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.