FRACTIONS — MULTIPLYING BY WHOLE NUMBERS - DIAGRAMS

This lesson is not about the algorithm, it’s about creating drawings — diagrams, illustrations — that help us SEE what’s happening when working with fractions.

THE ALGORITHM

Let’s multiply the fraction 1/16 by 4.

We know the algorithm, the mathematical process that shows us how to solve this kind of problem.

WHAT DOES THIS LOOK LIKE? — METHOD 1

Remember, we’re trying to illustrate 1/6 x 4 = ?

Perhaps the easiest way to visualize 1/16 times 4 is to start with what 1/16th looks like.

Now imagine that you have four of these.

Put all of the shaded areas in one rectangle and it looks like this.

It’s easy to see we have 4/16ths.

It’s also easy to see 1/4 of the rectangle is shaded.

So 1/16 x 4 = 4/16 which can be reduced to 1/4.

WHAT DOES THIS LOOK LIKE? — METHOD 2

Remember, we’re trying to illustrate 4 x 1/16 = ?

This method is a little harder to understand, but you need to know it.

You may find you need this method on the SSAT, on an entrance exam, or on IXL.

You may also need it for 5G math homework!

This method sees the problem starting with the whole number, not the fraction.

So let’s imagine you have 4 rectangular pizzas.

You can diagram these 4 pizzas like this:

Now, imagine you have 16 friends coming to your party.

You need to divide your 4 pizzas into 16 equal slices.

Imagine your best friend arrives early to the party. They’re hungry and they want their slice of pizza now.

If you give your best friend their piece of pizza, what do you give them?

You divided you 4 pizzas into 16 equal slices.

You give your friend one of those slices.

What did you give your friend?

You gave them 1/4 of one of the pizzas, right?

And 4 x 1/16 = 4/16 = 1/4

Method 1 and Method 2 give us the same answer.

You may like one method better than the other.

(FYI, Dr. Cat, Mr. Ching and other teachers prefer Method 1.)

And— we all need to know both methods.

HOMEWORK

You will need:

  • Math journal

  • Pencil & eraser

  • Ruler

For each of the problems below, please:

(a) Solve the fraction multiplication problem with numbers only, using the algorithm

(b) Use Method 1 to illustrate the fraction multiplication problem

(c) Imagine the whole number as rectangular pizzas. You need to divide your pizzas up so you can give each of your guests an equal slice. Use Method 2 to illustrate this.

Do as many problems as you can in 30 minutes, then you are done.

This is graded:

  • Does it look like you’ve worked hard for 30 minutes?

  • Is your work well formatted and neat?

  • Have you successfully solved the problem 3 ways (algorithm, Method 1 and Method 2)?

PROBLEMS

  1. 1/6 x 3 = a

  2. 1/12 x 3 = b

  3. 1/12 x 4 = c

  4. 1/9 x 3 = d

  5. 1/8 x 4 = e