A question gives a formula, a table, a total cost and several numbers. What do you do first so you do not use the wrong strategy?
Before calculating write a decision process.
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The strategy decision guide
+5 XP to read Before calculating, decide what the question is asking for.
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The strategy decision guide
+5 XP to read
Before calculating, decide what the question is asking for. Different algebra problems need different first moves.
Need an output? Substitute values into the formula.
Need an unknown input? Write and solve an equation.
Wrong subject? Rearrange first, then substitute.
Need a formula from data? Find starting value and rate.
Identify the required value first, then choose the strategy
Substitution
Used when the formula subject is already the required value. Replace each variable, then calculate.
Solve an equation
Used when a total is known and an unknown input must be found. Write the equation first, then solve.
Rearrange first
When the required variable is not the subject of the formula, rearrange to isolate it before substituting.
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What you'll master
Know
Key facts
Different algebra problems need different first moves.
Units, variable definitions and context guide the strategy.
A final answer should be checked for reasonableness.
Understand
Concepts
Substitution is used when the formula subject is already the required value.
Rearranging is useful when the required variable is not the subject.
Equations are needed when a total is known and an unknown input must be found.
Can do
Skills
Choose between substituting, solving, rearranging and building formulas.
Solve mixed practical algebra problems.
Explain whether an answer is reasonable in context.
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Key terms
SubstitutionReplacing a variable with its known value to calculate a formula output.
Solving an equationFinding the unknown value that makes an equation true, using inverse operations.
RearrangingRewriting a formula so a different variable becomes the subject, using inverse operations.
Subject of a formulaThe variable that is alone on one side of the formula. E.g. in $d = st$, $d$ is the subject.
Reasonableness checkConfirming the answer makes sense in context, correct magnitude, sign, and units.
Building a formulaWriting a formula from data by identifying the starting value and the rate of change.
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Choose the first move
Before calculating, decide what the question is asking for.
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Choose the first move
core concept
Before calculating, decide what the question is asking for. The table below summarises the key strategies:
Question asks for…
Best first move
Example
A total from known inputs
Substitute
$C = 12 + 4r$, find $C$ when $r = 7$
The input that produced a total
Solve an equation
$12 + 4r = 40$
A variable not currently alone
Rearrange
$d = st$, find $s$
A formula from data
Find starting value and rate
Table outputs increase by 5
Common error: Do not try to use every number immediately. Identify the required value first.
Decision table for worded formula problems: if given all values → substitute and evaluate; if asked for input → rearrange first then substitute; if comparing → calculate both and state the conclusion with a dollar/unit difference.
Pause, copy the three formula question types and their first moves: (1) all values given → substitute and evaluate; (2) asked for an input variable → rearrange first; (3) comparing two options → calculate both separately, then state the conclusion with a difference into your book.
Quick check: A total cost is known and the number of items is unknown. Which strategy should you use first?
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Check for reasonableness
We just saw the decision table for formula questions: if all values are given → substitute and evaluate; if asked for an input...
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Check for reasonableness
core concept
We just saw the decision table for formula questions: if all values are given → substitute and evaluate; if asked for an input → rearrange first; if comparing two options → calculate both, then state which is better and by how much. That raises a question: even after correctly choosing the first move and completing the calculation, how do you know the answer is actually right before writing it down? This card answers it → three reasonableness checks: magnitude (is the answer in the right ballpark?), sign (should it be positive or negative?), and units (does the unit match the question?).
After obtaining an answer, always ask: does this make sense?
A stopping-distance model is $D = 0.01v^2 + 0.3v$. If a student says the stopping distance at 60 km/h is 540 m, check: $D = 0.01(3600) + 18 = 36 + 18 = 54$ m. The student's answer is ten times too large.
Reasonableness habit: Large differences often signal a decimal, unit or substitution error.
After calculating, ask: does this answer make sense in context? Check magnitude (is it in the right ballpark?), sign (should the answer be positive?), and units (does the unit match the question’s requirement?). State a rejection reason if the answer is unreasonable.
Pause, copy the three post-calculation reasonableness checks: (1) magnitude, is the answer in the expected range?; (2) sign, should the answer be positive or negative?; (3) units, does the unit match what the question asked for? into your book.
True or false: If an answer seems unreasonably large, the error is always a sign mistake.
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Worked examples · 4 in a row, reveal as you go
Worked examples · 4 in a row, reveal as you go
Worked examples · 4 in a row, reveal as you go
PROBLEM 1 · SOLVE AN EQUATION
A printer charges $\$25$ setup plus $\$2$ per page. A job costs $81. How many pages were printed?
