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.