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

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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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.

## Why doesn’t Nike+ use math to encourage me to run?

Artifact

The Nike+ app, which at the end of my run the other day, looked like this:

(editor’s note: yes, I’m slow. Thank you for noticing. Also, along with some encouragement in data format, I had Tim Tebow give me words of encouragement for bettering my pace.)

Now, there are a lot of numbers here, but I’m primarily interested in that last piece.

“You ran 0.10 mi more and 0’56″/mi faster than the average of your past 7 runs”

Why did Nike+ choose my past 7 runs? Was there some sort of algorithm to maximize how good I feel about myself?

Sadly, I think not. Witness my previous run:

So it looks like it just takes your past 7 runs and compares your mean distance and pace. That’s not very good motivation, now is it Nike+? Can we improve (at least, in my opinion it would be an improvement) Nike+’s distance and pace comparison to help the runner feel better about his or her progress?

Guiding Questions

• Would a different measure of central tendency lead to a different, and perhaps more encouraging, data capture?
• Would averaging a different number of past runs lead to a different, and perhaps more encouraging data capture?
• Could we write an IF…THEN or other type of algorithm to encourage the runner?

Suggested Activities

• Give some runner data (either fabricated or authentically generated; shoot, you can use my data if you want) and ask students to describe after each run “what should the app say in order to give the runner a sense of accomplishment?”
• Once students have done that with individual data points, have students sketch out an algorithm or decision tree.
• Test that algorithm or decision tree against a new set of runner data.

• Compare decision trees and algorithms to see who’s is the “positivest”(?).
• Turn into algebraic expressions if you want, presumably to help out the coders.

There are few things more discouraging than seeing that I’m actually running slower than my seven previous runs averaged out. At least package the data so I don’t feel like I’m out of shape.

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

## Underground parking spots cost \$30000-\$50000 to build in D.C. This smells like a systems problem.

Artifact

The Washington City Paper has a (rather lengthy) post on parking in D.C. Fair warning: it’s pretty wonky with zoning rules, ordinances, etc. However, the numbers caught my eye:

An underground parking spot costs between \$30,000 and \$50,000 to build, and residents pay for it one way or another.

“Let’s say it’s \$100 per month. If you built the parking space, and it cost you \$40,000, \$1,200 per year doesn’t cover it,” says Four Points Development’s Stan Voudrie.”

….

On the supply side of the parking problem, costs are fixed: You can’t dig a hole and line it with concrete on the cheap. Demand is more dynamic, and to some extent, it responds to price. Unlike in suburban areas, most District landlords don’t pair spaces with the unit, which means that tenants pay between \$100 and \$300 more per month for their cars.

As these numbers appear, I suppose it’s more of a simple equation problem, but I feel like we could easily transform this into a nice, in-depth linear systems problem. For example you could contrive or find the cost of a garage parking spot and monthly fees, and figure out which one will pay itself off faster.

Basically, I’m just excited to finally find a potential systems problem that doesn’t involve cell phone plans.

Guiding Questions

• How much do underground parking spots cost in our area?
• How much do parking garage spots cost?
• And what are the monthly costs of renting a spot?
• How long would it take to pay off each spot?

Suggested Activities

• Depending on how in-depth you want to get into this, you could turn this into some sort of apartment building project, or you could restrict it to a few-day investigation, having student find when each type of spot will pay for itself.
• Again, if you really want to dig into this, you could ask student to develop a pricing system for their parking spots (i.e. would underground spots be worth more? what about spots closer in?).

Aside: this is like my fourth post on traffic or parking and mathematics. I’m starting to wonder if PBL could stand for “Parking Based Learning”.

## The FAA wants to “take a fresh look” at rewriting the rules on electronic gadget usage on planes. How many flights equals “a fresh look?”

Artifact

Check out this NYTimes article. Apparently there’s some encouraging news for those of us with e-devices, which is everyone: the F.A.A. is going to review the rules for takeoff and landing whilst using particular electronic devices. Surprisingly it appears as if airlines could start allowing electronic devices right away but would have to test the devices themselves. But not only the devices, but, well, I’ll let you read:

Abby Lunardini, vice president of corporate communications at Virgin America, explained that the current guidelines require that an airline must test each version of a single device before it can be approved by the F.A.A. For example, if the airline wanted to get approval for the iPad, it would have to test the first iPad, iPad 2 and the new iPad, each on a separate flight, with no passengers on the plane.

It would have to do the same for every version of the Kindle. It would have to do it for every different model of plane in its fleet. And American, JetBlue, United, Air Wisconsin, etc., would have to do the same thing. (No wonder the F.A.A. is keeping smartphones off the table since there are easily several hundred different models on the market.)

Emphasis mine. That sounds like a lot of combinatorics and permutations to me.

Guiding Questions

• Which kind of device would require the least/most testing?
• Which airlines could conceivably do this in the least amount of time, with their fleet size?

A bit of research on airline fleets, a bit of googling on the different types of electronic devices (e-readers, MP3 players, etc), and you’ve got a nice permutations problem.

## “Equation Ratscrew”

I used to love “Egyptian Ratscrew” (also called “Egyptian Rat Slap” for the more easily offended I suppose). We used to play it all the time in my Physics/Calculus class. It combined the speed of “Speed” with the ferocity of “Spoons” without all the injuries.

True, I’m guessing “Equation Ratscrew” won’t engender quite the excitement of the original, but nevertheless, here it is. Might make for some decent equation practice.

## 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.