g10-11-03 · Additional mathematics
Change one factor or both?
In joint variation, do two doublings still give a doubling?
1 MIN · PREDICT ON PAPER
For y=kxz, compare doubling x alone with doubling both x and z.
Discuss or write first. No typed answer is required to continue.
Doubling one factor gives 2y; doubling both gives 4y, including zero cases.
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
One doubled factor doubles y; two doubled factors multiply y by four. Zero inputs remain legitimate but can make ratios undefined. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: k,x,z are real; y/(xz) is used only when xz≠0.
Mathematical justification: k(sx)(tz)=st(kxz) by associativity and commutativity. This identity does not divide by any potentially zero output.
1 MIN · EXIT CHECK
y=2xz; initially x=3,z=4. What if x doubles and z halves?
Check after answering
y stays 24 because the two multipliers have product one.
Teacher guidance and extension
Before: Products and direct variation
Ask: Ask groups to alter just one factor before altering both.
Watch for: Joint variation requires specifying which variables are held fixed.
Challenge: Let x scale by s and z by t; determine the combined multiplier.
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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