Multiplication Models
By Michael Naylor

To read the related article click here.

"Two wrongs don't make a right," so why should two negatives make a positive? Multiplying negative numbers is counterintuitive at first, but it makes sense in many different ways.

With the above models, multiplying (–3) x (–4) can mean:

• staring at 0 , turning around walking backwards 4 steps, repeating 3 times


• removing 4 red (–1) chips, and doing this 3 times


• removing a debt of $4, and doing this 3 times

A very nice demonstration helps show how negative x negative = positive fits in very well with mathematics because of a pattern. First elicit answers as you write this series of equations in a column:

3 x 3 = 9

3 x 2 = 6

3 x 1 = 3

3 x 0 = 0

3 x –1 = –3

3 x –2 = –6

3 x –3 = –9

Ask your students to describe any patterns they see in the column, in particular that the product decreases by 3 each time. Now start a new column with the final equation and repeat, this time decreasing the multiplicand:

3 x –3 = –9

2 x –3 = –6

1 x –3 = –3

0 x –3 = 0

–1 x –3 = 3

–2 x –3 = 6

–3 x –3 = 9

The patterns of mathematics and the models support each other.

To read the related article click here.

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

January 2006, Vol.37, No.4