Mathematics Standard • Year 11 • Module 1 • Lesson 9
Coordinates, Tables and Linear Patterns
Apply ordered pairs and constant-difference checks to real Australian contexts, taxi fares, casual pay, fuel use and savings.
Problem 1, Sydney taxi fare table
A Sydney taxi charges $6 flagfall plus $3 per km. The fare for the first 4 km is shown.
Distance (km): 0, 1, 2, 3 Fare ($): 6, 9, 12, 15
Set up: What are we solving for?
(i) Write the four ordered pairs in the form (distance, fare). 1 mark
(ii) Show, using the difference in fares, that the table is linear. 2 marks
(iii) Predict the fare for a 7 km trip. 2 marks
Stuck on (iii)? Each extra km adds $3. From the 3 km row at $15, add 4 × $3.Problem 2, Casual pay slip
A retail casual is paid the same amount per hour worked. Her pay slip shows:
Hours: 0, 1, 2, 3, 4 Pay ($): 0, 27, 54, 81, 108
Set up: What are we solving for?
(i) Write the ordered pairs. 2 marks
(ii) Use the constant difference to state her hourly rate. 2 marks
(iii) Predict her pay for a 7.5-hour shift. 2 marks
Stuck? The pay starts at $0 with 0 hours, so the formula is simply pay = rate × hours.Problem 3, Fuel tank emptying
A car starts with a 60 L fuel tank. The driver records the litres remaining every 100 km on a road trip.
Distance (km): 0, 100, 200, 300 Fuel left (L): 60, 51.8, 43.6, 35.4
Set up: What are we solving for?
(i) Find the constant difference per 100 km and explain what it means in plain English. 2 marks
(ii) Predict the fuel remaining after 500 km, assuming the pattern continues. 2 marks
(iii) Explain in one sentence at roughly what distance the tank would empty (if no refuel happens). 2 marks
Stuck on (iii)? The car loses 8.2 L per 100 km. How many 100-km steps from 60 L until it hits 0?Problem 4, Compound-effect savings (not linear)
A savings account is opened with $100. After 1, 2, 3, 4 years the balance is:
Year: 0, 1, 2, 3, 4 Balance ($): 100, 110, 121, 133.1, 146.41
Set up: What are we solving for?
(i) Calculate the year-on-year differences. 2 marks
(ii) Decide whether the table is linear and justify your answer. 2 marks
(iii) Explain in one sentence why predicting year-10 balance by adding "$10 per year" is risky for this table. 2 marks
Stuck? Revisit lesson § Increasing Does Not Always Mean Linear.Problem 5, Comparing two phone plans
Plan A: $30 base + $5 per GB. Plan B: $0 base + $12 per GB. The monthly cost for 0, 1, 2, 3 GB is:
GB used: 0, 1, 2, 3 Plan A ($): 30, 35, 40, 45 Plan B ($): 0, 12, 24, 36
Set up: What are we solving for?
(i) Show that both Plan A and Plan B are linear (state the constant difference for each). 2 marks
(ii) Extend each table to 5 GB and 6 GB. 2 marks
(iii) At which GB usage does Plan B first exceed Plan A? Use your extended table to identify the cross-over GB. 2 marks
Stuck? Just keep adding +$5 to Plan A and +$12 to Plan B for each extra GB, then compare row by row.How did this worksheet feel?
What I'll revisit before next class:
Problem 1, Taxi fares
Set up. Pair input (distance) with output (fare), then check constant differences.
(i) (0, 6), (1, 9), (2, 12), (3, 15).
(ii) Fare differences are +3, +3, +3, equal, so the table is linear.
(iii) From 3 km ($15), add 4 × $3 = $12. 7 km fare = $27. (Or use $6 + $3 × 7 = $27.)
Problem 2, Casual pay
Set up. Each extra hour adds the hourly rate.
(i) (0, 0), (1, 27), (2, 54), (3, 81), (4, 108).
(ii) Constant difference = $27 per hour, so the rate is $27/h.
(iii) Pay(7.5) = 27 × 7.5 = $202.50.
Problem 3, Fuel emptying
Set up. Each 100 km uses the same number of litres if the rate is constant.
(i) Differences: 60 − 51.8 = 8.2 L, 51.8 − 43.6 = 8.2 L, 43.6 − 35.4 = 8.2 L. The car uses 8.2 L per 100 km (the published fuel economy).
(ii) 500 km is 2 more 100-km steps from 300 km (35.4 L). Fuel left = 35.4 − 2(8.2) = 35.4 − 16.4 = 19.0 L.
(iii) Total fuel ÷ rate per 100 km = 60 / 8.2 ≈ 7.32 (in 100-km units), so the tank empties at about 732 km.
Problem 4, Compound savings (not linear)
Set up. List year-on-year differences and check whether they are equal.
(i) +10, +11, +12.10, +13.31.
(ii) Differences are not equal, so the table is not linear. (It is compound growth at 10 % per year.)
(iii) Adding $10 each year would under-estimate later balances because each year's interest is calculated on a larger amount, the differences themselves grow.
Problem 5, Phone plans
Set up. Add the per-GB cost to each plan, then compare totals at each GB level.
(i) Plan A: +$5 each step (linear, rate $5/GB). Plan B: +$12 each step (linear, rate $12/GB).
(ii) Plan A at 5 GB = $55, at 6 GB = $60. Plan B at 5 GB = $60, at 6 GB = $72.
(iii) At 5 GB, Plan A = $55 and Plan B = $60. Plan B first exceeds Plan A at 5 GB. (At 4 GB, Plan A = $50 and Plan B = $48, so Plan B was still cheaper.)