Grade by Grade
B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

Multiplying Fractions by Fractions

B2.8 A1About 45 minutes
Learning goals
  • I can use an area model to show a fraction of a fraction.
  • I can multiply fractions by fractions, including mixed numbers.
  • I can simplify before or after multiplying.
  • I can predict whether a product will be greater or less than the numbers I multiply.

Words to know

product
The answer to a multiplication.
area model
A rectangle split into rows and columns to show a multiplication.
of
In fraction problems, "of" usually means multiply: 1/2 of 3/4 = 1/2 × 3/4.
improper fraction
A fraction like 9/4 where the numerator is greater than or equal to the denominator. Change mixed numbers to improper fractions before multiplying.

Remember from Grade 6

In Grade 6 you multiplied a whole number by a proper fraction: 3 × 3/4 means 3 groups of 3/4, which is 9/4 = 2 1/4. You only multiplied the numerator by the whole number. Now you will multiply a fraction by a fraction.

Let's start with leftover lasagna

After dinner, 3/4 of a pan of lasagna is left. You eat 2/3 of what's left for lunch the next day. What fraction of the whole pan did you eat?

You ate 2/3 of 3/4. The word of means multiply: 2/3 × 3/4.

The area model below shows it. Split the pan into 4 columns and shade 3 of them (3/4). Then split it into 3 rows and take 2 of the rows of that shaded part (2/3 of it). The double-shaded part is 6 of the 12 pieces, so you ate 6/12 = 1/2 of the pan.

3/42/36 of 12 parts= 6/12 = 1/2
An area model for 2/3 × 3/4. The pan has 4 columns and 3 rows, making 12 parts. Three columns are the 3/4 that was left. Two of the three rows of that part (the 6 green parts) are what you ate: 6/12 = 1/2.

The rule

To multiply fractions, multiply the numerators and multiply the denominators:

2/3 × 3/4 = (2 × 3)/(3 × 4) = 6/12 = 1/2.

Why does it work? The denominators multiply to give the number of parts in the whole (3 rows × 4 columns = 12). The numerators multiply to give the number of double-shaded parts (2 × 3 = 6).

Simplify first (optional): you may divide any numerator and any denominator by a common factor before multiplying. In 2/3 × 3/4, the 3s cancel and the 2 and 4 share a 2, leaving 1/1 × 1/2 = 1/2. Smaller numbers, fewer mistakes.

Mixed numbers: change them to improper fractions first. Do not multiply the wholes and the fractions separately.

Worked example

Example 1: Multiply and simplify

Find 5/6 × 3/10.

  1. Multiply numerators: 5 × 3 = 15. Multiply denominators: 6 × 10 = 60.
  2. 15/60 in lowest terms: GCF is 15, so 15/60 = 1/4.
  3. Or simplify first: 5 and 10 share 5 (→ 1 and 2); 3 and 6 share 3 (→ 1 and 2). Then 1/2 × 1/2 = 1/4.

5/6 × 3/10 = 1/4

Worked example

Example 2: Mixed numbers

Find 1 1/2 × 2 2/3.

  1. Change to improper fractions: 1 1/2 = 3/2 and 2 2/3 = 8/3.
  2. 3/2 × 8/3 = 24/6 = 4.
  3. Estimate to check: about 1.5 × 2.7 is about 4. ✓

1 1/2 × 2 2/3 = 4

Worked example

Example 3: In a recipe

A pancake recipe calls for 3/4 cup of milk. You are making 2/3 of the recipe. How much milk do you need?

  1. You need 2/3 of 3/4 cup: 2/3 × 3/4.
  2. Simplify first: the 3s cancel, and 2 and 4 share 2. 1/1 × 1/2 = 1/2.

1/2 cup of milk

Bigger or smaller?

When you multiply by a fraction less than 1, the answer is smaller than the number you started with. 2/3 × 3/4 = 1/2, which is less than 3/4. When you multiply by a number greater than 1 (like 2 2/3), the answer is bigger. Use this to check that your answer makes sense.

012343/41/21 1/2× 2/3smaller× 2 2/3bigger
Multiplying by a fraction less than 1 (2/3) moves you toward 0. Multiplying by a number greater than 1 (2 2/3) moves you away from 0: 1 1/2 × 2 2/3 = 4.
Hands-on activity

Fold-and-shade fractions

You need: a few sheets of scrap paper, a pencil and a pencil crayon or marker.

  1. To show 2/3 × 3/4, fold a sheet into 4 equal columns. Unfold and shade 3 columns lightly with the pencil.
  2. Fold the same sheet into 3 equal rows. Unfold and colour over 2 rows with the pencil crayon, but only inside the shaded part.
  3. Count the parts that have both markings, and count all the parts on the sheet. Write the fraction. (6/12 = 1/2.)
  4. Repeat for 1/2 × 3/5 (5 columns, 2 rows) and 3/4 × 2/3 (3 columns, 4 rows).

What to look for: 1/2 × 3/5 = 3/10. 3/4 × 2/3 = 6/12 = 1/2, the same as 2/3 × 3/4 (any order works for multiplying).

Math talk: "Why is the answer smaller than both fractions?"

Just for fun

Half of a half of a pizza is a quarter of a pizza. Multiplying fractions is the only time "more math" can mean less pizza!

Your turn

1

1/2 × 1/3 = ?

2

2/5 × 1/4 = ?

3

3/4 × 2/3 = ?

4

5/6 × 3/5 = ?

5

What is 1/2 of 3/8?

6

1 1/2 × 2/3 = ?

7

2 1/4 × 1 1/3 = ?

More practice: warm-up

1

1/2 × 1/2 = ?

2

1/3 × 1/4 = ?

3

2/3 × 1/2 = ?

4

3/4 × 1/3 = ?

5

4/5 × 1/2 = ?

More practice: on your own

1

3/8 × 4/9 = ?

2

7/10 × 5/14 = ?

3

3/5 of a backyard is garden. 2/3 of the garden is planted with tomatoes. What fraction of the backyard is tomatoes?

4

2 1/2 × 1 1/5 = ?

5

A recipe uses 3/4 cup of sugar. You make 1/3 of the recipe. How much sugar do you need?

Challenge

1

A rectangular placemat is 3/4 m long and 2/3 m wide. What is its area?

2

Without calculating, is 5/6 × 7/8 more or less than 7/8? Explain. Then check.

3

1 3/4 × 2 2/7 = ?

Check your understanding

Quiz

1.

What is 2/3 × 4/5?

2.

What is 3/4 × 2/9?

3.

What does 1/2 of 2/5 equal?

4.

What is 1 1/2 × 1 1/2?

5.

Which product is less than 3/5?

6.

You have 3/4 of a bag of flour and use 1/3 of it. What fraction of the full bag did you use?

For parents and teachers

Big idea: The Ontario curriculum (B2.8) asks Grade 7 students to multiply and divide fractions by fractions, using tools in various contexts. Tools like the area model help students see why the numerators and denominators are multiplied, rather than memorizing a rule.

Common difficulties:

  • Finding a common denominator first (not needed for multiplying) or adding numerators.
  • Multiplying mixed numbers part by part (1 1/2 × 1 1/2 is not 1 1/4).
  • Expecting a bigger answer: multiplying by a fraction less than 1 gives a smaller answer.

How to help: Do the fold-and-shade activity and scale recipes together ("we're making half the recipe"). This lesson builds A1 skills: using models to represent ideas and connecting a picture to a rule.