Students often get confused with negative numbers because the same symbol is used to mean two related but different things. N*egative quantities *and *subtraction *are not the same.

–7 can mean the opposite of 7
–7 can also mean |

Students have learned that the subtraction symbol means *remove *or *find the difference. *But the negative sign, though it looks the same, means something else. It means *the opposite of.*

Here’s another way to think of the distinction between negatives and subtraction:

- Subtracting involves two quantities: one subtracted from the other.
- Taking a negative involves only one quantity; the negative sign means take the opposite of the given quantity.
^{2}

Numbers below zero, especially in unfamiliar contexts, can seem abstract to students. It’s not surprising when students give up on understanding and fall back on memorizing rules. But rules without reason are easily forgotten. Physical and visual models for negative integers as well as everyday contexts such as temperature can help students make sense of computing with negative numbers.

**Real-Life Experience with Negative Numbers**

Compared to their experience with positive numbers, students’ experience with negative numbers is minimal. Most of their classroom workand real-world encounters with negative numbers is limited to specific contexts such as temperature.To achieve proficiency with integers, students must understand their meaning. They need to understand what is meant by the *opposite *of a number, that is, they need to understand the concept of *additive inverse. *They need to be able to think of negative 5 as a quantity –

the quantity that is the opposite of 5, the quantity that yields zero when added to 5.

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**Using a Model with Negatives and Subtraction**

Models can help students understand integers below zero, but no model seems to work perfectly in all instances.^{1}Coins or two–colored counters are particularly useful forhelping students understand the distinction between *negative quantities *and *subtraction*.

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^{1}Fuson, Wearne, Hiebert, Murray, Human, Olivier, Carpenter, and Fennema, 1997;

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, , Olivier, and Human, 1997

Cited in *Adding it Up*

^{2}There are at least two other ways to mean ‘take the negative’

Multiply the quantity by -1.

Find the number that can be added to get to zero, (That is, find the additive inverse.)