My main purpose for commenting is that I wanted to build off two points made above. First, @Dan’s point above asserting the importance of coherence over introducing a new factoring technique each day, and second, @Mark’s comment above on using a visual puzzle to teach factoring.

This year, I am teaching factoring as a visual/kinesthetic skill in which we use algebra tiles to re-arrange a polynomial into a rectangle. I use language like “Find a rectangular arrangement of x^2 + 7x + 5” and “Can you find all the rectangular arrangements of x^2 + ___x + 6”. I used Algebra tiles, and asked students to find what percent of the polynomials x^2 + ___x + ____ would make rectangles. We agreed to only allow the digits 0 – 9 to occupy the ___. This was a different way for me to approach this topic this year and I was surprised the other day when I realized that the rectangle analogy can extend and motivate the method of solving a polynomial equation by collecting the terms of the polynomial and using the zero-product property of multiplication. I think the biggest payoffs here are in terms of coherence. Students can think about factoring as rectangles, and they can easily see when the rectangle is improperly constructed.

When we factor x^2 + 5x + 4 = 0 to (x + 1)(x + 4) = 0, students can visualize a rectangle with sides of length x + 1 and x + 4. Since multiplying the sides gives the area, the equation (x + 1)(x + 4) = 0 means that our rectangle has zero area, and students can determine for themselves that means that one of the sides must have a length of zero, and that choosing x = -1 or x = -4 will make our rectangle have an area of zero to satisfy the equation.

This approach is simply trying to connect factoring to the geometric understanding that students have of rectangles. This is useful for relating polynomials which can be factored to composite numbers which can be made into rectangles, and prime numbers and prime polynomials which have no “interesting” rectangular arrangements.

I suppose I should not still be surprised that when I search for factoring polynomials on youtube I find only explanations that move directly into symbolic manipulation, factoring algorithms and provide almost no mention of a visual approach to factoring which might give students a chance to figure it out for themselves.

Here is a short animation I made to try and communicate the meaning of factoring. Potentially relevant to the discussion above

https://www.youtube.com/watch?v=nbUxSK6XTCc

Update: searching for “factoring algebra tiles” is where you can find the attempts to explain factoring visually. Not recommended for students — most videos are 15+ minutes and even the 2 minute videos make 2 minutes feel like more than 2 minutes and put us back at the beginning of this conversation — how do you generate the curiosity in students that would lead them to be interested in watching one of these videos to learn factoring? Also, I’m allowed to criticize these videos because I made one myself. For shame!

Some students who had previously had Algebra recognized this as factoring, but I told them not to announce it. Please note that this area concept helped them “see” factoring. Further, there is the relationship to the quadratic. We proceeded from this to the more traditional demonstrations of finding zeroes, etc.

I came to this because most students self-report as visual learners and I wanted to take this into account.

The Game

– The “grid” is divided into 4 sections. You have blue, green, red and yellow chips.

– You will take a specified number of chips of each color.

– You can place the colored chips only in the section with that designated color. For the section marked “green or red” you can’t place red and green; only red or green.

– The chips must be placed such that the edges of any configuration of chips in any section must make contact with both the horizontal and vertical axes (see example below).

– When all the chips have been placed in their designated section, the configuration of all the chips combined must be a rectangle or a square. (see example below).

For example, take 6 blue, 7 green and 2 yellow chips and build the rectangle & you get:

B B B G G
B B B G G
G G G Y Y

– After you have made your rectangle with the designated number of chips, please record the set used and a picture of your rectangle.

Use the following “sets” of chips to make your rectangles.

Set↓ blue green red yellow
1 2 1 6 3
2 6 7 ─ 1
3 10 5 6 3
4 6 7 ─ 2
5 12 ─ 11 2
6 4 6 6 9
7 4 4 3 3
8 16 8 6 3
9 6 4 ─ 4

Is the goal to create intellectual need so this day’s topic goes well, or so that the student creates connections between this topic and others and the learning persists?

I don’t think these objectives are exclusive, but it makes some difference in how we talk about different approaches.

“If a chessboard were to have wheat placed upon each square such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of wheat would be on the chessboard at the finish?”.

In doing so could we create a task with a clear goal stated, with a low entry and a high exit, a task that is iterative with timely feedback?

I am not sure if this strategy would work, maybe the need generated by this series of tasks is not strong enough to create interest in the students. However, at this stage my concern is not whether this task is a good one for exponentials but rather the far more general question of the applicability of the “theory of need” as guiding principle for our teaching. I have no doubt that Dan will have much success in using the factoring trinomial strategy he outlines. But is that due to the fact that Dan is Dan and he has characteristics that some others of us don’t? Could it be due to the fact that this lesson is so different to other mathematics lessons that the class has experienced? What would happen if every class used this strategy, would the novelty wear off and the students lose interest?

a) I am not against using the headache-aspirin structure, but I am against having this as a guiding principle. Why force yourself into that box when thinking about a lesson?

b) The finding a zero is not a very good lesson. The need from the student’s perspective seems weak to me, but maybe there is a clever way to highlight that need a little better. The devil is in the details with any lesson, not just the puzzle approach.

c) Short initiation is needed, I think, if the goal is factoring via find the zero. A stretched out initiation might make for a more interesting lesson unrelated to factoring.

Another thought riffing off your original initiation might be to ask how many got an “even” value and how many got an “odd” value. Everyone should have “even” which gives you a heads up regarding arithmetic issues. Have them try another number, ask for opinions/reasons on whether the expression can be made odd, etc. Factor, (x-9)(x+2) and continue the discussion. Prime the pump, if you wish, with questions like: is 16 x 91 even or odd. To be clear, I am not thinking of “is it always even” as a headache in need of the factoring aspirin — just kind of a puzzling situation that might catch a little interest.

Unfortunately at my school we tend to teach factorisation and finding the zero in distinct units as only the top level students have exposure to graphing quadratics so finding the zero loses a lot of its meaning to the rest of the cohort.

As for exponent rules I treat them in a similar way to Tom in the comment above. By showing the students the reason why we have a multiplication sign again (I don’t know why but at some stage they do disassociate it with successive additions) you can have the discussion about what we should do if there are successive multiplications. In my experience, students tend to think of squared as it’s own distinct thing and it doesn’t register straight away that it could be ^3 or ^8 etc. So by giving them the problem of coming up with their own symbol can lead to an interesting and often fun discussion with the class about why they chose their sign.

“Hidden” multiplication signs are the bane of my classroom. By writing everything out students can get a better understanding on the exponent rules. But maybe “exponent rules” are the wrong type of aspirin? Aren’t we trying to move away from rules that we can rote memorise and come up with methods of solving the “headache”? Better yet, how to avoid the headache in the first place? I look forward to seeing your idea on how to create this headache and the discussion that follows.

Lastly, another language issue. Having power, exponent and indice all mean the same thing is another barrier to this topic. Can’t we all just choose one and banish the rest?