Today I issued a challenge to my 8th period class: Which group can construct an open box with the largest volume?
I explained that the box was to be constructed out of an ordinary sheet of paper – 8.5×11 inches and that they would need to cut out a square to fold up the corners.
Those were the only instructions I gave. I then handed out paper, rulers, and scissors. The only question they had was if the winning group would get candy. I suggested bragging rights. They wanted candy.
Then I walked around and observed.
Some groups immediately began cutting out squares.
A few groups argued about how they should approach the problem.
Some groups started making tables — length, width, height and volume.
They worked for about 15 minutes on this (much longer than I thought it would take my honors students). One group who had been making tables came up with the formula y=(x)(11-2x)(8.5-2x). I asked them what x and y represented. They told me. I said “So you have an equation that represents volume and you’re trying to find the maximum volume. Huh.” and walked away.
Two minutes later I asked for each groups’s volume. One group reported 72 cubic inches. The group with the formula told them it was impossible. Then they explained why.
They got their candy and I handed out the homework. Max/min modeling problems. As students were reading it I overheard “Oh, I get it. We just did this with scissors.”
I’m hopefull that they will have less difficulty than classes have had in the past with modeling. We’ll see tomorrow.
Continuities 
Nice setup! I like it.
Do you get many kids get super mad when you “won’t tell them the answer”? I do.
“So you have an equation that represents volume and you’re trying to find the maximum volume. Huh.”
We really do need to teach a common prep. I love your approach there.
I’ll take a stab at answering Kate’s question. Kids who have never had you before get frustrated with your “non-answers” but by the time 4th quarter rolls around, they come to expect it. Kids who have been in your class before would be shocked for you to give a straight answer.
Do I get candy?
Bragging rights aren’t enough for you either David? I’ll provide the candy when we meet. (I’m hoping that is a when and not an if).
Kate – as David said by this time in the year, they aren’t mad. Heck, these guys were happy to get a hint. I’ve found that freshmen get used to it quicker. Upperclassmen… it depends on who their prior teachers were. They have more habits to unlearn.
Update: They did well with the modeling. A bit of cloudiness with the domain in the context of the problem – but this GeoGebra file really helped. (Thanks David)
I’m sure it’ll be a when. I’m wishing I was going to SD for the NCTM conference since it’s right down the road (about 3.5 hours) but time with the family is just way too precious right now–and $ is tight in the district.
Glad the GGB file helped.
And I like Peanut M&M’s. 😉
Your posts make me really miss teaching math. That’s a good thing.
If you keep writing, I’ll keep reading.
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