Ferris Wheel (Graph) Symmetries
Lesson 2 of 11
Objective: SWBAT determine which angles on the unit circle have the same sine ratio, the same cosine ratio, and the same tangent ratio (without using any of those terms!).
The first decision that I will make today is whether or not to give students some more time to work on yesterday’s investigation. If they seem motivated and eager to fully solve the problem, I will give them more time (I may even give some students more time and allow others to move on.) They don’t need to include this problem in their portfolios. We will also be coming back to it later in the unit, so I leave this decision up in the air until class starts.
Once I have students moving on to today’s new investigation, I begin by asking them the question following questions about riding on a Ferris wheel:
Assuming that all the cars on the Ferris wheel are occupied...
- Is there always somebody at your same height above the ground?
- Is there always somebody directly above or below you?
- Is there always somebody diagonally across from you?
The answers are no, no and yes. Why not?
When you are at the top or the bottom there is nobody across from you, and when you are at the sides, there is nobody above or below you.
After I distribute the Ferris wheel symmetry question resource, I give students time to think about the scenarios. I ask them to write justifications for their answers. I will ask students to share these whether they are right or wrong.
Once the class has developed some productive ideas to build on, I plan to give them the Symmetry Pairs task. Note that this is deliberately not heavily scaffolded, which will hopefully prompt students to start asking questions about how to talk about the points on the circle. If students are creative and make up their own way of referring to points, I will let them do the whole task that way. At some point, I will ask, “Do you want me to show you the official convention we use to talk about points on the circle?” I'll let them decide when they want to learn this convention. But, they will learn it before the end of the lesson.
As students work on the Symmetry Pairs task, the idea is that as they list pairs, they start to investigate the relationship between coordinates and hopefully make progress towards some generalizations. The Ferris Wheel Symmetries Problem Set will ask students to do the same kind of thinking in a more organized way. More heavily scaffolded, the problem set walks students through each problem quadrant by quadrant, leaving them with the question about the broader generalization (MP7, 8). It is not essential that they formulate an algebraic generalization—though that would be great. They may explain their generalizations quadrant-by-quadrant, which is fine. The point is just that they see the patterns visually, so that later they can use these observations to explain the symmetries in the graphs of the sine, cosine and tangent functions.
Students may or may not have time to finish the problem set in class. It is a portfolio piece, so they can work on it in future lessons.
Today's closing questions ask students to describe any methods they developed to find the pairs of points with the same heights, slopes and horizontal distances. This can be a good time to introduce some new terms. “Height” is very intuitive for students, but it can also be called the “rise” or the “vertical distance from the origin”. The “horizontal distance from the origin” can also be called the “run” or even the “base”. The best terms are vertical and horizontal distance because these will make more sense in future units, though they are perhaps less intuitive.
After letting students write for a few minutes on their Exit Ticket, I plan to ask students to explain their methods for finding matching pairs. At the most basic level, a student may talk about folding the circle horizontally or vertically. They may draw lines on the circle to find matching points. All these visual methods are great, but I want to push them to a more algebraic or quantitative method dealing with the numerical angle measure. The highest level of understanding would be to provide algebraic formulas involving the angles. The more different ways students can answer each question on this warm-up, the better, and for each method a student shares, I will ask: “Does this method work? Does it always work?” As today's lesson winds down, it is a good time to talk about the angles that don’t always have matches (0, 90, 180, and 270) and to talk about what is different about those angles.