**The Task**

This task comes from MathWorks, which, as I understand it, is intended for Canada’s vocational track math students. I purchased PDFs of the curriculum in Saskatchewan because it featured a lot of interesting applications of secondary math, even if the print medium did those applications no favors.

**What I Did**

**Not a lot.**The last makeover took it out of me and it’s summer. Let’s do something a little simpler.**Put students in the shoes of the person who might actually experience this problem.**Perhaps that person is a homeowner. The homeowner either doesn’t have a carpet or has a carpet in need of replacement. She knows only one thing at this point: “I want carpet.” She*wonders*several things at this point: “How much will it cost me?” and “How much time will it take me?” and “How will I do it?” are probably high on the list. What she*doesn’t*have yet are all these facts, figures, and dimensions the problem includes.**Add intuition.**Our homeowner might try to ballpark the cost of the installation before she does anything else. Let’s ask students to do that.**Raise the ceiling on the task.**We need to extend this task at the top end for students who need the challenge.

Let me run this by you.

Shoot some quick video of a room in a house that has a similar design —Â composite rectangles. If it’s emptied of furniture so much the better. (Anybody moving this summer? Get at me.) Tell students, “We need new carpet in this room. Can you give me a guess how much it’ll cost?” Some of them won’t have a clue, but we’d like them to take their intuition as far as it’ll go, even if that’s just to say, “It’s definitely going to cost less than $10,000.”

Then ask them to brainstorm in groups: “What information will be important here? What skills will you need?” Because that’s the question our homeowner is likely asking herself and we’re trying to put our students in her shoes. (Also because the first task in “modeling” according to the Common Core is to “identify essential variables.”)

I have no trouble imagining the student response here because my own knowledge base for home handiwork is pretty much comparable.

- What kind of carpet are we buying?
- How much does it cost?
- How much does it cost to install?
- How do you
*get*carpet? - Are there any other costs we’re forgetting?

I’m sure I’d be (pleasantly) surprised by what students ask for. At this point, offer them information they want. Teach them about carpet installation. Show a YouTube video. (Or have them research all of the above online, though I’m not inclined to sacrifice the time myself.) Basically give them *the same* information given in the task, only after they’ve had a minute to think about *why* they’d need it and *how* they’d get it.

I’d probably pass out a floor plan of the room without dimensions. An interesting observation the original task glides past is that you don’t *have* to measure every single side of the room. You can measure *some* and use them to find the others. So ask them what sides they’d *want* or what’s the fewest sides they’d *need*?

As students work, some will need more of your help and others will finish quickly. My first attempt at an extension problem for the latter group is to *switch the known and unknowns of the original problem*. So previously we gave students dimensions and we asked for cost. Now give them cost and ask for dimensions.

“Tell me about a scenario where the total bill for the carpeting job was $1,000,000.” They can change anything they want.

**What You Did**

Over on the blogs:

- Algebra’s Friend clues in on the mismatching units.
- Evan Weinberg asks his students to make a controversial choice between two options, which is a useful way to begin a problem.
- Jonathan Claydon has some fun with the shape of the room.
- Bob Lochel takes on the issue of domain-specific knowledge by having his students model the act of carpeting a room.
- Mark Leadbetter is comfortable with the mixed units but also tackles the domain-specific knowledge.
- Chris Robinson submits a problem for #MakeoverMonday that may be beyond repair. This made me realize I’m looking for “tragic” problems rather than “failed” problems. Which is to say, problems whose greatness has been overwhelmed by its inner demons, problems with wasted potential.

Over on Twitter:

- Stuart Price throws in the towel.
- Haiwen Chu links up a fascinating anthropological study on the mathematics carpet layers actually
*use*. - Chris Adams and I may be on the same page, but Twitter’s brevity has me unsure what “let them figure it out” means in practice.

**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 2**. Jennifer Orr sends in two pictures we can all use.

## 10 Comments

## suehellman

July 1, 2013 - 11:37 am -My suggestion: In Canada many trades people receive contractors’ deep discounts on materials. I might find a local carpet layer and ask for job bids on say 3 types of carpet, and then ask the students to work out the DIY costs (after a trip to the local carpet warehouse store where they can compare types of carpet). The problem could be to decide if the DIY route is worthwhile, and that could lead to conversations about other ‘costs’ that might be factored into the DIY price or about how trades people figure out job bids.

## Andrew Busch

July 1, 2013 - 11:45 am -I like the idea of taking a video of a room needing new carpeting and asking the students what it would cost. This allows students to pick out the important details needed to solve the problem.

