This lesson tackles the logic behind the challenging formulas for the surface area of cylinders, spheres and cones. As usual, students are prompted to develop the formulas for each on their own. The most challenging - the cone - is presented first. I recommend walking students through it, then letting them develop the sphere and cylinder on their own.
Note that I have an alternate lesson for regular track 7th graders that does not include the cone - but advanced students benefit tremendously from exposure to the intricate formula for the cone.
This is one of 12 lessons in my "In and Out" unit that targets area and volume for 7th graders in the advanced track. The goal of the unit is to build understanding of why the formulas for area and volume work, such that if students forget the formulas, they can cognitively reconstruct them on their own.