Year 7 — Adding and Subtracting Integers (Positive & Negative)
Integers are whole numbers and their negatives … …, -3, -2, -1, 0, 1, 2, 3, ….
Use a number line: right = bigger (add), left = smaller (subtract).
1) Key Ideas
- Add a positive: move right. Example: \(3 + 4 = 7\).
- Add a negative: move left. Example: \(3 + (-4) = -1\).
- Subtract a positive: move left. Example: \(5 - 7 = -2\).
- Subtract a negative: move right (two negatives make a plus).
Example: \( -2 - (-5) = -2 + 5 = 3\).
Tip: If you see “\(-\)(negative)”, change it to “\(+\)(positive)”.
2) Sign Rules (Quick Reference)
| Operation | Rule | Example |
|---|
| Add a positive |
Go right on the number line |
\( -3 + 6 = 3 \) |
| Add a negative |
Go left on the number line |
\( 4 + (-7) = -3 \) |
| Subtract a positive |
Go left |
\( 2 - 5 = -3 \) |
| Subtract a negative |
Change to add a positive |
\( -6 - (-4) = -6 + 4 = -2 \) |
3) Method A — Number Line
- Start at the first number.
- For “\(+\) \(k\)”, jump \(k\) steps to the right. For “\(-\) \(k\)”, jump \(k\) steps to the left.
Example 1: \( -4 + 7 \)
Start at \(-4\). Add \(+7\) → move right 7 steps → \(\boxed{3}\).
Example 2: \( 5 - 9 \)
Start at \(5\). Subtract \(9\) → move left 9 steps → \(\boxed{-4}\).
Example 3: \( -2 - (-6) \)
“Subtract a negative” → change to add: \( -2 + 6 = \boxed{4}\).
4) Method B — Same Sign vs Different Sign (for Addition)
Adding integers (including negatives) can be done by size comparison:
- Same sign: add the sizes, keep the sign.
\( (-3) + (-8) = -(3+8) = \boxed{-11} \)
- Different signs: subtract the smaller size from the bigger size, keep the sign of the bigger size.
\( (+9) + (-4) = 9 - 4 = \boxed{+5} \)
Example 4: \( -12 + 5 \)
Different signs → \(12-5=7\). Bigger size is 12 (negative), so answer \(\boxed{-7}\).
Example 5: \( -6 + (-7) \)
Same sign (both negative) → \(6+7=13\), keep negative → \(\boxed{-13}\).
5) Subtracting Integers = Add the Opposite
Change every subtraction into addition by flipping the sign of the next number:
\[
a - b \;=\; a + (-b).
\]
Example 6: \( 7 - (-3) \)
Change: \(7 - (-3) = 7 + 3 = \boxed{10}\).
Example 7: \( -9 - 4 \)
Change: \(-9 + (-4) = \boxed{-13}\).
Example 8: \( -3 - (-10) \)
Change: \(-3 + 10 = \boxed{7}\).
6) Column (Vertical) Layout (to keep signs clear)
Write the numbers under each other with signs clearly shown. Then combine.
Example 9 (Addition): \( -15 + 9 \)
-15
+ 9
-----
-6
\(-15 + 9 = -6\) (more negatives than positives).
Example 10 (Subtraction): \( 8 - (-11) \)
8
-(-11)
------
8 + 11 = 19
Subtracting a negative becomes addition → \(\boxed{19}\).
7) Word Problems (real-life)
Example 11: Temperature change
Morning: \(-3^\circ\text{C}\). It warms by \(+7^\circ\text{C}\).
New temperature = \(-3 + 7 = \boxed{4^\circ\text{C}}\).
Example 12: Bank balance
Balance \(-\$12\). You deposit \$20.
New balance = \(-12 + 20 = \boxed{\$8}\).
Example 13: Lift (elevator) floors
Start at floor \(-2\). Go up 5 floors → \( -2 + 5 = \boxed{3}\) (3rd floor).
8) Quick Practice (Show Answers)
1) \( 6 + (-9) = \;?\)
\(6 - 9 = \boxed{-3}\).
2) \( -8 + (-7) = \;?\)
Same sign → \(8+7=15\), keep negative → \(\boxed{-15}\).
3) \( -4 - 11 = \;?\)
\(-4 + (-11) = \boxed{-15}\).
4) \( 13 - (-5) = \;?\)
\(13 + 5 = \boxed{18}\).
5) \( -10 - (-3) = \;?\)
\(-10 + 3 = \boxed{-7}\).
9) Common Mistakes to Avoid
- Forgetting that subtracting a negative is the same as adding.
- Ignoring signs in word problems (e.g., “a loss of 5” means \(-5\)).
- Mixing up “move right/left” on the number line.