The adding and subtracting integers rules are some of the most important — and most easily forgotten — rules in middle and high school math. Get them wrong and you’ll lose marks across every other topic that uses negative numbers: algebra, equations, slope, coordinate geometry, even financial literacy. Get them right and they become automatic, freeing your brain to focus on the actual problem. This guide walks through the integer rules step by step with worked examples, explains why the rules work rather than just asking you to memorise them, and ends with a free practice worksheet covering everything from Grade 7 introduction problems through Grade 9 EQAO-level applications. If you’re preparing for the EQAO Grade 9 assessment, integer fluency underpins the entire algebra strand — our EQAO Grade 9 complete guide walks through the wider picture.
What is an integer?
An integer is a whole number — positive, negative, or zero — with no fractional or decimal part. The set of integers is usually written as:
…, −3, −2, −1, 0, 1, 2, 3, …
Integers extend in both directions infinitely. They include:
- Positive integers: 1, 2, 3, 4, … (also called the natural numbers, or counting numbers)
- Negative integers: −1, −2, −3, −4, …
- Zero: 0, which is neither positive nor negative
Numbers like 1/2, 0.75, or √2 are not integers because they have a fractional component. Numbers like −7, 0, and 142 are integers.
Why integers matter
Integers show up everywhere in real life. Temperature drops below zero. Bank balances go negative. Elevations measure height above and below sea level. Football teams gain and lose yards. The Ontario Grade 7 and 8 curricula introduce integer operations explicitly, and the Grade 9 MTH1W curriculum assumes fluency. The EQAO Grade 9 assessment tests integer operations directly and uses them as the foundation for almost every algebra question.
If you can do integer arithmetic in your head without thinking, the rest of high school math becomes dramatically easier.
The integer rules at a glance
Before getting into worked examples, here are the four core adding and subtracting integers rules in one place. We’ll unpack each below.
| Operation | Rule |
|---|---|
| Adding two positive integers | Add the numbers. The answer is positive. |
| Adding two negative integers | Add the absolute values. The answer is negative. |
| Adding a positive and a negative | Subtract the smaller absolute value from the larger. The answer takes the sign of the larger absolute value. |
| Subtracting an integer | Change the subtraction to addition and flip the sign of the second number. |
These four rules cover every possible integer addition and subtraction. Memorise them, but more importantly, understand why they work — because the “why” makes them stick under exam pressure.
Visualising integers on a number line
The fastest way to understand integer rules is to think of a number line.
A number line is a horizontal line with zero in the middle, positive numbers extending right, and negative numbers extending left. Every integer corresponds to a position on this line:
← -5 -4 -3 -2 -1 0 1 2 3 4 5 →Two key facts about the number line will save you from almost every common mistake:
Adding a positive number means moving right. If you start at −3 and add 5, you move 5 places to the right and land on 2.
Adding a negative number means moving left. If you start at 4 and add −7, you move 7 places to the left and land on −3.
That’s it. Every integer addition is just movement along the number line. Subtraction is the same thing in reverse, but there’s a faster way to handle it — covered below.
Adding and Subtracting Integers Rules
Rule 1 — adding two positive integers
This is the easy one. When both numbers are positive, just add them. The answer is positive.
Examples
5 + 3 = 8
12 + 27 = 39
100 + 250 = 350
No tricks here. This is regular addition, and you’ve been doing it since Grade 1. The reason it’s listed in the integer rules is that students sometimes overcomplicate problems with positives because they’re so used to expecting negatives.
Rule 2 — adding two negative integers
When both numbers are negative, add their absolute values and make the answer negative.
Why this works
Think of negative numbers as debts or temperature drops. If you owe $5 and then borrow another $3, you owe $8 in total — not $2. Two negatives stack to make a bigger negative.
On the number line, you start at a negative number and move further left, away from zero. The result is more negative than where you started.
Examples
(−4) + (−6) = −10
Start at −4 on the number line, move 6 places to the left, land on −10.
(−12) + (−8) = −20
Both negative, add the values (12 + 8 = 20), keep the negative sign.
