Teaching unit · Grade 5 (ages 10 to 11)

Order of operations: parentheses, brackets and braces

Work out calculations with groups inside groups, and write expressions that show a sequence of steps

About three lessons of 45 to 60 minutes

Start here · hook

Does 3 + 4 x 5 mean 35 or 23?

For 3 + 4 x 5, 23 is correct. The answer 35 is incorrect because it comes from adding before multiplying. We use an agreed order so the same written calculation has the same meaning for everyone.

Brackets can change the order: (3 + 4) x 5 correctly gives 35. In this unit, you will learn the order, work with groups inside groups, and write calculations from words without finding their answers.

3 + 4 x 5multiply first: 3 + 20 = 23
(3 + 4) x 5the parentheses force the addition first: 7 x 5 = 35
[(10 - 4) x 2] + 3innermost first: 6 x 2 = 12, then 12 + 3 = 15
Double 5, then add 9written as the expression (5 x 2) + 9, not computed yet

Learning objective

What students will be able to do

Students will work out expressions containing parentheses, brackets and braces, starting with the innermost group. They will follow the order of operations and write expressions that match a sequence of calculations without finding the answer.

Success criteria

I can evaluate the innermost grouping symbol first, then work outward.
I can tell parentheses, brackets, and braces apart and nest them correctly.
I can evaluate an expression with mixed operations in the correct order.
I can write a numerical expression for a word description without computing the answer.
I can explain what a grouping symbol changes about the order of a calculation.

Curriculum anchor

Standards this unit teaches

5.OA.A.1Common Core (US)
Evaluate expressions with grouping
Use parentheses, brackets, and braces in numerical expressions and evaluate them in the correct order.
5.OA.A.2Common Core (US)
Write numerical expressions
Write simple numerical expressions to record calculations without actually computing them.
AC9M6A02Australian Curriculum v9 (ACARA)
Unknowns with brackets (Year 6)
Find unknown values in equations involving brackets and combinations of operations using the properties of numbers. This is the direct Australian match for working correctly with grouping symbols.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Expression: numbers and operation signs that describe a value, such as 3 + 4 x 5; an expression does not include an equals sign
Evaluate: to work out the value of an expression by following the order of operations
Parentheses: round grouping symbols ( ), also called round brackets; work out the calculation inside them as a group
Brackets: square grouping symbols [ ]; they do the same job as round brackets and can make nested groups easier to read
Braces: curly grouping symbols { }; they do the same job as round and square brackets
Order of operations: for the expressions in this lesson: groups first, then multiply and divide from left to right, then add and subtract from left to right

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Why a fixed order matters

Concrete

Compare 3 + 4 x 5 with (3 + 4) x 5. For the first, make four groups of five counters and add three extra counters: 23 altogether. For the second, make five groups of seven counters: 35 altogether.

Step 1: Work inside grouping symbols first. If one group is inside another, start with the innermost group.

Step 2: Multiply and divide from left to right. They have equal priority; multiplication does not always come first.

Step 3: Add and subtract from left to right. They also have equal priority; addition does not always come first.

For example, 12 / 3 x 2 = 4 x 2 = 8 is correct. The answer 2 is incorrect: it comes from multiplying 3 x 2 first. Similarly, 10 - 3 + 2 = 7 + 2 = 9 is correct; 5 is incorrect because it comes from adding 3 + 2 first.

Grouping symbols tell us to keep part of a calculation together. In (3 + 4) x 5, the whole addition must be worked out before multiplying by 5.

Worked example

Evaluate 3 + 4 x 5, then evaluate (3 + 4) x 5. Explain why they differ.

3 + 4 x 5 = 3 + 20. Multiply 4 x 5 first; keep the 3 + unchanged.
3 + 20 = 23. Now add. Correct: 23. Incorrect: 35, because that adds before multiplying.
(3 + 4) x 5 = 7 x 5. Work inside the parentheses first.
7 x 5 = 35. Now multiply. Correct: 35. Incorrect: 23, because that ignores the parentheses.
The same numbers and operations give different answers because the parentheses change which part is worked out first.

Answer: 3 + 4 x 5 = 23, but (3 + 4) x 5 = 35.

Check for understanding, ask

In 3 + 4 x 5, which operation happens first, and why?
What single change turns 23 into 35 for these same three numbers?

2. Nesting parentheses, brackets and braces

Pictorial

A group inside another group is called nesting. Parentheses ( ), brackets [ ] and braces { } all group parts of a calculation. Different shapes help you see which opening and closing symbols belong together.

Start with the innermost group: the group with no other groups inside it. Then work outward. The position of a group decides the order, not the shape of its symbols.

