Efficient calculation and estimation | Year 6

01 / LEARN

Understand the idea.

Change the calculation without changing its value

Compensation replaces an awkward number with a nearby convenient one, then corrects the change. If you replace $398$ by $400$ in an addition, you have added $2$ too much and must subtract $2$ from the result.

Regrouping factors can also help. For $25 \times 16$ , write $16 = 4 \times 4$ and combine $25 \times 4$ first. The factors still represent the same product, but the intermediate calculation becomes $100 \times 4$ .

Estimate before trusting an answer

An estimate uses easy nearby numbers to check the size of an answer. Mark it as approximate. A close estimate supports a calculation, but does not prove that every digit of the exact answer is correct.

Check what the context requires as well as the arithmetic. If some passengers remain after filling several buses, another whole bus is needed. Rounding a quotient down would leave those passengers without seats.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Calculate 599+286 efficiently.

Worked solution

1
Make the first addend easier.
$600 + 286 = 886$
Replacing 599 by 600 adds one extra unit.
2
Compensate for the extra unit.
$599 + 286 = 886 - 1 = 885$
Subtracting the added unit restores the original total.
3
Check the size.
$600 + 300 = 900$
The exact answer 885 is slightly below this rounded estimate, as expected.

Answer

885.

EXAMPLE 02

Worked example 2

A school buys 18 boxes of 24 pencils. Find the total and check it.

Worked solution

1
Calculate the total for 20 boxes.
$20 \times 24 = 480$
Twenty boxes give an easy nearby calculation.
2
Find the pencils in the two extra boxes.
$2 \times 24 = 48$
Eighteen boxes are two fewer than twenty.
3
Remove those extra pencils.
$18 \times 24 = 480 - 48 = 432$
The subtraction leaves exactly the pencils in 18 boxes.
4
Check and label the result.
$20 \times 20 = 400$
An estimate of about 400 agrees with the size of 432 pencils.

Answer

432 pencils.

03 / YOUR TURN

Try it yourself.

Check your understanding

Choose a sensible estimate for 198×31: 600, 6,000 or 60,000.

Show answer & reasoning

6,000.

Round to $200 \times 30 = 6.000$ . This is close to the original calculation, so $6.000$ is the sensible estimate.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Use compensation to calculate $497 + 358$ .
2 marks
Calculate $25 \times 28$ by making a factor of 100.
2 marks
Calculate $1, 002 - 498$ efficiently.
2 marks
Estimate $61 \times 39$ using convenient tens.
1 mark
A calculator shows 7,200 for $48 \times 15$ . Explain why this is unreasonable, then correct it.
3 marks
A club has 178 people. Each bus seats 40. How many buses are needed?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Use compensation to calculate $497 + 358$ .
2 marks
$500 + 358 - 3 = 855$ .
855.
Calculate $25 \times 28$ by making a factor of 100.
2 marks
$25 \times 4 \times 7 = 100 \times 7 = 700$ .
700.
Calculate $1, 002 - 498$ efficiently.
2 marks
$1, 002 - 500 + 2 = 504$ .
504.
Estimate $61 \times 39$ using convenient tens.
1 mark
$60 \times 40 = 2, 400$ .
About 2,400.
A calculator shows 7,200 for $48 \times 15$ . Explain why this is unreasonable, then correct it.
3 marks
$50 \times 15$ is about 750. Exactly, $48 \times 10 + 48 \times 5 = 480 + 240 = 720$ .
It is about ten times too large; the correct answer is 720.
A club has 178 people. Each bus seats 40. How many buses are needed?
2 marks
Four seat 160, leaving 18 people, so one more is needed.
5 buses.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Prime/composite/square numbers; factors/multiples revision; efficient calculation and estimation.

Official curriculum source ↗

In this lesson

Understand the idea.

Change the calculation without changing its value

Estimate before trusting an answer

See the method in action.

Worked example 1

Make the first addend easier.

Compensate for the extra unit.

Check the size.

Worked example 2

Calculate the total for 20 boxes.

Find the pencils in the two extra boxes.

Remove those extra pencils.

Check and label the result.

Try it yourself.

Check your understanding

Keep practising