Fraction Multiplication: A Model Years 5 - 8

Summary The array model of multiplication is very powerful. In this activity it is used to explore multiplication of fractions in a visual and kinaesthetic way. Not only do we find the answer to the question, but the plugs can be rearranged into a new array that shows both the simplest equivalent fraction and the 'cancelling factor' that produces it. Suitable for threading. Materials One Poly Plug set per pair Acknowledgement While reviewing early editions of Set Two for the centenary of the Mathematical Association of Victoria (MAV), I came across this model for fraction multiplication. As reported in Set Two, October 1975, it is derived from Alistair McIntosh, Learning Their Tables - A Suggested Reorientation. Mushroom pegs were the suggested concrete expression of the model in the article but Poly Plug also offers a useful representation. (Doug. Williams)

Procedure This example speaks for itself. Exploring a few more together will develop sufficient confidence for the children to be able to choose their own challenges. As you move around the class encourage children to: explain in words how they did their challenge. record the explanation in words and pictures (...as if you had to send a fax to explain it to a kid in another school). Then, later, ask something like: Do you think you could work out one of these if you didn't have the plugs and couldn't draw a picture? One child answered: Sure, I'd put a picture of it in my head. To which the teacher, of course, replied: Okay, let's try.

Content fraction calculations fractions as an array fractions as a partition of a whole mathematical conversation multiples, factors & primes multiplication - array model multiplication problem solving recording - written times tables visual and kinaesthetic representation of number visual representation of fractions

Suppose the multiplication is 3/5 x 2/3. Multiplication can be represented as a rectangle, so make one side of the rectangle 2/3 yellow (two parts out of a whole of three parts)...

...and the other side 3/5 yellow (three parts out of a whole of five parts).

Now complete the new whole rectangle which will have a yellow part and a blue part to make a new whole array. The result is 6/15 yellow (six parts out of a whole of 15 parts)...

...and the only way to rearrange these plugs into equal rows is...

... which is the same as 2/5 yellow three times!

• This contribution was mounted in June 2005, thirty years after the same idea appeared in Set Two.
  Perhaps an 'old idea' is not always to be discarded just because of its age.

Extensions

• If your school is a member of Maths300 you can combine this concrete experience with a software version in the form of an Excel spreadsheet which was donated to the Classroom Contributions section of Lesson 77, Rectangle Fractions, by Kevin Butler, Newcomb Secondary College.
• What happens if we are given the product and told one of the factors? Can we work out the other factor?

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