Surface area and volume investigations

Today was a much better day. I guess the class just needed to settle back into the school routine.

I started off today with watching about 10 minutes of the movie October Sky. I'm using this as a follow-up to the ClassDojo videos on mindset. I asked the students to think about the ClassDojo videos and try to identify characters with fixed mindset versus growth mindset. The first 10 minutes establishes the setting and major characters. After viewing the clip I asked the class for observations and questions. We had a brief discussion about some of the behaviors they observed and about the historical setting of the film. My plan is to continue watching the film in 10 minute increments on our block days (we have eight more of these for the remainder of the semester).

While students would have liked to continue watching the video (anything but math, right?), I asked students if they had come up with estimates for the number of cookie boxes being held in the Nissan Rogue. No one had any answers, so I moved on to the next investigation. I intend to follow up with this problem later.

Next, I displayed the following statement: The surface area of a cube can never be less than its volume. I then asked whether or not this statement was true. I told students I would check class progress in 10 minutes and then started walking around and asking questions.

As I talked to students, they had some difficulty expressing their thoughts. Several times I heard that they knew what they were thinking but were having trouble articulating their thoughts. For these students, I asked if they could come up with an example that would help get their thoughts out. Others stated their belief about the truth of the statement. For these, I asked how they could convince the rest of the class about their belief. What evidence could they provide that would help them.

As time progress, students became even more involved in the task. I had small cubes that students could use to help them organize their thoughts. Others started drawing representations and examining side lengths and the resulting surface areas and volumes in a systematic manner. A couple of students honed in on the side length of 6 as a critical value to examine.

I let students work a bit longer than 10 minutes since I saw so much productive thinking and discussion taking place. I made note of who did what and how far they progressed and then pulled the class together for some share out. I asked groups who were still struggling with articulating their thinking to share a bit about what they were focused on. I progressed to students who had done some additional work and calculations share what they discovered. I then had students who had focused on the side length of 6 to share what they did and why they focused on that side length.

I then posed the question of why 6 was such a critical value. Students struggled some with expressing why this was a critical value. One student had actually written out an inequality x·x·6 > x·x·x. I asked this student to present this result and explain where it came from. In the presentation and resulting questions that arose, the class did a nice job trying to understand the relationship and importance of the 6 side length.

After this discussion, I wrote surface area above the left side and volume above the right side. I reminded students that they worked with similar inequalities in Algebra 1. Just as working with an equality, we could divide both sides with x·x and end up with 6 > x. This means that
surface area > volume as long as the side length is less than 6. When the side length equals 6 the two quantities are equal and when the side length is greater than 6 then the statement that surface area is greater than volume is no longer true.

We then pursued the next investigation; I displayed the following:

You are a burn victim in a hospital and need skin grafts for 70% of your body.
A doctor walks in, examines you and orders 162 cm² of skin for you.
Should you think "Thank goodness!" or "Oh, no!"

I asked students to respond as to whether or not they should be concerned or not. This problem proved more difficult for students to grasp. Some tried to look up how much skin covered a human body. As I walked around the room, I began asking students how tall they were. I also asked them what their waist size was. Could they use those to help estimate their skin coverage. They still struggled with this. I took a sheet of paper and demonstrated how rolling the paper to form a cylinder made the sheet act like a wrapper. I then asked how big a wrapper would they need to cover themselves. This seemed to break the logjam and students started calculating a surface area.

The next issue came up with having the their personal measurements in in² and having the given value in cm². Students started to ask the relation between centimeters and inches. I provided the conversion that 1 in = 2.54 cm. There were still some points of confusion about conversions which I helped students work through. I used desktops as a way to explain that in² represents a count of the number of grids that could be formed on the desktop. This seemed to help students differentiate between the measurement of one side of their desktop versus the grid cells covering their desktop.

As we came toward the end of the class, students started to see that 162 cm² of skin was not nearly enough skin. They then started to wonder how much would be covered by 162 cm²? I explained that this would be equivalent to around a 10 inch by 3 inch patch.

With that, I passed out a sheet explaining how to read three dimensional math diagrams. These often show up on standardized tests and I wanted to help prepare them for their upcoming PSAT test.

Next class, I am away at our annual math competition. I will have students get some practice calculating surface areas and volumes while I am gone.