Math 7 · Integers
Opposites that cancel, and a line to count on
An integer is a whole number with a direction: above or below zero, owed or owned, up or down. Grade 7 adds and subtracts them, using two pictures that always give the right answer — tiles that cancel in pairs, and a number line.
- 1. The words, first
- 2. Adding integers
- 3. Subtracting integers
- 4. Integers in real situations
- 5. What costs marks
The words, first
The idea: The zero pair is the idea the whole unit rests on: a positive and a negative together make nothing.
| Word | What it means |
|---|---|
| Integer | A whole number that can be positive, negative or zero: …, −2, −1, 0, 1, 2, … |
| Positive, negative | Greater than zero, or less than zero. −4 is read “negative four”. A number with no sign is positive. |
| Opposites | Two integers the same distance from zero on opposite sides, like 6 and −6. They add to zero. |
| Integer tiles | Counters for integers: one colour is +1 and the other is −1. |
| Zero pair | One positive tile and one negative tile. Together they are worth 0, so adding or removing a zero pair never changes a value. |
| Number line | Zero in the middle, positives to the right, negatives to the left. Further right is always greater: −2 > −7. |
Adding integers
The idea: Put both numbers down as tiles, cancel the zero pairs, and count what is left. On a number line, adding a positive moves right and adding a negative moves left.
Worked example: −5 + 8. Five negatives and eight positives make five zero pairs, and 3 positives are left, so −5 + 8 = 3. On a number line: start at −5 and move 8 right. Five steps reach 0 and three more reach 3.
Same signs. In −4 + (−3) there is nothing to cancel: 4 negatives and 3 more make 7 negatives. −4 + (−3) = −7.
Different signs, without tiles. The answer takes the sign of whichever there is more of, and its size is the difference. In −9 + 4 there are more negatives, and 9 − 4 = 5, so −9 + 4 = −5.
Subtracting integers
The idea: Subtracting means taking tiles away. If the tiles you need are not there, add zero pairs until they are. The answer always matches adding the opposite.
Worked example: 3 − (−4). Follow the right half of the picture above.
- Start with 3: three positive tiles.
- You need to take away 4 negatives, and there are none. Add 4 zero pairs. The value is still 3.
- Take away the 4 negatives. The 3 original positives and the 4 from the zero pairs are left: 7.
So 3 − (−4) is the same as 3 + 4. Taking away a negative helps, the way having a $4 debt cancelled leaves you $4 better off.
Another: −2 − 5. Start with 2 negatives. To take away 5 positives, add 5 zero pairs, then remove the positives. That leaves 7 negatives: −2 − 5 = −7, the same as −2 + (−5).
3 − (−4) = 3 + 4 = 7 · −2 − 5 = −2 + (−5) = −7 · −6 − (−8) = −6 + 8 = 2
On a number line, 3 − (−4) asks how far it is from −4 to 3: four steps up to 0 and three more, so 7 to the right.
Integers in real situations
The idea: Decide what zero means and which way is positive. Write the question as an addition or a subtraction, then ask whether the answer makes sense in the story.
Worked example: temperature. On a chinook day in Lethbridge it is −14 °C at 7 a.m. and 5 °C at noon. How much did it warm up?
- A change is later minus earlier: 5 − (−14).
- Subtracting a negative adds: 5 + 14 = 19.
- Check on a number line: 14 degrees up to zero, then 5 more. It warmed up by 19 degrees.
The tempting wrong answer, 5 − 14 = −9, says it got colder. The story says warmer, so the answer must be positive.
Elevation. A diver is at −18 m, 18 m below the surface. She rises 7 m, then sinks 4 m: −18 + 7 + (−4) = −15 m. Still underwater, and higher than where she started.
Money. Mia has $12 and owes her brother $20. Overall she is at 12 + (−20) = −8: $8 short even if she hands over everything. After earning $15 babysitting, −8 + 15 = 7, so she has $7 once he is paid.
What costs marks
The idea: Nearly all of these are a sign that went missing or went the wrong way.
- Using “two negatives make a positive” for adding. −3 + (−5) = −8. That rule is for multiplying, which is Grade 8.
- Subtracting a negative as if it were positive. 4 − (−2) = 6, not 2.
- Subtracting in the wrong order. A change is later minus earlier.
- Ordering negatives by size. −8 is less than −3.
- Dropping the sign of the answer. −9 + 4 = −5, not 5.