Divide a whole number or proper fraction by a proper fraction without a calculator

What you will learn

• Divide a whole number or proper fraction by a proper fraction without a calculator

Models used: number sentences, stories, tables.

Before you start

• Divide a proper fraction by a whole number

How many small pieces fit?

This ribbon is three quarters of a metre long.

1. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   This ribbon is three quarters of a metre long.

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

2. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   Each piece must be one eighth of a metre. Split every quarter into two eighths.

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

3. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   Six eighth-sized pieces fit in the ribbon. Dividing asks how many pieces fit.

   3/4 ÷ 1/8 = 6

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

Worked example: A new example, together

Calculate 1 ÷ 1/10.

1. Compare equal-sized fractional units. Read each value with the heading above its column.

   Ask how many of the smaller amount fit into the larger one.

   Dividing amounts expressed in the same fractional units compares their counts.

   Compare equal-sized fractional units

   Amount	One group size
   1	1/10

2. Compare equal-sized fractional units. Read each value with the heading above its column.

   Think: how many groups of size 1/10 fit into 1?

   Compare equal-sized fractional units

   Amount	One group size
   1	1/10

   Calculate 1 ÷ 1/10.

   10

3. Compare equal-sized fractional units. Read each value with the heading above its column.

   Rename into the same units: 10 parts shared into groups of 1 parts.

   Compare equal-sized fractional units

   Amount	One group size
   1	1/10

4. Compare equal-sized fractional units. Read each value with the heading above its column.

   10 ÷ 1 = 10.

   Compare equal-sized fractional units

   Amount	One group size
   1	1/10

5. Compare equal-sized fractional units. Read each value with the heading above its column.

   Check the result against the question we started with.

   10 × 1/10 = 1.

   We found 10. Dividing amounts expressed in the same fractional units compares their counts.

   Compare equal-sized fractional units

   Amount	One group size
   1	1/10

Common mistakes to discuss

• Expecting a smaller answer. Dividing by a number below one gives more, not fewer. 2 ÷ 1/4 = 8, because eight quarters fit into two wholes.

Check your understanding

• Divide a whole number or proper fraction by a proper fraction without a calculator
• count how many fractional parts fit into an amount
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.