Doesn’t that have it backwards. Consider Sudoku, word scrambles, crosswords — all addictive, none with any context. Even Chess puzzles are ridiculously contrived.

Tetris, what is the context there?

Meanwhile, I remember like it was yesterday the delight I took in Algebra back in the ninth grade: every problem is a puzzle!

Finally, consider DragonBox Algebra. Pure puzzle-solving, very popular, and without even the added context of “this is why this manipulation is valid”. (DBA just shows you the allowed transformation (such as distributing multiplication over division) and away you go.)

The only problem is when students sit down to work these puzzles and have no way of knowing if/when they have in fact solved one. But the puzzles are there, we just have to find a way to let them know how they are doing *while* they are solving them.

Meanwhile, sorry folks, the real-world meaningfulness just is not there. It would be great if it were, but that road to student motivation is dead end.

Which is OK: math in and of itself is a delight when made accessible.

After students start to become pretty good at this puzzle, I remove different pieces and encourage fluency with sums and products and keep the game interesting.

Factoring with a=1 seems to flow easily when students see the underlying mathematics as a game they already understand and find engaging.

Further discussion is made and the necessity of deeper thought is needed when a =/= 1. By this point, hopefully, students have developed an intrinsic desire to see how the game continues when a isn’t 1.

First, both of them (independent of the other) immediately said to make factoring quadratics into a game. To me this was an interesting response as it indicated that they both know that ” factoring trinomials with integer roots, ie. turning x2 + 7x + 10 into (x + 5)(x + 2)” is not a real world skill. It’s interesting to note that they had no problem with it not being a real-world skill and it reflected that they knew that finding integer roots can be a fun and satisfying puzzle to solve. (Now don’t make the mistake of thinking these kids are math whizzes. Even though they have great number sense, the dyslexia has a negative impact on math performance, but I digress. )

After a certain point in his math classes my son had made a decision regarding quadratic equations: he decided that he would always just use the quadratic formula every time to solve every quadratic equation. That way he didn’t have to think about strategies, keeping different methods in mind, or start then fail and have to start over again. This worked for him. But, knowing this about him, I questioned his opinion that the factoring could be a game, after all, it doesn’t sound fun to me to solve equation after equation with the quadratic formula!

The point, then, would be to know in advance that there are integer solutions, making the quadratic formula unnecessary.
This seems perfectly legitimate to him. He said, “Mom, in real life you would only ever have one problem that would need the quadratic formula. You’d never have a list of them.” Oh. Yeah, uh, that’s right.

My daughter was adamant about not using graphs to solve quadratic equations. It took some time teasing out what the problem was that she had with graphs. Turns out that she was reacting to her own experience of having to create a graph over and over again for each homework problem, having to label the points, having to precisely position the curves, in other words graphing was cumbersome to the point of being obnoxious. And what was even more obnoxious to her was if shewas given a an expression like say, x^2 +3x and she factored into x(x+3) she would get points marked off for not writing it as (x+0)(x+3). Her teacher once remarked that it was surprising that my daughter didn’t like graphing since it included drawing, and she knew my daughter likes to draw. I’ll let you imagine my daughter’s response.

The fact was that my daughter was fine with factoring, but she hated how cumbersome her teachers made it. She was only taught the long way of doing things. Once when we were going over her homework together she saw me do a typical shortcut, which was to first lay out the parentheses and fill in with x , then put in the appropriate signs so my work looked like this (x + )(x – )then, after some thought I filled in the integers. She was aghast. And angry. She said that she had continually asked her teacher, isn’t there a simpler way of solving quadratics, but her teacher insisted that, no, always you first you had to make a diamond shape, fill in the spaces, then make the square, then fill that in (I am going to assume you know the techniques I am referring to), and there were no shortcuts. She was furious that she had not been introduced to any simpler methods of thinking about her factoring.

So that’s my last bit of insight: don’t muddy up factoring with too many auxiliary steps. Sure, teach all the methods, but then let quadratic factoring be the fun puzzle that they are.

I’ll answer your question with some questions of my own: why should we learn to factor trinomials (or more general polynomials), period? Perhaps if we have a master plan for this set of skills, we can place the issue of “integer roots” into its proper context. For that matter, why should we learn to factor *numbers?* Surely the issue of factoring polynomials is related to the issue of factoring numbers.

In the CCSSM, the issue of “turning one expression into another,” as you have done in your example, is motivated by the desire to “reveal the zeros of the function.” So perhaps we should be discussing this: why do we care about the zeros of a function? The issue of integer roots, in my opinion, is just a matter of “keeping things simple.” i.e. we do this for the same reason that, when teaching students to add fractions, we are more likely to give 2/5 + 3/7 than 385/486 + 999/1002.

One lens through which to view the skill of factoring is “puzzle-solving.” Consider this sequence of learning experiences.

1. Students learn to multiply multi-digit expressions, e.g. 25 x 31.

2. Students pose challenges to each other to perform “multiplication in reverse.” For instance: “I have multiplied two 2-digit numbers together, producing 3 hundreds, 34 tens, and 63 ones. Find the original two numbers.”

3. Students learn to multiply multi-term variable expressions, e.g. (2x+5)*(3x+1).

4. Students pose challenges to each other to perform “polynomial multiplication in reverse.” For instance: “I have multiplied two binomial expressions together, producing 3 groups of x-squared, 34 groups of x, and 63 groups of 1. Find the original two binomials.”

I’ll list pros and cons to this approach, as I see them. Pro: ties between “number arithmetic” and “algebra” are clear in this approach — one is merely an extension of the other. Pro: ties between “multiplying” and “factoring” are clear — one is merely the reverse of the other. Pro: working in reverse serves to strengthen math skills, and also involves elements of reasoning. Con: why would I want to work in reverse in the first place?

I’ll repeat what I think are the main questions to resolve here:

1. How does learning to factor numbers make me a more powerful problem-solver?

2. How does learning to factor polynomials make me a more powerful problem-solver?

3. What are the connections between the two questions above?

4. Why is it important to “reveal the zeros of a function?” What do we gain from doing this?