2 = 16; 4 � 4 = 16c) 2 � 9 = 18; 9 � 2 = 18; 3 � 6 = 18; 6 � 3 = 18d) 3 � 8 = 24; 8 � 3 = 24; 4 � 6 = 24; 6 � 4 = 24e) 4 � 9 = 36; 9 � 4 = 36; 6 � 6 = 36

10. There are 7 days in one week. Emi walks her dog7 � 2 = 14 hours in 1 week. In 5 weeks, she walks her dog5 � 14 or 14 + 14 + 14 + 14 + 14 = 70 hours.

11. The ones digits start at 8 and decrease by 1 each time. Thetens digits start at 0 and increase by 1 each time. All thehundreds digits are 1. The sum of the digits in eachproduct is 9.

REFLECT: I remember 7 � 6 = 42 this way: I know 5 � 6 is 30and 2 � 6 is 12. I add 30 + 12 to get 42.



What to Look For

Understanding concepts ✔ Students can write multiplication facts,

and identify factors and theirproducts.

✔ Students can recall multiplication factsup to 81.

Communicating✔ Students can describe patterns in

multiplication clearly and precisely,using appropriate language.

What to Do

Extra Support: Allow students to use a completed multiplicationchart to find products.Students can use Step-by-Step 2 (Master 4.13) to completequestion 10.

Extra Practice: Students can do the Additional Activity,Multiplication Bingo (Master 4.8).Students can complete Extra Practice 1 (Master 4.25).

Extension: Students play the Target game. One player writes amultiplication question with an open frame, such as 4 � . Hechooses a target number, such as 27. The other player chooses anumber for the open frame so the product is as close as possible tothe target number. The player’s score is the difference between theproduct and the target number. The player with the lower score wins.


Numbers Every DaySome strategies that students may use are counting on andmaking ten.

56 days 63 days

70 hours

108126144162180

117135153171

21273551


L E S S O N 3

Other Strategies forMultiplying

Key Math LearningMany strategies and patterns can be used to master the basicmultiplication facts.

LESSON ORGANIZER

Curriculum Focus: Use strategies to multiply bydifferent numbers. (N15)Student Materials Optional� counters � Step-by-Step 3 (Master 4.14)� number cubes, � Extra Practice 2 (Master 4.26)

labelled 4, 5, 6, 7, 8, and 9� Cross-Out Product game boards (Master 4.7)Vocabulary: arrayAssessment: Master 4.2 Ongoing Observations:Multiplication and Division

40–50 min


Review some basic facts with students. Ask:• If you know 7 � 6, what else do you know?

(I know 6 � 7.)• How can you find the product of 9 � 7?

(I use the patterns for multiplying with 9s. I knowthat the tens digit of the product will be 6 and thesum of the digits in the product will be 9. Theproduct of 9 � 7 is 63.)

Present Explore. Have students look at theCross-Out Product game board. Ask:

• Suppose 18 was one of the products on thegame board. What different multiplicationfacts could you write that have a product of18? (3 � 6 = 18; 6 � 3 = 18; 2 � 9 = 18;9 � 2 = 18)

• Suppose 25 was a number on the gameboard. What number would you have to rollto cross out 25? (5)

Look for students who use strategies to win.One strategy is to first cross out the productswhich have 1 factor, such as 7 � 7 = 49, thenproducts with 1 pair of factors, such as5 � 7 = 35, and so on. Remind students torecord the facts they used when crossing outproducts on the game board.

DURING Exp lore


Ask questions, such as:• Suppose you had 42 and 49 left on your

game board and you rolled a 7. Whatmultiplication fact would you write? Why? (I would write 7 � 7 = 49 so I could cross off 49. I canuse a 6 or a 7 to write a multiplication fact for 42.)

What strategies did you use to write themultiplication facts? (Sometimes I used amultiplication chart to help me write the multiplicationfacts. Other times, I used patterns or skip counting.)


Watch and listen for students who are able to:• recall basic facts• construct more difficult basic facts from

known facts• use strategies to help them win

AFTER Connec t

Invite students to share the strategies they usedto decide which multiplication facts to write.Ask students who used novel strategies toexplain them to the class. Ask:

• Which multiplication facts do you find easyto remember? (I find the facts with 1 to 6 easy to remember.)

• Which facts do you find harder to remember?(I find the facts with 8 and 9, and some of the factswith 7 harder to remember.)

• What strategies do you use to help youremember the harder facts? (I use differentstrategies to remember the harder facts. Sometimes Iuse a multiplication chart and sometimes I skip count.For example, to find 7 � 8, I skip count by 7s until Ihave said 8 numbers: 7, 14, 21, 28, 35, 42, 49, 56.Sometimes I use patterns. For example, to find 9 � 9,I know the tens digit of the answer is 8 and the sumof the digits is 9. The product of 9 � 9 is 81.)

Discuss the strategies to multiply listed inConnect. Elicit from students that doubling canalso be used to multiply by 6 and by 8. Onpages 128 and 129, the first number in anyarray represents the number of rows, and thesecond number represents the number ofcolumns, or how many objects are in each row.However, the number of rows and columnscould be transposed to draw the array in adifferent way.

Alternative ExploreMaterials: 1-cm grid paper (PM 20)On grid paper, students draw rectangles of different lengths andwidths. They use different strategies to find the areas of therectangles, and then discuss the strategies they used.

Early FinishersStudents use the Cross-Out Product game board. They list all themultiplication facts for each number on the game board.

Common Misconceptions➤Students write the number of rows in an array, or the number

in each row, for both factors in a multiplication question.How to Help: Have students write the multiplication question inwords before using symbols. For example, 5 rows of 8 make 40,so 5 � 8 = 40.

➤Students do not understand the commutative rule; for example,9 � 8 = 8 � 9.

How to Help: Have students make an array for each product,then count to see they have the same number of counters.

ESL StrategiesModel the activity, incorporating verbal instructions and gestures.


Sample Answers2. a) I used patterns for multiplying with 9s to find 9 � 8 = 72.

b) To find 7 � 5, I found 5 � 5 = 25 and 2 � 5 = 10, thenadded to get 25 + 10 = 35.

c) To find 8 � 4, I used doubling. 8 � 2 = 16; double 16 is 32.d) To find 4 � 9, I used doubling. 2 � 9 = 18; double 18 is 36.

4. I could use 8 � 5 and 8 � 1, or I could double 8 � 3.

It is important that students understand thecommutative property; for example, 8 � 7 = 7 � 8. They can then rewrite any facta different way if it makes it easier to multiply.It also halves the number of facts they needto remember.

There are many facts that can be learned byrelating them to facts already known. Forexample: 6 � 7 is related to 5 � 7; think: 5 sevens then add 7 more. Alternatively, 6 � 7 is double 3 � 7.

A fact with 4 as a factor can be considered asdouble, then double again. For example, 4 � 8is double 8 to get 16, then double 16 to get 32.

A fact with 3 as a factor can be considered asdouble, then add. For example, 3 � 6 is double6 to get 12, then add 6 to get 18.

Prac t i ce

Have counters available for all questions.


Students should understand that 9 � 7 can bethought of as the combination of 9 � 5 and9 � 2. Students should either draw arrays toillustrate their explanation, or explain thatsince 5 + 2 = 7, the sum of 9 � 5 and 9 � 2 isequal to 9 � 7.


2124

2124

7232

3536

27

2434

Numbers Every DaySince the differences are small, counting on is the best strategy.Students recognize that in each statement, the numbersare close.

6. 9 � 5 = 45; 9 � 2 = 18; 45 + 18 = 637. There could be 1 stool and 7 chairs, 5 stools and 4 chairs, or

9 stools and 1 chair.8. There are many possible problems. I see 7 cars in a parking

lot. Each car has 4 tires. How many tires can I see in theparking lot? 7 � 4 = 28 ; I see 28 tires.

REFLECT: I know 7 � 6 and 6 � 7 are equal. I also know5 is less than 7. So, 6 groups of 5 (6 � 5) is less than 6 groups of 7 (6 � 7).



What to Look For

Understanding concepts ✔ Students can recall multiplication facts

to 81.

Applying procedures✔ Students can use different strategies to

recall more difficult facts.

What to Do

Extra Support: Have students model multiplication facts witharrays, and then locate the facts on a multiplication chart.Students can use Step-by-Step 3 (Master 4.14) to completequestion 6.

Extra Practice: Have students write all the multiplication factsthat have 8 as a factor, and then write about the patterns they see.Students can complete Extra Practice 2 (Master 4.26).

Extension: Students make up a story problem like the one inquestion 7, and solve their problem.


Making ConnectionsMath Link: We use the same process to find the number ofobjects in an array as we do to find the area of a rectangle. InLesson 6, students will draw arrays on grid paper to multiply a2-digit number by a 1-digit number.

70 cents

1 stool and 7 chairs; 5 stools and 4 chairs; 9 stools and 1 chair


L E S S O N 4

Exploring MultiplicationPatterns

Key Math LearningPatterns can be used to mentally multiply with multiples of10, 100, and 1000.

LESSON ORGANIZER

Curriculum Focus: Use a calculator to find patterns relatingmultiples of 10, 100, and 1000. (N13, N16, N18) (PR1)Student Materials Optional� calculators � Step-by-Step 4 (Master 4.15)� Base Ten Blocks � Extra Practice 2 (Master 4.26)Vocabulary: place valueAssessment: Master 4.2 Ongoing Observations: Multiplicationand Division

40–50 min


Have students multiply each number from 1 to7 by 10. Record the questions and theirproducts on the board.(1 � 10 = 10; 2 � 10 = 20; 3 � 10 = 30; 4 � 10 = 40;5 � 10 = 50; 6 � 10 = 60; 7 � 10 = 70)

Ask:• What patterns do you see when multiplying

by 10? (The tens digit of the product is the same asthe number that is being multiplied by 10. All theproducts have a ones digit of 0.)

• How can you use the pattern to find theproduct of 8 and 10? 9 and 10? (80; 90)

• How can you use the pattern for multiplyingby 10 to predict the product when youmultiply 5 by 100?(When I multiplied by 10, the tens digit of theproduct was the same as the number multiplied by10, and the ones digit was 0.

When I multiply 5 by 100, I think the hundreds digitwill be 5 and the tens and ones digits will both be zero.I predict the product of 5 � 100 will be 500.)

Present Explore. Ensure students understandthat they will only be using calculators to lookfor patterns. Once they have identified thepatterns, they multiply with multiples of 10,100, and 1000 mentally.

DURING Exp lore


Ask questions, such as:• What are the products in the first set of

numbers?

4404004000

9909009000

25250250025 000

6606006000


Alternative ExploreMaterials: Base Ten BlocksStudents work in pairs. They use Base Ten Blocks rather than acalculator to explore the patterns in multiplying with multiples of10, 100, 1000.

Early FinishersHave students write 3 different pairs of factors whoseproduct is 2400.

