36 x 25 Strategies versus recall Understandings mental? Sequence - - PDF document

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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 25 Halve x double 18 x 50 Halve 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 25 4 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?

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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 www.drpaulswan.com.au |

Connected Chart

• Factor Factor Product

x 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 10 20 30 40 50 60 70 80 90 100 Factor Factor Product

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Multiplication property

• f 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 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 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 one

19 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 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 10 20 30 40 50 60 70 80 90

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Commutative Property

• f Multiplication
• Each fact is related, that is 4 x 3 produces

the same result as multiplying 3 x 4

x 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 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 www.drpaulswan.com.au |

x5 and x10 facts

Ideal time to introduce:

• Halving
• Doubling

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

x 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 10 20 30 40 50 60 70 80 90

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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

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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.

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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 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 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 www.drpaulswan.com.au |

Multispin, Spindiv & Race Car Rally 2, 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 sharing

Sharing 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

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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

)

3 5 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”

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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

Pattern

Relate 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 Tables Page 43 Multispin / Spindiv 6 Race Car Rally 6 www.drpaulswan.com.au |

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 Tables Page 43 Multispin / Spindiv 6 Race Car Rally 6 www.drpaulswan.com.au |

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 Cards

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Link Problem Solving and Fluency with Multo

• 1 x 1 – 10 x 10
• Use products only once
• Download stickers from

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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. www.drpaulswan.com.au |

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 Chances

4 Chances 3 Chances 2 Chances 1 Chance 0 Chances 6 4 2 1 11 8 9 3 25 13 10 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
• 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 www.drpaulswan.com.au |

Extended Basic Facts

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

61 62 63 64 65 66

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Cut and Count

Partitioning

• DeNardi, E. (2004). Avanti Mental Maths, p. 45

Partitioning: 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 9 I 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.

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