B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Dividing Whole Numbers by Proper Fractions

B2.10 A1About 45 minutes

Learning goals

I can explain that 4 ÷ 2/3 asks how many groups of 2/3 fit in 4.
I can divide a whole number by a proper fraction using a model or a common denominator.
I can explain what a leftover part means in a division with fractions.

Words to know

proper fraction
A fraction less than 1, like 2/3 or 3/4.

quotient
The answer when you divide.

common denominator
A denominator two numbers share. To divide 4 by 2/3, write 4 as 12/3.

measurement division
Division that asks how many groups of this size fit?, like how many 3/4 m pieces fit in 6 m.

Remember from Grade 5

In Grade 5 you divided whole numbers by unit fractions. 3 ÷ 1/4 = 12, because there are 4 fourths in each whole, and 3 wholes hold 12 fourths. Now you'll divide by fractions like 2/3 and 3/4.

Let's start with scarves

A knitting club has 6 m of fabric. Each scarf needs 3/4 m. How many scarves can they make?

The question is 6 ÷ 3/4: how many 3/4 m pieces fit in 6 m?

Cut the 6 m into fourths. Each metre has 4 fourths, so 6 m has 24 fourths.
Each scarf uses 3 fourths. 24 fourths ÷ 3 fourths = 8.

They can make 8 scarves. So 6 ÷ 3/4 = 8.

The common denominator strategy

Write the whole number as a fraction with the same denominator as the divisor. (4 = 12/3.)
Now both numbers count the same size pieces. Divide the numerators. (12 thirds ÷ 2 thirds = 12 ÷ 2 = 6.)

Another way: first divide by the unit fraction, then by the numerator. 4 ÷ 1/3 = 12 (12 thirds). Each group needs 2 thirds, so 12 ÷ 2 = 6.

Bigger answer: dividing by a proper fraction gives an answer bigger than the whole number, because the groups are smaller than 1.

4 wholes cut into thirds make 12 thirds. Each numbered group (shaded in two alternating shades) is 2/3. There are 6 groups, so 4 ÷ 2/3 = 6.

Worked example

Example 1: Use a model

4 ÷ 2/3 = ?

Ask: how many groups of 2/3 fit in 4?
4 wholes = 12 thirds (see the picture).
Each group is 2 thirds. 12 ÷ 2 = 6 groups.
Check: 6 × 2/3 = 12/3 = 4. ✓

4 ÷ 2/3 = 6

Worked example

Example 2: Use a common denominator

9 ÷ 3/5 = ?

Write 9 as fifths: 9 = 45/5.
45 fifths ÷ 3 fifths = 45 ÷ 3 = 15.
Check: 15 × 3/5 = 45/5 = 9. ✓

9 ÷ 3/5 = 15

Worked example

Example 3: Something left over

You have 5 cups of flour. Each batch of cookies needs 2/3 cup. How many batches can you make?

5 ÷ 2/3. Write 5 as thirds: 5 = 15/3.
15 thirds ÷ 2 thirds = 15 ÷ 2 = 7 R1. That's 7 batches with 1 third of a cup left over.
The leftover 1/3 cup is half of the 2/3 cup you'd need for another batch. So you could also say 5 ÷ 2/3 = 7 1/2.

You can make 7 full batches (with 1/3 cup of flour left). As a number, 5 ÷ 2/3 = 7 1/2.

Multiply or divide?

Don't mix up 4 ÷ 2/3 and 4 × 2/3.

4 × 2/3 means 2/3 of 4. The answer is smaller than 4: 8/3 = 2 2/3.
4 ÷ 2/3 means how many 2/3 fit in 4. The answer is bigger than 4: 6.

Ask yourself which question you are answering.

Top: 4 × 2/3 takes part of 4, so the answer (2 2/3) is smaller than 4. Bottom: 4 ÷ 2/3 counts jumps of 2/3, so the answer (6) is bigger than 4.

Just for fun

Riddle: How many 3/4 m scarves can the knitting club make from 6 m of fabric?

Unless the cat finds the fabric first. Then the answer is 0 scarves and 1 very happy cat.

Hands-on activity

How many scoops?

You need: measuring cups (1 cup, 1/2 cup, 1/3 cup, 1/4 cup), a bowl, and water or dry rice. Work over a sink or tray.

Fill the 1-cup measure 3 times and pour into the bowl. You have 3 cups.
Predict how many 3/4 cup scoops are in 3 cups. (3 = 12/4, and 12 ÷ 3 = 4.)
Measure 3/4 cup by filling the 1/4 cup three times into a container. Count how many 3/4 cups you can scoop out of the bowl. Were you right?
Try 2 ÷ 2/3: put 2 cups in the bowl. How many times can you take out 2/3 cup (two 1/3 cups)? (3)

Math talk: "Why is the answer bigger than the number of cups you started with?"

Your turn

2 ÷ 1/4 = ?

3 ÷ 3/4 = ?

4 ÷ 2/5 = ?

6 ÷ 2/3 = ?

9 ÷ 3/5 = ? (Try it without looking at Example 2.)

2 ÷ 2/3 = ?

10 ÷ 5/6 = ?

More practice: warm-up

1 ÷ 1/2 = ?

2 ÷ 1/3 = ?

4 ÷ 1/5 = ?

Write 3 as a fraction with denominator 4.

How many halves are in 3?

More practice: on your own

8 ÷ 4/5 = ?

12 ÷ 3/8 = ?

5 ÷ 5/6 = ?

7 ÷ 7/10 = ?

You have 9 m of ribbon. Each bow needs 3/4 m. How many bows can you make?

Challenge

5 ÷ 3/4 = ? Give the answer as a mixed number and explain the leftover.

3 L of juice is poured into cups that hold 2/5 L. How many cups can be filled? How much juice is left?

Find 6 ÷ 2/3 and 6 × 2/3. Explain why one answer is bigger than 6 and the other is smaller.

Check your understanding

Quiz

What is 3 ÷ 1/2?

What is 4 ÷ 2/3?

What is 6 ÷ 3/5?

Which question matches 8 ÷ 2/3?

What is 2 ÷ 3/4?

When you divide 5 by a proper fraction, the answer is…

For parents and teachers

Big idea: B2.10 asks Grade 6 students to divide whole numbers by proper fractions, using appropriate tools and strategies. Grade 5 covered unit fractions only. We use the measurement meaning (how many groups of 2/3 fit in 4?) and the common-denominator strategy, which shows why it works. The "invert and multiply" rule is not required in Grade 6, so we don't teach it as a shortcut without meaning.

Common difficulties:

Mixing up 4 ÷ 2/3 and 4 × 2/3. Ask "How many fit?" versus "What part of?"
Misreading the leftover. In 5 ÷ 2/3 = 7 R1, the 1 is one third of a cup, which is half a group.

How to help: Measuring cups make this real. Ask your child to predict before scooping and to explain the leftover. Encourage them to check every answer by multiplying back (6 × 2/3 = 4).