Multiplication and Division€¦ · Multiply the remaining three digits together, what is the product of the three numbers? Is the product smaller or larger than 84? Make the target

4Fluency Teaching Slides

Multiplicationand Division

4Fluency & Reasoning Teaching Slides

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11 and 12Times Tables

2 lots of 10 and 2 lots of 1 is the same as 2 lots of __

3 lots of 10 add 3 lots of 1 = __ x 11

Fill in the blanks.

4 - Multiplication & Division

Activity 1 11 and 12 Times Tables

3

2 lots of 10 and 2 lots of 1 is the same as 2 lots of __

3 lots of 10 add 3 lots of 1 = __ x 11

Fill in the blanks.


11

3


4

4 x 11 = 4 x 12 =


Using maths equipment, build and record the 11 and 12 times tables, then complete the calculations.


5

4 x 11 = 44


Using maths equipment, build and record the 11 and 12 times tables, then complete the calculations.


6

4 x 12 = 48

2 x 11 =

33 ÷ 11 =

x 11 = 44

77 ÷ 11 =

66 ÷ 11 =

8 x 11 =

2 x 11 =

132 ÷ 11 =

x 12 = 48

7 x 12 =

÷ 12 = 8

60 ÷ 12 =

12 ÷ 12 =

5 x 12 =

9 x 12 =

180 ÷ 12 =


Answer the questions below.

Can you spot any patterns between the 11 and 12 times tables?Why do you think this happens??


7

2 x 11 = 22

33 ÷ 11 = 3

x 11 = 44 4

77 ÷ 11 = 7

66 ÷ 11 = 6

8 x 11 = 88

2 x 11 = 22

132 ÷ 11 = 12

x 12 = 484

7 x 12 = 84

÷ 12 = 8 96

60 ÷ 12 = 5

12 ÷ 12 = 1

5 x 12 = 60

9 x 12 = 108

180 ÷ 12 = 15


Answer the questions below.


8

Reasoning 1

Here is one box of crayons.


Malachi has 10 boxes of crayons in his bag. How many crayons does he have altogether?

In each box there are four red crayons, five blue crayons, four yellow crayons, and two pink crayons. How many of each crayon

does Malachi have in 10 boxes?

Malachi lost five boxes of crayons.How many crayons does he have left?

11 and 12 Times Tables

9

10

Reasoning 1

Here is one box of crayons.


Malachi has 10 boxes of crayons in his bag. How many crayons

does he have altogether?

In each box there are four red crayons, five blue crayons, four yellow crayons, and two pink crayons. How many of each

crayon does Malachi have in 10 boxes?

Malachi lost five boxes of crayons.

How many crayons does he have left?

Malachi has 150 crayons altogether.

Red: 40Blue: 50

Yellow: 40Pink: 20

150 − 75 = 75

Malachi has 75 crayons left.


11

Reasoning 2

Leanna uses a bar model to represent 77 divided by 11.


7711 11 11 11 11 11 11 11 11 11 11

Explain Leanna’s mistake.

Can you draw a bar model to represent 77 divided by 11 correctly?


12

Reasoning 2


Corrected bar model representing 77 divided by 11:

7711 11 11 11 11 11 11

Leanna has divided by grouping into 11s but has found 11 groups of 11 which is equal to 121.

To divide 77 by sharing into 11 equal groups, there would be 7 in each group.

To divide 77 by grouping into 11s, there would be 7 groups of 11.


Leanna uses a bar model to represent 77 divided by 11.

7711 11 11 11 11 11 11 11 11 11 11

Discuss

Which multiplication and division facts in the 11 and 12 times tables have not appeared before in other times tables?

Can you partition 11 and 12 into tens and ones?

What times tables can we add together to help us multiply by 11 and 12?

If I know 11 x 10 is equal to 110, how can I use this to calculate 11 x 11?

13

11 and 12 Times-table




Multiply Three Numbers

I have __ lot of __, 3 times. I have __ lot of __, twice.

___ x ___ x ___ ___ x ___ x ___

8 x 3 12 x __

Complete the table.

What calculation do the cubes represent? If you rearrange the cubes, what calculation do they represent? What is the same? What is different??


Activity 1 Multiply Three Numbers

15

I have __ lot of __, 3 times. I have __ lot of __, twice.

___ x ___ x ___ ___ x ___ x ___

8 x 3 12 x __


1 8 1 12

2

3 2 4 2 3 4

Complete the table.


