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Year 6 Maths Number Sentences & Order of Operations

Order of operations & grouping symbols - With brackets

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Theory

Brackets ( ) are grouping symbols. They say "work this part out first". \(9 + 6 \times 3 = 27\), but \((9 + 6) \times 3 = 45\). This Year 6 maths page shows how to work out number sentences with brackets, including brackets inside brackets, and how to place brackets to make a number sentence true.

Brackets group part of a number sentence. The part inside the brackets is worked out first, before any other operation.

After the brackets, use the usual order: \(\times\) and \(\div\) before \(+\) and \(-\); equal rank from left to right.

With brackets inside brackets, start with the innermost pair and work outwards.

Brackets can also just make the order clear: \(5 + (3 \times 6)\) has the same value as \(5 + 3 \times 6\), because the multiplication would be done first anyway.

Working out (9 + 6) × 3 The line (9 + 6) × 3 with the brackets highlighted, then 15 × 3, then the answer 45. (9 + 6) × 3 brackets first 15 × 3 then multiply 45
With brackets: \((9 + 6) \times 3 = 45\). Without them, \(9 + 6 \times 3 = 27\).
Brackets inside brackets The line 4 × (15 − (2 + 6)) with the inner brackets 2 + 6 highlighted, then 4 × (15 − 8), then 4 × 7, then 28. 4 × (15 − (2 + 6)) innermost brackets first 4 × (15 − 8) then the outer brackets 4 × 7 then multiply 28
Work from the innermost brackets outwards.

Brackets come first:

\[ \text{brackets} \;\rightarrow\; \times, \div \;\rightarrow\; +, - \]
brackets →×÷→+−
\[ (9 + 6) \times 3 = 45 \quad \text{but} \quad 9 + 6 \times 3 = 27 \]
(9+6)×3=45 but 9+6×3=27
Brackets change the answer. The same numbers and signs can give different answers with and without brackets.
Innermost first. In \(4 \times (15 - (2 + 6))\), do \(2 + 6\) first.

Working out a number sentence with brackets

  1. Work out the inside of each pair of brackets (innermost first).
  2. Rewrite the line with that answer in place of the brackets.
  3. Finish with \(\times\) and \(\div\), then \(+\) and \(-\).

Placing brackets to make a sentence true

  1. Try brackets around different pairs of numbers.
  2. Work out each try until one gives the target.
Example 1 — With and without
Work out
(i) \((9 + 6) \times 3\)
(ii) \(9 + 6 \times 3\)
Solution
\(15 \times 3\)\(=\)\(45\)
\(9 + 18\)\(=\)\(27\)
45 and 27

(i) \(45\)
(ii) \(27\)

Example 2 — Subtract in brackets
Work out \(6 \times (11 - 4)\).
Solution
\(6 \times (11 - 4)\)\(=\)\(6 \times 7\)
\(6 \times 7\)\(=\)\(42\)
6×7=42

\(42\).

Example 3 — Divide by a bracket
Work out \(72 \div (3 + 5)\).
Solution
\(72 \div (3 + 5)\)\(=\)\(72 \div 8\)
\(72 \div 8\)\(=\)\(9\)
72÷8=9

\(9\).

Example 4 — Take away a group
Work out \(60 - (14 + 9)\).
Solution
\(60 - (14 + 9)\)\(=\)\(60 - 23\)
\(60 - 23\)\(=\)\(37\)
60−23=37

\(37\).

Example 5 — Brackets inside brackets
Work out \(4 \times (15 - (2 + 6))\).
Solution
\(4 \times (15 - 8)\)\(=\)\(4 \times 7\)
\(4 \times 7\)\(=\)\(28\)
4×(15−8)=28

\(28\), as in the second diagram.

Example 6 — Two pairs of brackets
Work out \((30 + 6) \div (7 - 3)\).
Solution
\((30 + 6) \div (7 - 3)\)\(=\)\(36 \div 4\)
\(36 \div 4\)\(=\)\(9\)
36÷4=9

\(9\).

Example 7 — Place the brackets
Put brackets in \(5 + 3 \times 6\) so that its value is \(48\).
Solution
\((5 + 3) \times 6\)\(=\)\(48\)
(5+3)×6=48

\((5 + 3) \times 6 = 48\). Without brackets the value is \(23\).

Example 8 — Word problem
Tickets cost \(\$9\) each. A family buys tickets for \(4\) adults and \(3\) children. Write one number sentence for the total cost and work it out.
Solution
\(9 \times (4 + 3)\)\(=\)\(9 \times 7\)
\(9 \times 7\)\(=\)\(63\)
9×(4+3)=63

\(\$63\).

Example 9 — Find the mistake
Ana worked out \(4 \times (7 + 2)\) and wrote \(30\). What went wrong?
Solution
\(4 \times (7 + 2)\)\(=\)\(4 \times 9\)
\(4 \times 9\)\(=\)\(36\)
4×9=36

She ignored the brackets and did \(4 \times 7 + 2\). The brackets come first: the answer is \(36\).

Common pitfalls

Ignoring the brackets. \(4 \times (7 + 2) = 36\), not \(4 \times 7 + 2 = 30\).
Starting with the outer brackets. With brackets inside brackets, start with the innermost.
Forgetting the rules after the brackets. After the brackets, still do \(\times\) and \(\div\) before \(+\) and \(-\).
Brackets that change nothing. \(5 + (3 \times 6)\) is still \(23\); the brackets must group the addition to change it.

Frequently asked questions

What do brackets mean in a number sentence?

Work out the part inside the brackets first.

What is (9 + 6) × 3?

\(15 \times 3 = 45\).

What if there are brackets inside brackets?

Work out the innermost brackets first, then work outwards.

Do brackets always change the answer?

No. They only change it if they change which operation is done first.

How do I place brackets to get a target?

Try grouping different pairs and work each one out until you hit the target.

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