Download Unit 4: Multiplication and Division

ADDISON WESLEY
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Western Canadian
Teacher Guide
Unit 4: Multiplication and
Division
UNIT
4
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Multiplication and Division
Learning mathematics with
understanding is essential, and
makes subsequent learning
easier. Mathematics makes more
sense and is easier to remember
and to apply when students
connect new knowledge to
existing knowledge in
meaningful ways. Wellconnected, conceptually
grounded ideas are more
readily accessed for use in new
situations.
Mathematics Background
Principles and Standards of School
Students use grouping and sharing to model division. Using the
relationship between multiplication and division, students use arrays
and other models to organize and learn basic division facts. They explore
patterns to determine numbers that are divisible by 2, by 5, and by 10.
Mathematics, NCTM, 2000
FOCUS STRAND
Number: Number Operations
SUPPORTING STRAND
Patterns and Relations
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Unit 4: Multiplication and Division
What Are the Big Ideas?
• A multiplication sentence is another way to write a repeated
addition sentence.
• Multiplication and division are inverse operations.
• A variety of models (for example, arrays, groups, number lines) can be
used to illustrate multiplication and division.
How Will the Concepts Develop?
Students use repeated addition to develop an understanding of
multiplication. They use arrays, counters, skip counting, and patterns to
learn the basic multiplication facts. Patterns and rules related to factors
of 0, 1, 2, 5, and 10 are studied.
Why Are These Concepts Important?
Number relationships, and the meanings of the operations, play a
significant role in one’s ability to master basic mathematics and employ
mental math skills. All levels of mathematics require knowledge of
computation and patterning to reason numerically in number-related
situations.
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Curriculum Overview
Launch
Cluster 1: Understanding Multiplication
Here Comes the Band!
General Outcomes
Specific Outcomes
Lesson 1:
• Students apply an arithmetic
operation (..., multiplication ...) on
whole numbers, and illustrate its
use in creating and solving
problems.
• Students use and justify an
appropriate calculation strategy or
technology to solve problems.
• Students use manipulatives,
diagrams and symbols with
maximum products ... to 50, to
demonstrate and describe the
process(es) of multiplication, ...
(N15)
• Students recall ... multiplication
facts to 49 (7 7 on a
multiplication grid). (N16)
• Students calculate products ...,
using ... mental mathematics
strategies. (N20)
• Students use objects and concrete
models to explain the rule for a
pattern, such as those found on ...
multiplication charts. (PR2)
• Students make predictions based on
... multiplication patterns. (PR3)
Relating Multiplication and
Addition
Lesson 2:
Using Arrays to Multiply
Lesson 3:
Multiplying by 2 and by 5
Lesson 4:
Multiplying by 10
Lesson 5:
Multiplying by 1 and by 0
Lesson 6:
Using a Multiplication Chart
Lesson 7:
Strategies Toolkit
Cluster 2: Understanding Division
General Outcomes
Specific Outcomes
Lesson 8:
• Students apply an arithmetic
operation (..., division ...) on whole
numbers, and illustrate its use in
creating and solving problems.
• Students use and justify an
appropriate calculation strategy or
technology to solve problems.
• Students use manipulatives,
diagrams and symbols with
maximum products and dividends to
50, to demonstrate and describe the
processes of multiplication and
division. (N15)
• Students recall ... multiplication
facts to 49 (7 7 on a
multiplication grid). (N16)
• Students recognize and explain if a
number is divisible by 2, 5, or10.
(N12)
Modelling Division
Lesson 9:
Using Arrays to Divide
Lesson 10:
Dividing by 2, by 5, and by 10
Lesson 11:
Relating Multiplication and
Division
Lesson 12:
Number Patterns on a Calculator
Show What You Know
Unit Problem
Here Comes the Band!
Unit 4: Multiplication and Division
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Curriculum across the Grades
Grade 2
Grade 3
Grade 4
Students demonstrate the
processes of
multiplication and
division, using
manipulatives and
diagrams.
Students recognize and
explain if a number is
divisible by 2, 5, or 10.
They use manipulatives,
diagrams, and symbols
with maximum products
and dividends to 50, to
demonstrate and describe
the processes of
multiplication and
division.
Students use skip counting
(forward and backward)
to support an
understanding of patterns
in multiplication and
division. They demonstrate
and describe the process
of multiplication (3-digit
by 1-digit), using
manipulatives, diagrams,
and symbols.
Students recall
multiplication facts to 49
(7 7 on a
multiplication grid). They
calculate products and
quotients, using mental
mathematics strategies.
Students demonstrate and
describe the process of
division (2-digit by a
1-digit), using
manipulatives, diagrams,
and symbols. They recall
multiplication and division
facts to 81 (9 9 on a
multiplication grid). They
verify solutions to
multiplication and division
problems, using
estimation, calculators,
and the inverse operation.
Students justify the choice
of method for
multiplication and
division, using estimation
strategies, mental
mathematics strategies,
manipulatives, algorithms,
and calculators.
Materials for This Unit
In Lesson 11, pairs of students require a set of cards, each card containing a
multiplication fact or a division fact. Ensure that the set contains pairs of related
facts, for example one card could be 4 6 = 24 and another card could be
24 6 = 4. Students will use these cards to play Concentration.
Curriculum Focus
Your curriculum requires that students calculate products and quotients using
estimation and mental math strategies (N20). Many opportunities for students
to develop estimation and mental math strategies are interspersed throughout
the units that follow, both in the Practice exercises and in the Numbers Every
Day feature.
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Unit 4: Multiplication and Division
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Additional Activities
Amazing Arrays
Multiplication Tag
For Extra Practice (Appropriate for use after Lesson 2)
Materials: Amazing Arrays (Master 4.8),1-cm grid
paper (PM 21), scissors
For Extra Practice (Appropriate for use after Lesson 6)
Materials: Multiplication Tag (Master 4.9), scissors
Preparation: Students cut rectangles to represent each
product from 1 2 to 7 7. Students write the factors
on the grid side of each rectangle, and the product on the
other side.
The work students do: Students play in pairs. They
spread out the rectangles, then take turns selecting a
rectangle and naming either the factors or the product,
whichever is not visible. Students keep all rectangles
they identify correctly. The player with the most
rectangles wins.
Take It Further: Students leave the rectangles blank.
All rectangles are placed grid side up. Students take
turns to select a rectangle, then give both the factors and
the product.
Mathematical/Interpersonal
Partner Activity
The work students do: Students play in pairs. Each
student cuts out 8 number tags, then uses each tag only
once to make 4 multiplication problems. Students solve
their own problems, then add the products to get their
score. Students play 3 more rounds, each time adding
their score to their previous score. The player with the
highest score wins. Students should soon realize how to
arrange the tags to maximize their score.
Take It Further: Students place all 16 number tags
face down. Player A turns over 2 tags, then finds the
product and records the product as his score. The tags
are not replaced. Player B takes a turn. Players continue
to take turns until all tags have been used. The highest
score wins.
Kinesthetic/Interpersonal
Partner Activity
Magic Squares
Make A Sentence
For Extra Practice (Appropriate for use after Lesson 10)
Materials: Magic Squares (Master 4.10)
For Extra Practice (Appropriate for use after Lesson 11)
Materials: Make A Sentence (Master 4.11), a set of
cards numbered from 1 to 49
The work students do: Students work alone.
Students write the product for each of 9 given
multiplication facts. They write the products in the
matching squares in the magic square. Students add the
rows, columns, and diagonals. If all sums are equal, the
square is a magic square. If the sums are not equal,
students should check their addition and their
multiplication.
Take It Further: Students use the magic square to
create another magic square. They divide each entry by
2, then check to see that all sums are the same.
Mathematical
Individual Activity
The work students do: Students play with the class.
Each student draws 1 card from the deck and finds
2 classmates with numbers that, when used with her
number, make a multiplication or a division sentence.
When a sentence is formed, students write the sentence
and their names on the board. The object of the game is to
be a part of as many multiplication and division sentences
as possible.
Take It Further: Each student takes 1 card. They are
given 1 minute to write all the multiplication and division
sentences they know that contain their number. The player
with the most sentences wins.
Mathematical/Social
Group Activity
Unit 4: Multiplication and Division
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Planning for Unit 4
Planning for Instruction
Lesson
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Unit 4: Multiplication and Division
Time
Suggested Unit time: About 2–3 weeks
Materials
Program Support
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Lesson
Time
Materials
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Program Support
Planning for Assessment
Purpose
Tools and Process
Recording and Reporting
Unit 4: Multiplication and Division
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LAUNCH
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Here Comes the Band!
LESSON ORGANIZER
10–15 min
Curriculum Focus: Activate prior learning about multiplication
(as repeated addition).
ASSUMED PRIOR KNOWLEDGE
✓ Students can identify equal groups.
✓ Students can add equal groups or skip count.
ACTIVATE PRIOR LEARNING
Invite students to imagine watching a marching
band coming down the street in a parade.
Direct their attention to the marching band on
page 145 of the Student Book.
Ask questions, such as:
• How many people are in the first row of
the band?
(5 people)
• Does each row in the band have the same
number of people?
(Yes, each row has 5 people.)
• How many rows of people are marching in
this band?
(There are 6 rows of people.)
• How many people are in this band? (30) How
did you find out?
(I counted by 5s until I had said 6 numbers.)
• How else can you find out how many people
are in the band?
(I can add; 5 + 5 + 5 + 5 + 5 + 5 = 30.)
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Unit 4 • Launch • Student page 144
Discuss how many different ways students can
find how many people are in the band. Listen
for students to explain whether they could
count all the band members, count by 5s, use
addition, use multiplication, or use a different
strategy.
Tell students that, in this unit, they will learn
how to multiply and divide whole numbers. At
the end of the unit, they will demonstrate what
they have learned by solving problems related
to a marching band in the Unit Problem.
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LITERATURE CONNECTIONS FOR THE UNIT
The Doorbell Rang by Pat Hutchins. Harper Trophy, 1989.
ISBN 0688052525
Victoria and Sam learn about division when they must share
their cookies with more and more guests.
REACHING ALL LEARNERS
Some students may benefit from using the virtual
manipulatives on the e-tools CD-ROM. The
e-Tools appropriate for this unit include Place-Value Blocks and
Counters. Students can use these blocks to create concrete
models of numbers. Students can manipulate the models when
multiplying and dividing.
DIAGNOSTIC ASSESSMENT
What to Look For
What to Do
✔ Students can identify
equal groups.
Extra Support:
✔ Students can add
equal groups or skip
count.
Students may benefit from drawing circles around equal groups and then counting
the circles drawn.
Work on this skill during Lessons 2 and 9.
Students who demonstrate minimal skip-counting skills can use a hundred chart to
practise skip counting.
Work on this skill during Lessons 3 and 4.
Unit 4 • Launch • Student page 145
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LESSON 1
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Relating Multiplication
and Addition
LESSON ORGANIZER
40–50 min
Curriculum Focus: Show that multiplication is repeated
addition. (N15)
Teacher Materials
hundred chart transparency (PM 13)
Student Materials
Optional
counters
Step-by-Step 1 (Master 4.12)
Snap Cubes
Extra Practice 1 (Master 4.25)
Vocabulary: multiplication sentence, times, equal groups
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
Key Math Learnings
1. Multiplication is the same as adding equal groups.
2. For every multiplication sentence, there is a corresponding
addition sentence.
BEFORE
Get Started
Use a hundred chart transparency (PM 13) on the
overhead projector. Start at 5 and count by 5s.
Ask:
• What patterns did you notice when counting
by 5s?
(There is always a 5 or a 0 in the ones place.)
• What is the addition sentence if I land on 15?
(5 + 5 + 5 = 15)
• How can you put this sentence into words?
(3 groups of 5 added equals 15.)
• How could you model this addition sentence
with counters?
(I could make 3 groups of 5 counters, then add the
counters to get 15.)
Present Explore. Provide students with counters
or Snap Cubes to represent the stickers.
Remind students that they should record their
work and share their results with a classmate.
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Unit 4 • Lesson 1 • Student page 146
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• How many stickers does Kia have?
(Kia has 15 stickers.)
• How did you find out?
(I added 5 + 5 + 5 to get 15.)
• How else could you find out?
(I could model the stickers with counters, then count.)
• What problem did you make up that is like
the sticker problem?
(Meagan has 4 tulips. There are 2 leaves on each
tulip. How many leaves are there altogether?)
• How can you solve this problem?
(I can add; 2 + 2 + 2 + 2 = 8. There are 8 leaves.)
Watch to see that students are making equal
groups of 5. Are they finding the total by
counting, adding, or skip counting by 5s?
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REACHING ALL LEARNERS
Alternative Explore
Materials: Pattern Blocks (blue rhombuses) (PM 25)
Have students determine how many vertices are on
3 blue rhombuses.
Early Finishers
Students can make up and solve other “sticker strip” problems.
Encourage students to think of other items that are available
in groups.
Numbers Every Day
Students should recognize that they can mentally add 9 to a
number by adding 10 to the number, then taking 1 away.
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16
24
37
41
1+1+1+1+1=5
51=5
AFTER
2+2+2+2=8
42=8
Connect
Invite students to share with the class the
strategies they used.
Use Connect to show students how the problem
can be solved by modelling with Snap Cubes,
by adding, and by multiplying.
Write 4 5 = 20 on the board. Tell students
this is a multiplication sentence. It means
that 4 groups of 5 = 20, or that 4 times 5 equals
20. Introduce the term equal groups. Tell
students that equal groups have the same
number of things in each group. When you
have equal groups, you can add or multiply to
find how many altogether.
Ask questions such as:
• How many equal groups of stickers does Kia
have? (3)
• How many stickers are in each equal group?
(5 stickers)
• What addition sentence represents
Kia’s stickers?
(5 + 5 + 5 = 15)
• What multiplication sentence represents
Kia’s stickers?
(3 5 = 15)
• Can you always write a multiplication
sentence if you have equal groups? (Yes)
• Can you write an addition sentence for any
multiplication sentence? (Yes)
Practice
Have counters or Snap Cubes available for
all questions.
Assessment Focus: Question 7
Students should recognize that the groups must
be equal in order to write a multiplication
sentence. Some students may recognize that an
addition sentence is possible.