� = 2400

Common Misconceptions➤Students do not write the correct number of zeros

in the product.How to Help: Have students circle the digits in the basicmultiplication fact in each question. For example, 5 � 200 = 1000.


• What patterns do you see in the products?(When I multiply by 10, 100, or 1000, the first digitof the product is the same as the number that isbeing multiplied by 10, 100, or 1000. This digit is inthe same place as the 1 in 10, 100, or 1000. When Imultiply by multiples of 10, 100, or 1000, I use therelated basic fact, followed by 1, 2, or 3 zeros. Forexample, in 5 � 5000, the related basic fact is5 � 5 = 25. 5000 has 3 zeros, so 5 � 5000 is25 000. In words, 5 times 5 thousand is25 thousand, or 25 000.)

• What patterns did you use to find theproducts in the second set of numbers? (To find the products in the second set of numbers,I used the related basic fact, and added zeros tothe end.)

• What are the products in the second set ofnumbers?

Some students may benefit from using BaseTen Blocks to model each product to confirmthe calculated results.

AFTER Connec t

Have students share the patterns they foundwhen they multiplied by multiples of 10, 100,and 1000. Many students will state that theproduct will have the same number of zeros asthe multiple of 10, 100, or 1000. This is notalways true. Be sure students understand theplace-value concepts. Ask:

7707007000

8808008000

8808008000

18180180018 000

• How many zeros are in the product of8 � 500? How do you know? (There are 3 zeros in the product of 8 � 500. Theproduct of 8 � 5 is 40. The product of 8 � 500 willhave 2 more zeros, so 8 � 500 is 4000.)

Discuss the examples in Connect. Whenmultiplying by multiples of 10, 100, or 1000, asin the second example in Connect, some studentsmay prefer to think of the multiple of 10, 100,or 1000 as the product of a single-digit numberand 10, 100, or 1000. For example, studentsmay think of 40 as 4 � 10. So, 2 � 40 becomes2 � 4 � 10. They can multiply 2 � 4 to get 8,and then multiply 8 by 10 to get 80. Givestudents an opportunity to model the productswith Base Ten Blocks to reinforce the place-value concepts.

Prac t i ce

Have Base Ten Blocks available forall questions.Have students use a calculator to check theiranswers.


Students should provide at least one examplefor each situation. They should mention placevalue in their explanations. Students may useBase Ten Blocks to explain, or draw pictures ofthe blocks.


30704080

5090100

660600

212102100

242402400

300 + 80 + 4900 + 80 + 76000 + 700 + 90 + 38000 + 300 + 80 + 4

400100

600700

9000

480 cards

150720

280120

270240

1200 cents

Numbers Every DayStudents should use place-value concepts when they writenumbers in expanded form. Some students may benefit fromwriting the numbers in a place-value chart before they writethem in expanded form. If time permits, have students also saythe numbers aloud and/or write them in words.

Remind students that when they write or say whole numbers,they do not use “and” before the last digit. For example, 8384 iseight thousand three hundred eighty-four. The word “and” isused for a decimal, to indicate where the decimal point is. Forexample, 3.8 is three and eight-tenths.

Sample Answers9. a) 6 � 30 = 180 ; 6 � 40 = 240; 6 � 50 = 300

b) 6 � 300 = 1800; 6 � 400 = 2400; 6 � 500 = 3000c) 6 � 3000 = 18 000; 6 � 4000 = 24 000;

6 � 5000 = 30 000To multiply by a multiple of 10, find the basic fact, then write1 zero on the end. For multiples of 100, find the basic fact,then write 2 zeros on the end. For multiples of 1000, find thebasic fact, then write 3 zeros on the end.Explanations may vary. I know my strategies work becausethe products follow a pattern. Also, I can check my answersusing a calculator.

REFLECT: When you multiply by 10, by 100, or by 1000, youfind the related basic fact, and then put the appropriatenumber of zeros on the end. If you multiply by 10, you put 1zero. If you multiply by 100, you put 2 zeros. If you multiplyby 1000, you put 3 zeros.


6006000

42006000

10004000

6000 mL

450 cents

400 red800 yellow, 800

white

2000 blocks


What to Look For

Applying procedures✔ Students can use basic facts and

place value to mentally multiply 1-digit numbers by multiples of10, 100, and 1000.

Communicating✔ Students can use appropriate

language to clearly describe thepatterns they find when multiplying bymultiples of 10, 100, and 1000.

What to Do

Extra Support: Have students model each question with BaseTen Blocks.Students can use Step-by-Step 4 (Master 4.15) to completequestion 9.

Extra Practice: Students work in pairs. They take turns to makeup a multiplication question where one of the factors is a multipleof 10, 100, or 1000. Their partner then finds the product.Students can complete Extra Practice 2 (Master 4.26).

Extension: Students write multiplication questions with multiplesof 10, 100, and 1000 in which the number of zeros in theproduct is not the same as the number of zeros in the multiple of10, 100, or 1000.



L E S S O N 5

Estimating Products

Key Math Learnings1. Rounding can be used to estimate a product.2. In some number calculations, it is better to use an estimate.

In others, an exact answer is required.

LESSON ORGANIZER

Curriculum Focus: Use rounding to estimate a product. (N16, N18)Student Materials Optional

� Step-by-Step 5 (Master 4.16)� Extra Practice 3 (Master 4.27)

Vocabulary: roundAssessment: Master 4.2 Ongoing Observations: Multiplication and Division

40–50 min


Ask students for examples of situations inwhich you might ask for an exact product. Talkabout other situations in which an estimatemight be used.

If you need to know if $15 is enough to buy 3books, you may estimate the total cost.

Present Explore. Ensure students understandthey are to estimate, not calculate.

DURING Exp lore


Ask questions, such as:• How did you round 22?

(I rounded 22 to the nearest ten to get 20.)• How did you estimate the number of

passengers? (I multiplied 8 by 20.)• About how many passengers can travel on 8

Bombardier Challengers? (About 160 people can travel on 8 of these planes.)

AFTER Connec t

Have students share their estimates and thestrategies they used. If students got an estimateother than 160, ask them to share their strategywith the class.

Review the strategies in Connect, then ask:• How do you know when to use an estimate?

(When the question asks “about how many?”)

Prac t i ce


Students may make different assumptionsabout the number of days Kyle’s mother worksin one week. Students should know that theymay get different answers depending on theassumptions they make.

Sample Answers4. 77 cm is close to 80 cm; 5 � 80 cm = 400 cm6. My mom works 5 days each week. In 2 weeks, she works 10

days. 47 km is close to 50 km. One way is about 50 km. So,to work and back is 2 � 50 km = 100 km. 100 � 10 = 1000, so Kyle’s mother drives about 1000 km in2 weeks.

7. There are many possible questions. 60 � 6 = 360, so I chose62 � 6. I could have chosen any number from 56 to 64, andthen multiplied it by 6.

REFLECT: When I round to estimate, I usually use the rules forrounding. If I am rounding to the nearest ten, I find out whichten my number is closer to. When I round 82, I round down to80. When I round 87, I round up to 90.



What to Look For

Applying procedures✔ Students can use rounding to

estimate products.

Understanding concepts✔ Students round appropriately when

estimating products.

What to Do

Extra Support: When estimating, have students use a numberline to help them round numbers. Students can do the AdditionalActivity, Spinning Products (Master 4.19).Students can use Step-by-Step 5 (Master 4.16) to completequestion 6.

Extra Practice: Students can complete Extra Practice 3(Master 4.27).

Extension: Students make up a problem like Practice question 7.They trade problems with a classmate and solve their classmate’sproblem.


Early FinishersStudents predict whether the actual products in Practice questions1 and 3 are greater than or less than the estimated products,and explain their predictions. They can use a calculator to checktheir predictions.


60 120 200 490

490 cents

240 480 60360

7 � 66 is greater

500 km

808010060


L E S S O N 6

Strategies forMultiplication

Key Math LearningMany strategies can be used to multiply a 2-digit number by a1-digit number.

LESSON ORGANIZER

Curriculum Focus: Use different strategies to multiply a 2-digit number by a 1-digit number. (N13, N16)Student Materials Optional� Base Ten Blocks � Step-by-Step 6 (Master 4.17)� calculators � Extra Practice 3 (Master 4.27)Vocabulary: multiplication sentenceAssessment: Master 4.2 Ongoing Observations:Multiplication and Division

40–50 min


Have students draw an array for 4 � 6. Ask:• How many rows will there be? How many

counters will be in each row? (There will be 4 rows with 6 counters in each row.)

Have students model the number 24 with BaseTen Blocks. Ask:• What blocks do you need to model 24?

(2 ten rods and 4 unit cubes)

Present Explore. Encourage students to use BaseTen Blocks to solve the problem.

DURING Exp lore


Ask questions, such as:• How could you model the number 12 with

Base Ten Blocks? (I could use 1 ten rod and 2 unit cubes.)

• How many cartons are there? (6) • How could you show the number of eggs in

6 cartons with Base Ten Blocks? (I could show the number of eggs with 6 groups ofblocks. Each group would have 1 ten rod and2 unit cubes.)

• How many eggs are in 6 cartons? How didyou find out? (There are 72 eggs in 6 cartons. Iused Base Ten Blocks. There are 6 ten rods and 12unit cubes. I traded 10 unit cubes for 1 ten rod to get7 ten rods and 2 unit cubes; that is, 72.)

• How can you find how many eggs are in 6cartons without using blocks? (I can multiply 12 � 6.)

Numbers Every DayTo make the greatest difference, students subtract the leastnumber from the greatest number. Students should use place-value concepts to make the greatest 3-digit number and to makethe least 1-digit number, using the given digits.


AFTER Connec t

Invite students to share the strategies they usedto find how many eggs are in 6 cartons. Somestudents may have modelled the problem withBase Ten Blocks. Others may have added12 + 12 + 12 + 12 + 12 + 12. Ask students whoused a different strategy to present it tothe class.

Review the strategies in Connect, then ask:• How is using Base Ten Blocks to multiply a

2-digit number by a 1-digit number similarto breaking a number apart to multiply? (In both cases, I multiply the tens, then I multiplythe ones, and then I add.)

You could model the array in Connect as arectangle with length 36 cm and width 4 cm.Ask how many 1-cm squares would fit in therectangle. Students will learn about areameasured in square centimetres in Unit 9.

When students break a number apart tomultiply, they may find it helpful to write theproduct on 1-cm grid paper. This helps to keepthe digits aligned correctly.

Prac t i ce

Have Base Ten Blocks available for allquestions. If you have grid paper with 0.5 cmsquares, students could use it to draw arrays.For questions 2 and 3, have students estimatebefore multiplying and use their estimates tocheck whether their answers are reasonable.For questions 4 to 8, have students use acalculator to check their answers.


Students should be able to multiply 24 � 4 andcompare the product with 90. Students’strategies may vary. They could use a differentstrategy to multiply again, to check.

