**The Task**

**What I Did**

**Change the context.**I’m generally pessimistic about the effect of grafting different real-world contexts onto a task that is rotten at its core. But we have to correct pseudocontext when we see it or students will come to believe that this math thing is a lie. This isn’t the only change we’ll make, though.**Put students in the shoes of the person who might actually experience this problem.**It’s striking to me that the question “Which company should you choose?” only emerges at the*end*of the problem when that’s probably the*first*thing someone would wonder when presented with two competing plans.**Lower the floor on the task.**The task starts at a very abstract level with the construction of linear equations and then proceeds*down the ladder of abstraction*to a very concrete level by asking students to evaluate their plans for 20 hours of Internet use each month. It’s like asking someone to lift*less*weight the more they exercise. We need to turn that around.**Raise the ceiling on the task.**The task quits too early. We can develop the concept further.**Provoke an intellectual need for the solution.**The finale of the task asks students which plan they’d choose if they used the Internet for 20 hours each month, a question that requires none of the work preceding it. Seriously, you just evaluate both plans for x = 20 and you’re done. We need to provoke some kind of*need*for creating and graphing a system of equations.

When they come into class ask them to write down any number between 1 and 25. Then show them this flyer (courtesy Frank Noschese):

Tell them, “If you were going to workout for that many months, and all you cared about was cost, which plan should you pick?”

Now they’re doing the concrete step *first*, the easier evaluation, and we’re setting ourselves up to *need* a generalization.

As they finish, ask them to come up and write their answer on a number line above their number of months. If students finish quickly, ask them to double check a few of their classmates. Assessment should be fairly straightforward here, after which the board will look something like this:

The clumping of answers will be expected for some students but surprising to others. “It seems like there’s a point where the plans switch over.” Finding that point will make linear equations seem like more of a necessity than they do in the original task.

Graphing the equations is the least essential aspect of this task. (What purpose does it serve?) Those graphs become more *interesting*, however, once we’ve located the switch-over point.

I’ll try to position my students as hired experts for some consumer who needs their expertise. So when they write down “x = 12.3” and circle it, I can say, “Come on, man. They’re never going to understand that. You have to spell it *out* for them. And tell them why plan A will *never* be the better deal, also, or they’ll get confused.”

We can *extend* the task by asking students to come up with two plans that switch over only after two years. Now they’re exercising a little more creativity and working their algorithms in reverse.

**What You Did**

Over on the blogs:

**Beth Ferguson**swaps the context over to smartphones.**Glenn Waddell**makes the task over using a questioning technique called “What-If-Not.”**Phil Aldridge**personalizes the context of the problem.**Jonathan Claydon**lifts 3D coordinates off the printed page and into his classroom space.

Twitter is just the wrong medium for this kind of writing prompt, I’m finding. With only 140 characters, a lot of people default to “less helpful” stances like, “Well I wouldn’t give them anything except the subject of *internet cafes* and then see where they took it!” But the “unhelpfulness” I’m diagramming here is kind of a lot of work and takes more than 140 characters.

That said, I think Nicholas Chan managed to fit something actionable and important into the tight space:

@ddmeyer Have students first guess which cafe is cheaper for some hours (1, then 10, then 100?)? Change context to 2 different paying jobs?

— NC (@SergtPeppa) July 8, 2013

**Call for Submissions**

You should play along. I post Monday’s task on Twitter the previous Thursday and collect your thoughts. (Follow me on Twitter.)

If you have a textbook task you’d like us to consider, you can feel free to e-mail it. Include the name of the textbook it came from. Or, if you have a blog, post your own makeover and send me a link. I’ll feature it in my own weekly installment. I’m at dan@mrmeyer.com.

**2013 Jul 9**. Eric Biederbeck reminds me of Kyle Pearce’s Detention Buy Out lesson, which is a perfectly functional makeover in its own right.

**Featured Comments**

I think these comments illustrate what’s fun, useful, and difficult about modeling. I respond below.

The equation for plan A doesnâ€™t take into account the possibility of working out â€œMore than 24â€³ days in a month

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.

Note the fine print which says you canâ€™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.

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).

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, itâ€™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 thatâ€™s meaningful. Maybe thatâ€™s the next step for students?

## 18 Comments

## Matt H.

July 8, 2013 - 10:30 pm -The equation for plan A doesn’t take into account the possibility of working out “More than 24” days in a month (nevermind the possible implications of those quotation marks…). Even taking into account the fine print, it seems like you could only pay $31 down and $29/year if you can commit to that workout schedule. Admittedly, few people would, and a discussion about the gym’s pricing strategy could be an interesting addition to the task.

## Jonathan Claydon

July 8, 2013 - 10:54 pm -Is it just me or is the scenario presented in that word problem a textbook’s entire justification for systems of equations. I feel like I did this exact problem earlier in the day while tutoring.

I enjoy playing this game even if I’ve only done the assigned problem once.

## Stephanie Reilly

July 9, 2013 - 3:39 am -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.

If you pay $199 for C and go for 3 months and decide you don’t like it, plan C cost you $199 (I’m sure there is no prorating or money back!) but plan B would have only cost you $103 (70+33). Plan C doesn’t look like a good deal for any number of months. Your yearly cost is nearly the same ($199 for C and $200.88 for B) but you have lots more flexibility in B.

Am I looking at this wrong? Is my lifetime of failed fitness attempts jading my outlook on these plans? ;)

I think plan A if you commit to the consistent working out is by far the best deal. $1 down and then $20*6 months, $121 for year. Could figure out how many months you could go <24 times and still be better off with plan A.

I love the flyer. Leave it to @fnoschese! This is a great task with lots of interesting questions. Thx.