(−25) + (−35) = −60
Common mistake
Treating two negatives as “cancelling out” to make a positive. They don’t, when you’re adding. (−4) + (−6) is not 10 or 2 — it’s −10. The “two negatives make a positive” rule applies to multiplication and to subtracting a negative, not to adding two negatives.
Rule 3 — adding a positive and a negative integer
This is the rule students get wrong most often. When you add a positive and a negative integer:
- Subtract the smaller absolute value from the larger absolute value.
- Give the answer the sign of the larger absolute value.
What “absolute value” means
The absolute value of a number is its distance from zero on the number line, regardless of sign. The absolute value of −7 is 7. The absolute value of 4 is 4. Absolute value is written with vertical bars: |−7| = 7.
When you’re adding a positive and a negative, ignore the signs for a moment and ask: which number is bigger in absolute terms? That answer determines the sign of the final answer.
Examples
Example A: 8 + (−3)
Absolute values: 8 and 3. Larger is 8. Subtract: 8 − 3 = 5. The larger absolute value (8) is positive, so the answer is positive.
8 + (−3) = 5
Example B: 4 + (−9)
Absolute values: 4 and 9. Larger is 9. Subtract: 9 − 4 = 5. The larger absolute value (9) belongs to a negative number, so the answer is negative.
4 + (−9) = −5
Example C: (−15) + 7
Absolute values: 15 and 7. Larger is 15. Subtract: 15 − 7 = 8. The larger absolute value (15) is negative, so the answer is negative.
(−15) + 7 = −8
A simpler way to think about it
Some students find it easier to picture this as a tug of war. The positive number pulls right, the negative pulls left. Whichever side is bigger wins, and the answer ends up on that side by the difference between them.
8 + (−3): positive side wins by 5, so answer is +5. 4 + (−9): negative side wins by 5, so answer is −5. (−15) + 7: negative side wins by 8, so answer is −8.
Use whichever mental model works for you. The answer is the same.
Rule 4 — subtracting integers (the “keep, change, flip” rule)
Subtraction is where most students go wrong, because it adds an extra layer to the rules above. There’s a shortcut that makes every subtraction problem easier.
The rule
To subtract an integer, change the subtraction to addition and flip the sign of the second number. Then use the rules for adding integers.
This is sometimes called the “keep, change, flip” method, or “add the opposite”:
- Keep the first number the same
- Change the subtraction sign to an addition sign
- Flip the sign of the second number
Why this works
Subtracting a number is the same as adding its opposite. On the number line, subtracting 5 means moving 5 places to the left — which is the same as adding −5. So 8 − 5 and 8 + (−5) give the same answer: 3.
This works in both directions:
- Subtracting a positive becomes adding a negative
- Subtracting a negative becomes adding a positive
That second one is the famous “two negatives make a positive” rule — but it only applies to subtraction, not to addition.
Examples
Example A: 7 − 12
Keep 7. Change − to +. Flip 12 to −12. 7 + (−12)
Now use Rule 3. Absolute values 7 and 12; larger is 12. Subtract: 12 − 7 = 5. Larger absolute value is negative, so the answer is negative.
7 − 12 = −5
Example B: (−8) − 4
Keep −8. Change − to +. Flip 4 to −4. (−8) + (−4)
Now use Rule 2 (two negatives). Add absolute values: 8 + 4 = 12. Keep the negative sign.
(−8) − 4 = −12
Example C: 6 − (−9)
Keep 6. Change − to +. Flip −9 to +9. 6 + 9
Now use Rule 1 (two positives). Just add.
6 − (−9) = 15
Example D: (−3) − (−7)
Keep −3. Change − to +. Flip −7 to +7. (−3) + 7
Now use Rule 3. Absolute values 3 and 7; larger is 7. Subtract: 7 − 3 = 4. Larger absolute value is positive, so the answer is positive.
(−3) − (−7) = 4
The big takeaway
Once you’ve converted every subtraction to addition, you only ever need to think about adding integers. Three addition rules (Rules 1, 2, 3 above) cover everything. The keep-change-flip trick eliminates subtraction entirely from your mental workload.
