A few years ago I was teaching a 6/7 split class. We had been exploring surface area and volume of prisms (rectangular so far), but were quite early on in the learning cycle. At the beginning of my math class I drew a picture of a rectangular prism and wrote the dimensions on the diagram like below:
I asked everyone to calculate the Surface Area. Really this was just a check for me to see if we understood surface area… the problem for the day was coming up.
However, as I walked around, I realized something VERY unusual. EVERYONE got the answer of 250… but not everyone did it correctly. Let me show you:
Take a look at the 2 answers above. Some students calculated the Surface Area correctly at 250 units squared… and others calculated 250 units cubed (they found the volume).
I apologized to the class for giving them the worst problem ever… THE ONLY POSSIBLE rectangular prism that has the same surface area as volume. Then one student commented, “Is that the only possible prism?”
I didn’t know the answer, so we started seeing if it was the only possible prism.
Over the next 100 minutes, every student in the class, some in pairs, some on their own, started drawing prisms and calculating the surface area and volume.
Eventually, a student told me that one of the dimensions couldn’t possibly be a 1. I asked him to prove it… After a few minutes he showed me several examples with numbers that were small or large. He attempted to find the limits:
He explained to the class that if there is a 1 as any of the dimensions, the Surface Area would always be larger. We explored why that happens.
Minutes later, another student told me that it was impossible to have a 2 as one of the dimensions. I asked her to prove it… She showed me the work that she had completed and shared it with the class.
She explained that the numbers were getting close if she chose 2 of the numbers identical, but the end pieces of the shape would always add up to more than the volume would.
After lots of trial and error… guessing and refining ideas, several students started finding possible prisms that also had the same Surface Area and Volume…
The class started noticing a pattern between the dimensions and found limits between the smallest and largest possible prisms…
By the end of class, all students had calculated dozens of surface areas and volumes… all students were making conjectures or testing out the conjectures of others.
My original problem, which I intended as a quick warm-up was not a quality engaging problem. However, I want you to think about what made this lesson better?
WHO posed the problem that day? Did this have something to do with the shared responsibilities that happened later in the lesson?
What if I just moved on? would the learning have been as rich?
Think about how the students picked the shapes they were testing? Some students worked in teams to work strategically… others made their own conjectures and followed those patterns.
In this lesson, choice was key, but the choices didn’t come from me… Students were working together to reach an ultimate goal, not in competition with each other… Conjectures were made and tested, not because I told everyone to, but because it served our purpose!
I think about Dan Meyer’s “Real World vs Real Work” a lot. Why were my students so engaged here? There was NO real world connection. That wasn’t what was motivating my students at all!
I also think we need to reflect on the level of cognitive demand we ask our students to be engaged in:
“Doing Mathematics Tasks” seems like something that is hard to do, yet, every one of my students were engaged in this problem… everyone eagerly searched for patterns, many drew pictures, used snap cubes, visualized what was happening… all in the name of better understanding the relationships between the dimensions of a rectangular prism, and its surface area and volume!
By the end of the class, my students had found exactly 10 rectangular prisms that have the same value of its surface area as its volume (using only whole numbers), and could prove that these were the only 10 possible.
I’d love to hear your thoughts about our problem… or why the students were SO engaged… or about the conditions that must have been present in the class… or how problem solving can be used as a purposeful practice of procedures…..