## Jason Dyer

July 9, 2013 - 4:43 am -I’m agreeing with the comments above. The 24-day condition messes things up (in an interesting way, at least).

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.

## Gretel Johnson

July 9, 2013 - 5:26 am -You know, when you say that A is “never” the better deal, I feel like you’re missing something that makes math in the real world a little bit more difficult. Plan C is likely not an installment plan. Meaning just that a more accurate way to graph it might be with a y-intercept of $199 and a slope of zero for 12 months.

I mention this only because I’ve done a lot of lessons with my students on the upfront “opportunity costs” of different deals. Companies that reimburse expenses are different than companies that pay the expenses for their employees up front. People do choose plans like Plan A because on the first day, the out of pocket cost to them is only $21 instead of $79.99 or $199 which they might not be able to scrape together at that moment.

## l hodge

July 9, 2013 - 6:03 am -I really like the number line idea. I like the fitness problem as well, but it serves a different purpose than the internet exercise. 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).

If you want an unambiguous, heavily guided systems â€œexerciseâ€ or â€œpracticeâ€, maybe the internet problem is borderline ok. If you want a less structured exercise, donâ€™t break it into parts. Yes, the context is dubious and it probably makes sense to change the order to d), a), b) , c).

If you want a more authentic financial â€œbreakevenâ€ problem, like the fitness problem, there will probably be messy details that are harder to quantify and time consuming to think about. Some other possibilities that come to mind that are messy but more authenticare the following:

â€¢ Rent a car at the airport or take a shuttle into the city to get a better rate? Messy details: how much is time and convenience worth to you, how many people are you travelling with, how many days will you rent the car, will the cheapo agency at the airport be open if you arrive at 1:00 am (like me & no it wonâ€™t)?

â€¢ T-mobile no contract vs. Verizon share everything contract. T-mobile has lower monthly fee and no contract, but Verizon has a better network (in some cases) and big discounts on phones ($400 savings on an iphone 5 with two year contract). Messy details are how much is the better network worth to you, how many devices on the plan, what data plan do you want, what kind of phone do you want, how long will you keep your phone?

â€¢ Whether to pre-pay for gas on a rental car. Some real world details to deal with: will you drive the car enough to use a tank of gas, how close to empty can you reasonably expect to return the car, and actual gas prices?

â€¢ Should I take the low interest rate or the cash back deal when buying a car? Edmonds.com has a calculator for this â€“ how does it work?

I might not use a system or a graph or even an equation for some of these problems. So, they might not be so good if I am desperately focused on showing that systems of equations are of vital importance to our financial well being.

## Tracy

July 9, 2013 - 6:21 am -I found your twitter comment interesting. You’re doing something much more meaty than 140 characters will allow, which is why I haven’t chimed in on twitter. Any chance you can move it to a different platform?

## Timfc

July 9, 2013 - 10:49 am -I dunno if the comments on this task are really the place to broach this, but it’s already implicitly part of the conversation due to the thinking about plan C for the gym memberships…

But, maybe adding some sort of caveat for teachers and stuents that 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?

Maybe ask:

a) What are three or four values that you can substitute into the cost function that don’t give an output that matches what you’d actually pay?

b) What characterizes those points?

c) Why does that make sense?

d) What does it tell you about the cost functions that we’ve written? [I don’t like this last phrasing at all]

## Beth

July 9, 2013 - 12:09 pm -I appreciate having students select a random number to begin with to start the work more concretely.

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.

## Isaac D

July 9, 2013 - 12:54 pm -I strongly agree with Timfc’s comment above.

I’ve been struggling with how to explain to students that they will never encounter a “real-world” situation that is exactly equivalent to an equation, and that the equations are just approximations that can be used to roughly model a limited number of cases without making linear algebra sound like a pointless exercise in abstraction.

## William Carey

July 10, 2013 - 5:21 am -> So when they write down “x = 12.3â€³ and circle it, I can say, “Come on, man. They’re never going to understand that. You have to spell it out for them.

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.

## William Carey

July 10, 2013 - 5:24 am -Hmm – maybe a better way of saying what I was trying to say is that starting with the concrete and moving to the abstract is vital in mathematical modeling, and moving from an answer in the abstract to an answer that speaks to some actual concrete audience is a vital part of mathematical modeling too.

## Bill Funkhouser

July 10, 2013 - 11:09 am -One thing I rarely see in teaching the complexity of systems is emphasizing the simple question that is revealed half way through solving algebraically.

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.

## Dan Meyer

July 11, 2013 - 6:24 am -I promoted these lines from

Matt H., Stephanie Reilly, Jason Dyer, L Hodge, and Timfcup the main post because I think they illustrate what’s fun, useful, and difficult about modeling:Matt H:

Stephanie Reilly:

Jason Dyer:

l hodge:

Timfc:

Out of all of these, I think

Stephanie’sis the most devastating to my case. Elsewhere, I tried to frame the issue solely in terms ofprice, which was an attempt to mollify some of the unquantifiable perks (eg. “you can useeveryclub.”).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’sobservation that there’sanotherhidden 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.## Dina

July 12, 2013 - 9:33 pm -This is great. I just watched your math makeover video. I think you are so right. I do believe that CCSS will move to more patient problem solving as student won’t be looking to bubble an answer anymore. How do you think this could apply to Elementary math?

## Dan Meyer

July 13, 2013 - 10:24 am -It’s a good question,

Dina. I’m afraid it’s outside my expertise. Ben Rimes has been tackling the issue, though. Here’s his introductory post.## Kyle Pearce

July 14, 2013 - 9:15 pm -Great question as always, Dan. Just reading the comments will prove that questioning along these same lines will improve the discussion and debates in your classroom. Being comfortable with not always knowing the “answer” allows for your students to feel like their opinions count.

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