Mathematics Standard • Year 11 • Module 1 • Lesson 10
Gradient as Rate of Change
Apply gradient as a real-world rate — savings per week, average speed, water flow, temperature change and electricity use.
Problem 1 — Savings goal
Mia's savings balance is $120 at week 0 and $390 at week 18.
Set up: What are we solving for?
(i) Calculate the gradient m. Include units. 2 marks
(ii) Interpret m in one plain-English sentence about Mia's saving. 1 mark
(iii) If she keeps the same rate, predict her balance after 30 weeks. 2 marks
Stuck on (iii)? Add (30 − 0) × m to her starting balance.Problem 2 — M1 highway drive
A car records (0.5 h, 30 km) and (2.5 h, 180 km) on a drive between Newcastle and Sydney.
Set up: What are we solving for?
(i) Calculate the average speed (with units). 2 marks
(ii) The M1 speed limit is 110 km/h. State whether the average speed is below the limit and by how much. 2 marks
(iii) Explain in one sentence why an "average" speed of 75 km/h does NOT mean the car was always doing exactly 75 km/h. 2 marks
Stuck? Average means total distance divided by total time, regardless of the speed at any moment in between.Problem 3 — Backyard rainwater tank
A 600 L rainwater tank is draining. Volume readings: 600 L at 0 min, 540 L at 5 min, 480 L at 10 min, 420 L at 15 min.
Set up: What are we solving for?
(i) Use two points (0, 600) and (15, 420) to find m. 2 marks
(ii) State what the sign of m means for the tank. 1 mark
(iii) If the rate stays the same, predict the time (in minutes from start) when the tank empties (V = 0). 2 marks
Stuck on (iii)? Starting volume divided by the drain rate gives the time to empty.Problem 4 — Cooling oven
A kitchen oven cools after being switched off. Temperature is 220 °C at 0 min, 170 °C at 10 min, 120 °C at 20 min.
Set up: What are we solving for?
(i) Find the gradient between (0, 220) and (20, 120). 2 marks
(ii) Interpret the value of m in one plain-English sentence. 1 mark
(iii) Using the constant rate, predict the oven temperature at 30 min. State one reason the real oven might NOT match this prediction. 3 marks
Stuck on (iii)? Apply the per-minute drop for 10 more minutes. Real cooling slows as the oven approaches room temperature.Problem 5 — Household electricity meter
An electricity meter reads 45 230 kWh at the start of the month (day 0) and 45 530 kWh at the end (day 30).
Set up: What are we solving for?
(i) Calculate the average daily usage (m) in kWh/day. 2 marks
(ii) Electricity costs $0.28 per kWh. Calculate the average daily cost. 2 marks
(iii) Estimate the monthly bill from the rate. State whether your estimate is a reasonable approximation of the household's true bill, in one sentence. 2 marks
Stuck? Monthly bill = average daily use × days in month × price per kWh. Or just total kWh used × price.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Savings goal
Set up. Use the two points (0, 120) and (18, 390).
(i) m = (390 − 120) / (18 − 0) = 270 / 18 = $15 per week.
(ii) Mia is saving an average of $15 each week.
(iii) Week 30: 120 + 30(15) = 120 + 450 = $570.
Problem 2 — M1 drive
Set up. Use (0.5, 30) and (2.5, 180).
(i) m = (180 − 30) / (2.5 − 0.5) = 150 / 2 = 75 km/h.
(ii) 75 km/h is below the 110 km/h limit by 35 km/h — well within the legal speed.
(iii) Average is total distance / total time. The car might have travelled at 110 km/h on the open road, slowed for traffic, or stopped for fuel — the moment-to-moment speed varies even though the average works out to 75 km/h.
Problem 3 — Rainwater tank drain
Set up. Use (0, 600) and (15, 420).
(i) m = (420 − 600) / (15 − 0) = −180 / 15 = −12 L/min.
(ii) Negative sign means the volume is decreasing (the tank is draining at 12 L per minute).
(iii) 600 L ÷ 12 L/min = 50 minutes until V = 0 (starting from t = 0).
Problem 4 — Cooling oven
Set up. Use (0, 220) and (20, 120).
(i) m = (120 − 220) / 20 = −100 / 20 = −5 °C/min.
(ii) The oven is cooling by 5 °C per minute over the first 20 minutes.
(iii) Predicted at 30 min: 120 + 10(−5) = 120 − 50 = 70 °C. Real cooling usually slows as the oven gets closer to room temperature, so the actual reading at 30 min would probably be a bit higher than 70 °C.
Problem 5 — Electricity bill
Set up. Difference in meter readings ÷ days, then multiply by price.
(i) m = (45 530 − 45 230) / 30 = 300 / 30 = 10 kWh/day.
(ii) Daily cost = 10 × 0.28 = $2.80/day.
(iii) Estimated monthly bill = 300 kWh × $0.28 = $84.00. This is a reasonable approximation because it uses the actual total kWh used in the billing period; the real bill may differ slightly due to fixed daily supply charges and any time-of-use pricing.