Alternative ExploreMaterials: 1-cm grid paper (PM 20)Students draw several rectangles. The width of each rectanglemust be 9 units or less. The length of each rectangle must begreater than 10 units. Students find the area of each rectangle.

Early FinishersHave students solve this problem: Sometimes eggs are packaged18 to a carton. How many eggs are in 6 cartons of 18 eggs?


976� 4972

3 � 15 = 45

4 � 23 = 92

69 72 248 432 141

305 186 180 150 72

Sample Answers5. 7 � 23 is one group of 23 greater than 6 � 23.7. a) Tom needs 90 candles. Tom has 4 � 24 = 96 candles, so

he has enough.b) 96 � 90 = 6; Tom will have 6 candles left over.

8. Many problems are possible. Including jokers, a deck of cardshas 54 cards. How many cards are there in 5 decks of cards?54 � 5 = 270

9. Chris multiplied the ones: 6 � 2 = 12. He then multiplied thetens: 6 � 60 = 360. Finally, he added: 12 + 360 = 372.

REFLECT: There are 2 ways I can use addition to help memultiply. First, I can remember that multiplication is a quickway to add. I can find 4 � 23 by adding: 23 + 23 + 23 + 23 = 92. Second, I can use known facts andaddition to help me multiply. For example, to multiply 4 � 23,I can multiply 4 � 20 = 80 and 4 � 3 = 12, and then add to get 80 + 12 = 92.



What to Look For

Applying procedures✔ Students can multiply a 2-digit

number by a 1-digit number.

Understanding concepts✔ Students can use more than one

strategy to multiply a 2-digit numberby a 1-digit number.

✔ Students can use estimation to verifysolutions.

What to Do

Extra Support: Students model each multiplication problemwith Base Ten Blocks.Students can use Step-by-Step 6 (Master 4.17) to completequestion 7.

Extra Practice: Have students use Eva’s strategy from Practicequestion 6 to multiply 4 � 21, 7 � 39, and 6 � 79. (84, 273, 474)Students can complete Extra Practice 3 (Master 4.27).

Extension: Ask students to complete Practice questions 2 and 3using mental math.


105

23

Yes

Yes

Yes

At Home: When students bring in their work, give them timeto explain the strategy to the class, then post it on a bulletinboard for students to use if they wish.


Strategies Toolkit

Key Math LearningSolving simpler problems, then looking for patterns is anefficient way to solve many problems.

LESSON ORGANIZER

Curriculum Focus: Interpret a problem and select anappropriate strategy. (N12) (PR1)Teacher Materials� pennies or round counters (optional)� Colour Tiles transparency (optional)Student Materials Optional

� pennies or counters� Colour Tiles

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

40–50 min

L E S S O N 7


Present Explore. Ensure students understandthey are finding the total number of penniesneeded to make an 8-row penny triangle. Havepennies or counters available for students whowish to model the problem.

DURING Exp lore


Ask questions, such as:• How many pennies will be in the 4th row?

How do you know? (There will be 4 pennies in the 4th row. Each rowhas one more penny than the row before.)

• How many pennies are in a penny trianglewith 2 rows? 3 rows? (There are 3 pennies in a triangle with 2 rows and6 pennies in a triangle with 3 rows.)

AFTER Connec t

Work through the problem in Connect. Ask:• How many hands does the 4th person have

to shake to meet everyone else? (3)• What pattern do you see in the number of

handshakes? (The number of handshakes starts at1 and goes up by 2, then 3, then 4, then 5, and so on.)

• How many hands does the 7th person shake?How many handshakes are there altogether?(6, 21)

• Why is the number of handshakes not 7 � 6?(The product 7 � 6 counts Student A shaking handswith Student B, and vice versa. The same handshakeis counted twice. So, we divide by 2; 7 � 6 � 2 = 21.)

Prac t i ce

Have Colour Tiles available for question 1.

Sample AnswersExplore: The number of pennies is

8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36



What to Look For

Problem solving✔ Students can select an appropriate

strategy and use it to solve a problem.

Communicating✔ Students can describe their strategy

clearly, using appropriate language.

What to Do

Extra Support: For Practice question 2, have students usecutouts of 1 by 1, 2 by 2, and 3 by 3 squares to help themcount the total number of squares.Extra Practice: Have students solve this problem: There are 8teams at a basketball tournament. Each team will play every otherteam once. How many games will be played altogether? (28)

Extension: Have students count the total number of squaresthey can find on a checkerboard or a chessboard. (There are 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squareson a checkerboard.)

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

Early FinishersAsk students to find how many pennies are needed to make a12-row triangle. Challenge students to solve the problem withoutbuilding all the rows.

Common Misconceptions➤For questions like the penny triangle, students give the number

in the last row, rather than the total.How to Help: Have students record their work in a table withthese headings: “Row,” “Number in Row,” “Total Number.”


Figure 1 2 3 4 5Tiles in Figure 1 3 5 7 9Total Tiles 1 4 9 16 25

2. There are nine 1 by 1 squares, four 2 by 2 squares, and one3 by 3 square. There are 14 squares altogether.

REFLECT: For Practice question 1, I counted how many tiles werein the first figure and how many tiles were in the secondfigure. I then added to get the total number of tiles in the first2 figures. I did the same for the 3rd, 4th, and 5th figures. Irecorded my answers in a table:

I looked for patterns in the numbers, and then extended thepatterns to find the number of tiles in the 7th figure (13) andthe number of tiles in the first 7 figures (49).

1349

14 <<<<<<<<


Dividing by Numbersfrom 1 to 7

Key Math Learnings1. An array can show both multiplication and division facts.2. Multiplication facts can be used to help remember division

facts, and vice versa.

LESSON ORGANIZER

Curriculum Focus: Relate division to multiplication. (N14, N15, N17)Teacher Materials� overhead counters (optional)Student Materials Optional� counters � Step-by-Step 8 (Master 4.18)

� Extra Practice 4 (Master 4.28)Assessment: Master 4.2 Ongoing Observations:Multiplication and Division

40–50 min

L E S S O N 8


Scatter 20 counters on the overhead projector.Ask:• How could you find how many groups of 5

counters you could make? (I could arrange the counters into groups of 5, andthen count the number of groups.)

Arrange the counters into a 4 by 5 array. Ask:• How could you find how many groups of 5

counters you could make? (There are 20 counters arranged in rows of 5. Icould count the number of rows to find how manygroups of 5 I could make.)

• Which way was easier? Why? (It was easier to find how many groups of 5 I couldmake when the counters were arranged in an array.All I had to do was count the number of rows.)

Present Explore. Have counters or Base TenBlocks available for students who wish touse them.

DURING Exp lore


Ask questions, such as:• How can you find how many teams of

7 players can be made from a group of 56 students? (I can arrange 56 counters into rows of 7 and count the number of rows. I can alsodivide. I know 7 � 8 = 56, so I know 56 � 7 = 8.)

Listen for students who talk about dividing 56into groups of 7.

AFTER Connec t

Invite students to share their answers andstrategies. If any students said they usedmultiplication, have them explain their strategyto the class.

TEACHING TIPIn discussing the relationship

between division and multiplicationin this lesson, you may wish to

introduce the term inverse operation.

Sample Answers2. a) b)

Discuss the examples in Connect. As studentslook at the array for 30 � 6, ask:• How many counters are there in all?

(There are 30 counters in all.)• How many are in each row?

(There are 6 counters in each row.)• How many rows are there? (There are 5 rows.)• How does the array show division? (The total

number of counters divided by the number in eachrow gives the number of rows. The total number ofcounters divided by the number of rows gives thenumber of counters in each row.)

• How does the array show multiplication?(The number of rows times the number of countersin each row gives the total number of counters.)

• How can you use multiplication to checkyour answers to division problems? (I can multiply my answer by the number I divided by.)

There are many software programs that providedrill of basic facts. Most programs have gamesand/or exercises at various difficulty levels.However, practice with computer softwareshould only be done after students havedeveloped strategies and understand whatdivision is.

Prac t i ce

Have counters available for all questions.Encourage students to use multiplication tocheck their answers for questions 4 and 5.


Students should recognize that to share equallymeans to divide. Some students may use 6 � 7to find the total number of cubes required.Others may use 35 � 7 to find the number ofcubes each student needs.


Alternative ExploreMaterials: Snap CubesHave students find the number of 7-cube towers that can bemade from 56 cubes.

Early FinishersAsk students to find the number of different ways they couldmake equal groups from a class of 24 students.

ESL StrategiesAsk questions using simple, clear language. Give students timeto process the question and think about their response.


3 � 6 = 18; 18 � 6 = 3

7 � 4 = 28; 28 � 4 = 7

4 � 5 = 20; 20 � 5 = 4 2 � 7 = 14; 14 � 7 = 2

6 5 4

75

74

57

61

7. There are many possible problems. There are 42 students whowant to play basketball. How many teams of 6 can be made?

8. If all the students play on both teams, there could be 30students in the class. Any number that is a multiple of 30could also be an answer, but it is very unlikely that therewould be a class of 60 or more students.

9. There are 35 cubes. Joe shares them equally among 7students; 35 � 7 = 5. Each student gets 5 cubes, but eachstudent needs 6 cubes. Joe does not have enough cubes.

REFLECT: If I wanted to divide 42 by 7, I would ask myself,“what number times 7 is 42?” I know that 6 � 7 is 42, so Ialso know that 42 � 7 = 6.



What to Look For

Applying procedures✔ Students can recall division facts to 49.

Understanding concepts✔ Students can describe the relationship

between multiplication and division.

What to Do

Extra Support: Have students model each division problemwith counters.Students can use Step-by-Step 8 (Master 4.18) to completequestion 9.

Extra Practice: Students can complete Extra Practice 4(Master 4.28).

Extension: Have students write a story problem similar to theproblem in Practice question 8.


Numbers Every DayStudents should recognize that they need to put the smallestdigits in the thousands positions, the next-smallest digits in thehundreds positions, and the largest digits in the ones positions.Answers may vary. One solution is shown.

6

57

67

76

64

No

1357+ 2468

3825


L E S S O N 9

Dividing by Numbersfrom 1 to 9

Key Math Learnings1. For most division facts, there are 2 related multiplication

facts and 1 related division fact.2. For most multiplication facts, there are 2 related division

facts and 1 related multiplication fact.

LESSON ORGANIZER

Curriculum Focus: Use related multiplication facts to divide.(N15, N17)Teacher Materials� multiplication chart transparency (PM 15)Student Materials Optional� counters � Step-by-Step 9 (Master 4.19)� 1-cm grid paper (PM 20) � Extra Practice 4 (Master 4.28)� multiplication charts (PM 15)Vocabulary: related factsAssessment: Master 4.2 Ongoing Observations: Multiplication and Division

40–50 min


Have the class count by 2s from 2 to 20. Recordthe numbers on the board. Remind studentsthat these numbers are multiples of 2.

