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31/07/2019 1 www.drpaulswan.com.au | Developing a Whole School Approach to Mental Computation Day 2 – Basic Facts Milestones for Multiplication and Division Dr Paul Swan and David Dunstan www.drpaulswan.com.au | Key Ideas Strategies versus recall Understandings Sequence and pre-requisites www.drpaulswan.com.au | Expectations 36 x 25 mental? www.drpaulswan.com.au | Facts & Understandings 36 x 25 www.drpaulswan.com.au | Doubling and Halving 36 x 25 Halve x double 18 x 50 Halve x double 9 x 100 1 2 3 4 5 6

Developing a Whole School Computation · Dr Paul Swan and David Dunstan Developing a Whole School Approach22 | x5 and x10 facts Ideal time to introduce: •Halving •Doubling Dr

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Page 1: Developing a Whole School Computation · Dr Paul Swan and David Dunstan Developing a Whole School Approach22 | x5 and x10 facts Ideal time to introduce: •Halving •Doubling Dr

31/07/2019

1

www.drpaulswan.com.au |

Developing a Whole School Approach to Mental

Computation Day 2 – Basic Facts Milestones for Multiplication and Division

Dr Paul Swan and David Dunstan

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Key Ideas

•Strategies versus recall•Understandings•Sequence and pre-requisites

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Expectations

36 x 25 mental?

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Facts & Understandings

• 36 x 25

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Doubling and Halving

36 x 25Halve x double

18 x 50Halve x double

9 x 100

1 2

3 4

5 6

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Pre - requisites

• Trigger number (25)• Double and halve strategy• Number Sense

• Strategies• Bank of facts

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Factors and properties of number

36 x 254 x 9 x 25 (why not 6 x 6 x 25?)

9 x 4 x 25 (property?)9 x (4 x 25)

9 x 100

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36 x 25

• What is going wrong?

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Multiplication & Division

• How do you teach• How do children learn tables?• Teaching vs testing

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Learning a new set of multiplication facts• 18 x table• What do we know?

• 0 x 18 = 18• 1 x 18 = 18

• What strategies can we use to derive more?• Doubling

• 1 x 18 = 18• Double• 2 x 18 = 36• What would 4 x 18 =?• What about 8 x 18?

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Learning the 18 x Table

• What would 10 x 18 =?• Can I work out 5 x 18?

• What facts have I worked out• 1 x 18• 2 x 18• 4 x 18• 5 x 18• 8 x 18• 10 x 18

• Can I work out 3 x 18, 6 x 18, 9 x 18• What different strategies could I use?

• Could I become fluent?

7 8

9 10

11 12

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Plan - Milestones

• What are the expectations?• When do they need to know

them?• How will we know they’ve got

it?

Dr Paul Swan and David Dunstan Developing a Whole School Approach 13 www.drpaulswan.com.au |

AC: Basic facts x ÷ (Year 2)

1. “Lots of”

2. “Groups of”

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AC: Basic facts x ÷(Year 2)

3. Array model understanding of multiplication.

Dr Paul Swan and David Dunstan Developing a Whole School Approach 15

4 rows of 3

3 rows of 4

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AC: Basic facts x ÷(Year 3)Yr 3 ACMNA056• Recall multiplication facts of two, three,

five and ten and related division facts.

Dr Paul Swan and David Dunstan Developing a Whole School Approach 16

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Connected Chart

• Factor Factor Product x 0 1 2 3 4 5 6 7 8 9 10

0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10

2 0 2 4 6 8 10 12 14 16 18 20

3 0 3 6 9 12 15 18 21 24 27 30

4 0 4 8 12 16 20 24 28 32 36 40

5 0 5 10 15 20 25 30 35 40 45 50

6 0 6 12 18 24 30 36 42 48 54 60

7 0 7 14 21 28 35 42 49 56 63 70

8 0 8 16 24 32 40 48 56 64 72 80

9 0 9 18 27 36 45 54 63 72 81 90

10 0 10 20 30 40 50 60 70 80 90 100

Factor

Fact

or

Product

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Multiplication property of zero• 21 facts 0 x 0, 0 x 1, 0 x 2, 0 x 3, 0 x 4, 0 x 5,

