Students said they found the practice problems on volume difficult. I asked questions to get a better feel of what they struggled with. The first was just a conceptual understanding of volume. They recognize volume and how to calculate it for rectangular prisms. They don't translate the idea of volume to other shapes, even though I have tried to emphasize the idea that volume is area being stacked on top of each other. Thus, the volume of cylinders posed a road block.
After explaining the idea of volume as base area times height, I had students work the two volume of cylinder problems that I gave them. Things went better and they could at least apply the idea of calculating the base area (B) and then multiplying by height (h) to find volume (V), i.e. how much base area is being stacked (in essence).
The next problems that students got stuck on were ones in which they had to reverse the process. For example, you are given the volume of a cylinder (150π cm3) and the height of the cylinder (6 cm), what is the radius of the cylinder? I referenced students back to the idea of V = B x h. Students were able to find the base area of 25π cm2. I then asked what the radius of a circle would be if the area of the circle was 25π cm2? This was something the class could do and they found the radius was 5 cm. I asked them to reflect on this problem. What was in this problem that was so difficult? How could they overcome their fears and uncertainty to forge ahead and attempt something?
The next problem was similar in that it provided the volume of a cube as 2744 cm3. Students were asked to determine the side length of the cube. I asked students what they knew about cubes. The cube has 6 faces and the side lengths are all the same. For cubes, we can think of volume as
V = l x w x h. But in this situation, the side lengths are all the same, so V = l3 or, more specifically,
2744 cm3 = l3.
I asked students if they remembered how to find cube roots. None did. I then told them they were all capable of determining a number, when multiplied to itself three times would equal 2744. I then left them to their own devices. I walked around to check how they were doing. Many gravitated to using a calculator and guessing, with 10 being a popular first guess. The next guess tended to be 20 and students readily recognized the value they needed was a number between 10 and 20. After a couple of minutes, everyone in the class had found the cube side length was 14.
I used this as an opportunity to recognize that they didn't need to know a formula or algorithm to find a cube root, they just needed to think about the problem and have a way they could come to a reasonable answer. I think we have, too often, taught students to believe that math is about learning formulas and applying formulas rather than reasoning and problem solving. As a result, students find math confusing and boring. Who wouldn't if all they were walking away with were a bunch of memorized formulas that didn't make any sense?
I continued to use October Sky as a bridge for Mindset discussions and to provide students an opportunity to view high school students from a past age and to see what they were able to accomplish through effort and perseverance. We got to the point where the four teenagers had their first successful rocket launch. The discussion afterward was rich and provided many insights. We talked about characters and their mindsets. We discussed things that stood out or touched them. One student commented on how being successful took a lot of hard work. I am so glad I decided to bring this movie into the class room.
At this point, we have covered the core concepts in surface area and volume. I didn't cover ideas like apothems and surface area of cones because they are not something that the general student really needs and they just promote the idea that math is about memorizing meaningless stuff. It was time to unveil their performance-based assessment for the unit.
The "Car Project" is something that I thought up in my first or second year of teaching geometry. It requires students to consider surface area, volume, and the interplay of the two. It requires proportional thinking and ties back to dilations as they must use a scale factor to translate between their model design and reality.
Below are the instructions and rubric for the Car Project. I presented these to the class, let them absorb what the task was about and then ask questions. I received a lot of good, clarifying questions and was pleased to see they were actually thinking about what they needed to do and how they would need to approach it.
I limited groups to two or three people. A student can work on their own if they choose. For this project, students can form their own groups. This avoids personality conflicts that may arise with random groupings or groupings that I may put together.
Students took the last 15 minutes of class forming their groups and discussing initial designs. I made clear that groups needed to consider multiple designs as they had to convince the reader that their final design was indeed best. I also encourage students to think outside the box as to the shapes they would use. There is a tendency for students to stick with rectangular shapes.
I am giving students three class periods to work through their designs and to construct a model. They will then have two nights to complete their write-up of the project, which will be due at the start of the 4th class. I'll use this period to have students present their design and why they believe it is the best design.
From past experience, I will have to push students to consider other designs and other shapes, to challenge their thinking on convincing me their design choice is best, and to pose questions that force them to consider additional options. I'm looking forward to seeing this year's designs.
The next problem was similar in that it provided the volume of a cube as 2744 cm3. Students were asked to determine the side length of the cube. I asked students what they knew about cubes. The cube has 6 faces and the side lengths are all the same. For cubes, we can think of volume as
V = l x w x h. But in this situation, the side lengths are all the same, so V = l3 or, more specifically,
2744 cm3 = l3.
I asked students if they remembered how to find cube roots. None did. I then told them they were all capable of determining a number, when multiplied to itself three times would equal 2744. I then left them to their own devices. I walked around to check how they were doing. Many gravitated to using a calculator and guessing, with 10 being a popular first guess. The next guess tended to be 20 and students readily recognized the value they needed was a number between 10 and 20. After a couple of minutes, everyone in the class had found the cube side length was 14.
I used this as an opportunity to recognize that they didn't need to know a formula or algorithm to find a cube root, they just needed to think about the problem and have a way they could come to a reasonable answer. I think we have, too often, taught students to believe that math is about learning formulas and applying formulas rather than reasoning and problem solving. As a result, students find math confusing and boring. Who wouldn't if all they were walking away with were a bunch of memorized formulas that didn't make any sense?
I continued to use October Sky as a bridge for Mindset discussions and to provide students an opportunity to view high school students from a past age and to see what they were able to accomplish through effort and perseverance. We got to the point where the four teenagers had their first successful rocket launch. The discussion afterward was rich and provided many insights. We talked about characters and their mindsets. We discussed things that stood out or touched them. One student commented on how being successful took a lot of hard work. I am so glad I decided to bring this movie into the class room.
At this point, we have covered the core concepts in surface area and volume. I didn't cover ideas like apothems and surface area of cones because they are not something that the general student really needs and they just promote the idea that math is about memorizing meaningless stuff. It was time to unveil their performance-based assessment for the unit.
The "Car Project" is something that I thought up in my first or second year of teaching geometry. It requires students to consider surface area, volume, and the interplay of the two. It requires proportional thinking and ties back to dilations as they must use a scale factor to translate between their model design and reality.
Below are the instructions and rubric for the Car Project. I presented these to the class, let them absorb what the task was about and then ask questions. I received a lot of good, clarifying questions and was pleased to see they were actually thinking about what they needed to do and how they would need to approach it.
I limited groups to two or three people. A student can work on their own if they choose. For this project, students can form their own groups. This avoids personality conflicts that may arise with random groupings or groupings that I may put together.
Students took the last 15 minutes of class forming their groups and discussing initial designs. I made clear that groups needed to consider multiple designs as they had to convince the reader that their final design was indeed best. I also encourage students to think outside the box as to the shapes they would use. There is a tendency for students to stick with rectangular shapes.
I am giving students three class periods to work through their designs and to construct a model. They will then have two nights to complete their write-up of the project, which will be due at the start of the 4th class. I'll use this period to have students present their design and why they believe it is the best design.
From past experience, I will have to push students to consider other designs and other shapes, to challenge their thinking on convincing me their design choice is best, and to pose questions that force them to consider additional options. I'm looking forward to seeing this year's designs.
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