Grade by Grade
B. Number · Operations: Numbers to 10 000, Decimal Tenths, Multiplying and Dividing

Repeated Addition of Unit Fractions: Multiplying a Fraction by a Whole Number

B2.7About 40 minutes
Learning goals
  • I can write repeated addition of a unit fraction as a multiplication.
  • I can show a whole number times a unit fraction with drawings, fraction strips or a number line.
  • I can write the answer as a fraction, and as a whole or mixed number when it is more than 1.

Words to know

unit fraction
A fraction with 1 on top, like 1/2, 1/3, 1/4 or 1/10. It is one equal part of a whole.
numerator
The top number of a fraction. It counts how many parts.
denominator
The bottom number of a fraction. It tells how many equal parts make one whole.
mixed number
A whole number and a fraction together, like 1 2/3.

Remember from Grade 3

In Grade 3 you added groups of halves, thirds and fourths, like 1/4 + 1/4 + 1/4 = 3/4. You learned that the numerator counts how many unit fractions there are. In Grade 4 you will write this repeated addition as multiplication.

From adding to multiplying

Remember: multiplication is repeated addition. 4 + 4 + 4 = 3 × 4.

The same is true for fractions!

1/4 + 1/4 + 1/4 = 3 × 1/4 = 3/4

We read 3 × 1/4 as "3 groups of one fourth" or "3 times one fourth".

1/41/41/41/4
A bar cut into 4 equal parts. 3 parts are shaded: 3 × 1/4 = 3/4.

The rule

When you multiply a whole number by a unit fraction:

  • The whole number tells how many pieces. It becomes the numerator.
  • The denominator stays the same, because the size of each piece doesn't change.

5 × 1/6 = 5/6. 2 × 1/8 = 2/8. 7 × 1/10 = 7/10.

Worked example

Example 1: From addition to multiplication

Write 1/5 + 1/5 + 1/5 + 1/5 as a multiplication, then solve.

  1. Count the unit fractions: there are 4 of them.
  2. Each one is 1/5.
  3. So the multiplication is 4 × 1/5.
  4. 4 fifths: 4/5.

1/5 + 1/5 + 1/5 + 1/5 = 4 × 1/5 = 4/5

When you have more than a whole

What happens when you have more pieces than make a whole?

5 × 1/3 = 5/3. That's 5 thirds. But 3 thirds make 1 whole. So 5 thirds is 1 whole and 2 thirds more: 1 2/3.

1/31/31/31/31/31/3
Two bars, each cut into thirds. The first bar has all 3 thirds shaded (1 whole). The second has 2 thirds shaded. Together: 5/3 = 1 2/3.
01/32/33/3= 14/35/36/3= 2
A number line in thirds from 0 to 2. Five jumps of 1/3 land on 5/3, which is between 1 and 2. 5 × 1/3 = 5/3 = 1 2/3.
Worked example

Example 2: Making wholes

Solve 7 × 1/2.

  1. 7 halves: 7/2.
  2. 2 halves make 1 whole. 7 halves make 3 wholes (6 halves) and 1 half more.

7 × 1/2 = 7/2 = 3 1/2

Worked example

Example 3: Exactly a whole number

Solve 6 × 1/3.

  1. 6 thirds: 6/3.
  2. 3 thirds make 1 whole, so 6 thirds make 2 wholes.

6 × 1/3 = 6/3 = 2

Tenths: fractions and decimals

One tenth can be written as 1/10 or 0.1. So 6 × 1/10 = 6/10 = 0.6.

00.51
Six jumps of one tenth from 0 land on 0.6. So 6 × 1/10 = 6/10 = 0.6.
Worked example

Example 4: A baking story

A muffin recipe uses 1/4 cup of blueberries for each muffin. How many cups for 6 muffins?

  1. 6 groups of 1/4 cup: 6 × 1/4 = 6/4.
  2. 4 fourths make 1 whole cup. 6 fourths = 1 whole and 2 fourths.
  3. 2/4 is the same as 1/2.

