Another busy week in the Math classroom. However, this unit is SO much easier for me and the students than the Pythagorean Theorem was. Next year I definitely think I’ll start with the integer unit instead.
I began by reviewing (in most cases) how to add positive and negative integers. We used number lines to visualize the process and then manipulatives (two-sided red and yellow counters) to practice. Students eventually got the idea, but, of course, the subtraction of negative numbers from a negative number was the hardest concept to get across.
Next class, we were ready to move on to multiplying. Here I ran into a roadblock. It is almost impossible to illustrate the concept of negative times negative equals positive on a number line. In fact, the textbook even switches to the integer counters to explain it, rather than the number line. I knew that this wouldn’t be enough for my inquisitive crew of grade 8’s: if I can illustrate the other rules of integer multiplication with a number line, I should be able to show negatives times negatives.
After wracking my brain and scouring the internet, I did find one site that attempted a reasonable explanation:
Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y. Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps. So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.
It sounds so reasonable! As soon as I go to explain it however, the logic of it falls apart. I sound like a babbling idiot insisting on something the grade 8’s are sure is patently false. So I switched to the integer counters, but I felt like a bit of a failure. I really wanted to be able to use the number line consistently! Anybody have any suggestions? How do you explain this concept to your class?
On the bright side, I did find this video which was my ‘flipped instruction’ for the day. The students loved it (who wouldn’t love a singing ninja?) and it is catchy!