🔐 Teacher Only

ThunderPeak Roller Coaster

Interactive activity covering displacement-time graphs, gradients of curves, and tangent approximations in a real-world context. Approx 30 minutes. Students work individually or in pairs.

Phase 1: Analysis (15 min) Phase 2: Build (8 min) Phase 3: Launch (7 min)
Timing Lesson Plan
TimeActivityYour role
0–3 minIntroduction slide
Set the scene — students are junior engineers at ThunderPeak Rides. Explain the three phases.		Show intro slide, share student file link in Teams chat
3–18 minPhase 1: Analysis
Students read the height-distance graph, click to draw tangents at 4 marked points, enter gradient estimates.		Circulate. Key question: "What does a negative gradient tell us about the coaster here?"
18–22 minBrief whole-class check
Show the correct gradients. Discuss what each gradient means in context (speed of height change).		Use the slides to reveal answers. Ask: "Which point has gradient ≈ 0 and why?"
22–28 minPhase 2 & 3: Build and Launch
Students drag control points to reshape their track, then launch the cart and watch the physics play out.		Let them enjoy this. It's motivational payoff. Ask early finishers: "What track shape gives maximum flight distance?"
28–30 minSubmission
Students copy their submission box and paste into Google Classroom.		Collect via Classroom assignment
Mark Scheme Tangent Gradient Questions

The graph shows height (m) on the y-axis vs horizontal distance (m) on the x-axis. Gradient = change in height ÷ change in horizontal distance. Negative gradient = descending. Accept any answer within ±0.12 for full marks, within ±0.25 for partial.

Q1

Gradient at x = 120m — Top of first big descent

True gradient
≈ −0.93
The track drops steeply at this point — about 0.93m of height lost per metre of horizontal travel.
Full marks: −1.05 to −0.81  |  Partial (1pt): −1.18 to −0.68
Context answer
The coaster is descending steeply — height is decreasing rapidly relative to horizontal distance. The magnitude tells us how steep the drop feels to riders.
⚠ Common error: Students draw the tangent as horizontal (gradient = 0) because the curve looks nearly flat from a distance at this scale. Prompt them to look carefully at the slope direction.
Q2

Gradient at x = 300m — Bottom of first valley

True gradient
≈ 0
At the minimum point of the curve, the tangent is horizontal.
Full marks: −0.12 to +0.12  |  Partial: −0.25 to +0.25
Key teaching point
This is the most important conceptual question. At a minimum (or maximum), the gradient is zero because the tangent is horizontal. Students who understand this have grasped the link between stationary points and zero gradient.
⚠ Many students will say gradient = 0 but not be able to explain why. Push for: "The tangent at the bottom of a valley is flat, so rise = 0, therefore gradient = rise/run = 0."
Q3

Gradient at x = 440m — Near top of second hill

True gradient
≈ −0.02 (approximately 0)
Near a maximum, the gradient approaches zero. The track is almost flat at the crest.
Full marks: −0.14 to +0.10  |  Partial: −0.27 to +0.23
Context answer
Near the peak of a hill, the track briefly levels out. This is where riders feel momentarily weightless. The gradient is close to 0 because we are near a local maximum.
Q4

Gradient at x = 560m — Mid-descent on second drop

True gradient
≈ −0.88
Steeply descending again, similar magnitude to Q1.
Full marks: −1.00 to −0.76  |  Partial: −1.13 to −0.63
Context answer
The track is descending almost as steeply as the first drop. Riders are accelerating downward rapidly. Compare this to Q2 — the contrast between gradient 0 and gradient −0.88 demonstrates how sharply the slope changes over the ride.
Discussion Suggested class questions
After Q2 "Why is the gradient zero at the very bottom of the valley? What does this mean for the rider?"
After Q3 "Compare the gradient at the top of the hill to the gradient halfway down. What's happening to the speed of height change?"
After all 4 "If gradient represents rate of change of height — what would gradient represent if this were a speed-time graph instead?"
Build phase "What track shape gives the longest flight? Can you design the track so the cart flies rather than slides to a stop?"
Watch for Common misconceptions
⚠️
Confusing gradient with height — Students may say "the gradient is 80 because the height is 80m." Remind them gradient = rise ÷ run, not just the y-value.
⚠️
Forgetting the sign — A descending track has a negative gradient. Students who give positive answers for descending sections need to reconsider their rise direction.
⚠️
Drawing the tangent too short — A tangent that only spans 10–20m of horizontal distance will give an unreliable gradient estimate. Encourage students to extend their tangent line across a wider range.
⚠️
Gradient ≠ speed of cart — The gradient is the rate of height change with horizontal distance, not the speed of the cart. These are related but distinct. The cart's speed depends on the component of velocity along the track.
💡
Positive teaching moment at Q2 and Q3 — If students get ≈ 0 for the minimum and maximum, use this to preview stationary points and the second derivative test. "How could we tell whether this zero gradient is a peak or a valley?"
Scoring How points work
3 points — Within ±0.12 of true gradient. Full credit.
1 point — Within ±0.25 of true gradient. Partial credit — reasonable tangent drawn but reading slightly off.
0 points — Outside ±0.25. Worked answer and hint shown to student automatically.
Bonus: Flight distance — Students earn up to 50 bonus points (10 points per 50m flown) from the launch phase. This incentivises engagement with the build phase.