Worked examples — bringing it all together
These examples mix all four rules. Work through them carefully.
Example 1 — multi-step
Calculate: (−5) + 8 − (−3) + (−6)
Step 1: convert every subtraction to addition. (−5) + 8 + 3 + (−6)
Step 2: work left to right.
(−5) + 8 = 3 (Rule 3: 8 − 5 = 3, positive wins)
3 + 3 = 6 (Rule 1)
6 + (−6) = 0 (Rule 3: equal absolute values, answer is zero)
Final answer: 0
Example 2 — real-world context
The temperature in Toronto was −7°C at 6am. By 2pm, it had risen by 12°C. What was the temperature at 2pm?
This is integer addition. Start: −7. Change: +12.
(−7) + 12 = 5 (Rule 3: 12 − 7 = 5, positive wins)
Temperature at 2pm: 5°C
Example 3 — EQAO Grade 9 style
A diver is 18 metres below sea level. She ascends 7 metres, then descends another 15 metres. What is her final depth?
“Below sea level” = negative. “Ascends” = adds positive. “Descends” = adds negative.
Start: −18. After ascending 7m: (−18) + 7 = −11 (Rule 3) After descending 15m: (−11) + (−15) = −26 (Rule 2)
Final depth: 26 metres below sea level, or −26 m.
Example 4 — algebraic context (Grade 9 algebra strand)
Simplify: 4x − (−2x) + (−9x) − 3x
Convert subtractions: 4x + 2x + (−9x) + (−3x)
Combine like terms: (4 + 2 − 9 − 3)x
Apply integer rules: 4 + 2 = 6; 6 + (−9) = −3; (−3) + (−3) = −6.
Answer: −6x
This is exactly the kind of algebra question EQAO Grade 9 uses. Students who haven’t locked in integer rules lose marks here even when they understand the algebra perfectly.
Common mistakes to avoid
After marking thousands of integer problems, the same handful of errors come up over and over.
Treating “+ −” and “− +” as the same. They are — both mean “add a negative” or “subtract a positive,” which gives the same result. But students often see 5 − 3 and 5 + (−3) as different problems requiring different methods. They’re the same: both equal 2.
Forgetting to convert subtraction before applying rules. The “keep, change, flip” step is non-negotiable for subtraction. Skip it and you’ll mix up signs.
Confusing addition rules with multiplication rules. “Two negatives make a positive” is true for multiplication and for subtracting a negative. It is not true for adding two negatives — (−5) + (−3) is −8, not +8.
Sign errors in chains. A long expression like 4 − 7 + 3 − 2 + 6 trips up students who lose track of which sign goes with which number. The fix: rewrite as 4 + (−7) + 3 + (−2) + 6 first, then work left to right.
Mistaking absolute value for the number itself. |−9| = 9, not −9. The absolute value is always non-negative.
Calculator dependency. A scientific calculator handles adding and subtracting integers rules correctly if you input the signs properly — but students who rely on the calculator for every integer problem often make input errors and don’t catch the wrong answer. Build mental fluency first; use the calculator as a check, not a crutch.
Free adding and subtracting integers rules worksheet
The fastest way to lock in the adding and subtracting integers rules is targeted practice. Below is a worksheet of 30 problems covering all four rules, from basic Grade 7 problems to Grade 9 EQAO-style applications.
Section A — adding two positives or two negatives (Rules 1 and 2)
- 7 + 4 = ___
- 12 + 9 = ___
- (−5) + (−8) = ___
- (−14) + (−6) = ___
- (−25) + (−15) = ___
- 18 + 23 = ___
Section B — adding a positive and a negative (Rule 3)
- 9 + (−4) = ___
- 6 + (−11) = ___
- (−13) + 7 = ___
- (−18) + 25 = ___
- 14 + (−14) = ___
- (−32) + 17 = ___
Section C — subtraction (Rule 4: keep, change, flip)
- 8 − 12 = ___
- (−5) − 9 = ___
- 11 − (−6) = ___
- (−4) − (−10) = ___
- (−15) − 8 = ___
- 20 − (−13) = ___
Section D — mixed problems
- (−7) + 4 − (−6) = ___
- 12 − 18 + (−5) = ___
- (−10) − (−15) + (−8) = ___
- 25 + (−30) − (−10) = ___
- (−6) + (−9) − (−4) + 11 = ___
- 50 − 75 + (−10) + 35 = ___
Section E — word problems (Grade 9 EQAO style)
- The temperature in Sudbury was −12°C at sunrise. By noon, it had risen 15°C. What was the temperature at noon?
