g10-11-02 · Additional mathematics
A fixed product instead of a fixed ratio
What remains fixed in inverse variation?
1 MIN · PREDICT ON PAPER
Predict the new output when x is multiplied by t in y=k/x.
Discuss or write first. No typed answer is required to continue.
The fixed product is 4; y(tx)=y(x)/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 product xy=k is fixed; for nonzero t the new output is y/t. x=0 is excluded. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: x≠0 and t>0; even if k=0, the original expression k/x remains undefined at zero.
Mathematical justification: (tx)·[k/(tx)]=k and k/(tx)=(k/x)/t, under the nonzero-denominator conditions.
1 MIN · EXIT CHECK
y=12/x. Find y at x=3 and x=6.
Check after answering
4 and 2 respectively; both products xy equal 12.
Teacher guidance and extension
Before: Division and domain restrictions
Ask: Ask what survives algebraic simplification when the input is zero.
Watch for: Multiplying x by two does not subtract a fixed amount from y.
Challenge: Compare negative and positive x while retaining the same k.
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
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
Complete backup and restore