Negative Numbers: The 5 Wrong Rules Kids Invent (Grades 6–7)

When a sixth grader gets integer problems wrong, the errors are rarely random. Look at ten wrong answers and a pattern appears: the same wrong answer, produced the same way, over and over. The child is not guessing. They have built a rule, the rule is consistent, and it is wrong.

This is good news, because a consistent wrong rule can be found and replaced. A child who is guessing needs to learn the topic. A child running a faulty rule needs one conversation and some practice. Below are the five rules children most often invent with negative numbers, how to recognise each from the wrong answers alone, and what to do about it.

Why negatives are the first genuinely abstract idea#

Until sixth grade, nearly every number a child meets can be pointed at. Three apples, half a pizza, 0.75 of a mile. Negative numbers cannot be pointed at. There is no pile of minus four objects anywhere in the room.

Worse, negatives break rules that have held for years. Multiplying used to make things bigger. Subtracting used to make things smaller. Now $-4 > -9$, even though 9 is obviously bigger than 4, and $6 \times (-2)$ is smaller than either number that went into it. Every reliable instinct suddenly misfires.

This is also where working memory starts to buckle. Ashcraft and Krause (2007) documented how anxiety consumes the very working memory that multi-step problems require, and integer arithmetic is the first place where a child must hold a sign, a magnitude and an operation at once. A child who is slightly unsure about all three has nothing left over for the actual question.

Wrong rule 1: "The bigger number wins the sign"#

Looks like: $-8 + 3 = 11$, or $-8 + 3 = 5$.

The invented rule: ignore the signs, do something with the digits, then attach whichever sign belonged to the bigger digit.

This one is close enough to correct to survive for months. For $-8 + 3$, the true answer is $-5$, and the child who says $-5$ by this rule gets a tick. The rule surfaces only when it produces $+5$ or $11$, so it can hide through an entire unit.

The fix is movement. Start at $-8$ on a number line. Adding 3 means moving 3 to the right. You land on $-5$. Ask your child to narrate it rather than compute it: "I start at negative eight, I move three to the right, I am at negative five." Repeat with $-2 + 7$ and $-9 + 4$ until the narration is automatic. Do not mention the rule at all — you are replacing it, not correcting it.

Wrong rule 2: "Two negatives make a positive" (applied to subtraction)#

Looks like: $-5 - 3 = 8$, or $-5 - 3 = 2$.

The invented rule: a minus and a minus make a plus, always, everywhere.

This is the most damaging one, because the rule is genuinely true for multiplication and children are taught it with real conviction. They then export it to subtraction, where it is nonsense.

The fix is to distinguish the two minus signs. Say out loud that the symbol does two different jobs: it marks a number as negative, and it means subtract. In $-5 - 3$, the first minus is a label and the second is an instruction. Then go back to movement: start at $-5$, subtract 3, move 3 further left, land on $-8$.

The useful question when your child produces $+8$ is not "why did you do that?" but "you started at negative five and took away three more — did you end up with more or less than you had?" Every child knows that taking away leaves less. Anchoring to that intuition does more than restating the rule.

Wrong rule 3: "Negatives are just small positives"#

Looks like: $-3 > -7$ marked wrong because the child wrote $-7 > -3$. Or "which is colder, minus 3 or minus 7?" answered with minus 3.

The invented rule: compare the digits; more is more.

The fix is a thermometer or a very cold day. Temperature is the one context where nearly every child already has the right intuition: minus 20 is colder than minus 5, no argument. Build from there to the number line, where further left means smaller, and test with pairs: which is greater, $-12$ or $-2$? Then mix in a positive: $-100$ or $1$?

A useful drill is ordering five cards containing, say, $-7$, $3$, $-12$, $0$ and $-1$ from smallest to largest. It takes ninety seconds and exposes the misconception immediately.

Wrong rule 4: "Subtracting a negative is the same as subtracting"#

Looks like: $6 - (-2) = 4$.

The invented rule: two symbols, but do the obvious one.