0 x 6, 0 x 7, 0 x 8, 0 x 9 0 x 10 and related facts

x 0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0 0 0 0 0 0 01 0 1 2 3 4 5 6 7 8 9 102 0 2 4 6 8 10 12 14 16 18 203 0 3 6 9 12 15 18 21 24 27 304 0 4 8 12 16 20 24 28 32 36 405 0 5 10 15 20 25 30 35 40 45 506 0 6 12 18 24 30 36 42 48 54 607 0 7 14 21 28 35 42 49 56 63 708 0 8 16 24 32 40 48 56 64 72 809 0 9 18 27 36 45 54 63 72 81 9010 0 10 20 30 40 50 60 70 80 90 100

13 14

15 16

17 18

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Grid paper: Arrays

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Multiplication property of one19 new facts: • 1 x 1,• 1 x 2,• 1 x 3,• 1 x 4,• 1 x 5,• 1 x 6,• 1 x 7, • 1 x 8, • 1 x 9, • 1 x 10 • and related facts

x 0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0 0 0 0 0 0 01 0 1 2 3 4 5 6 7 8 9 102 0 2 4 6 8 10 12 14 16 18 203 0 3 6 9 12 15 18 21 24 27 304 0 4 8 12 16 20 24 28 32 36 405 0 5 10 15 20 25 30 35 40 45 506 0 6 12 18 24 30 36 42 48 54 607 0 7 14 21 28 35 42 49 56 63 708 0 8 16 24 32 40 48 56 64 72 809 0 9 18 27 36 45 54 63 72 81 9010 0 10 20 30 40 50 60 70 80 90 100

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Commutative Property of Multiplication• Each fact is related, that is 4 x 3 produces

the same result as multiplying 3 x 4

x 0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0 0 0 0 0 0 01 0 1 2 3 4 5 6 7 8 9 102 0 2 4 6 8 10 12 14 16 18 203 0 3 6 9 12 15 18 21 24 27 304 0 4 8 12 16 20 24 28 32 36 405 0 5 10 15 20 25 30 35 40 45 506 0 6 12 18 24 30 36 42 48 54 607 0 7 14 21 28 35 42 49 56 63 708 0 8 16 24 32 40 48 56 64 72 809 0 9 18 27 36 45 54 63 72 81 9010 0 10 20 30 40 50 60 70 80 90 100

4 rows of 3

3 rows of 4

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x2 Facts

• Relate to doubles addition facts (Year 2)

Dr Paul Swan and David Dunstan Developing a Whole School Approach 22

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x5 and x10 facts

Ideal time to introduce:• Halving• Doubling

Dr Paul Swan and David Dunstan Developing a Whole School Approach 23

x 0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0 0 0 0 0 0 01 0 1 2 3 4 5 6 7 8 9 102 0 2 4 6 8 10 12 14 16 18 203 0 3 6 9 12 15 18 21 24 27 304 0 4 8 12 16 20 24 28 32 36 405 0 5 10 15 20 25 30 35 40 45 506 0 6 12 18 24 30 36 42 48 54 607 0 7 14 21 28 35 42 49 56 63 708 0 8 16 24 32 40 48 56 64 72 809 0 9 18 27 36 45 54 63 72 81 9010 0 10 20 30 40 50 60 70 80 90 100

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Exposure to Doubling

Five rows of 2 Five rows of 4

Ten rows of 2

19 20

21 22

23 24

Page 5: Developing a Whole School Computation · Dr Paul Swan and David Dunstan Developing a Whole School Approach22 | x5 and x10 facts Ideal time to introduce: •Halving •Doubling Dr

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x3 Facts

• Page 40 Tackling Tables

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Array Game

• See Tackling tables p. 32 - 33

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Strategy: Relate to a known fact

• Implies that students have learned some facts• Askew, M. (1998). Teaching primary mathematics: A guide for newly

qualified and student teachers. London: Hodder & Stoughton

KNOWN NUMBER FACTS

DERIVE NUMBER FACTS

ARE USED TO HELP BUILD MORE

Askew, M. (1998). Teaching primary mathematics: A guide for newly qualified and student teachers. London: Hodder & Stoughton. www.drpaulswan.com.au |

