Our department had a meeting recently where we discussed the need for more investigative approaches in our teaching. We all use investigative approaches at least part of the time, but some of us disagreed about whether it was possible to approach every topic with an investigation.
One of the specific topics that came up was "the rules for multiplying positive and negative integers". Here are some ideas we came up with:
- Give students calculators and have them try out different calculations with different signs. They are likely to quickly discover the "rules" for multiplying and dividing positive and negative numbers, but I am not sure if they will understand why the rules work. Still, it’s a step in the right direction toward student discovery.
- Have students generate patterns when they are multiplying positive and negative numbers like so:
3 x 4 = 12
3 x 3 = 9
3 x 2 = 6
3 x 1 = 3
3 x 0 = 0
Prompt the students to see what patterns they notice about this list of multiplications. Ask them to extend the patterns another couple of rows. They will have hopefully noticed that the second number in the multiplication is decreasing by one, and that the answer is decreasing by three each time. They will need to have a good understanding of multiplication, subtraction, and negative numbers to be able to be successful in this investigation. This will be an excellent opportunity to formatively assess students on their understanding of negative numbers to see if they can extend this pattern to 3 x -1 = -3.
Have students repeat this process for another set of similar multiplications but perhaps with the pattern flipped around slightly. For example:
2 x 5 = 10
1 x 5 = 5
0 x 5 = 0
You can then set up other similar patterns which should result in the students developing similar rules for other forms of multiplication.
5 x -3 = -15
4 x -3 = -12
3 x -3 = -9
2 x -3 = -6
1 x -3 = -3
0 x -3 = 0
There are some advantages of this approach. First, the students will see that the rules for multiplying positive and negative integers come somewhere; they are in essence necessary to preserve the internal inconsistency of multiplication with these patterns. Second, the students will necessarily get some practice multiplying some of the numbers together with a purpose.
I’d like to try and figure out a visual investigation which doesn’t seem completely contrived. I could imagine some sort of animation involving positive and negative areas which could be useful, but it would require significant preparation ahead of time to ensure that students have a solid sense of multiplication as area of rectangle model before using it.