100% coverage of New Zealand Mathematics and Statistics Curriculum for Phases 1-3
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Strand(s):
• CB: p. 71 • Math Pro: Recall
• Multiply numbers within the multiplication table of 3 and 5
• Use related multiplication facts to divide by 7 and 9
Let’s Remember
Unit 1: Multiplication and Division Facts for 2s to 10s New Zealand Curriculum Year 5: Number — Operations
1.1 Practices ( ◆ ) ◆ Memorising multiplication and corresponding division facts for 2s to 12s
• CB: pp. 72–73
• PB: pp. 47 • Math Pro: Practice
• Recall multiplication facts up to 10 × 10 and corresponding division facts
1.1 Multiplication and division facts up to 10 × 10
Unit 2: Multiplying and Dividing by 11 New Zealand Curriculum Year 5: Number — Operations
2.1 Practices ( ◆ ) ◆ Memorising multiplication and corresponding division facts for 2s to 12s
• CB: pp. 74–76
• PB: p. 48 • Math Pro: Practice
• Observe the commutative and distributive properties of multiplication
2.1 Multiplying by 11
• Multiply numbers within the multiplication table of 11
• Build up the multiplication table of 11 and commit the multiplication facts to memory
2.2 Practices ( ◆ )
Memorising multiplication and corresponding division facts for 2s to 12s
• Divide numbers using the multiplication table of 11
• CB: p. 76 • PB: p. 49 • Math Pro: Practice Unit 3: Multiplying and Dividing by 12 New Zealand Curriculum Year 5: Number — Operations
• CB: pp. 77–80 • PB: p. 50 • Math Pro: Practice
2.2 Dividing by 11
3.1 Practices ( ◆ )
Memorising multiplication and corresponding division facts for 2s to 12s
• Connecting cubes
• Observe the commutative and distributive properties of multiplication
3.1 Multiplying by 12
• Multiply numbers within the multiplication table of 12
• Build up the multiplication table of 12 and commit the multiplication facts to memory
3.2 Practices ( ◆ )
Memorising multiplication and corresponding division facts for 2s to 12s
• Divide numbers using the multiplication table of 12
• CB: p. 80 • PB: p. 51 • Math Pro: Practice Unit 4: Problem Solving New Zealand Curriculum Year 5: Number — Operations
• CB: pp. 81–83
• PB: p. 52–53 • Math Pro: Practice
• CB: pp. 84–86
• Math Pro: Assessment
3.2 Dividing by 12
4.1 Practices ( ◆ )
Memorising multiplication and corresponding division facts for 2s to 12s
• Solve 1-step word problems involving multiplication or division
4.1 Word problems
• Use a part-whole bar model to represent a multiplication or division situation
• Solve a non-routine problem involving multiplication tables using the strategy of drawing a bar model
4.2 Mind stretcher
• Solve a non-routine problem involving multiplication table of 12 using the strategy of looking for a pattern
The suggested duration for each lesson is 1 hour.
Chapter 3 Multiplication Tables
Chapter Overview
Let’s Remember
Unit 1: Multiplication and Division Facts for 2s to 10s
Unit 2: Multiplying and Dividing by 11
Unit 3: Multiplying and Dividing by 12
Unit 4: Problem Solving
Let's Remember
Recall:
1. Multiplying numbers within the multiplication table of 3 and 5 (CB2 Chapter 8 and CB3 Chapter 7)
2. Using related multiplication facts to divide by 7 and 9 (CB4 Chapter 4)
EXPLORE
Have students read the word problem on CB p. 71. Discuss with students the following questions:
• Do you do charity?
• What things can you donate?
• How do you feel when you help someone?
Have students form groups to complete the tasks in columns 1 and 2 of the table. Let students know that they do not have to solve the word problem. Ask the groups to present their work.
Tell students that they will come back to this word problem later in the chapter.
EXPLORE
1.1 Multiplication and division facts up to 10 × 10
Let's Learn Let's Learn
Objective:
• Recall multiplication facts up to 10 × 10 and corresponding division facts
Resources:
• CB: pp. 72–73
• PB: p. 47
(a) Stages: Pictorial and Abstract Representations
Write: 2 × 5 = ÷ 5 = 2
Have students look at the multiplication chart shown on CB p. 72.
