Common Algebra Mistakes and Where They Come From

Different topics, the same mistakes

Every MathGem topic page has its own list of common mistakes, and the lists look different: a sign error when simplifying, a missing term when expanding, a wrong exponent rule in Algebra 2. Put the lists side by side, though, and the same few habits show up again and again. A student who writes $4(x - 5) = 4x - 5$ is likely to also subtract 3 from only one side of an equation and to multiply only one equation in a system by 2. These are not three separate mistakes but one habit in three places.

This guide groups the mistakes into five families. Fixing a habit is quicker than fixing every mistake one at a time, and it keeps working when the student moves on to topics they have not met yet.

Read the wrong answer, not just the mark

A wrong answer in algebra is rarely random. Most of them can be produced exactly by following one wrong rule, and finding that rule tells you what to practise. When marking a worksheet against the answer sheet, try to work out how the student got their answer before writing the correct one. If $3(2x + 5)$ came back as $6x + 5$, the arithmetic was fine and the 3 simply never reached the 5. A student who gets the same kind of wrong answer three times on one page has a habit, not a slip.

1. The sign stays behind

A minus sign belongs to the term that follows it. Students who see $7x + 2 - 3x$ as "$7x$, 2 and $3x$" will add the $x$-terms to get $10x$ instead of $4x$. The same habit appears as $-2(x - 4) = -2x - 8$ instead of $-2x + 8$, and in a system of equations as $2y - (-y) = y$ instead of $3y$.

Underneath, this is almost always a gap in integer rules rather than in algebra. The integer worksheets practise one operation at a time, so it is easy to see whether the trouble is with subtracting a negative, with multiplying, or with something else. A useful habit while the rules settle is to circle each term together with the sign in front of it before combining anything.

2. Only part of it changes

Many rules in algebra say "do this to everything": multiply every term in the brackets, change both sides of an equation, multiply every term of an equation in a system, including the number on the right. The mistake is doing it to the part that is easiest to see. Typical results are $5(x + 3) = 5x + 3$, or subtracting 4 from the left of $2x + 4 = 18$ but leaving 18 alone, or turning $x + 2y = 7$ into $3x + 6y = 7$.

Drawing an arrow from the multiplier to each term, or writing "$-4$ both sides" in the margin, makes the "everything" visible. The distributive property worksheets are the cleanest place to practise this, because on the Easy setting each problem is a single bracket such as $4(x - 5)$ with nothing else around it.

3. A rule that looks right but is not

Some wrong rules are tempting because they look like rules that do work. Multiplication spreads over addition, so it feels natural that squaring and square roots do too. They do not:

• $(x + 3)^{2}$ is $x^{2} + 6x + 9$, not $x^{2} + 9$.
• $\sqrt{9 + 16}$ is $\sqrt{25} = 5$, not $3 + 4 = 7$.
• $3x + 6$ cannot become $9x$, and $2^{3}$ is 8, not $2 \times 3 = 6$.

Telling a student that a rule is wrong rarely makes it go away. Testing it with numbers usually does: $(2 + 3)^{2} = 25$, but $2^{2} + 3^{2} = 13$. A student who checks a suspicious rule with small numbers, and sees it fail, remembers that much longer. These mistakes cluster in simplifying expressions early on and in polynomial multiplication, radicals and exponents in Algebra 2.

4. The right steps in the wrong order

Some answers are wrong because every step was correct but the steps were done in the wrong order: $2 \times 3^{2}$ worked out as $6^{2} = 36$ instead of 18, or $-3^{2}$ read as 9 instead of $-9$. Solving equations has the same problem in reverse. To build $4x - 1$ you multiply by 4 and then subtract 1, so to solve $4x - 1 = 19$ you undo those steps the other way round: add 1, then divide by 4, giving $x = 5$.

When order is the problem, a few pages of Order of Operations I and II help more than more equations. Writing one operation per line also makes the order visible, both to the student and to whoever is marking the page.

5. Stopping before the end

The last family is answers that are true but not finished: $5x - 2x - 7$ left as it is instead of $3x - 7$, $\sqrt{50}$ left as $\sqrt{50}$ instead of $5\sqrt{2}$, $2x^{2} + 6x + 4$ factored as $(2x + 2)(x + 2)$ instead of $2(x + 1)(x + 2)$, or a system of equations answered with $x = 3$ and no $y$. These are worth treating differently from the other four families. The student has usually understood the topic and just does not yet know what a finished answer looks like. Comparing their answer with the form on the answer sheet, and asking "what is still left to do?", tends to fix it within a page or two.

One check that catches most of them

Families 1 to 4 have one thing in common: the wrong answer is not equal to the original expression. That makes them easy to catch by putting a number in for $x$. For $5(x + 3)$ and the wrong answer $5x + 3$, try $x = 3$: the first is 30 and the second is 18, so something went wrong.

The choice of number matters. With $x = 0$ or $x = 1$, many different expressions give the same value, and even $x = 2$ cannot tell $2x$ from $x^{2}$, since both are 4. A number such as 3 or 5 is a safer test. For equations the check is even simpler: put the answer back into the original equation and see whether both sides match.

Fewer problems, more attention

Algebra mistakes are a good example of why. A student repeating a wrong habit on a long worksheet gets better at the wrong habit. A shorter page, marked straight away and with each wrong answer traced back to its family, does more. Once the habit is gone, longer pages make sense again.