- A submarine is at a depth of 250 metres below sea level. It ascends 80 metres, then descends another 120 metres. What is its final depth?
- Jasmine has a bank balance of −$45 (overdrawn). She deposits $130, then writes a cheque for $90. What is her final balance?
- A football team gained 8 yards, lost 12 yards, lost another 5 yards, then gained 15 yards on four consecutive plays. What is the team’s net yardage?
- Simplify: 5x − (−3x) + (−8x) − 4x
- Evaluate: (−6) − (−2) + (−9) − 5 − (−11)
Answer key
| # | Answer | # | Answer | # | Answer |
|---|---|---|---|---|---|
| 1 | 11 | 11 | 0 | 21 | −3 |
| 2 | 21 | 12 | −15 | 22 | 5 |
| 3 | −13 | 13 | −4 | 23 | 0 |
| 4 | −20 | 14 | −14 | 24 | 0 |
| 5 | −40 | 15 | 17 | 25 | 3°C |
| 6 | 41 | 16 | 6 | 26 | −290 m (290m below sea level) |
| 7 | 5 | 17 | −23 | 27 | −$5 |
| 8 | −5 | 18 | 33 | 28 | +6 yards |
| 9 | −6 | 19 | 3 | 29 | −4x |
| 10 | 7 | 20 | −11 | 30 | −7 |
Work through these in order, mark your own answers honestly, and circle any patterns in your mistakes. If you got problems 7–12 wrong, your Rule 3 (positive + negative) needs more work. If you got 13–18 wrong, the keep-change-flip step isn’t sinking in yet. Pattern recognition turns a worksheet into a diagnostic.
How integers connect to senior math and EQAO
Integer fluency isn’t a Grade 7 or 8 topic that’s done with — it’s the foundation for everything that follows.
In Grade 9 MTH1W algebra, almost every equation involves negative coefficients or negative constants. Solving 3x − 7 = −13 requires fluent integer arithmetic. The EQAO Grade 9 practice test tests these constantly.
The point slope formula for linear equations uses integers when points have negative coordinates: writing the equation through (−2, 7) involves several sign-handling steps that fall apart if integer rules aren’t automatic.
Coordinate geometry, surface area problems, financial literacy with debits and credits, and probability with positive and negative deviations all assume adding and subtracting integers rules fluency. The students who score Level 4 on EQAO Grade 9 are not the students who happen to know more algebra — they’re the students whose integer arithmetic is fast and accurate enough that they can focus on the interesting parts of each problem.
For Grade 9 students preparing seriously, adding and subtracting integers rules also underpin the geometry and algebra problems in Canadian math contests like the Pascal and Cayley. A typical Pascal Part A problem looks straightforward but punishes any sign error mercilessly.
How Think Academy Canada teaches integer rules
Think Academy is the international arm of TAL Education Group, one of the largest education companies in the world. Our Canadian programs build integer fluency the same way we build every other foundation topic: derivation first, then memorisation, then application across increasing difficulty levels.
We teach the adding and subtracting integers rules through the number-line model, not through “tricks.” Students who understand why the rules work — that adding a negative is the same as moving left, that subtracting is the same as adding the opposite — rarely forget them under pressure.
Our curriculum runs ahead of the Ontario standard. Grade 6 and 7 students at Think Academy meet integer operations with full fluency by the end of Grade 7, which means by the time they hit MTH1W in Grade 9, adding and subtracting integer rules are completely automatic.
