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.

Matt H:

The equation for plan A doesn’t take into account the possibility of working out “More than 24”³ 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?

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 — 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.

The Task

This task comes from the Mathematics Vision Project, a free curriculum from some sharp Utah educators. This is the first task in their “Secondary One” curriculum, a variation on the classic Pool Border task.

What I Did

• Reduce the literacy demand. There’s a lot of text here boxing out English Language Learners and obscuring the point of the task. Much of that text exists only to service the story surrounding the problem — this school administration replacing tile in the cafeteria. I’ll argue later this story buys us very little while the text it requires costs us quite a lot.
• Start the problem with a concise, concrete question. Let’s not bury the point of the problem. Let’s tell students what we’re doing here as soon as possible, as quickly as possible, with as little jargon and in as few words and syllables as possible. Keep it to a tweet.
• Add intuition. Let’s make that initial question something students can guess at. This is my go-to. I’d be surprised if it didn’t feature in every single makeover this summer. It’s the easiest, cheapest move I can make to get students to commit to a task.
• Motivate the move to generalization. Why should we move from counting by hand in part A (easy!) to generalizing to a formula in part B (hard!)? The problem just asserts that the contractor wants to. I’d rather put the student in a position to say to herself, “My word. I’d rather eat chalk than count up all these tiles. Does anybody have a faster way?”
• Scaffold the move to generalization. The task moves straight from “count the number of tiles in a single small instance of the pattern” to “generalize the pattern” and I’m not confident the pattern will be clear to the student. Bree Murray thinks the domain of the pattern is only even-numbered squares, for instance. She may be right but I interpret the problem to include all whole numbers. This is not the good kind of ambiguity.
• Open the problem up to more than one possible generalization. I can see at least two ways to generalize the pattern. One defines itself in terms of the large boundary square and the other defines itself in terms of the small inset square. Defining the pattern by the large boundary square felt most natural to me. The task authors force the student to define the pattern by the small inset square. We’re losing mathematical richness and student creativity there and, aside from easier grading, I’m not sure what we’re picking up on the other side of the trade.
• Raise the ceiling on this task. In addition to lowering the floor with all the guessing and scaffolds we’ve already added, if a student figures out how to generalize quickly, can we raise the ceiling with extra questions that will provoke her to develop the concepts further?

Let’s try this.

Tell your students, “I’m going to show you a picture for only a few seconds and I want to know how many blue squares you see. Just a guess. Don’t overthink it. Go from the gut.”

Show this image for a few seconds.

Ask them to think-pair-share their guess.

Ask them to write down a number of blue tiles they know is too high and a number they know is too low.

Tell them, “I know that if you had to, you could count up those squares one-by-one. But that sounds like a lot of work. Let’s look at some smaller versions of the pattern and see if we can think smart rather than working hard.”

Pass out this handout [pdf].

It has the four earliest iterations of the pattern. They have to count and/or construct each one, which should give them a better intuition for the pattern than the original task does.

Ask them to write down and share a fast way to figure out the number of blue tiles in the twentieth iteration of the pattern. (I initially wrote “Ask them not to draw it” here but I don’t think they’ll find drawing it all that enticing. That’s the point of using larger numbers.)

Ask them to share how they’re thinking about the number of blue tiles. They’ll help each other here.

Then show them the twentieth iteration and see if they were right.

Now we tackle the huge square. We give students the information they want. The huge square is the 100th version of the pattern.

Some may generalize to a formula now. I imagine most will still just apply the informal verbal method they created earlier. Say, “If you can turn this sentence into variables, then a computer can handle any problem of this sort instantly and we’ll be done with them forever.”

Ask them to take their answer for the huge square and reconcile it against the error boundaries they set up earlier.

If students generalize to a formula in different ways, show those different formulas and connect them to the ways of thinking they shared earlier. Ask students to reconcile the differences. How are the formulas different? How could they turn one formula into the other?

As students finish, offer some follow-up questions:

• Tell them “One version of this pattern has one million blue tiles. Tell me everything about that version of the pattern.”
• Ask them to find the number of white tiles in the huge square. (Computationally simple, but it’s a bit of conceptual leap to realize you’ve already done the hardest work.)
• Ask them what numbers of blue tiles they’ll never ever see.

