Inspect Changing Area Increments
Distinguish nonconstant first differences from a linear rule.
Supplied materials
- Paper or a note app.
- Supplied model A(w) = w(10 − w); widths 0,1,2,3,4,5 give areas 0,9,16,21,24,25. Zero is a collapsed boundary shape.
Learn the idea
A linear rule changes by the same output amount for every equal input step. This area model does not: moving width from zero to one adds nine square units, but moving from one to two adds only seven. Each width increase also reduces the other dimension.
Subtract consecutive outputs to form first differences. Here they are 9, 7, 5, 3, 1. Their own differences are all −2, consistent with the quadratic term −w². A supplied formula establishes the model; a short table alone cannot prove how all unseen inputs behave.
- Copy the width and area table in order.
- Subtract consecutive area values to get five first differences.
- Subtract consecutive first differences to see −2, then inspect the changed table below.
See the method in action
A(2) − A(1) = 16 − 9 = 7, while A(3) − A(2) = 21 − 16 = 5. Their difference is 5 − 7 = −2, so equal width steps do not give equal area steps.
Check your reasoning
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See the answer and explanation
9,7,5
Subtract in order: 20 − 11 = 9, 27 − 20 = 7, and 32 − 27 = 5. The −2 values are second differences, not the requested first differences.
Take it further
Replace the model with B(w) = 3w + 2. At widths 1,2,3,4 its outputs are 5,8,11,14. All first differences equal 3, so this rule is linear.
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