I used the diamond problems that you spoke of when I taught Algebra 1 several years ago. I remember not mentioning anything about factoring and merely telling the students that we were going to spend time solving some math puzzles. When the students though of it as a game, all of the students were engaged and it turned out to be a “race” of sorts to see who could solve the puzzles the quickest. In fact, kids would take great pride in being able to figure out the difficult ones. So whether it’s the idea of dice, diamonds, or whatever else, it’s good to show students any form of application of mathematics in various contexts.

Thanks for sharing!

Make it 3 dice, 4 dice, so much extension actually.

With 3 dice are there any problems which have more than one solution?

What I did last year (and I liked) was I just gave kids a set of factoring problems and told them to keep their eyes out for patterns. The changes from one problem to the next were subtle (e.g. change the value of c; change the sign) and they picked up the sum/product patterns on their own.

At the same time, I do love the activity in this post. I see myself using it in more or less the way that you describe, except that I don’t see myself explicitly making the connection between sum/product and factoring.

I was always dissatisfied with doing trial-and-error for quadratic equations where the leading coefficient wasn’t equal to one, and enjoyed learning how to attack those when I taught an Algebra I class for the first time (the book called it “the Master Product Method,” which sounds pretty heady, but I’ve not bumped into that name since first seeing it in the late ’80s).

That said, I agree with mr bombastic, but if students are exposed to an approach that demystifies the alleged magic, and if they have been able to see that one of the things that goes on a lot in algebra is “doing-undoing,” then maybe they’ll learn some things that are useful. Certainly a “black box” approach isn’t something I’d advocate.

But also, it shouldn’t be surprising that it’s easier to multiply polynomials together (if you’re systematic and careful) than to factor. After all, we know from cryptography that it’s a heck of a lot easier to multiply two large primes together than to factor their product without knowing either of the original prime factors.

I find a lot of the examples of factoring methods really distrurbing. Factoring by grouping, airplane method, slide method are all “magic” as far as most kids are concerned. The reason the methods work is almost never explained, and it is very unlikely the kids would be able to follow an explanation anyway.

Being able to factor a messy polynomial isn’t really that important, but it is important to understand that polynomials can be written in different forms. The guess and check method is easy to understand and reinforces that polynomials can be written in different forms.

“This chapter. . . returns to the topic of quadratic equations. Every child is not expected to have found the secrets by now. Some people never learn about quadratic equations, and quadratic equations are not among the minimum essentials for productive adulthood — or for promotion into the sixth grade.

We would suggest this attitude toward a student who has not yet found the secret: Quadratic equations are a game. If the students have fun at them (most students do) and if they are good at them (most students are), why, that is very nice. But if the students do not enjoy them and are not good at them, don’t worry. There are lots of other things — linear equations, signed numbers, graphs, identities, derivations, and so on — that they will find exciting and amusing.

Pedagogically, it is desired that the students get *experience* with mathematical material, learn from this experience, and enjoy as much of it as possible.

*We are convinced students do much better work when they are kept away from too much pressure.* They pull us along after them: we don’t push them. . . .

It is a mistake to expect that we will ever *know* exactly what should be taught in grade 4, or in grade 5, and so on. The teacher who is not surprised by his students’ ability is probably not observing them carefully (or sympathetically) enough.” (p. 67)

Those last two paragraphs make an interesting contrast to the current pressure-cooker philosophy that serves as educational policy in this country.

I was thinking more along the lines of this:

If I multiply (x+m)(x+n) I get x^2+(m+n)x+mn. (I chose 1 as the coefficient for x for simplicity). When I have to work the other way, do I just notice that the game I’ve been playing shows up and is useful here or is there a way to connect factoring to the puzzle “m+n=b and mn=c, what are m and n?”. What I’m looking for is a connection of the generalization of the game structure to the factoring process. Maybe there isn’t one, and it’s just something that shows up as a result of applying the distributive property.

What would be great is to somehow connect this further back to multiplying multi-digit whole numbers.

Sorry if I sucked the fun out of the activity for everyone :(

Another extension is to ‘fake the dice results’ and have students determine whether or not there IS a solution to Kaleb’s problem; can they prove they have exhausted all possibilities?

Ugh. That’s awesome.

There is a demonstration here and another one here. It may look intimidating (I worked with teachers who didn’t follow me on it), but the only tools used are Kaleb’s problem and the distributive property.

The method works for all integer polynomials with rational roots. Students practice the distributive property several times and finding greatest common factors after grouping, which are all good skills. It also introduces students to factoring by grouping, which is sometimes taught in Algebra or Precalculus and can occasionally be used to factor third-degree polynomials.

Another extension is to ‘fake the dice results’ and have students determine whether or not there IS a solution to Kaleb’s problem; can they prove they have exhausted all possibilities? Is it faster to satisfy the product first then search for the sum, or vice versa? As a teacher, thinking about Kaleb’s problem allows me to create examples of polynomials with rational roots on the fly.

http://misscalculate.blogspot.com/2011/12/factoring-ax2-bx-c.html
http://patternsinpractice.wordpress.com/2011/04/25/factoring/