Multiply proper and improper fractions by whole numbers

What you will learn

• Multiply proper and improper fractions by whole numbers

Models used: number sentences, stories, tables.

Before you start

• Add and subtract mixed numbers

Find a fraction of a group

Here are 12 counters. We want two thirds of them.

1. Each dot stands for one object. A full row holds five; two full rows make ten.

   Here are 12 counters. We want two thirds of them.

   The denominator determines one part’s size. The numerator tells us how many of those parts to take.

2. There are 3 groups, with 4 objects in each group. Count a group before counting them all.

   Thirds means three equal groups. Each group has 4 counters.

   4 + 4 + 4 = 3 × 4 = 12

3. There are 2 groups, with 4 objects in each group. Count a group before counting them all.

   Keep two of those groups. Two groups of four make eight.

   2/3 of 12 = 8

   4 + 4 = 2 × 4 = 8

Worked example: A new example, together

Find 13/8 × 5.

1. Share into denominator-many parts, then select numerator-many. Read each value with the heading above its column.

   Identify the amount that acts as the whole, then one part, then the selected parts.

   A fraction of a amount scales that quantity by the fraction.

   Share into denominator-many parts, then select numerator-many

   Equal groups	In each group	Parts needed
   8	0.625	13

2. Share into denominator-many parts, then select numerator-many. Read each value with the heading above its column.

   A fraction of an amount means divide by 8, then take 13 of those parts.

   Share into denominator-many parts, then select numerator-many

   Equal groups	In each group	Parts needed
   8	0.625	13

   Find 13/8 × 5.

   65/8

3. Share into denominator-many parts, then select numerator-many. Read each value with the heading above its column.

   A fraction of an amount means divide by 8, then take 13 of those parts.

   Share into denominator-many parts, then select numerator-many

   Equal groups	In each group	Parts needed
   8	0.625	13

4. Share into denominator-many parts, then select numerator-many. Read each value with the heading above its column.

   5 ÷ 8 × 13.

   Share into denominator-many parts, then select numerator-many

   Equal groups	In each group	Parts needed
   8	0.625	13

5. Share into denominator-many parts, then select numerator-many. Read each value with the heading above its column.

   The amount is 65/8.

   Share into denominator-many parts, then select numerator-many

   Equal groups	In each group	Parts needed
   8	0.625	13

6. Share into denominator-many parts, then select numerator-many. Read each value with the heading above its column.

   Check the result against the question we started with.

   The selected amount divided by 5 equals 13/8.

   We found 65/8. A fraction of a amount scales that quantity by the fraction.

   Share into denominator-many parts, then select numerator-many

   Equal groups	In each group	Parts needed
   8	0.625	13

Common mistakes to discuss

• Multiplying the denominator as well. Only the number of parts grows. 3 × 2/5 = 6/5, because the fifths stay fifths.

Check your understanding

• Multiply proper and improper fractions by whole numbers
• explain fraction multiplication as selecting a share of 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.