16

I have __ lot of __, 2 times I have __ lot of __, 3 times.

___ x ___ x ___ ___ x ___ x ___

8 x 2 6 x __

What calculation do the toys represent? If you rearrange the toys, what calculation do they represent? What is the same? What is different??

Complete the table.



17

I have __ lot of __, 2 times I have __ lot of __, 3 times.

___ x ___ x ___ ___ x ___ x ___

8 x 2 6 x __

Complete the table.


1 8

2 2 4

1 6

3 3 2

3


18

___ x ___ x ___

___ x ___

Complete the calculations.

___ x ___ x ___

___ x ___

=




19

___ x ___ x ___

___ x ___


___ x ___ x ___

___ x ___

=


3 2 2

3 4

2 3 2

2 6



20

How many different calculations can you make using your three digit cards?Which order do you find it the most efficient to calculate the product?

How have you grouped the numbers?

x x

Choose three single digit cards. Arrange them to create a multiplication calculation

and work out the answer.

Reasoning 1

=



21

How many different calculations can you make using your three digit cards?Which order do you find it the most efficient to calculate the product?

How have you grouped the numbers?

Choose three single digit cards. Arrange them to create a multiplication calculation

and work out the answer.

x x =


Possible answers using 2, 3 and 6:

2 x 3 x 6 = 362 x 6 x 3 = 363 x 2 x 6 = 363 x 6 x 2 = 366 x 2 x 3 = 366 x 3 x 2 = 36

Reasoning 1 Multiply Three Numbers

22

Multiply the remaining three digits together, what is the product of the three numbers?

Is the product smaller or larger than 84?

Make the target number of 84 using three of the digits below.


7 5 3 4 6 2

X X =

Reasoning 2 Multiply Three Numbers

23

Multiply the remaining three digits together, what is the product of the three numbers?

Is the product smaller or larger than 84?

Make the target number of 84 using three of the digits below.

Reasoning 2


7 5 3 4 6 2

X X =

Possible answers:

7 x 2 x 6 = 844 x 3 x 5 = 60

60 is smaller than 84

7 x 3 x 4 = 842 x 6 x 5 = 60

60 is smaller than 84


24

Discuss

Can you use concrete materials to build the calculations?

How will you decide in which order to do the multiplication?

What’s the same and what’s different about the arrays?

Which order do you find easier to calculate efficiently?

25





Factor Pairs

What factor pairs for 12 do these arrays show?

4 - Multiplication & Division 27

Activity 1 Factor Pairs




12 x 1

6 x 2 4 x 3







24 x 1

12 x 2 8 x 3

6 x 4

This is a factor bug.

Draw your own factor bugs for 16, 48, 56, 35.

Complete the factor bug for 36.

12

1 12

62

3 4

36

1 36

__2

__ 12

__4

__

6



This is a factor bug.


Complete the factor bug for 36.

12

1 12

62

3 4

36

1 36

__2

__ 12

__4

__

6


3

18

9

32


Factor bug for 16:


Factor bug for 48:

16

1 16

82

4 4


48

1 48

242

3 16

124

__

6

__

8

33


Factor bug for 56:


Factor bug for 35:

56

2 28

144

8 7


35

1 35

75

1 56

34


42 has _____ factors.

The factors of 42 are:

What methods do you use to find factors of a number?

Complete the sentences.

Do factors always come in pairs?Do whole numbers always have an even number of factors??



42 has _____ factors.


What methods do you use to find factors of a number?

Complete the sentences.


1 x 42 = 422 x 21 = 423 x 14 = 427 x 6 = 42

8



37

Factor PairsReasoning 1

Is Malachi correct?Use arrays to explain your answer.

The greater the number, the more factors it will have.


38


Is Malachi correct?Use arrays to explain your answer.

The greater the number, the more factors it will have.

Malachi is incorrect.For example:

15 has 4 factors: 1, 3, 5 and 15.17 has 2 factors: 1 and 17.


39



Some numbers are equal to the sum of all their factors(not including the number itself).

e.g. 6 6 has 4 factors, 1, 2, 3 and 6 Add up all the factors not including 6 itself. 1 + 2 + 3 = 6 6 is equal to the sum of its factors (not including the number itself)

How many other numbers can you find that are equal to the sum of their factors? Which numbers are less than the sum of their factors? Which numbers are greater than the sum of their factors?