Say: This is a multiplication chart. Let us use it to complete the related multiplication and division facts.
Have students point to 2 below the multiplication sign on the blue square.
Say: Let’s use the chart to multiply 2 by 5. We start from 2 . Since we are multiplying by 5, we move along the row from the number 2 to the column with the number 5 .
Have students move along the row and stop when they reach the column with the number 5
Ask: What is 2 × 5? (10)
Complete the multiplication sentence on the board.
Say: Since 2 multiplied by 5 gives 10, 2 and 5 are factors of 10, and 10 is a multiple of 2 and 5.
Ask: What number divided by 5 gives 2? (10)
Complete the division sentence on the board.
Point to students that we can multiply numbers in any order. So, 2 × 5 is the same as 5 × 2.
Say: We can also find the product by moving along the row from the number 5 to the column with number 2
6121824303642485460
7142128354249566370
102030405060708090100
Write: 5 × 2 = ÷ 2 = 5
Ask: What is 5 × 2? (10)
Complete the multiplication sentence on the board.
Say: Since 2 × 5 and 5 × 2 give 10, they are related multiplication facts.
Ask: Since 5 multiplied by 2 gives 10, what number divided by 2 gives 5? (10)
Say: When we want to find the answer to a division sentence, we can use a related multiplication fact to help us.
(b) Stages: Pictorial and Abstract Representations
Write: × 9 = 72
72 ÷ 9 =
Say: Besides finding the product of two numbers, we can also use the multiplication chart to find a factor of a number.
Refer students to the first thought bubble and explain the steps for using the chart to find a number that multiplies with 9 to give 72. Lead students to see that the answer is 8 and then complete the multiplication sentence on the board.
Say: Since 8 multiplied by 9 gives 72, 8 and 9 are factors of 72, and 72 is a multiple of 8 and 9.
Ask: What number do we get when we divide 72 by 9? (8)
Complete the division sentence on the board.
Write: × 8 = 72
72 ÷ 8 =
Lead students to similarly deduce the answers for this set of related multiplication and division facts by using the multiplication chart in a similar way as before.
Have students realise that both 8 × 9 and 9 × 8 give 72 and that they can use a multiplication sentence to find the answer to a related division sentence.
Let's Do
and
Let's Practise Let's Practise
Task 1 requires students to complete multiplication facts up to 10 × 10 and the corresponding division facts.
Unit 2: Multiplying and Dividing by 11
2.1 Multiplying by 11
Let's Learn Let's Learn
Objectives:
• Observe the commutative and distributive properties of multiplication
• Multiply numbers within the multiplication table of 11
• Build up the multiplication table of 11 and commit the multiplication facts to memory
Resources:
• CB: pp. 74–76
• PB: p. 48
(a) Stage: Pictorial Representation
Copy the figure in (a) on CB p. 74 on the board.
Ask: How many rows are there? (2) How many squares are there in each row? (11)
Say: There are 2 rows of 11 squares. There are 2 × 11 squares.
Ask: How many squares are there altogether? (22)
Stage: Abstract Representation
Write: 2 × 11 = 22
(b) Stages: Pictorial and Abstract Representations Have students look at the figure in (b) on CB p. 74.
Ask: How many rows are there? (9) How many squares are there in each row? (11)
Say: There are 9 rows of 11 squares. There are 9 × 11 squares.
Ask: How many squares are there altogether? (99)
Write: 9 × 11 = 99
Ask: How many columns are there? (11) How many squares are there in each column? (9)
Say: There are 11 columns of 9 squares. There are 11 × 9 squares.
Ask: How many squares are there altogether? (99)
Write: 11 × 9 = 99
Say: 9 × 11 = 99 and 11 × 9 = 99 are related multiplication facts. We can multiply the numbers in any order and get the same answer.
(c) Stages: Pictorial and Abstract Representations
Have students look at the dot card on the left in (c) on CB p. 75.
