NSW Maths (Years 7 to 10)
Adding and Subtracting Integers: The Number Line Method
Integer questions trip up more NSW students than almost any other topic in Years 7–10 Maths, and the reason is almost always the same: students try to rely on memory tricks instead of understanding what the signs actually mean. When a question like 5 - (-3) appears on a school assessment, a student who has memorised "subtracting moves left" will write 2 and lose the mark. A student who understands how to read both signs and move on the number line will write 8 and move on confidently. This post explains exactly how to do that.
What the syllabus asks
The NSW Mathematics 7–10 curriculum asks students to add and subtract integers. That means working with both positive and negative whole numbers, understanding what each sign in a calculation tells you about direction, and being able to cross zero on the number line without hesitation.
The idea, explained
The number line is your foundation
Every integer lives at a specific position on the number line. When you add or subtract, you are not just crunching symbols, you are making a move from a starting position. The first number in any calculation tells you where you stand. The operation and the sign of the second number together tell you which direction to walk and how many steps to take.
One sign: straightforward moves
When there is only one sign between the two numbers, the rule is simple:
- Adding a positive number means move right.
- Subtracting a positive number means move left.
For example, to calculate -3 + 7, start at -3 on the number line. Adding 7 means 7 steps to the right. Counting from -3, you pass through -2, -1, 0, 1, 2, 3 and land on 4. Passing through zero is completely fine; zero is just another tick mark. So -3 + 7 = 4.
Notice that the starting position being negative did not change the direction of the move. The direction depends on the operation and the sign of the second number, not the first.
Two signs: each negative flips the direction
This is where most students go wrong. When you see something like 5 - (-3), there are two signs sitting between the two numbers: a subtraction sign and a negative sign on the 3. Each negative flips the direction once.
- Subtracting alone would send you left.
- But the negative on the 3 flips that back to the right.
So from 5, you move 3 steps to the right and land on 8. The calculation 5 - (-3) = 8 gives exactly the same result as 5 + 3 = 8. Subtracting a negative is the same as adding the positive version of that number.
The same logic applies to adding a negative. In 4 + (-3), there are two signs: a plus and a negative on the 3. Adding alone would send you right, but the negative flips that to the left. From 4, move 3 steps left and land on 1. So 4 + (-3) = 1, which is the same as 4 - 3 = 1.
A useful way to remember the direction rule
Count the negative signs between the two numbers. If there is one negative sign, the move is in the "opposite" direction to plain adding (so you go left). If there are two negative signs, they cancel each other out and you go right. You can also think of it this way: taking away a debt leaves you better off, not worse, so removing a negative moves you to the right.
Zero pairs
A zero pair is any pair of numbers that add to zero, such as 3 and -3. Recognising zero pairs can speed up calculations. If you are adding -2 - (-2), the two values cancel and the answer is 0. This idea also helps you check your work: if your answer seems far from zero and the numbers involved are small, retrace your steps on the number line.
What the exam asks
School assessment questions on this topic typically come in a few forms:
- Straight calculations such as -6 + 2, 4 - 10, -2 - 5, 3 + (-8), or -2 - (-6).
- Word problems involving temperature changes, for example: the temperature at dawn was -4 degrees and rose 9 degrees by midday, giving a midday temperature of 5 degrees.
- Context problems involving money, for example: a bank account sitting at -20 dollars has a mistaken 5-dollar fee removed, which means the balance changes by -(-5), leaving the account at -15 dollars.
For every one of these, the method is the same: identify your starting position, count the signs between the two numbers, determine the direction, count the steps, and state the answer.
Common mistakes
- "Take the smaller from the bigger": For -2 - 5, some students compute 5 - 2 = 3. This is wrong because the first number is a position, not a size. Subtracting 5 moves left from -2, landing on -7.
- "Subtracting always moves you left": A student who stops reading after the subtraction sign will write 5 - (-3) = 2. The fix is to read both signs. The negative on the 3 flips the direction to the right, giving 5 - (-3) = 8.
- Ignoring the negative sign when adding: For 4 + (-3), some students write 7, treating it as plain addition. Adding a negative is a move to the left, so the correct answer is 1.
- Getting confused when crossing zero: Students sometimes stop counting at zero or treat it as a barrier. Zero is just another position on the number line; keep counting through it.
- Mixing up which number sets the starting position: The first number is always where you stand. The direction and distance of your move come from the operation and the second number.
If you want to practise these steps with immediate feedback, Avocado's AI-native tutoring service walks you through each calculation at your own pace. Try the Adding and Subtracting Integers lesson to work through number line moves, faded examples, and real assessment-style questions with hints whenever you need them.