g10-05-14 · Core topics
Cheapest at this quantity or most within budget?
Can the best plan change when the decision question changes?
1 MIN · PREDICT ON PAPER
At n=10, 30 and 50, predict the cheaper plan; then use budget 160 to predict the greatest affordable count.
Discuss or write first. No typed answer is required to continue.
Integer quantities only. Budget 200: P permits 60 booklets; Q permits 70. P = 50; Q = 50; P − Q = 0
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
P(n)=20+3n. Q(n)=5n for n≤20 and 2n+60 for n>20. Ties occur at integer counts 10 and 40. With budget 160, P allows 46 booklets and Q allows 50.
Conditions: Printing counts are integers 1–80. Rental time uses 0<t≤40 and different cost rules; t=0 is excluded.
Mathematical justification: On n≤20, P−Q=20−2n, giving n=10; on n>20, P−Q=n−40, giving n=40. Budget counts follow integer inequalities, not rounded real solutions.
1 MIN · EXIT CHECK
With budget 160, why is P limited to 46 rather than 47?
Check after answering
P(46)=158≤160, but P(47)=161>160.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: State the objective and domain aloud before students choose a quote token.
Watch for: The plan cheaper at one quantity must always be the best choice.
Challenge: Explain the two printing tie points using the two Q branches instead of just spotting crossings.
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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