Say: This dot card shows 10 groups of 11 or 10 × 11.
Ask: How many dots are there in this dot card?
(110)
Guide students to count by 10s to find the total number of dots in the 11 columns of the dot card.
Write: 10 × 11 = 110
Say: Let us find 11 × 11.
Write: 11 × 11 =
Ask: How many groups of 11 do we add to the dot card to show 11 × 11? (1 group of 11)
Have students look at the dot card on the right on the page.
Say: This dot card shows 11 groups of 11 or 11 × 11. 11 × 11 is 11 more than 10 × 11.
Explain that since 10 × 11 = 110, 11 × 11 = 11 + 110 = 121.
Write: 11 × 11 = 121
(d) Stages: Pictorial and Abstract Representations
Write: 12 × 11 =
Say: We want to multiply 12 and 11. Let us start with a multiplication fact that we know, 10 × 11. Have students look at the dot card on the left in (d) on CB p. 75.
Say: This dot card shows 10 groups of 11 or 10 × 11.
Ask: How many dots are there in this dot card? (110)
Guide students to count by 10s to find the total number of dots in the 11 columns of the dot card.
Write: 10 × 11 = 110
Ask: How many groups of 11 do we add to the dot card to show 12 × 11? (2 groups of 11)
Have students look at the dot card on the right on the page.
Say: This dot card shows 12 groups of 11 or 12 × 11. 12 × 11 is 2 groups of 11 more than 10 × 11.
Ask: What are 2 groups of 11? (22)
Explain that we can find 12 × 11 by adding 22 to 10 × 11. Since 10 × 11 = 110, 12 × 11 = 22 + 110 = 132.
Write: 12 × 11 = 132
Have students read aloud the multiplication table of 11 on the page. Have them observe the products of 1 × 11 to 9 × 11.
Ask: What pattern do you observe in the products of 1 × 11 to 9 × 11? (When we multiply 11 by a 1-digit number, the digits in the tens and ones places of the product are the same as the 1-digit number.)
Point out to students that they can use this pattern to help them remember the first nine multiplication facts of 11.
Let's Do Let's
Task 1 requires students to multiply numbers within the multiplication table of 11 and apply the commutative property of multiplication.
Let's Practise
Let's Practise
Task 1 requires students to multiply numbers within the multiplication table of 11.
2.2 Dividing by 11
Let's Learn Let's Learn
Objective:
• Divide numbers using the multiplication table of 11
Resources:
• CB: p. 76
• PB: p. 49
(a) Stage: Abstract Representation
Write: 66 ÷ 11 =
Say: Let us use a related multiplication fact to help us find the answer.
Write: × 11 = 66
Ask: What number multiplied by 11 is equal to 66? (6)
Complete the multiplication sentence.
Say: Since 6 × 11 = 66, we know that 66 ÷ 11 = 6.
Complete the division sentence on the board.
(b) Stage: Abstract Representation
Follow the procedure in (a).
Let's Do Let's and
Let's Practise Let's Practise
Task 1 requires students to divide numbers using the multiplication table of 11.
EXPLORE
Have students go back to the word problem on CB p. 71.
Ask: Can you solve the problem now? (Answer varies.) What else do you need to know? (Answer varies.)
Students are not expected to be able to solve the problem now. They will learn more skills in subsequent lessons and revisit this problem at the end of the chapter.
3.1 Multiplying by 12
Let's Learn Let's Learn
Objectives:
• Observe the commutative and distributive properties of multiplication
• Multiply numbers within the multiplication table of 12
• Build up the multiplication table of 12 and commit the multiplication facts to memory
Materials:
• Connecting cubes
Resources:
• CB: pp. 77–80
• PB: p. 50
(a) Stage: Concrete Experience
Have students work in groups. Distribute connecting cubes to each group.
Ask students to put 12 connecting cubes into equal groups in different ways. Invite students to share their answers. Guide students to realise that the 12 cubes can be put into 1 group of 12, 2 groups of 6, 3 groups of 4, 4 groups of 3, 6 groups of 2 or 12 groups of 1.
