Common algebra misconceptions, and the wrong answers that reveal them

Six misconceptions at a glance

Multiplying only the first term

3(x − 4) becomes 3x − 4 instead of 3x − 12. On 3(x − 4) = 2x + 5 this gives x = 9 instead of x = 17. Reteach with an area model: three rows of (x − 4) means three of each part, so the 4 is taken three times too.

Dropping the sign on a negative

−2(x + 3) becomes −2x + 6 instead of −2x − 6. Students who are fine with positive brackets fail every negative one, which makes this easy to spot in item analysis. Reteach by distributing the −2 to both terms out loud, and check by substituting the answer back into the original equation.

Combining unlike terms

3x + 2 becomes 5x. The student treats the variable as a label that can be ignored. Reteach with substitution: try x = 10 in both expressions, 32 against 50, and they are plainly not equal.

Undoing operations the wrong way

On x/4 + 3 = 7 a student divides 4 by 4 and writes x = 1 instead of subtracting 3 and multiplying both sides by 4 to get x = 16. Reteach inverse operations as undoing the last step first, like taking off shoes before socks, and always check by substituting back.

The equals sign as "the answer comes next"

Asked to fill in 8 + 4 = ☐ + 5, many younger students write 12, because they read the equals sign as an instruction to calculate. This one hides behind later algebra errors. Reteach the equals sign as "balances", with a balance picture, and use equations with operations on both sides from early on.

Swapping rate and fixed amount

A phone plan at $20 plus $5 per gigabyte is written C = 20g + 5 instead of C = 5g + 20. Reteach by asking what the cost is at zero gigabytes: that number is the fixed amount, and it stands alone.

Why these stick

Each of these misconceptions is a rule that worked somewhere before. Multiplying the first term works when the bracket has one term. Reading the equals sign as "the answer comes next" works for years of arithmetic worksheets. Combining 3x + 2 into 5x feels like the addition students have always done. Because the wrong rule once worked, students apply it confidently, which is why telling them the correct rule again rarely changes the answer. Showing where the old rule breaks does.

Common mistakes when reteaching them

• Marking the answer wrong without naming the rule the student used.
• Practising only questions where the wrong rule happens to give the right answer, such as brackets with a single term.
• Moving on once students can do the retaught example, without a new question a week later.
• Treating all wrong answers on a question as the same problem, when two different misconceptions may be mixed in.

Writing questions that reveal the misconception

Put the typical wrong answers in as options in multiple-choice questions, one per misconception. Then the wrong answer a student picks tells you which idea to reteach, without marking any working. When one wrong option dominates a question, reteach that idea to the whole class; when a few students pick it, they are a small group.