g10-07-05 · Core topics
Unroll the circle into a wave
How does a moving circle point produce a sine graph?
1 MIN · PREDICT ON PAPER
Predict the graph height after a quarter, half and full turn.
Discuss or write first. No typed answer is required to continue.
Green copies the circle y-coordinate; gold copies x.
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
The horizontal graph coordinate records angle; graph height copies the circle y-coordinate. Cosine similarly copies x.
Conditions: The graph axis uses radians even when the control reports degrees; a full turn is2π.
Mathematical justification: For every θ the point(θ,sinθ) is constructed by copying the unit-circle ordinate. Equal points after2π give periodicity.
1 MIN · EXIT CHECK
What graph point corresponds to one full turn?
Check after answering
(2π,0); the angle is not0 although the height is0.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: Which quantity keeps increasing while the circle point returns?
Watch for: Reading the graph horizontal coordinate as the circle x-coordinate.
Challenge: Locate matching sine values with different angle inputs.
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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