Unit 4 • Lesson 1 • Student page 147
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Sample Answers
1. a) Yes, the groups are equal; 2 4 = 8
b) No, the groups are not equal.
c) Yes, the groups are equal; 3 2 = 6
5. 3 ways: 6 + 6 + 6 = 18, 6 3 = 18, or I can count all
the legs. (Students may count 8 legs; in this case, there would
be 24 legs altogether.)
6. a) 12 straws; 4 3 = 12
b) More straws; she needs 4 straws for 1 square, so for
4 squares she would need 4 4 = 16 straws.
7. You can write an addition sentence, 1 + 3 + 4 = 8, but you
cannot write a multiplication sentence because the groups are
not equal.
REFLECT: I can use a multiplication sentence to find how many
when I have equal groups. For example, to find how many
mittens are in 5 pairs of mittens, I find 5 2 = 10; there are
10 mittens.
3 7 = 21
2 5 = 10
6 + 6 = 12
3+3+3=9
61=6
7 2 = 14
1+1+1+1=4
18
12
More
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students can write and solve
multiplication sentences when
combining equal groups.
Extra Support: Provide students with a hundred chart (or a
number line) to help them skip count.
Students can use Step-by-Step 1 (Master 4.12) to complete
question 7.
Applying procedures
✔ Students can write a multiplication
sentence for an addition sentence with
equal addends, and an addition
sentence for a multiplication sentence.
Extra Practice: Students can use a symbol of their choice to
draw “equal group” pictures, then write and solve multiplication
sentences for the pictures drawn.
Students can complete Extra Practice 1 (Master 4.25).
Extension: Give pairs of students a calculator. Students take
turns to choose a number. On the calculator, students press + ,
then the number they chose, then = 7 times. Each time, students
predict the numbers that will be displayed.
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
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Unit 4 • Lesson 1 • Student page 148
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ESSON 2
Using Arrays
to Multiply
40–50 min
LESSON ORGANIZER
Curriculum Focus: Use arrays to multiply. (N15)
Teacher Materials
counters for the overhead projector
Student Materials
Optional
counters
Step-by-Step 2 (Master 4.13)
Extra Practice 1
(Master 4.25)
Vocabulary: array, factors, product
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
Key Math Learnings
1. When using arrays to model multiplication sentences, the first
19
21
factor tells the number of rows, and the second factor tells
how many are in each row.
2. Arrays can help show that changing the order of the factors
does not change the product.
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BEFORE
Get Started
Place 6 counters on the overhead projector.
Arrange them into 3 rows of 2 counters each.
Ask questions, such as:
• How many rows of counters are there? (3)
• How many are in each row? (2)
• How many counters are there altogether? (6)
Write 2 + 2 + 2 = 6 and 3 2 = 6 on the board.
Now rearrange the 6 counters into 2 rows of 3.
Ask:
• What is different about this arrangement?
(There are 2 rows of counters, with 3 counters in
each row.)
• What is the same?
(There are 6 counters.)
Present Explore. Distribute 12 counters to each
pair of students. Suggest that one student in
each pair makes an arrangement with the
counters while the other student records the
work. Students switch roles. Students continue
to switch roles until all possible arrangements
have been made.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• What is one way that you arranged the
counters?
(I arranged them in 1 row of 12 counters.)
• What is a multiplication sentence for that
arrangement?
(1 12 = 12) An addition sentence?
(1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 12)
• How else can you arrange the counters?
(2 rows of 6, 3 rows of 4, 4 rows of 3, 6 rows of 2,
and 12 rows of 1)
• How many different ways can you arrange
the counters? (6 ways)
Unit 4 • Lesson 2 • Student page 149
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REACHING ALL LEARNERS
Alternative Explore
Materials: Colour Tiles
Provide pairs of students with 12 Colour Tiles. Have them make
as many different rectangles as they can using all the tiles.
Early Finishers
Have students find all the ways they can to arrange 4 counters,
9 counters, and 16 counters into equal groups. Have students
record what they notice. For example, one of the arrangements
for each of these numbers will be a square array.
Common Misconceptions
➤Students make arrangements that do not have equal rows.
For example, they use 2 rows of 5 and 1 row of 2 to
represent 12.
How to Help: Show students they cannot write a multiplication
sentence for this arrangement. Have students place the counters
on a grid so the rows and columns “line up.”
Numbers Every Day
For 40 – 21, students could use 40 – 20 = 20, then take away
1 more. For 40 – 19, students could use 40 – 20 = 20, then add
1 more. For 42 – 20, students could use 40 – 20 = 20, then add
2 more.
2 6 = 12
15=5
42=8
• How do you know you have found all the
ways to arrange the counters?
(I cannot arrange the counters in any other way so
that the rows are equal.)
• What patterns did you find?
(I found 1 12 = 12 and 12 1 = 12; 2 6 = 12
and 6 2 = 12; 3 4 = 12 and 4 3 = 12.)
ask questions, such as:
• If the multiplication 4 6 describes an array,
what does the first factor tell you?
(How many rows there are) The second factor?
(How many are in each row)
• What is the product? (24)
Practice
AFTER
Connect
Invite volunteers to share how they kept track
of their arrangements. Ask the class for ways to
ensure they have listed all possible
arrangements. As students share their results,
make a list of the arrangements found.
Use Connect to introduce the term array to
describe objects arranged in equal rows. Write
the multiplication sentence 3 6 = 18 on the
board. Tell students that the numbers you
multiply, 3 and 6, are factors, and the answer,
18, is the product. To reinforce the vocabulary,
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Unit 4 • Lesson 2 • Student page 150
Questions 2, 3, and 4 require counters.
Assessment Focus: Question 7
Students should have an organized way of
determining all the possible arrays. Students
should recognize that if the rows are not equal,
it is not an array.
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Sample Answers
2. a)
b)
166
616
2 7 14
=6
=6
=7
=7
7 2 14
3. a)
b)
c)
d)
20
4.
5.
6.
36
7.
18
10
4
Switching the order of the factors
does not change the product.
5 4 = 20; there are 20 tomato plants altogether.
6 6 = 36 OR 4 6 = 24; 2 6 = 12; 24 + 12 = 36
The products are the same, but the order of the factors is
switched.
a) 4 ways; 6 1, 1 6, 2 3, 3 2
You cannot form equal groups if you try to put 4 or 5
counters in a row.
b) 2 ways; 1 7 and 7 1
These are the only arrangements that work. If you try to
make rows using any other number of counters, you end up
with counters left over or rows that are not equal.
REFLECT: A classroom has 4 rows of desks with 6 desks in each
row. How many desks are there altogether? To find 4 6, I
can make an array of 4 rows of 6; 4 6 = 24. There are 24
desks in all.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students can identify the factors
and the product in a multiplication
sentence.
Extra Support: Provide counters for students who wish to use
them in question 7. Students should write the repeated addition
sentence for each multiplication sentence. Students create the
model by looking at the addition sentence to see how many rows
there are and how many counters are in each row. Finally,
students count the counters. As students do more problems in this
manner, multiplication patterns will become more apparent.
Students can use Step-by-Step 2 (Master 4.13) to complete
question 7.
✔ Students recognize that the order of
the factors does not change the
product.
Applying procedures
✔ Students can write a multiplication
sentence for an array.
Extra Practice: Students can play the Additional Activity,
Amazing Arrays (Master 4.8).
Students can complete Extra Practice 1 (Master 4.25).
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
Unit 4 • Lesson 2 • Student page 151
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LESSON 3
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Multiplying by 2 and
by 5
LESSON ORGANIZER
40–50 min
Curriculum Focus: Use different strategies to multiply by 2
and by 5. (N15)
Teacher Materials
hundred chart transparency (PM 13)
number line transparency (Master 4.6)
Student Materials
Optional
2-cm grid paper (PM 21) Step-by-Step 3 (Master 4.14)
spinners
Extra Practice 2 (Master 4.26)
paper clips
number cubes (labelled 1 to 6)
counters
hundred charts (PM 13)
number lines (Master 4.6)
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
Key Math Learnings
1. Multiplying by 2 is the same as skip counting by 2s.
Multiplying by 5 is the same as skip counting by 5s.
2. There are many ways to model multiplication (number lines,
hundred charts, pictures, etc.).
BEFORE
Get Started
Present the game in Explore. Students should
make a product card using 2-cm grid paper. As
students complete their product cards, ask
questions, such as:
• What factors might be multiplied to give a
product of 12?
(3 and 4 or 2 and 6)
• Can you get both of these multiplication
sentences using the spinner and the
number cube?
(No, I cannot get 3 4 = 12 because the spinner
does not have a 3 or a 4.)
Tell students that although there are many
possible multiplication sentences for some of
the numbers on the product card, in this Lesson
we are only using multiplication sentences that
have 2 or 5 as one of the factors.
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Unit 4 • Lesson 3 • Student page 152
Suggest students hold the open paper clip in
place with the point of a pencil before
spinning. Remind students that they should
play the game a second time, this time with the
spinner giving the first factor and the number
cube giving the second factor.
DURING
Explore
Ongoing Assessment: Observe and Listen
Observe how students are finding the products.
Are they using counters, adding, or counting
by 2s or 5s? Do they seem to have any
facts memorized?
Ask questions, such as:
• What factors were multiplied to give a
product of 15?
(3 and 5)
• What other factors could have been
multiplied to give a product of 15?
(5 and 3)
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REACHING ALL LEARNERS
Alternative Explore
Have students write the numbers from 1 to 20 on grid paper.
Students place one number in each square and write only
2 numbers in each row. When they reach 20, students should
recognize that the numbers in the right-most squares of their lists
are multiples of 2. Students repeat this exercise, this time writing
five numbers in each row. Students should recognize that the
numbers in the right-most squares of their lists are multiples of 5.
Early Finishers
Students play the game in Explore. This time, the first person to
cover the entire product card wins.
Common Misconceptions
➤Students do not identify numbers ending in 0 as being a
product of 2.
How to Help: Have students use a number line and skip count
by 2. Students will see that numbers ending in 0 are part of the
skip-counting pattern and are therefore a product of 2.
• What do you notice about the product of 3
and 5 and the product of 5 and 3?
(The product of 3 5 is the same as the product
of 5 3.)
• What do you notice about the order of the
factors in a multiplication sentence?
(Order does not matter.)
• What strategy did you use to multiply by 5?
(I skip counted by 5s on a hundred chart.)
• What other strategy could you have used?
(I could have added. For example, to find 3 5, I
could have added 3 groups of 5; 5 + 5 + 5 = 15.)
AFTER
Connect
Introduce Connect by showing two numbers
lines on a transparency on the overhead
projector. On the first number line, model how
to multiply 6 2, by starting at 0 and counting
on by 2s six times: 2, 4, 6, 8, 10, 12
6 2 = 12
On the second number line, model how to
multiply 2 6, by starting at 0 and counting
on by 6s two times: 6, 12
2 6 = 12
Ask questions such as:
• What do you notice about the product of
6 2 and the product of 2 6?
(The products are the same.)
• Do you think the product of 3 4 and the
product of 4 3 will be the same?
(Yes, order does not matter when multiplying.)
• How would you model 4 2 on the
number line?
(I would do 4 jumps of 2.)
• How would you model 2 4 on the
number line?
(I would do 2 jumps of 4.)
• Do you end up at the same number in
both cases?
(Yes, I end up at 8.)
• Are the products the same? (Yes)
Unit 4 • Lesson 3 • Student page 153
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Sample Answers
2. The products are the numbers you get when skip counting by 2s.
3. Multiplying by 2 is like counting by 2s. You will always get
a number ending in 2, 4, 6, 8, or 0. You will never get an
odd number.
4. The products are the numbers you get when skip counting by 5s.
5. When 5 is multiplied by an odd number, the product ends in
a 5 and is odd.
When 5 is multiplied by an even number, the product ends in
a 0 and is even.
7. 6 2 = 12
0
1
2
3
4
5
6
7
8
9
10
11
2+7=9
9–2=7
5 + 6 = 11
11 – 5 = 6
9 + 8 = 17
17 – 8 = 9
4+3=7
7–4=3
12
9. Barbara: 4 children: 4 $2 = $8
2 groups of 5
2 5 = 10
2 adults: 2 $5 = $10
Total: $8 + $10 = $18
Carlos: 2 children: 2 $2 = $4
3 adults: 3 $5 = $15
Total: $4 + $15 = $19
$19 – $18 = $1
Carlos spent $1 more than Barbara.
1 group of 2
12=2
=2
=4
=6
=8
= 10
= 12
No
Numbers Every Day
=5
= 10
= 15
= 20
= 25
= 30
Remind students to think of both addition and subtraction facts
for each set of numbers.
Demonstrate multiplying by 5 on the hundred
chart. For example, to multiply 5 5, we can
skip count on a hundred chart. We start at 5,
then count on by 5s until we have counted
5 numbers: 5, 10, 15, 20, 25. 5 5 = 25
Explain that a hundred chart can help to
multiply by any factor, but that for this lesson
it is being used to show multiplication by
2 and by 5.
Tell students that another way to multiply by
2 is to use doubles. For example, to find
5 2, we can use doubles; double 5 is 10.
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Unit 4 • Lesson 3 • Student page 154
Practice
Have counters available for all questions.
Question 2 requires a hundred chart (PM 13).
Question 4 requires a number line (Master 4.6).
Assessment Focus: Question 9
Students correctly multiply the number of
children by $2.00 and the number of adults by
$5.00, then add correctly to find the total.
Students subtract the smaller total from the
larger total to find how much more Carlos paid.
7+2=9
9–7=2
6 + 5 = 11
11 – 6 = 5
8 + 9 = 17
17 – 9 = 8
3+4=7
7–3=4
Home
Yes
=8
= 30
= 14
= 14
=5
=6
= 35
=2
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REFLECT: For example, if I want to multiply 6 5, I could count
by 5s six times:
5, 10, 15, 20, 25, 30. So, 6 5 = 30
I could also use a number line and do six jumps of 5, or
I could use a hundred chart and count on by 5s until I have
counted 6 numbers.
I would use the same strategies to multiply by 2.
Making Connections
15 stamps
At Home: Be sure students recall that a nickel is equal to
5 cents. Finding the number of cents in a group of nickels means
counting by 5s.
Carlos
$1.00
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand the relationship
between skip counting by 2s and 5s
and multiplying by 2 and by 5.
