SWBAT collect their thinking up to this point and begin moving toward algebraic generalization and abstraction.

Students work to abstract what theyâve been doing, and they see that Ï is only as precise as our rounding. Deep understanding ensues.

10 minutes

Students enter to see that they have 10 minutes to work with a partner to complete Part 3 of the Defining Pi Project. They received this handout at the end of the previous class, and I challenged them to try to finish it outside of class. Some get it done, some do not. Either way, this opening gives them a moment to collect their thoughts. If a pair of students are already done with Part 3, I give them Part 4 and tell them to “try to stay a step ahead of me throughout today’s class.”

For today’s class, **I am trying to strike a magical balance**:

I consistently toe the line between the urgency to move along and get things done, and the understanding that it really does take time to construct deep understanding. I circulate throughout the room making sure students are engaged, and I firmly but quietly nudge anyone who needs it to get to work. I tell the class that in 10 minutes we’re moving on, and write the time on the board indicating when that will be.

Part 3 of the project is fairly straightforward: it consists of a series of questions that will get kids thinking about what is happening in this project. Here are a few notes I’d like to share:

- The name of this part of the project is “Developing Generalizations.” When I name a handout (or choose the words in a learning target), I am intentionally trying to draw student attention to those words. I’ll spend a few moments checking in with kids and making sure they know what these words mean, and how they’re connected to the problems on the page.
- The first task is about the relationship between the central angle and the number of slices in each of the constructions from Part 2. This is meant to serve as the “quick” sort of formative assessment (toolbox link). When I circulate the room, I am taking a look at each student’s page, and (very informally!) noting what they have written here. We have discussed this question in the preceding days. I want to know whether they’ve got a handle on this relationship, and if they’re confident enough to express it algebraically. Have they written a sentence? An algebraic relationship? Are they stumped? Do they at least have the number 360 written somewhere? Do they at least have a multiplicative relationship? I hope not to see any “+” or “-“ signs here. In my class, most students are pretty set with putting the idea into words, but only a few venture the algebra. This will inform one of my moves in the next lesson (link), in which the algebraic abstraction takes center stage.
- The rest of the handout consists of five more questions that will help lend structure to student thinking. I have posted on the board, and I reiterate to students, that “It’s ok if you don’t know the “right” answers yet – put your best thinking on the page.” I tell them that we’re building a “time capsule” of what they understand right now, and that they will revisit their answers when they write a paper in Part 5 of the project. My hope is that even if a student leaves one of these blank, it will now be in the back of his or her mind when they continue along.

After 10 minutes are up, whether students are done or not, we move on to the next part of the lesson. If anyone needs more time here, they’ll get it at the end of class.

55 minutes

**Getting Started on Part 4 (5 Minutes) **

I hand out Part 4: The Big Chart to all students, and I elicit their acknowledgement that each column on this chart corresponds to a field on their construction sheets from Part 2. I tell them they have 5 minutes to transfer all of the information from their construction sheets to this chart. I walk around the room and I listen for observations kids are making as they engage in this work. When I hear what I’m hoping to hear, I quietly write student quotes on the board, so we can come back to them in a little while.

In terms of a building urgency, this is a really neat moment for the students who haven’t kept on the work. If they do not have the constructions done, they’re not ready to fill in this chart. I have seen that moment of “oh crap” realization as they watch their classmates quickly get this done. They key here is that I haven’t had to say a word about it. One piece leads into the next, and they’re seeing it for themselves. Whatever level of mathematical understanding students get out of this experience, hopefully that nudge toward developing a great habit of getting stuff done helps everyone.

I give** a more complete description of Part 4 in the following lesson**.

**Mini-Lesson - Example: 360 Slice (15 Minutes)**

Most students have three rows of their big chart filled in, because each of them completed three constructions. For the fourth row, we’re going to start by working together to consider a 360-sided polygon. I ask what this shape would be called, and wait for a brave (or better, the occasional wisecracking) student to say “a 360-gon?” (It always comes out as a question.) Yes, I say, and I write that on the board.

