I have seen a tremendous improvement in my classroom since applying your approach. Thanks as always!

Matt H:

The equation for plan A doesn’t take into account the possibility of working out “More than 24″ days in a month

Stephanie Reilly:

Plan C is a one-time charge of $199 and you get to go for 12 months, right? I think I would graph that as a straight line at y=199 (for 0-12 months) and then a straight line at $398 for 13-24 months.

Jason Dyer:

Note the fine print which says you can’t sign up for less than 12 months on plan A. All three plans also seem to have a $29 maintenance fee per-year and a $10 card fee.

l hodge:

The fitness decision depends a lot on how much value you place on the different add-ons for each plan. These details are harder to quantify and not incorporated into the graphical model (the model stinks).

Timfc:

we only graph them as nice continuous linear equations because it makes our lives easier, but we should really graph them as piecewise linear?

I mean, it’s not like you can pay for pi months (although how awesome would that be?) and so evaluating the equation at pi is not going to give an output that’s meaningful. Maybe that’s the next step for students?

Out of all of these, I think Stephanie’s is the most devastating to my case. Elsewhere, I tried to frame the issue solely in terms of price, which was an attempt to mollify some of the unquantifiable perks (eg. “you can use every club.”).

That mollification might’ve resulted in a lousy model, but that’s the point of modeling. You simply the world down to math. You work with the math. And then you recontextualize the math to the world. And you ask yourself, “Did we do damage?”

PS. I like Jason’s observation that there’s another hidden flat fee for every plan. (A fee that recurs yearly too, but never mind that for a second.) It’d be fun to drop that fine print on students who’d finished the original problem, let them get pissed and struggle for a second, and then find out it doesn’t change any of the outcomes.

To make my point, I’ll use the textbook values of $2.25 per hour and 2.95 per hour. Perhaps if was asked, “How would the situation be different if the prices were $9.50 and $10.20 per hour?” or “15.21 and 15.91”? The answer is, of course, they are no different because the differences in the rates are the same.

We want students to have “number sense” but don’t take the questioning to the common sense level of, “How much are you saving each month with one plan over another?”

You probably see my point, that the complexity of y1=2.25x + 9.95 and y2=2.95x really boils down to the savings equation of 0.7x=9.95 which is never brought to light in the textbook or your examples. All the complexity of solving systems and graphing multiple equations really boils down to the simpler question of “how much are you saving each month with one plan over the other?”

Perhaps instead of asking students to only justify the property used for each step of solving but also “what does this mean in terms of the situation?” would deepen understanding.

This might be a lot of concern over a small point, but it is a symptom of math education that seems to prevail: believing complex problems are difficult to solve rather than complex problems often disguise much simpler situations.

This is vital, and something we math teachers don’t often think about. Mathematical modeling is an exercise in rhetoric; the goal is to persuade someone (often yourself!) of some conclusion. The arguments we use are different from, say, an argument in history, but they’re arguments nonetheless. It’s vital to give students a real audience that they’re communicating to, and especially vital that the audience not always be “their math teacher”.

For this made-over problem, I’d make the audience their parents (or some other grown-ups), who a) control the money decisions and b) have not done school math in a long, long time. How are you going to persuade them to get the best plan? Saying “12.3” all by itself won’t work.

Then adding the number line provides a great visual. I had not thought of applying that to a problem like this. Will definitely add that strategy to my teaching plan! I can envision 2 – 3 colors of sticky notes … choose the color based on the “plan” you select.