🔄 Grade 7 · Lesson 1 of 14
Zero Pair Module: Adding and Subtracting Integers
One +1 and one −1 cancel to nothing — that simple idea unlocks every integer addition and subtraction problem.
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The big idea: A zero pair is a positive number and its negative twin — they cancel out because they add to zero. Once you can spot and cancel zero pairs, adding integers is just counting up whatever is left over. Subtraction turns out to be nothing new: subtracting a number is always the same as adding its opposite, so every subtraction problem can be rewritten as an addition problem you already know how to solve.
Pairs that cancel to nothing
Imagine you have some +1 tokens and some −1 tokens. Every time you match a +1 with a −1, they cancel out completely — together they are worth exactly 0. That matched pair is called a zero pair, and it is the key to adding integers without getting lost in the signs.
Whatever tokens are left over after all the pairs cancel are your answer, sign included.
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A zero pair sums to zero
A zero pair is any number paired with its opposite: +1 and −1, +5 and −5, +100 and −100. Add the two together and you always get 0, because they are the same distance from zero in opposite directions.
\[ (+1) + (-1) = 0 \]
The simplest zero pair. Any matching +n and −n pair works the same way.
📝 Worked example: Use zero pairs to find 5 + (−3).
- Picture 5 positive tokens and 3 negative tokens.
- Match 3 positives with the 3 negatives — that is 3 zero pairs, which cancel completely.
- Count what is left over: 2 positive tokens remain.
✓ 5 + (−3) = <strong>2</strong>. Three of the positives canceled the three negatives, leaving two positives.
Adding integers on the number line
Zero pairs explain why the answer comes out the way it does; the number line shows where it lands. Start at the first number. Adding a positive number moves you to the right; adding a negative number moves you to the left. Either way, you move that many steps.
🎮 Interactive: check where your answer lands LIVE
Set two integers with the sliders. Positive (+1) and negative (−1) chips pair up and cancel to zero, and whatever chips are left over are the sum. Watch how adding a negative moves you left and adding a positive moves you right.
\[ a - b = a + (-b) \]
Subtracting b is always the same as adding the opposite of b.
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Subtracting = adding the opposite
You never actually need a separate subtraction rule for integers. To compute a − b, flip the sign of b and add instead: 4 − 9 becomes 4 + (−9). This turns every subtraction problem into an addition problem you already know how to do with zero pairs or the number line.
📝 Worked example: Find −2 − 5.
- Rewrite the subtraction as adding the opposite: −2 − 5 = −2 + (−5).
- Both numbers are now negative, so add their sizes: 2 + 5 = 7.
- Keep the negative sign, since both addends were negative.
✓ −2 − 5 = <strong>−7</strong>.
📝 Worked example: Find 8 − (−3).
- Rewrite as adding the opposite of −3: 8 − (−3) = 8 + 3.
- Subtracting a negative flips it into a positive, so this is now simple addition.
- Add: 8 + 3.
✓ 8 − (−3) = <strong>11</strong>. Subtracting a negative moved the answer up, not down.
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Watch two signs sitting next to each other
When you see two signs in a row, like −(−3), do not just drop one. Minus a negative becomes plus: −(−3) = +3. Rewrite it before you calculate, or the sign of your final answer will come out backwards.
Check your understanding
1. Which pair of numbers forms a zero pair?
A zero pair is a number and its opposite: 3 + (−3) = 0.
2. Use zero pairs to find 6 + (−4).
Four positives cancel with four negatives (4 zero pairs), leaving 2 positives, so 6 + (−4) = 2.
3. On a number line, adding a negative number moves you…
Adding a negative number always moves you to the left, no matter where you started.
4. Rewrite 7 − 10 as an addition problem.
Subtracting means adding the opposite, so 7 − 10 becomes 7 + (−10), which equals −3.
5. Find −5 − (−8).
Subtracting a negative is the same as adding its positive opposite: −5 − (−8) = −5 + 8 = 3.
✅ Key takeaways
- A zero pair is a number and its opposite (like +1 and −1); together they add to 0.
- To add integers, cancel matching zero pairs — whatever is left over is your answer, sign included.
- On a number line, adding a positive moves right; adding a negative moves left.
- Subtraction means 'add the opposite': flip the sign of the second number, then add.
- Subtracting a negative is the same as adding a positive, so two signs together become a plus.
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