40



Some numbers are equal to the sum of all their factors (not including the number itself).

e.g. 6 6 has 4 factors, 1, 2, 3 and 6 Add up all the factors not including 6 itself. 1 + 2 + 3 = 6 6 is equal to the sum of its factors (not including the number itself)

How many other numbers can you find that are equal to the sum of their factors? Which numbers are less than the sum of their factors? Which numbers are greater than the sum of their factors?

Possible answers:28 = 1 + 2 + 4 + 7 + 14 28 is equal to the sum of

its factors.

12 < 1 + 2 + 3 + 4 + 6 12 is less than the sum

of its factors.

8 > 1 + 2 + 48 is greater than the sum

of its factors.

Discuss

Which number is a factor of every whole number?

Do factors always come in pairs?

Do whole numbers always have an even number of factors?

How do arrays support us in finding factors of a number?

How do arrays support us in seeing when a number is not a factor of another number?

41

Factor Pairs




Efficient Multiplication

We have __ lots of __ x __

The question becomes 3 x 5 x 6

How could you use this to work out the answer?

15 x 6

3 x 5 x 6

We can use our knowledge of factors to help us solve 15 x 6.


Activity 1 Efficient Multiplication

43



How could you use this to work out the answer?

15 x 6

3 x 5 x 6


3 5 6



44



12 x 7

3 x 4 x 7




45



12 x 7

3 x 4 x 7


3 4 7



46

Ten lots of 4 = _____ One lot of 4 = _____

Eleven lots of 4 = _____

10 x 4 1 x 4

11 x 4 = ____ x 4 + 4

____ x 4 + ____ x 4 = 11 x 4

11 x 411

10 1

Use this method to solve:

21 x 5 31 x 6 7 x 22



47

Ten lots of 4 = _____ One lot of 4 = _____

Eleven lots of 4 = _____

11 x 4 = ____ x 4 + 4

____ x 4 + ____ x 4 = 11 x 4

11 x 411

10 1 10 x 4 1 x 4


21 x 5 31 x 6 7 x 22


40 4

44

10

10 1


48

21 x 5 = ____ x 5 + 5

____ x 5 + ____ x 5 = 21 x 520 1

Twenty lots of 5 = _____ One lot of 5 =_____

Twenty-one lots of 5 = _____

100 5

21 x 521

20 1


21 x 5 31 x 6 7 x 22


105

20

20 x 5 1 x 5


49

31 x 6 = ____ x 6 + 6

___ x 6 + ____ x 6 = 31 x 6

Thirty lots of 6 = _____ One lot of 6 =_____

Thirty-one lots of 6 = _____

180 6

30 1

31 x 631

30 1


21 x 5 31 x 6 7 x 22


186

30

30 x 6 1 x 6


50

7 x 22 = ____ x 7 + 14

____ x 7 + ____ x 7 = 7 x 2220 2

Twenty lots of 7 = _____ Two lots of 7 =_____

Twenty-two lots of 7 = _____

140 14

7 x 2222

20 2


21 x 5 31 x 6 7 x 22


154

20

20 x 7 2 x 7


51

11 x 2

Ten lots of 2 = __ One lot of 2 = __ Eleven lots of 2 = __

10 x 2 1 x 2

11 x 2 = __ = __ x 2 + 2

__ x 2 + __ x 2 = 11 x 2



52

11 x 2

Ten lots of 2 = __ One lot of 2 = __ Eleven lots of 2 = __

10 x 2 1 x 2

11 x 2 = __ = __ x 2 + 2

__ x 2 + __ x 2 = 11 x 2


20 2 22

22 10

10 1


53

Use this method to solve:22 x 519 x 448 x 68 x 33

19 x 7 = 20 x 7 − 1 x 7



54


22 x 5 = 30 x 5 − 8 x 5



55


19 x 4 = 20 x 4 − 1 x 4



56


48 x 6 = 50 x 6 − 2 x 6



57


8 x 33 = 40 x 8 − 7 x 8



58

59

Efficient MultiplicationReasoning 1

Can you explain his mistake and correct the diagram?


20 x 3 = 6060 − 1 = 5919 x 3 = 59

Zach has calculated 19 x 3.

60



20 x 3 = 6060 − 1 = 5919 x 3 = 59

Zach has calculated 19 x 3.

Zach has subtracted 1, rather than one group of 3. He should have calculated,

20 × 3 = 60 60 − 1 × 3 = 57

61



Here are three number cards.

