geometry – emergent math

More math food blogging: I may need some help from my Southern friends.

Posted by Geoff in food, geometry, tasks, triangles

I think I may have an eating problem. Or just a eating mathematically problem. Here’s my problem today.

Delicious, delicious pigs-in-a-blanket (from pillsbury.com):

Pigs-in-a-blanket, for the uninitiated, are little hot dog/sausage type things warmly embraced by crescent rolls dough. In fact, that’s the ingredient list:

Little sausages.
A can of crescent rolls dough.

Cooking instructions: Wrap those little buggers up and toss them into an oven until you can’t stand it any longer.

At least, that’s how I’ve always made them. Maybe I could get super-ambitious and make my own dough but that sounds a lot of work for breakfast (side note: yes, this is a breakfast food).

Here’s the problem. How am I supposed to cut this triangular piece of dough to ensure proper sausage coverage?

Like this, this, or this? Or none of the above?

I can’t seem to get congruent triangles out of this thing. So I end up with mismatched pigs-in-blankets. Some have too much dough, some have too little. Many don’t wrap properly.

Awful. Just awful.

Like I said, I can’t get the triangles to come out congruent.

Not only are the triangles not congruent, they’re not similar at all. They’re not even the same type of triangle. So I need advice on a few levels.

How can I cut the initial right triangle dough in order to get:

The most congruent-like triangles?
The most similar-like triangles?
Obtain congruent and similar triangles that make for easy sausage-wrapping?

Here’s what I start with.

I want to end with those perfectly covered pigs-in-blankets above. How to I get from start to finish? Please let me know in the comments or tweet me a picture of the proper triangle-slicing orientation.

Sprinklers, Circles, Sectors and such (my first real foray into Geogebratubing)

Posted by Geoff in area, circles, geometry, sectors of circles, tasks

First off, I basically stole this from @NatBanting. And when I say “basically”, I mean “entirely.” Here’s the original blog post where you can see his frustratingly boob-shaped backyard. I tried to video my backyard but I didn’t have a good ladder to get a good vantage point. And my roof is terrifyingly steep. And I’m scared of heights. So I let him do the grunt work and I sat back and tried to figure out Geogebra on the fly. I’ve used Geogebra before, but not quite in this capacity, and to this degree. Huge thanks to John Golden (@mathhombre) for his Geogebra session in Grand Rapids a few months ago for encouraging us to create and share. I created the shape of the yard using inkscape (inkscape.org).

Now, I have a sprinkler system, rather than a hook-up-to-the-hose kind of sprinkler. So once you place them in the ground, you’re sort of stuck. Therefore, it’s much more imperative that you place your sprinkler heads correctly to minimize the amount of water overlap and the sprinkler cost and maximize the amount of water coverage. That’s what this task is intended to do.

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Geogebratube: Sprinkler task

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As for facilitating a task like this, I’d consider having students attempt this using paper and pencil first, make predictions, share their work, before giving them this Geogebra worksheet. In fact, you could start with Nat’s original post using a mobile sprinkler, then throw in the installation of the sprinkler system as an extension (or twist).

Also, you could easily modify this task with several different yard shapes (possibly by someone who isn’t afraid of heights). You could also scaffold nicely by starting off with just the 360 sprinklers and then slowly bringing in the sector sprinklers. If you want to go crazy, you could extend this into a project with PVC pipe distance and such.

How would you use or modify this task for your classroom? Let everyone know in the comments!

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Epilogue: It Takes A Village (To Develop Curriculum)

This is curriculum-creation in the 21st Century: Nat created the task. John taught me how to better use Geogebra. I just took their stuff and ran with it. This may warrant an entire blog post, but for now, I just wanted to highlight this.

Red Solo Cup / I Lift You Up / Let’s Find the Volume! / Let’s Find the Volume!

Posted by Geoff in geometry, proportion, tasks, volume

Hat tip to @mathhombre.

Artifact

The red Solo cup, apparently.

Guiding Questions

What are those little markings between 12 oz and “Top Me Off, Bro”?

Can you use proportions to find it? Similar polygons? Volume? Help me out here. Can I have a ruler? Can I use Geogebra to diagram this? Wait, can’t I just physically get my hands on one?

