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 two multiplication four 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, two multiplication four equals eight.

Suppose you wanted to determine open four minus close multiplication two 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, equation left hand side open four minus close multiplication two equals right hand side open eight minus close.

What happens if you need to determine open four minus close multiplication open two minus close? 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 symbol Activity: Multiplication on the Number Line

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

(a) open three minus close multiplication three

Discussion

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

Solution symbol

Answer

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

Thus, equation left hand side open three minus close multiplication three equals right hand side open nine minus close.

(b) five multiplication open two minus close

Solution symbol

Answer

(b) First, we must rewrite five multiplication open two minus closeas open two minus close multiplication five, 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, equation sequence five multiplication open two minus close equals open two minus close multiplication five equals open one minus times zero close.

5.5 Extensions and Further Exploration

5.5.2 Calculator Exploration—Exponents with Negative Numbers