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Strategy: write an equation
The total is known and the number of pages is unknown, so write an equation. Let $p$ = number of pages.
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$25 + 2p = 81$
Setup cost ($\$25$) plus $\$2$ for each page equals the total ($81).
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$2p = 56$, so $p = 28$
Subtract 25 from both sides: $2p = 56$. Divide by 2: $p = 28$. Answer: 28 pages were printed.
PROBLEM 2 · REARRANGE BEFORE SUBSTITUTING
A cyclist travels 135 km in 3 hours. Use $d = st$ to find the average speed.
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Strategy: rearrange first
Speed is not the subject of $d = st$, so rearrange to make $s$ the subject.
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$d = st$ becomes $s = \dfrac{d}{t}$
Divide both sides by $t$ to isolate $s$.
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$s = \dfrac{135}{3} = 45$ km/h
Substitute $d = 135$ and $t = 3$. The average speed is 45 km/h.
PROBLEM 3 · BUILD A FORMULA FROM A TABLE
Hours, $h$
0
1
2
3
Cost, $C$
$30
$42
$54
$66
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Starting value: $C = 30$ when $h = 0$
Read the table at $h = 0$. This is the starting value (y-intercept).
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Rate of change: $+\$12$ per hour
Each time $h$ increases by 1, $C$ increases by $42 - 30 = 12$. This is constant, so the formula is linear.
Formula confirmed. The starting value is 30 and the rate is 12 per hour.
PROBLEM 4 · CHECK REASONABLENESS
A stopping-distance model is $D = 0.01v^2 + 0.3v$. A student says the stopping distance at 60 km/h is 540 m. Is this correct?
1
$D = 0.01(60)^2 + 0.3(60)$
Substitute $v = 60$ to check the student's claim.
2
$D = 0.01(3600) + 18 = 36 + 18 = 54$ m
The correct answer is 54 m, not 540 m.
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Error: student likely used 0.1 instead of 0.01
The student's answer is ten times too large. A decimal error: $0.1 \times 3600 = 360$ vs $0.01 \times 3600 = 36$. Always check reasonableness.
Fill the gap: A table has outputs 8, 13, 18, 23 for inputs 0, 1, 2, 3. The starting value is and the rate of change is , so the formula is $y =$ .
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Common errors · the 3 traps that cost marks
Common errors · the 3 traps that cost marks
Common errors · the 3 traps that cost marks
Trap 01
Using every number before identifying the goal
A question may give many numbers, not all are needed. Read the question, identify what is required, then select the appropriate numbers and strategy. Don't calculate before you know what you're calculating.
Trap 02
Substituting before rearranging
If the required variable is not the subject (e.g. finding $s$ from $d = st$), rearrange first. Substituting into the original formula and then manipulating the result usually leads to errors.
Trap 03
Not testing a formula built from a table
After writing a formula from a table, always substitute a known input to verify it gives the correct output. This one step can catch errors before they cost marks.
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Revisit the decision process
Quick-fire practice · 4 calculations
Quick-fire practice · 4 calculations
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A gym charges $\$20$ plus $\$15$ per class. Find the number of classes if the total is $110.
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Use $A = s^2$ to find the area of a square with side length 9.5 m.
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Rearrange $A = bh$ to find $h$ when $A = 72$ cm² and $b = 9$ cm.
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Write a formula for outputs 8, 13, 18, 23 for inputs 0, 1, 2, 3.
Odd one out: Three of these statements about the strategy guide are correct. Which one is wrong?
In your own words: Describe the correct first move when you are given a formula and need to find the value of a variable that is not the subject.
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Revisit the decision process
A reliable first move is: identify the required value, define the variable if needed, choose the strategy, calculate, then interpret the answer with units.
Earlier you wrote a decision process before seeing the lesson. Compare it to what you now know.
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Show what you have learned
Multiple choice, then short answer under exam conditions.
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Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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Short answer
Choose a strategy, show working and interpret the answer.
ApplyBand 44 marks
Q1. A hire company charges $\$35$ plus $\$18$ per hour. The total cost is $143. Find the hire time and explain your strategy. (4 marks)
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ApplyBand 33 marks
Q2. Use $d = st$ to find time when $d = 210$ km and $s = 70$ km/h. Rearrange before substituting. (3 marks)
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AnalyseBand 44 marks
Q3. A table has outputs 14, 20, 26, 32 for inputs 0, 1, 2, 3. Write a formula and test it using input 3. (4 marks)
Q1 (4 marks): Strategy: write and solve an equation [1]. Let $h$ = hire time in hours. $35 + 18h = 143$ [1]. $18h = 108$, $h = 6$ [1]. The hire time is 6 hours [1].
