Calculate a decimal quotient of whole numbers without a calculator

What you will learn

• Calculate a decimal quotient of whole numbers without a calculator

Models used: number sentences, stories, tables.

Before you start

• Multiply and divide decimals

Keep sharing into smaller decimal units

Three litres are shared equally among four containers. How many litres go in each?

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

   Three litres cannot give each of four containers a whole litre. We need smaller units.

   3 ÷ 4

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

   3

   ?container 1?container 2?container 3?container 4

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

   Rename three litres as 300 hundredths of a litre. Now share the 300 hundredths equally into four groups.

   300 ÷ 4 = 75 hundredths

   300

   7575 hundredths7575 hundredths7575 hundredths7575 hundredths

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

   Each container receives 75 hundredths of a litre. Write this as 0.75 litre.

   3 ÷ 4 = 0.75

   Ones0

   Tenths7

   Hundredths5

   0.75 = 7 tenths + 5 hundredths

4. Start at the marked number. A jump to the right adds; a jump to the left takes away. Read the size of each jump.

   If an answer to one decimal place is needed, round 0.75 to 0.8. Keep the exact quotient separate from its rounded value.

   Exact: 0.75 · To 1 decimal place: 0.8

Worked example: A new example, together

Calculate 12 ÷ 6, rounded to 2 decimal places.

1. Round only at the final step. 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.

   Each decimal column names a unit ten times smaller than the one before it. Matching those units keeps the value clear.

   Round only at the final step

   Unrounded value	Requested decimal places
   2.0000	2

2. Round only at the final step. Read each value with the heading above its column.

   Keep digits through the requested 2 decimal places.

   Round only at the final step

   Unrounded value	Requested decimal places
   2.0000	2

   Calculate 12 ÷ 6, rounded to 2 decimal places.

   2.00

3. Round only at the final step. Read each value with the heading above its column.

   Keep digits through the requested 2 decimal places.

   Round only at the final step

   Unrounded value	Requested decimal places
   2.0000	2

4. Round only at the final step. Read each value with the heading above its column.

   Inspect the next digit: 5 or more rounds the retained value up.

   Round only at the final step

   Unrounded value	Requested decimal places
   2.0000	2

5. Round only at the final step. Read each value with the heading above its column.

   The rounded answer is 2.00.

   Round only at the final step

   Unrounded value	Requested decimal places
   2.0000	2

6. Round only at the final step. Read each value with the heading above its column.

   Check the result against the question we started with.

   Check neighbouring rounded values and the halfway point.

   We found 2.00. Each decimal column names a unit ten times smaller than the one before it. Matching those units keeps the value clear.

   Round only at the final step

   Unrounded value	Requested decimal places
   2.0000	2

Common mistakes to discuss

• Rounding partway through. Work to one more place than the answer needs, then round once at the end. 2 ÷ 3 is 0.666…, which is 0.67 to two decimal places.

Check your understanding

• Calculate a decimal quotient of whole numbers 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.