Stages: Pictorial and Abstract Representations
Have students look at the first figure in (a) on CB p. 77.
Ask: How many rows are there? (1) How many squares are there in each row? (12)
Say: There is 1 row of 12 squares. There are 1 × 12 squares.
Ask: How many squares are there altogether? (12)
Write: 1 × 12 = 12
Ask: How many columns are there? (12) How many squares are there in each column? (1) Say: There are 12 columns of 1 square. There are 12 × 1 squares.
Ask: How many squares are there altogether? (12)
Write: 12 × 1 = 12
Say: 1 × 12 = 12 and 12 × 1 = 12 are related multiplication facts.
Unit 3 Multiplying and Dividing by 12
3.1 Multiplying by 12
Let's Learn
Have students look at the second figure on the page and guide them to conclude that 2 × 6 = 12 and 6 × 2 = 12 are related multiplication facts. Explain that 2 and 6 are factors of 12 so the multiplication table of 12 is related to the multiplication tables of 2 and 6. Have students look at the third figure on the page and guide them to conclude that 3 × 4 = 12 and 4 × 3 = 12 are related multiplication facts. Explain that 3 and 4 are factors of 12 so the multiplication table of 12 is related to the multiplication tables of 3 and 4. Summarise that the multiplication table of 12 is related to the multiplication tables of 2, 3, 4 and 6.
(b) Stage: Abstract Representation
Say: 6 is a factor of 12, and 12 is double 6. So, when we double the products in the multiplication table of 6, we get the products in the multiplication table of 12.
Have students look at the multiplication tables in (b) on CB p. 78. Point out to students that the products in the multiplication table of 12 are double the corresponding products in the multiplication table of 6.
Say: Let us find 5 × 12. We can find 5 × 12 by doubling the product of 5 and 6.
Ask: What is 5 × 6? (30) What is double 30? (60) So, what is the product of 5 and 12? (60)
Write: 5 × 12 = 60
Say: Let us find 9 × 12.
Ask: How can we find the product of 9 and 12 using the multiplication table of 6? (Double the product of 9 and 6.) What is 9 × 6? (54) What is double 54? (108) So, what is the product of 9 and 12? (108)
Lead students to see that it is easier to multiply by 2 than it is to multiply by 12. So, if we know the multiplication table of 6 well, we can simply take the product of 6 and a number and multiply it by 2 to get the product of 12 and that same number.
(c) Stage: Abstract Representation
Say: 4 is a factor of 12, and 12 is triple or three times of 4. So, when we triple the products in the multiplication table of 4, we get the products in the multiplication table of 12. Have students look at the multiplication tables in (c) on CB p. 78. Point out to students that the products in the multiplication table of 12 are triple the corresponding products in the multiplication table of 4.
Say: Let us find 5 × 12 again. We can find 5 × 12 by multiplying the product of 5 and 4 by 3.
Ask: What is 5 × 4? (20) What is 3 × 20? (60) So, what is the product of 5 and 12? (60)
Write: 5 × 12 = 60
Say: Let us find 9 × 12.
Ask: How can we find the product of 9 and 12 using the multiplication table of 4? (Multiply the product of 9 and 4 by 3.) What is 9 × 4? (36) What is 3 × 36? (108) So, what is the product of 9 and 12? (108)
Lead students to see that it is easier to multiply by 3 than it is to multiply by 12. So, if we know the multiplication table of 4 well, we can simply take the product of 4 and a number and multiply it by 3 to get the product of 12 and that same number.
(d) Stages: Pictorial and Abstract Representations
Write: 11 × 12 = ______
Say: We want to multiply 11 and 12. Let us start with a multiplication fact that we know, 10 × 12. Have students look at the dot card on the left in (d) on CB p. 79.
Say: This dot card shows 10 groups of 12 or 10 × 12.
Ask: How many dots are there in this dot card? (120)
Guide students to count by 10s to find the total number of dots in the 12 columns of the dot card.
Write: 10 × 12 = 120
Ask: How many groups of 12 do we add to the dot card to show 11 × 12? (1 group of 12)
Have students look at the dot card on the right on the page.
