Today's lesson is essentially an emphasized repeat of part of the mini-lesson from Day 5 of this project. I’m running through it one more time to help establish a key idea of this project.
To begin today’s class, I want the whole class to take closer look at the second and third questions on the back of Part 4 of the Defining Pi Project. Many students have attempted these questions already, and almost all have already recorded the necessary calculations on the front side of this handout. I want to spend a few moments discussing rounding error with them. We'll do this by revisiting the calculation of Perimeter/diameter for a specific example.
As students arrive and get situated, I just do my own work on the board, writing some now-familiar notes. I say that today I’m going to let the length of my radius be 100, and I quickly sketch an isosceles triangle on the board with two sides of 100 and a central angle of 45 degrees. I run through the calculations very informally, casually remarking to the kids that they have done this already. I know that some students are still in the process of solidifying their grasp on this knowledge, but I do this to illustrate our familiarity with the topic. Either way, any student who has completed an octagon construction already has the answer (for their radius) recorded on Parts 2 and 4.
I set up the calculations and I get to the point where I’ve got the length of the outer side, a = 100 * 2 * sin(22.5) written on the board. In addition to helping us accomplish our stated task, the way I have written this in one step will also help as students move to generalize an algebraic expression and Excel formula for the length of the outer side. I don’t mention that to kids right now, because I’m just setting up some scaffolds, but I know it will come in handy. I put out the call for students to make my calculation for me, and the answer comes back “76.53669”. I ask students to refer back to question #2, which asks them to round this value to 2 decimal places. I ask what is meant by “a few important steps in this calculation”. I’m looking for students to multiply this length by the number of sides to get the perimeter, and then to divide that perimeter by the diameter. After we run this example for both a = 76.54 and a = 76.537, we see that P/d comes out to either 3.0616 or 3.06148. I ask students if this is a big deal, and no matter what they say I withhold my opinion. There’s not actually much to see here.
The example of the octagon is really just there so we can compare it to the 360-gon, which is really what we want to look at. I ask what I’ll have to change about my calculations to work through that example. Again, I’m laying the scaffolding for the abstractionthat is to happen today when I point to the outer side calculation for n = 8 and ask what I’ll have to change here and what will stay the same.
I want to elicit that he radius is still the same, that I still have to divide the isosceles triangle slice in half to use the sine ratio, and so the only thing I have to to do is erase the "22.5" and replace it with of the new central angle. I’m watching faces closely here to see who is with me. The only other thing I have to change is the n value, from 8 to 360. I erase with my hand and just replace numbers on the chalkboard.
a = 100 * 2 * sin(22.5) becomes a = 100 * 2 * sin(0.5), and
P = 8 * a becomes P = 360 * a
I find that this is a good way to help students move toward generalized expressions – later, I’ll just tell them to look for the eraser marks (see rounding Opener). We get a = 1.75 or a = 1.745, which in turn translate to P/d values of 3.15 or 3.141, respectively. Now, I ask if they notice anything here. Some students will say that both of these values are “better,” because they’re both closer to π than what we got for the octagon. I pace around and scratch my head and say, “ok, but something troubles me about that 3.15…does it bother anyone else?” Students point out that it’s greater than π, and I press them on this: “Is that possible? We’re talking about a polygon that is inscribed inside of a circle. Is it ok for P/d to be greater than π? What does π mean anyway?” Now they’re getting it.
To help us move toward closure, I point out that in different situations, different levels of precision are necessary. If someone owed me $3.15 and they gave me $3.14, would I really make a stink about it? But here, the difference between the two is the difference between the possible and the impossible; it just doesn’t make sense for P/d = 3.15. Now, we’re doing an exercise in calculating π as precisely as we can. In order to do that, we’re going to have to pay attention to precision (MP6). Fortunately, we have a tool to help us with that, and we’re going to pick up on our Excel sheets where we left off.
The middle of today’s class is a workshop period. Students are given computers, and they have time to finish up their Big Charts on Excel. I’ll take this space to touch on some aspects of what I’m looking for as we move through the completion of this project. As with most workshop periods, my big move here is giving students time to engage in mathematical practices (see my reflection on Coherence).
