Tuesday, January 22, 2008

Lesson Idea: Hands-On Volume and Surface Area

Year after year, the students at my campus have trouble with Objective 8, which deals with basic geometry, measurement and similarity. Students have to find measurements, surface area and volume for various 3D shapes and their nets, and then apply those skills in faux real life situations.

I'm still not sure what it is about this that confounds students so much, but it could be any number of things: Too many formulas (and thus variables)? Lack of practice and training in visualizing drawings in 3D or putting together and taking apart nets of objects? Too abstract (not enough concrete, real life examples)? I think it's a mix of all these things, along with their poor preparation in even the most basic of mathematic concepts (times tables, anyone?).

So the first thing we did this year was to deconstruct actual objects, draw their nets and measure their dimensions before even thinking about formulas and advanced problems. I tried to use products that I could pull apart or unfolds and then put back together again:
• Cylinder: For volume and total surface area, I used a Pringles can, emptied and cut vertically and 2/3 of the way around the circumference of the base (so it stays attached). I taped the plastic top to the box to be the top or bottom (depending on your perspective). It was something the kids could unroll and roll up again, and that I could refer to repeatedly throughout the unit. The label of any soup can is a clear example of lateral surface area if you need it.
• Rectangular box: I used a box that had some extra tabs that made it easy to view flat or in 3D, but any rectangular box that you can cut so that the net is easy to see is good. I told students that when I saw a problem with a tall, skinny box, I thought of a cereal box. While I didn't use one, it would be another good example to cut and show. I would also suggest cutting a box that is a common sight in the room (tissue box, printer paper box, etc) which is easiest for you to get and easy for students to use as a reference.
• Triangular prism: Toblerone, the oddly-shaped Swiss chocolate bar, is one of the few triangular-shaped retail boxes that is both widely available and immediately recognizable. I had two boxes, one taped together (after I removed the chocolate to avoid any distractions) and another cut to form a easy-to-sketch net. A large 3-ring binder is another potential example hiding in your classroom.
I couldn't think of or find any common items in cone, pyramid or cube-shaped boxes, so I used a kit of durable plastic models borrowed from my co-chair. These were adequate, but since they couldn't be flattened out, it forced students to rely solely on their imagination (for better or worse).

Before we started this unit, we had worked on orthographic (top, side and front) views of 3D objects and drawings. On our state test these problems usually have stacked cubes in various arrangements, so I used Jenga blocks and a document camera to demonstrate what each view would look like. We did a lot of predicting, verifying, and modeling of 3D shapes from different perspectives, which made the next unit a bit easier.

On the first day of this unit, students sketched nets, measured the dimensions (using the rulers on my state's TAKS math formula chart) and labeled the drawings of each of the shapes listed above. They worked in groups to help each other work efficiently. I didn't tell them what exactly to measure, just to measure what they thought would be important and later, we would use these to find surface area and volume. I did tell them to focus on trying to visualize these shapes, because it was a skill needed for every word problem--figuring out what exactly was being described so that we could then figure out what to do.

The next step was creating a "better" version of the state formula chart, because the chart was far too generalized. This would, I explained, make applying these formulas a lot easier.

For example, the volume of every prism and cylinder is V = Bh, where B is the area of the base, which the student then has to figure out and then look up, making what is supposed to be an aid into more work on an already intense test. Things get quite hairy when they have to find the volume and surface area of other shapes. So we made our better chart more explicit; for example; the volume of a cylinder is πr2h and the surface area of a rectangular prism is 2(lw + lh + hw). We practiced on some nets we had drawn as a Do Now problem to bring the second lesson full circle.

Finally, we went back to the six nets and their measurements from the first day, and completed the volume and surface area of each one. With all of the knowledge built up over the previous days, this was a piece of cake for the vast majority of my students.

We wrapped up the week by working on challenging test prep questions that required a lot of visualization, sketching and multiple steps to complete. Students had to apply everything we had done to complete the problems, which we corrected, analyzed and reviewed afterwards.

So as tomorrow's assessment on this half of the objective looms (the other half deals with similarity and proportional change in these figures), I'm feeling confident in how my students will perform. I still worry, however, that this knowledge will quickly be replaced by other short term knowledge, so I am planning a project where students will once again be forced to apply all of these skills, finding their own objects to "deconstruct" (sketching nets, completing measurements, calculating volume and surface area, etc). I don't see much value in doing the same exact thing we did in class, but perhaps that practice done completely independently will be more memorable.