Additive versus multiplicative growth: adding the same amount and multiplying by the same factor are different worlds

01The mistake

Given a table where a quantity goes 3, 6, 12, 24, students describe the pattern as “adding 3, then 6, then 12” and struggle to write a rule, or they average the differences and fit a line. The multiplicative structure — each term is twice the last — is available and not looked for, because the only pattern question they have ever been trained to ask is about differences.

The tell is asking for a prediction far outside the data. Over three or four points a linear fit to exponential data looks respectable. Ask for the value at $t = 30$ and the two models differ by orders of magnitude. A student who is untroubled by giving a linear extrapolation for a doubling process has the misconception; short-range questions will never reveal it.

It runs the growth-rate errors too. Students confuse the growth factor with the growth rate — treating a 5% increase as multiplying by $0.05$ rather than by $1.05$, or reading $b = 1.05$ as 105% growth. That is U2-PR4, and it is what this misconception looks like once students have accepted multiplication but not the structure around it.

Watch also for it in reverse: students who have learned that exponential means “fast” will label any steeply increasing data exponential, including quadratics. Speed is not the distinguishing property. Constant ratio is.

02Why it makes sense to the student

Linear models are almost everything students have seen. Slope, rate of change, unit conversion, direct proportion — years of mathematics in which the answer to “what is the pattern” is a constant difference. That is a well-earned habit and it is correct nearly everywhere it has been applied.

Over short intervals the two are hard to tell apart. Any smooth increasing function looks roughly linear over a small enough window, which is a real mathematical fact and not a student error. The data sets in early examples are often short enough that both models fit acceptably.

Percentage language is genuinely ambiguous. “Growing by 5% a year” sounds additive — you add 5% — and students who compute 5% of the original value each year produce linear growth from an exponential description. The phrase does not signal that the base changes.

And difference tables are taught as the tool for finding patterns. Nobody hands students a ratio table. If the only instrument you have measures differences, a constant-ratio pattern reads as a mess of unequal differences rather than as a different structure.

03The correction

Give them the second question and make it routine: compute the ratios as well as the differences. For 3, 6, 12, 24 the differences are 3, 6, 12 — not constant — while the ratios are 2, 2, 2. Constant differences means linear; constant ratios means exponential. Two rows under every table, every time, until it is automatic.

State the two forms side by side so the structural difference is visible in the notation. Linear: $f(x) = a + bx$, where $b$ is added once per unit step. Exponential: $f(x) = ab^x$, where $b$ is multiplied once per unit step. Same letter, different operation, and the operation is the whole distinction.

Fix the percentage language explicitly, because it causes independent damage. A 5% increase multiplies by $1.05$, not by $0.05$: the 1 preserves what was already there and the $0.05$ adds the increase. A 5% decrease multiplies by $0.95$. Have students state the growth factor before writing any model.

Then make the long-range consequence concrete, since that is what students remember. A quantity starting at 100 and growing by 10 per year reaches 400 after 30 years. Growing by 10% per year it reaches about 1,745. Same starting point, same-sounding description, and after thirty steps one is four times bigger and the other is seventeen. The gap is the point.

A useful classroom test: hand over a table of four values that fits neither cleanly and ask “is this linear, exponential, or neither, and how do you know?” The reasoning is what is assessed. A student who checks only differences will call anything without a constant difference “not linear” and stop, which tells you the second question has not been installed.

04A sample question

05What each wrong answer reveals

• A Linear fitted to the first step. The student computed one difference, assumed it was constant, and never checked the second. This is the misconception at its most direct, and it is worth noticing that the model is exactly right at $t=0$ and $t=1$ — it fails from the third point onward. A student who checks only two points has no evidence against it, which is why the ratio row matters more than a warning.
• B Correct. The ratios are $6/3 = 12/6 = 24/12 = 2$, constant, so the growth is exponential with growth factor 2 and initial value 3: $P(t) = 3 \cdot 2^t$.
• C Linear fitted more carefully, and still linear. This student noticed the differences were not constant — a real observation that A missed — and responded by averaging them into a single slope. The instinct to accommodate the data is good; the model class is wrong. They need the ratio question, not more care, and telling them to check their work will not help since their work is fine.
• D Non-linear recognised, wrong family. The student has correctly rejected linear growth and reached for a quadratic because the values accelerate. This is the “exponential means fast” confusion inverted: they know it grows faster than linearly and are using speed rather than constant ratio to choose the family. Testing the model against the data takes ten seconds and $3t^2$ gives 0, 3, 12, 27 — visibly wrong at every point.

A, C and D are three different distances from the answer. A never questioned linearity. C questioned it and had no alternative structure available. D abandoned linearity and picked the wrong replacement. Only A is fixed by “check more points”; C and D need the constant-ratio test, and D additionally needs to hear that acceleration does not identify a family.

06Try it in Mistake Master