21 42 38

Tia, Rosie and Esin choose one of the number cards each.

I did 40 x 5 and then subtracted 2 lots of five.

Tia says, I multiplied my number by 10 and then divided 210

by 2 Rosie says,

Esin says, I halved my 2-digit number and doubled 5 so I calculated 21 x 10

Which number card did each child have? Would you have used a different method to multiply the numbers by 5?

62



Here are three number cards.

21 42 38

Tia, Rosie and Esin choose one of the number cards each.

I did 40 x 5 and then subtracted 2 lots of five.

Tia says, I multiplied my number by 10 and then divided 210

by 2 Rosie says,

Esin says, I halved my 2-digit number and doubled 5 so I calculated 21 x 10

Which number card does each child have? Would you have used a different method to multiply the numbers by 5?

Tia has 38Rosie has 21Esin has 42

Discuss

Which method do you find the most efficient?

Can you see why another method has worked?

Can you explain someone else’s method?

Can you think of an effective way to multiply by 99?

63

Efficient Multiplication




Written Methods

Use this method to solve:36 x 451 x 210 x 18

Multiplication and Division - 4

22

22

22

T OT O

2 2

3x

6 (3 x 2)

(3 x 20)6 0

6 6

There are 22 cartons of drink in each box. How many drinks are there in three boxes?

Activity 1 Written Methods

65


36

36

36

T O T O

3 6

4x

4 (6 x 4)

(4 x 30 + 2 x 10)1 4 0

1 4 4


36 x 4

36

2


66



51

T O T O

5 1

2x

2 (1 x 2)

(50 x 2)1 0 0

1 0 2

36 x 4

51


67

Use this method to solve:36 x 451 x 210 x 18 Multiplication and Division - 4

18

T O

T O

1 0

1 8x

0 (0 x 8)(8 x 10)8 0

1 8 0

10 x 18

18

18

18

18

18

18

18

18

18

0 (1 x 0)

1 (1 x 1)


68


10 10 1 1 1 1

10 10 1 1 1 1

10 10 1 1 1 1

10 10 1 1 1 1

10 10 1 1 1 1

10 10100

2 4x 5

2 0 (5 x 4)

(5 x 20)1 0 0


Use place value counters to calculate 5 x 24.


69

There are 87 sweets in a bag.I buy four bags.

How many sweets do I have in total?



70

There are 87 sweets in a bag.I buy four bags.

How many sweets do I have in total?


4 bags x 87 sweets = 348 sweets in total


71

72

Written MethodsReasoning 1

Which of the multiplications would you calculate mentally? Which of the multiplications would you use a written method for?

Explain your choices to a partner. Did your partner choose the same methods as you?


Here are six multiplications.

43 x 5 54 x 6 38 x 6

33 x 2 19 x 7 84 x 5

73


Which of the multiplications would you calculate mentally? Which of the multiplications would you use a written method for?

Explain your choices to a partner. Did your partner choose the same methods as you?


43 x 5 54 x 6 38 x 6

33 x 2 19 x 7 84 x 5

Children will sort the

multiplications in different

ways.

Here are six multiplications.

74


Zach is calculating 36 multiplied by 4 using the part-whole model.


36 x 4= 1,224

30 x 4= 120

6 x 4= 24

Can you explain Zach’s mistake?

75


Zach is calculating 36 multiplied by 4 using the part-whole model.


36 x 4= 1,224

30 x 4= 120

6 x 4= 24

Can you explain Zach’s mistake?

Zach has multiplied the parts correctly, but added them up

incorrectly.120 + 24 = 144

Discuss

Why are there not 26 jumps of 8 on the number line?

Could you find a more efficient method?

Can you calculate the multiplication mentally or do you need to write down your method?

Can you partition your number into more than two parts?

76

Written Methods




Multiply Two Digits by One

Digit

T O

110 1 1

110 1 1

110 1 1

Calculate 13 x 3. Use place value counters and the formal method.

1 3x 3


Activity 1 Multiply Two Digits by One Digit

78

T O

110 1 1

110 1 1

110 1 1

Calculate 13 x 3. Use place value counters and the formal method.

