Multiply and divide three-place decimals by 10, 100 and 1000 without a calculator

What you will learn

• Multiply and divide three-place decimals by 10, 100 and 1000 without a calculator

Models used: number sentences, stories, tables.

Before you start

• Add and subtract two-place decimals without a calculator

The digits change their value

0.35 has three tenths and five hundredths.

1. The number at the top of each column is the value of one counter in that column.

   0.35 has three tenths and five hundredths.

   0.35

   The same digits can describe very different values. An estimate and the inverse operation help check the scale of the answer.

2. The number at the top of each column is the value of one counter in that column.

   Make the amount ten times as large. Each tenth becomes a one; each hundredth becomes a tenth.

   0.35 × 10

   A digit tells us how many units there are. Its position tells us what size those units are; that is why a zero can matter.

3. The number at the top of each column is the value of one counter in that column.

   The new amount is 3.5. Dividing by ten takes us back.

   3.5 ÷ 10 = 0.35

   A digit tells us how many units there are. Its position tells us what size those units are; that is why a zero can matter.

Worked example: A new example, together

Calculate 6.7 × 10.

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.

   Every bottle holds the same amount. Multiplication combines that amount once for each bottle.

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

   Starting amount	Scale factor
   6.7	10

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
   6.7	10

   Calculate 6.7 × 10.

   67

3. 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
   6.7	10

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

   6.7 × 10.

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

   Starting amount	Scale factor
   6.7	10

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

   The result is 67. Check the reverse operation.

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

   Starting amount	Scale factor
   6.7	10

6. 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.

   67 ÷ 10 = 6.7. Undo the scaling to recover the original amount.

   We found 67. Every bottle holds the same amount. Multiplication combines that amount once for each bottle.

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

   Starting amount	Scale factor
   6.7	10

Common mistakes to discuss

• Adding or deleting a zero. Each digit moves place. 6.7 × 10 = 67, and 4.2 ÷ 100 = 0.042, where a zero must hold the empty tenths place.

Check your understanding

• Multiply and divide three-place decimals by 10, 100 and 1000 without a calculator
• 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.