In {[(10 - 4) x 2] + 3}, draw a box around 10 - 4, a larger box around (10 - 4) x 2, and a final box around the whole expression. Match each box to its pair of grouping symbols.

After each step, rewrite the whole expression. Replace only the part you have worked out and copy everything else unchanged.

Inside each group, the usual order still applies. Correct: (4 + 2 x 3) = (4 + 6) = 10. Incorrect: 18, because that adds 4 + 2 before multiplying.

Worked example

Evaluate {[(10 - 4) x 2] + 3}.

{[(10 - 4) x 2] + 3} = {[6 x 2] + 3}. Work out the innermost group: 10 - 4 = 6.
{[6 x 2] + 3} = {12 + 3}. Work out the next group: 6 x 2 = 12.
{12 + 3} = 15. Finish the outermost group: 12 + 3 = 15.

Answer: {[(10 - 4) x 2] + 3} = 15.

Check for understanding, ask

Which grouping symbol do you evaluate first in a nested expression: the innermost or the outermost?
In {[(10 - 4) x 2] + 3}, name the three grouping symbols in the order they get evaluated.

3. Writing an expression without computing it

Abstract

To write an expression, turn the words into numbers and operation signs. Do not work out the final answer. This lets you record a calculation before you carry it out, a skill you will use in formulas and algebra.

Read one instruction at a time. For 'add 8 and 7, then multiply the result by 2', start with 8 + 7. Put that addition in parentheses because the whole result must be multiplied: (8 + 7) x 2.

Correct: (8 + 7) x 2. Incorrect: 8 + 7 x 2, because that doubles only 7 before adding 8. The parentheses make the expression match the words.

Double means multiply by 2; triple means multiply by 3. Sum means the result of adding; difference means the result of subtracting. 'Triple the sum of 6 and 2' is 3 x (6 + 2).

The word 'then' does not always require brackets. For 'double 5, then add 9', both 5 x 2 + 9 and (5 x 2) + 9 are correct: the usual order already does the multiplication first.

Worked example

Write a numerical expression (do not evaluate it) for: 'Subtract 3 from 10, then multiply the result by 6.'

'Subtract 3 from 10' is 10 - 3.
'The result' means the whole subtraction. Group it so it happens before multiplying: (10 - 3).
'Then multiply ... by 6' attaches x 6 after the grouped subtraction.

Answer: Correct: (10 - 3) x 6. Incorrect: 10 - 3 x 6, which multiplies before subtracting. The value 42 is numerically correct, but giving only 42 does not answer the instruction to write an expression.

Check for understanding, ask

Why does '10 - 3' need parentheses once you multiply the result by 6?
Write an expression, without solving it, for 'double 5, then add 9'.

Watch for

Common misconceptions and how to address them

MisconceptionOperations are always worked strictly left to right, so 3 + 4 x 5 is (3 + 4) x 5 = 35.

Why it happens: Reading left to right is how students read everything else, so it feels natural to compute that way too.

How to address it: Correct: 3 + 4 x 5 = 3 + 20 = 23. Incorrect: 35, because it comes from adding first. Check groups first, then multiply/divide from left to right, then add/subtract from left to right.

MisconceptionParentheses, brackets and braces mean different math operations.

Why it happens: Three different-looking symbols suggest three different jobs, especially since brackets are used for other things elsewhere (like coordinates).

How to address it: Correct: all three shapes group parts of a calculation. Start with the innermost group. Incorrect: always do round brackets before square brackets regardless of position. In ([2 + 3] x 4), the square brackets are innermost, so work out 2 + 3 first.

MisconceptionOnce inside a grouping symbol, you still work strictly left to right instead of following the order of operations.

Why it happens: Students treat 'do the parentheses first' as 'do every symbol inside in reading order', forgetting the order of operations still applies inside the group.

How to address it: Correct: (4 + 2 x 3) = (4 + 6) = 10. Incorrect: 18, because it comes from adding first. Apply the usual order inside the group: multiply before adding.

Misconception'Write an expression' means write the final numeric answer.

Why it happens: Students are so used to every math question ending in a number that they cannot resist computing, even when told not to.

How to address it: Correct response: (8 + 7) x 2. Giving only 30 does not follow the instruction to write an expression, even though 30 is its correct value. Distinguish an incorrect calculation from an answer that does not complete the requested task.