Use the multiplication chart transparency. Havestudents find the multiplication facts that have2 as a factor. Ask:• How can we use the multiplication facts that

have 2 as a factor to write the division factsfor dividing by 2? (The product divided by 2gives the other factor. For example, for 5 � 2 = 10,we could write the division fact 10 � 2 = 5.)

• How would you draw an array to show4 � 2? 8 � 2? (For both 4 � 2 and 8 � 2, Iwould draw an array with 4 rows of 2.)

Present Explore. Have counters available forstudents who wish to use them.

DURING Exp lore


Ask questions, such as:• What are the division facts for dividing by 8?

(72 � 8 = 9; 64 � 8 = 8; 56 � 8 = 7; 48 � 8 = 6;40 � 8 = 5; 32 � 8 = 4; 24 � 8 = 3; 16 � 8 = 2;8 � 8 = 1)

• How would you draw an array to show56 � 8 = 7? (I would draw an array with 8 rows of 7.)

• What are the division facts for dividing by 9?(81 � 9 = 9; 72 � 9 = 8; 63 � 9 = 7; 54 � 9 = 6;45 � 9 = 5; 36 � 9 = 4; 27 � 9 = 3; 18 � 9 = 2;9 � 9 = 1)

• How would you draw an array to show36 � 9 = 4? (I would draw an array with 9 rows of 4.)

Sample Answers2. a) 3 � 7 = 21; 21 � 7 = 3; 21 � 3 = 7

b) 6 � 8 = 48; 48 � 8 = 6; 48 � 6 = 8c) 9 � 5 = 45; 45 � 5 = 9; 45 � 9 = 5d) 7 � 9 = 63; 63 � 9 = 7; 63 � 7 = 9

3. a) 9 � 6 = 54; 6 � 9 = 54; 54 � 9 = 6; 54 � 6 = 9b) 5 � 8 = 40; 8 � 5 = 40; 40 � 8 = 5; 40 � 5 = 8c) 4 � 7 = 28; 7 � 4 = 28; 28 � 7 = 4; 28 � 4 = 7d) 1 � 7 = 7; 7 � 1 = 7; 7 � 7 = 1; 7 � 1 = 7

4. a) 7 � 9 = 63; 9 � 7 = 63; 63 � 9 = 7; 63 � 7 = 9b) There are many possible answers. 8 could be a factor or it

could be the product; for example, 2 � 4 = 8; 4 � 2 = 8;8 � 2 = 4; 8 � 4 = 2.


AFTER Connec t

Elicit from students that the 9 � 9 multiplicationchart shows 9 different division facts for eachnumber. Review the examples in Connect. Ensurestudents understand the importance of relatedfacts. Ask:• How do you know the array for 9 � 8 has

the same number of counters as the array for8 � 9? (The array for 9 � 8 has 9 rows of 8. If Iturn this array one-quarter turn, it becomes anarray with 8 rows of 9. This is the array for 8 � 9.)

• How do you know when a multiplicationfact has only 1 related division fact? (A multiplication fact has only one related divisionfact when the factors in the fact are the same.)

• How can you use the relationship betweenmultiplication and division to check youranswers to division problems? (I can multiply the answer by the number I divided by.)

Prac t i ce

Have counters and grid paper available for allquestions. You may wish to have multiplicationcharts available for those students havingdifficulty to reinforce the connection betweenmultiplication and division.

Encourage students to check their answers forquestions 5, 7, 9, and 10 by multiplying.


Students should recognize that if two differentnumbers are divided into the same number ofgroups, the larger number yields the largergroups. They should also recognize that if thesame number is divided into groups ofdifferent sizes, the larger the size of each group,the smaller the number of groups.

Alternative ExploreMaterials: countersStudents use counters to model, then write, all the multiplicationand division facts for 8 and 9.

Early FinishersHave students create a problem, similar to Practice question 12,and then solve their problem.


9 � 6 = 546 � 9 = 5454 � 6 = 954 � 9 = 6

6 � 7 = 427 � 6 = 4242 � 7 = 642 � 6 = 7

21 48 45 63

6. 1 � 1 = 1; 2 � 2 = 4; 3 � 3 = 9; 4 � 4 = 16; 5 � 5 = 25;6 � 6 = 36; 7 � 7 = 49; 8 � 8 = 64; 9 � 9 = 81These products are on a diagonal.1 � 1 = 1; 4 � 2 = 2; 9 � 3 = 3; 16 � 4 = 4; 25 � 5 = 5;36 � 6 = 6; 49 � 7 = 7; 64 � 8 = 8; 81 � 9 = 9

11. There are many possible problems. Pencils come in boxes of8. How many boxes are needed so that every student in aclass of 32 has a pencil? (4)

12. a) 48 is greater than 40. If you have 48 things to shareamong 8 people, each person will get more than if youonly have 40 things to share; 48 � 8 = 6 and40 � 8 = 5.

b) 9 is greater than 8. If I make more groups from the samenumber of things, each group will be smaller. I can makemore groups of 8 out of 72 things than I can make groupsof 9 out of 72 things; 72 � 8 = 9 and 72 � 9 = 8.

REFLECT: For the same array, I can multiply the number of rowsby the number of things in each row to find the total numberof things. I can also divide the total number of things by thenumber of rows to find the number of things in each row.



What to Look For

Applying procedures✔ Students can write the related

multiplication and division facts for aset of numbers.

Communicating✔ Students can clearly describe the

relationship between multiplicationand division using appropriatelanguage.

Understanding concepts✔ Students can verify solutions to

division problems by multiplying.

What to Do

Extra Support: Students may benefit from using a multiplicationchart and counters to multiply and divide.Students can use Step-by-Step 9 (Master 4.19) to completequestion 12.

Extra Practice: Have students choose one row of themultiplication chart and write all the related facts for each numberin that row.Students can complete Extra Practice 4 (Master 4.28).

Extension: Students can do the Additional Activity, Division Tic-Tac-Toe (Master 4.10).


Numbers Every DayThere are 6 different ways students could arrange the digits.They should try all 6 arrangements to find the one that gives theproduct closest to 100.

39

47

79

91

28

85

54

96

37

63 � 7 = 9, 7 � 9 = 63, 9 � 7 = 63

4

6

26� 4104


Division withRemainders

Key Math Learnings1. Division does not always result in a whole number answer.2. Remainders are dealt with in different ways, depending on

the context of the problem.

LESSON ORGANIZER

Curriculum Focus: Use counters to divide with remainders.(N14)Teacher Materials� overhead counters (optional)Student Materials Optional� counters � Step-by-Step 10 (Master 4.20)

� Extra Practice 5 (Master 4.29)Vocabulary: remainder, division sentenceAssessment: Master 4.2 Ongoing Observations:Multiplication and Division

40–50 min

L E S S O N 1 0


Place 11 counters on the overhead projector.Ask:• How many groups of 2 can we make?

(We can make 5 groups of 2. There will be 1 counterleft over.)

Present Explore. Have counters or grid paperavailable for students who wish to use them.

DURING Exp lore


Ask questions, such as:• Does Monica have enough oranges to make 6

baskets? 7 baskets? (If Monica makes 6 baskets, she will need 36oranges. She has enough oranges to make 6 baskets.If she makes 7 baskets, she will need 42 oranges.She does not have enough oranges to make7 baskets.)

• Will there be any oranges left over? (Monicaonly has enough oranges to make 6 baskets.41 � 36 = 5; she will have 5 oranges left over.)

AFTER Connec t

Invite students to share their answers. If anypair of students had an answer other than 6,ask them to explain their answer. Encouragestudents to discuss how they treated theremainder, and explain why.

Review the examples in Connect. Ask:• How many apples does Monica need to put

1 apple in each basket? 2? 3? 4? (4, 8, 12, 16)

Ensure students understand the notationfor remainder.

Sample Answers5. 15 � 2 = 7 R16. The greatest remainder when you divide by 7 is 6. Any

number greater than 6 can be divided into at least one moregroup of 7.a) 2 b) 2 R1 c) 2 R2 d) 2 R3e) 2 R4 f) 2 R5 g) 2 R6 h) 3

Elicit from students the different ways we treatremainders. Have them give an example foreach way.• Sometimes we ignore the remainder. For

example, in Explore, Monica has 41 orangesand therefore, she can make 6 fruit basketswith 6 oranges in each basket. There are notenough oranges for a 7th basket.

• Sometimes we round up. For example, in thesecond example in Connect, all students mustgo on the trip. Only 24 students will fit in 4 vans, so we need one more van for the25th student.

• Sometimes we can write the remainder as afraction. For example, if 2 children share 5cookies, each child will get 2 and one-halfcookies. Since this is a new concept,illustrate this with pictures and havestudents do the same.

Divisions with remainders are more common inreal situations than divisions with noremainders. To find 40 � 7, most people willthink about multiplication facts. Five times 7 is35, that’s too low; 6 times 7 is 42; that’s toohigh but close. So, 40 � 7 is 5 with 5 left over.This process can and should be drilled.Students should be able to solve problemsinvolving 1-digit divisors and 1-digit answers,and remainders, mentally.

Prac t i ce

Have counters available for all questions.


Students should recognize that when dividing,the remainder must be smaller than the numberyou are dividing by.


Alternative ExploreMaterials: Snap CubesStudents find how many 6-cube towers can be made with 41Snap Cubes.

Early FinishersHave students solve this problem: Monica has 41 oranges. Howmany fruit baskets can she make if she puts 4 oranges in eachbasket? 5 oranges? 7 oranges? Is there any number of orangesthat Monica can put in each basket for which there will be noremainder? Explain.


19 � 3 = 6 R1

17 � 5 = 3 R2

8 R1 5 R3 7 3 R2

53

7. There are many possible problems. You put 5 pears in eachfruit basket. How many baskets can you make with 22 pears?(I can make 4 baskets, with 2 pears left over.)

8. Tyler is incorrect because the remainder is greater than thenumber he is dividing by. He can make one more group of 4.Amina is correct.

9. b) Six cartons are needed for 32 bottles, but there are only 2bottles in the 6th carton. There is room for 4 more bottles inthe 6th carton. 35 is 3 more than 32. The extra 3 bottleswill fit in the 6th carton.

REFLECT: Sometimes you ignore the remainder in a divisionproblem. For example, in question 5, Elizabeth can only take2 apples to school each day for 7 days, whether she has 14apples, or 15 apples. Sometimes you add one more to theanswer. For example, in the second problem in Connect, youneed to add one more van so that all the students can go onthe field trip. Sometimes, you can write the remainder as afraction. For example, when 2 people share 3 sandwiches,each person gets 1 and one-half sandwiches.



What to Look For

Applying procedures✔ Students can solve division problems

with remainders, and writedivision sentences.

Understanding concepts✔ Students recognize that the remainder

has to be smaller than the numberthey are dividing by.

What to Do

Extra Support: Have students model each division questionwith counters.Students can use Step-by-Step 10 (Master 4.20) to completequestion 8.