Extra Support: Have students use counters, a number line,
and a hundred chart. Students use each method several times,
then decide which method helps them the most and is easiest
to use.
Students can use Step-by-Step 3 (Master 4.14) to complete
question 9.
Applying procedures
✔ Students can use number lines and
hundred charts to multiply by 2 and
by 5.
Extra Practice: Have students skip count by 2s to 50, then skip
count by 5s to 50. Students find the numbers common to both
lists and describe the pattern in these numbers.
Students can complete Extra Practice 2 (Master 4.26).
Extension: Students use the pattern they found in Extra Practice
to predict which numbers greater than 50 will be in both lists.
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
Unit 4 • Lesson 3 • Student page 155
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LESSON 4
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Multiplying by 10
40–50 min
LESSON ORGANIZER
Curriculum Focus: Use patterns to multiply by 10.
(N15, N20)(PR3)
Optional
Base Ten Blocks
Step-by-Step 4 (Master 4.15)
play money (coins) (PM 27) Extra Practice 2 (Master 4.26)
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
Student Materials
40
Key Math Learnings
1. Skip counting by 10 is the same as multiplying by 10.
2. Patterns and place value can be used to multiply numbers
by 10.
BEFORE
Get Started
Have students skip count by 10s to 100. Ask:
• How would you find how many things are
in 2 bundles of 10?
(I would skip count by 10s two times.)
• How would you find how many things are
in 3 bundles of 10?
(I would skip count by 10s three times.)
Present Explore. Explain that skip counting by
10s to 100 is the skill required to solve this
problem. Remind students that the problems
they make up should involve multiplying by 10.
• What strategy did you use to multiply by 10?
(I skip counted by 10s four times on a number line.)
• What other strategy could you have used?
(I could have modelled the candles with counters,
then counted the counters.)
• What multiplication sentence can you use to
find the total number of candles?
(I can use the multiplication sentence 4 10 = 40.)
• What similar problem did you make up?
(Jenna has 5 packages of hockey cards. There are
10 cards in each package. How many hockey cards
does Jenna have? (Answer: 50 cards; 5 10 = 50))
AFTER
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• What addition sentence can you use to find
the total number of candles? (I can use the
addition sentence 10 + 10 + 10 + 10 = 40.)
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Unit 4 • Lesson 4 • Student page 156
Connect
Have a pair of students share the problem they
made up with the class. Ask:
• Is this problem similar to the problem in
Explore? (Yes) In what way?
(It involves multiplying by 10.)
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REACHING ALL LEARNERS
Alternative Explore
Materials: play money (dimes) (PM 27)
Students can use dimes to help count by 10s. Have students
determine the value of a set of 4 dimes, 6 dimes, etc.
Early Finishers
Students can make a chart to show how coins are used to count
by 2s, 5s, and 10s. The chart should include illustrations.
Common Misconceptions
➤Students add instead of multiply. For example, students think
that 3 10 = 13.
How to Help: Remind students that when one factor is 10, the
product always has a zero in the ones place.
➤Students want to use pencil and paper instead of becoming
comfortable with multiplying by 10 mentally.
How to Help: Remind students that when multiplying by 10
mentally, the product has the factor that is not 10 in the tens
place and a 0 in the ones place.
ESL Strategies
3 10 = 30
5 10 = 50
2 10 = 20
Pennies, nickels, and dimes may be unfamiliar to some students.
Have students for whom English is a second language talk to the
class about the money used in their country of birth. Encourage
them to bring in some of their country’s coins for the class to see.
6 10 = 60
Introduce Connect and distribute Base Ten
Blocks. Ask:
• What does 1 rod represent? (10)
• How can we use Base Ten Blocks to multiply
5 10?
(We can use 5 rods to model the multiplication, then
skip count by 10s five times; 5 10 = 50.)
• How can we use a number line to multiply
5 10?
(We can start at 0, then count on by 10s five times:
10, 20, 30, 40, 50; 5 10 = 50.)
• What do you notice about the ones digit of
the product when we multiply by 10?
(The ones digit of the product is always 0.)
• What do you notice about the tens digit of the
product when we multiply a number by 10?
(The tens digit of the product is the same as the
number being multiplied by 10.)
• How can you use this pattern to multiply
8 10? 9 10?
(8 10 is 80 because the tens digit of the product is
8 and the ones digit is 0. The product of 9 10 has
a 9 as the tens digit and a zero as the ones digit. So,
9 10 = 90)
Practice
Encourage students to use number lines or
hundred charts until they are comfortable
multiplying by 10. Have play money (coins)
(PM 27) available for questions 4 and 5.
Assessment Focus: Question 4
Students recognize that they multiply by 10 to
find the value of a group of dimes, and by 5 to
find the value of a group of nickels. Students
should then add the results to get the total
value of the money.
Unit 4 • Lesson 4 • Student page 157
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Sample Answers
4. 5 dimes is: 5 10¢ = 50¢
=50
6 nickels is: 6 5¢ = 30¢
50¢ + 30¢ = 80¢
5. b) 10 cups will cost: 10 4¢ = 40¢
35¢ will not be enough to buy 10 cups because 35¢ < 40¢.
6. Party invitations come in packages of 10. How many
invitations will there be in 3 packages?
(Answer: 3 10 = 30; there will be 30 invitations.)
7.
=70
=10
=60
=10
=20
=40
=30
80¢
1 20 20
2 10 20
4 5 20
5 4 20
60¢
20¢
20 1 20
50¢
10¢
10 2 20
No
REFLECT: When you multiply a number by 10, you write the
number multiplied by 10 in the tens digit of the product, and
you write a 0 in the ones digit of the product.
Numbers Every Day
To find 5 + 3, students could
4 + 3 + 1 = 4 + 4 = 8.
To find 2 + 9, students could
1 + 9 + 1 = 1 + 10 = 11.
To find 6 + 4, students could
5 + 4 + 1 = 5 + 5 = 10.
To find 7 + 5, students could
6 + 5 + 1 = 6 + 6 = 12.
take from one to give to the other:
take from one to give to the other:
take from one to give to the other:
take from one to give to the other:
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that multiplying
by 10 is the same as skip counting
by 10.
Extra Support: Have students use Base Ten rods to model a
number, then work backward to write its multiplication sentence.
Students can use Step-by-Step 4 (Master 4.15) to complete
question 4.
Applying procedures
✔ Students can use patterns to mentally
multiply numbers by 10.
Extra Practice: Students play in pairs. Each student has a set
of digit cards numbered from 0 to 9. Students shuffle the cards
and place them face down on the table. Each student turns over
1 card at the same time. The first player to correctly say the
product of her or his card and 10 scores a point. Play continues
until all cards have been turned over.
Students can complete Extra Practice 2 (Master 4.26).
Communicating
✔ Students can explain how dimes are
related to multiplying by 10.
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
16
Unit 4 • Lesson 4 • Student page 158
8
11
10
12
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ESSON 5
Multiplying by 1 and
by 0
40–50 min
LESSON ORGANIZER
Curriculum Focus: Use patterns to multiply by 1 and by 0.
(N15, N20)(PR3)
Optional
paper plates
Step-by-Step 5 (Master 4.16)
counters
Extra Practice 3 (Master 4.27)
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
Student Materials
Key Math Learnings
5
1. When 1 is a factor, the product is always the other factor.
2. When 0 is a factor, the product is always 0.
0
Numbers Every Day
4
40
400
BEFORE
Get Started
Have four volunteers stand at the front of
the class.
Ask the class:
• How many eyes can you count on these
students? (8)
• What multiplication sentence can you write
to show this?
(4 2 = 8)
• How many noses can you count on these
students? (4)
• What multiplication sentence can you write
to show this?
(4 1 = 4)
• How many tails can you count on these
students? (0)
• What multiplication sentence can you write
to show this?
(4 0 = 0)
Students recognize that 9 – 5 = 4. They use this fact and
place value to help them find 9 tens – 5 tens = 4 tens, and
9 hundreds – 5 hundreds = 4 hundreds.
Present Explore. Distribute paper plates and
counters to each student. Tell students that a
counter will represent a waffle.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask:
• How many waffles does Mark make? (5)
• How many waffles (counters) will you put
on each plate? (1)
• How can you use a multiplication sentence
to show this? (5 1 = 5)
• How many empty plates does Mark have? (3)
How many waffles are on these plates? (0)
• How can you use a multiplication sentence
to show this?
(3 0 = 3)
Unit 4 • Lesson 5 • Student page 159
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REACHING ALL LEARNERS
Alternative Explore
Materials: pennies
Students determine the value of a given number of pennies. For
example, 6 pennies is 6 1¢ = 6¢.
Students also determine, for example, that if 3 children each
have 0 pennies, they have 0 pennies altogether.
Common Misconceptions
➤Students confuse multiplying by 0 and adding 0. For example,
they might say 3 0 = 3.
How to Help: Use counters to model how many candies are in
5 empty bags, or how much money is in 3 empty pockets.
Students should soon recognize that when 0 is a factor, the
product is always 0.
=0
AFTER
Connect
Invite volunteers to show what they noticed
about multiplying by 1 and by 0.
Ask:
• What is special about multiplying by 1?
(When 1 is a factor, the product is always the
other factor.)
• What is special about multiplying by 0?
(When 0 is a factor, the product is always 0.)
Use Connect to look at multiplying other factors
by 1 and 0. Some students may confuse
multiplying by 1 and by 0 with adding 1 and
adding 0. For example, the fact 4 + 0 stays the
same, but 4 0 is always 0. When adding 4 + 1,
students add 1 more, but 4 1 stays the same.
Write 4 0 on the board. Tell students they
should think about 4 groups of 0. Four groups
of 0 is always 0; 0 + 0 + 0 + 0 = 0. Write 4 1
on the board. Tell students they should think
18
Unit 4 • Lesson 5 • Student page 160
=5
=0
=3
about 4 groups of 1. Four groups of 1 is 4;
1 + 1 + 1 + 1 = 4.
Ask:
• What is 358 0? (0) How do you know?
(When 0 is a factor, the product is always 0.)
• What is 525 1? (525) How do you know?
(When 1 is a factor, the product is always the
other factor.)
Practice
Have counters available for all questions.
Assessment Focus: Question 6
Students start with one strawberry, then multiply
by 2 to find the number of raspberries. Students
then multiply the number of raspberries by 3 to
find the number of blueberries. Some students
might extend the problem and record the total
number of pieces of fruit on the waffle. Students
should explain their thinking.
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Sample Answers
2. Jessica buys 6 scoops of ice cream; 6 1 = 6.
6
0
30=0
1
0
1
0
2
6
5. Cathy has 4 empty piggy banks. How much money does
Cathy have in her piggy banks? (Answer: 4 0 = 0;
Cathy has no money in her piggy banks.)
6. I know there are twice as many raspberries as strawberries.
Since there is 1 strawberry, there are 2 raspberries;
2 1 = 2. I know there are three times as many blueberries
as raspberries. Since there are 2 raspberries, there are
6 blueberries; 3 2 = 6.
7. a) It is easier to solve 24 1 because when 1 is a factor, the
product is always the other factor, 24.
b) They are both easy to solve. When 0 is a factor, the
product is always 0. In both cases, the product is 0.
REFLECT: When you multiply by 0, the product is always 0; for
example, 73 0 = 0. When you multiply by 1, the product is
always the other factor; for example, 73 1 = 73.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that when 1 is
a factor, the product is always the
other factor.
Extra Support: Students can use different coloured cubes to
represent the strawberries, raspberries, and blueberries in
Practice question 6.
Students can use Step-by-Step 5 (Master 4.16) to complete
question 6.
✔ Students understand that when 0 is a
factor, the product is always 0.
Applying procedures
✔ Students can use patterns to mentally
multiply by 1 and by 0.
Extra Practice: Students play in pairs. Students take turns to
roll a number cube, then multiply the number rolled by 1.
Students get 1 point for each correct answer. The player with the
most points after 1 minute wins. Students play again, this time
multiplying the number rolled by 0.
Students can complete Extra Practice 3 (Master 4.27).
Extension: Students write multiplication sentences where 2-digit
and 3-digit numbers are multiplied by 0 and by 1.
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
Unit 4 • Lesson 5 • Student page 161
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LESSON 6
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Using a Multiplication
Chart
40–50 min
LESSON ORGANIZER
Curriculum Focus: Use patterns in a multiplication chart to
multiply. (N16)(PR2, PR3)
Teacher Materials
transparency of the Explore Multiplication Chart (Master 4.7)
Student Materials
Optional
multiplication charts (PM 15) Step-by-Step 6 (Master 4.17)
Explore Multiplication
Extra Practice 3 (Master 4.27)
Chart (Master 4.7)
decks of playing cards
spinners
2-cm grid paper (PM 21)
counters
calculators
Vocabulary: row, column
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
9 12
12 16
18 21 24 27
24 28 32 36
18 24
21 28
24 32
27 36
36
42
48
54
42 48 54
49 56 63
56 64 72
63 72 81
Key Math Learnings
1. A variety of patterns and strategies can be used to help
recall the basic multiplication facts.
2. Changing the order of the factors in a multiplication sentence
does not change the product.
BEFORE
Get Started
Have students recall some of the patterns they
saw in a hundred chart.
Ask questions, such as:
• What is one pattern that you saw in a
hundred chart?
(I saw that counting by 2s gave all even numbers.)
• What is another pattern that you saw?
(I saw that counting by 5s gave numbers ending
in a 5 or a 0.)
• How do we use skip counting when we
multiply?
(For example, when we multiply 6 2, we start at
0 and count by 2s six times;
2, 4, 6, 8, 10, 12. So, 6 2 = 12)
Present Explore. Distribute copies of the Explore
Multiplication Chart (Master 4.7) to pairs of
students. Encourage students to use patterns to
fill in the missing products.
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Unit 4 • Lesson 6 • Student page 162
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• What row has the same number in
every square?
(The row for 0)
• Is there a column with the same number in
every square?
(Yes, the column for 0)
• What do you notice about the row and the
column for 1?
(The numbers start at 0 and increase by 1 each time.)
• What do you notice about the row and the
column for 2?
(The numbers start at 0 and increase by 2 each time.)
• How did you complete the chart?
(I used patterns. For example, to complete the row
for 8, I added 8 to the previous number each time.)
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REACHING ALL LEARNERS
Common Misconceptions
➤Students think that the order of the factors in a multiplication
sentence affects the product.
