Understanding Integer Operations
Adding and subtracting negative numbers trips up many students because the rules feel backwards. Why does subtracting a negative make the answer bigger? Why does adding a negative make it smaller? The number line makes it visual: every operation is a direction and a distance.
How the Number Line Shows Operations
Every integer operation can be shown as two arrows on a number line. The first arrow goes from 0 to the first number. The second arrow starts where the first ended and moves in a direction determined by the operation and the sign of the second number.
- Adding a positive: The second arrow goes right.
- Adding a negative: The second arrow goes left.
- Subtracting a positive: The second arrow goes left.
- Subtracting a negative: The arrow starts going left, then reverses to go right. This is the double negative!
The Double Negative
The expression 5 − (−3) is the hardest case for students. The visualiser shows the second arrow starting to go left (because of the minus sign), then flipping direction (because the number is negative). Two negatives cancel out, and the arrow ends up going right — the same as adding 3.
Context Views
Thermometer: Temperature drops below zero. Adding −5 degrees means the temperature falls by 5.
Elevator: Going below ground level. Floor −2 is two levels underground.
Sea Level: Diving below the surface. −10 metres means 10 metres underwater.
Difficulty Levels
Easy (Year 6): Positive numbers only. Students see that subtraction means moving left and addition means moving right before negatives are introduced.
Medium (Year 7): Mix of positive and negative numbers. Students learn what happens when you add or subtract a negative.
Hard (Year 8): Both operands can be negative. Double negatives and expressions like (−7) − (−4) are explored.
Australian Curriculum Alignment
This tool supports the Australian Curriculum: Mathematics content descriptions for Number and Algebra — Integers across Years 6-8, including ordering integers, adding and subtracting positive and negative integers, and understanding the effect of operations on integers.
Frequently Asked Questions
What are integers?
Integers are whole numbers that can be positive, negative, or zero. Examples include -5, -1, 0, 3, and 12. They do not include fractions or decimals.
How do you add a negative number?
Adding a negative number is the same as subtracting the positive version of that number. For example, 5 + (-3) means start at 5 and move 3 to the left, landing on 2. The arrow goes left because the number being added is negative.
What does subtracting a negative mean?
Subtracting a negative is the same as adding a positive. For example, 5 - (-3) = 5 + 3 = 8. On the number line, the arrow starts going left (subtract) but then reverses direction (negative) and goes right instead.
Why does subtracting a negative give a bigger number?
Think of it as removing a debt. If you owe someone 3 dollars (that is -3) and someone takes that debt away (subtract -3), you are 3 dollars richer. So the number gets bigger.
What is a double negative?
A double negative occurs when you subtract a negative number. The two negatives cancel out and become a positive. For example, 4 - (-2) = 4 + 2 = 6.
How do the context views help?
The thermometer, elevator, and sea level views show real-world situations where negative numbers appear naturally. A thermometer dropping below 0, going below ground level in an elevator, or diving below sea level all help make negative numbers feel concrete rather than abstract.