Last week we started working with 3D figures, surface area, volume, etc. We went through developing all of the necessary formulas, practiced using them, and the whole class was a set of all stars. So the natural thing to do was move apply these beautiful concepts. Initially I thought of starting this unit with application and developing everything from examples. Instead, since we just came off the area/perimeter unit, we used those skills to develop everything necessary for this. In my opinion, these two units are the easiest to find applications because everyone has to paint a room or put something together at some point in their lives. The natural progression was to throw some 3 Act problems at my students and watch what happened.
They started with Dan Meyer's world's largest coffee mug. The questions started flying, I provided information, and they were off to the races.
I like the image below. When I noticed the approach they were taking, it allowed me to generate some decent discussion among the students.
They took their mug and split it into 2D boxes, or so it seemed. In speaking with the group, they were thinking in terms of 3D cubes but they had not taken into account the fact that cubes would not completely fill in the mug. They actually were having trouble finding a way to make up for the curved walls of the mug. For some reason, thinking in terms of a cylinder did not occur to them. In our dialogue, they did have that 'Ah ha' moment and realized an easier take on the problem. I could tell that they were thinking in terms of volume, and they were keeping in the 3D realm, but breaking a round surface into cubes creates a few problems. The good news was that they recognized something was wrong and were problem solving in how to make up for the missed area, they were just having issues coming up with a clear solution. This kind of outside-the-box thinking makes me happy because even though there was a much easier solution, they didn't give up. They got stuck, but they kept the discussion going and tried new ideas. I believe if I wouldn't have come over, they would've continued working and come up with something that brought them closer to the answer, but not quite a fully correct one.
From this point we moved onto Dan's water tank. After showing Act 1, I actually got the question, "Do we have to watch this thing fill up? Its going to take forever!" I had never done this problem with my students, but thinking back to Dan TED talk, I got the exact reaction he described. Before Act 1 was even over, they were complaining about how long it would take and some actually starting focusing their attention elsewhere. When the video stopped, everyone erupted into, "Hey wait, isn't it going to fill up?" "Uh, Mr. Brandt, the video broke." "I wanted to see how much water their was!" The questions flowed naturally and they got to work. Below is a mashed up, kind of, gross, panoramic-type view of my board when all was over.
I had the students present their solutions because I noticed many different approaches. Throughout their work, I got the question of "How do you calculate the area of an octagon?" numerous times, to which I responded, "I don't know, but I can tell you its a gross formula." Some students wanted to look it up, which I allowed (most were not successful in finding it and those that did couldn't use it correctly). Others broke it into familiar shapes and went from there. Some went for 16 right triangles, some 8 isosceles triangles (both recognized the need for trig without prompting in these processes, which was sweet)., and some broke it into a rectangle and two isosceles trapezoids (which for some reason I never thought of). There was actually only one group that did this last thought process and it came from a group that I did not expect, which added to the awesomeness. Each group explained their thoughts and why they approached it how they did and everyone learned something. I could see on the students' faces surprised looks of enlightenment as other groups were presenting; it was a beautiful thing.
As I explained to my students after this problem, this is what pushed me towards math. I love the fact that every problem can be solved in numerous ways, some more creative than others, but they all take different aspects of math and combine them to form a solution. Its wonderful how all of these numbers and concepts fit together into a nice, complex package that is eerily consistent. This is exactly what I want to push onto my students. I want them to see where it all comes together and to see that any problem can be solved, even if you don't have the knowledge that you think is required. Through these 3 Act problems and allowing the students the run the curriculum through questioning, I believe I'm on my way. After my speech to them, I didn't have any students saying "Yeah, but you're a nerd. You're a math teacher, you have to care about this stuff." Instead, I saw heads nodding in understanding, as if to say, "Yeah, this is kinda cool."
I loved the way this day went. After these two problems, the students saw me project Dan Meyer's google doc list of 3 Acts, and they started requesting them. They were begging me for more problems! As a strict teacher that sticks to the curriculum, I said "Nope." NOT! We did some more! I teach geometry, but we were doing problems that weren't geometry related. Why? Because the desire to learn was present. No one in the room cared that we were moving on to topics outside of our current area of study because they wanted to be perplexed and solve some problems. If this isn't the goal of the teacher, I don't know what is.
All in all, it was a pretty sweet day.
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