One day in a College Prep Precalculus class (I call it CPPC, which is a palindrome), we worked on a problem involving volumes of three different objects. Find a relationship for a sphere, a cylinder, and a cone that all have the same radius and volume. The students’ objection was that there were “all variables, no numbers!” It looked like this:
Solving the first pair of equations we got h1=4r3. The second pair yielded h2=4r. After that minor struggle for the day, I waited until the next class for us to continue the problem. I posed the question, what if I want that volume to be a specific number? We decided upon 150 mL as a “nice” number, representative of a small amount of liquid that I might want to give away as a sample of my new perfume! (fake news) Most students knew that 1 mL converted to 1 cubic centimeter, so our measurements would be in centimeters. They set about solving a new problem, like 150 = 43r3 to get r = 3.2961. Now, it became easy to find the heights, since we had just solved for h1 and h2! So, h1=4(3.2961)=34.3948 and h2=13.1844. A ha! Now, I query, which one of these shapes would be the most cost effective in terms of a “product giveaway”? (material needed to fabricate it) I asked the kids to go home and calculate the surface area of each shape. In the meantime, I asked the Robotics / Computer science/ CAD teacher and 3-d printing Ninja Ryan McMonigle if he could have a student make these objects for me with our constraints. Very cleverly, he even asked if I wanted the dimensions of the inside of the figures or the outside to match our numbers. Wow, I never would have thought of that! It turns out that the student who made them was one of my own students from a different class! It took a few days, but he did it, with his own modifications. He left the cylinder open so that we could pour water into it, he left the cone open and gave it a base to stand on, and the sphere has a tiny hole that probably could use a bigger drill bit to open it up some more! The sphere also has one “flat” side so that it sits still on my desk! (aren’t they pretty?) The students discovered that the smallest surface area would be the sphere.
Rumor has it that is known already by everybody but me. But it’s cool that the kids figured it out. Mr. McMonigle encouraged me to utilize his students’ skills again if I could….I actually did last year, too.
Here is a mini story about that: I was getting ready to do a presentation about some of my favorite real world problems, including one about the safe angle at which a ladder should be leaned against a wall. I really enjoy this lesson in class because I send out a teaser to the kids the night before: a video from OSHA about ladders in the workplace. The kids have NO idea what is going on! The next day, I send them a TI-nspire document with an interactive graphic of a sliding ladder, and the angles at which it is leaning. I ask them to slide it back and forth until they come up with a “range” of safe angles.
After that, I wish I could have the students interact with a real ladder, but I think it’s against a rule somewhere. I wasn’t allowed to do it at the presentation in a hotel/conference center, either. I guess nobody wants a lawsuit. So, I bought a mini ladder online that comes from a set of WWF Wrestling dolls and accessories. But then I wanted another. So I asked a student to make one for me on the 3-d printer. I guess it was a stretch for this kid to do it, and he tried very hard. When he brought me his finished product the day before I left for the conference, he was a little embarrassed, but I was so proud! I showed it around during the presentation, and gave it away to a teacher to take home.
The question I posed was do folding ladders come at a pre-determined safe angle? Extension ladders must be placed against a wall or surface at a safe angle by the human who wished to climb it. I have several pictures from my summer Mission Trips where kids are up on ladders at a worksite, and I am responsible for their safety. We “measure” the slope and angles using the TI-nspire calculators to determine if the ladders are safe. I will continue to use my mini ladder and the technology to engage students (or teachers) in this “real world” activity.