g10-11-04 · Additional mathematics
The fixed part changes the ratio
Does a straight graph always mean direct proportion?
1 MIN · PREDICT ON PAPER
Compare y=a+bx with y=bx while keeping the same rate b.
Discuss or write first. No typed answer is required to continue.
Doubling difference: y(2x)−2y(x)=-3=−a
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 fixed part a gives y(0)=a. Equal x increments give equal output increments, but output ratios are not generally fixed. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: x is nonnegative; a and b are model parameters, not measured empirical facts.
Mathematical justification: y(2x)−2y(x)=−a, so doubling outputs holds for all x iff a=0. Also y(x+h)−y(x)=bh.
1 MIN · EXIT CHECK
A charge is 20+3x. Compare x=2 and x=4.
Check after answering
26 and 32; doubling x does not double the fixed 20.
Teacher guidance and extension
Before: Affine functions and ratios
Ask: Ask students to identify the initial amount before comparing per-unit rates.
Watch for: Linearity with an intercept is not the same as direct proportionality through the origin.
Challenge: From two input-output pairs, identify a and b and test a third pair.
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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