Teaching unit · Grade 5 (ages 10 to 11)

Order of operations: parentheses, brackets and braces

Evaluating expressions with nested grouping symbols, and writing a numerical expression without computing it

About three lessons of 45 to 60 minutes

Start here · hook

Does 3 + 4 x 5 mean 35 or 23?

Two students both work out 3 + 4 x 5. One gets 35 (adds first, then multiplies). The other gets 23 (multiplies first, then adds). They cannot both be right, and if maths let everyone choose their own order, every calculator, every recipe, and every shared spreadsheet would give a different answer to the same expression.

Mathematicians settled this centuries ago with one fixed order: grouping symbols first, then multiplication and division, then addition and subtraction, always working left to right within a step. Parentheses ( ), brackets [ ], and braces { } let you override that order on purpose, and nesting them (brackets around parentheses, braces around brackets) lets you build up a whole calculation as one expression before anyone computes it.

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 evaluate numerical expressions containing parentheses, brackets, and braces by working the innermost grouping symbol first and following the fixed order of operations, and will write a numerical expression that correctly records a sequence of calculations without evaluating it.

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.
AC9M5A02Australian Curriculum v9 (ACARA)
Unknowns with brackets (Year 5)
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: a string of numbers and operations, such as 3 + 4 x 5, with no equals sign and no single answer written yet
Evaluate: to work out the value of an expression by following the order of operations
Parentheses: the round grouping symbols ( ), the innermost and most common grouping symbol
Brackets: the square grouping symbols [ ], used to group a parentheses expression with more
Braces: the curly grouping symbols { }, used to group a brackets expression with more
Order of operations: the fixed order every expression is evaluated in: grouping symbols, then multiply/divide, then add/subtract

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

Start with the conflict. Write 3 + 4 x 5 on the board and let students compute it their own way; some will add first, some will multiply first. Reveal that they get different answers, then reveal the rule: multiplication and division always happen before addition and subtraction, unless grouping symbols say otherwise. So 3 + 4 x 5 means 3 + (4 x 5), the multiplication first.

This is not an arbitrary rule to memorise, it is an agreement that makes every written expression mean exactly one thing to everyone who reads it. Without it, a recipe scaled '2 cups + 3 cups x 4 batches' would be ambiguous.

Grouping symbols are how you override the default order on purpose. Parentheses say 'do this part first, no matter what it is'.

Worked example

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

3 + 4 x 5: no grouping symbols, so multiply first. 4 x 5 = 20. Then add: 3 + 20 = 23.
(3 + 4) x 5: the parentheses force the addition first. 3 + 4 = 7. Then multiply: 7 x 5 = 35.
Same three numbers, same two operations, different order, different answer: 23 versus 35.

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

When an expression needs more than one layer of grouping, mathematicians switch symbols so the layers are easy to see: parentheses ( ) on the inside, brackets [ ] around those, braces { } around those. Always evaluate the innermost symbol first and work outward, like peeling an onion from the centre.

Take {[(10 - 4) x 2] + 3}. The innermost layer is (10 - 4) = 6. Substitute that in: [6 x 2] + 3 inside the braces becomes [12] + 3. Now the bracket layer, which is just 12, is done, so evaluate the addition: 12 + 3 = 15.

Worked example

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

Innermost first, the parentheses: 10 - 4 = 6.
Next layer, the brackets: 6 x 2 = 12.
Outermost layer, the braces: 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

The second half of this standard runs the process in reverse: turn a spoken calculation into a written expression, and stop, without finding the answer. This matters because in algebra and in real formulas, you often need to record a calculation exactly before you are ready or able to compute it.

Translate the words in order, phrase by phrase, and add grouping symbols wherever the words say to do something 'first' or 'then'. 'Add 8 and 7, then multiply by 2' becomes (8 + 7) x 2, because the parentheses force the adding to happen before the multiplying, matching the word 'then'.

Watch for the trap words: 'double' and 'triple' mean multiply by 2 and by 3, and phrases like 'the sum of' or 'the difference of' name an addition or subtraction that usually needs its own parentheses if another operation follows.

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' refers to that whole subtraction, so it needs parentheses when another operation follows: (10 - 3).
'Then multiply ... by 6' attaches x 6 after the grouped subtraction.

Answer: (10 - 3) x 6. (Left as an expression, not computed -- though for your own check, it is 42.)

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: Multiplication and division always outrank addition and subtraction, regardless of reading order. In 3 + 4 x 5, the 4 x 5 is 'glued together' and must be done first: 3 + 20 = 23. Only grouping symbols can force a different order.

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: In an order-of-operations expression they all mean exactly the same thing: 'do this part first'. They are only nested in different shapes so a reader can tell which layer is which at a glance.

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: Inside any grouping symbol, the normal order of operations still applies: multiply and divide before you add and subtract, even inside the parentheses. (4 + 2 x 3) is 4 + 6 = 10 inside the parentheses, not (4 + 2) x 3 = 18.

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: An expression is the recipe, not the meal. (8 + 7) x 2 is a complete, correct answer to 'write an expression for add 8 and 7, then multiply by 2' -- computing it to 30 answers a different question.

Do it together

Guided practice (with answers)

1. Evaluate 3 + (4 x 5).
Answer: 23. The parentheses just confirm the usual order: 4 x 5 = 20, then 3 + 20 = 23.
2. Evaluate (3 + 4) x 5.
Answer: 35. The parentheses force the addition first: 3 + 4 = 7, then 7 x 5 = 35.
3. Evaluate 20 / (2 + 3).
Answer: 4. The parentheses first: 2 + 3 = 5. Then 20 / 5 = 4.
4. Evaluate 2 x [(6 + 4) / 5].
Answer: 4. Innermost first: 6 + 4 = 10. Then the bracket: 10 / 5 = 2. Then 2 x 2 = 4.
5. 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.
6. 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

Set the matching ChalkBee order-of-operations worksheets for independent practice. Answer keys are computed in code, so they are never wrong. Start with single grouping symbols, then move to nested expressions once the basic rule is automatic.

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.
Use a simple mnemonic phrase for the order (grouping, multiply/divide, add/subtract) posted where students can see it while they work.
Start with expressions that have only one grouping symbol and no nesting before introducing brackets and braces.
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.
Find two different grouping placements that give two different but both 'valid' expressions for a real situation, and discuss which matches the situation.
Explore why some calculators give 23 and others 35 for 3 + 4 x 5 typed without parentheses (a good discussion of why the order-of-operations convention exists at all).

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).
Language to keep saying: innermost first, grouping symbols before multiply and divide before add and subtract, an expression is a recipe not a meal. These target the four misconceptions directly.
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 5 (AC9M5A02), which finds unknown values in bracketed expressions rather than requiring students to write new expressions from words -- the writing half of this unit runs a little ahead of the ACARA Year 5 descriptor.
Present mode and print both work: project the nested-expression peeling for a whole-class walkthrough, then print for independent practice with the Print button.

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