Example 1
Substitute into a taxi formula
- Use C=4.50+2.20d with d=8 km.
- Substitute: C=4.50+2.20×8.
- Multiply first: C=4.50+17.60.
- Add to obtain C=$22.10.
Final answer: The taxi cost is $22.10.
Check: Substitute d=8 back into the original formula.
What students learn in AC9M7A01
We are learning to recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown.
A variable is a symbol representing a quantity that may change or be unknown. A formula states a general relationship among quantities, such as taxi cost C = 4.50 + 2.20d.
The fixed term and variable term have different roles: $4.50 is charged once, while $2.20 is multiplied by distance d. Units help explain each symbol and coefficient.
Substitution evaluates the formula for a particular case. Replace each variable with its given value, retain grouping, apply operation order and report the resulting quantity with its unit.
Success criteria
- I can define variables and represent an everyday relationship with a formula.
- I can explain the fixed and variable terms in a formula.
- I can substitute values, calculate accurately and report the correct units.
Key vocabulary
- variable
- A symbol representing a numerical quantity that may change or be unknown.
- formula
- A general mathematical relationship between quantities.
- coefficient
- A numerical factor multiplying a variable.
- fixed term
- A term whose value does not change when the variable changes.
- variable term
- A term containing a variable, so its value can change.
- substitution
- Replacing each variable with a given value before calculating.
Visual models and representations
Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.
4 worked numerical & application examples
Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.
Example 2
Represent and use a perimeter formula
- A rectangle has length l=8 cm and width w=5 cm.
- Use P=2l+2w.
- Substitute P=2×8+2×5.
- Calculate P=16+10=26 cm.
Final answer: The perimeter is 26 cm.
Check: Adding 8+5+8+5 also gives 26 cm.
Example 3
Application problem 1
Problem: A taxi fare is C = 4.50 + 2.20d, where d is kilometres travelled. Find the cost of a 13 km trip and explain what each number in the formula represents.
- Plan: Represent the information first, then calculate, interpret and independently check the result.
- Work: C = 4.50 + 2.20×13 = $33.10. The 4.50 is the fixed charge and 2.20 is the cost per kilometre.
- Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Final answer: C = 4.50 + 2.20×13 = $33.10. The 4.50 is the fixed charge and 2.20 is the cost per kilometre.
Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Example 4
Application problem 2
Problem: Create a formula for a $40 setup fee plus $12 per item, then find the cost of 15 items.
- Plan: Define the variable, write the formula, then substitute 15.
- Work: Let n be the number of items. Then C=40+12n, and for 15 items C=40+12×15=$220.
- Interpret: A formula represents the relationship before evaluating a case.
Final answer: Let n be the number of items. Then C=40+12n, and for 15 items C=40+12×15=$220.
Check: A formula represents the relationship before evaluating a case.
Common misconceptions
Common mistake: Variable treated as a label only.
Correction: It represents a numerical quantity with possible units.
Common mistake: Adjacent symbols added.
Correction: In algebra 2d means 2×d.
Common mistake: Units mixed during substitution.
Correction: Convert quantities first.
10 important problems to solve
Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.
1. In C=4.50+2.20d, identify the variable and its meaning.
Check answer
Answer: d is the variable and represents distance travelled in kilometres.
Hint: Use the context and units.
Why: A variable represents a quantity that can change.
2. Use P=2l+2w when l=8 cm and w=5 cm.
Check answer
Answer: P=2×8+2×5=26 cm.
Hint: Use brackets or multiplication signs clearly.
Why: Substitution replaces each variable with its value.
3. Use A=lw when l=12 m and w=7 m.
Check answer
Answer: A=12×7=84 m².
Hint: Substitute both dimensions before multiplying.
Why: The formula represents rectangle area.
4. A taxi costs C=4.50+2.20d. Find C for d=8 km.
Check answer
Answer: C=4.50+2.20×8=$22.10.
Hint: Multiply the rate by distance, then add the fixed charge.
Why: The fixed charge and distance charge are both included.
5. The formula T=3n+5 gives points for n completed tasks. Find T when n=9.
Check answer
Answer: T=3×9+5=32 points.
Hint: Replace n with 9 and follow operation order.
