Grade by Grade
B. Number · Operations: Numbers to 100 000, Decimals, Fractions, Ratios and Rates

Dividing Whole Numbers by Unit Fractions

B2.8 A1About 40 minutes
Learning goals
  • I can divide a one-digit whole number by a unit fraction, like 3 ÷ 1/4.
  • I can explain that 3 ÷ 1/4 asks "How many fourths are in 3?"
  • I can show the answer with fraction strips, drawings and number lines.
  • I can tell when dividing by a unit fraction gives a bigger number.

Words to know

unit fraction
A fraction with 1 as the numerator, like 1/2, 1/4 or 1/5.
quotient
The answer when you divide.
how many groups
One meaning of division: how many groups of a size fit into an amount.

Let's start with sandwiches

You have 3 sandwiches. You cut each one into fourths for a party. How many pieces do you get?

Each sandwich gives 4 fourths. 3 sandwiches give 4 + 4 + 4 = 12 pieces.

In math, that's 3 ÷ 1/4 = 12. It asks: How many fourths are in 3?

1/41/41/41/41/41/41/41/41/41/41/41/4
Three strips, each cut into 4 fourths. Count all the pieces: 4 + 4 + 4 = 12 fourths. So 3 ÷ 1/4 = 12.

What does dividing by a fraction mean?

You already know this meaning of division: 12 ÷ 3 asks how many groups of 3 fit in 12.

Dividing by a fraction works the same way:

  • 2 ÷ 1/2 asks: how many halves fit in 2? Each whole has 2 halves, so 4.
  • 3 ÷ 1/3 asks: how many thirds fit in 3? Each whole has 3 thirds, so 9.
  • 5 ÷ 1/4 asks: how many fourths fit in 5? Each whole has 4 fourths, so 20.

See the pattern? Each whole has as many unit-fraction pieces as the denominator. So you can multiply: 5 ÷ 1/4 = 5 × 4 = 20.

01/22/2= 13/24/2= 25/26/2= 3
A number line from 0 to 3 marked in halves. Six jumps of 1/2 reach 3. So there are 6 halves in 3, and 3 ÷ 1/2 = 6.
Worked example

Example 1: Count the pieces

Solve 3 ÷ 1/2.

  1. Ask: how many halves fit in 3?
  2. Each whole has 2 halves. 3 wholes have 2 + 2 + 2 = 6 halves.
  3. Check: 6 × 1/2 = 6/2 = 3. ✓

3 ÷ 1/2 = 6

Worked example

Example 2: Scoops of flour

You have 2 cups of flour. Your scoop holds 1/3 cup. How many scoops can you fill?

  1. This is 2 ÷ 1/3: how many thirds fit in 2?
  2. Each cup holds 3 thirds. 2 cups hold 3 + 3 = 6 thirds.
  3. Check: 6 × 1/3 = 6/3 = 2. ✓

6 scoops

Worked example

Example 3: Cutting ribbon

A ribbon is 4 m long. You cut it into pieces that are each 1/5 m long. How many pieces do you get?

  1. This is 4 ÷ 1/5: how many fifths fit in 4?
  2. Each metre has 5 fifths. 4 metres have 4 × 5 = 20 fifths.

20 pieces

Bigger or smaller?

Compare these two:

  • 3 × 1/4 = 3/4. Three groups of a fourth. The answer is smaller than 3.
  • 3 ÷ 1/4 = 12. How many fourths in 3? The answer is bigger than 3.

When you divide by a unit fraction, you're counting small pieces, so you get lots of them.

Check it

Always check a fraction division with multiplication. If 3 ÷ 1/4 = 12, then 12 × 1/4 should equal 3. 12/4 = 3. ✓ Multiplication and division are still partners, even with fractions!

Just for fun

Strange but true: dividing by 1/2 makes numbers bigger. 3 ÷ 1/2 = 6. Imagine if dividing up your chores worked like that... oh no, that means more chores!

Hands-on activity

How many pieces?

You need: paper strips (or paper you can fold), scissors (ask a grown-up to help), and a 1/4 cup measure and a 1 cup measure with some rice or water, over a tray.

  1. Take 2 strips of paper. Fold each into 4 equal parts and cut them apart. Count the pieces. What is 2 ÷ 1/4? (8)
  2. Take 3 strips and cut each into halves. What is 3 ÷ 1/2? (6)
  3. Fill a 1 cup measure using a 1/4 cup scoop. How many scoops? Then fill 2 cups. What is 2 ÷ 1/4? (8)
  4. Make up your own: pick a number of strips and a unit fraction, predict the number of pieces, then cut and count to check.

Math talk: "Why do we get more pieces than strips?"

Your turn

1

2 ÷ 1/2 = ?

2

3 ÷ 1/3 = ?

3

4 ÷ 1/4 = ?

4

5 ÷ 1/2 = ?

5

1 ÷ 1/8 = ?

6

6 ÷ 1/3 = ?

7

You have 3 pizzas. Each person gets 1/6 of a pizza. How many people can you feed?

More practice: warm-up

1

1 ÷ 1/2 = ?

2

1 ÷ 1/3 = ?

3

1 ÷ 1/4 = ?

4

2 ÷ 1/4 = ?

5

1 ÷ 1/10 = ?

More practice: on your own

1

7 ÷ 1/2 = ?

2

9 ÷ 1/3 = ?

3

A 4 m ribbon is cut into 1/5 m pieces. How many pieces?

4

8 ÷ 1/4 = ?

5

Which is bigger: 6 ÷ 1/2 or 6 × 1/2?

Challenge

1

Find the missing number: □ ÷ 1/4 = 20.

2

Ellie has 6 cups of trail mix. She fills bags with 1/2 cup each. She gives away 5 bags. How many bags does she keep?

3

Explain why 5 ÷ 1/2 is bigger than 5.

Check your understanding

Quiz

1.

What is 3 ÷ 1/4?

2.

What is 2 ÷ 1/3?

3.

How many half-hours are in 4 hours?

4.

What does 5 ÷ 1/2 mean?

5.

What is 6 ÷ 1/5?

6.

Which is the biggest?

For parents and teachers

Big idea: B2.8 also asks Grade 5 students to divide one-digit whole numbers by unit fractions, using tools and drawings. Dividing by a unit fraction asks "how many of these pieces fit?", so 3 ÷ 1/4 = 12 because there are 12 fourths in 3. Students do not need a memorized rule; they should be able to show it with strips, number lines or measuring cups.

Common difficulties:

  • Confusing 3 ÷ 1/4 with 3 × 1/4, or with "a fourth of 3". Ask "How many fourths fit in 3?"
  • Expecting division to give a smaller number. Cut paper strips and count the pieces.

How to help: Kitchen scoops are ideal: "How many 1/4 cup scoops make 2 cups?" Cutting a sandwich or a strip of paper also works well. Ask your child to predict first, then check by counting.