g10-10-06 · Syllabus supplement
Double a sphere: four or eight times?
Do surface area and volume scale at the same rate?
1 MIN · PREDICT ON PAPER
Double a sphere radius. Predict its area ratio and volume ratio.
Discuss or write first. No typed answer is required to continue.
Area uses the square factor; volume uses the cube factor.
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 sphere has surface area4πr² and volume4πr³/3. Scaling radius by k changes area by k² and volume by k³.
Conditions: r>0,k>0 for spheres; k=0 collapses to a point, not a sphere. Area and volume use different units.
Mathematical justification: Substitute kr into the formulas and divide by the original positive area or volume; constants cancel and leave k²,k³.
1 MIN · EXIT CHECK
A sphere radius triples. State the area and volume factors.
Check after answering
Area9,volume27.
Teacher guidance and extension
Before: Plane area, Pythagoras and units.
Ask: Which power of radius belongs to each physical dimension?
Watch for: Doubling area or volume merely because radius doubles.
Challenge: If surface area grows by9, find the volume factor.
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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