I will make this a very brief introduction. I will tell my students there are two very important questions to be answered today: 1) What shapes make up the net of a cylinder? 2) What is a formula for the surface area of a cylinder?
One thing I love about this lesson is how a simple piece of paper becomes such a powerful tool (MP5) for students to understand the surface area formula of a cylinder. I will first guide students through a series of questions that should help students answer everything they’ll need to know about the surface area of a cylinder (See video). I won’t require students to write anything. They just need to be prepared to listen to and discuss each question. I want to give students a chance to discuss and make sense of the task (MP1) before I require them to write anything. I find that doing the discussion first opens the way to understanding for the majority of my students. They will not feel constrained by the need to get everything right.
After the discussion, students will work through a similar set of questions with their partners. Once we debrief those questions, students will be asked to label the parts of their net to find the surface area. In other words, they’ll put the formula for the area of a circle on each of the bases, they label the length of the lateral surface (rectangle) using the circumference formula and the height of the lateral surface using the height of the prism. At this point I will show students the formula for the surface area of a cylinder . I will ask students to explain what the parts of the formula mean. If they are stuck I may ask: “What part of the formula finds the area of the bases? Which part finds the area of the lateral faces?”. As students construct and deconstruct the formula they are developing their ability to look for and make use of structure (MP7).
After this (and if time permits), students will find the surface area of a few shapes.
We finish by summarizing our findings. I will pose the question: "We found that the surface area of all prisms is the sum of the area of the bases plus the areas of the lateral faces. Does this apply to cylinders too?" This is another turn and talk. Hopefully students conclude that yes, the generalization applies. Some students might suggest that instead of lateral faces we say lateral surface. Students then take an exit ticket.