Extra Practice: Have students write a story problem for eachpicture in Practice question 1.Students can complete Extra Practice 5 (Master 4.29).

Extension: Ask students to write a story problem for 26 � 3,in which:• 9 is an appropriate answer.• 8 is an appropriate answer.


Numbers Every DayStudents should recognize that an estimate is sufficient for thefirst, third, and fourth questions, but an exact answer is neededfor the second question.

a, b, c, d, f, g, h

7 days

6

Amina

6

No

10396

14

25


L E S S O N 1 1

Using Base Ten Blocksto Divide

Key Math Learnings1. Base Ten Blocks can be used to model dividing a 2-digit

number by a 1-digit number.2. Remainders are dealt with in different ways, depending on

the context of the problem.

LESSON ORGANIZER

Curriculum Focus: Use Base Ten Blocks to divide a 2-digitnumber by a 1-digit number. (N14, N17)Teacher Materials� overhead Base Ten Blocks (optional)Student Materials Optional� Base Ten Blocks � Step-by-Step 11 (Master 4.21)

� Extra Practice 5 (Master 4.29)Assessment: Master 4.2 Ongoing Observations:Multiplication and Division

40–50 min


Use Base Ten Blocks on the overhead projector.Ask:• How would you show 24 using Base Ten

Blocks? (I would use 2 ten rods and 4 unit cubes.)• How can we divide 24 into 2 equal groups?

(I can divide the blocks into 2 equal groups, with1 ten rod and 2 unit cubes, or 12, in each group.)

Place 1 more unit cube on the overheadprojector.Ask:• Can I divide 25 into 2 equal groups? (No)• How do you know you can’t divide 25 into 2

equal groups? (There are 12 in each group, with1 left over. If I put the leftover cube in one of thegroups, the groups will not be equal.)

Present Explore. Have Base Ten Blocks availablefor students who wish to use them.

DURING Exp lore


Ask questions, such as:• Which Base Ten Blocks could you use to

model 76? (I could use 7 ten rods and 6 unit cubes.)

• How could you use Base Ten Blocks to show76 � 4? (I could make 4 equal groups. I would put1 ten rod and 1 unit cube in each group. I wouldhave 3 ten rods and 2 unit cubes left over. I couldtrade each ten rod for 10 unit cubes to get a total of32 unit cubes. I would then add 8 unit cubes to eachgroup to get 4 groups of 19; 76 � 4 = 19.)

• How can you tell if Felipe can divide 78books equally among 4 boxes? (78 is 2 more than 76. Two books cannot be sharedequally among 4 boxes, so 78 books cannot beshared equally among 4 boxes.)


AFTER Connec t

Ask students to share their answers andstrategies. If any students solved the problemwithout using Base Ten Blocks, have themexplain their strategies to the class.

Remind students of the relationship betweenmultiplication and division. Ask: • How can you use multiplication to check

your answer to the first problem? (I can multiply 19 by 4. The answer is 76, so Iknow that 76 ÷ 4 = 19.)

• How can you check your answer to thesecond problem? (I can multiply 19 by 4. The answer is 76. Then Ican add the remainder, 2, to get 78.)

In all other operations, students usually startwith the ones place. Some students may chooseto do so when dividing. This is a validapproach. However, when students learn the

standard algorithm for division, they willdivide starting with the highest place value.

Review the examples in Connect. Ask:• When we find 57 � 4, why do we trade the

ten rod for 10 unit cubes? (We cannot share one ten rod equally among 4groups. If we trade the ten rod for 10 unit cubes, weget 13 unit cubes. We can share 12 of these cubesequally among 4 groups.)

Prac t i ce


Students should write a story problem with acontext in which the numbers 78 and 6 arereasonable. Division should be required as onemethod of solving the problem. Students coulddraw simple pictures of the Base Ten Blocks toillustrate their solutions.

Early FinishersHave students write a division sentence for each statement. Canthey write more than one division sentence for each statement?• When 37 is divided by a number, the remainder is 1.• When 28 is divided by a number, the remainder is 4.• When 84 is divided by a number, the remainder is 3.

Common Misconceptions➤Students complete a question with a remainder that is greater

than the number they divide by (the divisor).How to Help: Have students check to see if they can continueto divide some of the remaining blocks among the groupsthey made.


Sample Answers4. There are many possible problems. You have 78 books. You

want to divide the books equally among 6 boxes. How manybooks will go in each box? (13)

5. If the ones digit is odd, there will be a remainder when youdivide by 2.a) 20 b) 20 R1 c) 21 d) 21 R1e) 22 f) 22 R1 g) 23 h) 23 R1

6. If the ones digit is not 0 or 5, there will be a remainder whenyou divide by 5.a) 8 b) 8 R2 c) 9 d) 9 R1e) 10 f) 10 R4 g) 11 h) 11 R2

7. Chin-Tan could use 2 boxes. 52 � 2 = 26, so 26 actionfigures could go in each box.He could also use 4 boxes. 52 � 4 = 13, so 13 action figurescould go in each box.Chin-Tan cannot use 3 boxes because 52 � 3 = 17 R1.

REFLECT: Even numbers are multiples of 2, so they have noremainder when divided by 2. Numbers with 0 or 5 in theones place are multiples of 5, so they have no remainderwhen divided by 5.


Numbers Every DayFor the first two sets, students should compare the hundredsdigits, then the tens digits, and then the ones digits. For the lasttwo sets, students should compare the thousands digits, then thehundreds digits, then the tens digits, and then the ones digits.


What to Look For

Applying procedures✔ Students can use Base Ten Blocks to

model dividing a 2-digit number by a1-digit number.

✔ Students can divide a 2-digit numberby a 1-digit number.

Understanding concepts✔ Students can deal with remainders

reasonably.

✔ Students can use multiplication toverify answers.

Communicating✔ Students can write division sentences

for division story problems.

What to Do

Extra Support: Have students use Base Ten Blocks for allquestions.Students can use Step-by-Step 11 (Master 4.21) to completequestion 4.

Extra Practice: Students can do the Additional Activity,Multiplication and Division Puzzle Squares (Master 4.11).Students can complete Extra Practice 5 (Master 4.29).

Extension: Have students create their own division puzzlesquares as on Master 4.11b.


23 17 43 R1 16 12 R3

6

31 12 R1 12 11 R1 13

753, 735, 573, 357654, 564, 546, 4567532, 7352, 5732, 57238021, 8012, 2801, 1802


Another Strategy forDivision

Key Math LearningThere are many strategies that can be used to divide numbers.

LESSON ORGANIZER

Curriculum Focus: Relate the use of Base Ten Blocks toshort division. (N14)Teacher Materials� overhead Base Ten Blocks (optional)Student Materials Optional� Base Ten Blocks � Step-by-Step 12 (Master 4.22)

� Extra Practice 6 (Master 4.30)Vocabulary: short divisionAssessment: Master 4.2 Ongoing Observations:Multiplication and Division

40–50 min

L E S S O N 1 2

This lesson introduces short division as a symbolic way torecord division with manipulatives. If your students arenot ready for this, delay its introduction until a latergrade.

Math Note


Use Base Ten Blocks on the overhead projector.Ask:• How can I use Base Ten Blocks to show

33 � 2? (Use 3 ten rods and 3 unit cubes. Make 2groups, each with 1 ten rod and 1 unit cube. Therewill be 1 ten rod and 1 unit cube left over. Trade theten rod for 10 unit cubes to get 11 unit cubes. Dividethe unit cubes equally between the 2 groups. Eachgroup then has 1 ten rod and 6 unit cubes, or 16.There is 1 left over.)

Present Explore. Have Base Ten Blocks availablefor students who wish to use them.

DURING Exp lore


Ask questions, such as:• What Base Ten Blocks would you use to

model 63? (I would use 6 ten rods and 3 unit cubes.)

• How did you find how many trees will be ineach row? (I divided 63 by 4.)

• How do you know if there will be any treesleft over? (If there is a remainder when I divide 63by 4, I know there will be trees left over. Theremainder tells me how many trees will be left over.)


Early FinishersHave students answer these questions:• Will you get a remainder when you divide an even number

by 4? Explain. (Sometimes. If the number is a multiple of 4, there will not bea remainder. Numbers such as 6 and 10 are not multiples of4. When I find 6 � 4 and 10 � 4, I get a remainder.)

• Will you ever get a remainder of 3 when you divide an evennumber by 4? (No, when I divide an even number by 4, theremainder is either 0 or 2.)


a, b, f

• How could you use Base Ten Blocks to show63 � 4? (I could make 4 equal groups. I would put1 ten rod in each group. I would have 2 ten rodsand 3 unit cubes left over. I could trade each ten rodfor ten unit cubes to get 23 unit cubes. I would put 5unit cubes in each group to get 4 groups of 15 with3 left over; 63 � 4 = 15 R3.)

AFTER Connec t

Invite students to share the strategies they usedto find the number of trees in each row. If anystudents used a novel approach to solve theproblem, have them present their strategies tothe class. Ask students to explain how theydealt with the 3 leftover trees.

Review the examples in Connect. Emphasizethat short division is one method students canuse to divide. Other methods are also valid. Touse the short division notation, students beginby sharing the tens, and then trading any

remaining tens before sharing the ones.Students are familiar with trading in this wayfrom the strategies they used for subtraction.You might wish to do another example toensure students are comfortable with shortdivision notation.

Prac t i ce

Have Base Ten Blocks available for allquestions.


Most students will divide twice, once to findthe number of cans of cat food needed each day,and again to find the number of days for whichTrenton has cat food. Some students may saythere is enough cat food to feed 2 cats on the12th day.

Sample Answers5. Trenton needs 4 cans of cat food each day; 8 � 2 = 4. Since

45 � 4 = 11 R1, Trenton has 11 days of cat food.6. If the number divided by 3 is not a multiple of 3, there will be

a remainder. A number is a multiple of 3 if the sum of itsdigits is a multiple of 3; for example, 45 is a multiple of 3because 4 + 5 = 9, and 9 is a multiple of 3.

REFLECT: I prefer to use short division to divide because it ismuch faster than using Base Ten Blocks.



What to Look For

Applying procedures✔ Students can divide a 2-digit number

by a 1-digit number.

Understanding concepts✔ Students can deal with

remainders reasonably.

Communicating✔ Students can write division sentences

for division story problems.

What to Do

Extra Support: Have students model each division questionusing Base Ten Blocks and record their results in shortdivision notation.Students can use Step-by-Step 12 (Master 4.22) to completequestion 5.

Extra Practice: Have students complete Practice questions 1and 3 from Lesson 11, using short division notation.Students can complete Extra Practice 6 (Master 4.30).

Extension: Have students write a story problem in which therewill be a remainder, and then use short division to solve theproblem.


Numbers Every DayStudents should estimate to see if each question has an answerthat is close to 42. If an answer is close to 42, they can calculateto check.

