While there are a whole lot of indicators of “Algebra Readiness”, including both mathematical skills and habits of mind, a solid understanding of integers undoubtedly fits somewhere on that list.
My Algebra Scholars (8th grade Pre-Algebra) class is nose-deep into a study of integers and I am trying to find ways to lead them to mastery that doesn’t involve memorizing rules.
Here’s how we’re doing multiplication and division. This was neither the first nor last day in our study, just somewhere in the middle.
We spent a lot of time talking about what taking the opposite means in a mathematical context. Although I didn’t follow the integer chapter exactly as written, I got the ideas for these opposite discussions from the Mind Research Institute’s Algebra Readiness Textbook.
They debated the meaning of – – 6.
Or even – – – 6
Each of these discussions were framed in the context of taking the opposite a certain number of times.
Then, we moved onto multiplication and division. Each student got a sheet of paper with a number written on one side and its opposite written on the other side.
FRONT & BACK OF PAPER:
We went outside and lined up. I brought my magic wand and the idea was each time I waved my wand, it represented taking the opposite of their number. Three waves of the wand and you took the opposite of your number 3 times. I had them all start with the same color (sign) facing forward so it was easy for me to see if they were doing it right (since after the waves of the wand, the same color should be facing forward for all of them).
Then, I asked for 2 volunteers to step forward. We multiplied their two numbers by first finding the product of the absolute value and then counting how many times we’d take the opposite.
-4 * 5 went something like this: the product of the absolute value is 20. Now, show me on your fingers how many times we’re taking the opposite…once…so the answer?
-4 * -7: the product of the absolute values is 28. How many times do we take the opposite…twice…so think of 28, take it’s opposite once (mentally flip a piece of paper), then take its opposite again (mentally flip again). Where do you end up?
For the next few days, every time we did integer multiplication or division problems, I had them picture that their product was on a sheet of paper and think about how many times they would flip it to its opposite.
So far…I have about 90% of my struggling 8th graders very proficient at integer multiplication and division. And so far, there has been no mix up with addition and subtraction (“A negative plus a negative = a positive” ) I’m curious to hear from others if this is a mathematically sound way to approach the concept. Quite honestly, I made it up and have never seen it approached in quite this way before. I’m sure I’m not the first to think of this, and am curious if others have had success teaching these concepts in similar ways.