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Section 27.1 Through the Looking Glass

When we introduced the number line, we showed that the negative numbers were the numbers to the left of zero. But we have not yet worked with negative numbers in our arithmetic.

Activity 27.1.1. Addition with Negative Numbers.

It turns out that if we add to a positive number or subtract from a positive number, there are no conceptual changes to how we work with the number line. Here is an example for addition. Notice that the process has not changed.

Try it!

Calculate \(-8 + 4\) using a number line.

Solution.

Activity 27.1.2. Subtraction with Negative Numbers.

Here are two subtraction calculations. Notice that it does not matter whether we start from a negative value or simply cross over from positive values to negative values. In both situations, the process remains exactly the same.

Try it!

Calculate \(2 - 7\) using a number line.

Solution.

At this point, it is important that the number we are adding or subtracting is positive, and that all we are doing is allowing negatives as the starting position. We will look at the other possibilities in another section. But as long as that restriction is met, we can expand our ideas to allow us to think about addition and subtraction with larger numbers.

Activity 27.1.3. Number Line Addition with Negative Numbers.

Let's consider the calculation \(-75 + 27\text{.}\) When we try to draw this on a number line, there are two observations we need to make. The first is that we are starting at a negative value, so we must begin our movement to the left of zero. The second observation is that the movement to the right is smaller than the distance to zero. This means that we will stay to the left of zero when we are done.

From here, it doesn't matter how you get to your answer. If you need to do the calculation in two steps, then do it in two steps. If you can perform the calculation mentally, that's also fine. But avoid using a calculator. It is worth the time to practice your mental arithmetic skills, as this feeds directly into your general mathematical confidence. The important thing is to think about the relative locations of the numbers.

This is mostly about the practice of thinking through the diagram to understand the underlying logic, not about following rules. It would be possible to write out a set of 5 or 6 rules for how to do all of these problems, but if you simply think about the picture, many of those rules will work out naturally and intuitively.

Try it!

Calculate \(-53 + 38\) using a number line.

Solution.

Activity 27.1.4. Number Line Subtraction with Negative Numbers.

Sometimes, the calculation takes us past zero. In this case, we just need to look at the picture and think about what that means. Consider \(22 - 54\text{.}\) If we start at 22 and move 54 spaces to the left, we have definitely gone past zero.

Conceptually, it's easiest to think of this as simply returning to zero and then taking the remaining number of steps to finish the movement.

If you need to move a total of steps to the left, and you've already moved steps, how many steps are left? You should hopefully be able to think that through and figure out that there are steps remaining. And that leads us to the final result.

Try it!

Calculate \(34 - 78\) using a number line.

Solution.

Activity 27.1.5. Arithmetic Practice with the Number Line.

You do not want to think about these calculations as rules. You just want to think about them as following the natural logic of the problem. Draw the diagram and let the diagram guide your thinking.

Try it!

Calculate \(-15 - 32\) using a number line.

Solution.