1 3x 3

93 03 9

(3x3 = 9)

(3x10 = 30)



79

7 2

3x

2 6

7x

4 1

6x

1 9

5x

Calculate:



80

Calculate:


7 2

x 3

6

2 1 0

2 1 6

(3 x 2 = 6)

(3 x 7 = 21)

4 1

x 6

6

2 4 0

2 4 6

(6 x 1 = 6)

(6 x 4 = 24)

2 6

x 7

2

1 8 0

1 8 2

(6 x 7 =42)

(7 x 2 + 4 = 18)

1 9

x 5

5

9 0

9 5

4 4

(9 x 5 =45)

(5 x 1 + 4 = 9)


81

Each horse eats 36 carrots a day. How many do they eat altogether?



82

Each horse eats 36 carrots a day. How many do they eat altogether?


3 horses x 36 carrots = 108 total carrots eaten by three horses


83

84

Reasoning 1

Correct the multiplications.


Here are three incorrect multiplications.

T O

5 1

x 5

2 5

T O

6 4

x 7

4 2 8

T O

3 5

x 2

6 1 0

Multiply Two Digits by One Digit

85

Reasoning 1


Here are three incorrect multiplications.

T O

5 1

x 5

2 5

T O

6 4

x 7

4 2 8

T O

3 5

x 2

6 1 0

Corrected Multiplications

T O

5 1

x 5

2 5 5

2T O

6 4

x 7

4 4 8

2T O

3 5

x 2

7 0

1


86

Reasoning 2


Always, Sometimes, Never

Prove it.

• When multiplying a two-digit number by a one-digit number, the product has three digits.

• When multiplying a two-digit number by 8 the product is odd.

• When multiplying a two-digit number by 7 you need to exchange.


87

Reasoning 2


Always, Sometimes, Never

Prove it.

Sometimes: 12 x 2 has only two digits;23 x 5 has three

digits.

Never: all multiples of 8 are even.

Sometimes: most two-digit numbers need exchanging, but not 10 or 11.


• When multiplying a two-digit number by a one-digit number, the product has three digits.

• When multiplying a two-digit number by 8 the product is odd.

• When multiplying a two-digit number by 7 you need to exchange.

Discuss

Which column should we start with, the ones or the tens?

Can we create a list of key things to remember when multiplying using the column method?





Multiply Three Digits by One

Digit

H T O1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

100 100

100 100

100 100

Complete the calculation.

How is multiplying a three-digit number by one digit similarto multiplying a two-digit number by one digit??


H T O

2 0 6

X 3

Activity 1 Multiply Three Digits by One Digit

90

H T O1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

100 100

100 100

100 100



H T O

2 0 6

X 3

6 1 8

1


91

H T O1 1 1

1 1

1 1 1

1 1

1 1 1

1 1

100 100 10 10 10

100 100 10 10 10

100 100 10 10 10

1 1 1

1 1

1 1 1

1 1

100 100 10 10 10

100 100 10 10 10


H T O

2 3 5

X 5



92

H T O1 1 1

1 1

1 1 1

1 1

1 1 1

1 1

100 100 10 10 10

100 100 10 10 10

100 100 10 10 10

1 1 1

1 1

1 1 1

1 1

100 100 10 10 10

100 100 10 10 10


H T O

2 3 5

X 5

1 1 7 5



93

H T O1 1 1

1 1

1 1 1

1 1

1 1 1

1 1

100 100

100 100

100 100

10 10 10100

100

100

10 10 10

10 10 10

10 10 10

10 10 10

10 10 10

Use the place value counters to solve the problem.Remember, if there are 10 or more counters in a column, to

make an exchange.

A shop has 365 packets of sweets.Each packet contains three sweets.

How many sweets are there altogether?

H T O

3 6 5

X 3



94

H T O1 1 1

1 1

1 1 1

1 1

1 1 1

1 1

100 100

100 100

100 100

10 10 10100

100

100

10 10 10

10 10 10

10 10 10

10 10 10

10 10 10


make an exchange.

A shop has 365 packets of sweets.Each packet contains three sweets.

How many sweets are there altogether?

H T O

3 6 5

X 3

1 0 9 5



95

What multiplication calculation is represented?

H T O

100 100

100 100

100 100

10 10 10100

100

100


make an exchange.

100

100

100

10

10 10 10 10

10 10 10 10



96

H T O

100 100

100 100

100 100

10 10 10100

100

100


make an exchange.

100

100

100

10

10 10 10 10

10 10 10 10

H T O

4 4 0

X 3

1 3 2 0



97

What multiplication calculation is represented?

