g10-06-03 · Core topics
Equal additions or equal ratios?
Does adding 20 each round match growing by 20% each round?
1 MIN · PREDICT ON PAPER
Starting at 100, predict both quantities after two rounds before reading the graph.
Discuss or write first. No typed answer is required to continue.
Round 2: fixed increment 140, fixed ratio 144.
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
Fixed increments give L(n)=100+20n. Fixed ratios give E(n)=100×1.2ⁿ. They agree after one round but differ after two: 140 and 144.
Conditions: Rounds n are nonnegative integers. A constant ratio is an assumption. A negative linear-model prediction is not a meaningful remaining quantity.
Mathematical justification: The recurrence E(n+1)=rE(n), E(0)=100 gives E(n)=100rⁿ by induction. L(n+1)−L(n)=100(r−1) is constant, while exponential differences depend on n.
1 MIN · EXIT CHECK
After three rounds retaining 80% of 100, what remains?
Check after answering
100×0.8³=51.2.
Teacher guidance and extension
Before: Real-number operations and reading coordinates; write a prediction on paper.
Ask: Ask for the next increment and next ratio in each model, not just the final values.
Watch for: A fixed percentage means the same absolute change each round.
Challenge: Set multiplier 0.8 and compare repeated 20% removal with subtracting 20 each round.
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.
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