Another way we often think about numbers is as abstract quantities that can be measured: length, area, and volume are all examples.

In a measurement model, you have to pick a *basic unit*. The basic unit is a quantity — length, area, or volume — that you assign to the number one. You can then assign numbers to other quantities based on how many of your basic unit fit inside.

For now, we’ll focus on the quantity length, and we’ll work with a number line where the basic unit is already marked off.

## Addition and Subtraction on the Number Line

Imagine a person — we’ll call him Zed — who can stand on the number line. We’ll say that the distance Zed walks when he takes a step is exactly one unit.

When Zed wants to add or subtract with whole numbers on the number line, he always starts at 0 and faces the positive direction (towards 1). Then what he does depends on the calculation.

If Zed wants to *add* two numbers, he walks forward (to the right of the number line) however many steps are indicated by the first number (the first *addend*). Then he walks forward (to your right on the number line) the number of steps indicated by the second number (the second *addend*). Where he lands is the *sum* of the two numbers.

##### : 3 + 4

If Zed wants to add 3 + 4, he starts at 0 and faces towards the positive numbers. He walks forward 3 steps, then he walks forward 4 more steps.

Zed ends at the number 7, so the sum of 3 and 4 is 7. 3 + 4 = 7. (But you knew that of course! The point right now is to make sense of the *number line* *model*.)

When Zed wants to *subtract* two numbers, he he walks forward (to the right on the number line) however many steps are indicated by the first number (the *minuend*). Then he walks *backwards* (to the left on the number line) the number of steps indicated by the second number (the *subtrahend*). Where he lands is the *difference* of the two numbers.

##### : 11 – 3

If Zed wants to subtract 11 – 3, he starts at 0 and faces the positive numbers (the right side of the number line). He walks forward 11 steps on the number line, then he walks backwards 3 steps.

Zed ends at the number 8, so the difference of 11 and 3 is 8. 11 – 3 = 8. (But you knew that!)

##### Think / Pair / Share

- Work out each of these exercises on a number line. You can actually pace it out on a life-sized number line or draw a picture: $$4 + 5 \qquad 6 + 9 \qquad 10 - 7 \qquad 8 - 1$$
- Why does it make sense to walk forward for addition and walk backwards for subtraction? In what way is this the same as “combining” for addition and “take away” for subtraction”?
- What happens if you do these subtraction problems on a number line? Explain your answers. $$6 - 9 \qquad 1 - 7 \qquad 4 - 11 \qquad 0 - 1$$
- Could you do the subtraction problems above with the dots and boxes model?

## Multiplication and Division on the Number Line

Since multiplication is really repeated addition, we can adapt our addition model to become a multiplication model as well. Let’s think about 3 × 4. This means to add four to itself three times (that’s simply the definition of multiplication!):

\[3 \times 4 = 4 + 4 + 4 \ldotp \nonumber \]

So to multiply on the number line, we do the process for addition several times.

To multiply two numbers, Zed starts at 0 as always, and he faces the positive direction. He walks forward the number of steps given by the second number (the second *factor*). He repeats that process the number of times given by the first number (the first *factor*). Where he lands is the *product* of the two numbers.

##### : 3 × 4

If Zed wants to multiply 3 × 4, he can think of it this way:

\[\begin{split} 3& \qquad \qquad \qquad \times \\ \downarrow & \\ \text{how many times}\; & \text{to repeat it} \end{split} \begin{split} 4& \\ \downarrow & \\ \text{how many steps}\; & \text{to take forward} \end{split} \nonumber \]

Zed starts at 0, facing the positive direction. The he repeats this three times: take four steps forward.

He ends at the number 12, so the product of 3 and 4 is 12. That is, 3 × 4 = 12.

Remember our quotative model of division: One way to interpret 15 : 5 is:

How many groups of 5 fit into 15?

Thinking on the number line, we can ask it this way:

Zed takes 5 steps at a time. If Zed lands at the number 15, how many times did he take 5 steps?

To calculate a division problem on the number line, Zed starts at 0, facing the positive direction. He walks forward the number of steps given by the second number (the *divisor*). He repeats that process until he lands at the first number (the *dividend*). The number of times he repeated the process gives the *quotient* of the two numbers.

##### : 15 ÷ 5

If Zed wants to compute 15 : 5, he can think of it this way:

He starts at 0, facing the positive direction.

- Zed takes 5 steps forward. He is now at 5, not 15. So he needs to repeat the process.
- Zed takes 5 steps forward again. He is now at 10, not 15. So he needs to repeat the process.
- Zed takes 5 more steps forward. He is at 15, so he stops.

Since he repeated the process three times, we see there are 3 groups of 5 in 15. So the quotient of 15 and 5 is 3. That is, 15 : 5 = 3.

##### Think / Pair / Share

- Work out each of these exercises on a number line. You can actually pace it out on a life-sized number line or draw a picture: $$2 \times 5 \qquad 7 \times 1 \qquad 10 : 2 \qquad 6 : 1$$
- Can you think of a way to interpret these multiplication problems on a number line? Explain your ideas. $$4 \times 0 \qquad 0 \times 5 \qquad 3 \times (-2) \qquad 2 \times (-1)$$
- What happens if you try to solve these division problems on a number line? Can you do it? Explain your ideas. $$0 : 2 \qquad 0 : 10 \qquad 3 : 0 \qquad 5 : 0$$