## Mathepragmatics

Posted by Neal on April 1, 2012

We hear a lot about partisan gridlock in Congress these days, so it’s nice to hear about times when Congress (or at least the House of Representatives) can lay politics aside and just get the job done, as they did in 2009 when they declared March 14 National Pi Day. Adam’s math teacher has taken the resolution to heart, and spent a couple of weeks in March teaching his class lots of formulas involving π, including both surface area and volume of spheres, cones, and cylinders. Adam asked for some help on that cylinder worksheet, and I found that the poor guy had actually been doing the multiplication of π by hand, instead of just doing like you get to do once you hit pre-calculus and leaving all the answers in terms of π. Furthermore, he was doing them by hand, and worst of all, he was doing it twice in the formula (π*dh*)+(2π*r*^{2}). The first thing I did was convince him that (π*dh*)+(2π*r*^{2}) was equal to 2πr(*h*+*r*), so at least he’d only have to multiply by π once.

He did the first six problems, which all had labeled diagrams of cylinders to go by, but got confused when he got to the last three problems, which moved from diagrams to verbal descriptions of various cylinders. In particular, he was uneasy about #8, which asked for

the surface area of the outside of a cylindrical barrel with a diameter of 10 inches and a height of 12 inches.

Just looking at the words, I figured Adam would have to start by finding the surface area of the side: 120π square inches. Then there would be the ends to consider. Assuming the thickness of the side was negligible, the surface area on the inside would be the same as on the outside. So two ends, each with a surface area of 25π square inches. In other words, the exact same procedure as finding the surface area of any cylinder.

So why did the worksheet creators go to the trouble of asking about the **outside** of a barrel, when the outside and inside surface area were going to be the same anyway? That seemed like a pretty clear violation of the Maxim of Relevance.

Or was it? As seeming violations of rules of conversation will do, this one made me look a little closer at the situation. I looked at problem #7, and found that it was asking for the

surface area of a can with a radius 4cm and a height of 11cm.

I looked at #9, and found that it was asking for the

curved surface of a D battery with a diameter of 3.2cm, and a height of 5.6cm.

And with that, the task in problem #8 became clearer. In #7, you had to find the complete surface area of a cylinder; in #9, the surface area of a cylinder without its top or bottom. If #8 were about a cylinder with a bottom but without a top, that would make a nice progression, and I was convinced that that was what the worksheet creators had been after.

## Glen said

I hope Adam wouldn’t have been counted wrong for failing to assume the top of the barrel was open. The wording is highly ambiguous.

## Philip Whitman said

Would he have been allowed to say something like, “Assuming the bottom of the barrel is closed, and assuming that the diameter given to be 10 inches is the outside diameter, and assuming that the top of the barrel is open, then the outside surface area would be 145 pi; but, assuming the top is closed, other things being the same, then the outer surface area would be 170 pi.”?

## the ridger said

Yeah, the words “can, barrell, battery” (especially with that “curved part”) are the keys. Someone’s trying to get the kids to think about what they read instead of blindly applying the same formula three times in a row. Nice.