I’d probably already have the carpeting picked out. That decision can take weeks for people actually going through the process. So, with an eye to the clock, that’s a decision I would make for them. When they ask how much the carpeting costs, I would show them a picture and say the homeowner decided on this, which is $_.__ per square foot.

We could add an extra step of complexity by comparing modular carpeting sold by the square (19.7″ by 19.7″) rather than the roll.

I think students will be surprised at how much carpeting a room can cost (close tiling it with dollar bills).

Getting this done in one 45 minute class period–priceless.

## l hodge

July 1, 2013 - 12:23 pm -The most intriguing part of this to me is that you will have to buy more carpet than the area of the room (because carpet only comes in 12 foot width). How much more? Somewhere I read that the rule of thumb is to buy 10% more than the area of room. This is my attempt. Minimizing the number of cuts might be interesting for larger rooms that would require several sections of carpet.

## suehellman

July 1, 2013 - 1:06 pm -(1) Re vocabulary: ‘bolt’ can easily be switched to ‘roll’ but ‘nap’ is important so the carpet layer doesn’t rotate left over pieces to fit them in. Bringing a couple of samples in would make this point. Pattern match is part if the reason for adding 10%.

(2) Re units: Canadian trades people have to be able to transpose between metric and imperial. There’s some realism to someone measuring their room in metres and then finding to their surprise that carpet is sold by the sq. yd. Lesson to learn? Go home and measure again or know that a meter is 3″ longer than a yd.

(3) Re complexity: To me those are authentic complications of unravelling the problem, but has anyone mentioned that the sketch badly misrepresents the room’s proportions?

I think this might be an exercise in giving students practise in sorting through the information to reconstruct the problem for themselves into manageable chunks. They might not even have to solve it, just find and deal with all the red herrings.

## Patti

July 1, 2013 - 4:23 pm -As I am currently in the process of getting new carpet for a house that desperately needs it, I’ll tell you what some of my students answered when I posed a similar problem to them using my own downstairs:

“Don’t the Lowe’s* guys take care of all the math for you?”

And, they’d be right. The Lowe’s guys WILL take care of all the math for you. Of course, that takes the fun out of it.

You’ve inspired me to try to get pictures or video of my house once we get the furniture moved. We have tons of stuff, so there may be a lot of shifting while the installers are here. If I get anything usable, I’ll share it. I’m guessing it will be too much of a mad scramble for me to do a decent job, but we’ll see.

*Lowe’s is a large home improvement chain, for those who don’t have them.

## Roger Gemberling

July 1, 2013 - 8:21 pm -Carpet is sold in a variety of widths. Twelve feet is the most common width. Fifteen feet is next most popular width. Finally 13 feet 6 inches in the third most common width sold.

A carpet company in Georgia (Georgia Carpet Industries) has a variety of widths available. They are 6 feet, 8 feet, 12 feet, 13 feet 2 inches, 13 feet 6 inches, 15 feet, and 15 feet 4 inches.

My students have completed a similar problem. I would provide carpet with widths of 12 feet and 15 feet.

## Scott

July 1, 2013 - 8:23 pm -Don’t the guys from Lowes…..

I know of two math teachers and a very astute science teacher (Jeff) who caught mistakes by these people who *will do* the math for you…

Even the sneaky kids (sorry, maybe I know my kids better than you) perk up when it comes to money….

I want to call 3 contractors to do something myself tomorrow, now…

:).

## Sarah Miller

July 2, 2013 - 8:03 am -I really like Dan’s addition of watching ‘how to lay carpet’ videos. I know something like that would do a lot for my kids and would be worth the time. It builds on my relationship with the class when we can talk about non-math stuff, even something “boring” like laying carpet. (well, non-math to them, anyway). Plus, I like the message of “You could take on a big project like this if you wanted to. There are free resources (YouTube) to help you figure out how to do it. And you can figure it out”

## Dan Meyer

July 2, 2013 - 8:43 am -Sue Hellman:I’m not convinced that practice has a lot of application outside of math textbooks and math tests. In life, we tend to start with a goal state (eg. the room gets fully carpeted) and then we work towards that state by deciding what information is relevant. It’s unusual to start with the goal state and a pile of information someone has given you, some of which may be relevant, some of which may not be.

PattiandScottare onto something useful here. What do you do when students realize other people will do the math for you? One option is to make those people incompetent or liars.“Right. The carpet guys came by and said they’ll do it for ten thousand dollars. They insist it won’t cost any less than that. Can you prove they’re liars?”