maths the modern way!! multiplication and division st teresa’s primary school paul hargreaves...

Maths the Modern Way!!Maths the Modern Way!!Multiplication and DivisionMultiplication and DivisionSt Teresa’s Primary SchoolSt Teresa’s Primary School

Paul HargreavesPaul HargreavesPrimary Strategy Consultant – Primary Strategy Consultant –

MathematicsMathematicsEssex County CouncilEssex County Council

Mental Starter – Bunny EarsMental Starter – Bunny Ears

Total Recall!Total Recall!

Select pairs of numbers from the target board on Select pairs of numbers from the target board on your table. Add these using one of the methods your table. Add these using one of the methods from last session. Will you use a number line? Will from last session. Will you use a number line? Will you partition and add mentally? Do you need to you partition and add mentally? Do you need to make jottings? make jottings? BE BRAVE – try to avoid using the standard BE BRAVE – try to avoid using the standard method!method!Now try with three numbers! Or try pairs of Now try with three numbers! Or try pairs of numbers and carry out a subtraction!numbers and carry out a subtraction!

The Primary National The Primary National StrategyStrategy

Basis of teaching since 1999 – based on Basis of teaching since 1999 – based on extensive research and proven successextensive research and proven success

Daily entitlement to maths lessonDaily entitlement to maths lesson Key featuresKey features

Progression carefully set outProgression carefully set out Interactivity – use of models, images, games, Interactivity – use of models, images, games,

practical activitiespractical activities Focus on mental skills as well as writtenFocus on mental skills as well as written Vocabulary, problem solving, communication, Vocabulary, problem solving, communication,

explanation and reasoningexplanation and reasoning

There is no “right way” to There is no “right way” to work!!work!!

Children exposed to a range of methods.Children exposed to a range of methods. Methods selected will depend upon the Methods selected will depend upon the

situation and the numbers involved, situation and the numbers involved, including when to use calculators.including when to use calculators.

Children make decisions about methods and Children make decisions about methods and draw on a range of strategies and draw on a range of strategies and approaches when applying Maths is context.approaches when applying Maths is context.

Children in same class could be using Children in same class could be using different methods to others depending on different methods to others depending on their ability, confidence and stage of their ability, confidence and stage of mathematical development.mathematical development.

Describing ShapesDescribing Shapes

The Importance of The Importance of VocabularyVocabulary

Key to success in mathematicsKey to success in mathematics Can be confusion between school Can be confusion between school

and homeand home Children need opportunities in class Children need opportunities in class

and in homework to use and in homework to use mathematical vocabulary – games, mathematical vocabulary – games, collaborative work, open ended collaborative work, open ended investigationsinvestigations

Mathematical Vocabulary Mathematical Vocabulary BookletBooklet

Guide to which words and phrases Guide to which words and phrases are introduced to each year groupare introduced to each year group

Schools many make decisions Schools many make decisions regarding vocabularyregarding vocabulary

It is not a checklistIt is not a checklist Check with children and teachers if Check with children and teachers if

there are unfamiliar wordsthere are unfamiliar words

MultiplicationMultiplication

Calculate the answer to this…Calculate the answer to this…

5 65 6

xx 3 3

Did you do this?Did you do this?

5 65 6

xx 3 3

11 6 86 811

Or did you use a mental method?Or did you use a mental method?Why did you choose the method Why did you choose the method

you used?you used?

Repeated Addition (Year Repeated Addition (Year 2/3)2/3)

5 added to 5 added to 55 added to 5 added to 5

5 + 5 + 55 + 5 + 5

3 lots of 53 lots of 5

5 x 35 x 3

3 x 53 x 5

Lots of practical experiences and use of Lots of practical experiences and use of number lines. Children will begin to use x and number lines. Children will begin to use x and

= signs.= signs.

Multiplication as an ArrayMultiplication as an Array

4 x 2 = 8

2 lots of 4 = 8

2 x 4 = 8

4 lots of 2 = 8

Arrays are quite common – ice cube trays, egg boxes, chocolate boxes, medicine wrapping, tiles etc.

Multiplication by 10Multiplication by 10

7 x 10 = 707 x 10 = 70


7 x 10 = 707 x 10 = 70

BUT WE DIDN’T JUST ADD A 0!BUT WE DIDN’T JUST ADD A 0!


7 x 10 = 707 x 10 = 70

BUT WE DIDN’T JUST ADD A 0!BUT WE DIDN’T JUST ADD A 0!

77

7.07.0

Both of these numbers are worth the Both of these numbers are worth the same!same!

7 add a 0 is 7 + 0 = 77 add a 0 is 7 + 0 = 7

We haven’t multiplied here!We haven’t multiplied here!