98

Reasoning 1

Malachi and Tia both completed the same multiplication.


Spot the mistake.

H T O

2 3 4

X 6

1 2 0 4

H T O

2 3 4

X 6

1 4 0 4

Malachi Tia

Who has the correct answer?

What mistake has been made by one of

the children?

2 2 2 2

Multiply Three Digits by One Digit

99

Reasoning 1

Malachi and Tia both completed the same multiplication.


Spot the mistake.

H T O

2 3 4

X 6

1 2 0 4

H T O

2 3 4

X 6

1 4 0 4

Malachi Tia

2 2 2 2

Tia has the correct answer.

Malachi has forgotten to add the two hundreds he exchanged from the tens column.


100

Reasoning 2


Zach and his mum were having a reading competition.In one month, Zach read 715 pages.

His mum read five times as many pages as he did. How many pages did they read altogether?

How many fewer pages did Zach read? Use the bar model to help.

715

715 715 715 715 715

Zach

Mum


101

Reasoning 2


Zach and his mum were having a reading competition.In one month, Zach read 715 pages.

His mum read five times as many pages as he did. How many pages did they read altogether?

How many fewer pages did Zach read? Use the bar model to help.

715

715 715 715 715 715

Zach

Mum

715 x 5 = 3,575

3,575 + 715 = 4,290They read 4,290 pages altogether.

715 x 4 = 2,860

Zach read 2,860 fewer pages than his mum.


Discuss

How is multiplying a three-digit number by one digit similar to multiplying a two-digit number by one digit?

Would you use counters to represent 84 multiplied by 8? Why?





Divide Two Digits by One

Digit (1)

TENS ONES

10

10

10

10

10

10

10

2. One tens and four ones left..1. Share the tens. 3. Exchange for 10 ones add remaining 4 ones.

11

11

1 1

11

11

10

1 1 11

1 1

1 1

Leanna solves 84 ÷ 7 like this.


Lesson 1 Divide Two Digits by One Digit (1)

104

TENS ONES

10

10

10

1

1

1

1

10

10

10

11

1 1

10

11

11

11

2. One tens and four ones left..1. Share the tens. 3. Exchange for 10 ones add remaining 4 ones.

11

11

1 1

11

11

4. Share the ones.

10

1 1 11

1 1

1 1



Lesson 1 Divide Two Digits by One Digit (1)

105

Step 1: Build the number Step 2: Share the tens Step 3: Share the ones

36

30 6

36

30 6

36

30 6

10

10

10

1 1

1 1

1 1

TENS ONES

÷ 3 = 10

10

10

10

TENS ONES

÷ 3 = 2

1 1

1 1

1 1

10

10

10

Use this approach to solve: 55 ÷ 5 120 ÷ 6 124 ÷ 4



Activity 1 Divide Two Digits by One Digit (1)

106

107

Reasoning 1

Tia is calculating 72 ÷ 3. Before she starts, she says the calculation

will involve an exchange.


Do you agree? Explain why.

Divide Two Digits by One Digit (1)

108

Reasoning 1

Tia is calculating 72 ÷ 3. Before she starts, she says the calculation

will involve an exchange.


Do you agree? Explain why.

Tia is correct because 70 is not a multiple of 3 so when you divide seven tens between three groups there will be one remaining which will be exchanged.


109

Reasoning 2


Use <, > or = to complete the statements.

63 ÷ 3 96 ÷ 3

96 ÷ 4 96 ÷ 3

91 ÷ 7 84 ÷ 6


63 ÷ 3 96 ÷ 3

110

Reasoning 2


Use <, > or = to complete the statements.

96 ÷ 4 96 ÷ 3

91 ÷ 7 84 ÷ 6

<

<

<


111

Reasoning 3

Leanna has 96 sweets.She shares them into equal groups.

She has no sweets left over.Into how many groups could Leanna have

shared her sweets?



112

Reasoning 3

Leanna has 96 sweets.She shares them into equal

groups.

She has no sweets left over.Into how many groups could

Leanna have shared her sweets?


Possible answers:96 ÷ 1 = 9696 ÷ 2 = 4896 ÷ 3 = 3296 ÷ 4 = 2496 ÷ 6 = 1696 ÷ 8 = 12


Discuss

How can we partition 84?

How many rows do we need to share equally between?

If I cannot share the tens equally, what do I need to do?

How many ones will I have after exchanging the tens?

