Oblique Cylinder Connections to Graphing Sine/Cosine Functions

This is the start of a lesson for precalculus. Will update with more pictures and materials soon!

Create a truncated cylinder in Tinkercad or find mine to use: Wan - Truncated Cylinder Lab

Print enough for each student or student pairs. It might help to print multiples. Thankfully, my printers have internet connected live feeds, so I cancelled a print at 3am in the library from home.

Materials:

1. Enough paper for each cylinder to be wrapped around 2-3 times. Adding machine paper works well for this. (You know you have extra rolls in your supply closet from the 90s)
2. Tape
3. Scissors (one per cylinder)
4. Graph paper
5. Truncated cylinder

Preparation tips:

1. Having enough tape pieces if you only have one spool of tape is a mess. Precut pieces for students either on their tables or bins
2. If you're using adding machine paper, precut those as well.
3. Have I mentioned bins? Presetting bins with all the materials is a good idea to save time.
4. It will take extra time for students to set up the x- and y- axis on the graph paper. Your decision to either have the axis pre-printed for them or have them draw them. You know what you want to convey and your students best.

Instructions:

1. Engage:
   • Ask students to hold the truncated cylinder in their hand and define the top, bottom, and side of the cylinder.
   • Ask students to hypothesize about the shape of the finished piece of paper if were wrapped around the cylinder multiple times, and the top and bottom were cut off along the shape. (have students draw this shape and explain why). Students should not be wrapping paper around yet for this step.
   • Students will share with each other their hypothesis. Engage students in a discussion about various ideas. Do not verify correctness at this point.
2. Explore:
   • Students take the truncated cylinder and wrap the paper around it. Make sure it is not loose. Tape the end. Without cutting the paper yet, ask students to revisit their conception of what the cut off piece of paper would look like. Students will share with each other their hypothesis. Engage students in a discussion about various ideas. Do not verify correctness at this point.
   • Students cut off the excess top and bottom paper.
   • Have students hypothesize after the the excess paper is cut off.
3. Explain:
   • Unwrap the paper and have students lay it on top of the graph paper.
   • Have students label the graph paper.
   • Students determine the equation graph of the top of their cut paper based on their labeled graph and explain their equation.

   • Question regarding the parts of the truncated cylinder that would correspond to a change in the graph. What would change on the cylinder to change the period? Amplitude? What things would change on the cylinder that would not?

4. Elaborate
   • Students can create cylinders based on equations in class and 3D print to test
5. Evaluate
   • Students create a portfolio of this project on Tinkercad, word document, and or physical models and papers.