As students enter, I frame the task simply, and let it unfold for students. I make sure that everyone knows where to find a ruler, a protractor, and a compass. Look at the first slide of today's Prezi.
Students are eager to jump into this task because it's immediately accessible. Each of them falls into one of the following groups:
I'll give the class between 5 and 10 minutes to do what they can. I circulate throughout the room to see how everyone is doing, and I fight the urge to say anything! This is really important. I don't want to influence what students are doing, because I want them to make their own connections to the task.
Source Url for Prezi: (accessed Oct 2, 2014)
Individual thinking on index cards
I distribute index cards and tell the class that we're going to discuss our experiences constructing equilateral triangles. In order to prepare for that conversation, I say, everyone should start by responding to the prompts posted on Slide #2 of today's Prezi. There is a prompt for each side of the index_card. First, students are asked to write something that was either "difficult" or "interesting" about this task. I like to give choice like this, so everyone has an entry point. Most students are game to think of something to write, and if they insist that this task was both easy and boring, I ask them to write about that instead. On the other side of the card, they should explain why they've chosen what they did.
Save the Last Word: Small group discussions
After I give students a few minutes to write on their index cards, I change to Slide #3 on the Prezi, which outlines a modified version of the "Save the Last Word" protocol. (You can see the original protocol, written for groups of teachers, by using the link here.) For today's purposes, I've stripped the protocol to just 2.5 minutes, and I love how much conversation it enables kids to pack into a short time.
Whole Class Discussion: Who made the most perfect equilateral triangle?
This is a natural place to introduce a little competition by asking who made the "most perfect" equilateral triangle. There will invariably be a few students who are eager to prove why they're the best, and it's fun to get a brave volunteer to pit their triangle against the rest of the class. What naturally happens here is that students have the opportunity to make and critique each other's arguments (MP3) about the merits of each triangle. I try to get someone who measured sides and another who measured angles, and to have them show their process. As the teacher, it's important for me to listen closely here: students will share the most astute observations about the properties of the shape, and there are bound to be comments from which the ensuing mini-lesson can begin.
Construction: Euclid's equilateral triangle
To begin the mini-lesson, I post Mathematical Habit #4: I can construct mathematical models (MP4). I start by pointing out that there are different sorts of mathematical models. In the Fall semester, we did a lot of statistical and algebraic modeling; now we're going to construct a geometric model. To begin the task, we want to make sure to understand the word "construct" in the first place.
The point I make is that if we're making a constuction, we're not going to measure anything. I compare this to some of the examples made by students who measured angles or sides. These might have turned out well, but it's possible to make an equilateral without measuring anything -- this is what is meant by construction. I talk about Euclid a little bit, and how he was able to prove all sorts of geometric theorems with little more than a straightedge (not a ruler!) and a compass. "What's amazing to me," I add, "is that by avoiding measurement, we're actually able to make an even more precisely perfect equilateral triangle. At the end, we'll measure its angles to see that I'm telling you the truth!"
We run through the notes. Students may need a little help remembering how to use a compass, this provides time to review the use of that tool (MP5). It's important to remind students not to change the radius of their compass after making the first circle, because in the next step we're going to make another one with the same radius.
The language of Step 2 is precise, although confusing at first glance to many students. This is a nice opportunity to touch on the idea of how slowly one should read mathematical statements. One way to attend to precision (MP6) is to stop and make sure that you're really understanding the precise meaning (and the order!) of each carefully chosen word.
At Step 3, I point out that we're using the ruler as a straightedge -- we're not concerned about the lengths of the lines, we just want to make sure they're straight.
Discussion: Did it work?
As students connect points A, B and C, they often exclaim in delight at the result. It's quite clear that this is an equilateral-looking triangle. Because of the way they grappled with the task to begin our class, they are more invested in seeing this successful result.
Despite what our eyes seem to indicate, however, I model some healthy skepticism. "How can we say for sure that this is an equilateral triangle?" I ask. The key word sits at the bottom of Slide #10: radius. I ask the class what the radius has to do with anything, and we see pretty quickly that all three sides of this triangle are the same length because each is a radius of at least one of these two congruent circles. Just for fun, students can also measure the angles of their triangles, to confirm that everything worked out, but this measurement is not necessary; the proof lies in the fact that each side of the triangle is a radius.
In order to foreshadow our Circles and Unit Circle units a bit, I mention that this is not the last time that we'll see the radius of a circle corresponding to the side of a triangle.
As the semester moves along, I'll give more ownership of the SLTs to students. One way I begin that process is by posing this question and seeing where students go with it.
What SLTs did we work on today?
My high-fliers will take out their syllabus and look for a learning target -- I publicly commend these students and suggest this as a good practice.