# 5.5.1 Using the Number Line to Multiply

At the beginning of Unit 2, we learned about number lines. In Unit 5, we took advantage of this tool and used it to add and subtract, especially when negative numbers were involved. Did you know that you can also use the number line to multiply and divide? Let’s check it out!

Suppose you wanted to determine using the number line. You take the first factor, 2, and
draw an arrow that is 2 units long. Because the 2 is positive, we will draw
the arrows to the **right**. The second factor tells us how many arrows
to draw. So, starting from zero, the arrow (of length 2) is drawn end-to-end
4 times, as shown below.

Thus, .

Suppose you wanted to determine using the number line. First, we will be drawing arrows
that are 4 units long, but to the **left** since the 4 is negative.
Starting at zero, we will draw an arrow that is 4 units long, 2 times, end
to end, as shown below.

Thus, .

What happens if you need to determine ? The first factor tells us that the arrow will be 4 units long and go to the left, but we can’t draw the arrow a negative number of times. Unfortunately, our lovely number line model fails for this example.

Not to worry! In the calculator exploration earlier, we discovered that the reason a negative number times another negative number is positive is because it is following a pattern.

Let’s try a few more examples on the number line.

## Activity: Multiplication on the Number Line

In your math notebook, determine the answer using a number line for each problem below.

(a)

### Discussion

If you have a negative **factor**, write it first.
Remember that both length and direction are given by the
first factor.

### Answer

(a) Our arrow will be 3 units long and go to the
**left**. We will need to draw it three
times.

Thus, .

(b)

### Answer

(b) First, we must rewrite as , which is an application of the
commutative property of multiplication. If you can’t quite remember what commutative means check your math notebook! So, our arrow will
be two units long and go to the **left**. We will draw it
five times.

Thus, .

5.5 Extensions and Further Exploration