Transcript The Array Model


Visual Models for Multiplying
Fractions (and Decimals)
2(x  4)
3 4

1
1
1 2
3
4
1 1

3 4
0.30.4
Jim Hogan
(x  1) 2

1.32.4
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Secondary Mathematics Advisor
SSS, Waikato University

(x  2)(x  4)
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Purpose
To learn how to use the array to
model multiplication
- with whole numbers
- fractions
- decimals
To re-learn how important robust
mental models are to learning.
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3x4
Draw 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.
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Why the Array?
By Year 6 students are
developing
multiplicative ideas –
or should be…
3 4
The “repeated
addition” model is
common.
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Useful Array
This model connects
to factors, multiples,
primes, fractions,
decimals …
3 4
This model obstructs
connections to CL Level 4
mathematics.
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3x4
This is the array model for 3 x 4.
- seeing the 3 and the 4 at the same time
- more complex thinking than adding
- transfers from and to other subjects
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Revision of One
Make a model of 1.
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Flexible One
One can be anything I choose it to be!
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Make a model of 1/3 x 1/4
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1 1

3 4
1
The sides have been
divided into thirds
and quarters.
1
There are 12 parts. Each part is 1 twelfth.
How would an “adder” see this problem?
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1 1

3 4


1
3
 1
3
1
3
1
4
1
4
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1
4
1
4
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The array clearly shows that multiplication of the two fractions.
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The array is intact. The rectangular shape is preserved.
The answer is the orange square.
Is it important that the “ones” are the same size?
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2 3

3 4
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Notice…NO RULES!  
1
3
 1
3
1
3
1
4
1
4



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2 3 6
 
3 4 12
1
4
1
4
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1
1
1 2
3
4
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
1
3
 1
3
1
3
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Make a model of this problem
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1
4
1
4
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1
4
1
4
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1
1 4 9 36
1 2   
3
3
4 3 4 12
Does your model or drawing show every number,
every equals and the answer 3?
Where is the 4?
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1
1 4 9 36
1 2   
3
3
4 3 4 12
1
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1
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1
4
1
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
1
3
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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
0.30.4
Make a model.
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Do you need help?
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0.30.4
Essential knowledge
1
 0.1
10
1
0.1
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The hundreds board is a
very useful device.
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.
x
4
x
2
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.
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2(x+4)
Curiously many teachers do not use the array model here.
x
4
2
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.
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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And so we move to yet
another place…
[email protected]
All files are located at
http://schools.reap.org.nz/advisor
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