Say: This dot card shows 11 groups of 12 or 11 × 12. 11 × 12 is 1 group of 12 more than 10 × 12.
Explain that since 10 × 12 = 120, 11 × 12 = 12 + 120 = 132.
Write: 11 × 12 = 132
(e) Stages: Pictorial and Abstract Representations
Write: 12 × 12 =
Say: We want to multiply 12 and 12. Let us start with a multiplication fact that we know, 10 × 12. Have students look at the dot card on the left in (e) on CB p. 79.
Say: This dot card shows 10 groups of 12 or 10 × 12.
Ask: How many dots are there in this dot card? (120)
Guide students to count by 10s to find the total number of dots in the 12 columns of the dot card.
Write: 10 × 12 = 120
Ask: How many groups of 12 do we add to the dot card to show 12 × 12? (2 groups of 12)
Have students look at the dot card on the right on the page.
Say: This dot card shows 12 groups of 12 or 12 × 12. 12 × 12 is 2 groups of 12 more than 10 × 12.
Ask: What are 2 groups of 12? (24)
Explain that we can find 12 × 12 by adding 24 to 10 × 12. Since 10 × 12 = 120, 12 × 12 = 24 + 120 = 144.
Write: 12 × 12 = 144
Remind students that the commutative property of multiplication also applies to the multiplication table of 12. So, we can derive multiplication facts of 12 using known facts of other multiplication tables. For example, we can find the 2 × 12 by finding 12 × 2.
Let's Do Let's Do
Task 1 requires students to multiply numbers within the multiplication table of 12 and apply the commutative property of multiplication.
Let's Practise
Let's Practise
Task 1 requires students to multiply numbers within the multiplication table of 12.
3.2 Dividing by 12
Let's Learn Let's Learn
Objective:
• Divide a number using the multiplication table of 12
Resources:
• CB: p. 80
• PB: p. 51
(a) Stage: Abstract Representation
Write: 36 ÷ 12 =
Say: Let us use a related multiplication fact to help us find the answer.
Write: × 12 = 36
Ask: What number multiplied by 12 is equal to 36? (3)
Complete the multiplication sentence.
Say: Since 3 × 12 = 36, we know that 36 ÷ 12 = 3.
Complete the division sentence on the board.
(b) Stage: Abstract Representation
Follow the procedure in (a).
Let's Do Let's and Let's Practise Let's Practise
Task 1 requires students to divide numbers using the multiplication table of 12.
3.2
Dividing by 12
Unit 4 Problem Solving
4.1 Word problems
4. Check if your answer is correct. Guide students to check their answer using the related division fact. Say: Since 36 ÷ 12 is equal to 3, the answer is correct.
Solve the word problems. Show your work clearly.
1. Wiremu bought 6 boxes of marbles. There were 11 marbles in each box. How many marbles did he buy altogether?
2. Ariki saved $4 in a week. His sister saved 12 times as much money as him. How much money did Ariki’s sister save? 3. Oliver jogged a total distance of 44 kilometres over 11 days. He jogged the same distance each day. How many kilometres did he jog each day?
144 children were divided into teams of 12. How many teams were there?
CREATE
YOUR OWN Grace cousin 3 years
Read the given words.
Use the words and the bar model to write a multiplication word problem. How did you come up with the word problem?
Answer varies. Sample: Grace is 3 years old. Her cousin is 12 times as old as her. How old is Grace’s cousin? 12 × 3 = 36 Grace’s cousin is 36 years old.
Next, solve the word problem. Show your work clearly. What did you learn?
has five times as many toy cars as Jack. How many toy cars does Noah have?
5. + Plus Solve the problem in another way. Have students try to solve the problem in a different way. Have 1 or 2 students share their methods. If students are unable to solve the problem in a different way, explain the method shown on CB p. 82. Ask: Which method do you prefer? Why? (Answers vary.)
Let's Do Let's
Task 1 requires students to solve a 1-step word problem involving division.
Task 2 requires students to solve a 1-step word problem involving multiplication.
