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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 25Halve x double
18 x 50Halve x double
9 x 100
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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?
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9 10
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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
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21 22
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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
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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
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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)
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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
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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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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
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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
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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
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