Our practice problem library includes thousands of adding and subtracting integers rules problems, organised by rule and difficulty, scaling from basic Grade 7 questions up to Fermat and AMC contest problems that hide sign-handling traps inside more complex setups.
Our teachers mark every homework set personally, with feedback on the types of integer mistakes a student is making. Auto-graded software flags wrong answers but doesn’t explain whether the issue is conceptual or careless — and that distinction determines what to practise next.
Our free Grade 7–9 math assessment is the fastest way to find out where your child stands. They complete a short online test aligned to the Ontario curriculum, and you get a detailed feedback report on strengths and gaps by topic, plus free practice resources tailored to their level. No commitment, no sales pressure — just the diagnostic information most parents otherwise have to wait until report card season to see.
Frequently asked questions
What are the adding and subtracting integers rules?
Four rules cover every case. Adding two positives: just add. Adding two negatives: add the absolute values and keep the negative sign. Adding a positive and a negative: subtract the smaller absolute value from the larger, and take the sign of the larger. Subtracting an integer: change the subtraction to addition and flip the sign of the second number (the “keep, change, flip” rule).
Why does subtracting a negative make a positive?
Because subtracting a number is the same as adding its opposite. So 6 − (−9) becomes 6 + 9 = 15. On the number line, moving “left by negative 9” is the same as moving right by 9.
What’s the difference between adding two negatives and subtracting a negative?
Adding two negatives gives a more negative answer: (−4) + (−3) = −7. Subtracting a negative gives a more positive answer: (−4) − (−3) = −1. The signs in front of the numbers tell you which rule applies. “Two negatives make a positive” only works for subtraction (and multiplication), not addition.
How do I add a positive and a negative integer?
Find the absolute value of each number (ignore the sign). Subtract the smaller from the larger. The answer takes the sign of whichever number had the larger absolute value. For example: (−12) + 5. Absolute values are 12 and 5. Larger is 12. Subtract: 12 − 5 = 7. The 12 was negative, so the answer is −7.
What’s the “keep, change, flip” rule?
It’s a shortcut for subtracting integers. Keep the first number the same, change the subtraction sign to an addition sign, and flip the sign of the second number. Then use the rules for adding integers. Example: 8 − (−5) becomes 8 + 5 = 13.
Is “subtracting a positive” the same as “adding a negative”?
Yes, exactly. 7 − 5 and 7 + (−5) give the same answer: 2. This is why the keep-change-flip rule works — it just converts every subtraction into an addition of the opposite.
Are integer rules tested on the EQAO Grade 9 assessment?
Yes, heavily. Integer operations appear directly in the Number strand and indirectly across every other strand. The Algebra strand in particular uses integer rules constantly when solving equations and simplifying expressions.
My child keeps getting integer questions wrong even after weeks of practice. What’s the issue?
The most common cause is incomplete understanding of why the rules work — they’ve memorised the rules without the number-line intuition. Re-teach using a number line first, then return to abstract problems. The free assessment at Think Academy can also pinpoint exactly which of the four rules your child is struggling with.
What’s the best way to practise integer rules?
Daily short sessions beat long weekend sessions. Twenty minutes a day for two weeks produces dramatically more fluency than three hours one weekend. The worksheet above is a good starting point; pattern recognition in mistakes is more useful than total problems solved.
Can my child just use a calculator for integer problems?
A calculator handles integer arithmetic correctly if the input is correct — but students who can’t do integers mentally often make input errors and don’t catch the wrong answer. Build mental fluency first; the calculator is for checking, not crutching. EQAO Grade 9 provides an on-screen calculator, but most integer problems at the Grade 7–9 level are designed to be done faster mentally than by typing.
About Think Academy Canada
Think Academy Canada, part of TAL Education Group, supports K–12 students with structured math programs built around an online interactive platform, gamified learning, and teachers who personally mark every homework set. Our curriculum runs ahead of the provincial standards and is designed to prepare students for both school excellence and Canadian math competitions, including the Gauss, Pascal, Cayley, Fermat, and Euclid contests.
