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.

And I’m not talking about the fact that he was a math cult leader. The Pythagorean Theorem is a tough one to teach in a “student-driven” or “discovery” fashion. It’s not very intuitive. It’s not like a high school student will be walking along one day, see a ladder propped up against a building and go,

Aha! A²+B²=C²!

This is ironic considering how widely used it and its parent, the Law of Cosines may be applied. So under the suggestion of a colleague, we’ll be looking at Pythagorean Theorem explicitly for the next couple posts. If you have any awesome ideas or stuff that your students have enjoyed in the past, please comment below or email me.

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.

This is a blog dedicated to math instruction and curricula. You always hear “Math is Everywhere!” from Math teachers and wall posters that came with their new set of precious textbooks, but often it’s more difficult to than you’d hope. And it’s even more difficult to get students thinking about Math outside of the Math classroom. With any luck, we’ll hopefully be able to provide some inspiration for interesting math projects and problems found as we travel through life. Think of this blog as a Math brainstorming site. Ideas are just going to be thrown out there. A lot of them will be discardable, but hopefully a few of them will inspire teachers to more fully develop them and use them in their class.

Some ideas/posts will be more fleshed out than others. Some posts will just be links to or pictures of something we think might be Math relevant, but we’re not quite sure how. It’ll be up to you to fill in the blanks. Sometimes we’ll even provide solutions to problems.

Typically, we like to adhere to student-driven learning. That is, we like for students to come up with the relevant questions upon seeing an artifact. Often it will take a little teacher guidance, but when students come up with the questions, good things happen. Therefore, a typical Emergent Math post will look like this.

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[Artifact: i.e. a picture, a video, a fact, etc. Just basically anything that exists.]

[Guiding Questions (GQs): questions that your students have about the artifact, possibly some teacher questions sprinkled in.]

At what age does the kink in the graph occur?

Would the slope eventually decrease?

What are the units for “eyebrow density”?

[Suggested activities (optional): what will the students do once the GQs are made manifest?]

Discuss.

Estimate eyebrow density of people of various ages.

Take measurements?

[Solutions to GQs (optional)]

47.

No.

Hairs / m²?

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Lastly, we’ll try to keep the commentary to a minimum. Let the administrators worry about that. We’ll also try to keep it PG or PG-13 at the most mature, but no promises there. After all, I used to be a teacher and now work with teachers.

Still in its infantile state, Emergent Math will certainly evolve as it gets older, we hope for the better. See more about the blog here and the author here.