g10-11-01 · Additional mathematics
Which power controls the multiplier?
Does doubling x always double y?
1 MIN · PREDICT ON PAPER
For y=kxⁿ, predict y(2x)/y(x) for n=1,2,3.
Discuss or write first. No typed answer is required to continue.
Output multiplier: 2; identity multiplier 2ⁿ=2.
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 fixed invariant is y/xⁿ=k. Doubling x multiplies y by 2ⁿ when a ratio is defined. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: Here x>0 and n∈{1,2,3}. If k=0, both outputs vanish and their ratio is undefined.
Mathematical justification: k(2x)ⁿ=2ⁿkxⁿ by exponent laws. Dividing requires y(x)≠0; the identity itself also holds at k=0.
1 MIN · EXIT CHECK
y=3x². If x is tripled, what happens to y?
Check after answering
y becomes nine times as large.
Teacher guidance and extension
Before: Powers and ratios
Ask: Distinguish an output identity from a ratio that may be undefined.
Watch for: “Directly proportional to x²” means kx², not (kx)².
Challenge: Use a noninteger positive scale t and derive the multiplier tⁿ.
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.
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