Next I ask if anyone has completed this construction already, and they see the humor here. “Would anyone want to?” I ask, and everyone is in agreement that this task would be pretty annoying. A few students pull out a circle and a protractor and draw up what a few “one-degree slices” would look like, just to validate the point. So now we’ve made it to the point where we’re going to able to think about something without actually drawing it: this is one meaning of the word “abstraction” I tell them, and I point to the second Mathematical Habit.

Next I ask, “Even if we’re not going to draw this thing, who can imagine what it would look like?” This year, my favorite answer to that question was, “Wouldn’t it basically be a circle?” You’ll see this question written on the board in this photograph of today’s notes.

As a result of previous lessons, students are pretty comfortable with the idea that this 360-gon will consist of 360 very skinny isosceles triangles whose sides are the same length as the radius they have chosen. I sketch an isosceles triangle that I say is “definitely not to scale,” and I say that anyone who wants to see it would really look like should use a protractor to make one on paper. Students are ok with this now, and we once again move through the computations of finding the length of the “outer side” of this triangle. I choose a radius randomly, to prove the point that I can, and on this day I choose 79, “because I’m pretty sure that none of you used this length for your radius.” As I write my work on the board, I am trying to structure it in such a way that variables could easily replace the numbers I’m writing, and that will help us continue to move toward generalization.

With all of this in mind, however, what I really want to get at here is precision and the role of rounding error. Against the well-developing instincts of many of my students, I initially push that we can just round the value of a (please see the photo) to 0.7. This means the outer side will have a length of 1.4, and the perimeter of the 360-gon will be 504. Many students are already catching the drift of this project, and they know what to expect when we divide 504 by the diameter of my circle. 504/158 turns out to be 3.18987, which is a) not as close to π and we would have hoped, and b) it’s actually bigger than π – which should be impossible if this polygon is inscribed inside of the circle. I point back to that great question: “Wouldn’t it basically be a circle?” and ask the class, “Well, is it?” Everyone agrees that our value for π is pretty close, but everyone also wishes that it were closer. You might be able to make out the eraser marks in this photograph. We go back and try successively less rounded values for a: 0.69, 0.689, and finally all of the precision that the TI-83 allows for. Eventually we see that if we don’t round, P/d matches π to four decimal places, and is indeed ever-so-slightly smaller than π.

**Continuing on the Big Chart (10 Minutes), leads into work for the remainder of the period**

Through the preceding mini-lesson, we made some of the calculations for the 360-gon, but there remain some columns of the chart to complete. As we move into the work period, I tell students that they might want to start here.

Some students are excited about this challenge, and others have some work to finish. This time is available for anyone to do exactly what they need. If they’re not done with Part 3, they can finish up those generalizations. I point out that putting everything in the chart on Part 4 might help people see what’s going on. If they’re not done with Part 2, I encourage them to look at the work that’s on the board, and see if helps them develop any shortcuts.

For anyone who is up to date, there are two more rows to fill out on the big chart, and ask students how quickly they think they can do this, now that they’ve been practicing. It’s a pretty easy sell, and they jump into the challenge, only arguing about which central angle values would be the most interesting to pursue. Our developing ideas about rounding errors continue to bounce around conversations. I circulate, and as I said in the previous lesson, it’s all about really listening to kids to see where they’re at. No matter how much (or how little) a student has done so far, this project gives them opportunities to learn frequently unintended lessons. Our job as teachers is to take these opportunities and work with them!

10 minutes

With ten minutes left in class, I ask all students to turn to the back of Part 4, no matter what they have done so far. Everyone should answer the first question, I say. “Write a few sentences about how the front side of this handout made you feel,” the prompt says, and I like to have a little fun with the idea of writing about our feelings in math class before reiterating how serious I am about this question.

Next is a quick reminder about staying organized. Everyone should take a moment to make sure they have four parts of the Defining Pi Project, and everyone should remember that Problem Set #10 is due at the end of the week. Once students are organized, I tell them, they should write down some next steps for what they will accomplish tonight.

Finally everyone should complete their Record Sheet prompts before the bell rings.