g10-06-08 · Core topics
The threshold between two integer rounds
When does 100rⁿ first reach a target, and why might rounding a logarithm fail?
1 MIN · PREDICT ON PAPER
For growth r=1.2 and target 150, predict the first whole round that reaches the target.
Discuss or write first. No typed answer is required to continue.
First round: 3. Previous value 144; at round 3: 172.8.
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
For growth r>1 and target above 100, the real crossing is n*=log(target/100)/log r, but the first integer round is the least integer ≥n*. For decay 0<r<1, use the first round at or below the target.
Conditions: n is a nonnegative integer and the target is positive. The question is at-or-above for growth and at-or-below for decay. At r=1 only a target already met at n=0 is reachable.
Mathematical justification: Monotonicity converts the threshold to an inequality in n; evaluate the candidate round and its predecessor to certify first attainment. A logarithmic approximation alone is insufficient.
1 MIN · EXIT CHECK
100×1.2ⁿ reaches at least 150 first at which integer n?
Check after answering
3: round 2 gives 144 and round 3 gives 172.8.
Teacher guidance and extension
Before: Real-number operations and reading coordinates; write a prediction on paper.
Ask: Require both a failed preceding round and a successful first round in the written explanation.
Watch for: Round the real logarithmic answer to the nearest integer.
Challenge: For r=0.8 and target 50, check the two integer rounds on either side of the real crossing.
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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