6/4 cups, which is 1 2/4 or 1 1/2 cups.

Don't change the denominator!

A common mistake is to multiply both numbers in the fraction: 3 × 1/4 = 3/12. Picture it! Three fourths of a granola bar is almost a whole bar, but 3/12 is only a small piece. The pieces stay fourths, so the answer is 3/4.

Just for fun

What if you ate 1/4 of a pizza, 4 times? 4 × 1/4 = 4/4 = 1 whole pizza. Your stomach just did multiplication!

Hands-on activity

Measuring-cup fractions

You need: a 1-cup measuring cup, a 1/4-cup measure and a 1/3-cup measure, a big bowl, and water or dry rice. (Do this at the kitchen counter with a grown-up.)

  1. Fill the 1/4 cup and pour it into the 1-cup measure. Repeat until the 1-cup measure is full. How many fourths? Write: 4 × 1/4 = 4/4 = 1.
  2. Empty it. Now pour 6 scoops of 1/4 cup into the bowl, one at a time. Say "one fourth, two fourths, ...". Write 6 × 1/4 = 6/4. Scoop it back into the 1-cup measure with the 1/4 cup: 4 scoops fill it, and 2 scoops (1/2 cup) are still in the bowl. So 6/4 is 1 full cup and 1/2 cup more.
  3. Try the 1/3 cup: How many scoops fill 1 cup? What is 5 × 1/3?

Answers to check: 3 scoops of 1/3 fill 1 cup. 5 × 1/3 = 5/3 = 1 2/3 cups.

Your turn

1

Write 1/6 + 1/6 + 1/6 + 1/6 as a multiplication. Solve it.

2

3 × 1/5 = ?

3

2 × 1/8 = ?

4

5 × 1/10 = ? Write it as a fraction and as a decimal.

5

4 × 1/2 = ?

6

9 × 1/3 = ?

7

Each muffin uses 1/4 cup of oats. How much for 3 muffins?

More practice: warm-up

1

1/3 + 1/3 = ?

2

2 × 1/3 = ?

3

4 × 1/4 = ?

4

1/10 + 1/10 + 1/10 = ?

5

5 × 1/6 = ?

More practice: on your own

1

7 × 1/4 = ?

2

6 × 1/5 = ?

3

10 × 1/2 = ?

4

Write 7/8 as a repeated addition and as a multiplication.

5

Ella walks 1/10 km to the mailbox and back each day. How far does she walk in 8 days?

Challenge

1

Which is bigger: 3 × 1/4 or 3 × 1/2? Explain.

2

Find the missing number: □ × 1/6 = 2

3

A recipe uses 1/3 cup of milk for each batch of pancakes. Nico makes 5 batches. He has 2 cups of milk. Is that enough?

Check your understanding

Quiz

1.

What is 4 × 1/5?

2.

Which is the same as 1/3 + 1/3 + 1/3 + 1/3?

3.

What is 6 × 1/2?

4.

What is 3 × 1/8?

5.

What is 5 × 1/4?

6.

What is 10 × 1/10?

For parents and teachers

Big idea: The curriculum (B2.7) asks Grade 4 students to represent the relationship between repeated addition of a unit fraction and multiplying that unit fraction by a whole number, using tools, drawings and fraction notation. 1/4 + 1/4 + 1/4 = 3 × 1/4 = 3/4.

Common difficulties:

  • Multiplying the denominator too (3 × 1/4 = 3/12). Use measuring cups or paper strips so your child sees the pieces stay the same size.
  • Not recognizing wholes in answers like 6/3. Ask "How many thirds make a whole?"
  • Mixed numbers. Writing 7/4 as 1 3/4 is a nice extra; the key Grade 4 idea is the repeated-addition link.

How to help: Bake together and use the measuring cups: "We need 3 scoops of 1/4 cup. How much is that?" Folding paper strips into halves, fourths and eighths is another great tool.