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Section 29.1 Turn Around and Walk Backwards

In the previous sections, we have discussed the idea of both numbers and arithmetic being represented by movement. We're going to formalize this idea a bit more in this section.

We will begin by thinking about numbers as arrows that indicate movement. When we were looking at integer chips, we talked about taking a certain number of steps to the left or to the right. It is actually more useful to think about this as taking a certain number of steps forward or backward. Imagine that you are standing on the number line facing to the right:

Every number represents movement. Positive numbers represent forward steps, negative numbers represent backwards steps, and zero represents taking no steps. It is important to focus on the idea that this is an instruction about movement, not location. This means that 3 means 3 steps forward no matter where you are on the number line.

Similarly, means steps backwards no matter where you are on the number line.

It is very important that the orientation remain consistent. The default direction is to face to the right, so that the default for "forward" is to the right. From this idea, we can actually reduce the entire picture to just arrows (though we'll stick with including the figure for emphasis).

Activity 29.1.1. Adding Numbers on a Number Line.

Addition of two numbers means to perform two sets of movements in sequence. So \(3 + 4\) means to move 3 steps forward followed by 4 steps forward. The end result is the same as 7 steps forward, so \(3 + 4 = 7\text{.}\)

Try it!

Draw a movement diagram to represent \(2 + 4\) and compute the result.

Solution.

Activity 29.1.2. Subtraction on a Number Line.

Subtraction is the same idea, except that we face to the left before moving for subtracted values. This fits in with the concept of subtraction undoing addition, since if you take a certain number of steps forward then turn around and do it again, you will end up right back where you started.

When reading a calculation, you want to be able to read it as a series of steps from left to right.

In the diagram, it is important to draw the figure facing the correct direction to really emphasize the point.

Try it!

Draw a movement diagram to represent \(2 - 5\) and compute the result.

Solution.

Activity 29.1.3. Arithmetic on a Number with Negative Numbers.

If we replace the values with negative numbers, the only thing that changes is that instead of forward steps we use backward steps. This does not change the direction that we're facing.

Try it!

Draw a movement diagram to represent \(3 - (-4)\) and compute the result.

Solution.

With this representation in place, we can very quickly understand and visualize a particular relationship between addition and subtraction. Notice that facing right and walking backward results in the same type of movement as facing left and walking forward. Similarly, facing right and walking forward is the same as facing left and walking backward.

These diagrams show us the following relationships:

\begin{equation*} \begin{array}{lrll} \textit{(Top-left)} \qquad \phantom{.} \amp a - b \amp = a + (-b) \amp \qquad \textit{(Bottom-left)} \\ \textit{(Top-right)} \qquad \phantom{.} \amp a - (-b) \amp = a + b \amp \qquad \textit{(Bottom-right)} \end{array} \end{equation*}

These ideas give us a clear representation of "subtraction is addition of the opposite." Walking forward (no matter which way you're facing) gives you the same result as turning around and walking backward. At a very deep level, this is just another way to see that addition and subtraction are fundamentally related concepts.