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.
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.
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:
- 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.
- 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!
- 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.
- 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!