6

46

3 R29 R218 R213

27 R2

8

1214

1315

13 R115 R1

20151210

4242

6 R5

4 R6

12 R2

14 R1

Unit 4 • Game • Student page 157 41

G A M E

Array, Array!

LESSON ORGANIZER

Student Materials� 2-cm grid paper (PM 21)� scissors

20 min


Give students time to make the arrays. Partnersshould share the work. For each array, ensurestudents know to write the products of factorson the grid side of the paper and the product onthe other side. Invite a student to read throughthe rules for Game 1 and Game 2. Each pair ofstudents should play each game at least once.

DURING Game

As students play, ask questions, such as:• On a multiplication chart, where would you

find the products for the arrays that aresquare? (I would find the products for squarearrays on a diagonal of the multiplication chart.)

• How can you tell, without looking at theproduct, which array represents the greaterproduct? (The array with the greater arearepresents the greater product.)

AFTER

After students have played both games, givethem an opportunity to talk about which gamethey liked better, and why.

Challenge students to a new game. Players taketurns to choose one array, and then find a pairof arrays that, together, have the same value asthe first array. For example, suppose a playerchooses the array for 3 � 6. The product of3 � 6 is 18. The player could then choose 3 � 2and 3 � 4 (3 � 2 = 6 and 3 � 4 = 12;6 + 12 = 18). The player with the most sets ofequivalent arrays is the winner.

Sample Answers1. a) 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

b) 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42,45, 48

c) 12, 24, 36, 48; These numbers are multiples of 12.2. a) I used known facts.

b) When I multiply by 1, the product is the other factor.c) When I multiply by 0, the product is 0.d) I found 7 � 5, and then added 7 � 1.e) I used patterns to multiply by 9. The tens digit of the

product is 8. The sum of the digits is 9, so the productis 81.

f) I used patterns to multiply by 10. The basic fact is 8 � 1 = 8. When I multiply by 10, I write one 0 onthe end, so 8 � 10 = 80.

g) I used patterns to multiply by 9. The tens digit of theproduct is 7. The sum of the digits is 9, so the productis 72.

h) I found 7 � 5, and then added 7 � 3.4. There are many possible questions; 9 � 500.9. a) 7 � 6 = 42; 42 � 7 = 6; 42 � 6 = 7

b) 1 � 8 = 8; 8 � 1 = 8; 8 � 8 = 1c) 9 � 5 = 45; 45 � 9 = 5; 45 � 5 = 9

10. a) 5 � 6 = 30; 6 � 5 = 30; 30 � 6 = 5; 30 � 5 = 6b) 9 � 7 = 63; 7 � 9 = 63; 63 � 7 = 9; 63 � 9 = 7c) 8 � 6 = 48; 6 � 8 = 48; 48 � 6 = 8; 48 � 8 = 6d) 7 � 8 = 56; 8 � 7 = 56; 56 � 7 = 8; 56 � 8 = 7

11. 72 � 8 = 913. 25 � 4 is 6 R1; 7 tapes are needed to record

the series.14. a) 15 R1 b) 21 R2

c) 10 R1 d) 5 R2e) 14 f) 6 R6g) 9 R6 h) 10 R3

42 Unit 4 • Show What You Know • Student page 158

LESSON ORGANIZER

Student Materials� counters� Base Ten Blocks� 1-cm grid paper (PM 20)Assessment: Masters 4.1 Unit Rubric: Multiplication andDivision, 4.4 Unit Summary: Multiplication and Division

40–50 min

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

4081

880

072

4256

42001000

72008000

3002800

4000180

150 420 480 280

7 � $300 = $2100

58 219 204 380

Unit 4 • Show What You Know • Student page 159 43


What to Look For

Accuracy of procedures✔ Question 7: Student can multiply a 2-digit number by a 1-digit number.

✔ Question 14: Student can divide a 2-digit number by a 1-digit number.

Reasoning; Applying concepts✔ Question 6: Student understands that the question can be answered by multiplying.

✔ Question 13: Student understands that, in this context, the answer to the problem is 1 more than theanswer to the division statement.

Recording and ReportingMaster 4.1 Unit Rubric: Multiplication and DivisionMaster 4.4 Unit Summary: Multiplication and Division

5 4 8 9 1

42 8 45

9 students

1413

1311

1311

1317

7 tapes

11 rides

Be AccurateEncourage students to work carefully and precisely whencalculating. They should show all of their work. This will helpthem to find, and then correct, any errors. Students shouldestimate each answer before calculating, so they know if theiranswer is reasonable.

SHOW YOUR BEST

Have students turn to the Unit Launch on page 118 of the Student Book.

Invite volunteers to read aloud the LearningGoals for the unit. Discuss each goal brieflywith students.

Present the Unit Problem. Have volunteers readthe instructions. Answer any questionsstudents have about the problems.

Have a volunteer read the Check List. Ensureall students know what is expected. Askstudents to think about the strategies theycould use to solve the problems before theysolve them.

44 Unit 4 • Unit Problem • Student page 160

At the Garden Centre

Sample Response1. Jean can deliver the order 3 different ways.

He can fill the order with boxes of 4. Jean needs 72 � 4, or 18 boxes of 4.Jean can also fill the order with boxes of 9. Jean needs 72 � 9, or 8 boxes of 9.Jean can also fill half the order with boxes of 4 and theother half with boxes of 9.Jean needs 9 boxes of 4 (36 � 4 = 9), and 4 boxes of 9 (36 � 9 = 4).

2. I can fill the order with 12 boxes of 4 and 3 boxes of 9. 12 � 4 = 48 and 3 � 9 = 27; 48 + 27 is 75.Six boxes of 4 fit on a tray, so two trays will hold 12 boxes.Three boxes of 9 will fit on 1 tray. Three trays are needed.

LESSON ORGANIZER

Student Grouping: 2Student Materials� calculatorsAssessment: Masters 4.3 Performance Assessment Rubric:At the Garden Centre, 4.4 Unit Summary: Multiplication andDivision

40–50 min

U N I T P R O B L E M

3.

If the customer wants to buy exactly 180 pots, there is onlyone way she can do this. She can buy 6 packages of 30 for$60. If the customer does not mind buying more than 180pots, the cheapest way is to buy 2 packages of 100 for $54.

4. May-Lin could plant 80 trees in equal rows 10 different ways:1 row of 80 1 � 80 = 80 80 � 80 = 12 rows of 40 2 � 40 = 80 80 � 40 = 24 rows of 20 4 � 20 = 80 80 � 20 = 45 rows of 16 5 � 16 = 80 80 � 16 = 58 rows of 10 8 � 10 = 80 80 � 10 = 810 rows of 8 10 � 8 = 80 80 � 8 = 1016 rows of 5 16 � 5 = 80 80 � 5 = 1620 rows of 4 20 � 4 = 80 80 � 4 = 2040 rows of 2 40 � 2 = 80 80 � 2 = 4080 rows of 1 80 � 1 = 80 80 � 1 = 80

Reflect on the Unit

Multiplication and division are opposites. Here is an array of 21objects arranged into 3 rows, with 7 objects in each row:

For this array, I can write 2 multiplication sentences and 2division sentences:3 groups � 7 things in each group = 21 things7 things in each group � 3 groups = 21 things21 things � 3 groups = 7 things in each group21 things � groups of 7 things = 3 groups

Packages Packages Number Total Costof 30 of 100 of Pots

6 0 180 $600 2 200 $543 1 190 $57

Unit 4 • Unit Problem • Student page 161 45


What to Look For

Applying procedures✔ Students multiply and divide

accurately.

Understanding concepts✔ Students choose the appropriate

operation for each problem.

✔ Students use multiples of 4 and 9 tofind different ways to fill orders.

Communicating✔ Students present their calculations in a

clear and organized fashion.

What to Do

Extra Support: Make the problem accessible.

Some students may have difficulty finding all the ways to fill anorder of 72 or 75 petunias. Suggest students list all the multiplesof 4 and all the multiples of 9 to 80, and then look for pairs ofmultiples that add to 72 or 75.

For question 3, tell students to consider combinations of packagesthat give more than 180 pots.

Recording and ReportingMaster 4.3 Performance Assessment Rubric: At the Garden CentreMaster 4.4 Unit Summary: Multiplication and Division

Students should use a protractor (Master 3.7) orPattern Blocks to measure the angles inquestion 8. In question 10, they should tracethe angles on tracing paper, and use thetracings to compare angles.

Sample Answers1. a) This is a growing pattern.

The pattern rule is: Start at 3. Add 2 each time.b) This is a repeating pattern.

The pattern rule is: repeat the core 3, 5, 7.c) This is a shrinking pattern.

The pattern rule is: Start at 99. Subtract 5 each time.2. The pattern in question 1b has a core.4. There are many possible 4-digit numbers.

a) 6078 b) 6000 + 70 + 8c) six thousand seventy-eight

5. 5643, 5634, 5463, 5436, 5364, 5346, 4653, 4635,4563, 4536, 4365, 4356

6. a) About 20 b) About 800 c) About 350d) About 600

7.

9. Kite: 2 pairs of equal adjacent sides; 1 pair ofequal anglesSquare: 2 pairs of parallel sides; all sides equal;all right anglesTrapezoid: 1 pair of parallel sidesParallelogram: 2 pairs of parallel sides; opposite sidesequal; opposite angles equal

10.Has opposite sides equal

B

D

C

A

Has some angles equal

46 Cumulative Review • Units 1–4 • Student page 162

Cumulative Review

LESSON ORGANIZER

Student Materials Optional� dot paper (PM 22) � tracing paper or Pattern Blocks� rulers� protractors (Master 3.7)

U N I T S 1 – 4

15, 17, 193, 5, 774, 69, 64

3, 5, 7

1111

1614

5643, 5634, 5463, 5436, 5364, 5346,4653, 4635, 4563, 4536, 4365, 4356

About 20 813 About 350 591

For Your Notes

Unit 4 • Units 1–4 • Student page 163 47

About 2 and a halftan Pattern Blocks

About 4 and a halftan Pattern Blocks

kite

square

trapezoid

parallelogram

424204200

242402400

454504500

202002000

12 R1 11 R3

Copyright © 2004 Pearson Education Canada Inc. 48

Evaluating Student Learning: Preparing to Report: Unit 4 Multiplication and Division This unit provides an opportunity to report on the Number Concepts and Number Operations strands. Master 4.4: Unit Summary: Multiplication and Division 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

Strands: Number Concepts and Number Operations

Reasoning; Applying concepts

Accuracy of procedures

Problem solving

Communication Overall

Ongoing Observations 3 3 2 2 2/3 Strategies Toolkit 2 3 2 3 2/3 Work samples or portfolios; conferences

3 3 3 3 3

Show What You Know 3 3 3 3 Unit Test 2 3 3 3 3 Unit Problem At the Garden Centre

3 3 2 3 3

Achievement Level for reporting 3

Recording How to Report Ongoing Observations Use Master 4.2 Ongoing Observations: Multiplication and Division 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 7). 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 4.1 Unit Rubric: Multiplication and Division to guide evaluation of collections of work and information gathered in conferences. Teachers may choose to focus particular attention on the Assessment Focus questions. Work from late in the unit should be most heavily weighted.