We’re trying to challenge students who finish quickly here and buy ourselves some time to help students who need it.

Show the answer. Find out who had the closest original guess. Have the class give that student one clap.

What You Did

Over on Twitter:

• The usually stalwart Chris Lusto threw in the towel immediately.
• James Cleveland thought the problem could stand as is.
• Jennifer Silverman created a Geogebra applet. We had a split decision, she and I, on whether students would get better intuition for the pattern by having Geogebra create it dynamically or drawing it themselves.

Over on the blogs:

• Beth Ferguson motivates the move to generalization by posing a school administration that quickly and repeatedly changes its mind on the dimensions of the cafeteria. That’s great. I’m curious what sending students to the cafeteria buys us here in exchange for the time it costs.
• Mary Dooms pulls a nifty trick by taking a task that’s fully prescribed and makes it both harder (students have to ask themselves “What information is important here?”) and more fun (students get to be creative and autonomous) at the same time. That isn’t easy.
• Jonathan Claydon modifies another task that moves to generalization way too quickly.

Open Question

What does the real world buy us here? I know what the cafeteria context costs us. It costs us an extra page to include all that text. It costs us some participation from our English Language Learners. That’s fine if we get something worth all that cost, but I can’t see what we’re getting.

Lately, when I work with math teachers, I encounter a particular theory of student engagement that says, “Make the problem about something the students are familiar with or which is close to their everyday lived experience.” On Twitter, for instance, Sarah Lowe suggests students may be more intrigued by stones or gardens.

I don’t think this is wrong. But I think we overestimate the effect of swapping the context from checkerboard cafeteria tiles to a checkerboard dance floor or a checkerboard stone walkway or a checkerboard pizza platter while leaving the rest of the task intact. I just don’t think students are that easily placated, so I haven’t focused on context at all in my makeover. We’re just counting up abstract blue tiles in my task. Instead I’ve tried to start the task easy enough to bring in a lot of students and end hard enough to be worth their time, with or without any context.

I’d like someone to convince me that adding a context would raise a student’s heart rate over this problem, but I have a hard time seeing it.

Featured Comment

Bethany:

I just did this activity in my College Algebra course (I teach at a two-year college) as an introduction to sequences and series.

19 students in class, 18 participated. (1 student walked out and came back when the activity was over.) I teach in an ‘active learning classroom’ with desks set up in groups. 4 groups of students working the problem together and individually.

3 different models for the pattern were given to me. I had already created my own model, and the 3 in class were all different from mine.

Each group explained their thinking. Most had not generalized their approach into algebra.

I helped them put it into algebra. I also showed that we could simplify the algebra for each approach and end up with the same thing!

Comments on the lesson:

1. I was surprised how long it took them to come up with the pattern. I had to ‘nudge’ one table along and point out an error in their thinking at another table. The process of coming up with 88 tiles for the 20th iteration took over 10 minutes.
2. I was happy to see so many different approaches to solve the problem. Things also ‘felt different’ in the classroom. Working to figure out a problem like this is much different than working on the process of completing a square. Different people were taking the lead and speaking up to help their group mates. It was great!
3. I was able to use this task as a reference right away. For example, when introducing the idea about the domain of a sequence, which normally is very confusing, I referenced the iteration # in this task and the students seemed more able to make the connection.
4. I didn’t realize the importance of telling the students the handout referred to the first, second, third and (draw-in) fourth iteration until after I handed it out. I had to go to each table numerous times and show them which one was #1, #2, etc… In the future, I would label the handout first.

This activity took about 20 minutes from start to finish. I definitely think it was time well spent! Thanks!

The Task

That’s from Connected Math.

What I Did

• Simplify the prompt. It’s already pretty spare, but I’m going to get rid of the information for a minute.
• Add intuition. A choice between two items like this lends itself really well to a guess. But when the information is already included, students will start to calculate right away.
• Justify the constraints. One set of dimensions is given in feet and the other in meters. Why? Is that just a contrivance for the sake of a math problem?

That’s everything. If I’m teaching this material tomorrow, I don’t have time to whip up a video or a photo.

