Today I was observed on a lesson on finding the surface area formula for cylinders and rectangular prisms. The class is mostly based on lecture, so this lesson I wanted them to discover the formula on their own. I had already done this with a previous class and it worked great. I was warned by my coordinating teacher that this is one of the toughest lessons for students to grasp.
I began the lesson linking back to previous lessons. I had students find the circumference and area of a carnival ride which was a circle. I had students go up to the board and explain their work. I then posed a question to them. I took out a piece of paper and wrapped it like a cylinder. I stated that the circumference of the folded paper is 11 inches. If I were to unroll it, would the length also be 11 inches? I had them think, pair, share and after some discussion we decided that yes the length of the paper would still be 11 inches. We then talked about what exactly surface area is. I then handed out a net of a cylinder and explained that using everything we just did, determine how to find the surface area of a cylinder. After some discussion with the groups, some groups got an answer, but two of the groups ended up getting the right formula. We then discussed as a class the surface area of a cylinder.
Then, we moved on to rectangular prisms. I handed out the net of a rectangular prism and said determine the surface area of this now using how you solved the surface area of a cylinder. After some time of visiting groups, we came together as a class. Two students were able to get an answer. Upon discussing, it was difficult for the rest of the students to see the surface area of this prism. They kept wanting to say four sides are congruent instead of only two. I drew pictures on the board, used their net, and drew on another prism. I then had one girl explain it in her own words and the class went “ohhhh, now I see”. I was waiting on that moment for a while. I still feel that only about half the class really seen where the formula came from. We were then out of time and the lesson was over.
Talking with my observer later, I feel the first part of the lesson, finding the surface area of the cylinder went pretty well. The students seem to understand where the terms came from, and when I did some sort of formative assessment with it, most seemed to get it. I feel I could have done a little more assessment such as talking to pairs about the formula, or addressing it to more students. The surface area for a rectangular prism did not go as well. There seemed to be a lot of confusion, and some misconceptions that I had not previously planned to address. There were a lot of confusing looks on students faces and I attempted to explain it with multiple manipulatives, and it still fell short. It was not until I had a student explain it in her own words (student language) was when more students seemed to grasp the concept of surface area of rectangular prisms.
I feel most of the students got hung up on the net part of the rectangular prism. They were more focused on seeing the two rectangles on the end, then finding the area of the long rectangle down the middle like shown below.
They were also finding it hard to see how the sides mirrored each other when it was folded up into a prism. In other words, the length on the bottom face was the same as the length on the top face. They were finding this difficult to grasp.
I think I need to come more prepared with possible misconceptions students may have. I had already taught this lesson before, and they got it very well. However, every class is different. One issue that does not come up in one class, may in fact come up in a different class. One suggestion that was brought up during the meeting was the fact of using concrete examples then move to the abstract. I really like this approach. Students can come up with values by measuring first, then seeing the surface area formula. I think it would have worked great with the rectangular prism since many students could not see the sides mirroring each other were congruent, so this would have solidified their understanding. There would be a set procedure the students would be following which may have made it easier for students to understand. Also, a suggestion was made to use a bigger manipulative compared to my hand held prism. Then as we reasoned about the prism, we could color each face a the same color as the face that is congruent to it. Students can have a visual then of which face is congruent to each one. The formula may have come easier to some of the students then who were just not getting it.
Debriefing afterwards and writing up this blog post has some great benefits as I continue to grow as a teacher. The smaller things that I would have forgotten about we touched base on and how to correct it. One example was one student asked “why aren’t all the sides the same?” and I answer “because they are all rectangles”. I feel this could have been handled much differently. Also, I would have never considered just the way I was holding the smaller prism in my explanation could cause confusion for students since I was moving it around and not keeping it in one place. Students could lose track of where the faces lined up were. Reflecting on the process has also given me ideas on how I would have attacked the lesson differently and what changes I could have made. Some ideas were mentioned by both of us that would have strengthened the lesson all together. I have mentioned earlier about the larger, colorful manipulative and using concrete numbers first. In the end, if the students are struggling badly, maybe throwing in a short lecture would benefit and tie up any loose ends. It is always great to reflect on any lesson. This is where the most learning about yourself as a teacher can take place.
After the lesson and communicating with my observer I feel the first part of the lesson with cylinders went pretty well and the second part with a few tweaks could be an effective lesson.