Making and Testing Conjectures: The Diagonal Problem

I have been thinking about MP3 from the Common Core, specifically about how to get students to make conjectures, to test those conjectures and to refine their conjectures when it turned out they were not always true. I was also thinking about student perseverance and helping them not get too frustrated. I’ve done some activities like Marilyn Burns’ consecutive sums problem (see additional resources below), but I want something that feels messier and a little more unwieldy.

I came across the diagonal problem, spent some time with it and it seemed like a good fit. My own solution process was messy, filled with frustrations (first that there were too many possibilities and then that I kept finding counterexamples for my conjectures), breakthroughs and satisfaction. I am particularly interested to see what kinds of systems teachers (and students) use to make sense of all the possibilities.

I really like this problem – it incorporates Common Core Math Practices #1, 3, 7 & 8.

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Launch

Warm-Up: Sometimes, Always, Never

This is a great instructional routine that works well with students. The basic structure is that students are given a statement(s) and they need to decide (usually in pairs) if the statement is sometimes true, always true or never true.

I used them here to bring the idea of example and counterexamples into the room. At the meeting I had teachers discuss them in pairs. With students I have them cut each statement out and make a poster with three columns and have them place each statement in the appropriate place, with examples and counterexamples.

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Problem-Posing

To get things started, I showed the following video, pausing it each time there was a number to ask, “What does that number mean?”

Eventually everyone agreed that the ordered pairs – like (10,7) – referred to the number of spaces to the right and then up from where the blue diagonal begins and that the larger bolder numbers referred to the number of squares (shaded) touched by the diagonal.

Then I showed the video uninterrupted. And asked everyone to come up with some questions, which they shared in pairs and then all together.

Here were our collective questions:

• How can you predict the number of shaded squares based on the dimensions of a rectangle? How could you find the number of shaded squares touched by the diagonal without counting them?
• What is the pattern/relationship between the length/side/area/perimeter of each rectangle and the number of shaded squares touched by the diagonal?
• Which kinds of rectangles produce the most shaded squares as a percentage of their total area? Which produce the fewest?
• Will the number of shaded squares in a square always be equal to the length of a side? If so, why?
• Given different rectangles with the same perimeter, which dimensions would give you the most shaded squares? How about for different rectangles with the same area?
• Why were there two rows that had three shaded squares in the 10 x 7 rectangle? Is there a pattern in the number of shaded squares in each row?
• How many shaded squares will there be in that last rectangle in the video?

I said, “There are a lot of questions here and there is certainly room for larger and more prolonged explorations. Since we’re limited by time, I wonder if we could all work on the same one. The one I’ve been struggling with and could use your help with is the first one.”

How can you predict the number of shaded squares based on the dimensions of a rectangle? How could you find the number of shaded squares touched by the diagonal without counting them?

Then I broke the group up into pairs and gave out large sheets of square-inch newsprint for teams to draw on. I also shared an applet developed by David Cox which allows you to create rectangle and calculate the number of shaded squares quickly and precisely. (The applet was an especially important tool for me to check the number of shaded squares. Even the slightest imprecision in your line and you can miss a square or add one that doesn’t belong.)

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Making and Testing Conjectures

No one found a way to predict the number of shaded squares at the meeting, but here is some of the work that each group created.

Ruben and Linda

Solange and Cynthia

Eric

Mark

My initital strategy was just to draw a whole bunch of rectangles and get a whole bunch of numbers and then see what patterns emerged. My big a-ha moment was to realize that you could figure out the number of shaded squares by adding the two sides and subtracting something. Sometimes you subtracted one, sometimes you subtracted 3. Then I used a strategy from our Pytagorean Triple meeting and put the rectangles I had drawn into families. From there I was able to make a connection between the number you subtract and the sides of the rectangles.

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Hint Cards

Last winter, Eric, Solange and I attended a lesson study open house at the Greenwich Japanese School. One of the things that really struck me was the use of hint cards, which were very effective in the class we observed. I wanted to see how it felt to use hint cards, and also to start getting better at them by getting my first, inevitably not as helpful ones, under my belt. Here is the sequence of hint cards I prepared. Diagonal Problem – Hint Cards.

Even though no one took me up on them at the meeting, having prepared them helped me come up with some of the counterexamples I posed to the groups as they shared their conjectures with me.

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“Who Wore It Best?”

I first came across this problem in a blogpost by Dan Meyer called, “Redesigned: Follow that Diagonal“. Meyer shares four versions of the question including his own, which is the video I used for the meeting. I am curious about how the different variations might impact how students work on the problem.

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Final Thoughts

I think this problem could be used with students, but I think I want to have it come after smaller problems, maybe even a few where students are given conjectures written by “students” and they need to evaluate them. Check out the additional resources below to see what I’m looking at to scaffold the diagonal problem.

I also wonder if a launch more connected to the math would help – like a Which One Doesn’t Belong that got the words “factors” or “multiples” into the room, but would not need to be explicitly connected to the diagonal problem.

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Additional Resources

1. Sometimes, Always, Never
   • Fawn Nguyuen – Always, Sometimes, Never – A blog post from Fawn, with photos of student work and a set of statements she found
   • Malcolm Swan – Convincing and Proving Tasks
2. Lessons on Student Conjectures
   • The Consecutive Sums Problem  (from About Teaching Mathematics – Marilyn Burns) This is a low entry/high ceiling conjecture activity that any students who can add up to 25 can handle. I almost used this as the launch, but I was pretty sure folks were familiar with it.
   • Evaluating Statements: Consecutive Sums (Classroom Challenge from the Mathematics Assessment Project in which students have to make conjectures and evaluate the conjectures of other “students” and then revise their own.)
3. Reasoning and Proof (Teaching and Professional Development resources from Annenberg Learner)
4. I don’t usually share external answers to problems from CAMI meetings, but I came across this write-up of the solution from NCTM. I appreciate the way it models a progression of realizations (called “ponderings”) and I thought teachers might find the progression helpful as they plan questions to support student explorations and conjectures. Interior Crossings – Solution and Progression

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In attendance: Cynthia, Solange, Linda, Ruben, Eric and Mark

Programs represented: BMCC, the Literacy Assistance Center, CUNY Adult Literacy PD, Hostos, LaGuardia ESL