I begin this lesson by projecting an image of two real cylindrical Sinkholes. I tell the class that a sinkhole is a depression or hole in the ground caused by some form of collapse of the surface layer. Sinkholes are often shaped like a cylinder. The dimensions of these two holes are given below each picture. I ask:
Assuming the sinkholes are perfect right cylinders which sinkhole is larger? Which sinkhole holds more soil if they were filled completely?
I ask my students to write down their reasoning before they share their ideas. I say, "I'll call on a few of you to make a prediction, but I want to hear a clear explanation of your prediction." Explaining their conjectures helps students think about the math that is inherent in the situation. It also encourages students to realize that there is more to a mathematical solution than a confident guess.
My experience is that quite a few students will conclude that the holes will hold the same amount of soil because the dimensions of both holes are the same, just flipped. This would work for a rectangle, for example, so it is good reasoning based on prior experience. Others will conclude that the deeper sinkhole held more soil, because it is deeper. I am looking to see how many students consider the radius as the more significant dimension.
As students write, I walk around listening, being careful not to share information. I want students to feel that I think they can do this on their own. I also ask students to try to work without technology, particularly technology that contains (or searches for) formulas. The objective here is for students to explain their reasoning. It is important practice with a skill that will be used over and over in this unit.
Next I pair up my students with their shoulder partner and hand them two sheets of printer paper (8.5" by 11"). I ask that they roll one paper the long way to make a cylinder and tape it. The other paper should be rolled the short way and taped at the edges as well. After the cylinders are done, I hand each pair the following Cylinder Activity Instructions, and ask that they read and discuss each instruction carefully.
I walk around listening and observing students working. The following are short narratives for each question.
Question 1: Many of my students may not recall the Lateral Surface Area formula 2πrh. I tell these students to search for it. Some students will realize or recall, that the lateral surface area is the length times the width of the sheet. I ask that they be discrete and allow the other students to work on their own. Even after obtaining the formula, some students may complain that they don't know how to find the radius. I don't give any info and motivate students to continue figuring it out. I may say..."you really don't need the radius here, to get the LSA".
After students answer Question 1, I stop and address the class. Some may not have answered or may have done so incorrectly, so I call on one of the students who figured the formula quickly without the radius and ask that he/she explain their reasoning to the class. I expect the student to show how the circumference of the base is actually the width of the rectangular paper.
It is important for students to fully understand why the Surface Area is the same for both cylinders, and, to be able to explain it concretely and using mathematical abstractions (i.e., a formula). This dual approach is a key to the lesson, and in fact, the unit.
Question 2: For Question 2, I expect students who said that both sinkholes were the same size will once again conclude that both cylinders have the same volume because the same piece of paper is used for both cylinders. I believe that this is a misconception, a stubborn one! Following the Warmup, a multiple choice question can be used productively to survey the results and once again ask for an explanation. I usually ask for explanations from students who favor a, b, and c when we discuss this question.
Question 3: Here, I ask students to search for the formula, Πr2h. But, I allow students to struggle with figuring out how to obtain the radius when they know the volume and the height. If students get stuck, and can't seem to figure things out, I will use the following hint: Can you use the circumference as in Question 1 to find the radius. An important conversation occurs after the formula is applied as students react to the discovering that the volumes are different.
Question 4: This question gives students the opportunity to write an explanation of their current understanding in their own words.
Question 5: I will take some time here to help students formulate a clear explanation. Some, but not all, will be thinking about the fact that the radius is squared, so it is more of a factor in the numerical value of the volume than the height. So, one can expect a wider sinkhole to hold more soil, if one is asked for an estimate.
I close this lesson with an EXIT QUIZ. I tell my students that it is a formative assessment. I say, "I'll make corrections for you and I will not give you a number grade." I want to take the pressure off students, but I encourage them to give their best explanation. They try their best knowing that they will eventually receive feedback on whatever they don't understand. Please read my After teaching this lesson... reflection for more details.
After teaching this lesson here are some thoughts and about what occurrenced:
1. Not one student stated that the long thin cylinder with the small radius had more volume. Most actually said they had the same volume due to having the same values for their dimensions, despite one value being the radius of the base and the other being the height of the cylinder. A few predicted correctly but with an inaddequate reason.
2. The lesson actually went well, but mostly with respect to the actual math work that was required. Students had little difficulty, if any, with calculations to find areas, the radius, or volume of the cylinders, but...
3. Despite it being a lesson where students had to explain their predictions and reason at certain points, I felt towards the end that they only had a vague idea if any, of why the difference in volumes between cylinders. They knew that volume meant the space inside but I felt they did not grasp the idea. This made me think that maybe part two of the lesson where volume is defined based on the unit cube, could be taught first. Then again, it is true that students don't necessarily need to grasp a concept in its entirety immediately.
These were some of the comments by students.