Why: Substitution evaluates one case of a general formula.
6. Write a formula for total cost C of x tickets at $7 each plus a $3 booking fee.
Check answer
Answer: C=7x+3.
Hint: Multiply ticket count by 7, then add 3.
Why: The variable charge and fixed fee form different terms.
7. A rectangle has area A=lw. Find A when l=6.5 m and w=4 m.
Check answer
Answer: A=6.5×4=26 m².
Hint: Write the substituted formula before calculating.
Why: Area retains square units after substitution.
8. A hire cost is H=25+6h. Compare 3 hours and 7 hours.
Check answer
Answer: H(3)=25+18=$43; H(7)=25+42=$67; the difference is $24.
Hint: Evaluate the formula twice, then subtract.
Why: The fixed fee is unchanged while the hourly term changes.
9. Create a formula for a $40 setup fee plus $12 per item, then find the cost of 15 items.
Check answer
Answer: Let n be the number of items. Then C=40+12n, and for 15 items C=40+12×15=$220.
Hint: Define the variable, write the formula, then substitute 15.
Why: A formula represents the relationship before evaluating a case.
10. A taxi fare is C = 4.50 + 2.20d, where d is kilometres travelled. Find the cost of a 13 km trip and explain what each number in the formula represents.
Check answer
Answer: C = 4.50 + 2.20×13 = $33.10. The 4.50 is the fixed charge and 2.20 is the cost per kilometre.
Hint: Represent the information first, then calculate, interpret and independently check the result.
Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Curriculum coverage and elaborations
The content description and elaborations below show the curriculum ideas taught in this unit. Items marked as teaching context support lesson planning.
- Content description: recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown
- E1: linking variables to attributes and measures being modelled when using formulas, such as the area of a rectangle is equal to the length x width as A\;=\;l\;\times\;w or using p\;=\;6g\;+\;b to describe a total of points expressed as goals (worth 6 points) and behinds (worth one point)
- E2: interpreting and using formulas obtained from other sources; for example, maximum heart rates and target heart rates for moderate exercise
- E3: substituting numerical values for variables when using formulas and calculating the value of an unknown in practical situations; for example, calculating weekly wage W given base wage b and overtime hours h at 1.5 times rate r, W\;=\;b+1.5\times h\times r, using values for mass m and volume v to determine density d of a substance where d\;=\;\frac mv
- E4: using everyday formulas and their application to contexts on Country/Place, investigating the relationships between variables
International curriculum mapping
This table gives closest-topic mapping for search and planning. The Australian Curriculum code is exact; overseas entries are broad equivalents because each jurisdiction structures outcomes differently.
| Region | Curriculum | Closest mapping |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M7A01 — recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown |
| Victoria | Victorian Curriculum F-10 | Year 7 Maths: closest match in Algebra. Use this page as a VIC-aligned practice and homework reference. |
| NSW | NSW Curriculum | Stage 4 Maths: closest content focus for Recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown and related outcomes. |
| United States | Common Core / NGSS | Grade 7 Common Core Mathematics/ELA closest topic match for Recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown. |
| England / UK | National Curriculum | Key Stage 3 / Year 7: closest programme-of-study match for Recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown. |
| Canada | Provincial and territory curricula | Grade 7 closest topic match. Canada varies by province, so use this as a broad Ontario/BC-style learning outcome reference. |
| New Zealand | New Zealand Curriculum | Level 4 Maths: closest achievement-objective topic for Recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown. |
| India | NCERT / CBSE | Class 7 closest NCERT/CBSE topic match for Recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown. |
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Recommended: What Is Algebra?
Math Antics — Understand how letters represent quantities and how expressions describe calculations.
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for And use variables to represent everyday formulas algebraically and substitute...
Mapped skill: recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown
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 | AC9M7A01 · Year 7 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M7A01 · Level 7 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA4-EQU-C-01 + MA4-LIN-C-01 · Stage 4 |
| United States (USA) | Common Core State Standards for Mathematics | Grade 7 |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 7 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 8, Key Stage 3 |
| India | NCERT / CBSE — Mathematics | Class 7 |
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: AC9M7A01 — Variables, Formulas and Substitution
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