You Win Some, You Lose Some

Recently, I have been doing Estimation 180 activities ( for part of my warm ups. For the uninformed, they look like this:

How many cheeseballs will fit in the tray?

Of course, there are video answers. Somehow, the medium hooks the students. I can see the anticipation and hear the groans as the video plays.

Since my Enrichment classes were starting on the concept of volume, I figured that this particular activity could be a good tie-in. After watching the video, I asked them, “What if there were two layers of cheeseballs? How many would fit in two layers?” Most of my students doubled the solution and shouted out an answer. I did have a couple students in each class say, “You can’t just assume that the cheeseballs will line up perfectly. It would be close to _______ but maybe not exactly.” This brought a smile to my face. At the beginning of the year, I (jokingly) shamed my students for making assumptions about triangles. “It LOOKS like a right triangle? So it HAS to be a right triangle? That guy LOOKS dangerous, so he HAS to be a criminal? Stop stereotyping triangles!” Over the course of the year, my students have gotten better at realizing when they are making assumptions. I have gotten more questions and statements such as “Can we assume that line is a diameter?” and “I’m assuming that this triangle is equilateral. Then…oh wait. It can’t be equilateral because…” This is a win.

Moving on from assumptions, I introduced the concept of volume as taking a 2-D shape (or one layer of cheeseballs) and stacking it on itself multiple times. We started off with simple prisms (rectangular, triangular, and circular bases). Eventually, we moved on to prisms whose bases were regular hexagons, pentagons, etc. Not once (across all 3 classes!) did I hear anyone ask for a formula. In fact, the next day, I heard a student explaining the concept to a student who was absent: “You just find the area of the circle. Since the circle is being stacked up 7 times, which is the height, you then multiply the area of the circle by 7 to get the volume.”

Volume of a rectangular prism: length-times-width-times-height

Volume of a cylinder: πr²h

Volume of a triangular prism: base-times-height-divided-by-two-times-height

Throughout the year, I have stressed to my students that understanding the concept would trump memorizing formulas. (Note: I am not against formulas. I have told my students that formulas make simple problem-solving more efficient. I am against memorizing formulas without any conceptual understanding.) After some practice, I finally did flash the formulas on the TV screen and explain where each piece came from ( πr² is the area of the circle, h is the height, etc.). By then, my students had already internalized the stacking of a base figure. I don’t think they are relying on plugging numbers into a formula. I could give them a heptagonal prism and they would be able to find the volume. They would not ask me for a formula (although they probably know my answer is “I don’t have one.”). This is a win.

I was just told that as of this week, there aren’t enough math classes to go around. I’m the odd one out. Last-hired, first-fired or something like that. This is (maybe not) a win.

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