We've been talking about making our own rules, so we have this sequence of products and I ask students to discuss what patterns they notice.
-3 (3) = -9
-3 (2) = -6
-3 (1) = -3
-3 (0) = 0
-3 (-1) = ??
Stuff we noticed:
"It starts with a -3 every time."
"It goes down by 1."
"It changes by 3."
I zero in to the apparent contradiction in going down by 1 and changing by 3 so we can clean up the language a bit. This starts an nice little exchange about whether or not going from -9 to -6 is an increase or decrease. We conclude it's actually an increase. I have to remember to take my time here because this isn't an insignificant point: Kids seem to think in absolute value.
So what comes next?
I wrote down everything I heard.
"3". "-3". "4". "-4".
"Wow!" I say. "We've got a great argument about to happen. This is awesome! So many different opinions. So which is it?"
Some minds change when groups start to discuss. The students who thought 4 or -4 were thinking of sums and not products. That leaves 3 or -3.
"Ok, so which is it?"
If I had a dollar for every time a student said "A negative times a negative is a positive" followed by "because my teacher told me", I'd have all the dollars.
But then Isaac offers a reason worth looking at.
"I think it's -3, because positive 3 times positive 1 is positive 3, so negative 3 times negative 1 is negative 3."
So I write the following on the board:
(pos) (pos) = pos
(neg) (neg) = neg
We talk about this pattern Isaac. has noticed. "Does this work for you all?"
Jordan speaks up, "I don't think so. It has to be positive three so that it doesn't break the pattern."
"Which pattern is that?"
"The pattern goes from -9 to -6 to -3 to 0. It's increasing by 3 each time so the next answer has to be 3."
"Why would that be so?" I ask.
Then Vanessa chimes in.
"Because math don't break."