# Limited Area, Maximum Space

Simple questions with not so simple answers have been growing in the classroom. I’ve seen multiple takes on the classic “which glass has more” from Piaget, to Meyer, Timon Piccini, and I’m sure plenty of others. I figured we could also do the same type of thing lo tech style.

Hook:

You have to use **half of a standard size sheet of paper**. What is the **largest** **cylinder** you can make ignoring the top and bottom lids (bases).

In other words, paper = lateral area.

There are at least 2 ways of splitting a sheet of paper. Many teachers refer to them in similar foods:

versus , versus, or whatever else the students can come up with. I started this prompt with my first class and quickly had to apply some planned scaffolding.

*Mr. Butler we don’t know the radius? Can we use a ruler? What is the height? *and of course…*I don’t get it.*

We rebooted and brainstormed what we did know about what we had:

- The base is a circle
- The wall (lateral area) rolls out to be a rectangle
- Area of a Circle = Pi*r^2
- Circumference of a Circle = 2*Pi*r

- I can use the grid to measure the height but how do I find the radius?
- Do I even need to find the radius? Can I answer the question with just the height and the size of the paper?
- Can I use the grid to measure the radius?
- Mr. Butler, I can just stand up the cylinder like this and look down. I can see the radius across the inside. Can I just use this to measure it? (He was referring to the diameter, but I got the point, more importantly this caused me to determine the point of the whole activity:

- increase fluency and familiarity with the formulas for surface area and volume of a sphere? …or…
- practice using a tool (grid paper) appropriately and apply problem solving?

Procedural Thinking | Problem Solving |

Practice with the formulas. The abstract concept will apply to any situation. Students can’t use grid paper on the standardized test |
Encourage and support practical application. Using tools appropriately is a universal skill. Life beyond high school will depend more on tools than abstract concepts. |

**Side note: This question was pretty elementary, with more of an intention to rule out one edge and focus on the other. There’s probably a better way to set that up with answering my own question.*

*Crickets. Wait time. Wait for it.*

*Crickets. Wait time. Wait for it. Wait…*

**What matters more in the volume of a cylinder?**

**The radius or the height?**

**Is it the same for Surface Area?**

Posted on April 13, 2014, in Geometry, Good Teaching and tagged geometry, graph paper, optimization, simple question, surface area, volume. Bookmark the permalink. Leave a comment.

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