In first grade, pupils learn that 100 – 92 means take away 92 from 100. The minus sign (-) means take away or subtract.
After two or three birthdays, pupils learn that 100 – 92 means the difference between 100 and 92. The minus sign (-) means difference. The lucky ones will have a teacher that would line up numbers on a number line to show that the difference is the distance between the two numbers.
small number – big number = negative number
The negative numbers are the opposites of the counting numbers they already know which turn out to have a second name, positive. The positive and the negative numbers can even be arranged neatly on a line with 0, which is neither a positive nor a negative number, between them. The farther left a negative number is from zero the smaller the number. Of course, the pupils already know that the farther right a positive number is from zero the bigger it is. It goes without saying that negative numbers are always lesser than positive numbers in value.
Now, what is 92 – 100 equal to? The difference between 92 and 100 is 8. But because we are taking away a bigger number from a smaller number, the result must be a negative number. That is 92 – 100 = -8. Notice that the meaning of the sign, -, before 8 is different from that between 92 and 100.
What about -100 – 92? Because -100 is 100 units away from the left of 0 and 92 is 92 units away from the right of 0, the total distance or difference between them is 192. But because we are taking away a bigger number, 92, from a smaller number, -100, the answer must be negative (-). That is, -100 – 92 = -192.
We shouldn’t have a problem with 100 – -92. These numbers are 192 units apart and because we are taking away a small number from a bigger number, the answer must be positive. That had always been the case since first grade.
Who says we need rules for subtracting integers?
Click the links for other ideas for teaching integers with conceptual understanding
- A problem solving approach for introducing positive and negative numbers
- Assessing operations involving integers
- Algebraic thinking and subtracting integers – Part 1
- Algebraic thinking and subtracting integers – Part 2
- Subtracting integers using numberline – why it doesn’t help the learning