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

(b)  (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, .