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.

I love your way of teaching integer multiplication and division. Really interesting!

That sounds like a good approach that makes the kids think and its somewhat of a contest where others see how each person does. I always give our fifth graders a math problem that is related but slightly beyond their lesson for the day and then ask each at the end of the regular lesson to go to the board and write their answer and the key clue (e.g., when the 10th aniversary of 9/11 came on Sunday explain why 9/11 was on a Tuesday). Three out of six correctly noted that there were two leap years in the decade.

I like thinking about negatives as “taking the opposite of.” You might even have the students line up in numerical order and when you wave your wand and they take opposites, they’ll have to all completely reverse order. (That order reversing property is an important concept and one students sometimes forget.) As for multiplying the absolute values and then figuring out how many times you took the opposite, it’s not at all obvious why that should be the procedure for multiplying and yet not for addition. On the one hand it’s nice that students aren’t making the mistake of doing that for addition, but on the other hand maybe they should be asking, “Why not?” What is it about multiplication versus addition that allows us to do first deal with absolute values and taking opposites separately with one, but not the other? I think the key boils down to making the connection that “taking the opposite of” IS “multiplying by -1” (Then the associative and commutative properties of multiplication justify treating them separately, whereas the distributive property of multiplication over addition applies when adding negatives). So how does one make the connection that “taking the opposite of” IS “multiplying by -1”? One can prove that it must be true using the distributive property, but I don’t know how to translate that into something that intuitively makes sense to 8th graders.

Sorry, but thinking about this a little bit more, here is a nice way that I’ve seen that might make intuitive sense of the “taking opposite of” is “multiplying by -1”. 4 times 3 is 12. 3 times 3 is 9. 2 times 3 is 6. 1 times 3 is 3. 0 times 3 is 0. What should -1 times 3 be? This idea of the relation between multiplication and addition given rise to this property is important as it is the same idea as applied to exponentiation and multiplication that justifies that “raising to the -1” should be “take the reciprocal of.”

Just in case you were wondering how the internet works: Just saw the MIND Research Institute Math Without Words TED talk. Looked at site. Saw the Alg Readiness book. Remembered seeing same pic on your site. Came back here.

Anyway, what are your thoughts on the curriculum?

Jason: I really (!) like the curriculum. Unfortunately, I only have a sample copy and it’s not really a manipulative-based program where I can have students do it w/out the text. I have copied some problem sets for students, but mostly I just consult it for ideas. My hope is that this spring I can convince someone to give me a grant or the district to find the money to purchase it. In the meantime, I’m eager to watch that TED talk. Thanks for sharing it.