Interpret a fraction as division

What you will learn

• Interpret a fraction as division

Models used: number sentences, stories, tables.

Before you start

• Add and subtract fractions with denominators up to 12 using at most two different denominators

Sharing explains the fraction

Three whole cakes are shared equally among four children. How much cake does each child get?

1. Each strip is one cake. There are three cakes altogether.

   These are our three cakes. Cut each cake into four equal parts, one for each child.

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

   4/4 = 4/4 · 4/4 = 4/4 · 4/4 = 4/4: the same length, so the same amount.

2. Only one child’s share is highlighted. The other pieces go to the other three children.

   Give one child a quarter from each cake. The highlighted pieces are that child’s share.

   1/4 + 1/4 + 1/4 = 3/4

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

   1/4 = 1/4 · 1/4 = 1/4 · 1/4 = 1/4: the same length, so the same amount.

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

   The three pieces together make three quarters of one cake. Every child receives the same share.

   3 ÷ 4 = 3/4

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

4. Each dot is one quarter of a cake. Four groups of three quarters use twelve quarters: three whole cakes.

   Four equal shares of three quarters put all three cakes back together. The numerator is the amount shared; the denominator is the number of equal shares.

   4 × 3/4 = 3

   3 + 3 + 3 + 3 = 4 × 3 = 12

Worked example: A new example, together

Share 2 whole cakes equally among 12 children. What fraction of a cake does each receive?

1. Division can have a fractional quotient. Read each value with the heading above its column.

   Name the amount being shared and the number sharing it, then read the fraction as that division.

   A whole-number division can be represented by a fraction: dividend over divisor.

   Division can have a fractional quotient

   Cakes	Equal shares
   2	12

2. Division can have a fractional quotient. Read each value with the heading above its column.

   Each whole cake can be split into 12 equal parts.

   Division can have a fractional quotient

   Cakes	Equal shares
   2	12

   Share 2 whole cakes equally among 12 children. What fraction of a cake does each receive?

   1/6

3. Division can have a fractional quotient. Read each value with the heading above its column.

   Each child receives 2 such parts.

   Division can have a fractional quotient

   Cakes	Equal shares
   2	12

4. Division can have a fractional quotient. Read each value with the heading above its column.

   2 ÷ 12 = 1/6.

   Division can have a fractional quotient

   Cakes	Equal shares
   2	12

5. Division can have a fractional quotient. Read each value with the heading above its column.

   Check the result against the question we started with.

   1/6 × 12 = 2.

   We found 1/6. A whole-number division can be represented by a fraction: dividend over divisor.

   Division can have a fractional quotient

   Cakes	Equal shares
   2	12

Common mistakes to discuss

• Dividing the wrong way round. The numerator is shared. 3/4 means 3 ÷ 4, so three pizzas shared among four children give 3/4 each.

Check your understanding

• Interpret a fraction as division
• read a fraction as a division and a division as a fraction
• 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.