Some students still need a little scaffolding on the path toward generalizing formulas on Part 4. Today’s opener details one way in which I do that. Another way I demonstrate to students the relationship between quantities and abstracting them is by looking at a few parts of the project at once. I’ll tell students to take out their constructions from Part 2. I say they should ask themselves what they did to get each of these values, and I explain that this is the first step in writing a formula. I might have some students use their finger to point to the number of slices on each page. Then I have them point to the same column on Part 4. We remember that tables are one way to look at a function, and that another representation is an equation. So how are we going to write that equation? We develop the algebra. Once we have such an abstraction, we can translate it into Excel language, which then us helps list all the quantities quickly, and see relationships. By laying out all three of these parts (the Construction Sheets, the table of values, and the generalized formulas) at once I help my students to persevere at this point in the project. My students are able to build connections and to better develop their algebraic thinking.
For the majority of students, here is what will happen: from an Excel perspective, the length of the outer side and the area of the triangle are the only real challenges here. I don’t tell students this, I just say it’s a workshop period, and I let the class move along. In reality, I have a pretty predictable mini-lesson ready for them after I see a few things happen. I want students to really chew on the algebra of finding the length of the outer side in terms of the radius and the central angle. To the greatest extent possible, I want them to recognize on their own that sine is part of this calculation. I make the point that this probably involves looking at their notes from all the isosceles triangles we’ve practiced over the last few weeks, and that it’s still up on the board from the opener. For many students, this is the first time they’re seeing a trig ratio in a formula, and this novelty can make for a leap the first time around. I want them to generate the generalization that we need to take half of the central angle, and then we need to double the measurement again. I also want them to search the web to figure out how to use the sine function on Excel.
If all of this happens, there is still the question of radians, which we haven’t yet studied in this course. But I don’t just tell them this, I let them see it for themselves. Watch this excel and radians screencast to see what I mean.
Once we have the outer side and the triangle’s area down, it’s really interesting to watch students realize that the spreadsheet technology is actually easier than writing an algebra formula. They’re all very comfortable with the idea that the perimeter of the polygon is simply the length of the outer side times the number of sides. On the Excel sheet, they just have to get the product of columns E and G. Furthermore, it’s a fine move to assign a variable like a to the length of the outer side, and then give the algebra asP = na. I pursue it further, explaining that a better formula would be written only in terms of the radius and the central angle. Can we do that. In this context, algebraic substitution makes more sense than ever, thanks to the modularity of the formulas in this list.
The same goes for the area of the triangle and of the polygon. In their charts on the back of Part 4, many students will write A = 1/2bh in the “Area of Triangle” row on their chart, which is true, of course. I tell students they’re right, but now they have to figure out where b and h are defined in this table, and to try substitute these expressions into the area formula. For the polygon, we could just multiply the area of a triangle by the number of sides, and that’s what we do on Excel, but how about a generalized formula in terms of just the radius the central angle? Earlier in the project, some students found the word “apothem” online, but couldn’t explain what it meant. As an extension, now I send students to go see if they can explain that. I challenge them to try to find other formulas and see if they work. “What is the least number of inputs we could take in a formula?” I might ask, or, “Compare the formula you have developed to others that you have found online.”
After about 45 minutes of work time, students will be at a variety of different points of progress. Tomorrow’s class is an even more open-ended workshop period than this one. Before that happens, I take a little time today to introduce the final part of the project.
I hand out Defining Pi Project Part 5, which consists of a few things. First is a chart listing all five parts of the project and related learning targets. Next is an outline of the Defining π Paper. The assignment consists of a listing of four sections and a series of guiding questions for each section. The last question on each section is an extension that requires some research. I explain this to students and tell them they should try to write paragraphs that provide answers to each question. I give them a few minutes to read through the questions and then to discuss these questions with their classmates. I walk around listening for insights, so I can identify a few students from whom to elicit expertise tomorrow. I am also making myself available for clarifying questions.
Teacher's Note: As a note to teachers using this project, please take a look at these questions. If you have any questions about my thinking, please let me know. For the most part, I hope that the moves I have made throughout this project make sense as you take a look at this list of questions.
Next, I ask students to turn the page over, and to take a look at the self-reflection that is in place of a rubric on this project. It’s up to each student, I explain, to tell me what they think they’ve learned on this project, and I want them to connect SLTs to different parts of the project as evidence. The chart on the front of this handout will help along these lines, but in the end I’m leaving it to students to reflect and interpret their work.
After seeing this handout, students have a picture of what they have left to accomplish. I tell them that the next class is a final workshop period for this project, and anything they can’t finish in that time will be homework. They should spend a few moments now making sure they have all parts of the project, and taking stock of what remains to be done.