Discovery through Orange Peels

Today was by far the most excited I've been all semester about giving a lesson. The lesson was covering the surface area and volume of spheres. Kind of a tricky lesson if you want to do it correctly and not simply give students the formulas, which I think is ridiculous and would much rather take the risk of students not following the 'discovery' of the formula versus simply giving it to them. To me, it's important that students see where the formula is coming from and make connections from the formulas they already know to the new ones in the lesson of the day. That being said, before yesterday I had no idea how to do this with the formulas for surface area and volume of spheres. I decided my best option was to google activities on the discovery of the formulas. By doing this I found what I wanted to do; for the volume formula, which I chose to do first.

I decided to lead the discovery by starting with what we had used the previous two days to find the formulas for other 3D figures; the volume formula for a cylinder. From here I constructed an example on the board of a cylinder and a sphere with the same height and radius. I then label the height of the cylinder, and the radius of both the cylinder and sphere. My students then came up with the formula they already knew for the cylinder which I wrote on the board. Next, I told students to keep this in mind as we watched the following YouTube video dealing with actual objects representing the cylinder and sphere I drew in my example. Throughout the video I stopped to ask students questions to make connects to our example on the board. Initially we stopped to talk about what the black tick marks on the cylinder were breaking the cylinder into and after pausing here I told students to also keep this in mind as we continued.

We watched as the volume (water) of the sphere was poured into the cylinder. Students then noted that the volume filled up two of the three sections the cylinder was broken into. I asked students what this would mean for the formula we already know for the volume of a cylinder. They said this mean that the volume of the sphere was 2/3 the volume of the cylinder (yayy! exactly what I wanted students to conjecture/discover). Next, together we worked through the algebra of replacing what was the value h in the cylinder formula in terms of the radius of the sphere. Students got a little tripped up on the more complex math of these steps but once I had a student come up and explain the steps most (if not all) of the class caught on.

Link to the video: http://www.youtube.com/watch?v=aLyQddyY8ik

It should be noted that initially I was confused on how to go from the video to the formula I knew for the volume of the sphere. Once I presented this to my placement partner, Josh and we talked through what I was thinking, Josh had an Epiphany and realized the connection (the algebra portion of the discovery). There also was a joke along the way in the algebra about "how many R's ARE there?" or something of the sorts which I caught and laughed at and students caught on and we all enjoyed a good laugh together.

After I felt students understood this, we moved onto the really fun part of my lesson, which I was super stoked about all morning, surface area. This activity involved an orange, a piece of paper, a marker, and team work. Before even handing out the materials I first reviewed with students the formula for the area of a circle. Once we decided that the formula was pi*r^2 and not 2*pi*r (after much debate) I decided students were ready to move forward with the activity, students were told to work with their table partners on this activity. The basic steps of the activity are covered in the pictures below but generally it goes as follows: trace the circle four times onto your sheet of paper (one student holds the orange while one traces), next talk about the area of the circles and what relationship the circle has to the sphere (same radii), after this is solidified I had students peel their oranges. It's recommended that students break the peels into the smallest pieces they can (not ridiculously small but not large), this helps with the next step which is placing the peels into the circles that the students traced from their oranges. At this point some groups started working faster/slower than others so I gave students the directions for the next step and had them try and use what they knew (area formula for circles) to find a formula about the surface area of our 'sphere'. I walked through the class to check the progress of my students whether that be in fitting the peels into the circles or finding the formula.

I told students that they needed to remember their circles they traced from the orange were not going to be perfect, thus how the peels fit into the circles was not going to perfect but should be very close. Students were also told that their four circles 'should' be the same size (or very close) since they're coming from the same sphere. Once every group was done, every group was able to come up with the formula for the surface area of the sphere, 4*pi*r^2. I had one student come up to the projector and use my 'set of peels' to walk her classmates through WHY this works and how it worked. I could not believe how well my students understood this portion of the lesson and was extremely impressed with the students' explanation of the activity, how it worked, and why it worked. Needless to say but the end of the activity I was feeling like a proud mama bear because my students not only took away from the activity exactly what I wanted them to but they gained a deeper understanding of the why behind the math, had fun doing so, and were able to do all of this without their safety net of a PowerPoint which simply gives them the formulas and examples. I was able to create a lesson where students worked together, discovered, had fun, and still took notes**! What an accomplishment!

**For this chapter, the students are putting together flip-books with drawings of each 3D shape, the names, and the formulas for surface area and volume all in one place. This is where students took their notes for this lesson along with a separate sheet of paper where they wrote the examples I gave them once the formulas were found.

The following are the step-by-step pictures of the second activity:
Step One:

Step Two: 

Step Three:

Note that once you peel half the orange you should be able to fill two of your circles with peel.

Step Four: 

Ta-dah! Once the whole orange is peeled, all four circles should be filled with the peel.

Overall, I'm so proud of my students, myself, and how well this lesson went and am happy to share any lesson materials with anyone who would like them. I hope if you choose to do this lesson that it goes as smoothly for you as it did for me! :)