Do it together

Guided practice (with answers)

1. Evaluate 12 / 3 x 2. Explain why 2 is incorrect.
Answer: Correct: 8. Work left to right: 12 / 3 = 4, then 4 x 2 = 8. Incorrect: 2, because that multiplies 3 x 2 first.
2. Evaluate 10 - 3 + 2. Explain why 5 is incorrect.
Answer: Correct: 9. Work left to right: 10 - 3 = 7, then 7 + 2 = 9. Incorrect: 5, because that adds 3 + 2 first.
3. Evaluate 3 + (4 x 5).
Answer: 23. The parentheses just confirm the usual order: 4 x 5 = 20, then 3 + 20 = 23.
4. Evaluate (3 + 4) x 5.
Answer: 35. The parentheses force the addition first: 3 + 4 = 7, then 7 x 5 = 35.
5. Evaluate 20 / (2 + 3).
Answer: 4. The parentheses first: 2 + 3 = 5. Then 20 / 5 = 4.
6. Evaluate 2 x [(6 + 4) / 5].
Answer: 4. Innermost first: 6 + 4 = 10. Then the bracket: 10 / 5 = 2. Then 2 x 2 = 4.
7. Write a numerical expression (do not evaluate) for: 'Subtract 3 from 10, then multiply the result by 6.'
Answer: (10 - 3) x 6, because the subtraction must be grouped to happen before the multiplication.
8. Write a numerical expression (do not evaluate) for: 'Triple the sum of 6 and 2.'
Answer: 3 x (6 + 2), because the sum must be grouped to happen before the tripling.

On their own

Independent practice worksheets

Use the linked worksheets for independent practice; answer keys are included. Start with one pair of grouping symbols. Move to nested groups once students can explain which part to work out first. Ask students to show each step so you can identify where an error begins.

Grade 5 order of operations worksheet (Set A)evaluating expressions with grouping symbols Grade 5 order of operations worksheet (Set B)more practice, including nested grouping Grade 6 order of operations worksheet (Set A)a stretch set once this unit is secure All order of operations worksheetsthe full set across Grades 4 to 6

Reach every student

Differentiation

Support

Colour-code each layer of a nested expression (parentheses in one colour, brackets in another) so the innermost-first order is visible, not just remembered.
Display the three steps: groups first; multiply and divide from left to right; add and subtract from left to right. Keep each equal-priority pair together.
Start with expressions that have one pair of grouping symbols and no nesting before introducing groups inside groups.
Rewrite the expression after each step so students see the shrinking expression rather than trying to track everything mentally.

Extension

Introduce exponents into grouped expressions, previewing 6.EE.A.1's place in the order (after grouping symbols, before multiply/divide).
Write a numerical expression for a genuinely multi-step word problem (three or more operations) and swap with a partner to evaluate each other's.
Place brackets in two different ways to make two valid expressions with different values. Write a situation that matches each one, then explain why changing the brackets changes the meaning.
Compare a calculator that follows the order of operations with one that calculates immediately after each operation is entered. For the written expression 3 + 4 x 5, 23 is correct; a display of 35 reflects a different input process, not another correct value for that expression.

Check it stuck

Assessment: exit ticket

A four-question exit ticket for the last five minutes: two evaluations and two expression-writing tasks.

1. Evaluate (9 - 5) x 3.
Answer: 12, because the parentheses force 9 - 5 = 4 first, then 4 x 3 = 12.
2. Evaluate 4 + (18 / 3).
Answer: 10, because the parentheses give 18 / 3 = 6 first, then 4 + 6 = 10.
3. Write a numerical expression (do not evaluate) for 'triple the sum of 6 and 2'.
Answer: 3 x (6 + 2).
4. Write a numerical expression (do not evaluate) for 'add 4 to 9, then double the result'.
Answer: (9 + 4) x 2.

For the teacher

Teacher notes and timings

Rough timing across three lessons: Lesson 1 why order matters plus single grouping symbols (section 1), Lesson 2 nesting (section 2), Lesson 3 writing expressions plus the exit ticket (section 3 and assessment).
Repeat: innermost group first; multiply and divide from left to right; add and subtract from left to right. When correcting work, name the first incorrect step, explain the rule, and let the student try again.
Accept equivalent expressions that record the requested calculation, such as (6 + 2) x 3 for 'triple the sum of 6 and 2'. Optional brackets do not make an answer wrong. Ask students to explain how each part matches the words.
This unit deliberately keeps every expression to whole numbers with straightforward answers so the order-of-operations rule itself, not the arithmetic, is what is being taught. Save decimals and larger numbers for once the rule is secure.
Curriculum note: the US names this Grade 5 standard, requiring parentheses, brackets and braces plus writing expressions. ACARA's closest match is Year 6 (AC9M6A02), which finds unknown values in bracketed equations rather than requiring students to write new expressions from words, so this unit runs about a year ahead of the ACARA placement.
Use Student view for a whole-class walkthrough. Print the teacher unit for planning; it includes answers. Use the linked worksheets for independent student practice.

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