Strategies – Multiplication and Division. From repeated addition to multiplication as an array...

Strategies – Multiplication and Division

From repeated addition to multiplication as an array

3+3+3+3 4+4+4

4 groups of 3 = 4 x 3

3 groups of 4 = 3 x 4

4 x 3 does not represent the same as 3 x 4 but it has the same value.

3

4

4 x 3

Multiplication represented as an area

4

3

3 x 4

3

4

4 x 3 = 12

Multiplication represented as an area

4

3

3 x 4 = 12

14

30

Calculate 30 x 14

Multiplication represented as an area

14

10

4

30

30 x 10 = 300

30 x 4 = 120

30 x 14 = (30 x 10) + (30 x 4) = 300 + 120 = 420

Multiplication as an area

Modelling 38 x 14. Find ?

14

10

4

3038

?

38 x 14 = (30 x 10) + (? x 10) + (30 x 4) + (? x 4) = 300 + ? + 120 + ? = ?

30 x 10 ? x 10

30 x 4 ? x 4

Compare these models!

10

4

30

30 x 10 = 300

30 x 4 = 120

14 x 38 as an area

8 x 10 = 80

8 x 4 = 32

8

30 x 10 = 300+ 8 x 10 = 80+ 30 x 4 = 120+ 8 x 4 = 32 14 x 38 = 532

Area model for multiplication 38 x 14

38X 14 152 380 532

14 x 38 as a calculation

+

Modelling 38 x 142.

142

100

40

30

388

2

30 x 100 8 x 100

30 x 40 8 x 40

30 x 2 8 x 2

Progression in the area model for multiplication

2

x

2 x x

2 (x + 3) = 2x + 6

2 x 3

3

“I know the 2 times and 3 times table. My teacher tells me I can

work out the rest.”

Discuss !

Multiplication facts from the 2 times and 3 times tables.

Using x2 table x4 and x8 (by doubling)

Using x3 x6 (x2 x3 or x3+x3) and x9 (x3 x3)

Using x2 and x3 tables x5 (x2+x3)

Using x3 and x4 x7 (x3+x4)

x10 (x5+x5) deserves special attention.So, I can work out all the times tables up to x10???

you can derive the

Learning multiplication facts

Hints and tips for helping with 7x8, 8x9, 9x9 etc

When they have mastered the early multiplication facts eg x2, x3, x4, x5, x10, children have only 10 more

x facts to learn!

Discuss!

Learning multiplication facts

Use commutative properties,

e.g. 3 x 4 = 4 x 3

So, when learning 6x table children already know facts up to 6 x 5 (from the x5 table) which leaves only four facts to learn:

That leaves only six more facts to master on the x7, x8 and x9 tables, ie :

6x6, 6x7, 6x8, 6x9 Why?

Three facts of x7 table ie 7x7, 7x8, 7x9

=

Two facts of x8 table ie 8x8, 8x9One fact of x9 table ie 9x9

20 4 20 4 20 4

20 24 44 48 68 72

3 X 24 = 24 + 24 + 24 Multiplication as repeated addition

20 20 20 4 4 4

20 40 60 64 68 72

3 X 24 = 3 x 20 + 3 x 4 Distributive property of multiplication

Development & Progression 2nd level – ‘ using their knowledge of commutative, associative and distributive properties to simplify calculations’

The ‘distributive’ property for X and ÷

21p21p 21p

3 x 21p= (3 x 20p) + (3 x 1p)

Using money to illustrate the distributive property

Progression – Level 2: ‘use their knowledge of ... distributive properties ...’

Multiplication

Show 3 x 24p

= (3 x 20) + (3 x 4)

Using money to illustrate the distributive propertyProgression – Level 2: ‘use their knowledge of ...

distributive properties ...’Multiplication

Show 87p ÷ 3

Using money to illustrate the distributive propertyProgression – Level 2: ‘use their knowledge of ...

distributive properties ...’Division