Mathematics Standard • Year 11 • Module 1 • Lesson 1
Algebraic Language, Variables and Substitution
Practise HSC-style writing on substitution into formulas — three multi-mark short answers and one extended response with marking criteria.
1. Short-answer questions
1.1 The cost of hiring a small van is given by C = 75 + 0.85k, where C is the cost in dollars and k is the number of kilometres travelled. Use the formula to find the cost for an 84 km trip and explain in one sentence what each number in the formula represents. 3 marks Band 3
1.2 The volume of a rectangular tank is given by V = lwh. A backyard water tank has l = 1.8 m, w = 1.2 m and h = 1.5 m. Calculate the volume in cubic metres, then convert to litres (1 m³ = 1000 L). 3 marks Band 3-4
1.3 Evaluate the expression x² − 5x + 2 when x = −3.
(a) Show the substitution with brackets.
(b) Calculate the value step-by-step.
(c) State your final answer. 4 marks Band 4
2. Extended response
2.1 Two electricians charge for callouts using the formulas below.
Electrician A: C = 90 + 65h, where C is the cost in dollars and h is the number of hours on site.
Electrician B: C = 45 + 80h, with the same meanings for C and h.
(a) Calculate the cost charged by each electrician for a 2-hour job.
(b) Calculate the cost charged by each electrician for a 5-hour job.
(c) Explain in one sentence what the constant ($90 vs $45) and the coefficient ($65/h vs $80/h) represent for each electrician.
(d) For a job of length h hours, write down the inequality that says "Electrician A costs less than Electrician B" and solve it to find the cut-off number of hours. State, in a conclusion sentence, which electrician is cheaper for short jobs and which is cheaper for long jobs. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — both 2-hour costs calculated correctly: A = $220, B = $205.
Part (b) — 1 mark
• 1 mark — both 5-hour costs calculated correctly: A = $415, B = $445.
Part (c) — 2 marks
• 1 mark — identifies the constant as a fixed callout/booking fee charged regardless of time.
• 1 mark — identifies the coefficient as the hourly rate (cost per hour on site).
Part (d) — 3 marks
• 1 mark — writes the inequality 90 + 65h < 45 + 80h (or equivalent).
• 1 mark — correctly solves to find h > 3 (cut-off at exactly 3 hours).
• 1 mark — clear conclusion sentence: Electrician B is cheaper for jobs under 3 hours; Electrician A is cheaper for jobs longer than 3 hours.
Your response:
Stuck on (d)? Set the two cost formulas equal first to find the cross-over: 90 + 65h = 45 + 80h gives h = 3. Then decide which side is cheaper before and after that cut-off.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Van hire cost (3 marks)
Sample response.
C = 75 + 0.85(84) = 75 + 71.40 = $146.40 for the 84 km trip.
The 75 is the fixed hire fee (charged regardless of distance). The 0.85 is the cost per kilometre.
Marking notes. 1 mark — correct substitution shown. 1 mark — correct arithmetic with $146.40. 1 mark — both numbers' real-life meanings stated. A response giving only the dollar answer scores 2/3.
1.2 — Tank volume (3 marks)
Sample response.
V = lwh = 1.8 × 1.2 × 1.5 = 3.24 m³.
In litres: 3.24 × 1000 = 3240 L.
Marking notes. 1 mark — substitutes all three measurements correctly. 1 mark — correct V = 3.24 m³. 1 mark — correct unit conversion to 3240 L. A response that forgets to convert to litres scores 2/3.
1.3 — Substituting a negative value (4 marks)
(a) Sample response. x² − 5x + 2 with x = −3 becomes (−3)² − 5(−3) + 2.
(b) Sample response. = 9 − (−15) + 2 = 9 + 15 + 2.
(c) Sample response. = 26.
Marking notes. 1 mark — brackets used in the substitution. 1 mark — (−3)² = +9 (not −9). 1 mark — −5(−3) = +15 (sign handled). 1 mark — correct final answer 26 with arithmetic shown. A bare "26" without working scores 1/4.
2.1 — Electricians A vs B (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) 2-hour job.
A: C = 90 + 65(2) = 90 + 130 = $220. B: C = 45 + 80(2) = 45 + 160 = $205. [1 mark — both 2-hour costs.]
(b) 5-hour job.
A: C = 90 + 65(5) = 90 + 325 = $415. B: C = 45 + 80(5) = 45 + 400 = $445. [1 mark — both 5-hour costs.]
(c) Meaning of the constants and coefficients.
The constant ($90 for A, $45 for B) is a fixed callout fee charged for showing up regardless of time. [1 mark.]
The coefficient ($65/h for A, $80/h for B) is the hourly labour rate — the amount added per hour on site. [1 mark.]
(d) Cross-over inequality.
A cheaper than B: 90 + 65h < 45 + 80h. [1 mark — correct inequality.]
Subtract 45 and 65h from both sides: 45 < 15h. Divide by 15: 3 < h, i.e. h > 3. [1 mark — correct cut-off h = 3.]
Conclusion: Electrician B is cheaper for jobs shorter than 3 hours (lower callout); Electrician A is cheaper for jobs longer than 3 hours (lower hourly rate). At exactly 3 hours both cost the same ($285). [1 mark — clear conclusion with both ranges named.]
Total: 7/7.
Band descriptors for marker.
Band 3: Substitutes correctly for one electrician at one duration; no comparison. ≈ 2-3 marks.
Band 4: All four cost calculations correct in (a) and (b); identifies callout vs hourly but does not attempt the inequality. ≈ 4-5 marks.
Band 5: Sets up and solves the inequality correctly but conclusion is incomplete (e.g. names only the cut-off, not which is cheaper on each side). ≈ 5-6 marks.
Band 6: Complete, correct, conclusion sentence names both ranges (under 3 h vs over 3 h) and which electrician wins. 7/7.