Show What You Know Master 4.1 Unit Rubric: Multiplication and Division may be helpful in determining levels of achievement. #4, 6, 9, 10, 11, 13, and 15 provide evidence of Reasoning; Applying concepts; #1–3, 5, 7, 8, 12, and 14 provide evidence of Accuracy of procedures; all provide evidence of Communication.

Unit Test Master 4.1 Unit Rubric: Multiplication and Division may be helpful in determining levels of achievement. Part A provides evidence of Accuracy of procedures; Part B provides evidence of Reasoning; Applying concepts; Part C provides evidence of Problem solving; all parts provide evidence of Communication. Teachers can also enter raw scores.

Unit performance task Use Master 4.3 Performance Assessment Rubric: At the Garden Centre. The Unit Problem offers a snapshot of students’ achievement. In particular, it shows their ability to synthesize and apply what they have learned.

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

Comments Analyse the pattern of achievement to identify strengths and needs. In some cases, specific actions may need to 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.


Name Date

Unit Rubric: Multiplication and Division Not Yet

Adequate Adequate Proficient Excellent


• shows understanding by demonstrating, explaining, and applying concepts of:

– multiplying and dividing whole numbers

– relating multiplication and division facts

– estimating products

may be unable to demonstrate, explain, or use concepts of: – multiplying and

dividing whole numbers



partially able to demonstrate, explain, and use concepts of: – multiplying and




appropriately demonstrates, explains, and uses concepts of: – multiplying and




in various contexts, appropriately demonstrates, explains, and uses concepts of: – multiplying and





• accurately multiplies 2-digit whole numbers by 1-digit numbers

• accurately divides 2-digit whole numbers by 1-digit divisors

• verifies solutions using inverse operations, estimation, and calculators

• recalls multiplication and division facts to 81

often makes major errors in: – multiplying a 2-digit

whole number by a 1-digit number

– dividing a 2-digit whole number by a 1-digit divisor

– verifying solutions – recalling facts to 81

makes frequent minor errors in: – multiplying a 2-digit




makes few errors in: – multiplying a 2-digit




rarely makes errors in: – multiplying a 2-digit




Problem-solving strategies

• chooses and carries out a range of strategies to solve and create whole number problems

may be unable to use appropriate strategies to solve and create whole number problems

with limited help, uses some appropriate strategies to solve and create whole number problems; partially successful

uses appropriate strategies to solve and create whole number problems successfully

uses appropriate, often innovative, strategies to solve and create whole number problems successfully

Communication • explains reasoning

and procedures clearly

unable to explain reasoning and procedures clearly

partially explains reasoning and procedures

explains reasoning and procedures clearly

explains reasoning and procedures clearly, precisely, and confidently

• presents calculations clearly; uses appropriate symbols

frequently presents calculations unclearly; rarely uses appropriate symbols

presents calculations with some clarity; uses some appropriate symbols

presents calculations clearly and precisely; uses appropriate symbols

presents calculations clearly and precisely; uses a range of appropriate symbols

Master 4.1


Name Date

Ongoing Observations: Multiplication and Division 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: Multiplication and Division Student Reasoning;

Applying concepts Accuracy of procedures

Problem solving Communication

Explains and applies concepts related to multiplication and division; makes connections

Multiplies and divides whole numbers accurately Verfies results Recalls facts

to 81

Uses appropriate strategies to solve and create problems involving whole numbers

Communicates clearly Presents

calculations clearly, including appropriate symbols

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

Master 4.2


Name Date

Performance Assessment: At the Garden Centre

Not Yet Adequate Adequate Proficient Excellent

Reasoning; Applying concepts • shows

understanding and ability to apply concepts of multiplication and division by selecting appropriate operations and determining whether answers are reasonable

shows little understanding; may be unable to: – choose appropriate

operations – determine whether

answers are reasonable

– give appropriate explanations

shows partial understanding by: – usually choosing

appropriate operations

– sometimes determining whether answers are reasonable

– giving some appropriate explanations

shows understanding by: – choosing appropriate

operations, in most cases

– determining whether most answers are reasonable

– giving clear and appropriate explanations

shows thorough understanding by: – choosing appropriate

operations in a variety of contexts

– determining whether all answers are reasonable

– giving clear, appropriate, and detailed explanations

Accuracy of procedures • multiplies and

divides accurately to find: – the number of

boxes and trays needed

– the cost of 180 pots

– the different ways to plant 80 trees in equal rows

often makes major errors in finding: – the number of boxes

and trays needed – the cost of 180 pots – the different ways to

plant 80 trees in equal rows

makes frequent minor errors in finding: – the number of boxes



makes few errors in finding: – the number of boxes



makes no errors in finding: – the number of boxes



Problem-solving strategies • uses appropriate

strategies to find: – the number of

ways to deliver an order of 72 petunias

– the number of trays needed for an order of 75 petunias

– the cheapest way to buy 180 pots

unable to develop workable plans that meet the criteria given

uses some appropriate strategies to find: – the number of ways

to deliver an order of 72 petunias



uses appropriate and effective strategies to find: – the number of ways

to deliver an order of 72 petunias



uses innovative and effective strategies to find: – the number of ways to

deliver an order of 72 petunias



Communication • uses

mathematical terminology, numbers, and symbols correctly

uses few appropriate mathematical terms and symbols

uses some appropriate mathematical terms and symbols

uses appropriate mathematical terms and symbols

uses a range of appropriate mathematical terms and symbols clearly and precisely

• shows thinking clearly

unable to show thinking clearly

shows thinking with some clarity

shows thinking clearly shows thinking clearly, precisely, and confidently

Master 4.3


Name Date

Unit Summary: Multiplication and Division 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: Number Concepts and Number Operations



Problem solving

Communication OVERALL

Ongoing Observations

Strategies Toolkit (Lesson 7)

Work samples or portfolios; conferences

Show What You Know

Unit Test

Unit Problem: At the Garden Centre

Achievement Level for reporting

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

Comments: (Strengths, Needs, Next Steps)

Master 4.4


Name Date

To Parents and Adults at Home … Your child’s class is starting a mathematics unit on multiplication and division. Children will develop strategies for multiplying and dividing with whole numbers. They will use mental math, estimation, concrete materials, and pencil-and-paper calculations. In this unit, your child will:

• Skip count. • Recall basic multiplication and division facts. • Use different strategies to multiply and divide. • Relate multiplication and division. • Identify patterns in multiplication and division. • Multiply by 10, 100, and 1000. • Multiply and divide a 2-digit number by a 1- digit number. • Pose and solve problems using multiplication and division.

We use multiplication and division in many day-to-day situations. Encourage your child to practise basic multiplication and division facts from 1 × 1 to 10 × 10. Talk with your child about the strategies you use to recall these facts. Here is an activity you can do at home: Multiplication Challenge

• Remove the jokers and face cards from a deck of playing cards. • Shuffle the cards. Divide them into two equal piles.

Keep 1 pile and give the other pile to your child. • Each player turns over two cards and multiplies the numbers on the cards.

An ace counts as 1. The player with the greater answer takes the cards.

• Continue playing until one player runs out of cards.

Master 4.5


Name Date

Multiplication Chart

× 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10

2 2 4 6 8 10 12 14 16 18 20

3 3 6

4 4 8

5 5 10

6 6 12

7 7 14

8 8 16

9 9 18

10 10 20

Master 4.6


Name Date

Cross-Out Product Game Board

14

32

49

81

12

27

45

72

9 24

42

64

6 20

40

63

4 18

35

56

Master 4.7


Name Date

Additional Activity 1: Multiplication Bingo

Work in a group. You will need blank bingo cards, a multiplication chart, and counters. Choose one student to be the caller. The other students in the group are players.

How to play: • Each player fills in the squares of her blank Bingo card with different

products from the Multiplication Chart.

• The caller calls two factors from a multiplication fact, for example, 6 × 7.

• Find the product of the factors.

Look for the product on your Bingo card.

If you have the product on your card, cover it with a counter.

• The first player to cover a row, column, or diagonal is the winner. Take It Further: Play Multiplication Bingo again. This time, write two factors in each square of your Bingo card, for example, 6 × 7. The caller calls out the products.

Master 4.8


Name Date

Blank Bingo Card

Master 4.8b


Name Date

Additional Activity 2: Spinning Products

Play with a partner. You will need Spinner A, Spinner B, and a calculator. Use an open paper clip as a pointer. Hold it in place with the point of a pencil.

How to play: • Players take turns to spin the pointers on Spinner A and Spinner B.

• Record the numbers the pointers land on.

Estimate the product of these numbers.

• Use a calculator to find the product of the numbers.

• The player whose estimate is closer to the product scores a point.

• The first player to score 10 points wins. Take It Further: Take turns to spin Spinner A. Choose two numbers whose product is close to the number the pointer landed on. The player whose product is closer to this number scores a point.

Master 4.9


Name Date

Spinners

Spinner A

Spinner B

Master 4.9b


Name Date

Additional Activity 3: Division Tic-Tac-Toe

Work with a partner. You will need counters in two different colours.

How to play: • Each player chooses one counter colour.

• Player 1 chooses a number from List 1.

• Player 2 chooses a number from List 2.

Player 2 divides Player 1’s number by the number he chose from List 2. If the number divides exactly and the answer is on the Tic-Tac-Toe grid, he covers it with a counter.

• Players switch roles.

• The game continues until one player wins by getting 3 counters in a row,

column, or diagonal. List 1: 3 8 12 16 21 24 35 36 40 45 List 2: 1 2 3 4 5 6 7 8 9 10

Tic-Tac-Toe

1 2 3

4 5 6

7 8 9

Take It Further: Create a new Tic-Tac-Toe game. Choose your own numbers for List 1, List 2, and the Tic-Tac-Toe grid.

Master 4.10


Name Date

Additional Activity 4: Multiplication and Division Puzzle Squares

Work on your own. You will need Puzzle Squares and scissors.

• Cut out the 16 puzzle squares.

• Answer each multiplication and division question. Write each answer next to the question, or below the question.

• Fit the squares together. Match each question with its answer.

The questions and answers should match all around each square.

• When you have finished, you will have a 4 by 4 square. Take It Further: Create your own puzzle squares. Trade puzzles with a classmate. Solve your classmate’s puzzle.

