NonlinearSIMathematics Learning

Mathematics Learning

Equal input steps can produce unequal output steps

Consider squares with side lengths of one, two and three centimetres. Their areas are calculated by multiplying each side length by itself. This original worksheet compares the three cases using exact arithmetic. It describes ideal squares and does not include measurements from physical samples, a fitted model or an experiment that has been carried out.

Three ideal squares for the NonlinearSI worksheetArea = side × side1 cm2 cm3 cm1 cm²4 cm²9 cm²

01

Write the quantities and units before the values

Label the input side length in centimetres and the output area in square centimetres. State the rule as area equals side length squared. Use nonnegative lengths for this geometric example. The table then records one centimetre with one square centimetre, two with four, and three with nine. Keep the units in the column headings even when the cells contain only numbers. A reader should not need to infer whether a value refers to an edge, a perimeter or an area from the shape of the diagram.

Successive changes for the NonlinearSI worksheetCompare neighbouring rowsSide changeArea change1 → 2: +1 cm1 → 4: +3 cm²2 → 3: +1 cm4 → 9: +5 cm²Same input step; different output step

02

Compare changes between neighbouring rows

The first step increases side length from one to two centimetres, adding one centimetre. Area increases from one to four square centimetres, adding three. The next step also adds one centimetre to the side, while area increases from four to nine, adding five square centimetres. Record both differences beside their rows. Equal input steps have produced unequal output steps. Doubling the side from one to two also makes the area four times as large. These statements follow from the chosen geometric rule, not from a claim about every growing quantity.

Scope of the example for the NonlinearSI worksheetThis worksheetIdeal squaresExact arithmeticNot establishedReal-world forecastMeasured behaviourA new application needs its own basis

03

Carry the boundary of the example into the conclusion

The reusable sheet contains quantity names, units, rule, input range, value table and successive differences. Add a final line stating what the example does not represent: it is not a model of revenue, population or system intelligence. If a later lesson uses another relationship, give it a new rule and its own evidence where relevant. A chart can illustrate the calculated values, but three plotted points alone do not establish the behaviour of an unknown process. Keeping that distinction visible makes the worksheet useful without overstating what it demonstrates.

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