Efficient strategies | Year 4

01 / LEARN

Understand the idea.

Choose a strategy to suit the numbers

Choose a method that makes the numbers easier. For $19 \times 7$ , count twenty groups of seven and remove one group. For $35 \times 8$ , pair groups: eight groups of thirty-five become four groups of seventy. Both descriptions count the same objects.

Doubling and halving keeps the product unchanged only when the changes balance. Double one factor and halve the other. Doubling just one factor doubles the product. Use partitioning as another way to check that your new calculation has the same total.

Use friendly parts for division

For division, split the total into parts that divide exactly by the divisor. To share $96$ among four groups, share $80$ and then $16$ . Each group receives a share from both parts, so add those two shares.

Look for facts that turn work into friendly tens. Multiplying by five can mean multiplying by ten and halving that total, because five groups are half of ten groups. Explain why your chosen shortcut counts the same amount.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Double and halve

Calculate 35 × 8 efficiently.

Worked solution

1
Pair the eight groups.
$35 \times 8 = 70 \times 4$
Each pair contains seventy, and eight groups make four pairs.
2
Calculate the easier product.
$70 \times 4 = 280$
Four groups of seven tens contain twenty-eight tens.
3
Check by partitioning the original factor.
$\begin{aligned} 30 \times 8 & = 240 \\ 5 \times 8 & = 40 \\ 240 + 40 & = 280 \end{aligned}$
The independent calculation includes both parts of 35 and gives the same product.

Answer

280.

EXAMPLE 02

Divide using friendly parts

Find 96 ÷ 4.

Worked solution

1
Choose parts that divide by four.
$96 = 80 + 16$
Both eighty and sixteen contain a whole number of groups of four.
2
Share each part equally.
$\begin{aligned} 80 \div 4 & = 20 \\ 16 \div 4 & = 4 \end{aligned}$
Each of the four shares receives twenty from the first part and four from the second.
3
Combine the two parts of each share.
$20 + 4 = 24$
The quotient includes the share of the whole ninety-six.
4
Check the total by multiplication.
$24 \times 4 = 96$
Four shares of twenty-four reproduce the original amount.

Answer

24.

03 / YOUR TURN

Try it yourself.

Check your understanding

Find 62 × 5 by multiplying by 10 then halving.

Show answer & reasoning

310.

First find ten groups: $62 \times 10 = 620$ . Five groups are half as many, so $620 \div 2 = 310$ . Therefore $62 \times 5 = 310$ .

Keep practising

6 worksheet questions · 9 marks

Open worksheet Worked solutions

Read the questions online

Use a near-round number to find $19 \times 7$ .
2 marks
Use doubling and halving to find $25 \times 8$ .
1 mark
Find $84 \div 4$ using $80 + 4$ .
1 mark
Find $2 \times 7 \times 5$ by grouping useful factors.
1 mark
A school buys 18 packs of 5 pencils. Show two different ways to find the total.
2 marks
A learner changes $16 \times 6$ into $32 \times 3$ . Is this valid? Explain and calculate.
2 marks

Read the worked solutions

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

Use a near-round number to find $19 \times 7$ .
2 marks
$20 \times 7 = 140$ .
Remove one group of 7: $140 - 7 = 133$ .
133.
Use doubling and halving to find $25 \times 8$ .
1 mark
$25 \times 8 = 50 \times 4 = 100 \times 2 = 200$ .
200.
Find $84 \div 4$ using $80 + 4$ .
1 mark
$80 \div 4 = 20$ ; $4 \div 4 = 1$ .
$20 + 1 = 21$ .
21.
Find $2 \times 7 \times 5$ by grouping useful factors.
1 mark
$2 \times 5 = 10$ , so the product is $7 \times 10 = 70$ .
70.
A school buys 18 packs of 5 pencils. Show two different ways to find the total.
2 marks
Method 1: $10 \times 5 + 8 \times 5 = 50 + 40 = 90$ .
Method 2: $18 \times 10 \div 2 = 180 \div 2 = 90$ .
90 pencils.
A learner changes $16 \times 6$ into $32 \times 3$ . Is this valid? Explain and calculate.
2 marks
One factor is doubled and the other halved, so the product stays the same.
$32 \times 3 = 96$ ; check $16 \times 6 = 10 \times 6 + 6 \times 6 = 60 + 36 = 96$ .
Yes; both products equal 96.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Facts to 10×10; related division; larger calculations; multiples/powers of ten; efficient strategies.

Official curriculum source ↗

In this lesson

Understand the idea.

Choose a strategy to suit the numbers

Use friendly parts for division

See the method in action.

Double and halve

Pair the eight groups.

Calculate the easier product.

Check by partitioning the original factor.

Divide using friendly parts

Choose parts that divide by four.

Share each part equally.

Combine the two parts of each share.

Check the total by multiplication.

Try it yourself.

Check your understanding

Keep practising