So I’ll tell students, “Two students drew pictures of their rooms. Which is bigger or are they both the same?” (Good catch from Chris Lusto on Twitter: “‘How much greater?’ omits the possibility they have equal area. Why do that?”)

I’ll ask them to write down their guesses, then share with a neighbor. Then we’ll take a quick poll.

A student may ask if the two drawings are at the same scale, which would make for a nice, quick discussion (“Good eye. That’s really important to know. They are.) but it isn’t an essential moment.

I’ll say, “Okay. I’m going to give you the width and height of the rectangles and we’ll find out who guessed correctly. But first I have some bad news. Rodney is from the United States and Emile is from France. Do you know why that’s bad news?”

Here’s what I expect to be pretty interesting as students work with the fact that there are 3.28 feet in a meter:

• I imagine most students will convert the meters to feet. But some may run the other way. Do they arrive at the same conclusion?
• I imagine most students will convert the linear dimensions and then multiply. But will other students multiply the linear dimensions and then convert the area? Will they arrive at the same conclusion?
• If they convert in their last step, will they multiply by the conversion factor twice as they should? (ie. 2.5m – 3.5m – 3.28 f/m – 3.28 f/m) or just once? If no one makes that error, I’m for sure going to throw it out there that “a student from another class got a different answer.” Then they’ll construct an argument for or against.

What You Did

Good ideas from the blogs:

• Andrew Shauver made Rodney and Emile brother and sister and brought in some realia with a floor plan.
• Caren Hickman makes over a different task, one about food, with the goal of using real data and giving students some choice over the constraints of the problem.

Good ideas from the Twitters:

@ddmeyer instead of asking whose room is bigger, maybe ask which room would need more carpet (or other type of flooring)

— Erin Wade (@ewade4) June 13, 2013

@ddmeyer tie in linear equat & $$ to purchase/install carpet. Also if every so many sq ft requires an elec outlet how many would there be

— Sarah Lowe (@spilarcik_lowe) June 13, 2013

https://twitter.com/mathdiva77/status/345330572098433024

@ddmeyer Picture of soccer field vs picture of football field. Soccer data in m and football in yards. Which one is bigger?

— Ignacio Mancera (@Ignacio_Mancera) June 14, 2013

@ddmeyer I'd remove the numbers, the last sentence, and maybe add "They want to know if their rooms are fair." #mathchat

— David Wees (@davidwees) June 14, 2013

@davidwees The word "sketch" is misleading. Measurements suggest precision, "sketch" does not. Would add they're siblings. @ddmeyer

— Jenn Borgioli Binis (@JennBinis) June 14, 2013

@davidwees or that they are exchange students, one in US, one in Canada. Will rooms during exchange be similar to home rooms? @ddmeyer

— Jenn Borgioli Binis (@JennBinis) June 14, 2013

Every problem at it's heart is "measuring squareness"@mpershan @ddmeyer

— eddi vulić (@zidaya) June 14, 2013

@ddmeyer 1) Draw a picture of your bedroom. Estimate the dimensions (don't specify which units); 2) Who has the biggest bedroom?

— Jessica Drake (@jess_drake_) June 14, 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 Jun 26. See every edition of Makeover Monday.

Here is a “high-leverage teaching practice,” according to Deborah Ball:

Teachers appraise and modify curriculum materials to determine their appropriateness for helping particular students work towards specific learning goals. This involves considering students’ needs and assessing what questions and ideas particular materials will raise and the ways in which they are likely to challenge students. Teachers choose and modify materials accordingly, sometimes deciding to use parts of a text or activity and not others, for example, or to combine material from more than one source.

So every Monday this summer, I’ll post a problem from a textbook and start a conversation about how we could modify it. The details of that makeover may take the form of a loose sketch or something more formal. In either case, I’m going to be explicit about the goal of the makeover.

Fawn Nguyen, who’s been on an absolute tear lately, illustrated this process recently. She took this task:

And then she showed how she implemented it with her students. Her goal wasn’t something formless along the lines of, “Well this sucks and I want to make it more engaging.” In the title of her post, she says explicitly she wanted students to have some personal, creative input on the constraints of the problem. So she had her students start by drawing their own golf course. She set a high bar for the rest of us.

You should play along. You can feel free to e-mail me a textbook task you’d like us to consider. 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.