Do Two Minuses Make a Plus? (2)

For me, getting students to understand what happens when we multiply integers depends on them:
1) Understanding what happens when we add integers
2) Understanding the commutative property and
3) Extending patterns.
I start by getting them to model ¯3 + ¯3 + ¯3 + ¯3 concretely:

 I ask how else could this be written and the usual replies are 4ׯ3=¯12 or ¯3×4=¯12

Next, I ask them to model ¯4 + ¯4 +¯4 concretely and typically I get something like this:

although some students will just the use the first array. Either way when I ask how this could be written, the usual replies are 3ׯ4= ¯12 or ¯4×3=¯12. After a few more examples of this type I then ask the students to complete this statement:

Multiplying a positive and a negative gives...

Here I want the students to see that because of commutativity, it doesn't matter if we multiply a negative by a positive, or a positive by a negative: the product will always be negative. I deal with division in a similar way as well as connecting to fact families or related facts.

Multiplying two negatives is a bit more tricky to represent concretely though (at least in my opinion). I can tell the students that multiplying by ¯1 is equivalent to flipping the two-colour counters:

but this of course begs the question: Why?

So I prefer using patterns to make sense of multiplying two negatives. Firstly, I ask them to continue this pattern which will confirm the negative multiplied by a positive gives a negative result:

Next, I ask them to extend and complete this pattern:

As the students notice that the product of each row increases by 2 each time (i.e. from ¯6 to ¯4 to ¯2 to 0) then the next two rows must be 2 and 4 respectively. In other words, a negative multiplied by a negative gives a positive.

This is mindblowing for some students: they think that if you have a negative number and multiply it by another negative number, then the product will be somehow even more negative! So I make the point of assuring them that it's OK to be perplexed by this as this was how I felt learning this result. But, like it or not, we cannot escape with the beautiful mathematical logic shown above. So negative multiplied by negative will give a positive. This is much better than saying 'two minuses make a plus'.

Some Thoughts on Terminology and Notation

In the UK, I remember there being an emphasis on careful use of terminology with integers. For example, for the expression:

¯3–¯5

I say "negative 3 subtract negative 5". I notice that here in Canada, the tendency is to say "minus 3 minus minus 5". I often see it written as (–3)–(–5). To me this cumbersome and could lead to some misconceptions. So when writing expressions like these, I make a point of using superscript symbols when I went to indicate negative.

Also, I give a little cheer whenever I hear a weather reporter say something like, "Today's low will be negative 7."