Let's Practise
Tasks 1 and 2 require students to solve 1-step word problems involving multiplication.
Tasks 3 and 4 require students to solve 1-step word problems involving division.
CREATE YOUR OWN
Have students work in pairs. Get students to create a word problem and exchange the word problem with their partner. Ask students to solve the word problem from their partner. Have a few pairs of students present their work. They should first explain how they came up with the word problem and their partner has to explain the solution.
Students are expected to write a word problem associated with the comparison bar model such that one quantity is 12 times as many as the other quantity. ‘3 years’ should be used such that the problem can be solved by recalling multiplication facts of 12.
Let's Learn Let's Learn
Objectives:
• Solve a non-routine problem involving multiplication tables using the strategy of drawing a bar model
• Solve a non-routine problem involving multiplication table of 12 using the strategy of looking for a pattern
Resources:
• CB: pp. 84–86
Have students read the word problem on CB p. 84.
1. Understand the problem.
Pose the questions in the thought bubble in step 1.
2. Plan what to do.
Say: We can draw a bar model to help us solve the problem.
3. Work out the Answer.
Say: Viliami is 3 years old this year and his mother is 11 times as old as him.
Draw a comparison bar model as shown in the first bar model on the page.
Ask: How do we find the age of Viliami’s mother this year? (Multiply Viliami’s age this year by 11.)
Write: 3 × 11 = 33
Say: Viliami’s mother is 33 years old this year. We want to find the number of years when she will be 3 times as old as Viliami.
Draw a comparison bar model as shown in the second bar model on the page.
Explain to students that the difference in age always remains the same.
Say: Let us find the difference in their ages.
Write: 33 – 3 = 30
Ask: How much older is Viliami’s mother than Viliami? (30 years)
Say: Viliami’s mother will always be 30 years older than Viliami.
Label the difference between the two bars in the second bar model as ‘30’.
4.2 Mind stretcher
Viliami
years
3 × 11 =
33 − 3 = 30
30 ÷ 2 = 15 Viliami will be 15 years old when his mother is 3 times his age.
15 − 3 = 12 In 12 years’ time, Viliami’s mother will be 3 times his age.
Say: From the second bar model, we can see that 2 units represent 30 years and 1 unit represents Viliami’s age. So, we can find Viliami’s age in the future by dividing 30 years by 2.
Write: 30 ÷ 2 = 15
Say: Viliami will be 15 years old when his mother is 3 times his age.
Ask: How old is Viliami now? (3 years old) So, in how many years’ time will Viliami’s mother be 3 times his age? (12)
Write: 15 – 3 = 12
Say: In 12 years’ time, Viliami’s mother will be 3 times his age.
4. Check if your answer is correct. Say: We can check the answer by finding the age of Viliami’s mother in 12 years’ time and then subtracting her present age to see if it is also 12 years.
Write: 15 × 3 = 45
Say: Viliami’s mother will be 45 years old when Viliami is 15 years old.
Write: 45 – 33 = 12
Say: Viliami’s mother will be 45 years old in 12 years’ time. So, the answer is correct.
5. + Plus Solve the problem in another way. Have students try to solve the problem in a different way.
Have 1 or 2 students share their methods. If students are unable to solve the problem in a different way, explain the method shown on CB p. 85.
Ask: Which method do you prefer? Why? (Answers vary.)
Let's Do Let's
Students have to figure out the rule of the number pattern to find the two missing numbers. To do this, they should first find the difference between each pair of adjacent numbers. Next, they should observe that the differences between each pair of adjacent numbers is 12. Hence, students should subtract 12 from 964 to get 952, and another 12 from 952 to get 940.
EXPLORE
Have students go back to the word problem on CB p. 71. Get them to write down in column 3 of the table what they have learnt that will help them solve the problem, and then solve the problem.
Have a student present his/her work to the class.
Maths Journal Journal
Have students work on the tasks in Maths Journal (BM3.1) independently to check and reinforce their understanding.
Use the rubric provided on page 340 of the blackline masters to score students’ work.
1,000, 988, 976, 964, , 928, 916, 904
