The main struggle with integers comes, not with the numbers themselves, but with some of the operations. There seem to be so many little rules to remember (though less than with fractions).
Some good real-life MODELS for integers are:
- temperature in a thermometer
- altitude vs. sea depth
- earning money vs. being in debt.
When first teaching integer operations, tie them in with one of these models.
I'll take for example the temperature.
Assuming n is a positive integer, the simple rules governing this situation are:
* x + n means the temperature is x° and RISES by n degrees.
* x − n means the temperature is x° and DROPS by n degrees.
It's all about MOVEMENT — moving either "up" or "down" the thermometer n degrees.
For example:
- 6 − 7 means: temperature is first 6° and drops 7 degrees.
- (-6) − 7 means: temperature is first -6° and drops 7 degrees (it's even colder!).
- (-2) + 5 means: temperature is first -2° and rises 5 degrees.
- 4 + 5 means: temperature is first 4° and rises 5 degrees.
These simple situations handle adding or subtracting a positive integer. Practice those first, until kids are familiar with these cases.
The remaining cases to handle ar adding or subtracting a negative integer:
- (-2) + (-5) would mean: temperature is first -2° and you "add" more negatives so it gets even colder.
The last case is least intuitive one:
- 1 − (-5) or subtracting a negative integer. I personally just remember the little rule of "two negatives turns into a positive".
Some people explain it this way. In (-7) − (-3) you can think that you have 7 negatives at first, and you "take away" three of those negatives, leaving -4.
This rule of "two negatives makes a positive" might seem counterintuitive at first, but it is needful so that many principles of mathematics can continue to apply (for example distribuitive property).
See also an
excellent treatise of integers vs. submarine depth at Text Savvy. Excerpt:
"When you add or subtract with integers, you are NOT combining collections or extracting from collections; you are moving in certain directions."
The "collections" idea does work nicely, for ADDITION:
- 7 + (-4) means you have a collection of 7 red balls and 4 blue balls. A pair of one red and one blue ball "cancels" or becomes zero. Total therefore will be 3 red balls.
- (-3) + (-9) means you have 3 blue balls and 9 blue balls more. Total 12 blue balls, or -12.
However, this "moving" idea is exactly how I have always intuitively done simple integer problems — except adding (negative) + (negative) and subtracting a negative, which I change to adding a positive.
Some books might present the rules for adding integers this way:
- If you add two integers that have the same sign, add the absolute values and put the same sign as what the numbers had.
- If you add two integers that have a different sign, subtract their absolute values and the answer will have the same sign as the number wiht greater absolute value.
Then they instruct to change the subtraction to addition; for example 5 − 7 becomes 5 + (-7) and (-4) − 2 becomes (-4) + (-2), and 5 − (-3) becomes 5 + 3.
While these are technically totally correct, I find it SO much easier to use the "moving" idea for most quick integer calculations. It is easier to start that way, and then learn these other rules to be used with more complex expressions, such as when adding many integers, or with negative and positive decimals.
Please feel free to
download a simple integer addition and subtraction worksheet that practices some of these ideas (from
Math Mammoth Grade 6-B Worksheets collection).
You will also benefit greatly from reading
my previous article on teaching integers, which goes through all four integer operations.
