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.