Powers do not distribute over sums: the single most productive wrong rule in algebra

01The mistake

Students write $(a+b)^2 = a^2 + b^2$, $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$, and $\frac{1}{a+b} = \frac{1}{a} + \frac{1}{b}$. These look like three separate errors on three separate topics. They are one belief: that operations pass through addition, term by term.

The tell is that the error is consistent rather than occasional. A student who genuinely forgot the binomial expansion will produce it correctly some of the time and sometimes get stuck. A student holding the linearity rule produces the wrong answer confidently, quickly, and every time — because from where they are standing there is nothing to check.

It shows up beyond algebra too, which is worth knowing when you are deciding how much class time it deserves. Students apply the same reasoning to $\sin(a+b) = \sin a + \sin b$, to function notation with $f(a+b) = f(a) + f(b)$, and to scaling problems where doubling a length is expected to double an area. Every one of those is the same move.

De Bock, Van Dooren, Janssens and Verschaffel have documented this as the illusion of linearity across a long research programme, finding that secondary students apply the linear model very strongly in situations where it does not hold, and that the tendency is resistant to ordinary instruction. Their teaching experiments found that even targeted intervention often leaves it partly intact.

02Why it makes sense to the student

The distributive property is real, useful, and taught early. $a(b+c) = ab + ac$ works, is drilled, and is one of the first tools that makes hard problems tractable. Generalising a rule that has always worked is not a failure of care — it is the same inductive habit that makes students good at mathematics elsewhere.

The notation invites it. $(a+b)^2$ has the exponent sitting outside a parenthesis in exactly the position the distributing coefficient occupies in $2(a+b)$. Visually the two expressions are near-identical, and one of them does distribute.

Nothing in the wrong answer looks wrong. $a^2 + b^2$ is a clean, symmetric, plausible expression. Compare it to $a^2 + 2ab + b^2$, which has a stray middle term that appears from nowhere if you do not know where it comes from. The correct answer is the uglier one, and students have no reason to prefer it.

And the error is invisible on the special cases students most often check. If $b = 0$, both give $a^2$. If a student spot-checks with zero, or with an expression where one term vanishes, the wrong rule survives the test.

03The correction

Kill it with numbers, immediately, before any algebra. $(3+4)^2 = 49$. $3^2 + 4^2 = 25$. Not close, not a rounding issue, not a special case. Do the same for radicals: $\sqrt{9+16} = 5$, while $\sqrt{9}+\sqrt{16} = 7$. Students who have seen two numerical counterexamples stop trusting the rule in a way that no amount of algebraic instruction achieves.

Then show where the middle term comes from, so the correct expansion is not just another thing to memorise. $(a+b)^2 = (a+b)(a+b)$, and multiplying out gives four products: $a^2$, $ab$, $ba$, $b^2$. The $2ab$ is not an extra rule; it is the two cross terms that the distributing model never generated because it never wrote the product out.

Name the general principle so students can transfer it. Squaring, square-rooting, taking reciprocals, and applying a function are not distributive over addition. Multiplication is. That is a short list worth putting on the wall, because it converts a dozen separate error types into one thing to watch for.

Give them a check they can run under time pressure: substitute small numbers. Any time a student is unsure whether a manipulation is legal, plugging in $a=1, b=2$ takes ten seconds and settles it. This is the single most useful transferable habit in the algebra units, and it works on rules they have never been taught.

A useful classroom test: put $(a+b)^2$, $\sqrt{a+b}$, $\frac{1}{a+b}$ and $2(a+b)$ on the board and ask which can be split into two terms. The correct answer is only the last. A student who splits all four has the general belief; a student who splits three has it too and has memorised one exception.

04A sample question

05What each wrong answer reveals

• A The distributed exponent. $3^2 + 4^2 = 9 + 16 = 25$. This is the misconception in its purest form, and using concrete numbers makes it unmistakable — the student has not slipped on algebra, they have applied a rule. Worth noting that 25 is also $5^2$, so a student may double-check by taking a square root and find something that looks reassuringly clean. The correction is the numerical comparison, side by side, not a re-derivation.
• B Correct. $(3+4)^2 = 7^2 = 49$. Evaluate inside the parentheses first, or expand as $a^2 + 2ab + b^2 = 9 + 24 + 16 = 49$.
• C Exponent read as a multiplier. $3 \times 4 = 12$, or possibly $(3+4) \times 2 \ne 12$ — either way the squaring has been replaced by multiplication. This is a different error from A: the student is not distributing, they have lost what the exponent means. It needs the definition of a power, which is a more basic repair and should be handled first.
• D The exponent dropped entirely. $3 + 4 = 7$. The student evaluated the parenthesis correctly and stopped, which usually indicates rushing rather than a misconception — and on a timed test that matters, since it is fixed by process rather than by instruction. Related to U6-SM1, stopping at an intermediate value.

A is the conceptual error and the only one of the three that will generalise to radicals, reciprocals and trigonometry. C and D are more basic and more local: C has lost the meaning of the exponent, D has stopped early. If a class is mostly on A, the linearity lesson is worth a full period, because it will pay off in four other units. If it is mostly on D, the fix is a checking habit, not a concept.

06Try it in Mistake Master