Build and annotate the model
Represent unknown values with brackets and mixed operations and label the important parts, quantities or choices.
Learning goals
Learning goal
Students interpret equality as balance, follow grouping, identify the role of the unknown and use inverse operations or number properties before substituting to check.
Success criteria
- I can represent or identify the concept.
- I can explain the underlying relationship.
- I can select an appropriate strategy or feature.
- I can apply it in a new context.
- I can justify and verify the response.
Teaching routine
- Represent
- Reason
- Calculate
- Interpret
- Verify
Curriculum focus: find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations
Key concept
Solve 4(□ + 7) = 100
Use the visual model first. Ask students to identify the quantities, structure or conditions before calculating or explaining.
1Interpret4 equal groups total 100
2Undo ×4□ + 7 = 25
3Undo +7□ = 18
4Substitute4(18 + 7) = 100
5Confirmboth sides equal
Undo operations in reverse order while respecting brackets. The method follows the structure of the expression rather than moving symbols without meaning.
Worked examples
Use properties to simplify before solving
Connect the central relationship to a new context, then verify the conclusion with a second representation, estimate, inverse operation or reasonableness check.
6(□ + 4) = 180divide by 6, then subtract 4
3□ + 5□ = 96combine like multiplicative groups: 8□ = 96
5(20 − □) = 35divide, then solve 20 − □ = 7
(□ + 8) ÷ 4 = 9multiply by 4, then subtract 8
The unknown may appear in a subtracted position, so operation order and meaning matter. Always check in the original equation.
Curriculum elaborations explicitly taught
AC9M6A02: find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations
- E1: using brackets and the order of operations to write number sentences and appreciating the need for an agreed set of rules to complete multiple operations within the same number sentence; for example, for 40 ÷ 2 \times (4 + 6) = \square, you solve what is in the brackets first then complete the number sentence from left to right as there is no hierarchy between division and multiplication
- E2: constructing equivalent number sentences involving brackets and combinations of the 4 operations; explaining the need to have shared agreement on the order of operations when solving problems involving more than one operation to have unique solutions
- E3: finding pairs of unknown values in numerical equations that make the equation hold true; for example, listing possible combinations of natural numbers that make this statement true: 6+4\times8\;=\;6\times\bigtriangleup+\square
- E4: applying knowledge of inverse operations and number properties to create equivalent number sentences; removing one of the numbers and replacing it with a symbol, then swapping with a classmate to find the unknown values
Use the central and application models above to connect each elaboration to the same underlying concept.
Common misconceptions
Operations undone in written orderUndo the outer operation first.
Bracket ignoredTreat the grouped expression as one quantity.
Subtraction assumed commutative20 − x is not x − 20.
Check performed on a simplified but altered equationSubstitute into the original.
Classroom activities
Compare and reason
Use the application model to compare two cases, explain the relationship and identify a likely error.
Transfer and verify
Apply the idea in an unfamiliar context and use a second method, evidence source or text feature to check it.
Revision Notes and quick mastery check
Check understanding
- Solve 4(x+7)=100.
- Solve 5(20−x)=35.
- Combine 3x+5x.
- Explain reverse order.
- Substitute and check.
Evidence of mastery
- Represent or identify the concept
- Explain the underlying relationship
- Select an appropriate strategy or feature
- Apply it in a new context
- Justify and verify the response
Decision: continue when students can explain the model, apply it to a new example and justify their check. Otherwise return to the central model and reduce the numerical or representational load.
Curriculum wording and references
Australian Curriculum v9.0
AC9M6A02: find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations
The Australian Curriculum code and wording are exact. International teachers can use the underlying mathematical concept while matching the lesson to their local grade or year outcomes.
Resources and next steps
Teacher Slides
Project the fixed, branded four-page teaching sequence.
Open Classroom ViewHomework
Use the printable activity for written practice.
Open HomeworkPractice and Test
Use supported Practice first, then the separate Test when ready.
Open PracticeOpen Test
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Evaluating Expressions With and Without Parentheses
Khan Academy — Use grouping symbols to work out an expression before solving an equation with a missing value.
As you watch: How do brackets change the order in which calculations are completed?
Load video player Loads YouTube in this lesson. See the video notice below.
Try it: Find the missing value in 4 × (□ + 3) = 28. Show how you can check the answer by evaluating the bracket first.
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Find unknown values in numerical equations involving brackets and combinations...
Mapped skill: find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M6A02 · Year 6 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M6A02 · Level 6 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA3-MR-02 · Stage 3 |
| United States (USA) | Common Core State Standards for Mathematics | Grade 6 |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 6 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 7, Key Stage 3 |
| India | NCERT / CBSE — Mathematics | Class 6 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M6A02 — Unknown Values with Brackets and Mixed Operations
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