HH TT UU


HH TT UU

77


HH TT UU

77

77


HH TT UU

77

77 0 0

PartitioningPartitioning

15 x 515 x 5


15 x 515 x 5 This is 10 x 5 and 5 x 5 added

together. 10 x 5 = 505 x 5 = 25

50 + 25 = 75


36 x 436 x 4


36 x 436 x 4

36 x 4 is 30 x 4 and 6 x 4 added together. I know that 30 is three

lots of 10, so 30 x 4 is 10 x 4 added to 10 x 4 added to 10 x 4.

10 x 4 = 4010 x 4 = 4010 x 4 = 406 x 4 = 24

40 + 40 + 40 + 24 = 144


36 x 436 x 4

36 x 4 is 30 x 4 added to 6 x 4I know that 30 x 4 is 10 times

bigger than 3 x 43 x 4 = 12, so 30 x 4 = 120

6 x 4 = 24120 + 24 = 144

Grid MethodGrid Method

23 x 823 x 8


23 x 823 x 820

8

3x


23 x 823 x 820

8

3x

160


23 x 823 x 820

8

3x

160 24


23 x 823 x 820

8

3x

160 24+ 24

160

184

Have a Go!!Have a Go!!

26 x 526 x 5

32 x 432 x 4


26 x 526 x 520

5

6x

100 30+ 30

100

130


32 x 432 x 430

4

2x

120 8+ 8

120

128


346 x 4346 x 4

xx

44

300300 4040 66


346 x 4346 x 4

xx

44

300300 4040 66

12001200


346 x 4346 x 4

xx

44

300300 4040 66

12001200 160160


346 x 4346 x 4

xx

44

300300 4040 66

242416016012001200


346 x 4346 x 4

xx

44

300300 4040 66

242416016012001200

12001200160160

+ 24+ 24

13841384


72 x 3872 x 38xx

88

3030

7070 22


72 x 3872 x 38xx

88

3030

7070 22

560560

21002100 6060

1616

21002100

560560

6060

+ 16+ 16

27362736

Standard MethodStandard Method

23 x 823 x 820

8

3x

160 24+ 24

160

184


23 x 823 x 8

1 8 41 8 4

3 x 83 x 82 42 4

20 x 820 x 81 6 01 6 0

x 8x 82 32 3


23 x 823 x 8

1 8 41 8 4

3 x 83 x 82 42 4

20 x 820 x 81 6 01 6 0

x 8x 82 32 3

x 8x 82 32 3

4422


23 x 823 x 8

1 8 41 8 4

3 x 83 x 82 42 4

20 x 820 x 81 6 01 6 0

x 8x 82 32 3

x 8x 82 32 3

1 8 41 8 422

Why Not Just Teach the Why Not Just Teach the Standard Method?Standard Method?

1 0 0 0 7

x 3 x 4 25 6


1 0 0 0 7

x 3 x 4 25 6

0 0 0 2 1


1 0 0 0 7

x 3 x 4 25 6

0 0 0 2 1 1 2

2 0

3 2

Squashy BoxesSquashy Boxes

DivisionDivision

Share 8 sweets between two children.Share 8 sweets between two children.

DivisionDivision

Share 8 sweets between two children.Share 8 sweets between two children.

4 sweets in each pile

Repeated Subtraction Repeated Subtraction (Grouping)(Grouping)

8 8 2 can be thought of as 2 can be thought of as

8 – 2 = 68 – 2 = 6

6 – 2 = 46 – 2 = 4

4 – 2 = 24 – 2 = 2

2 – 2 = 02 – 2 = 0

I’ve taken 2 away 4 times, so the answer

is 4!!

0 2 4 6 8

-2-2-2 -2

13 13 3 3

13 – 3 = 1013 – 3 = 10

10 – 3 = 710 – 3 = 7

7 – 3 = 47 – 3 = 4

4 – 3 = 14 – 3 = 1I cannot make

anymore groups of 3 out of 1, so there is one

left over.

13 13 3 = 4 r 1 3 = 4 r 1

13 – 3 = 1013 – 3 = 10

10 – 3 = 710 – 3 = 7

7 – 3 = 47 – 3 = 4

4 – 3 = 14 – 3 = 1I cannot make

any more groups of 3 out of 1, so there is one

left over.

72 72 55

72 – 5 = 6772 – 5 = 67 37 – 5 = 3237 – 5 = 32

67 – 5 = 6267 – 5 = 62 32 – 5 = 2732 – 5 = 27

62 – 5 = 5762 – 5 = 57 27 – 5 = 2227 – 5 = 22

57 – 5 = 5257 – 5 = 52 22 – 5 = 1722 – 5 = 17

52 – 5 = 4752 – 5 = 47 17 – 5 = 1217 – 5 = 12

47 – 5 = 4247 – 5 = 42 12 – 5 = 712 – 5 = 7

42 – 5 = 37 42 – 5 = 37 7 – 5 = 2 7 – 5 = 2

72 72 55

72 – 5 = 6772 – 5 = 67 37 – 5 = 3237 – 5 = 32

67 – 5 = 6267 – 5 = 62 32 – 5 = 2732 – 5 = 27

62 – 5 = 5762 – 5 = 57 27 – 5 = 2227 – 5 = 22

57 – 5 = 5257 – 5 = 52 22 – 5 = 1722 – 5 = 17

52 – 5 = 4752 – 5 = 47 17 – 5 = 1217 – 5 = 12

47 – 5 = 4247 – 5 = 42 12 – 5 = 712 – 5 = 7

42 – 5 = 37 42 – 5 = 37 7 – 5 = 2 7 – 5 = 2

Too long winded!!