If we know 96 ÷ 4 = 24, what will 96 ÷ 8 be?

What will 96 ÷ 2 be?

Can you spot a pattern?





Divide Two Digits by One

Digit (2)

115

Step 1: Build the number Step 2: Share the tens Step 3: Share the ones

86

80 6

T O

÷ 4

Use this approach to solve: 85 ÷ 3 67 ÷ 3 81 ÷ 4

110

10

10

10

10

10

10

10

1

1

1

1

1

86

80 6

80 ÷ 4 = 20

10 10

10 10

10 10

10 10

86

80 6

T O

10 10

10 10

10 10

10 10

1

1

1

1

1

1

6 ÷ 4 = 1 r 2

86 ÷ 4 = 21 r 2


Zach solves 86 ÷ 4 like this.


116

T O

Use this approach to solve: 97 ÷ 7 85 ÷ 3

10

110 10

10 10

10 10

77 ÷ 31 1 1

1 1 1 1

1 1 1 1 1 2 left over

Exchange for ten ones and share.


Zach uses the same approach but now the calculations involve an exchange.

1

1

1 1


117

Reasoning 1

She says 65 must be 2 away from a multiple of 3.

Do you agree?


65 ÷ 3 = 21 r 2

Esin writes,


118

Reasoning 1

She says 65 must be 2 away from a multiple of 3.

Do you agree?


65 ÷ 3 = 21 r 2

Esin writes,

I agree, remainder 2 means there

are 2 left over. 65 is two

more than 63 which is a

multiple of 3.


119

Reasoning 2

Tia says it’s 1Esin says 11Malachi says 11 r 1Rosie says 10 r 4

Can you explain who is correct and the mistakes other people have made?


45 sweets are shared between four friends. How many sweets are left over?

Four children attempt to solve this problem.


120

Reasoning 2

Tia says it’s 1Esin says 11Malachi says 11 r 1Rosie says 10 r 4

Can you explain who is correct and the mistakes other people have made?


45 sweets are shared between four friends. How many sweets are left over?

Four children attempt to solve this problem.

Tia is correct as there will be one remaining sweet.

Esin has found how many sweets each friend will

receive.Malachi has written the

answer to the calculation.Rosie has found a remainder that is equal to the divisor

so it is incorrect.


121

Reasoning 3


Zach is thinking of a two-digit number that is less than 50.

When it is divided by 2, there is no remainder.

When it is divided by 3, there is a remainder of 1.


What number is Zach thinking of?


122

Reasoning 3


Zach is thinking of a two-digit number that is less than 50.

When it is divided by 2, there is no remainder.



What number is Zach thinking of?

Zach is thinking of 28.


Discuss

If we are dividing by 3, what is the highest remainder we can have?

If we are dividing by 4, what is the highest remainder we can have?

Can we make a general rule comparing our divisor (the number we are dividing) to our remainder?





Divide Three Digits by One

Digit

Step 1: Build the number Step 2: Group the hundreds Step 3: Group the tens and ones

H T O

10100 1100

100 100

100 100

100 100

1

1 1

1 1

H T O

10100 1100

100 100

100 100

100 100

1

1 1

1 1

81642

H T O

10 1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

Exchange the ten for ten ones and then group the ones.

8164204

Use this method to solve: 678 ÷ 3 791 ÷ 7 216 ÷ 4

Zach solves 816 ÷ 4 like this.


Activity 1 Divide Three Digits by One Digit

100 100

100 100

100 100

100 100

125


235 ÷ 5 147 ÷ 7 432 ÷ 8

124

100 24

25 6

÷ 4

= 31

124

100 420

÷ 2÷ 2

÷ 2

21050

÷ 2÷ 2

÷ 2

255 1

25 + 5 + 1 = 31Multiplication and Division - 4

Esin uses partitioning and the part-whole model to help her calculate 124 ÷ 4.

Activity 2 Divide Three Digits by One Digit

126

127

Reasoning 1


Rosie is calculating 208 ÷ 8 using part-whole models.Can you complete each model?

208 ÷ 8

80 ÷ 8 80 ÷ 8 48 ÷ 8

208 ÷ 8

96 ÷ 8 96 ÷ 8 16 ÷ 8

208 ÷ 8

160 ÷ 8 48 ÷ 8

Divide Three Digits by One Digit

128

Reasoning 1


Rosie is calculating 208 ÷ 8 using part-whole models.Can you complete each model?

