Multiplying Integers: Why is -3 x -4 = +12?

Here are

‘The Rules’ and ‘The Reasons’,

‘The How’ and ‘The Why’

for Multiplying Integers.

I uploaded a couple pages of my Math Model Book for a ‘Pair-a-Dimes’ post, “Assessment & Rote Learning: Math Conundrums“… and thought I would share these very practical resources here.

The first page has The Rules for Multiplying and Dividing Integers.

Next, using counters, I look at Why the Rules for Multiplying Integers Work*. I call this lesson “Why is a negative times a negative a positive?” and slowly build up to this at the end of the lesson. I enjoy seeing the a-hah moments in students when they finally understand this concept.

*It is very important to have pre-taught the concept of zero before this lesson, (the same negative and positive number together cancel each other out: together -4 and +4 = 0).

But what about division you might ask? I find this harder to show with counters so I usually explain that every multiplication question has two equivalent, related division questions:

If               3 x 4 = 12       Then        12 ÷ 4 = 3         and         12 ÷ 3 = 4

So if,     -3 x -4 = +12     Then     +12 ÷ – 4 = -3     and     +12 ÷ -3 = -4

This makes further sense to students when they realize that multiplying two integers with opposite signs = negative, and they can see that the same rings true for division as well.