How to Help: Use counters. Make an array to model 4 3 and
an array to model 3 4. Students total the counters in each
array to see that they have the same number of counters.
Students should now “see” that order does not matter.
Early Finishers
Students continue the patterns in the multiplication chart to
extend the chart beyond 9 9.
• What do you notice about the rows and
columns of the chart?
(For each factor, the matching row and column are
the same.)
Watch how students complete the chart. Do
they use skip counting? Do they use facts that
they are familiar with to help them with other
facts?
AFTER
Connect
Have students share the strategies they used to
find the products with the class. Have them
share any patterns they found. Ensure students
understand that rows go horizontally across
the page and that columns go vertically up
and down.
Use a transparency of the Explore Multiplication
Chart (Master 4.7) on the overhead projector.
Complete the chart together with the class so
that students can check their own charts.
Ask questions, such as:
• Which column has the same numbers as the
row for 3?
(The column for 3)
• Which rows (columns) have only even
numbers?
(The rows and columns for 0, 2, 4, 6, and 8)
Use Connect to introduce some of the patterns in
a 7 7 multiplication chart. Tell students that
the row and the column for the same factor
have the same numbers.
Point out that to fill in a column or a row,
students can skip count by the first number in
the column or row. For example, to fill in the
row or column for the factor 3, start at 0, then
count on by 3s: 0, 3, 6, 9, 12, 15, 18, 21.
Tell students they can use doubles to help them
complete the chart. For example, to find 6 6,
students can find 3 3, then double it. This
works because 6 is double 3.
Unit 4 • Lesson 6 • Student page 163
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Sample Answers
1. The numbers that are products for both the factor 2 and factor
3 are the products for the factor 6.
4. There are 7 days in 1 week. So, in 4 weeks there are
4 7 = 28 days.
5. a) ♥ = 10; I counted on from 8 by 2.
Ο = 9; I counted on from 6 by 3.
∆ = 8; I counted on from 6 by 2.
" = 15; I counted on from 10 by 5.
b) 10 + 9 = 19
c) 8 + 9 = 17
d) ∆, Ο, ♥, " (8, 9, 10, 15)
6.
The multiplication sentences for my
design are: 2 2 = 4,
2 3 = 6, 2 4 = 8,
3 3 = 9, 4 2 = 8,
4 3 = 12, 4 4 = 16
= 18
=9
= 18
= 36
28
19
17
The winner is the player with the most cards at the end of
the stated time.
Practice
Have multiplication charts (PM 15) available
for all questions.
Question 7 requires a deck of cards with the
8s, 9s, 10s, and face cards removed.
22
Unit 4 • Lesson 6 • Student page 164
= 35
= 42
24
7. Note: You might also play the game with a time limit.
Ask:
• What do you notice about the product of
2 7 and the product of 7 2?
(I notice that in both cases, the product is 14.)
• What do you notice about the product of
1 4 and the product of 4 1?
(I notice that in both cases, the product is 4.)
• What does this tell you about multiplication?
(It tells me that the order of the factors does not
matter. This is why the row and the column for the
same factor have the same numbers.)
= 35
= 49
∆, Ο, ♥, "
Question 8 requires a spinner (8 equal sections
numbered from 0 to 7), a paper clip as a
pointer, 2-cm grid paper (PM 21), and counters.
Assessment Focus: Question 5
Some students could use patterns in the
multiplication chart to determine what each
figure represents, while others might simply
use multiplication facts. Students then add to
find the required sums. Students could use
place value to order the numbers from least
value to greatest value.
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REFLECT: If I wanted to find 3 6 on a multiplication chart, I
would find where the row for the factor 3 and the column for
the factor 6 meet. They meet at the number 18;
3 6 = 18.
Numbers Every Day
Encourage students to experiment with their calculators.
Students could find several ways to display 68 using only
the numbers 3 and 5. For example, students could add
33 + 35, or they could use both multiplication and addition;
3 3 5 + 5 + 5 + 5 + 5 + 3.
33 + 35 = 68
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that a variety of
patterns and strategies can be used to
help recall the basic multiplication facts.
Extra Support: Allow students to use a completed copy
of the multiplication chart when playing the games in
Practice questions 7 and 8.
Students can use Step-by-Step 6 (Master 4.17) to complete
question 5.
✔ Students understand that changing the
order of the factors in a multiplication
sentence does not change the product.
Applying procedures
✔ Students can use patterns in a
multiplication chart to multiply.
Communicating
✔ Students can explain how to use a
chart to multiply.
Extra Practice: Students can play the Additional Activity,
Multiplication Tag (Master 4.9).
Students can complete Extra Practice 3 (Master 4.27).
Extension: Students combine the games in Practice questions 7
and 8. Students play in pairs. One student turns over two cards
and finds the product. The other student spins the spinner and
finds the product of the number and itself. The player with the
greater product scores a point. Play for a set amount of time.
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
Unit 4 • Lesson 6 • Student page 165
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LESSON 7
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Strategies Toolkit
40–50 min
LESSON ORGANIZER
Curriculum Focus: Interpret a problem and select an
appropriate strategy. (N16)
Student Materials
6
2-column charts (PM 17)
Assessment: PM 1 Inquiry Process Check List, PM 3 SelfAssessment: Problem Solving
Key Math Learning
Making a table is a good strategy to help solve many problems.
Sample Answers
12
1. 8 different snacks: celery and apple; carrot and apple;
celery and kiwi; carrot and kiwi; celery and pear; carrot
and pear; celery and orange; carrot and orange
2. Menus will vary.
Foods
Drinks
hot dog
milk
hamburger
orange juice
chicken nuggets
apple juice
pizza
BEFORE
Get Started
Present Explore. Students can draw pictures or
use models to help them find the different
combinations of pants and T-shirts.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• How many different outfits can Karlee make
with 1 T-shirt?
(Two, she can wear the T-shirt with 2 different pairs
of pants.)
• How many T-shirts does Karlee have? (3)
• How many different outfits can Karlee
make? (6) How do you know?
(3 2 = 6)
• How did you keep track of the outfits you
made?
(I made a table.)
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Unit 4 • Lesson 7 • Student page 166
AFTER
Connect
Have students share their solutions and
strategies with the class.
Work through the problem in Connect. Ask:
• What are the headings in the table?
(Colour and bike)
• How many choices does Ben have for a
blue bike? (3)
• How many colour choices does Ben have? (4)
• How can multiplication help you solve
this problem? (I can multiply the number of bikes
by the number of colours; 3 4 = 12. Ben can
choose 12 different bikes.)
• What other strategy could you use to solve
this problem? (I could draw a picture.)
Practice
Encourage students to refer to the Strategies
list. Have 2-column charts (PM 17) available
for all questions.
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REACHING ALL LEARNERS
Early Finishers
Have students repeat Explore. This time Karlee has 4 T-shirts and 3
pairs of pants. How many different outfits can Karlee now make?
(Answer: 12)
Common Misconceptions
➤Students use combinations of items from the same category. For
example, in Practice question 1, students choose 2 vegetables.
How to Help: Encourage students to make a table, listing all
fruits in one column and all vegetables in a second column.
Students can only choose 1 item from each column.
8
12 different meals are possible: hot dog and milk; hot dog
and apple juice; hot dog and orange juice; hamburger and
milk; hamburger and apple juice; hamburger and orange
juice; chicken nuggets and milk; chicken nuggets and apple
juice; chicken nuggets and orange juice; pizza and milk;
pizza and apple juice; pizza and orange juice.
REFLECT: Making a table helps me to organize my work and to
make sure that I do not miss any combinations of items. I can
solve a problem without completing a table by drawing
pictures. I would draw a celery stick with each fruit and then a
carrot with each fruit to do Practice question 1.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Problem Solving
✔ Students can interpret a problem
involving multiplication.
Extra Support: In Practice questions 1 and 2, give students a
partially completed table. Have students fill in the table to find
how many snacks Zakia can make.
✔ Students can organize their work in a
table to help solve a problem.
Extra Practice: Students write their own “combination”
problems. Students trade problems with a classmate and solve
their classmate’s problem.
Communicating
✔ Students can describe their strategy
clearly, using appropriate language.
Extension: Challenge students to repeat Practice question 1,
this time adding the drink category. For the drink, Zakia can
choose milk or orange juice. How many different combinations of
1 fruit, 1 vegetable, and 1 drink can Zakia make?
(Answer: 16)
Recording and Reporting
PM 1: Inquiry Process Check List
PM 3: Self-Assessment: Problem Solving
Unit 4 • Lesson 7 • Student page 167
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LESSON 8
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Modelling Division
40–50 min
LESSON ORGANIZER
Curriculum Focus: Use grouping and sharing to divide. (N15)
Teacher Materials
counters for the overhead projector
Student Materials
Optional
counters
Step-by-Step 8 (Master 4.18)
Extra Practice 4 (Master 4.28)
Vocabulary: division sentence
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
3
5
Key Math Learnings
1. Grouping and sharing can be used to divide.
2. To divide is to separate into equal parts.
BEFORE
Get Started
Write the multiplication sentence 3 6 = 18 on
the board. Remind students that this sentence
tells us there are 3 groups, with 6 in each
group, for a total of 18. Use counters to model
this sentence on the overhead projector.
Tell students that in this lesson, we will work
in the opposite order. We will begin with a
total and make equal groups.
Present Explore. Distribute 18 counters to each
pair of students. Remind students that they are
to record their work with pictures and numbers.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• When you arrange 18 counters into groups of
6, how many groups are there?
(There are 3 groups.)
26
Unit 4 • Lesson 8 • Student page 168
How did you find out?
(I made piles of 6 counters until I had no
counters left.)
• When you arrange 15 counters into 3 equal
groups, how many counters are in each
group? (5) How did you find out?
(I made 3 groups. I put one counter in each group,
then continued to add 1 counter to each group until
all of my counters were used.)
AFTER
Connect
Invite volunteers to share their work and the
strategies they used with the class.
Use Connect to introduce how we use grouping
and sharing to divide.
Use counters on the overhead projector to
model division by grouping. Use 16 counters.
Tell students that we want to arrange the
counters into groups of 2. How many groups
will there be? Put 2 counters in each group.
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REACHING ALL LEARNERS
Alternative Explore
Materials: 18 Snap Cubes
Students use Snap Cubes to find how many 6-cube towers they
can make from 18 cubes.
Students then find how many towers of equal height they can
make from 21 cubes.
Early Finishers
Have students write “equal group” questions. Students trade
questions with a classmate and solve each other’s question.
Common Misconceptions
➤Students arrange the counters into unequal groups.
How to Help: Demonstrate how to distribute the counters into
each group, one at a time, until all counters have been used.
6 groups of 1
61=6
5 groups of 4
20 4 = 5
Count how many groups. There are 8 groups.
Write 16 2 = 8 on the board. Tell students
this is a division sentence. We say, “16
divided by 2 is 8.”
Ask:
• What does 16 represent in the division
sentence?
(The total number of counters)
• What does 2 represent?
(The number of counters in each group)
The 8? (The number of groups)
Tell students that now we want to arrange the
counters into 4 equal groups. How many
counters will be in each group? Share the
counters among 4 groups. Count how many
counters are in each group. There are 4 groups
of 4. We write: 16 4 = 4. We say, “16 divided
by 4 is 4.” This is division by sharing.
Ask:
• What does 16 represent in the division
sentence?
(The total number of counters)
• What does the first 4 represent?
(The number of groups) The second 4?
(The number of counters in each group)
• What are the two kinds of division problems?
(Grouping and sharing)
Practice
Have counters available for all questions.
Assessment Focus: Question 4
Students recognize that to make equal groups
of more than 1 stamp, Omar can put 3, 7, or 21
stamps in each group. Students understand that
to make the greatest number of groups, they
would put the least number of stamps into
each group.
Unit 4 • Lesson 8 • Student page 169
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Sample Answers
4. Omar can make 1, 3, 7, or 21 groups with 21 stamps. If he
makes 21 groups, each group would only have 1 stamp and
this cannot be. So, the greatest number of groups Omar can
make is 7 groups. There would be 3 stamps in each group;
21 7 = 3.
5. a) Rachel has 20 marbles. She shares them equally among
4 people. How many marbles will each person get?
(Answer: 20 4 = 5; each person will get 5 marbles.)
b) Each page of a photo album holds 4 pictures. How many
pages are needed to hold 24 pictures?
(Answer: 24 4 = 6; 6 pages are needed to hold 24 pictures.)
REFLECT: There are two kinds of division problems: sharing and
grouping.
An example of a sharing problem is: Three people share
12 cookies. How many cookies does each person get?
(Answer: 12 3 = 4; each person gets 4 cookies.)
An example of a grouping problem is: Six bottles are
placed in a carton. How many cartons do you need to hold
18 bottles? (Answer: 18 6 = 3; you need 3 cartons.)
7 in the group
7 in each group
3 in each group
14 2 = 7
93=3
71=7
4
7
3
Numbers Every Day
In the first pair, students understand that 6 groups of 3 are
greater than 6 groups of 2. In the second pair, students
understand that order does not matter when multiplying, so the
products are equal. In the third pair, students understand that
when 1 is a factor, the product is always the other factor and, in
this case, 5 > 4. In the last pair, students understand that when 0
is a factor, the product is always 0; the products are equal.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that grouping
and sharing can be used to divide.
Extra Support: Provide students with counters and containers.
This enables students to use concrete locations when sharing is
called for. Students can distribute the counters equally among the
containers to determine how many are in each group.
Students can use Step-by-Step 8 (Master 4.18) to complete
question 4.
✔ Students understand that to divide is
to separate into equal parts.
Applying procedures
✔ Students can use grouping and
sharing to divide.
Extra Practice: Have students find how many different ways
they can make equal groups from 24 counters.
Students can complete Extra Practice 4 (Master 4.28).
Communicating
✔ Students can describe two kinds of
division problems.
Extension: Have students write a story problem for each part of
Practice question 2.
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
28
Unit 4 • Lesson 8 • Student page 170
equal
equal
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ESSON 9
Using Arrays to Divide
LESSON ORGANIZER
40–50 min
Curriculum Focus: Use arrays to divide. (N15)
Student Materials
counters for the overhead projector
Student Materials
Optional
counters
Step-by-Step 9 (Master 4.19)
Extra Practice 4 (Master 4.28)
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
Key Math Learning
Arrays can be used to model division.
