Three Trigonometry Openers give students a chance to refresh their knowledge of percent error (7.RP.A.3). The problems are projected on a screen as students enter, and I ask them to solve the problems in their notebooks. As they get to it, some will need a brief refresher on how to calculate percent error. When I write the phrase "percent error" on the board, I lightheartedly ask if anyone has seen these two words before; students get the gist, chuckle, and reply that of course they know the words, but it's seeing them together that gives them a hard time. I ask which word is "easier," and most shout, "error," informally defining it as "when you mess up."
Taking that cue, I talk about how in the first problem, my buddy messed up by not quite paying me enough. He owes me $11.57, but he gave me $10. I then propose the following: what if I was helping my little brother buy a car, and he owed me $1001.57, but he paid me back with $1000 - would that be ok? Immediately, students note that the error would be the same, but these two situations feel very different. So that's where the percent comes in, I tell them: it gives a way to compare numbers in a relative way. I then review an informal formula for percent error (amount of error/correct amount), and we make the calculation. Today's discussion of precision begins as we talk about how to round the solution.
After guiding my students through the first problem, the second one is a fairly simple chance to practice what we just reviewed; a student may want to come to the board to show what they did, but sometimes I like to save time by just talking through the solution.
With the third problem, our work really begins: if the circumference of a circle is 12π cm, what is the percent error when someone says it's 36 cm? A few things here: first of all, it will be important to use exact value here. Most students will solve this problem in two steps, but I lead them toward writing just one line on their TI-83's:
(12π - 36)/(12π)
Part of attending to precision and strategically using tools is understanding how those parentheses work. When some students ignore the parentheses aroud the denominator, it's imoportant to note how and why their answer is different.
As we work on this problem, I also ask what value of π is assumed if 12π = 36. Students note that it's 3, and as we move through today's activities, this value will appear again when we inscribe a hexagon in a circle.
The Defining Pi Project (DPP) begins with a problem solving gallery walk about the history of π. This activity serves two specific purposes:
This activity provides historical grounding for the mathematical practice of attending to precision (MP6). We are able to talk about how the improvement in the tools available to us over time have helped us attend to greater degrees of precision throughout history.
Before class, I set up this part of the lesson by printing and posting each of the 13 gallery items around the room in chronological order (See Section Resources). The items consist of 12 references to mathematicians who offered values for π, and one note about the symbol π itself. Each of the 12 items consist of a brief historical context and a problem that involves calculating a decimal value for π based on the given definition. It is important to note that at the times most of these mathematicians were working, decimals had not yet been invented: we are fortunate to have the handy tools we do, providing us the comfortable position of being able to fairly easily assess the work of the ancients!
Immediately before the gallery walk begins, students have the chance to record their current “working definition” of pi at the top of the Gallery Walk Note Catcher handout. A few lessons ago, students were asked to write their definitions of pi (if you haven’t yet done this activity, I recommend using it as an opener today), and some of them have already refined those definitions over the last few lessons. I want to cultivate in my students a habit of reflection on how they construct their understanding of big ideas. As students successively rework their knowledge of π, this activity will provide a parallel to the grand narrative of human understanding about the same concept.
The handout provides students with a place to record the person, time, and place in which a value for π was determined, then to calculate a solution to the given problem and the percent error. After a brief explanation of the handout and a moment for clarifying questions, I give students about 20 minutes to collect, and make calculations for, any five gallery items on their note catcher (see gallery walk 1).
Each gallery item indicates a “level of difficulty” for the stated problem. I determined difficulty levels based on the amount of algebra students will need to solve these problems. In practice, my stating these difficulty levels helps students self-differentiate. I pass no judgment on students for choosing to solve the problems they do, other than a quiet, occasional barb at stronger students if they’re taking it too easy on themselves.
As the gallery walk happens, students will congregate around some specific problems. It’s always a pleasure to see these impromptu problem solving sessions take root. Because students are moving around the room, groups are fluid, and I have found that the conversations that happen are exactly the ones that students need. Some students will end up involved in discussions of fractions (see Anthonisz or Apollonius), while others will get deeper into challenging algebra (see Ahmes or Chang Hong). Depending on how things go, it’s up to your discretion to decide whether to lead a class discussion about any particularly provocative problems!
This is a structure students have seen previously, and we’re setting up new one for this project. The purpose of this record sheet is to give students a place to reflect on their experience of working on this project; in their paper about this project, students can reference the Record Sheet as evidence of what they've learned. I provide legal paper, ask the students to fold it half – forming a little four-page booklet – and then to write “RECORD SHEET for Defining Pi Project,” and their name at the top of the front cover.
Students write the date and respond to today's two-part prompt:
I end by making a big deal of reminding students to take good care of this record sheet and the first two parts of the project. It’s all evidence and we’ll be jumping right back into it next class!