Continuing from last week, we have another potential Pythagorean’s Theorem Project/Problem. This one was sent in by Steve.

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AUNT BITTY’S GARDENS

Launch: My Aunt Bitty has a business creating “designer gardens”. These are beautiful little triangular gardens that fit into a particular space–usually the corner of a yard. You tell her your space, and she tells you just how to put in the rocks and plants to make it beautiful.

She is making models of different sized gardens that she wants to sell–and she has a problem.

There’s about a hundred different ways you could use the following artifacts to construct a lesson around Pythagorean’s Theorem. So I’ll just toss out all the artifacts and let you, esteemed teacher, take it from there. I’d love to get feedback and suggestions on how to implement these materials in the comments below.

The screen shot of the girl holding up one of the 5 most beautiful right triangles I have ever seen. (note: before math geeks go berserk, I know it’s technically not a right triangle with the extra bit off to the side, but still.)

So I had this simple little post in mind about the Tour de France and biking and elevation and distance and it would wrap up neatly into a little Pythagorean Theorem lesson. I start looking around for nice little diagrams much like this one, showing the distance and elevation. The idea was going to be simply, “find the absolute distance the bikers travel in this particular stage.”

But it turns out these diagrams are a sham. An absolute sham. You see, the x-axis already does represent the absolute distance. Take this stage, for example. Above is the “cross section” of the stage. And here’s the actual route.

Note the exact same distance peddled. So x-axis is in actual distance peddled, not simply the horizontal distance traveled, as any proper diagramer should do. I suppose it’s more helpful for the bikers to know the absolute distance they have to travel, but it’s …. it’s….. it’s… just wrong. In retrospect, I did sort of think these slopes seemed a tad steep….

As I mentioned last time, the Pythagorean Theorem is a difficult concept to have students discover intuitively. So we’re focusing on it specifically this week. If you have any activities or ideas, please let me know. Or tweet it to me.

Here’s one sent in by Kevin.

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A math nerd (Let’s say her name is Tara) decides that she is going to get a tattoo, and she draws up the following design for the artist to put on her back:

The colored ink costs $3 per square centimeter. Outlining in black is free if the color costs $150 or more. If the color is less than $150, then black outlining costs 50 cents per cm.

Tara wants those squares to form a right triangle like above, where the vertical leg is 5 cm long, and the horizontal leg is 7 cm long.

How much will the tattoo cost Tara? Make sure to write out all of your calculations.

Remember when in elementary school when teachers would instruct you to sit “boy-girl-boy-girl?” Well, it appears as if the United States Congress needs to be reprimanded for their childish behavior and must sit “Democrat-Republican-Democrat-Republican.” You see, Colorado Senator Mark Udall made a proposal that during the 2011 State of the Union (SOTU) speech members of congress would sit amongst members of the opposite party. This is in contrast to previous SOTU addresses where the house of congress was firmly divided and after each line half of the house would cheer like their favorite team is trying to make a crucial 3^{rd} down stop and the other half would sit on their hands and scowl. So members of congress accepted Udall’s proposal and immediately members of congress began trying to decide whom to sit next to so they could appear bi-partisan but so as not to sit next to anyone that’s an anathema to that politician’s base supporters (Seriously, they are children).

Anyway, as you may also know, Democrats lost a lot of seats in the 2010 mid-term elections after four years of steady gains. For the 2011 SOTU there were 242 Republicans (Rs) and 193 Democrats (Ds) in the House of Representatives; 53 Ds (and Independents who caucus with the Democrats) and 47 Rs in the Senate. As far as I know, the Senate and Congress sit together during the SOTU, so that’s a total of 289 Rs and 246 Ds.

So upon hearing that Ds and Rs were to sit together at the 2011 SOTU, I wondered a couple things:

Is it possible for every member of congress to sit next to a member of the opposite party?

If it is possible, how lopsided would the party split have to be before it becomes impossible?

I was hoping this wouldn’t be a simple discrete math problem. And as luck would have it, it isn’t (at least, I don’t think it is). Behold the floor-plan of the House of Representatives.

(note: not actually the floor plan of House of Representatives. At least I don’t think it is. And it certainly isn’t the seats used for the SOTU seeing as there are more members of congress than seats in this diagram. If anyone out there can find a better, more-correct diagram of the seats of Congress during a SOTU please email me. Below, I’ll start out using the entire congress, but eventually I’ll switch to just the House, and then I’ll declare that it doesn’t really matter.)

What a wonderful disparity of seats! There are rows of 15 and rows of two! There are things called the “Republican and Democratic Committee Tables”. And I’ve always wanted to know where the Tally Clerk sits.

Scene from Mel Brooks’ classic, Spaceballs. Start off by simply showing it (or any combination of the split scenes) to your students (apologies if there are advertisements, you can skip them in like 5 seconds):

Now, I’m not sure if you’d want to show the entire scene in a classroom setting, both because it’s rather long and because there are some, *ahem* cruder moments (“sir, she’s gone from suck to blow“). Personally, I think starting off the first five minutes of class with the entire scene might engage the kids, make them laugh, wake up, etc. But then, I don’t have any administrators or parents to answer to at the moment. Regardless, I’ve broken up the scene into five pieces, which I share below.

Guiding Questions (GQs)

Personally, this scene brings up a ton of questions for myself. Hopefully after watching the scene there will be several GQs from your students. Here are the two primary GQs (a.k.a. “Need to Knows” for you PBL types) that will lead to the mathematics behind this scene that I have.

How much air is in Planet Druidia?

How big is Planet Druidia?

How far above the surface is the “air shield”?

How quickly can Mega Maid suck the air out of a planet?

Did Mega Maid blow out the air faster than she sucked it in?

Solutions to GQs

Here is Planet Druidia again.

The two questions we really can’t answer for certain are “how big is Druidia?” and “how far above the surface is the ‘air shield’?”. But, you know what? Planet Druidia looks a lot like Earth to me, so let’s run with that.

What we need to find for the volume of air is the volume of a spherical shell. Or, the volume of two concentric spheres of differing radii.

So for this particular problem, we have

or,

where R=radius of Druidia/Earth+altitude of air shield and r=radius of Druidia/Earth.

The radius of the earth is about 6300 km (or 4000 miles). But how far up is the shell ceiling? Or, where does space begin?

According to famed astro-physicist and Nova Science Now host Neil de Grasse Tyson,

So, that’s 100,000 additional meters, or 100 km. So now the volume equation is

Which comes out to a volume of 50671795107 cubic km of “air”, which is how big Mega Maid’s vacuum bag would have to be.

However, I did leave out one potentially crucial fact that would add a whole extra level which we’ll revisit in the future: air is less dense as you ascend in the Earth’s (and presumably, Druidia’s) atmosphere. So I gave the volume of “air” in quotation marks, but as any good Coloradoan knows, there’s a lot less air at higher altitudes. At this point it becomes a density problem involving integration of the shell. As I said, we’ll revisit this problem in the future, but for now, since we’re just starting out, let’s stick with the volume problem. We’d hate to make a critical math mistake already.

Also unanswered is the rate of sucking/blowing portion of the problem. A stopwatch and some division should do the trick. Although, once we tackle the calculus portion of the problem we could get a nice interesting plot of Air in Mega-Maid’s Vacuum Bag vs. Time.