Developing a Whole School Approach to Mental Computation Day 2 - - PowerPoint PPT Presentation

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Developing a Whole School Approach to Mental Computation Day 2 - - PowerPoint PPT Presentation

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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 www.drpaulswan.com.au | Key Ideas Strategies versus recall Understandings

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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 2 x 25 men ental? al?

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Facts & Understandings

  • 36 x 25

25 is related to 100 36 is 4 x 9 4 x 9 = 9 x 4 Commutative property 9 x 4 x 25 I can do the 4 x 25 bit first 9 x 100. That is easy!

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Doubling and Halving

36 x 25 Halve x double 18 x 50 Halve 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 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

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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 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 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 10 20 30 40 50 60 70 80 90

100

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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 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 10 20 30 40 50 60 70 80 90

100

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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 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 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 1 2 3 4 5 6 7 8 9 10 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 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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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

2010 Yr 5 q 24

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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 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 10 20 30 40 50 60 70 80 90

100

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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 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 Pr Prob

  • blem
  • 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) Quotiti tion Pr Prob

  • blem:
  • 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

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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

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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)
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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

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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 Tables Page 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 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

www.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 Chances

4 C Chances 3 Ch Chances 2 Ch Chances 1 C Chance 0 Ch Chances 6 4 2 1 11 11 8 9 3 25 13 13 10 10 16 16 5 49 17 17 . 36 7 64 . . . 81 . . . 10 100 . .

9 4 23 23 6 58 58

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Positioning on the board

  • Where numbers

have been positioned makes a difference.

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Factors

  • Factor trees

60 10 x 6 2 x 5 x 2 x 3 60 12 x 5 3 x 4 x 5 3 x 2 x 2 x 5

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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

7 x 8 = 8 = 56 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

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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

30 7 20 5 30 x 20 = 600 7 x 20 = 140 30 x 5= 150 7 x 5= 35

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

3a 7 2b 5 3a x 2b = 6ab 7 x 2b = 14b 3a x 5= 15a 7 x 5= 35

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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 12 x x 18 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 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.