5
Numbers Every Day
Use counters, buttons, or Base Ten Blocks to model each
description.
3
0
20
6
30
BEFORE
Get Started
Invite six volunteers to come to the front of the
class. Have them line up in equal rows. Ask:
• How are your classmates arranged?
(They are arranged in 1 row of 6.)
• How else could they arrange themselves in
equal rows?
(They could arrange themselves in 2 rows of 3, 3
rows of 2, or 6 rows of 1.)
Present Explore. Distribute 16 counters to pairs
of students. Tell students they should let 1
counter represent 1 child.
DURING
Explore
Ongoing Assessment: Observe and Listen
•
•
•
•
(I can make 1 row of 16, 2 rows of 8, 4 rows of 4,
8 rows of 2, or 16 rows of 1.)
How many different ways are there altogether?
(There are 5 different ways.)
What is the division sentence for 8 rows of 2?
(16 8 = 2)
What does the division sentence 16 1 = 16
mean?
(It means that 16 children can be arranged in 1
row, with 16 children in the row.)
Can you make rows of 3 children?
(No, the rows would not be equal. There would be 5
children in each row, with 1 child left over.)
Watch as students work. Do they recognize
that they can turn an array to make another
array? Do they approach the problem in an
organized way?
Ask questions, such as:
• How can you arrange the children in
equal rows?
Unit 4 • Lesson 9 • Student page 171
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REACHING ALL LEARNERS
Alternative Explore
Materials: Colour Tiles
Give students 18 Colour Tiles. Students make all possible
rectangles using all 18 tiles.
(Answer: 1 by 18, 2 by 9, 3 by 6, 6 by 3, 9 by 2, 18 by 1)
Early Finishers
Students find all numbers between 2 and 16 that can be arranged
in equal rows in only 2 ways. (Answer: 2, 3, 5, 7, 11, 13)
Common Misconceptions
➤Students make equal rows but they do not use all of their
counters.
How to Help: Have students add the counters in their array. Tell
students that this total must match the total number of objects in
the question.
Sample Answers
3. a)
b)
c)
d)
6
5. a)
b)
AFTER
Connect
Invite volunteers to share their answers and
their division sentences.
Use Connect to show students how an array can
be turned to make another array. In each case,
the number of rows becomes the number of
columns, and the number of columns becomes
the number of rows.
Ask questions, such as:
• How do we know all possible arrays for
8 have been shown?
(These are all the factors that have a product of 8.
Any other number of rows would not be of
equal length.)
• What does the division sentence
8 2 = 4 mean?
(It means that when 8 dancers are arranged in
2 rows, there are 4 dancers in each row.)
30
5
Unit 4 • Lesson 9 • Student page 172
• What do you notice about the number of
rows and the number in each row?
(The number of rows multiplied by the number in
each row is always 8.)
Practice
Have counters available for all questions.
Assessment Focus: Question 6
Students could try to arrange the drummers
and horn players separately into equal rows, or
they could add the drummers and horn players,
then try to form equal rows. Students who add
first will have to work with the factors of 27,
and this goes beyond the 7 7 multiplication
chart. Students should show all possible ways
to arrange the musicians in equal rows.
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6. a) No, I cannot make an array of equal rows of 2 to show 27.
There would be one person left over.
b) Yes; they can form 9 rows of 3.
c) They can form 27
18 6 = 3
3
16 4 = 4
63=2
3
2
rows of 1, 9 rows
of 3, 3 rows of 9,
or 1 row of 27.
6
4
7. There are 30 students in class. They sit in 6 equal rows. How
6
many students are in each row?
(Answer: 30 6 = 5; there are 5 students in each row.)
REFLECT: An array helps show equal groups. I model the objects
with counters and arrange the counters in equal rows. From
my arrangement, I can make a division sentence.
For example, 12 4 = 3 can be thought of as 12 objects
divided into 4 equal rows, with 3 objects in each row.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that arrays can
be used to model division.
Extra Support: Provide students with grid paper on which to
make their arrays. This should make it easier for students to see
when the rows are equal and when equal rows are not possible.
Students can use Step-by-Step 9 (Master 4.19) to complete
question 6.
Applying procedures
✔ Students can write a division sentence
for a given array.
✔ Students can use arrays to model
division.
Extra Practice: Students make an array for each of the division
sentences they completed in Practice question 4.
Students can complete Extra Practice 4 (Master 4.28).
Extension: Have students list places where people or objects
are arranged in arrays. Have them think about why they are
arranged this way. For example, egg cartons are arranged in 2
rows of 6 eggs.
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
Unit 4 • Lesson 9 • Student page 173
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LESSON 10
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Dividing by 2, by 5,
and by 10
LESSON ORGANIZER
40–50 min
Curriculum Focus: Identify numbers that can be divided by
2, by 5, or by 10. (N12)
Teacher Materials
counters for the overhead projector
Student Materials
Optional
counters
Step-by-Step 10 (Master 4.20)
hundred charts (PM 13)
Extra Practice 5 (Master 4.29)
Venn diagrams (PM 28)
Vocabulary: divisible
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
6
3
15
No
No
No
Key Math Learnings
1. A number is divisible by 2, by 5, or by 10 if that number of
counters can be divided into 2, 5, or 10 equal groups
respectively.
2. Arrays can be used to model division.
BEFORE
Get Started
Use counters on the overhead projector to
review division by sharing.
Draw 3 large squares on an overhead
transparency and write the name of a student
in each square. Have a volunteer suggest a
way to share the 12 counters equally among
the 3 students.
Ask:
• How many counters does each student get? (4)
• What division sentence represents this way
of sharing?
(12 3 = 4)
• How could we share the counters equally
among 2 students?
(Each student would get 6 counters.)
• Can we share the counters equally among 5
students? (No) Why not?
(12 is 5 groups of 2, with 2 left over.)
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Unit 4 • Lesson 10 • Student page 174
Present Explore. Distribute 31 counters to pairs
of students. Remind students to draw arrays to
show their work.
DURING
Explore
Ongoing Assessment: Observe and Listen
Watch to see how students work. Do they
distribute more than 1 counter to each group
at a time? Do they start over again to make
10 equal groups, or do they split each of the
5 equal groups in half?
Ask questions, such as:
• Suppose there are 30 children. How many
children will be in each group if there are
5 equal groups? (6) 10 equal groups?
(3) 2 equal groups? (15)
• How did you find how many children were
in each equal group? (I modelled the children
with counters, then placed 1 counter in each group
until all my counters were gone.)
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REACHING ALL LEARNERS
Alternative Explore
Materials: linking cubes
Students use 30 linking cubes to make rectangles with 2 equal
rows, 5 equal rows, and 10 equal rows. Students find how many
cubes are in each row. Students then use 31 linking cubes to try
to make rectangles with 2 equal rows, 5 equal rows, and 10
equal rows. Students find they always have 1 cube left over.
Early Finishers
Students find numbers that are divisible by 2, by 5, and by 10.
(Answer: 10, 20, 30, 40, etc.)
Common Misconceptions
➤When making groups with counters, students do not leave
enough room between the groups, and they lose track of how
many groups they have.
How to Help: Provide students with small containers, cups, or
paper plates so they have concrete locations in which to
distribute their counters.
Numbers Every Day
Students should recognize that the first factor remains the same
but that the second factor is double. Students use doubling as a
strategy. Since 3 4 is double 3 2, 3 4 = 6 + 6 = 12.
• Suppose there are 31 children. Can you
make 5 equal groups? (No) Why not?
(31 is 5 groups of 6, with 1 left over.)
• Can you make 10 equal groups from
31 children? (No) Why not?
(31 is 10 groups of 3, with 1 left over.)
• Can you make 2 equal groups from
31 children? (No) Why not?
(31 is 2 groups of 15, with 1 left over.)
AFTER
Connect
Invite students to share their answers and the
strategies they used. If any students said they
used multiplication, have them explain their
strategy to the class.
Ask:
• How did you know you could not make
equal rows with 31 children?
(I always had 1 counter left over.)
Use Connect to review modelling groups with
arrays.
Place 10 counters on the overhead projector.
Have volunteers make 5 equal groups, 2 equal
groups, and 10 equal groups.
Ask:
• How many counters will be in each group if
there are 5 equal groups? (2) 2 equal groups?
(5) 10 equal groups? (1)
• If there are 5 rows of 2, what is the division
sentence?
(10 5 = 2)
• From our lesson on using arrays to multiply,
what else does this array tell you?
(It tells me that 5 2 = 10.)
Tell students, for example, that we say a
number is divisible by 5 if that number of
counters can be divided into 5 equal groups.
We can divide 10 into 5 equal groups of 2, so
10 is divisible by 5. Because we can also divide
Unit 4 • Lesson 10 • Student page 175
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Sample Answers
5. a)
b)
4
c)
7
5
d)
6
3
1
3
7
9
7
4
6. There are 3 baskets of strawberries, with 10 strawberries in
each basket; 3 10 = 30. There are 30 strawberries
altogether. To share 30 strawberries among 5 people, each
person would get 6 strawberries; 30 5 = 6.
=6
=3
=1
=7
=4
=5
=2
=2
7
7. a)
6
b) The pattern in the ones digits is:
2, 4, 6, 8, 0, 2, 4, 6, 8, 0, . . .
c) A number is divisible by 2 if it ends in 2, 4, 6, 8, or 0.
10 into 2 equal groups of 5 and into 10 equal
groups of 1, 10 is also divisible by 2 and by 10.
Place 11 counters on the overhead projector.
Have volunteers try to make 5 equal groups,
2 equal groups, and 10 equal groups.
Ask:
• Can we make 5 equal groups with
11 counters?
(No, 11 is 5 groups of 2, with 1 left over.)
• Is 11 divisible by 5? (No) Why not?
(I cannot make 5 equal groups.)
• Is 11 divisible by 2? (No) By 10? (No)
Why not? (I cannot make 2 equal groups or
10 equal groups.)
Tell students, for example, that a number is not
divisible by 5 if that number of counters cannot
be divided into 5 equal groups. If there are any
counters left over, we say the number is not
divisible by 5.
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Unit 4 • Lesson 10 • Student page 176
Practice
Have counters available for all questions.
Questions 7 to 9 require a hundred chart
(PM 13). Question 10 requires a Venn
diagram (PM 28).
Assessment Focus: Question 6
Students recognize they must first multiply the
number of baskets by 10 to find the total
number of strawberries. Students then arrange
the total into 5 equal groups to find how many
strawberries each person gets.
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8. a)
b) The pattern in the ones digits is: 5, 0, 5, 0, . . .
c) A number is divisible by 5 if it ends in 5 or 0.
9. a)
25
45
30
50
4 12
24 32
17
b) The ones digit is always 0.
c) A number is divisible by 10 if it ends in 0.
10. The numbers where the loops overlap are divisible by 2, 5,
12
and 10.
REFLECT: I will choose 40. 40 is divisible by 2 because it ends
in an even number. It is divisible by 5 because it ends in 0 (to
be divisible by 5 it must end in 5 or 0), and it is divisible by
10 because it ends in 0.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that arrays can
be used to model division.
✔ Students understand that a number is
divisible by 2, by 5, or by 10 if that
number of counters can be divided into
2, 5, or 10 equal groups respectively.
Extra Support: Use Snap Cubes to model arrays for 2, 4, 6, ...,
30; 5, 10, 15, ..., 50; and 10, 20, 30, ..., 100. This will help
students to identify numbers that can be divided by 2, by 5, or
by 10.
Students can use Step-by-Step 10 (Master 4.20) to complete
question 6.
Applying procedures
✔ Students can identify numbers that
can be divided by 2, by 5, or by 10.
Extra Practice: Students can do the Additional Activity, Magic
Squares (Master 4.10). Students can complete Extra Practice 5
(Master 4.29).
Extension: Students use patterns to discover which numbers on
a hundred chart are divisible by 5, but not divisible by 10.
Communicating
✔ Students can describe how to tell if
a number is divisible by 2, by 5, or
by 10.
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
Unit 4 • Lesson 10 • Student page 177
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LESSON 11
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Relating Multiplication
and Division
40–50 min
LESSON ORGANIZER
Curriculum Focus: Find related multiplication and division
facts. (N15, N16, N20)
Teacher Materials
counters for the overhead projector
1-cm grid transparency (PM 20)
Student Materials
Optional
multiplication and
Step-by-Step 11 (Master 4.21)
division fact cards
Extra Practice 5 (Master 4.29)
counters
1-cm grid paper (PM 20)
Vocabulary: related facts
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
Key Math Learnings
1. 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.
BEFORE
Get Started
DURING
Explore
Scatter 8 counters on the overhead projector.
Ongoing Assessment: Observe and Listen
Ask:
• How might you arrange these counters to
show a multiplication sentence?
(I could arrange the counters into 4 groups of 2;
4 2 = 8.)
• What division sentence can you write about
this arrangement? (8 2 = 4)
Ask questions, such as:
• What product did you choose? (18)
• How did you make an array to show 18?
(I coloured grid squares to make an array with
3 rows of 6.)
• What multiplication sentence did you write?
(3 6 = 18)
• What division sentence did you write?
(18 3 = 6)
• How is the multiplication sentence related to
the division sentence?
(All of the numbers are the same. They are just in a
different order.)
Place a 1-cm grid transparency on the overhead
projector. Have students suggest how we could
use the grid to show 4 2. Make an array of 4
rows of 2 by shading in grid squares.
Present Explore. Distribute 1-cm grid paper to
pairs of students.
AFTER
Connect
Invite pairs of students to share their arrays
and their multiplication and division sentences.
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Unit 4 • Lesson 11 • Student page 178
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REACHING ALL LEARNERS
Early Finishers
Students use decks of cards with the face cards removed. This
game can be played with up to 5 students. Split the deck of
cards into two equal piles. Play rotates around the table in a
clockwise direction. Player A turns over the top card in each
pile, then gives a multiplication sentence and a division sentence
that use these 2 numbers. If the students around the table agree
that both sentences are correct, Player A keeps the cards. If
either of the sentences is incorrect, the cards go into a discard
pile. Player B takes a turn. Players continue to take turns until the
original piles have been used. Players count their cards. The
player with the most cards wins.
