Total Time: 75 minutes

Image: Finished Construction on the Chalkboard

Why is this Lesson Awesome?

Students delve deeply into high levels of precision as they learn to compute a value of pi.

Extensions and Scaffolds

As students delve deeper into the Defining Pi Project, they have choices about setting the parameters for their next two constructions. For some students, this means inscribing the next two even-numbered n-gons: an octagon and a decagon. Other students may be up for the challenge of inscribing a 9-gon or, say, a 72-gon, and all the implications that follow. (differentiation by choice of value of n)
The History of Pi Gallery walk continues to offer students options about how they focus their work.

Environment

Please see the Defining Pi Project, Day 1 for an initial description of the work we’re doing today. This lesson takes both parts of that lesson a step further, clarifies misconceptions, and gives students a chance to get to the work on their own.

Tools/Instructional Resources/Software: TI-83 (or similar) Graphing Calculator

Section #1: Lesson Opener – Example Problem from Gallery Walk

Time: up to 30 minutes, depending on amount of individual work time

Resources: fibonacci.pdf

Narrative

In the previous lesson, students began their work on the Defining Pi Project (DPP). For part of the class, they got started on DPP Part 1, the History of Pi Gallery Walk. For approximately 20-30 minutes in the previous class, students grappled with different representations of numbers (radicals, fractions, algebraic expressions), and to began consider very small margins of error at high levels of precision. Today’s lesson begins with a guided example from the History of Pi Gallery Walk, that will help them gain clarity around this work.

I choose to use the Fibonacci example for two reasons: first, because the value Fibonacci proposes for π has a mixed number within a fraction, and it’s annoying to write it on a calculator, so we can practice a few calculator tricks. Secondly, Fibonacci’s margin of error ends up being just 0.007%, and I want students to become comfortable working with such a tiny error. This discussion takes us through two Mathematical Practices: attention to precision and use of the calculator as a tool; it also serves as a brief review of scientific notation. When you’re teaching this project, pay attention to the interesting students thinking and misconceptions that arise when students begin the gallery walk, and use what you notice to choose the most appropriate problem to use a full-class example.

When class begins, I give students a few minutes to take out their Gallery Walk sheets and to get situated with regards to the problem. Many of them tried this problem yesterday. I write the fraction on the board, and, just like I did yesterday with one of the openers, remind students how to make careful use of parentheses as they write this entire number on the calculator:

1440/(458+(1/3))

The discussion of precision is ongoing through this project, and this gives another chance to talk about it. On the screen of the calculator, we see the decimal value 3.141818182, leading some students to say that Fibonacci got π right – it’s 3.14 and he got 3.14. But others see the error. Fibonacci was correct to 3 decimal places (although we note here that he didn’t yet have access to the idea of decimal notation of fractions). I ask: “Was he 99% right?” to a chorus of different answers, and to the yesses I say, “How about 99.9% right? 99.99% right?” Students are engaged in the question.

We work through the example, and I review the calculator feature of storing a number. After carefully entering the fraction, we store its value as X, and I make the point that we’re not writing anything on paper. Because everything is stored in the calculator, we have not introduced rounding to any part of this problem. This idea will continue to come up throughout the project. We can then calculate percent error by typing this:

(π – X)/π

I make sure students recognize yesterday’s definition of percent error in this calculation. When the answer pops up in scientific notation, we have the opportunity to review that concept on the way to carefully converting this value to approximately 0.007%. So how right did Fibonacci turn out to be? 99%? 99.9%? 99.99%? Better?

One more question I like to ask at this point is, “Why do they call it scientific notation?” I point out that many scientists make measurements of objects much bigger and smaller than everyday objects. Scientific notation gives us access to measurements of things tiny and enormous. You or I might not notice an error of 0.007%, but might there be cases when such an error would matter?

After working together through this example, I give the students some time to work individually through another problem on their Gallery Walks.

Section #2: Lesson Middle – More on the First Construction, and Laying Groundwork for the Second
Total time: ~25 minutes

Resources: DPP Part 2: Constructions

Narrative

We left off yesterday with a hexagon inscribed inside of a circle, and today we will work through more details from the Part 2: Construction handout. I am prepared to teach a series of mini-lessons to help students understand this handout - it just depends on the amount of background knowledge they can recall. I want to continue to build their facility with exact values in terms π and in radical form, and I look for opportunities to continue to build on our discussion of precision. It's fine if a lot of this contruction is worked as a full-class example, because the next step is for students to complete at least two more constructions on their own. I want to make sure they understand what they're getting into.

Specifically, my focus is on three ideas today:

I refer to several of the fields on this first construction sheet as "common sense" - the number of slices, the number of triangles, and the polygon's number of sides are all the same. I check in with the students to make sure they're all confident with this, and I try to cultivate some ownership in them over these ideas: the more I can help them see that mathematics comes from a place of common sense rather than mystery, the better.
Once we establish this idea of common sense, an extension to arc length is a sensible next step. There are six 60 degree angles in a circle, so students can see quite clearly that the arc subtending this angle is 1/6 of the circumference of the circle. For many of my students, this is their first exposure to arc length, and I like that this gives the topic a nice grounding. Later in the course, we'll spend more time on it, but this is a great place to start. I circulate the room and make sure that students a writing an exact value for this number, in terms of π. For any student whose radius is a multiple of 3 (or 6 or 12, like many of them chose) this is a simple calculation. Other students will need to be reminded to write the arc length as a fraction as a fraction in lowest terms.
We next get to looking at what the construction sheet calls "Length of Outer Side." This refers to the length of a side of the inscribed polygon, and in terms of each triangle, it is the "outer side," or the one that is not the radius. In this particular example, this involves an informal proof that this hexagon is made up of 6 equilateral triangles. I try to elicit as much as possible from the students. Earlier in the course, we used Euclid's method of constructing an equiliateral triangle with two circles, so students have seen the idea of using equal radii. After noting that each triangle is "at least isosceles" because two of its sides are the radius of the circle, and that the angle in between them measures 60 degrees, we show that the other two angles must also measure 60 degrees, and that therefore the triangle is equilateral and the outer side has the same length as the radius. Students then see that the perimeter of the inscribed hexagon is six times the radius, and they fill in this part of their construction sheets.

Finally, we are able to see that the distance around the hexagon is three times the diameter of the circle. I ask the students: If the circle and the hexagon were actually the same shape, then what value of π would that imply? I remind them of yesterday's opener, and how that problem also implied a value for π.

In my enactment of this lesson, we're now out of time, which leaves the area and analysis for tomorrow's lesson. Please see the Defining Pi Project, Day 3 for details on how I continue this project.

To close out our classwork, I post today's Record Sheet prompts. Please see the closing from the Defining Pi Project, Day 1 for more about record sheets.

Record Sheet Prompts:

Write today's date, then:
Write a sentence about what you did today.
Write a question you have from today’s work.
Set a goal for how you are going to demonstrate perseverance while working on this project.

Section #3: Closing – Check In Quiz

Time: 15 minutes

Resources: isos triangles check in quiz

Narrative

For this check in quiz, students are asked to sketch and find the missing values of an isosceles triangle. The measurements of the triangle are based on the numbers in a student's birth date, so everyone’s is different.

Students had a chance to practice making calculations on isosceles triangles two lessons ago, so now I’m checking in on what they’ve retained. Hopefully this quiz feels easy for them. As a general rule of thumb, I find that there’s great efficiency in providing several exercises that allow for feedback, but when it comes to grading something, I only need one problem.