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:
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:
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.
This historical discussion of Fibonacci's approximation for pi takes us through two Mathematical Practices: attention to precision (MP6) and use of the calculator as a tool (MP5). It also serves as a brief review of scientific notation. There is a lot of good mathematical thinking to be had, even as students are being exposed the history of the subject (or of a number).
Planning note: When I am teaching this lesson, I pay attention to student thinking and the misconceptions that arise when students begin the Gallery Walk. Fibonnaci is often a good choice for my students at my school. However, I my choose to use what I observe to choose the most appropriate problem to share as a full-class example, based on something interesting that occurs on Day 1.
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:
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:
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.