Mathematics Standard • Year 11 • Module 1 • Lesson 10
Gradient as Rate of Change
Practise HSC-style writing on gradient and rates — three multi-mark short answers and one extended response with marking criteria.
1. Short-answer questions
1.1 A tank volume increases from 15 L to 75 L over 4 minutes. Find the gradient and interpret it in one sentence including units. 3 marks Band 3
1.2 A car travels from 20 km at 0.25 h to 140 km at 1.75 h. Calculate the average speed in km/h. Show the gradient formula in your working. 3 marks Band 3-4
1.3 A water tank's volume changes from 200 L to 80 L in 20 minutes.
(a) Calculate the gradient and include units.
(b) Explain what the sign of the gradient tells you about the tank.
(c) State, in one sentence, what a gradient of −6 L/min would mean in the same context. 4 marks Band 4
2. Extended response
2.1 A driving instructor compares two trips a learner driver took along the same 240 km coastal route between Sydney and the south coast.
Trip 1 (Saturday morning): distance vs time recorded at (0.5 h, 30 km) and (3.5 h, 240 km).
Trip 2 (Sunday afternoon): distance vs time recorded at (0.5 h, 25 km) and (3.0 h, 240 km).
(a) Calculate the gradient (average speed in km/h) for Trip 1 using the two recorded points.
(b) Calculate the gradient (average speed in km/h) for Trip 2 using the two recorded points.
(c) State which trip had the higher average speed and by how much (with units).
(d) The 240 km route has a posted maximum speed of 100 km/h on the open road plus 50 km/h in towns. The instructor says "Trip 2's higher average doesn't necessarily mean the learner broke the limit." Explain in 2-3 sentences why an average speed below 100 km/h is consistent with safe driving even on a faster trip, and what additional information the instructor would need to be sure no limit was exceeded. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — Trip 1: m = (240 − 30)/(3.5 − 0.5) = 210/3 = 70 km/h.
Part (b) — 1 mark
• 1 mark — Trip 2: m = (240 − 25)/(3.0 − 0.5) = 215/2.5 = 86 km/h.
Part (c) — 2 marks
• 1 mark — names Trip 2 as faster average.
• 1 mark — quotes the difference: 86 − 70 = 16 km/h faster with correct units.
Part (d) — 3 marks
• 1 mark — explains that average speed hides the moment-to-moment variation.
• 1 mark — links 86 km/h average to "could be higher at some moments, lower at others".
• 1 mark — names the additional info needed (e.g. GPS log, dashcam, or speed at each moment in time) to check whether any single instant exceeded 100 km/h.
Your response:
Stuck on (d)? Think about how average speed can hide brief bursts above the limit — what data would tell you the actual speed at each second?How did this worksheet feel?
What I'll revisit before next class:
1.1 — Tank filling rate (3 marks)
Sample response.
m = (75 − 15) / 4 = 60 / 4 = 15 L/min.
Interpretation: the tank is filling at 15 litres per minute.
Marking notes. 1 mark — correct rise/run setup. 1 mark — correct m = 15. 1 mark — units (L/min) and interpretation sentence.
1.2 — Average speed (3 marks)
Sample response.
m = (y₂ − y₁) / (x₂ − x₁) = (140 − 20) / (1.75 − 0.25) = 120 / 1.5 = 80 km/h.
Marking notes. 1 mark — formula or rise/run shown. 1 mark — correct substitution. 1 mark — final 80 km/h with units.
1.3 — Water tank decreasing (4 marks)
(a) Sample response. m = (80 − 200) / 20 = −120 / 20 = −6 L/min.
(b) Sample response. The negative sign means the volume is decreasing — the tank is draining (or being used).
(c) Sample response. A gradient of −6 L/min means the tank is losing 6 litres every minute.
Marking notes. 1 mark — correct gradient with sign. 1 mark — units L/min. 1 mark — sign interpretation. 1 mark — −6 L/min plain-English meaning.
2.1 — Two driving trips (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Trip 1 average speed.
m = (240 − 30) / (3.5 − 0.5) = 210 / 3 = 70 km/h. [1 mark.]
(b) Trip 2 average speed.
m = (240 − 25) / (3.0 − 0.5) = 215 / 2.5 = 86 km/h. [1 mark.]
(c) Comparison.
Trip 2 had the higher average speed. [1 mark — Trip 2 named.]
Difference = 86 − 70 = 16 km/h faster on Trip 2. [1 mark — quoted with units.]
(d) Speed-limit discussion.
An average speed is just total distance ÷ total time, so it smooths out every moment of the trip. [1 mark.] Even with a Trip 2 average of 86 km/h (below the 100 km/h limit), the learner could have driven at 100 km/h for long open stretches and then slowed for towns or traffic — the average alone does not prove the speed limit was always respected. [1 mark.] To be sure no limit was broken, the instructor would need moment-to-moment data such as a GPS log, dash-cam footage, or the car's speedometer history showing the actual speed at each second. [1 mark.]
Total: 7/7.
Band descriptors for marker.
Band 3: Calculates one trip's average correctly; little or no comparison or interpretation. ≈ 2-3 marks.
Band 4: Both averages correct; difference stated without explanation about averages hiding moments. ≈ 4-5 marks.
Band 5: Calculations correct; (d) explains the "average hides peaks" idea but doesn't name what extra data is needed. ≈ 5-6 marks.
Band 6: Complete, correct, with explicit recommendation of GPS / dash-cam / per-second data to verify legality. 7/7.