developing mathematical thinking in number : focus on multiplication

Developing Mathematical Thinking In Number : Focus on Multiplication

Aim of presentation 

To encourage staff reflection on approaches to teaching number.

To stimulate professional dialogue.

To use as a CPD activity for staff individually or collegiately.

Experiences and Outcomes

I can use addition, subtraction, multiplication and division when solving

problems, making best use of the mental strategies and written skills I have

developed.   MNU 1-03a

Having determined which calculations are needed, I can solve problems

involving whole numbers using a range of methods, sharing my approaches

and solutions with others.   MNU 2-03a

I can use a variety of methods to solve number problems in familiar contexts,

clearly communicating my processes and solutions. MNU 3-03a

Having recognised similarities between new problems and problems I have

solved before, I can carry out the necessary calculations to solve problems

set in unfamiliar contexts. MNU 4-03a

Progression

Building up times tables

How many cubes?

What would be efficient ways of finding out

how many cubes there

are?

What would be efficient ways of finding out

how many cubes there

are?

Group in 2s and Count in

2s?

Group in 2s and Count in

2s?

Group in 5s and Count in

5s?

Group in 5s and Count in

5s?

9

When children have mastered the facts ofeg x2, x3, x4, x5, x10,

children have only

10 more x facts to learn!

Multiplication Facts

Discuss!

Multiplication Facts

Using commutative property.

The 10 more facts to learn are

ie 6x6, 6x7, 6x8, 6x9 Why?7x7, 7x8, 7x9

8x8, 8x9

9x9

=

How well do children calculate?

6x6

Square numbers

5x54x43x32x2

Any other

patterns?

Any other

patterns?

Why are they called

square numbers?

Why are they called

square numbers?

How do we

encourage pupils to investigat

e?

How do we

encourage pupils to investigat

e?

What is the most sensible order for

teaching times tables?

What is the most sensible order for

teaching times tables? How can we

help children see the links between the times tables?

How can we help children see the links between the times tables?

“I know the 2x and 3x table. My teacher tells me I know the rest.”

Discuss !

From x2 x4 and x8 (doubling)From x3 x6 (x2x3) and x9 (x3x3)From x2 and x3 x5 (x2+x3)

From x3 and x4 x7 (x3+x4)

Making the links between the tables

What about x10? What tables does this help with?

What about x10? What tables does this help with?

From repeated addition to multiplication as array and as area

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

4 rows of 3 = 4 x 3

3 rows of 4 = 3 x 4

How do these images help children’s

understanding?

How do these images help children’s

understanding?

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)

Using the distributive property of multiplication

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

24p

Illustrating the distributive law using money 3 x 24p = (3x20p) + (3x4p)

How do these images help children’s

understanding?

How do these images help children’s

understanding?

What might be an added challenge in

this example?

What might be an added challenge in

this example?

24p

24p

14

30

Area = 30 x 14

Multiplication as area

1410

4

30

30 x 10 = 300

30 x 4 = 120

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

Area models for multiplication

14

10

4

30

30 x 10 = 300

30 x 4 = 120

38 x 14

8 x 10 = 80

8 x 4 = 32

8

30 x 10 = 300 8 x 10 = 8030 x 4 = 120 8 x 4 = 32

38 x 14 = 532

Area models for multiplication

What is the explanation for the algorithm

values ?Why

include the zero?

A challenge ...

Draw a similar diagram

to explain what is happening

in the calculation

48 x 34 ?

Solution

34

30

4

40

40 x 30 = 1200

30 x 4 = 120

8 x 30 = 240

8 x 4 = 32

8

2

x

2 x x

2 (x + 3) = 2x + 6

3 x 2

3

Area models for multiplication

x

2

x

X2

2x

(x + 3) (x + 2) = x2 + 3x + 2x + 3x2 = x2 + 5x + 6

3x

3 x 2

3

Area models for multiplication

y

b

x

xy

bx

(x + a) (y + b) = xy + ay + bx + ab

ay

ab

a

Area models for multiplication

Further support for progression in mathematics

http://www.ltscotland.org.uk/curriculumforexcellence/mathematics/outcomes/moreinformation/developmentandprogression.asp

Make the links

3x4=12

12÷3=4

12÷4=3

¼ of 12 = 330 x 4= 120

30 x 40 = 1200

0.3x 4= 1.2

0.4x 3= 1.2

25% of 120 = 30

Next stepsWhat

information will you

share with

colleagues?

What might you or your

staff do differently in

the classroom?

What else can you do as to improve learning and

teaching about number

What impact will this have on your

practice?

What impact will this have on your

practice?

Developing Mathematical Thinking In Number : Focus on Multiplication