g10-07-01 · Core topics
What does one radian measure?
Does doubling a circle radius double its angle or its arc?
1 MIN · PREDICT ON PAPER
For a 90° turn, predict the arc lengths when r=1 and r=2.
Discuss or write first. No typed answer is required to continue.
The radian ratio is independent of radius.
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
A radian angle is the ratio s/r. A full turn has angle 2π, independently of radius; arc length is s=rθ when θ is in radians.
Conditions: Use r>0 and the magnitude of a turn for arc length; a zero turn has zero arc.
Mathematical justification: A full circumference 2πr gives s/r=2π. Proportional arcs give θ=πd/180 and s=rθ.
1 MIN · EXIT CHECK
Find the arc of a 60° sector with radius 3.
Check after answering
θ=π/3 and s=π.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: Which quotient remains unchanged when both radius and arc length double?
Watch for: Substituting 90 into s=rθ while θ is still measured in degrees.
Challenge: Explain why the definition s/r cannot use r=0.
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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