The Array Model

7/30/2019 The Array Model

1/23

The Array Model

Jim HoganMathematics Advisor

SSS, Waikato University

1

31

4

0.3 0.4

1132 1

4

3 4

1.32.4

2(x 4)

(x2)(x 4)(x 1)2


2/23

PurposeTo learn how to use the array to

model multiplication

- with whole numbers

- fractions

- decimals

- algebra expansions

To re-learn how important robust

mental models are to learning.


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3 x 4Draw a picture of 3 x 4

Make a model of 3 x 4

What does 3 x 4 look like?

Ask your classes and staff to do this task

and see what mental models are

established.


4/23

The repeated

addition model is

common.

Why the Array?

3 4

By Year 6 students aredeveloping

multiplicative ideas.

See the NZC.


5/23

This model obstructs

connections to CL Level

4 mathematics.

Useful Array

3 4

This model connectsto factors, multiples,

primes, fractions,

decimals


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This is the array model for 3 x 4.

Where is 4 x 3?

Show 3 x 4 + 3 x 2 = 3 x (4 + 2)

Explain and generalise

3 x 4


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Make a model of 1.

Revision of One


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One can be anything I choose it to be!

Flexible One


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Make a model of 1/3 x 1/4


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1

3

1

4

1

1The sides have been

divided into thirds

and quarters.

There are 12 parts. Each part is 1 twelfth.

The adders of the world see this problem as one line of 3

is a quarter and 1 third of these is 1 a third.

Notice this causes the destruction of the array.


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1

3

1

4

The array clearly shows that multiplication of the two fractions.

The array is intact. The rectangular shape is preserved.

The answer is the orange square.

13

1

3

1

3

1

4

1

4

1

4

1

4


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2

33

4

13

1

3

1

3

1

4

1

4

1

4

1

4

2

33

4

6

12


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11

32

1

4

13

1

3

1

3

1

4

1

4

1

4

1

4

Make this one.


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11

3 21

4 4

3 9

4 36

12 3

Does your model or drawing show every number,

every equals and the answer3?

Where is the 4?


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1

1

3

1

4

11

32

1

44

39

436

12 3

1

1

This model tips out everything. There are 4x9=36 parts. Twelve

parts make up the 1. Joining the scattered parts makes another 1.

What is the meaning of 1 complete row?


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and so to 0.3 x 0.4

Do you need help?

0.3 0.4

Make a model.


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0.3 0.4

1

10 0.1

0.1

Essential knowledge

The hundreds board is a

very useful device.

1

1

The answer is the 12 orange squares.

A little reflection makes this 12 hundredths

and now the problem moves to how we write that answer.


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and so to 1.3 x 2.4

Draw a picture of the

answer of 1.3 x 2.4


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(x+2)(x+4)

Curiously, many teachers know and use the array model

to expand quadratics.

The square of x is clearly visible! The 4 groups of x blue squares

and the 2 groups of x yellow squares makes 6x. The 2 groups of

4 green squares makes 8.

So (x + 2)(x + 4) = x2 + 4x + 2x + (2x4) and everything is visible.

x

2

4x


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2(x+4)

Curiously many teachers do not use the array model here.

The x is represented by a clear line of 3 squares.

There are 2 groups of an (x and 4) blue squares.

So 2(x + 4) = 2x + 2 x 4 = 2x + 8 and everything is visible.

2

4x

Provided the -4 is seen as a number, 2 (x 4) is the same model.


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(x+1)2

This is nearly the end of this presentation and the

beginning of squares

Notice the coloured squares and the extra 1 can be transformed to two

the same and one more, making an odd number.

Between any two consecutive squares is an odd number.

What are the pair of squares that are different by 25?

Can you see an infinite number of Pythagorean Triples here?


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Generate random pairs (x,y) on [0,1]

Use the test

If x2 + y2


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And so we move to yet

another place

[email protected]

All files are located at

http:schools.reap.org.nz/advisor

Thank you for you patience, attention and input!
mailto:[email protected]:[email protected]

The Array Model

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