Common Misconceptions
4
4 5 = 20
20 5 = 4
3 7 = 21
21 7 = 3
➤Students write a related division fact for a given multiplication
fact in the wrong order. For example, they write 4 24 = 6
as a related fact for 4 6 = 24.
How to Help: Use counters to make an array for the
multiplication fact. Tell students that the total number of counters
is 24, and that 24 must be the first number in the division
sentence.
17=7
77=1
Draw an array of 3 rows of 7 on a 1-cm grid
transparency. Ask:
• What multiplication sentence can you write
for this array? (3 7 = 21)
• What division sentence can you write for this
array? (21 3 = 7)
Rotate the array one-quarter turn. It now
becomes an array with 7 rows of 3. Ask:
• What multiplication sentence can you write
for this array? (7 3 = 21)
• What division sentence can you write for this
array? (21 7 = 3)
Use Connect to introduce the term related facts.
Tell students that the 4 number sentences we
just found are related facts. Ensure students
understand the importance of related facts.
Draw an array of 4 rows of 4 on a 1-cm grid
transparency. Ask:
• What multiplication sentence can you write
for this array? (4 4 = 16)
• What division sentence can you write for this
array? (16 4 = 4)
Rotate the array one-quarter turn. Elicit from
students that because this array is square, the
multiplication and division sentences do not
change when the array is turned. In this case,
there are only 2 related facts. Ask:
• How do you know when a multiplication
fact has only 1 related division fact?
(A multiplication fact has only 1 related division
fact when the factors are the same.)
Practice
Have counters available for all questions.
Question 5 requires a set of related
multiplication and division fact cards.
Assessment Focus: Question 6
Students realize they need to look for numbers
that have 1 left over when divided by 6 and by
4. Some students will draw arrays to help them
solve the problem, while others will work with
numbers and facts.
Unit 4 • Lesson 11 • Student page 179
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Sample Answers
3. a) 3 5 = 15 b) 4 7 = 28 c) 7 5 = 35 d) 3 6 = 18
5 3 = 15
7 4 = 28
5 7 = 35
6 3 = 18
15 3 = 5
28 4 = 7
35 5 = 7
18 3 = 6
15 5 = 3
28 7 = 4
35 7 = 5
18 6 = 3
4. I can think of related facts that I know. For example, I know
that 3 3 = 9. Therefore, I know that 9 3 = 3.
6. Mrs. Bowski might have 13 or 25 children in her class. I
needed to find numbers that when divided by 6 and by 4 had
1 left over. I found that 7, 13, 19, and 25 had I left over
when divided by 6. I then checked to see which of these
numbers had 1 left over when divided by 4. The numbers 13
and 25 had 1 left over when divided by 4, and both these
numbers are less than 30.
7
4
=1
=6
=7
=2
=5
=2
=3
=5
4
49
7
5
REFLECT: I can find 21 7 by thinking about the related
multiplication fact. I know that 3 7 = 21, so 21 7 = 3.
I can use counters to make an array of 7 rows of 3 counters.
I can find 21 on a multiplication chart to see that the other
factor is 3.
Numbers Every Day
Students should use place value to order the numbers from least
to greatest.
21, 42, 55, 87, 99
26, 30, 72, 80, 91
20, 47, 53, 63, 68
18, 36, 42, 57, 85
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that for every
multiplication fact there is at least one
related division fact, and for every
division fact there is at least one
related multiplication fact.
Extra Support: For Question 6, help students make a chart.
Applying procedures
✔ Students can find and write related
multiplication and division facts.
Students can use Step-by-Step 11 (Master 4.21) to complete
question 6.
✔ Students can use mental math
strategies to multiply and divide.
Numbers that make groups of 6
6
12
18
24
30
Numbers that make groups of 6
with 1 left over
7
13
19
25
31
(too big)
Numbers that make groups of 4
4
8
12
16
20
24
28
Numbers that make groups of 4
with 1 left over
5
9
13
17
21
25
29
Extra Practice:
Students can do the Additional Activity, Make A Sentence
(Master 4.11).
Students can complete Extra Practice 5 (Master 4.29).
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
38
Unit 4 • Lesson 11 • Student page 180
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ESSON 12
Number Patterns
on a Calculator
LESSON ORGANIZER
40–50 min
Curriculum Focus: Use a calculator to create number
patterns. (N15)
Student Materials
Optional
4 function calculator,
Step-by-Step 12
such as the TI-108
(Master 4.22)
Extra Practice 6
(Master 4.30)
Assessment: Master 4.2 Ongoing Observations:
Multiplication and Division
Key Math Learnings
1. Calculators can be used to create and extend number patterns.
2. Calculators can be used to show multiplication as repeated
addition, and division as repeated subtraction.
BEFORE
Get Started
Distribute calculators to all students. Review
with students how to turn the calculator on,
how to clear the display, and how to use the
various operation keys. You may wish to work
through some examples as a class.
Present Explore.
DURING
Explore
Ongoing Assessment: Observe and Listen
Ask questions, such as:
• By following the first set of keystrokes,
what are you doing?
(I am adding 7 repeatedly.)
• What numbers do you see on the screen?
(7, 14, 21, 28, 35, 42, 49)
• What pattern do you see?
(The pattern is: Start at 7. Add 7 each time.)
• How would you write what the calculator is
doing on paper?
(7 + 7 + 7 + 7 + 7 + 7 + 7 = 49)
• How does what you see on the screen relate
to multiplication?
(It is 7 multiplied by 1, 2, 3, 4, 5, 6, and 7.)
• By following the second set of keystrokes,
what are you doing?
(I am subtracting 9 repeatedly.)
• What numbers do you see on the screen?
(72, 63, 54, 45, 36, 27)
• What patterns do you see?
(The ones digit starts at 2 and increases by 1. The
tens digit starts at 7 and decreases by 1. The digits in
each number add to 9.)
Watch to see how students work. Do they
record the results as they enter the keystrokes
or do they complete the keystrokes, then record
the results? Are students relating the repeated
addition to multiplication?
Unit 4 • Lesson 12 • Student page 181
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REACHING ALL LEARNERS
Early Finishers
One student enters the first 4 keystrokes for a repeated addition
of his choice. A classmate presses the equal sign as many times
as necessary to determine what number is being added
repeatedly.
Common Misconceptions
➤Students assume that all calculators do repeated addition,
subtraction, and multiplication in the same way.
How to Help: Explain to students that some calculators work
differently. Have students enter 2 + = = = on their
calculators. If the screen shows 2, 4, 6, and 8, their calculator
works the same way as described in the Student Book.
Sample Answers
1. 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
The pattern in the ones digits is: 5, 0, 5, 0, …
The pattern in the tens digits is: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5
2. 90, 80, 70, 60, 50, 40, 30, 20, 10, 0
The ones digit is always 0. The tens digit decreases by
1 each time. The numbers are decreasing by 10 each time.
3. 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
The ones digit starts at 9 and decreases by 1 each time.
The tens digit starts at 0 and increases by 1 each time.
The digits in each number add to 9.
4. 4, 8, 16, 32, 64, 128, 256, 512
The pattern in the ones digits is: 4, 8, 6, 2, 4, 8, 6, 2
AFTER
Connect
Invite students to share their patterns with
the class. Ask:
• What did you do to create your own pattern?
(I started with 88 and subtracted 8 repeatedly.)
• What did you see on the screen?
(80, 72, 64, 56, 48, 40, 32, 24, 16, 8, 0)
• What pattern did you see?
(There is a pattern in the ones digits: 0, 2, 4, 6, 8, 0,
2, 4, 6, 8, 0)
• What division sentence does this show?
(88 8 = 11)
Use calculators and work through the example
of repeated addition in Connect as a class.
Tell students that repeated addition relates to
multiplication. Have students enter the number
5 in their calculators, then add repeatedly.
Ask:
• What do you see on the screen?
(5, 10, 15, 20, 25, 30, 35)
40
Unit 4 • Lesson 12 • Student page 182
6
42 7 = 6
• What pattern do you see? (There is a pattern in
the ones digits: 5, 0, 5, 0, 5, 0, 5)
• What do you know about numbers that end
in 5 or 0? (They are all multiples of 5 and they are
all divisible by 5.)
• How can you relate the numbers on the
screen to multiplication?
(They are the numbers you get when you multiply 5
by 1, 2, 3, 4, 5, 6, and 7.)
Work through the other examples in Connect.
Practice
Calculators are required for all questions.
Assessment Focus: Question 9
Students recognize they are looking for all the
factors of 16. Students should approach the
question in an organized manner to ensure they
find all possible answers.
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9. 1 and 16, 2 and 8, 4 and 4, 8 and 2, 16 and 1
There are 5 different answers. I know I have found all the
answers because I worked in an organized way. I used 1 as
the first factor, then increased the first factor by 1 each time.
Each time, I tried to make a product of 16.
10. a) There are 3 dog walkers. They have 12 dogs to walk.
If each dog walker walks the same number of dogs,
how many dogs will each walker walk?
(Answer: 12 3 = 4; each dog walker will walk 4 dogs.)
b) There are 5 skipping ropes for 15 children to share.
The same number of children must share each rope.
How many children will share each rope?
(Answer: 15 5 = 3; 3 children will share each rope.)
Divide; 14
Multiply; 96
REFLECT: Randy is correct. I can divide 24 4 this way:
24 – 4 = 20
20 – 4 = 16
16 – 4 = 12
12 – 4 = 8
8–4=4
4–4=0
I subtracted 4 six times, so 24 4 = 6.
11
10
13
14
Numbers Every Day
Students should recognize that 6 + 5 is 1 less than 6 + 6,
6 + 4 is 2 less than 6 + 6, 6 + 7 is 1 more than 6 + 6, and
6 + 8 is 2 more than 6 + 6.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students recognize that calculators
can be used to create and extend
number patterns.
Extra Support: Have students write a multiplication fact, such
as 4 3, then write its associated repeated addition sentence.
Students first use the calculator to find 4 3. Students then enter
the repeated addition sentence in the calculator and record the
result. Finally, students enter the first factor, 4, then press
+ = = = and record the results. Students see that all results
are the same. Students repeat with other multiplication facts.
Students can use Step-by-Step 12 (Master 4.22) to complete
question 9.
Applying procedures
✔ Students can use a calculator to
create number patterns.
Communicating
✔ Students can describe how repeated
addition shows multiplication and
repeated subtraction shows division.
Extra Practice: Have students use their calculators to answer
questions such as, “Which is greater, the 15th number when I
count by 5s, or the 40th even number?”
Students can complete Extra Practice 6 (Master 4.30).
Recording and Reporting
Master 4.2 Ongoing Observations:
Multiplication and Division
Unit 4 • Lesson 12 • Student page 183
41
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S H O W W H AT Y O U K N
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40–50 min
LESSON ORGANIZER
Student Materials
Venn diagrams (PM 28)
4-function calculators
counters
multiplication charts (PM 15)
Assessment: Masters 4.1 Unit Rubric: Multiplication and
Division, 4.4 Unit Summary: Multiplication and Division
6 2 = 12 12 6 = 2
4 3 = 12 12 4 = 3
=0
=4
= 35
=8
= 18
= 28
=7
= 36
= 70
= 24
= 12
=0
$21
=6
=3
=2
=6
=4
=6
=7
=8
5
c) 5 6 = 30
Sample Answers
2. a)
b)
c)
d)
e)
7. a) 2 4 = 8
42=8
84=2
82=4
42
b) 3 7 = 21
7 3 = 21
21 3 = 7
21 7 = 3
Unit 4 • Show What You Know • Student page 184
d) 5 5 = 25
6 5 = 30
25 5 = 5
30 5 = 6
30 6 = 5
8. a) 1 12 = 12, 2 6 = 12, 3 4 = 12,
4 3 = 12, 6 2 = 12, 12 1 = 12
There are 6 different answers.
b) 7 7 = 1, 6 6 = 1, 5 5 = 1, 4 4 = 1,
3 3 = 1, 2 2 = 1, 1 1 = 1
Any number divided by itself will equal 1.
9. Divisible by 5: 5, 10, 15, 20; numbers that end in
0 or 5 are divisible by 5.
Divisible by 2: 2, 10, 14, 20, 26; numbers that end in
2, 4, 6, 8, or 0 are divisible by 2. All even numbers
are divisible by 2.
11. a) 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
There is a pattern in the ones digits:
8, 6, 4, 2, 0, 8, 6, 4, 2, 0
The numbers start at 8 and increase by 8 each time.
These numbers are the products of 8 and
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
b) 64, 56, 48, 40, 32, 24, 16, 8, 0
There is a pattern in the ones digits:
4, 6, 8, 0, 2, 4, 6, 8, 0
The numbers start at 72 and decrease by 8 each time.
= 50
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12. Addison could buy: 6 packages of 4 toys; 6 4 = 24
5
15
21
10
20
2
14
26
9
4 packages of 6 toys; 4 6 = 24
3 packages of 4 toys, and 2 packages of 6 toys;
3 4 = 12, 2 6 = 12 , 12 + 12 = 24
13. Number of rocks in large bags: 6 4 = 24
Number of rocks in small bags: 5 3 = 15
Total number of rocks: 24 + 15 = 39
Connor collected 39 rocks in all.
7
7 6 = 42
42 6 = 7
3 ways
39
ASSESSMENT FOR LEARNING
What to Look For
Accuracy of procedures
✔ Question 1: Student can write a multiplication sentence and a division sentence for an array.
✔ Question 3: Student can multiply two 1-digit numbers.
✔ Question 5: Student demonstrates knowledge of division facts.
✔ Question 12: Student understands that more than one pair of numbers can have the same product.
Reasoning; Applying concepts
✔ Questions 4 and 6: Student can apply multiplication and division concepts to solve problems.
✔ Question 9: Student can apply relationships and patterns to identify numbers as divisible by 2, by 5, and
by 10, and place them in a Venn diagram.
Problem solving
✔ Question 13: Student can solve a problem that involves multiple steps.
Recording and Reporting
Master 4.1 Unit Rubric: Multiplication and Division
Master 4.4 Unit Summary: Multiplication and
Division
Unit 4 • Show What You Know • Student page 185
43
UNIT PROBLEM
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Here Comes the Band!
