Mr. Lazur and the Volume of a Cone

I will never forget my favorite math teacher. Mr. Lazur taught ninth grade CAS Geometry (my school's version of AP) and also twelfth grade IB Calculus, so I was fortunate enough to have him as a high school freshman and then again as a senior. I'm incredibly thankful for Mr. Lazur because his fun and informal teaching style got even the most anxious students to actually enjoy math. In his classes I learned to think about math on a more “macro” scale, thinking about the concepts and how they related to each other rather than getting bogged down in numbers. He also knew exactly when and how to give a practical demonstration of a confusing concept so that none of us would ever forget it again.

One of these demonstrations has stuck with me ever since, and I don't think I'll ever lose the knowledge it provided. We were in Geometry, working on volumes of solids. The previous day we'd learned the formulas for volume for cubes and cylinders, and today we were supposed to be learning volume of pyramids and cones. Mr. Lazur came in from the hallway with a small plastic tub of water he'd filled up at the water fountain. He sat on the empty desk at the front of the room with his feet on the seat of the chair (his favorite posture for informally teaching a new concept) and set the tub beside him.

Wordlessly, Mr. Lazur reached into his pocket and pulled out four small hollow solids made of clear plastic. He set two of them aside and held the other two up. One was a cube, and one was a four-sided pyramid. He placed the bases of the two shapes together and then the heights so that we could see they had the same dimensions. Then, he began to dip the pyramid into the water, fill it up, and pour that water into the cube, using the pyramid to fill the cube. He did this three times, then held up the cube to show us it was exactly brimful of water. “So, three pyramids fit into this cube,” he said simply. Then he set them aside and repeated the process with the cylinder and the cone, showing us they had the same size base and height and then using three cones' worth of water to exactly fill the cylinder. There were murmurs of comprehension and quiet scratchings of pencils as some of us figured out his next move before it happened. Mr. Lazur went up to the blackboard and wrote the formula for the volume of a cylinder. As we watched, he inserted a “one-third” at the beginning of the formula and said “This is the formula for the volume of a cone. It's just one-third the volume of the cylinder with the same dimensions.” And we'd all just seen it proven, so no-one was confused.

Mr. Lazur always knew which topics would be hard to accept without proof, and never resorted to saying “because it is” when answering a student's question. He encouraged us to never accept rote memorization of a concept when there was an explanation we could learn. His classes often followed this format; beginning with a concept we knew, then walking us step by step through a process of reasoning, stopping along the way to make sure we understood the thinking behind each step. Eventually, he'd stop, circle an equation, and tell us “That's the formula for ___.” And we didn't need to ask “Why?” because he'd already shown us the proof.


That's beautiful.
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