8 comments on “Can we make an even “edgier” brownie pan? What about the “perfect” brownie pan?”

Can we make an even “edgier” brownie pan? What about the “perfect” brownie pan?

Artifact

This, my friends, is part math, part food, part art, all deliciousness:

It’s the all edge brownie pan, which I found from my new Favorite Website of All Time, Reasons to Go Broke. Here’s the description from the Amazon page (perfect 5-star rating):

“For corner brownie fans and chewy edge lovers, it’s a dream come true — a gourmet brownie pan that adds two chewy edges to every serving!”

2012 just became the best year ever.

Guiding Questions

  • How can we measure the “edginess” of this brownie pan?
  • What would happen if you added a couple more horizontal partitions?
  • What if you liked the center brownies? Could we make a pan to cater to these monsters?
  • Similarly, what if you like brownies with three or four edges?
  • Can we make an even “edgier” brownie pan by adjusting the partitions?
  • Does the edginess change if we increase or decrease the dimensions of the pan?

Suggested activities

  • Develop a metric for the “edginess” of a brownie pan. I’m thinking surface area-to-volume ratio should do the trick.
  • Plot the number of partitions against the “edginess”.
  • Use Google Sketch Up to make a model of this brilliance.
  • (Just go with me on this one) Take a poll. Figure out how many people like 1-, 2-, 3-, 4-, or zero-edged brownies, then challenge the class to make the “ideal” brownie pan.
  • Make awesome brownies.

I’d also be willing to bet that someone more skilled than I at Geogebra could make a construction of this, complete with a diagram and a plot of partitions vs. edginess.

The more I think about it, the more I like that “ideal” brownie pan idea. But here’s my question: are there people out there than think two is not the ideal number of brownie edges? My fear is that the “ideal” brownie pan has already been made. And it’s available for $34.95 at Amazon.

6 comments on “How can we measure the egregiousness of gerrymandering? Geometry, Perimeter, and Area”

How can we measure the egregiousness of gerrymandering? Geometry, Perimeter, and Area

Artifacts

This NY Times article/interview conducted by FiveThirtyEight.com’s Nate Silver and David Wasserman, House editor of the Cook Political Report. Particularly this snippet:

And/or this slideshow from Slate showing the most gerrymandered congressional districts in the nation.

Here’s my favorite from Illinois:

It’s worth noting that by federal law, congressional districts have to be “contiguous.” That means that (apparently) you can have a sidewalk connecting two blobs and you can call that contiguous.

Guiding Questions

  • What is gerrymandering?
  • Why do political parties do this?
  • Do the political parties that gerrymander egregiously like this ever get punished?
  • Is this really legal?
  • So, how can we measure “compactness”?
  • Would our measure of “compactness” yield the same top gerrymandered districts that Slate came up with?
  • What are the least gerrymandered districts in the U.S.?
  • How do we find the area of irregular shapes?

Suggested activities

  • Time to get you maps, compasses, colored pencils, and rulers out, just like Lewis and Clark.
  • Have students develop a metric or method to measure the “compactness” of a congressional district, particularly their own.
  • Students may want to research a history of gerrymandering, or the gerrymandering that has taken place in their state. I say, go for it. It might cross over into History and Political Science, but that’s definitely a good thing when it comes to HS and MS students.
  • The intended audience for such activities could range from politicians to political scientists to community board members. (aside: if you really want to rankle people in the community, I suggest looking at gerrymandering for school zones)
  • Assign each group a congressional district and ask them to develop a case for why theirs is the most egregious example of gerrymandering using geometry.

Potential solutions

  • While there isn’t necessarily one true, correct solution, it seems to me you could come up with a perimeter-to-area metric to measure the compactness, sort of analagous to the surface area-to-volume ratio that dictates how quickly, say, ice melts.
  • Or the centrality of or distance between the population centers. Perhaps analogous to the center of gravity, you can imagine if you were to cut out the shape of a congressional district and add mass by population and location, the more unstable it is, perhaps the less compact it is. (note: this is not necessarily true, but worth investigating)
  • Other potential metrics for students to measure: The distance between a population center and the borders of the distrct. Is there a way we could assign a penalty for districts who have conspicuous gaps or, as in the case in Illinois above, a tiny stretch of road (or something) connecting two large areas?
  • Ask each group to come up with a congressional district map for their state that is less gerrymandered than it currently is and demonstrate why mathematically. You could certainly envision a scenario in which an actual politician or political scientist may want to sit in on a panel. Certainly, your students can come up with a more mathematically appropriate map of congressional districts than this:


  • Note: as for the least gerrymandered congressional districts, that has to be a tie for first place between Wyoming, Montana, etc. because, well, you know….

How else could we measure the compactness and/or the egregiousness of gerrymandering?

2 comments on “A single robot-made rainbow ; what does it mean?”

A single robot-made rainbow ; what does it mean?

Artifact

This amazing video of a rainbow-painting robot. (h/t: Science Friday.)

Guiding Questions

  • How much paint did this guy need of each color?
  • What’s the radius of the rainbow? or, What’s the length of the arm that moves in a semi-circle that paints the rainbow?
  • Which color will run out soonest? And how much sooner?
  • What would the rainbow look like if he peddlded while the spraypainting arm was in action?
  • Could we do this for our integrated Math-Science Engineering final project?

Suggested Activities

To be decided, but here’s a screen shot if it helps.