g10-07-12 · Core topics
A wheel needs a starting position
Can amplitude and period alone determine a height model?
1 MIN · PREDICT ON PAPER
A wheel has radius5, centre height6 and period40s. Predict heights at0,10 and30s when starting on the right and moving upwards.
Discuss or write first. No typed answer is required to continue.
Check starting position, direction and units before trusting the model.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
With an upward start at centre height, h(t)=H+Rsin(2πt/T). Changing the starting phase changes the height at the same time.
Conditions: Use T>0,R>0 and consistent time units. A physical wheel above ground additionally needs H≥R.
Mathematical justification: Uniform rotation gives angle2πt/T+φ; the circle y-coordinate is Rsin of that angle, then add centre height H.
1 MIN · EXIT CHECK
For the stated wheel, find h(10) andh(30).
Check after answering
11 and1 metres.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: What must the formula predict at t=0 before using it later?
Watch for: Using cosine automatically without checking the starting point.
Challenge: Modify the model for a start at the lowest point.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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