Calculation in NAPLAN

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Start of Yr 4

• 2 – 4 weeks review of:• addition and subtraction facts• 2, 3, 5 and 10 facts• Assess

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What to do in Year 4

• Facts to be learned in Yr 4

Dr Paul Swan and David Dunstan Developing a Whole School Approach 30

x 0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0 0 0 0 0 0 01 0 1 2 3 4 5 6 7 8 9 102 0 2 4 6 8 10 12 14 16 18 203 0 3 6 9 12 15 18 21 24 27 304 0 4 8 12 16 20 24 28 32 36 405 0 5 10 15 20 25 30 35 40 45 506 0 6 12 18 24 30 36 42 48 54 607 0 7 14 21 28 35 42 49 56 63 708 0 8 16 24 32 40 48 56 64 72 809 0 9 18 27 36 45 54 63 72 81 9010 0 10 20 30 40 50 60 70 80 90 100

25 26

27 28

29 30

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What number facts Year 4?

• Recall multiplication facts to 10 x 10 (ACMNA075)• Use known multiplication facts to calculate related division facts

• Develop efficient mental … strategies for x and ÷ (no remainder) (ACMNA076)

• Using known facts and strategies such as commutativity, doubling and halving and connect to division

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Connection to Division

• Factor Factor Product Cards

Dr Paul Swan and David Dunstan Developing a Whole School Approach 32

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Multispin, Spindiv & Race Car Rally2, 3, 5

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Division - Sharing (Partition)

• The number of groups is known• The size of each group is found by a process of sharingSharing Problem• There are 18 bananas in a bunch• Three people will share them• How many for each person?

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Quotition (repeated subtraction)

• The size of each group is known• The number of groups is found by a process of repeated subtraction

(quotition)Quotition Problem:• There are 18 sunflowers• Three flowers are to be placed in each vase• How many vases are needed?

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Recording the operation-uses arrays

)18 divided by 3

3

6

31 32

33 34

35 36

Page 7: Developing a Whole School Computation · Dr Paul Swan and David Dunstan Developing a Whole School Approach22 | x5 and x10 facts Ideal time to introduce: •Halving •Doubling Dr

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Array for Division

)3

6

18 divided by 6

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Language

• Sharing language eventually replaced by the more formal language of ‘divided by’

• ‘goes into’ (gzinta) and ‘how many … in’ typically link to the repeated subtraction idea of division.

• Note ÷ symbol and ) symbol read in different ways. (read left to right, right to left)

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Thinking about the recording

)Number sharing

Number to be shared

Number each gets

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Introducing remainders• Share 17 among 3

)35 r 2

• 17 shared among 3 is 5 each; 2 remain

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Division with and without remainders

Dr Paul Swan and David Dunstan Developing a Whole School Approach 41

See Pocket Dice Book B pages 28/29 – “Diviso”

See Pocket Dice Book C page 22 – “Diviso

Remainders”www.drpaulswan.com.au |

Division Decision Game

Dr Paul Swan and David Dunstan Developing a Whole School Approach 42

37 38

39 40

41 42

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x9 Facts

PatternRelate to a known fact:

• 1 x 9 = 1 x 10 - 1

• 2 x 9 = 2 x 10 - 2

• 3 x 9 = 3 x 10 - 3

• 4 x 9 = 4 x 10 - 4

• 5 x 9 = 5 x 10 - 5

• 6 x 9 = 6 x 10 - 6

• 7 x 9 = 7 x 10 - 7

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Doubling

Five rows of 2 Five rows of 4

Dice Games for Tables

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x4 Facts

• Relate to x2, x4• Teach as a cluster

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x8 Facts

• Relate to x 2 , x 4• Teach as a cluster• Includes hardest table fact

Five rows of 2 Five rows of 4 Five rows of 8

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Known - unknown

• 7 x 8 hard table to learn

• 6 x 8 = 48 and one more 8 is 56

6 rows of 8

1 more row of 8

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Another way• 7 x 8 = 56• 56 = 7 x 8 (5 6 7 8)

43 44

45 46

47 48

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x6 Facts

• Further Resources

Dr Paul Swan and David Dunstan Developing a Whole School Approach 49

Networking Tables x6 Book

Tackling TablesPage 43

Multispin / Spindiv 6 Race Car Rally 6

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x7 Facts

• Single fact left to learn: 7 x 7

Dr Paul Swan and David Dunstan Developing a Whole School Approach 50

Networking Tables x6 Book

Tackling TablesPage 43

Multispin / Spindiv 6 Race Car Rally 6

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Doubling and Halving Yr 5 and 7 NAPLAN, 2008

• 8 x 3 = 4 x 6

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Square Numbers

Square numbers form squares.