LESSON ORGANIZER
40–50 min
Student Grouping: 2
Student Materials
counters
calculators
Assessment: Masters 4.3 Performance Assessment Rubric:
Here Comes the Band, 4.4 Unit Summary: Multiplication
and Division
Sample Response
Part 1
The band can be arranged in 10 different ways:
1 row of 48
1 48 = 48
2 rows of 24
2 24 = 48
3 rows of 16
3 16 = 48
4 rows of 12
4 12 = 48
6 rows of 8
6 8 = 48
8 rows of 6
8 6 = 48
12 rows of 4
12 4 = 48
16 rows of 3
16 3 = 48
24 rows of 2
24 2 = 48
48 rows of 1
48 1 = 48
Have students turn to the Unit Launch on pages
144 and 145 of the Student Book. Remind
students of the questions they answered about
the marching band at the beginning of the Unit.
Have one student read aloud the Check List to
ensure all students understand what their work
should include. Discuss possible ways students
could present their work.
Invite volunteers to read aloud the Learning
Goals for the unit. Discuss each goal briefly
with students. Tell students that they will use
the skills they have learned in this unit to
complete the Unit Problem.
Before students begin Part 3, remind students
that they should think carefully about the
number of band members they choose. For
example, if there are 23 band members, they
could only be arranged in 1 row of 23, or 23
rows of 1. This would not be an ideal
arrangement for a marching band.
Present the Unit Problem. Invite volunteers to
read the instructions for each part of the
problem aloud. Answer any questions students
might have.
44
Unit 4 • Unit Problem • Student page 186
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Part 2
I can set up the chairs in 7 ways:
1 row of 30
2 rows of 15
3 rows of 10
5 rows of 6
6 rows of 5
10 rows of 3
15 rows of 2
30
30
30
30
30
30
30
1 = 30
2 = 15
3 = 10
5=6
6=5
10 = 3
15 = 2
Thirty rows of 1 chair is not
acceptable because there
must be at least 2 chairs in
each row.
If the band has 31 members, the only way to arrange them on
stage is 1 row of 31. They probably would not fit across the
stage if they were arranged this way. All other arrangements
would result in rows that are not equal. There would always be
at least 1 member left over.
Part 3
I chose 36 band members.
I can arrange my band members in 9 different ways:
1 row of 36
2 rows of 18
3 rows of 12
4 rows of 9
6 rows of 6
9 rows of 4
12 rows of 3
18 rows of 2
36 rows of 1
1 36 = 36
2 18 = 36
3 12 = 36
4 9 = 36
6 6 = 36
9 4 = 36
12 3 = 36
18 2 = 36
36 1 = 36
36
36
36
36
36
36
36
36
36
1 = 36
2 = 18
3 = 12
4=9
6=6
9=4
12 = 3
18 = 2
36 = 1
Reflect on the Unit
Multiplication and division are opposites. I can arrange an array
of 18 counters into 3 rows, with 6 counters in each row.
I can write 2 multiplication sentences and 2 division sentences.
3 6 = 18
18 3 = 6
6 3 = 18
18 6 = 3
Multiplication joins objects together in equal rows or groups.
Division separates objects into equal rows or groups.
ASSESSMENT FOR LEARNING
What to Look For
What to Do
Understanding concepts
✔ Students understand that arrays can
be used to multiply and divide.
Extra Support: Make the problem accessible.
Some students may have difficulty identifying all the possible
arrangements in Parts 1, 2, and 3. Make multiplication
charts available.
Applying procedures
✔ Students can make an array and write
a corresponding multiplication and
division sentence.
Communicating
✔ Students can explain why 31 band
members might be a problem.
Some students may have difficulty arranging the members in
equal rows. Encourage these students to model the band
members with counters, then use the counters to make arrays.
This will allow students to visualize the different arrangements of
the band members. Encourage students to turn their arrays to find
other possible arrangements.
✔ Students present their work in a clear
and organized way.
Recording and Reporting
Master 4.3 Performance Assessment: Here Comes the Band!
Master 4.4 Unit Summary: Multiplication and Division
Unit 4 • Unit Problem • Student page 187
45
UNITS 1 – 4
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Cumulative Review
LESSON ORGANIZER
Student Materials
calculators
Base Ten Blocks
hundred charts (PM 13)
addition charts (PM 14)
square dot paper (PM 22)
Venn diagrams (PM 28)
Assessment: PM 9 Work Sample Records
<
>
<
>
3
9
= 660
8
= 976
15
= 158
159 balloons
Students can use this review to evaluate their
progress in the mathematical content of
Units 1 through 4.
Remind students that the unit containing the
content required to answer each question is
listed in red beside the question.
Encourage students to refer to the Connect
sections of the relevant units.
Encourage students to draw on each other’s
skills as they work through the review.
Students who have identified areas in which
they are strong may wish to act as volunteer
tutors for students who require assistance.
46
Unit 4 • Cumulative Review • Student page 188
= 102
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Sample Answers
1. 5, 10, 15, 20, 25, . . .
The pattern is: Start at 5. Add 5 each time. This is the same
as starting at 5 on a hundred chart and counting on by 5s.
3. a)
b)
c)
Cones
Square pyramids
Spheres
Triangular pyramids
Cubes
d)
Triangular prisms
= 18
= 49
=0
=2
= 35
= 24
=3
=6
=6
=5
=7
= 10
7 packages
7 8 = 56; 56 8 = 7
4. a) I used subtraction. I found 12 – 9 = 3, so the missing
number is 3.
b) I used addition. I found 9 + 9 = 18, so the missing
number is 9.
c) I used subtraction. I found 13 – 5 = 8, so the missing
number is 8.
d) I used addition. I found 7 + 8 = 15, so the missing
number is 15.
6. I could not subtract 9 ones from 8 ones, so I traded 1 ten
for 10 ones, making 2 tens and 18 ones. I subtracted the
ones; 18 – 9 = 9. I could not subtract 7 tens from 2 tens,
so I traded 1 hundred for 10 tens, making 6 hundreds
and 12 tens. I subtracted the tens: 12 – 7 = 5. I then
subtracted the hundreds; 6 – 5 = 1. 738 – 579 = 159
7.
angles greater than a right angle. Figure B has 3
right angles, and 2 angles greater than a right angle.
c) Figure A is a trapezoid. Figure B is a pentagon.
8. A: hexagon; B: rectangle; C: square; D: right triangle;
E: square; F: rectangle; G: triangle; H: right triangle;
I: hexagon; J: octagon
Figures A and I are congruent. Figures B and F are
congruent. Figures D and H are congruent. I know these
figures are congruent because they are exactly the same
size and shape.
9.
Triangular faces
G
H
5 faces
E
F
K
L
A B C D I J
The sorting rule is: Solids with triangular faces and solids
with 5 faces.
Figure A
Figure B
a) Figure A has 4 sides, with 2 of the sides parallel.
Figure B has 5 sides, with 2 of the sides parallel. No
sides are the same length in either figure.
b) Figure A has 2 angles less than a right angle and 2
Unit 4 • Cumulative Review • Student page 189
47
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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 strand.
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
Strand:
Number Concepts/
Number Operations
Reasoning;
Applying
concepts
Accuracy of
procedures
Problem
solving
Communication
Overall
Ongoing Observations
3
4
3
4
3/4
Strategies Toolkit
not assessed
Work samples or
portfolios; conferences
3
4
3
4
3/4
Show What You Know
4
4
4
4
4
Unit Test
3
4
4
Unit Problem
Here Comes the Band!
4
4
3
Achievement Level for reporting
4
4
4
4
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.
#1, 3, 5, and 12 provide evidence of Accuracy of procedures; #4, 6, and 9 provide evidence
of Reasoning; Applying concepts; #13 provides evidence of Problem solving; 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.
Unit performance task
Use Master 4.3 Performance Assessment Rubric: Here Comes the Band! 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
Ongoing Records
PM 4: Learning Skills Check List
PM 10: Summary Class Records: Strands
PM 11: Summary Class Records: Achievement Categories
PM 12: Summary Record: Individual
Use to record and report throughout a reporting period, rather
than for each unit and/or strand.
Use to record and report evaluations of student achievement over
several clusters, a reporting period, or a school year.
These can also be used in place of the Unit Summary.
48 Copyright © 2005 Pearson Education Canada Inc.
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Name
Master 4.1
Date
Unit Rubric: Multiplication and Division
Not Yet
Adequate
Adequate
may be unable to
demonstrate, apply, or
explain:
– divisibility by 2, 5, 10
– processes of
multiplication and
division
– choice of operations
partially able to
demonstrate, apply, or
explain:
– divisibility by 2, 5, 10
– processes of
multiplication and
division
– choice of operations
able to demonstrate,
apply, and explain:
– divisibility by 2, 5, 10
– processes of
multiplication and
division
– choice of operations
in various contexts,
appropriately
demonstrates, applies,
and explains:
– divisibility by 2, 5, 10
– processes of
multiplication and
division
– choice of operations
limited accuracy;
omissions or major
errors in:
– multiplication and
division to 50
– recalling multiplication
facts to 49
– recognizing which
numbers are divisible
by 2, 5, 10
– skip counting
– writing multiplication
and division
sentences
partially accurate;
omissions or frequent
minor errors in:
– multiplication and
division to 50
– recalling multiplication
facts to 49
– recognizing which
numbers are divisible
by 2, 5, 10
– skip counting
– writing multiplication
and division
sentences
generally accurate; few
errors in:
– multiplication and
division to 50
– recalling multiplication
facts to 49
– recognizing which
numbers are divisible
by 2, 5, 10
– skip counting
– writing multiplication
and division sentences
accurate; no errors in:
– multiplication and
division to 50
– recalling multiplication
facts to 49
– recognizing which
numbers are divisible
by 2, 5, 10
– skip counting
– writing multiplication
and division
sentences
may be unable to use
appropriate strategies
to solve and create
problems involving
multiplication and
division of whole
numbers
with limited help, uses
some appropriate
strategies to solve and
create problems
involving multiplication
and division of whole
numbers; partially
successful
uses appropriate
strategies to solve and
create problems
involving multiplication
and division of whole
numbers successfully
uses appropriate, often
innovative, strategies to
solve and create
problems involving
multiplication and
division of whole
numbers successfully
• explains reasoning and
procedures clearly,
including appropriate
terminology (e.g.,
multiply, divide, times,
array, factor, product,
divisible)
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 work clearly
work is often unclear
presents work with
some clarity
presents work clearly
presents work clearly
and precisely
Proficient
Excellent
Reasoning: Applying
concepts
• shows understanding
by applying and
explaining:
– divisibility by 2, 5, 10
– processes of
multiplication and
division, using
manipulatives,
diagrams, and
symbols
– which operation(s)
can be used to solve
a particular problem
Accuracy of
procedures
• accurately:
– multiplies and
divides to 50
(calculates products
and quotients)
– recalls multiplication
facts to 49 (7 x 7 on
a multiplication grid)
– counts by 2s, 5s,
and 10s
– recognizes which
numbers are
divisible by 2, 5, 10
– writes multiplication
and division sentences
Problem-solving
strategies
• chooses and carries
out a range of
strategies (e.g., using
manipulatives,
pictures, diagrams,
arrays, number lines,
patterns, charts,
tables, calculators) to
create and solve
problems involving
multiplication and
division
Communication
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Master 4.2
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
Demonstrates and
explains concepts
related to the
multiplication and
division of whole
numbers
Accuracy of
procedures
Multiplies, divides,
compares, and
orders whole
numbers accurately
Problem solving
Uses appropriate
strategies to solve
and create problems
involving the
multiplication and
division of whole
numbers
*Use locally or provincially approved levels, symbols, or numeric ratings.
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Communication
Presents work clearly
Explains reasoning and
procedures clearly,
using appropriate
terminology
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Master 4.3
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Performance Assessment Rubric:
Here Comes the Band!
Not Yet
Adequate
Adequate
Proficient
Excellent
does not apply the
required concepts of
multiplication and
division appropriately;
may be incomplete or
indicate misconceptions
applies some of the
required concepts of
multiplication and
division; may indicate
some misconceptions,
particularly in explaining
why 31 would be
difficult
applies the required
concepts of multiplication
and division
appropriately; may be
minor flaws in
explanation of why 31
would be difficult
applies the required
concepts of
multiplication and
division effectively
throughout; indicates
thorough understanding
omissions or major
errors in:
– multiplication
sentences for 48
– division sentences for
30
– multiplication and
division sentences for
chosen number
omissions or some
minor errors in:
– multiplication
sentences for 48
– division sentences for
30
– multiplication and
division sentences for
chosen number
few minor errors in:
– multiplication
sentences for 48
– division sentences for
30
– multiplication and
division sentences for
chosen number
accurate and precise; no
errors in:
– multiplication
sentences for 48
– division sentences for
30
– multiplication and
division sentences for
chosen number
uses few effective
strategies; does not
adequately find all
possible arrangements
for:
– 48 band members
– 30 band members
– chosen number of
band members
uses some appropriate
strategies, with partial
success, to find all
possible arrangements
for:
– 48 band members
– 30 band members
– chosen number of
band members
uses appropriate and
successful strategies to
find all possible
arrangements for:
– 48 band members
– 30 band members
– chosen number of
band members
uses innovative and
effective strategies to
find all possible
arrangements for:
– 48 band members
– 30 band members
– chosen number of
band members
• uses mathematical
terminology correctly
(e.g., multiply, divide,
times, array, factor,
product)
uses few appropriate
mathematical terms
uses some appropriate
mathematical terms
uses appropriate
mathematical terms
uses a range of
appropriate
mathematical
terminology with
precision
• explains the need for
band members to be
divided into equal
rows/columns clearly
does not explain
reasoning clearly
partially explains
reasoning; may be
vague and somewhat
unclear
explains reasoning
clearly
explains reasoning
clearly, precisely, and
confidently
Reasoning;
Applying concepts
• shows understanding
by applying the
required concepts of
multiplication and
division to each step,
and explaining why
arranging 31 band
members would be a
problem
Accuracy of
procedures
• writes accurate
multiplication sentences
for 48 (Part 1)
• writes accurate
division sentences for
30 (Part 2)
• writes accurate
multiplication and
division sentences for
the number chosen
(Part 3)
Problem-solving
strategies
• uses appropriate
strategies (e.g.,
drawing, making a
table) to identify all
possible ways to:
– arrange 48 band
members in equal
rows (arrays)
– arrange 30 band
members in equal
rows (arrays)
– arrange chosen
number of band
members in equal
rows (arrays)
Communication
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Master 4.4
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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/Number
Operations
Reasoning;
Applying
concepts
Accuracy of
procedures
Problem
solving
Ongoing Observations
Strategies Toolkit
(Lesson 7)
Work samples or
portfolios; conferences
Show What You Know
Unit Test
Unit Problem
Here Comes the Band!