Master 4.11


Name Date

Puzzle Squares

27 ÷ 3 78 ÷ 6 27

16

96 ÷ 3 40 11

18

10

84 ÷ 2

54 ÷ 9 36 ÷ 6 13

32 12

81 ÷ 9

48 ÷ 3 55 ÷ 5 150

16

8

19 × 10

56 ÷ 4 17

75 ÷ 5 5 × 30

14

64 ÷ 4 70 34 ÷ 2

36 ÷ 2 9 10 × 7

20 × 4

80 ÷ 2

96 ÷ 8 42

6

80

100 ÷ 10 8 × 20

6

190 54 ÷ 2

15

160 56 ÷ 7

9

Master 4.11b


Name Date

Step-by-Step 1 Lesson 1, Question 5 Step 1 Write all the multiples of 2 to 24. ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____

Write all the multiples of 4 to 24. ____ ____ ____ ____ ____ ____ Step 2 Compare the lists in Step 1. Answer true or false: A multiple of 4 is also a multiple of 2. _________ A multiple of 2 is also a multiple of 4. _________ Step 3 Choose 2 different even numbers.

Write the first 8 multiples of each number. Number ____ Multiples ____ ____ ____ ____ ____ ____ ____ ____

Number ____ Multiples ____ ____ ____ ____ ____ ____ ____ ____

Answer true or false: When a number is even, all its multiples are even. _________ Step 4 Choose 2 different odd numbers.

Write the first 8 multiples of each number. Number ____ Multiples ____ ____ ____ ____ ____ ____ ____ ____ Number ____ Multiples ____ ____ ____ ____ ____ ____ ____ ____

Answer true or false:

When a number is odd, all its multiples are odd. _________

Master 4.12


Name Date

Step-by-Step 2 Lesson 2, Question 10

Emi walks her dog every day for 2 hours. How many hours does Emi walk in 5 weeks?

Step 1 How many days are there in 1 week? ___________________________ Emi walks for 2 hours every day.

How many hours does Emi walk in 1 week? Show your work. ________________________________________________________ Step 2 How many hours does Emi walk in: 2 weeks? ________________________________________________________

3 weeks? ________________________________________________________

4 weeks? ________________________________________________________

5 weeks? ________________________________________________________

Master 4.13


Name Date

Step-by-Step 3 Lesson 3, Question 6 Step 1 Draw an array with 5 rows, and 9 counters in each row. What is the product of 9 × 5? _________ Step 2 Add counters to the array in Step 1 to show 9 × 7. How many rows are in the array? __________________ How many counters are in each row? __________________ What is the product of 9 × 7? __________________ Step 3 Explain how you can use the product of 9 × 5

to find the product of 9 × 7. ________________________________________________________ ________________________________________________________ ________________________________________________________ ________________________________________________________

Master 4.14


Name Date

Step-by-Step 4 Lesson 4, Question 9 Step 1 Write the multiples of 10 to 100. _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ Choose any 1-digit number and any multiple of 10. ______ ______ Multiply these two numbers. What strategy did you use? ___________________________________________________________ Step 2 Write the multiples of 100 to 1000. _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ Choose any 1-digit number and any multiple of 100. ______ ______ Multiply these two numbers. What strategy did you use? ___________________________________________________________ Step 3 Write the multiples of 1000 to 9000. _____ _____ _____ _____ _____ _____ _____ _____ _____ Choose any 1-digit number and any multiple of 1000. ______ ______ Multiply these two numbers. What strategy did you use? ___________________________________________________________ Step 4 How do you know your strategies for multiplying with

multiples of 10, 100, and 1000 work? ___________________________________________________________

Master 4.15


Name Date


Kyle’s mother drives 47 km to work every day. About how far does she drive in 2 weeks?

Step 1 Round 47 to the nearest 10. _____________________________________ Step 2 Your answer in Step 1 is a one-way trip.

About how far does Kyle’s mother drive to and from work?

_______________________________________________________________ Step 3 Estimate how many days Kyle’s mother might work in 1 week.

_______________________________________________________________ Step 4 Multiply your answers from Steps 2 and 3 to estimate how far Kyle’s

mother drives in 1 week. Show your work.

_______________________________________________________________ _______________________________________________________________ _______________________________________________________________ Step 5 Multiply your answer in Step 4 by 2. About how far does Kyle’s mother drive in 2 weeks? _______________________________________________________________ _______________________________________________________________

Master 4.16


Name Date


Tom is buying candles for his great grandmother’s 90th birthday.

There are 24 candles in a box. Tom buys 4 boxes of candles.

Step 1 Use the grid to show a 24 by 4 array.

How many candles does Tom have? __________________ Step 2 How many candles does Tom need? __________________ Will Tom have enough candles? How do you know? _______________________________________________________________

_______________________________________________________________ Step 3 Will Tom have any candles left over?

Look at how many candles Tom has and how many he needs. What will you do to find the answer?

_______________________________________________________________

_______________________________________________________________

Master 4.17


Name Date


Joe has 35 cubes. He shares the cubes equally among 7 students.

Each student needs 6 cubes. Does Joe have enough cubes? Explain.

Use grid paper or Base Ten Blocks if they help.

Step 1 Joe has 35 cubes. He shares them equally among 7 students. How many cubes will each student get?

_______________________________________________________________ _______________________________________________________________ Step 2 Each student needs 6 cubes.

Look at your answer to Step 1. Does each student have 6 cubes?

_______________________________________________________________ _______________________________________________________________ Step 3 Does Joe have enough cubes to give each student 6 cubes?

Explain. _______________________________________________________________ _______________________________________________________________

Master 4.18


Name Date


How do you know that 48 ÷ 8 is more than 40 ÷ 8?

Use counters or grid paper when it helps. Step 1 Use multiplication to help you.

8 × ______ = 48, so 48 ÷ 8 = ______

8 × ______ = 40, so 40 ÷ 8 = ______

Explain your answer. _______________________________________________________________ _______________________________________________________________

How do you know that 72 ÷ 8 is more than 72 ÷ 9?

Step 2 Use multiplication to help you. 8 × ______ = 72, so 72 ÷ 8 = ______ 9 × ______ = 72, so 72 ÷ 9 = ______ Explain your answer.

_______________________________________________________________ _______________________________________________________________

Master 4.19


Name Date


Amina solves a division problem this way: 21 ÷ 4 = 5 R1 Tyler solves the problem this way: 21 ÷ 4 = 4 R5

Who is correct?

Step 1 This is what a remainder means: There is not enough left to divide equally among the groups.

Step 2 What is Amina’s remainder? ____________ Can Amina’s remainder be divided equally among 4 groups?

How do you know? _______________________________________________________________ _______________________________________________________________ Step 3 What is Tyler’s remainder? ____________

Can Tyler’s remainder be divided equally among 4 groups? How do you know?

_______________________________________________________________ _______________________________________________________________ Step 4 Who is correct? How do you know? _______________________________________________________________ _______________________________________________________________

Master 4.20


Name Date


Write a story problem that can be solved using 78 ÷ 6.

Step 1 List some things that you might have 78 of. _______________________________________________________________ Choose one of the things from your list.

Choose to divide the things into 6 groups, or into groups of 6. _______________________________________________________________ _______________________________________________________________ Step 2 Write a story problem using the things and the numbers from Step 1. _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ Step 3 Solve your problem.

Show your work. _______________________________________________________________ _______________________________________________________________ _______________________________________________________________

Master 4.21


Name Date


Trenton has to feed 8 cats. Each large can of cat food feeds 2 cats per day.

Trenton has 45 large cans of cat food. How many days of cat food does Trenton have?

Step 1 In one day, one can of cat food feeds 2 cats. How many cans will feed 4 cats? 6 cats? 8 cats? Draw pictures to show your work.

_______________________________________________________________ Step 2 Your answer to Step 1 tells how many cans of cat food Trenton needs

each day. Trenton has 45 cans of cat food. Use counters or Base Ten Blocks. How many days of cat food does Trenton have? Are there any cans left over? Explain your answer.

_______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________

Master 4.22


Name Date

Unit Test: Unit 4 Multiplication and Division Use Base Ten Blocks, counters, or grid paper when they help. Part A 1. a) List the multiples of 3 to 50. b) List the multiples of 5 to 50. c) What numbers are in both lists? d) What can you say about these numbers? 2. Multiply. a) 8 × 5 ______ b) 4 × 9 ______ c) 7 × 8 ______ d) 6 × 100 ______ e) 80 × 9 ______ f) 400 × 5 ______ 3. Estimate each product, then multiply. a) 74 b) 49 c) 82 × 6 × 5 × 7 4. Divide. a) 54 ÷ 9 ______ b) 72 ÷ 8 ______ c) 35 ÷ 7 ______

Master 4.23


Name Date

Unit Test continued 5. Write 4 related facts for each set of numbers. a) 9, 8, 72 b) 5, 9, 45 6. Divide. a) 78 ÷ 4 ______ b) 87 ÷ 3 ______ c) 39 ÷ 5 ______ Part B 7. Fill in the missing number. a) 4 × = 36 b) × 300 = 600 c) 54 ÷ = 6 8. Farah has $35. One T-shirt costs $8.

a) Does Farah have enough money to buy 4 T-shirts? Explain.

b) If your answer to part a is yes, will Farah have any money left over? How much?

9. Francoise and 5 of her friends went to a movie. One ticket costs $7. Each person spent $2 on a drink.

How much did Francoise and her friends spend in all?

Part C 10. The answer to a multiplication question is 240.

What might the question be? How many different questions can you find?

Master 4.23b


Name Date

Sample Answers Unit Test – Master 4.23 Part A 1. a) 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36,

39, 42, 45, 48 b) 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 c) 15, 30, 45 d) They are multiples of 15. 2. a) 40 b) 36 c) 56 d) 600 e) 720 f) 2000 3. a) About 420; 444 b) About 250; 245 c) About 560; 574 4. a) 6 b) 9 c) 5 5. a) 9 × 8 = 72; 8 × 9 = 72; 72 ÷ 9 = 8;

72 ÷ 8 = 9 b) 5 × 9 = 45; 9 × 5 = 45; 45 ÷ 5 = 9;

45 ÷ 9 = 5 6. a) 19 R2 b) 29 c) 7 R4

Part B 7. a) 9 b) 2 c) 9 8. a) Yes b) She will have $3 left over. 9. $54

Part C 10. There are many possible questions; 1 × 240;

2 × 120; 3 × 80; 4 × 60; 5 × 48; 6 × 40; 8 × 30; 10 × 24; 12 × 20; 15 × 16

Master 4.24


Extra Practice Masters 4.25–4.31 Go to the CD-ROM to access editable versions of these Extra Practice Masters.

Program Authors

Peggy Morrow

Ralph Connelly

Bryn Keyes

Jason Johnston

Steve Thomas

Jeananne Thomas

Angela D’Alessandro

Maggie Martin Connell

Don Jones

Michael Davis

Linden Gray

Sharon Jeroski

Trevor Brown

Linda Edwards

Susan Gordon

Copyright © 2004 Pearson Education Canada Inc., Toronto, Ontario

All Rights Reserved. This publication is protected by copyright,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.

This book contains recycled product and is acid free.

Printed and bound in Canada

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