Today's Warm-Up sets up several big ideas that arise during the next several units. Each problem is important, so I like to frame this work time by telling students:
Think as much as you can about each of these problems because the ideas behind these problems are important for the next several units. If there is something you don’t understand, ask about it — whatever doesn’t make sense to you will be really helpful so that you can understand the ideas more fully.
I say over and over again things like:
I want to make sure that my students don’t slog through all of trigonometry with glaring misconceptions. I want our study of trig to be meaningful.
Problem (1) asks students to think about what happens to a person’s height above ground as they ride a Ferris wheel. In my classroom, I have a really simple model Ferris wheel that I made out of cardstock, tape, and a brad. This turns out to be quite useful.
Teaching Note: In the past I have allowed students who are struggling make their own Ferris wheel to help them visualize the situation. Even if this seems like a waste of time in the beginning, this is a really helpful object to ground their understanding because we will keep coming back to this.
The big idea behind Problem (1) is that this function will be periodic (it will repeat each time the wheel completes a revolution) and students should take enough time to think about this problem and make a good sketch. Whether their first attempt is right or wrong, I will ask them: “Why do you think the graph would make that shape? Can you describe how your graph represents this situation?” These questions are vague enough that students might not be sure how to set up their axes. If I observe that students are struggling with this, I will ask them: “What do the axes represent? Why did you decide this?”
Problem (2) is very different. Like Problem (1), students may feel like they are missing some important prior knowledge. Because we have recently reviewed the Pythagorean Theorem, I use that as a clue: “Could we use the Pythagorean Theorem to tackle this problem? How?”
Problem (2) is a good opportunity to talk with students about how they could tackle a problem with or without numbers (MP2). Numbers are provided, but students could look for generalized relationships. If students struggle to get started, however, I will encourage them to use the lengths provided. For students who easily tackle this problem, I plan to ask them: “How do you think this problem relates to Ferris wheels?” I would be pleased if they make a connection on their own. But, I think that if I keep asking this same question throughout the day’s investigation, some of my students may make the connection. I want to give my students the chance to discover a trigonometric function on their own.
Problem (3) sets up some of the key questions related to the Ferris wheel. The big idea here (and a big part of MP1) is for students to make sense of these ideas without much input from me. I will encourage them to refer to the context to make sense of the problem. The task introduces students to the idea of looking at the Ferris wheel as a clock-face, so in our digital world some students will need a reminder about how the clock is set up.
Throughout this warm-up, I will try to get as much sharing and cross-table talk going as possible. I plan to ask students to write their answers on white boards, or project their work using a document camera so that I can ask other students whether or not they agree. I believe that the more that I ask students to look critically at each other’s work, the more I enable students to trust their own reasoning. (This is really at the heart of reaching the goals of MP1 and MP3.)
I believe that the more open-ended the lesson, the more important the closing becomes. For this reason, today's Exit Ticket is more structured than usual. Students have been doing many different things during the day’s investigation. In order for students to synthesize their explorations and view them as worthwhile, I need to help them bring closure to their work in a way that enables them to leave the room confident about “what they are supposed to know.”
As I introduce the Exit Ticket, I will tell the class:
It is so great to see all your brilliant thinking about this problem—we are going to spend a lot more time thinking about this, so it is great that you started out with such great ideas. The exit ticket questions are showing you exactly what I want you to understand over the next few days, so answer as many of them as you can and think about them as deeply as you can.
By this point, I expect students should be able to sketch a graph that looks like a wave for the first question. If they are ready to be pushed further, I may start by asking them questions like:
I want to get them thinking about the parameters that will result in changes to this graph.
Question (b) is important, but students still might not have made this connection. That is totally fine— I would prefer them to make the connection when it really makes sense for them, than to tell them and for them to not understand it deeply. If students do have answers to my questions, I will ask them to share without evaluating their responses. Instead, I will ask other students to evaluate their peers contributions. This strategy will allow me to assess more students' understanding as we close the first day of this unit.
Question (c) raises many of the big ideas that students will need in order to understand these functions. I don't plan to discuss all of these questions today, but I include them here to see how well students understand this function that we have not yet named or found an equation for.