The Array Model

Jim HoganMathematics Advisor

SSS, Waikato University

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1

3×1

4

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0.3×0.4

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11

3×21

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3× 4

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1.3×2.4

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2(x + 4)

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(x + 2)(x + 4)

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(x +1)2

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.

3 x 4Draw a picture of 3 x 4Make a model of 3 x 4What does 3 x 4 look like?

Ask your classes and staff to do this task and see what mental models are established.

The “repeated addition” model is common.

Why the Array?

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3× 4

By Year 6 students are developing multiplicative ideas. See the NZC.

This model obstructs connections to CL Level 4 mathematics.

Useful Array

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3× 4

This model connects to factors, multiples, primes, fractions, decimals …

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

Make a model of 1.

Revision of One

One can be anything I choose it to be!

Flexible One

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.

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1

3

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3

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1

4

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1

4

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1

4

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2

3×3

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1

3

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1

3

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1

3

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1

4

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1

4

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1

4

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1

4

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2

3×3

4=6

12

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11

3×21

4

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1

3

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1

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1

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1

4

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1

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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 answer 3?

Where is the 4?

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1

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1

3

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1

4

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11

3×21

4=4

3×9

4=36

12= 3

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1

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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?

…and so to 0.3 x 0.4

Do you need help?

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0.3×0.4

Make a model.

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0.3×0.4

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1

10= 0.1

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0.1

Essential knowledge

The hundreds board is a very useful device.

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1

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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.

…and so to 1.3 x 2.4

Draw a picture of the answer of 1.3 x 2.4

(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

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.

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4x

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

(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?

πGenerate random pairs (x,y) on [0,1]

Use the test If x2 + y2 <1, true is another HIT

After a few attempts use this calculation to estimate π

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π =4 ×HITS

TOTAL

This is probably a connection to “π in the sky!”

And so we move to yet another place…

jimhogan@clear.net.nzAll files are located at http:schools.reap.org.nz/advisor

Thank you for you patience, attention and input!