!!

7272-- 50 (10 x 5)50 (10 x 5)

2222-- 5 5 (1 x 5)(1 x 5)

1717-- 5 5 (1 x 5)(1 x 5)

1212-- 5 5 (1 x 5)(1 x 5)

77 5 (1 x 5)5 (1 x 5)

22

72 72 5 = 14 r 5 = 14 r 22

7272-- 50 (10 x 5)50 (10 x 5)

2222-- 5 5 (1 x 5)(1 x 5)

1717-- 5 5 (1 x 5)(1 x 5)

1212-- 5 5 (1 x 5)(1 x 5)

77 5 (1 x 5)5 (1 x 5)

22

72 72 5 = 14 r 5 = 14 r 22

5 )5 )14 r 214 r 2

7272

-- 50 (10 x 50 (10 x 5)5)

2222

-- 2020 ( 4 x 5)( 4 x 5)

22

72 72 5 = 14 r 5 = 14 r 22

5 5 ) )14 r 214 r 2

Try it!!!Try it!!!

93 93 4 4 256 256 7 7

Why not use the way that we Why not use the way that we were taught?were taught?

The method that we are used The method that we are used to looks like this…to looks like this…

6 ) 1 3 36 ) 1 3 3


6 ) 1 3 36 ) 1 3 30

1


6 ) 1 3 36 ) 1 3 30 2

1 1


6 ) 1 3 36 ) 1 3 30 2 2 r 1 1 1

It does work, but many It does work, but many children make the following children make the following

errors…errors…


6 ) 1 3 36 ) 1 3 3

Hmmm! I can’t make any groups of 6 out of 1,

so…


6 ) 1 3 36 ) 1 3 3


so…

0


6 ) 1 3 36 ) 1 3 3


so…

0 0


6 ) 1 3 36 ) 1 3 3


so…

0 0 0


6 ) 1 3 36 ) 1 3 3

Great! The answer is 0!

0 0 0


6 ) 1 3 36 ) 1 3 3

OK – 6s into 1 don’t go, so…..


6 ) 1 3 36 ) 1 3 31


6 ) 1 3 36 ) 1 3 31

Now, 6s into 13. I know that two 6s

are 12 and I’ll have 1 left over.


6 ) 1 3 36 ) 1 3 31

121


6 ) 1 3 36 ) 1 3 31

121

Oh, look! 6s into 13 again! I know that!


6 ) 1 3 36 ) 1 3 31

121

12 r 1


6 ) 1 3 36 ) 1 3 31

121

12 r 1

The answer is 1212 r 1!


18 )1 1 0 18 )1 1 0


18 )1 1 0 18 )1 1 0 0

1


18 )1 1 0 18 )1 1 0 0 0

1 11


18 )1 1 0 18 )1 1 0 0 0

1 11

Back to square one! Lots more learning and understanding is needed here. To successfully tackle this problem, you need to know how to use repeated subtraction!

Multiplication TablesMultiplication Tables

Year 1 – begin to learn 2x, 5x and Year 1 – begin to learn 2x, 5x and 10x10x

Year 2 – know 2x, 5x and 10x. Year 2 – know 2x, 5x and 10x. Year 3 – know 2x, 3x, 4x, 5x, 6x and Year 3 – know 2x, 3x, 4x, 5x, 6x and

10x. 10x. Year 4 – know all facts to 10 x 10Year 4 – know all facts to 10 x 10

Multiplication TablesMultiplication Tables

Three for free!Three for free!

If you know 3 x 5 = 15, you also knowIf you know 3 x 5 = 15, you also know

5 x 3 = 155 x 3 = 15

15 15 5 = 3 5 = 3

15 15 3 = 5 3 = 5

Maths the Modern Way!!Maths the Modern Way!!Multiplication and DivisionMultiplication and DivisionSt Teresa’s Primary SchoolSt Teresa’s Primary School

Paul HargreavesPaul HargreavesPrimary Strategy Consultant – Primary Strategy Consultant –

Mathematics Mathematics Essex County CouncilEssex County Council

maths the modern way!! multiplication and division st teresa’s primary school paul hargreaves...

Documents

range of methods

home children

reasoning slide

mathematical vocabulary

different methods

written vocabulary

vocabulary schools

reasoning vocabulary