Actually, let’s do this in a manner that A) won’t get us fired, and B) doesn’t have that obnoxious “BroBible” stuff down on the bottom right.

Or even better than that, have an actual one on hand with some rulers hanging about, just for good measure.

The good news is, this problem has a real nice, testable solution: measure out the ounces for the attempted solutions and fill her up. Were your students right? If not, were we too low? Too high? Where did you go wrong?

Just make sure that when your students are working your next school fundraiser, they don’t overfill the red Solo cups with too much orange drink. That’s where the money is made and you want to be precise, after all.

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

Posted by Geoff in food, geometry, perimeter, surface area, tasks, volume

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.

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

Posted by Geoff in area, civis, geometry, perimeter, tasks

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?

So, what exactly am I supposed to eat? The new MyPlate icon vs. the classic Food Pyramid vs. Geometry.

Posted by Geoff in area, geometry, projects, tasks

A month ago, I was considering writing a post on the old (now “old, OLD”) food pyramid – you know, the one we all grew up with – and the new (now “old”) food pyramid, unleashed in 2005. It would be about area of triangles and trapezoids and Geometry and possibly graphic design.

See, here’s my tweet about it:

I was all set to contest that the new (“old”) food pyramid, adopted in 2005, was garbage mathematically and visually. And the challenge was for students to come up with a better, more mathematically accurate, food pyramid.

Then the United States government dropped the new MyPlate diagram in my lap.

Visually, and graphic design-ally, I think it’s miles better than the new (“old”) food pyramid, where you had no idea what each of the 137 slivers meant and exactly how much area was each of the 137 slivers. Also, it was unclear why it was a triangle at all? Why is that person climbing that pyramid of food? Are you supposed to eventually eat less and less until you eventually eat nothing?? Not sure if it’s mathematically better. Area of a sector of a circle whose vertex doesn’t meet in the exact center? Now THAT’s a mathematical investigation!

So I think, without having done any calculating, the new MyPlate is better. But you know what? That’s probably something for the students to decide, right?

Artifacts

Food Pyramid #1 (Classic)

Food Pyramid #2 (Post-modern?)

MyPlate (2011)

Guiding questions

Which of these government sponsored food diagram is the “best” and why?
So, how much of each food group are we supposed to eat again?

Suggested Activities

I would start by hosting a class discussion on what would make a diagram the “best.” Have students develop a rubric before you even begin. Potential categories: mathematical accuracy, ease of understanding, etc.
Have students find the area of each piece of each diagram and report back what the heck it means. You might want to assign different groups a different diagram, or not.
Have students attempt to craft their meals according to the three food diagrams for a week. You might want to assign different groups a different diagram to emulate, or not.
Compare the diagrams to each other: are they trending in a healthier or unhealthier direction?
The areas are representative of a 2,000 calorie diet. What if you had a 2,500 calorie diet? How would we dilate the pieces of the diagram?
Have students use their local and regional produce to create a meal based on the MyPlate (and contrast it with the Food Pyramid?).*

There are lots of different ways I think you could go with this as far as guiding questions and activities, but invariably it will come back to the area of these shapes. And tasks that have several entry points and investigation/exploration opportunities for students that require a mathematical understanding are always worth the time.

How else might we use the Food Pyramids and MyPlate diagrams to better math understanding?

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* Update: Commenter Sneha suggested this activity. I love the idea. Connection to Geography, anyone?

The Pizza Casbah 30-inch pizza challenge

Posted by Geoff in area, geometry, proportion, tasks

This is a picture of a single slice of pizza from my favorite pizza place in Fort Collins, Pizza Casbah.

My god that looks amazing. I’m getting hungry just looking at it.

This is a picture of an entire Pizza Casbah, 18-inch pizza (presumably, that means the diameter of the pizza is 18 inches):

It seriously is amazing pizza. And I can eat a lot of it.

A couple weeks ago, a friend and I were ordering a pizza from the online Pizza Casbah menu and we saw this (click to enlarge):

Now, I don’t mean to brag too much, but I really think I can eat anyone under the table when it comes to Pizza Casbah. I’ve never really gone head-to-head, or truly pushed my limits, but I’m fairly confident I can eat an entire 18-inch, 5-topping pizza by myself.

I really want my picture up on their wall-of-fame. I also really want a gift card for more Pizza Casbah.