Factor repeated.

Pattern

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Continued Practise

• COMBO Cardswww.drpaulswan.com.au |

Link Problem Solving and Fluency with Multo

• 1 x 1 – 10 x 10• Use products only once

• Download stickers fromwww.drpaulswan.com.au

49 50

51 52

53 54

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Student 1’s Multo Board

12 5 29 36

28 46 87 50

81 54 14 8

63 10 7 35 • Idea from Mathematics Assessment for Learning: Rich Tasks & Work Samples by Clarke et. al.

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Student 2’s Multo Board

1 2 3 4

20 81 90 49

18 25 9 10

32 35 36 28

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Student 3’s Multo Board

16 9 18 24

5 21 6 30

14 40 72 45

12 10 8 20

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Multo 1x 1 – 10 x 10 Chances4 Chances 3 Chances 2 Chances 1 Chance 0 Chances

6 4 2 1 118 9 3 25 1310 16 5 49 17. 36 7 64 .. . 81 .. . 100 .

.

9 4 23 6 58

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Positioning on the board

• Where numbers have been positioned makes a difference.

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Factors

• Factor trees

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55 56

57 58

59 60

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Factors

• Divide by prime numbers and continue as much as possible• 60 ÷ 2 = 30• 30 ÷ 2 = 15• 15 ÷ 3 = 5 (5 is a prime number)• Thus 60 = 2 x 2 x 3 x 5.• Knowledge of prime and composite numbers is handy.

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Divisibility: Ending rules

Multiples of:• 2• 5 and• 10

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

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Divisibility: Sum of Digits• Multiples of:

• 3 and• 9

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

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Develop Fact Families

Learn one thing, get five things free:• 7 x 8 = 56• 8 x 7 = 56• 56 ÷ 7 = 8• 56 ÷ 8 = 7• 1/7 of 56 = 8 (yr 6)• 1/8 of 56 = 7 (yr 6)• Make the links explicit

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Fractions of

Dr Paul Swan and David Dunstan Developing a Whole School Approach 65

Pocket Dice Book C

Page 39

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Extended Basic Facts

Dr Paul Swan and David Dunstan Developing a Whole School Approach 66

61 62

63 64

65 66

Page 12: Developing a Whole School Computation · Dr Paul Swan and David Dunstan Developing a Whole School Approach22 | x5 and x10 facts Ideal time to introduce: •Halving •Doubling Dr

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Cut and Count

Partitioning• DeNardi, E. (2004). Avanti Mental Maths, p. 45Partitioning: Multiplication• DeNardi, E. (2004). Avanti Mental Maths, p. 136

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Area model

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Area model (3a + 7)(2a + 5)

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Routine: If I know … then I also know…

10 x 5 = 50

11 x 5 =

9 x 5 =

5 x 5 =

50 ÷ 5 =

10 x 50 =

10 x 0.5 =

Explain why you know.

Show how each calculation is related.

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I can also see …

12 x 18

2 x 2 x 3 x 18

12 x 2 x 9 12 x 3 x 6

2 X 6 x 18

3 x 4 x 3 x 6

Are some calculations easier to complete that the original? Explain.

3 x 72

6 x 9 x 2 x 2

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I can also see… strategies

• Use of factors• Doubling and halving• Properties of number

• Commutativity• Associative property of multiplication

67 68

69 70

71 72

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Take it Easy

• If you had one wish and could change one number in the following question which one would you change and why?

17 x 9I would change … because

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Take it Easy

• Students might choose the ‘relate to a known fact strategy • 17 x 10

• Leads to the opportunity to discuss compensation 17 x 10 - 7• Or maybe doubling

• 18 x 9• 2 x 9 x 9 • 2 x 81 = 162

• Then discuss compensation need to subtract 9 from 162.

73 74