Of all five, this is the one children find genuinely unreasonable. Subtracting is supposed to make things smaller, and yet:

$6 - (-2) = 6 + 2 = 8$

Insisting on "two negatives make a positive" here teaches a child to distrust their own sense-making, which is a bad trade even when it produces right answers.

The fix is a story where removing a negative is plainly good. The temperature version works well. It is 6 degrees. A forecast that was going to take away 2 degrees is cancelled. Are you warmer or colder than you expected? Warmer — 8. Removing something bad leaves you better off. Or with money: you owe a friend two dollars, and they forgive the debt. Your subtraction of a negative made you richer.

Then formalise: subtracting a negative is adding. Once the story is in place, the rule feels like a description rather than a decree.

Wrong rule 5: "The sign rules apply to everything"#

Looks like: $-3 \times -4 = -12$ from a child who gets $-3 - (-4)$ right, or the reverse.

The invented rule: there is one sign rule, and it covers all four operations.

By seventh grade a child is juggling sign rules for addition, subtraction, multiplication and division simultaneously, and the rules genuinely differ. Multiplication is the one where "two negatives make a positive" is simply true:

$(-3) \times (-4) = 12$

The fix is to stop teaching addition and multiplication signs in the same session. Separate them by days. For multiplication, patterns work better than rules. Write the sequence and ask what comes next:

$$\begin{aligned} 3 \times (-4) &= -12 \\[2pt] 2 \times (-4) &= -8 \\[2pt] 1 \times (-4) &= -4 \\[2pt] 0 \times (-4) &= 0 \\[2pt] (-1) \times (-4) &= \;? \end{aligned}$$

Every step up has added 4. The pattern says the next answer is 4, and the child has derived the rule rather than received it. Rittle-Johnson and Schneider (2015) describe exactly this mutual reinforcement: a procedure that arrives with a reason attached is far more durable than one that arrives alone.

A ten-minute diagnostic#

Give these six, untimed, with no help, and watch which wrong rule appears:

1. $-8 + 3$
2. $-5 - 3$
3. Which is greater, $-12$ or $-2$?
4. $6 - (-2)$
5. $(-3) \times (-4)$
6. $-10 \div 2$

The wrong answers name the misconception: 11 or 5 on question 1 is rule 1; 8 or 2 on question 2 is rule 2; $-12$ on question 3 is rule 3; 4 on question 4 is rule 4; $-12$ on question 5 is rule 5.

Fix one at a time, ten minutes a day. Fixing two at once reliably produces a child who has both confused by Friday.

Why this matters more than it looks#

Integers are not a self-contained sixth-grade unit. Every one of these misconceptions resurfaces, magnified, in algebra. A student who cannot confidently evaluate $-5 - 3$ cannot solve $2x - 8 = -5 - 3$, and a student who thinks subtracting a negative shrinks things will mangle every equation that requires moving a term across an equals sign.

The National Mathematics Advisory Panel (2008) identified fluency with whole numbers, fractions and certain aspects of geometry and measurement as the critical foundations for algebra. Signed-number reasoning is where several of those foundations meet, and it is cheap to repair in sixth grade and expensive to repair in ninth.

If your child's difficulty spans several topics rather than just this one, it is worth checking whether the gap starts earlier. Our post on the five signs a child is falling behind describes what that pattern looks like, and ratios and proportional reasoning covers the other great sixth-grade leap, which lands in the same year and competes for the same attention.

How to help without taking over#

When your child produces one of these five answers, the instinct is to say "no, remember, two negatives make a positive." Resist it. Restating a rule to a child who has just misapplied a rule adds one more rule to a pile that is already too tall.

Ask instead where they are on the number line, and which way they are moving. Then wait. The pause before the answer is where the learning is — see why productive struggle works for the evidence, and the Socratic method in math for a script you can use tonight.

And if you want to know which of the five is actually in play, the free Math Check-Up runs ten grade-appropriate questions and tells you which skills are solid and which are not, in about ten minutes.