Mathematics Standard • Year 11 • Module 1 • Lesson 5
Building Formulas from Patterns and Tables
Apply the starting-value-plus-rate model to real Australian scenarios — gym packs, parking, water tanks, salary increases and matchstick patterns.
Problem 1 — Gym session pack
A gym offers a "pay-as-you-go" pack with the following pricing table.
Sessions (s): 0, 1, 2, 3, 4, 5
Cost ($C): 25, 38, 51, 64, 77, 90
Set up: What are we solving for?
(i) Identify the starting value and the repeated change per session. 1 mark
(ii) Write the formula for C in terms of s. Define both variables. 1 mark
(iii) Test the formula at s = 5 using the table, then use it to find the cost of 12 sessions. 2 marks
Stuck? The starting value is the cost at s = 0 (read directly off the table); the rate is the constant +$ change per row.Problem 2 — Sydney parking meter
A multi-storey car park has the following weekday rates:
Hours (h): 1, 2, 3, 4, 5
Cost ($C): 8.50, 14.00, 19.50, 25.00, 30.50
Set up: What are we solving for?
(i) Identify the repeated change per hour and the starting value (the entry fee for 0 hours, found by working backwards from h = 1). 2 marks
(ii) Write the cost formula C in terms of h. 1 mark
(iii) Use your formula to find the cost of 8 hours of parking, then explain in one sentence what the starting value represents in real life. 2 marks
Stuck? Change per hour: 14.00 − 8.50 = 5.50. Working back from (1, 8.50): C(0) = 8.50 − 5.50 = 3.00.Problem 3 — Draining rainwater tank
A rainwater tank starts with 800 L and is drained for garden watering at a constant rate. The volume is recorded each minute:
Time t (min): 0, 1, 2, 3, 4, 5
Volume V (L): 800, 785, 770, 755, 740, 725
Set up: What are we solving for?
(i) State the starting volume and the rate of change per minute (with sign). 1 mark
(ii) Write the formula for V in terms of t. 1 mark
(iii) Use your formula to find: (a) the volume after 30 minutes; (b) how long until the tank is empty (V = 0). 3 marks
Stuck? The rate is −15 L/min (the volume DECREASES). For (iiib), solve 800 − 15t = 0.Problem 4 — Graduate salary steps
A graduate teacher's salary by year of service is given by the table:
Year (y): 1, 2, 3, 4, 5
Salary ($S): 76 000, 80 200, 84 400, 88 600, 92 800
Set up: What are we solving for?
(i) Calculate the annual salary increase and find S at y = 0 (the "starting" base, by working back from y = 1). 2 marks
(ii) Write the salary formula S in terms of y. 1 mark
(iii) Predict the salary in year 10. State, in one sentence, why this is a model and may not match reality (think about salary scale ceilings). 2 marks
Stuck? Increase = 80 200 − 76 000 = $4,200/year. S(0) = 76 000 − 4 200 = $71 800.Problem 5 — Matchstick squares (pattern)
The diagram below shows a row of squares made from matchsticks. Each new square shares one side with the previous square.
Number of squares (n): 1, 2, 3, 4, 5
Matchsticks (M): 4, 7, 10, 13, 16
Set up: What are we solving for?
(i) Identify the change in matchsticks per extra square, and the starting value at n = 0 (by working backwards). 2 marks
(ii) Write the formula M in terms of n, and test it for n = 5. 2 marks
(iii) How many matchsticks are needed for 100 squares? Use your formula. 1 mark
(iv) Explain in one sentence why M(0) = 1 looks strange (one matchstick before any square exists) yet still gives the correct formula. 1 mark
Stuck? Each new square adds 3 sticks (since one side is shared). The "+1" is the leftmost vertical stick that you start with before any squares are completed.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Gym session pack
Set up. Read starting value and rate from the table, write a linear formula, then extend.
(i) Starting value = $25 (cost at s = 0). Change = +$13 per session.
(ii) Let C = cost in dollars and s = number of sessions. C = 25 + 13s.
(iii) Test s = 5: C = 25 + 13(5) = 25 + 65 = 90 ✓. For 12 sessions: C = 25 + 13(12) = 25 + 156 = $181.
Problem 2 — Parking meter
Set up. Find the per-hour rate, work back to the entry fee, then extend.
(i) Change per hour = $14.00 − $8.50 = +$5.50/h. Working back from (1, $8.50): C(0) = 8.50 − 5.50 = $3.00.
(ii) C = 3.00 + 5.50h.
(iii) 8 hours: C = 3.00 + 5.50(8) = 3.00 + 44.00 = $47.00. The $3.00 starting value represents the flat entry fee charged just for driving into the car park before any time is added.
Problem 3 — Draining rainwater tank
Set up. Identify a negative rate of change, write a linear formula, then use it for prediction and to find when V = 0.
(i) Starting volume = 800 L. Rate = −15 L/min (volume decreases).
(ii) V = 800 − 15t.
(iii) (a) V(30) = 800 − 15(30) = 800 − 450 = 350 L.
(b) 800 − 15t = 0 ⇒ 15t = 800 ⇒ t = 800/15 ≈ 53.3 min (or 53 min 20 s) until empty.
Problem 4 — Graduate teacher salary
Set up. Identify the constant annual rise, work back to year 0, write the formula, then predict.
(i) Increase = $80,200 − $76,000 = $4,200/year. S(0) = 76,000 − 4,200 = $71,800.
(ii) S = 71,800 + 4,200y.
(iii) Year 10: S = 71,800 + 4,200(10) = 71,800 + 42,000 = $113,800. This is a model — most teaching pay scales hit a maximum step (e.g. around year 8-9), so the linear formula will overpredict beyond that ceiling.
Problem 5 — Matchstick squares
Set up. Identify a constant +3 per extra square, write a linear formula, extend it.
(i) Change = +3 matchsticks per new square (since one side is shared). M(0) = 4 − 3 = 1.
(ii) M = 1 + 3n. Test n = 5: M = 1 + 15 = 16 ✓.
(iii) n = 100: M = 1 + 3(100) = 301 matchsticks.
(iv) The "1" represents the single leftmost vertical matchstick that always sits on the left of the first square — it is the part of the pattern that exists before any complete square has been formed. The formula stays correct because each new square then adds exactly 3 sticks to that starting stick.