Grouping with brackets | Year 7

01 / LEARN

Understand the idea.

A bracket creates one quantity

Brackets identify one complete quantity. In $4 (x + 3)$ , each of four groups contains $x + 3$ . In $4 x + 3$ , only $x$ is repeated four times and the extra $3$ occurs once. Removing brackets without preserving this grouping changes the expression.

When evaluating, work inside the brackets first. For $x = 2$ , $4 (x + 3)$ gives $4 \times 5 = 20$ , while $4 x + 3$ gives $8 + 3 = 11$ . This numerical comparison reveals the difference; the group interpretation explains why it occurs.

Count groups after writing one group

Write the contents of one group before counting several groups. If a set costs $p + 2$ dollars, three identical sets cost $3 (p + 2)$ dollars. Brackets make it clear that both items are purchased three times.

Grouping also matters in division and subtraction. The expression $\frac{a + b}{2}$ halves the combined amount, while $20 - (x + 4)$ removes both quantities in the bracket from twenty. Identify the complete group before performing the surrounding operation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Three sets

A pen costs $p$ dollars and a notebook costs 4 dollars. Write the cost of three sets, each with one pen and one notebook.

Worked solution

1
Find the cost of one complete set.
$p + 4 dollars$
Each set contains both a pen and a four-dollar notebook.
2
Multiply the whole set cost by three.
$3 (p + 4) dollars$
Three identical sets repeat both items, so the entire sum belongs inside brackets.
3
Check that every item is counted.
$3 (p + 4) = 3 p + 12$
The equivalent form contains three pens and three notebooks, confirming the grouping.

Answer

$3 (p + 4)$ dollars.

EXAMPLE 02

Two similar-looking expressions

Evaluate $2 (n + 5)$ and $2 n + 5$ when $n = 4$ .

Worked solution

1
Substitute into the bracketed expression.
$2 (n + 5) = 2 (4 + 5) = 2 \times 9 = 18$
Add inside the brackets before multiplying the resulting group by two.
2
Substitute into the unbracketed expression.
$2 n + 5 = 2 \times 4 + 5 = 8 + 5 = 13$
Without brackets, multiplication is completed before the addition.
3
Explain the difference.
$18 - 13 = 5$
The bracketed expression includes two copies of five; the other expression includes only one.

Answer

18 and 13, respectively.

03 / YOUR TURN

Try it yourself.

Check your understanding

Which means five times the sum of $x$ and 2: $5 x + 2$ or $5 (x + 2)$ ?

Show answer & reasoning

$5 (x + 2)$

“The sum of $x$ and two” is the whole group $x + 2$ . Multiplying that group by five gives $5 (x + 2)$ ; $5 x + 2$ multiplies only $x$ .

Keep practising

6 worksheet questions · 10 marks

Open worksheet Worked solutions

Read the questions online

Write four times the sum of $a$ and 6.
1 mark
Evaluate $3 (n + 2)$ for $n = 5$ .
2 marks
Evaluate $3 n + 2$ for $n = 5$ .
1 mark
Each pack has $x$ red counters and 3 blue counters. Write the total in six equal packs.
2 marks
For $x = 4$ , compare $20 - (x + 5)$ and $20 - x + 5$ .
2 marks
A student writes $2 p + 7$ for two boxes, each with $p$ pencils and 7 pens. Explain the error and correct it.
2 marks

Read the worked solutions

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

Write four times the sum of $a$ and 6.
1 mark
The sum is $a + 6$ . Multiply all of it by 4.
$4 (a + 6)$
Evaluate $3 (n + 2)$ for $n = 5$ .
2 marks
$3 (5 + 2) = 3 \times 7 = 21$ .
21
Evaluate $3 n + 2$ for $n = 5$ .
1 mark
$3 \times 5 + 2 = 17$ .
17
Each pack has $x$ red counters and 3 blue counters. Write the total in six equal packs.
2 marks
One pack contains $x + 3$ .
Six packs contain six copies of that sum.
$6 (x + 3)$
For $x = 4$ , compare $20 - (x + 5)$ and $20 - x + 5$ .
2 marks
$20 - (4 + 5) = 11$ .
$20 - 4 + 5 = 21$ .
11 and 21, respectively.
A student writes $2 p + 7$ for two boxes, each with $p$ pencils and 7 pens. Explain the error and correct it.
2 marks
Each box contains $p + 7$ items.
$2 p + 7$ counts the pens only once.
The pens must be counted in both boxes. Correct expression: $2 (p + 7)$ .

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Curriculum reference & lesson scope

Australian Curriculum v9.0 / SA Curriculum Mathematics

Year 7 variables, expression notation, substitution, simple equivalence and formula tables. Digital methods are explained with supplied data; the topic test works by hand.

Official curriculum source ↗

In this lesson

Understand the idea.

A bracket creates one quantity

Count groups after writing one group

See the method in action.

Three sets

Find the cost of one complete set.

Multiply the whole set cost by three.

Check that every item is counted.

Two similar-looking expressions

Substitute into the bracketed expression.

Substitute into the unbracketed expression.

Explain the difference.

Try it yourself.

Check your understanding

Keep practising