Multiply and divide decimals

What you will learn

• Multiply and divide decimals

Models used: number sentences, stories, tables.

Before you start

• Express decimals as fractions

Multiply and share decimal pieces

Three ribbons are each 0.4 metre long. What length do they make altogether?

1. Each strip is a one-metre reference. The shaded four tenths show one ribbon.

   Each ribbon is four tenths of a metre. We have three equal amounts.

   0.4 × 3

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

   4/10 = 4/10 · 4/10 = 4/10 · 4/10 = 4/10: the same length, so the same amount.

2. Read the place heading before the digit. A counter in a different column has a different value.

   Three groups of four tenths give twelve tenths. Ten tenths make one whole metre.

   4 tenths × 3 = 12 tenths = 1.2

   Ones1

   Tenths2

   1.2 = 1 one + 2 tenths

3. The complete bar is the whole. Each labelled section is one part of it.

   The total length is 1.2 metres. The decimal point records the unit size; it is not an extra object.

   0.4 × 3 = 1.2

   1.2

   0.40.4 m0.40.4 m0.40.4 m

4. The complete bar is the whole. Each labelled section is one part of it.

   Reverse the story: share the 1.2 metres into three equal lengths. Each is 0.4 metre again.

   1.2 ÷ 3 = 0.4

   1.2

   0.40.4 m0.40.4 m0.40.4 m

Worked example: A new example, together

Calculate 5.9 × 0.8.

1. Track the value of the units, not just the digit strings. Read each value with the heading above its column.

   Name the units before and after grouping. Compare the answer’s size with a sensible estimate.

   The scale factor multiplies every length. A factor below one makes the copied length smaller.

   Track the value of the units, not just the digit strings.

   Starting amount	Scale factor
   5.9	0.8

2. Track the value of the units, not just the digit strings. Read each value with the heading above its column.

   Track the value of every digit when scaling.

   Track the value of the units, not just the digit strings.

   Starting amount	Scale factor
   5.9	0.8

   Calculate 5.9 × 0.8.

   4.72

3. Track the value of the units, not just the digit strings. Read each value with the heading above its column.

   5.9 × 0.8.

   Track the value of the units, not just the digit strings.

   Starting amount	Scale factor
   5.9	0.8

4. Track the value of the units, not just the digit strings. Read each value with the heading above its column.

   The result is 4.72. Check the reverse operation.

   Track the value of the units, not just the digit strings.

   Starting amount	Scale factor
   5.9	0.8

5. Track the value of the units, not just the digit strings. Read each value with the heading above its column.

   Check the result against the question we started with.

   4.72 ÷ 0.8 = 5.9. Undo the scaling to recover the original amount.

   We found 4.72. The scale factor multiplies every length. A factor below one makes the copied length smaller.

   Track the value of the units, not just the digit strings.

   Starting amount	Scale factor
   5.9	0.8

Common mistakes to discuss

• Losing the decimal point. Place the point by the size of the answer. 0.8 × 3 is 2.4, close to 3 lots of 1, not 24.

Check your understanding

• Multiply and divide decimals
• explain the scale of a decimal product or quotient
• 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.