208 ÷ 8

80 ÷ 8 80 ÷ 8 48 ÷ 8

208 ÷ 8

96 ÷ 8 96 ÷ 8 16 ÷ 8

208 ÷ 8

160 ÷ 8 48 ÷ 8

208 ÷ 8 = 2680 ÷ 8 = 1048 ÷ 8 = 6

160 ÷ 8 = 2096 ÷ 8 = 1216 ÷ 8 = 2


129

Reasoning 2


You have 12 counters and the place value grid. You must use all 12 counters to complete the following:

Hundreds Tens Ones

Create a 3-digit number divisible by 2. Create a 3-digit number divisible by 3. Create a 3-digit number divisible by 4. Create a 3-digit number divisible by 5. Can you find a 3-digit number divisible by 6, 7, 8 or 9?


130

Reasoning 2


You have 12 counters and the place value grid. You must use all 12 counters to complete the following:

Hundreds Tens Ones

Create a 3-digit number divisible by 2. Create a 3-digit number divisible by 3. Create a 3-digit number divisible by 4. Create a 3-digit number divisible by 5. Can you find a 3-digit number divisible by 6, 7, 8 or 9?

2: Any even number3: Any digit number (as the digits add up to 12, a multiple of 3)

4: A number where the last two digits are a multiple of 4

5: A number with 0 or 5 in the ones column.

Possible answers: 6: Any even number

7: 7148: 840

9: impossible


Discuss

What is the same and what’s different when we are dividing three-digit number by a one-digit number and a two-digit

number by a one-digit number?

Do we need to partition 609 into three parts or could it just be partitioned into two parts?

Can we partition the number in more than one way to support dividing more efficiently?





Correspondence Problems

Find the total number of vertices for these sets of shapes in the same way:

4 x 6 + 3 x 3 = 33

Malachi says he can represent the total number of vertices of his shapes like this:


Activity 1 Correspondence Problems

133

Find the total number of vertices for these sets of shapes in the same way:

4 x 6 + 3 x 3 = 33

Malachi says he can represent the total number of vertices of his shapes like this:

6 x 3 + 1 x 5 + 5 x 4 = 43

6 x 4 + 3 x 8 = 48



134

23 22 21

Use circles, pentagons and squares to represent thefollowing vertices:



135

Represent this with manipulatives.

1 10


Activity 3 Correspondence ProblemsUsing the 6 and 4 times tables, how many different ways can you

make a total of 40?

136

137

Correspondence ProblemsReasoning 1


Here are the meal choices in the school canteen.

Starter Main Dessert

SoupGarlic Bread

PastaChicken

BeefSalad

CakeIce creamFruit salad

There are two starter options, four main options and three dessert options.

How many meal combinations can you find? Can you use a systematic approach? Can you represent the combinations in a multiplication?

If there were 20 meal combinations, how many starters, mains and desserts might there be?

138



Here are the meal choices in the school canteen.

Starter Main Dessert

SoupGarlic Bread

PastaChicken

BeefSalad

CakeIce creamFruit salad

There are two starter options, four main options and three dessert options.

How many meal combinations can you find? Can you use a systematic approach? Can you represent the combinations in a multiplication?

If there were 20 meal combinations, how many starters, mains and desserts might there be?

There are 24 meal combinations altogether.

2 × 4 × 3 = 24

20 combinations 1 × 1 × 20 1 × 2 × 10 1 × 4 × 5

2 × 2 × 5 etcAccept all other

variations of these four multiplications

e.g. 1 × 20 × 1

139



Zach has six T-shirts and four pairs of shorts.

Malachi has 12 T-shirts and two pairs of shorts.

Who has the most combinations of T-shirts and shorts? Explain your answer.

140



Zach and Malachi have the same number of combinations of T-shirts and shorts.

6 x 4 = 24 12 x 2 = 24

Zach has six T-shirts and four pairs of shorts.

Malachi has 12 T-shirts and two pairs of shorts.

Who has the most combinations of T-shirts and shorts? Explain your answer.

Discuss

Can you use a table to support you to find all the combinations?

Can you use a code to help you find the combinations? E.g. VS meaning Vanilla and Sauce

Can you use coins to support you to make all the possible combinations?

141

Correspondence Problems


Multiplication and Division€¦ · Multiply the remaining three digits together, what is the product of the three numbers? Is the product smaller or larger than 84? Make the target

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