Mathematics Standard • Year 11 • Module 1 • Lesson 5
Building Formulas from Patterns and Tables
Practise HSC-style writing on building formulas — three multi-mark short answers and one structured extended response with marking criteria.
1. Short-answer questions
1.1 A printing company charges $25 plus $2 per page. Write a formula for total cost C in terms of p pages, then find the cost of 18 pages. State your final answer in a sentence. 3 marks Band 3
1.2 A number pattern produces the values 9, 14, 19, 24 for term numbers n = 1, 2, 3, 4. Write a formula for the nth term, then verify it works for n = 4. 3 marks Band 3-4
1.3 A table shows outputs 5, 10, 20, 40 for inputs 1, 2, 3, 4.
(a) Calculate the change between each pair of rows.
(b) Explain in one sentence why a simple linear formula of the form y = a + bx cannot model this table.
(c) Describe in one sentence what kind of rule does fit. 4 marks Band 4
2. Extended response
2.1 A community council uses two cleaning contractors for park rubbish collection.
Contractor A: Charges follow this table.
Visits (v): 0, 1, 2, 3, 4
Cost ($C): 120, 165, 210, 255, 300
Contractor B: Charges follow this table.
Visits (v): 0, 1, 2, 3, 4
Cost ($C): 60, 120, 180, 240, 300
(a) For each contractor, identify the starting value and the rate per visit, and write a cost formula in v.
(b) Verify each formula is correct by testing it on a different row of its own table.
(c) Use both formulas to find the number of visits at which the two contractors cost the same.
(d) Write a conclusion sentence recommending which contractor the council should choose if the park needs about 8 visits per quarter. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark — Contractor A: starting $120, rate +$45/visit → C = 120 + 45v.
• 1 mark — Contractor B: starting $60, rate +$60/visit → C = 60 + 60v.
Part (b) — 1 mark
• 1 mark — both formulas tested at a non-trivial row (e.g. v = 3) and shown to match the table.
Part (c) — 2 marks
• 1 mark — sets up 120 + 45v = 60 + 60v.
• 1 mark — solves correctly to v = 4 (the value already visible at the bottom of each table — both give $300 at v = 4).
Part (d) — 2 marks
• 1 mark — calculates the cost for 8 visits with both contractors (A: $480, B: $540).
• 1 mark — clear recommendation sentence: choose Contractor A because they are $60 cheaper at 8 visits (and remain cheaper for any number of visits beyond 4).
Your response:
Stuck on (c)? Set 120 + 45v = 60 + 60v and collect like terms. The answer (v = 4) is already visible in both tables — both cost $300 at v = 4.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Printing cost (3 marks)
Sample response.
Let C = total cost in dollars and p = number of pages. C = 25 + 2p.
For 18 pages: C = 25 + 2(18) = 25 + 36 = $61. The cost of 18 pages is $61.
Marking notes. 1 mark — correct formula with both variables defined. 1 mark — correct arithmetic to $61. 1 mark — interpretation sentence. A bare "$61" without context loses the sentence mark.
1.2 — Pattern 9, 14, 19, 24 (3 marks)
Sample response.
Change = +5 per term. Starting value (at n = 0) = 9 − 5 = 4. T = 4 + 5n.
Test n = 4: T = 4 + 5(4) = 4 + 20 = 24 ✓ matches the table.
Marking notes. 1 mark — identifies +5 constant change. 1 mark — correct formula T = 4 + 5n (with working-back to find the starting value). 1 mark — verification step shown. Common error: writing T = 5n without the +4 gives T(4) = 20, not 24.
1.3 — Doubling table (4 marks)
(a) Sample response. Changes between rows: 5 → 10 (+5), 10 → 20 (+10), 20 → 40 (+20).
(b) Sample response. A linear formula y = a + bx requires the change per input step to be constant. Here the change doubles each step (+5, +10, +20), so no single rate b can fit all rows.
(c) Sample response. The output doubles for each unit increase in the input, so the pattern is exponential, e.g. y = 5 × 2^(x−1).
Marking notes. 1 mark — three correct change values. 1 mark — explains "change is not constant" in (b). 1 mark — identifies "doubling" in (c). 1 mark — uses correct terminology (exponential/non-linear).
2.1 — Two contractors (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Cost formulas.
Contractor A: starting cost $120 (at v = 0), increase $45/visit (165 − 120 = 45). C = 120 + 45v. [1 mark.]
Contractor B: starting cost $60 (at v = 0), increase $60/visit (120 − 60 = 60). C = 60 + 60v. [1 mark.]
(b) Verify on a non-trivial row.
Contractor A test, v = 3: C = 120 + 45(3) = 120 + 135 = $255 ✓.
Contractor B test, v = 3: C = 60 + 60(3) = 60 + 180 = $240 ✓. Both match the table. [1 mark — both formulas verified at a row other than v = 0.]
(c) Break-even.
Set costs equal: 120 + 45v = 60 + 60v. [1 mark.]
Subtract 45v and 60: 60 = 15v ⇒ v = 4 visits. (Confirmed by both tables — both give $300 at v = 4.) [1 mark.]
(d) Recommendation for 8 visits.
A at v = 8: C = 120 + 45(8) = 120 + 360 = $480. B at v = 8: C = 60 + 60(8) = 60 + 480 = $540. [1 mark — both numerical values.]
Conclusion: the council should choose Contractor A — they are $60 cheaper at 8 visits per quarter, and Contractor A remains cheaper for any number of visits above 4. Contractor B is only cheaper for 0-3 visits per quarter. [1 mark — clear, context-aware recommendation.]
Total: 7/7.
Band descriptors for marker.
Band 3: Writes one formula correctly; no verification or break-even. ≈ 2-3 marks.
Band 4: Both formulas correct and verified, but no break-even calculation or no recommendation. ≈ 4-5 marks.
Band 5: Break-even v = 4 found; recommendation only states which is cheaper at 8 visits without addressing the wider pattern. ≈ 5-6 marks.
Band 6: Complete, both formulas with verification, break-even = 4 found, AND a recommendation sentence noting Contractor A wins above 4 visits (with the 8-visit dollar comparison). 7/7.