Making an Illusion with a 45-degree Mirror
After collecting the check-in quizzes, I tell the class we're going to watch a short video. This is a high-engagement video from the BBC about a house-sized optical illusion that relies on a mirror being placed at a 45-degree angle to the ground.
As the video ends, I'll ask students what they noticed. How did this illusion work? What did the artist do? If none of my students mention the 45-degree angle, I ask if they heard mention of any angles while they watched the video. Once we establish the fact of the angle that's involved, I ask what it is about this angle that makes the illusion work. Most of my students lack the background knowledge in physics to fully understand what's going on here, but that's ok - I'm looking for a way to engage them in the idea that something special happens when we have such an angle. I try to sneak the word "special" in here, because soon we'll be using it in a more formal sense.
I do not spend too long on this, but I distribute a mirror to each table and I ask the question, "what would happen to the illusion if the mirror made a larger angle with the ground? What if it were a smaller angle?" I'm not running a physics lab today, even though I'd love to. Instead, I just leave the mirrors lying around and look for opportunities to have a conversation about this phenomena when we get to work in a few minutes.
Next, I tell students to find their "fascinating charts" from the Similar Triangles Project. There are features of this chart that students have noticed in their reflections on the project, but not many have yet developed a satisfactory explanation for why these things happen. When I say that, I mean that my students are not satisfied, and they're actually excited to figure out what's going on. This is how the narrative of this class is constructed. I ask if there are any particularly interesting rows in this chart that anyone would like to talk about, and students are immediately drawn to the 45 degree row, because we've just been talking about that angle. "What's special about 45 degrees?" I ask. Students point out that it's the only place where sine and cosine are the same. It's also one of the places where we see a "1," which amidst all those messy-looking decimal numbers looks pretty mysterious.
I frame today's class by saying that 45-degrees is one of the special angles we're going to talk about during this and our next class. "By looking at this chart," I propose to the class, "what other special angles do you think we might be able to discuss?"
Finally, I post this question on the board: What's the only rectangle that's so special that it gets its own name?
Earlier in the unit, we refined our definition of similiarity enough to realize that all rectangles are not similar. Squares are a special kind of rectangle, however, and squares are always similar. "Today, we're going to use squares and the equilateral triangles we constructed yesterday to explore some special right triangles," I say. Then, as a carefully-placed, seemingly afterthought, I add: "What is it that squares and equilateral triangles have in common?" And I give students space to ruminate on that as we transition to work time.
URL for the House Illusion Video: (accessed August 7 2013)
From Equilateral Triangles to 30/60/90 Triangles
Students pick up laptops, and they open up their Geogebra constructions of equilateral triangles from the previous class. As they're getting set up, I ask, "How many lines of symmetry are there in an equilateral triangle?" We have an informal discussion of these lines of symmetry, and I show students that the three lines of symmetry in an equilateral triangle each do the same thing: they split it into two congruent right triangles. I ask, "How can we add this line of symmetry to our construction?"
It's pretty straightfoward to see that by connecting the two points where the circles intersect, we'll get a line that bisects the equilateral triangle. I don't spend time proving this during today's class - I'm simply putting a lot of ideas on the table. As students add this line to their constructions, I post the new learning target, Triangles 4:
As usual, I ask for a volunteer to read the learning target, then I ask students to share the key words in this SLT. After a few rounds of that, I say that today we're really going to focus on the word "special." I ask students to consider the construction in front of them (here it doesn't really matter whether they're looking at Geogebra or their work on paper), and to look for any angle measurements with the angles indicated in the learning target. It doesn't take too long for them to see the right angles and 60 degree angles that we've made. Once they have those, there are two reasons to think that we also have a 30 degree angle. First of all, our line of symmetry bisects a 60 degree angle, yielding a 30 degree angle. Secondly, in the new right triangle that we've made, there are 60 and 90 degree angles, so what must the third angle be?
I summarize by saying that one of the Special Right Triangles is the 30-60-90 triangle, and that such triangle is half of an equilateral triangle. We will continue to apply this idea and to see its consequences throughout the semester.
From Squares to...
To transition to our next challenge, I say, "The other special right triangle comes from a square. I'd like each of you to try to construct a square in Geogebra."
Now: the Euclidean construction of a square is not nearly as straight-forward as that of an equilateral triangle, and Geogebra has a variety of tools that make it easy to "cheat" the task a little bit by going beyond just the use of a compass and straight edge. I don't specify that students have to constrain themselves to any particular tools. They should use whatever tools they wish to construct a square.
Teachers: I recommending reading the Geogebra tutorial about the "Drag Test". You may choose to run this tutorial with your students. I prefer to reveal the drag test as today's lesson goes. When a student thinks they have a square, I run the drag test on their work, and the challenge evolves each time.
The role of today's work on Geogebra is twofold. I want to give kids a place to play with Geogebra as a set of tools (MP5), and to familiarize themselves with the software. I also want them to think critically about what makes a square, and to consider how by attending precisely to the wording of the definition of a square (MP6), they might gain insight into the shape. I want them to learn firsthand some of the features and behaviors of these mathematical objects. This activity creates a terrific place for improvisation, because there are a variety of directions the exploration can take. All from trying to make some squares.
I circulate, facilitating conversations and showing students how to use the drag test. Some squares will pass the drag test on some vertices but not on others, so it's important to check all four. Some students will successfully construct squares today and others will not. My role is to help them describe what they see as they're working and to show them that journey of this task is worthwhile, even if they don't reach the end goal. Indeed, part of perseverance in problem solving (MP1) is to realize that the work is worth it even when it feels like a solution is hard to find.
When students do complete their squares, I ask them to describe the symmetry of a square and to consider which lines of symmetry divide a square into right triangles and which do not. At the end of today's class, students are ready for tomorrow's lesson, which is to formalize the knowledge of special right triangles.
With 10 minutes left in class, I distribute Problem Set 4.
Problem Set #4 is similar in content to Problem Set #3; both are opportunities to practice solving problems involing triangles. There are right triangle trig problems, and problems that require students to review some ideas from past studies of geometry.
Although this is certainly a chance to gain experience on SLT T3, I only grade this assignment on Habits #1 and #4.
Today's closing is an Index Card Essay. The prompt is:
What "special" characteristics do squares and equilateral triangles have in common?