Achievement Level for reporting
*Use locally or provincially approved levels, symbols, or numeric ratings.
Self-Assessment:
Comments: (Strengths, Needs, Next Steps)
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Communication
Overall
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Master 4.5
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To Parents and Adults at Home …
Your child’s class is starting a mathematics unit on multiplication and division.
Multiplication and division are basic computational skills that children will use
often, and skills that children must master to succeed in higher levels of
mathematics. Your child will develop strategies for multiplying and dividing
whole numbers. They will use multiplication charts, counters, grid paper, and
calculators.
In this unit, your child will:
• Relate multiplication sentences to repeated addition.
• Use arrays to multiply and divide.
• Discover patterns for multiplying and dividing by 2, by 5,
and by 10.
• Learn the rules for multiplying by 1 and by 0.
• Create multiplication and division fact families.
• Relate division sentences to repeated subtraction.
Talk with your child about the importance of learning her or his multiplication
and division facts. Find opportunities to practise these facts with your child to
build confidence and to establish a solid foundation.
Here are some suggestions for activities you can do with your child.
When walking down the street or riding on a bus, have your child multiply the
digits of 2-digit house numbers. For example, for a house numbered 36, your
child would multiply 3 × 6 to get 18. (Multiplication beyond 7 × 7 is not
required.)
Find items that are arranged in equal rows. Have your child give a
multiplication sentence and a division sentence for each arrangement. For
example, egg cartons are arranged in 2 equal rows of 6 eggs. Two possible
sentences are: 2 × 6 = 12 and 12 ÷ 2 = 6.
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Master 4.6
Number Lines
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Master 4.7
Date
Explore Multiplication Chart
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Master 4.8
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Additional Activity 1:
Amazing Arrays
Work with a partner.
You will need 1-cm grid paper and scissors.
How to prepare for play:
Cut rectangles to represent each product from 1 × 2 to 7 × 7.
Each rectangle should have a length equal to one factor and a width
equal to the other factor, in centimetres.
On the grid side of each rectangle, write the factors; for example, 3 × 2.
On the other side, write the product; for example, 6.
How to play:
Spread out the rectangles on a table.
Some rectangles should be grid side up.
Some rectangles should be grid side down.
Take turns selecting a rectangle.
If the product is showing, name the factors.
If the factors are showing, name the product.
If you are correct, keep the rectangle.
Continue to play until all rectangles have been won.
The player with the most rectangles wins.
Take It Further: Cut out the rectangles, as above, but leave them blank. Place
all rectangles grid side up.
Players select a rectangle, then give both the factors and the product.
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Additional Activity 2:
Multiplication Tag
Master 4.9
Play with a partner.
You will need 2 of these sheets and scissors.
Each player cuts out a set of number tags.
How to play:
Use each tag only once.
Make 4 multiplication problems with your tags.
Find the product for each problem.
Add the products. This is your score.
Play the game 3 more times.
Add your score to your previous score each time.
The player with the highest score wins.
Take It Further: Place all 16 number tags face down. Player A turns over 2
tags, then finds and records their product as his score. The tags are not
replaced. Player B takes a turn. Continue taking turns until all tags have been
used. The highest score wins.
0
1
2
3
4
5
6
7
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Master 4.10
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Additional Activity 3:
Magic Squares
Work on your own.
Write the product for each multiplication fact below.
Write the products in the matching squares.
a) 4 × 2 =
a
b
c
d
e
f
g
h
i
b) 3 × 6 =
c) 2 × 2 =
d) 2 × 3 =
e) 5 × 2 =
f) 2 × 7 =
g) 4 × 4 =
h) 1 × 2 =
i) 6 × 2 =
This is a magic square.
In a magic square, each row, column, and diagonal have the same sum.
Check your magic square by adding in any direction.
The sums should all be the same.
Take It Further: Use the magic square to create another magic square.
Divide each entry by 2.
Check your new magic square.
The sums should all be the same.
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Master 4.11
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Additional Activity 4:
Make A Sentence
Play with the class.
The object of the game is to be a part of as many multiplication and division
sentences as you can.
How to play:
Each student takes 1 card. Do not show your card until play begins.
When the teacher says “Start,” find 2 classmates with numbers that,
when used with your number, will make a multiplication or a division
sentence.
Write your sentence and your names on the board.
Continue to make different sentences with other classmates.
Each time, write your sentence and your names on the board.
If the card you took at the beginning of the game is a number whose only
factors are 1 and itself, write your number and your name on the board.
Then, take another card and begin making sentences with your
classmates.
When the teacher says “Stop,” no more sentences may be written on
the board.
The player whose name appears with the most number sentences is
the winner.
Take It Further: Each player takes 1 card. You have 1 minute.
Write all the multiplication and division sentences you know that use
your number.
The player with the most correct sentences wins.
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Master 4.12
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Step-by-Step 1
Lesson 1, Question 7
Step 1 Draw a picture for 3 × 1.
Step 2 Draw a picture for 3 × 3.
Step 3 Draw a picture for 3 × 4.
Step 4 Look at your pictures in Steps 1 to 3.
Can you write a multiplication sentence for the picture below?
Explain your answer.
________________________________________________________
________________________________________________________
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Master 4.13
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Step-by-Step 2
Lesson 2, Question 7
Use these counters for Steps 1 to 4.
Step 1 Draw an array with 1 counter in each row.
Step 2 Draw an array with 2 counters in each row.
Step 3 Draw an array with 3 counters in each row.
Step 4 Draw an array with 6 counters in each row.
Step 5 How many different arrays can you draw for 6 counters? _______
Step 6 How many arrays can you make with 7 counters? __________
Draw the arrays.
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Master 4.14
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Step-by-Step 3
Lesson 3, Question 9
Step 1 A child’s ticket costs $2.
How much do 4 tickets cost? ____________
Step 2 An adult’s ticket costs $5.
How much do 2 tickets cost? ____________
Step 3 Add the money in Steps 1 and 2.
How much did Barbara pay? ____________
Step 4 A child’s ticket costs $2.
How much do 2 tickets cost? ____________
Step 5 An adult’s ticket costs $5.
How much do 3 tickets cost? ____________
Step 6 Add the money in Steps 4 and 5.
How much did Carlos pay? ____________
Step 7 Who spent more money? How do you know?
________________________________________________________
________________________________________________________
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Master 4.15
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Step-by-Step 4
Lesson 4, Question 4
Step 1 How many cents in 1 dime? ______________
Step 2 How many cents in 5 dimes? _______________
Step 3 How many cents in 1 nickel? _______________
Step 4 How many cents in 6 nickels? _______________
Step 5 Add the money in Steps 2 and 4. ______________
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Step-by-Step 5
Lesson 5, Question 6
Step 1 Draw a picture to show the strawberries on the waffle.
Step 2 There are 2 times as many raspberries as strawberries.
Draw a picture to show the raspberries on the waffle.
How many raspberries are on the waffle? ________________
Step 3 Use the answer from Step 2.
There are 3 times as many blueberries.
Draw a picture to show the blueberries on the waffle.
How many blueberries are on the waffle? _____________
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Step-by-Step 6
Lesson 6, Question 5
Step 1 Skip count to find the number that each figure represents.
♥ = _________________
∆ = _________________
○ = _________________
‫_________________ = ڤ‬
Step 2 Use the numbers from Step 1. Add the numbers for ♥ + ○. ______
Step 3 Use the numbers from Step 1. Add the numbers for ∆ + ○. ______
Step 4 Order the numbers, in Step 1, from least to greatest.
Use a number line if it helps.
_____________________________________________________
Step 5 Match each number in Step 4 with a figure.
Draw the figures to match the order of the numbers.
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Master 4.18
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Step-by-Step 8
Lesson 8, Question 4
Use 21 counters.
Step 1 Try to make groups of 2. Draw what you find out.
Step 2 Try to make groups of 3. Draw what you find out.
Step 3 Try to make groups of 4. Draw what you find out.
Step 4 Try to make groups of 5. Draw what you find out.
Step 5 Continue to try to make groups of 6, 7, 8, …, up to 10.
Draw what you find out each time.
Step 6 Look at your pictures where the groups are equal,
with no counters left over.
Which picture has the most groups? ____________
How many counters are in each group? ____________
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Master 4.19
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Step-by-Step 9
Lesson 9, Question 6
Use 12 blue counters and 15 green counters.
Step 1 Put the counters together.
Try to make an array with 2 in each row.
Draw a picture.
Step 2 Try to make an array with 3 in each row.
Draw a picture to show the results.
Step 3 Try to make an array with 4 in each row.
Draw a picture to show the results.
Step 4 Continue to try to make an array.
Put 1 more counter in each row each time.
Draw a picture to show each result.
Step 5 Look at your pictures.
Which pictures have arrays with equal rows? ____________________
How many counters are in each row? __________________________
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Master 4.20
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Step-by-Step 10
Lesson 10, Question 6
Step 1 There are 10 strawberries in 1 basket.
How many strawberries are in 3 baskets? ______________________
Step 2 Use the strawberries in Step 1.
Share the strawberries among 5 people.
Draw a picture to show how you did this.
Step 3 How many strawberries did each person get? ___________________
How do you know? ________________________________________
________________________________________________________
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Master 4.21
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Step-by-Step 11
Lesson 11, Question 6
Step 1 Use 29 counters.
Make groups of 6. How many are left over? __________
Make groups of 4. How many are left over? __________
Step 2 Use 28 counters.
Make groups of 6. How many are left over? __________
Make groups of 4. How many are left over? __________
Step 3 Take away 1 counter.
Make groups of 6. How many are left over? __________
Make groups of 4. How many are left over? __________
Step 4 Repeat Step 3 as many times as you can.
Each time, tell how many are left over.
Step 5 In which of Steps 1 to 4 was 1 left over for groups of 6 and groups of 4?
________________________________________________________
Step 6 How many children might be in the class? ______________________
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Master 4.22
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Step-by-Step 12
Lesson 12, Question 9
Use counters to make arrays, if they help.
Step 1 16 ÷ 1 =
Write a related multiplication fact, if possible.
Step 2 16 ÷ 2 =
Write a related multiplication fact, if possible.
Step 3 16 ÷ 3 =
Why can you not write a related multiplication fact?
________________________________________________________
Step 4 Continue to divide 16 by 4, 5, 6, and 7.
That is, divide 16 by all the numbers from 4 to 7.
When you can, write a related multiplication fact.
Step 5 How many different facts did you write? ________________________
How do you know you have all the facts?
________________________________________________________
________________________________________________________
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Master 4.23a
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Unit Test: Unit 4
Multiplication and Division
Part A
1. For this picture:
a) Write an addition sentence.
______________________
b) Write a multiplication sentence.
_________________________
2. Multiply.
a) 6 × 4 ______
b) 2 × 7 ______ c) 5 × 1 ______
d) 7 × 10 ______
e) 9 × 0 ______
f) 1 × 8 ______ g) 4 × 3 ______
h) 5 × 5 ______
3. For this array:
Write 2 division sentences.
______________________________________
______________________________________
Write 2 multiplication sentences.
______________________________________
______________________________________
4. Divide.
a) 30 ÷ 5 _____
e) 24 ÷ 3 _____
5. 12
4
33
b) 14 ÷ 2 _____ c) 30 ÷ 10 _____
d) 28 ÷ 7 _____
f) 9 ÷ 9 _____
h) 60 ÷ 10 _____
20
7
30
g) 25 ÷ 5 _____
18
15
a) Which numbers are divisible by 2? ______________________________
b) Which numbers are divisible by 5? ______________________________
c) Which numbers are in both lists? _______________________________
d) What are the numbers in part c also divisible by? ___________________
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Unit Test continued
Part B
6. Ali has 21 photos. He wants to put 3 photos on each page of his album.
How many pages does he need?
____________________________________________________________
7. A junior hockey team has 6 children on each team.
How many teams can be made with 42 children?
____________________________________________________________
8. Stella has 4 dimes. Ian has 7 nickels.
Who has more money? How much more?
Show your work.
Part C
9. Nasrin has fewer than 40 hockey cards.
When she divides them into groups of 5, she has 3 left over.
When she divides them into groups of 6, she has 2 left over.
How many cards does Nasrin have? How do you know?
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Sample Answers
Unit Test – Master 4.23
Part A
1. a) 6 + 6 + 6 = 18
b) 3 × 6 = 18
2. a) 24
b) 14
c) 5
d) 70
e) 0
f) 8
g) 12
h) 25
3. 21 ÷ 7 = 3; 21 ÷ 3 = 7; 3 × 7 = 21; 7 × 3 = 21
4. a) 6
b) 7
c) 3
d) 4
e) 8
f) 1
g) 5
h) 6
5. a) 4, 12, 18, 20, 30
b) 15, 20, 30
c) 20, 30
d) 10
Part B
6. 21 ÷ 3 = 7; Ali needs 7 pages.
7. 42 ÷ 6 = 7; there can be 7 teams.
8. Stella: 4 × 10 cents = 40 cents
Ian: 7 × 5 cents = 35 cents
40 cents – 35 cents = 5 cents
Stella has 5 cents more than Ian.
Part C
9. 38 cards
I needed to find numbers that, when divided by
5 had 3 left over, and when divided by 6 had 2
left over. This number also had to be less than
40. I found 38.
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Extra Practice Masters 4.25–4.31
Go to the CD-ROM to access editable versions of these Extra Practice Masters
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Program Authors
Peggy Morrow
Ralph Connelly
Steve Thomas
Jeananne Thomas
Maggie Martin Connell
Don Jones
Michael Davis
Angie Harding
Ken Harper
Linden Gray
Sharon Jeroski
Trevor Brown
Linda Edwards
Susan Gordon
Manuel Salvati
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All Rights Reserved. This publication is protected by copyright,
and permission should be obtained from the publisher prior to
any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical,
photocopying, recording, or likewise. For information regarding
permission, write to the Permissions Department.
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