When students arrive for class, the opening problem is posted on the board:
What is the area of an equilateral triangle that has three sides 100 cm in length?
As they set to work, I divide the chalkboard into two columns, labeled Problem_Solving and Background_Knowledge. After a few minutes, I ask for the first step in solving this problem, and if students have been following along in this course, they know that they should start by sketching the and labeling the triangle. I then ask what background knowledge we might need. The first thing I'm looking for is the formula for the area of a triangle. When students come up with this, I make a bullet point in that column, and write "I need to know the formula for the area of a triangle," followed by the formula.
We notice that we already know the length of the base of the triangle, so now we just need the height. I ask how we're going to find that - can we do it without a calculator? What background knowledge will we need here? This gets us to review the idea that the height of this triangle will split it into two congruent halves, and that our knowledge of 30/60/90 triangles can help us find the exact value of the height without making any calculations. Some students want a little review of this, and I encourage them to find it in their notes or to ask a neighbor for help - these are both habits I want to cultivate, and I want to build coherence between tasks throughout the year. I want students to notice that they're using something they studied a month ago, and I want them to feel comfortable writing the height, and therefore area in terms of root 3. By the end of the conversation the board looks like this.
The next step on our first Constructions Sheet is an opportunity to apply what we just saw in the opener. If they couldn't already, students should now be able to find the area of a component equilateral triangle inside of their hexagon. I circulate and make sure they're able to do it, and I'm looking to make sure that everyone feels comfortable writing this number in simplest radical form and not as a rounded decimal. When I see that they've done it, I ask them if they can find the area of the entire hexagon (with reference to yesterday's conversation about common sense).
One question students might have is whether they have to multiply 6 by the coefficient, root 3, or both. Depending on how many students ask the question, I might put a few examples on the board with variables where root 3 was or I might just ask them to check for equivalence by entering expressions into their calculators.
Other students are able to see that it wouldn't be too hard to develop a formula for the area of an equilateral triangle, and I encourage this extension into abstraction! (This algebraic generalization plays a big role in Part 4 of the project.)
With all fields of the Construction Sheet now complete, students can move on to the Analysis Chart at the bottom of the page. Here are some of ideas I try to share with students, either through whole class discussion or by circulating table by table:
As I mentioned above, students are usually very surprised and excited to see the same values as their neighbors in the last three rows of this chart. This leads to a discussion of relative values (which are the same) vs. absolute differences (which are different). Over the years, students have called this cool, crazy, or creepy. When kids say “that’s crazy” about numbers appearing in everyone’s work, I make sure to point out that now they’re “doing analysis,” and our work here as teachers is to make that analysis richer. When you say "that's crazy," I explain, you're doing half of the work of analysis. The other half is to ask "why?" and figure out what's happening. (As a general principle, I like to create opportunities for students to say that something is crazy as often as possible.)
As for that explanation here, I ask students to think back to the first day of the semester. What was one of the key words we studied? It was similarity - and similarity explains why the percentage is the same: no matter how big or how small the circle, no matter what unit of measurement I use, or what I decide the radius should be (as we did here), the relationship between the hexagon and circle is going to be the same. That's percentages and ratios do - the capture the relationships between measurements, and as we have seen before, when shapes are similar, we know that these relationships are going to be the same. It's so satisfying to come back to a word like this, for both me and my students!
Finally, we're done with the first construction, and hopefully it pays off in students comfort levels with their next two constructions. Before moving on, we read briefly through one man's (slightly snarky) definition of π, and I tell students that we're going to see a method for this "harmless exercise" works.
See the Defining Pi Project, Day 1 for an overview of Record Sheets. As students find their Record Sheets and get started, I note that when they write their paper about this project, I am going to ask them to quote their Record Sheets as evidence of what they were thinking as they worked.