Adding and subtracting integers

Until now, numbers may have stopped at zero. Integers extend them in the other direction, into the negatives. Temperatures below freezing, money owed, floors below ground level, points lost in a game: negative numbers describe everyday situations all the time. This year you learn to compare, order and calculate with integers, and the number line is once again the clearest guide.

An integer is any whole number, positive, negative or zero. The negatives mirror the positives on the other side of zero, so just as 3 is three steps to the right, negative 3 is three steps to the left. Once you picture them on a line, adding and subtracting become simple movements rather than rules to memorise.

Moving along the number line

Adding and subtracting integers is a matter of direction. Adding a positive number moves you to the right, and subtracting moves you to the left. Starting at negative 2 and adding 5 means moving five steps right, passing through zero and arriving at 3. Starting at 2 and subtracting 6 means moving six steps left, dropping below zero to negative 4, exactly like a temperature falling below freezing.

This movement picture also makes ordering integers clear. A number further to the left is always smaller, so negative 5 is less than negative 1, which is less than 0. It can feel strange at first that negative 5 is smaller than negative 1, since 5 is larger than 1, but on the line negative 5 sits further left, and that settles it. Comparing integers is just reading their positions from left to right.

Subtracting a negative

The one rule that puzzles most students is subtracting a negative number. The result is the same as adding a positive: 5 minus negative 3 equals 5 plus 3, which is 8. The clearest way to feel why is to think of a negative as a debt. If you owe 3 dollars and that debt is taken away, you are 3 dollars better off, exactly as if someone had given you 3 dollars.

Adding a negative works the opposite way, moving you further left, so negative 3 plus negative 4 is negative 7. A simple way to keep track is to watch the signs: two minus signs next to each other turn into a plus, while adding a negative keeps you heading down. With the number line in mind and the debt picture to fall back on, integer arithmetic stops being a set of tricks and becomes a sensible story of moving left and right, which prepares you for working with negative numbers throughout algebra and beyond.

Ordering integers

Comparing integers trips students up because their instinct from positive numbers misfires: negative 5 feels bigger than negative 1 because 5 is bigger than 1. The number line settles it. Place each value where it belongs and read from left to right; whatever sits furthest left is the smallest. Dropping a scrambled handful of integers onto the line sorts them at a glance, turning ordering from a rule about signs into a simple matter of position.

Counters and zero pairs

A second picture of addition uses counters: a positive one and a negative one together make zero, a zero pair. To add a positive group and a negative group, line them up and cancel every pair you can. Whatever counters are left over, all of one sign, are the answer. Three positives and five negatives leave two negatives once three pairs cancel, so 3 plus negative 5 is negative 2. Two groups of the same sign have nothing to cancel, so they simply combine.

Subtraction as the gap

Subtracting two integers measures the distance between them, which is easiest to see on a vertical scale like a thermometer. The difference between a high of 3 and a low of negative 4 is the gap from one mark to the other, which is 7 units. Written out, 3 minus negative 4 is 3 plus 4, because crossing zero adds the two distances together. Reading subtraction as a gap is why taking away a negative makes the result larger.

Teaching tip: a real thermometer or a drawn vertical number line makes integers tangible. Ask the student to start at a temperature and count up or down through zero, since the physical act of moving past zero builds the intuition far better than a rule about signs ever could.

The debt analogy is worth keeping close for subtracting negatives. Owing money is a negative, and having a debt removed makes you richer, so subtracting a negative increases the total. Returning to this picture clears up the confusion almost every time it appears.