(Continuing with the perimeter of 60 and units omitted.)

As I have returned to this question many times over the years, it seems to me remarkable how the way of denoting the sides is presented as if it is “canonical” — that is, call one side x and the other 30-x [because it is the Right Thing To Do].

Now you have a function: Area(x) = x(30-x) = 30x – x^2.

Since x > 0, this is a downward facing parabola; i.e., its maximum occurs at the vertex.

Calculus? d/dx 30x – x^2 = 30 – 2x, which is 0 when x=15.

Algebra II? Vertex is at x = -b/2a = -30/2(-1) = 15.

Etc.

The suggestion from earlier is *not* to pick the same way of labeling sides always used, i.e., *not* to pick x and 30-x. If you want, you can think of the suggested approach as a substitution: let n=15-x, which means x=15-n.

Now adjacent sides are 15-n and 15+n.

So, this time, Area(n) = 225 – n^2.

Only “sense-making” (if forced to classify by topic, I would call this Real Analysis) is needed to realize that the area is now maximized when n=0.

Thus, adjacent sides are 15-0 and 15+0, i.e., 15×15 square.

From the standpoint of integrated algebra and geometry, this “substitution” is shifting the parabolic function that represents the area by 15, so that it is now symmetric about the origin.

I’m not sure why the problem’s history involves this marriage between x and 30-x; I find parabolas generally *much* easier to deal with when they are symmetric about the origin, i.e., when the coefficient of x is 0.

In fact, this idea of shifting — which is a frequent if confusing topic at the (graphical) intersection of algebra and geometry — is how I consider questions about quadratics more generally, e.g., how to find the x coordinate of the vertex and how to uncover the quadratic equation; cf. http://matheducators.stackexchange.com/a/9709

]]>It explains what a photon is, and then it shows ball bearings and light bouncing off a flat surface.

]]>Since n^2 is non-negative, subtracting it off will either lower the total area (uh oh) or leave it alone (iff n=0). The latter case means we oughtn’t vary the 15 at all. That is, the square yields maximal area.

OMG. All of the light bulbs in my head.

]]>**l hodge**:

Which of the questions, if any, require reasoning? Why not a question or two providing the destination and requesting the initial path?

The contrasting cases require an attention to structure and the ability to generate and defend a hypothesis. The initial exercise requires spatial intuition. The final exercise requires attention to precision. Your question is nice, though.

**Chester**:

After teaching about pool tables, I’m still going to have to cover the material more conventionally anyway, except I now have less time to do it in. I don’t see that helps.

I don’t mind you repeating yourself in every post, **Chester**. But clearly we come at the project of math education from very different angles. I tend to chalk your frustration with individual activities, lessons, and posts to *those* differences, not to the activities, lessons, and posts themselves.

That said, I *am* curious which material you’ll have to cover “more conventionally” anyway. The “material” I’ve focused on here is a particular understanding about angles and their application to pool. That material is “covered” in this activity already. It’s just sandwiched in between a couple of other productive and interesting activities.

One could make the argument that those extra activities are an unconscionable expense of time and I will shrug in response and move along. How a teacher spends her finite classroom time isn’t really my business.

One can’t argue that the material isn’t covered, though.

Just because I don’t subscribe to a program of straight explicit instruction, no chaser, *doesn’t* mean I subscribe to discovery learning, the usual boogeyman in your comments.

My question isn’t “should or shouldn’t we explain?” Rather, “What can we do *before* we explain, both to interest students in that explanation and prepare them to learn from it?”

OH YISSSS. Such a huge improvement on the originals!

I love this partly because the fake ones LOOK fake, and students have to think about why and are given materials to test their hypothesis. You’re making students refine their intuition to include mathematical precision, which they can then use to solve the rest. I feel like this honors and builds on the knowledge they already have in a way that’s far more motivational than throwing out some big-words statement about angles of incidence and reflection.

Actually, that last statement applies to the whole lesson.

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