Ferris Wheels and Triangles

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Objective

SWBAT use special right triangles to find the coordinates of more points on their Ferris Wheel graphs.

Big Idea

Giving students many days to make this connection enables them to understand it much more deeply--and to start asking questions about how to find more coordinates.

Warm-Up

30 minutes

More Accurate Ferris Wheel Graphs

40 minutes

The conceptual leap that students are asked to make when they undertake the work asked of them on Ferris Wheel Graph Practice Variations turns out to be really challenging. For this reason, I allow a lengthy amount of time for students to on it. I also teach by asking questions, rather than direct instruction or answering questions. I'll ask question like:

  • What does height mean?
  • How would a diagram help us understand this situation?
  • How can we apply the tools we have to this situation?
  • What exactly are we trying to find?
  • What do we know?

I find that my students struggle to set up a diagram that models the problem and makes sense to them. Furthermore, even if they have one, it is not intuitive to draw in the radius connecting the vertical line to the center of the circle. It is often helpful and interesting to ask students to think about their diagram by trying to identifying helpful triangles. Given time, my students do realize that they actually know the length of the hypotenuse of a right triangle, because it is a radius of the circle. The two resources listed below help my students to gain traction with this set of tasks:

Overall, there are numerous aspects of this activity that challenge students. If students muddle through it, they will be rewarded with much deeper understanding than they can gain through direct instruction.  In particular, there are many patterns my students discover in these data tables (MP7 and MP8). It really is a really great teaching strategy help students stick with the problem and regularly ask them if their answers seem reasonable.

As they work, students should be able to describe in words how the numbers in the data table increase and decrease in a regular pattern. This is a great conversation to have with groups or with the class:

  • If we know the maximum and minimum, what does that tell us about the other numbers in the table?
  • Does the rider’s height increase faster as the wheel moves from 3 o’clock to 2 o’clock or from 1 o’clock to 12 o’clock? Why? How should this show up in the data table? The graph?
  • How do the symmetries in the wheel show up in the data table? The graph?

As students work, it turns out that they when they have enough time to leverage their knowledge of data tables and the graphs, they can use this knowledge to make sense of this situation and make discoveries that help them to understand it deeply. 

Trigonometry Review

20 minutes

Once my students have made the connection between Ferris wheels and triangles, I want to take a little bit of time to review some right triangle trigonometry accumulate confidence in their application of their existing knowledge. This confidence will prove crucial tomorrow.  so that students are ready to make this leap tomorrow.

Solving Triangles Different Tools is both review and an opportunity to figure out where students' gaps are, because some students may have really solid background knowledge of trigonometry while others may have almost totally forgotten it. I use this activity to get everyone started thinking about different kinds of triangles so that they will be ready for the next day.

Closing

10 minutes

The key question of this lesson was:

  • How can you use special right triangles to find the exact heights of riders on all key points around the Ferris Wheel?

I like to restate this question as part of the closing. I ask students to make sure they they can answer it, by giving them a chance to discuss it with their partner. 

It helps to remind them of key terms they might use:

  • Vertical/horizontal
  • Hypotenuse/radius
  • Angle
  • Ratio
  • Isosceles triangle
  • Equilateral triangle
  • Side ratio

The idea is for them to put the pieces together. I like to give them a few minutes to discuss and write some ideas about how to answer the key question on a whiteboard, and then construct a cohesive explanation as a whole class.