Monday, March 14, 2016

Pi Day (Day 122)

The best day of the year in a Geometry class is finally here! Today is Pi Day -- March 14th, to be precise, since the number pi begins 3.14. Today is all about the magic of pi -- mathemagic, since it's both mathematical and magical. And last year, I considered Pi Day to be the ceremonial launch date of this blog, making today the blog's ceremonial first anniversary.

As you may expect, today's Pi Day post is action-packed and full of many ideas:

1. Round It Up to Pi?
2. Lessons 8-8 and 8-9 on Pi
3. Subbing on Pi Day
4. A Final Comment on Area
5. The Third -- and Fourth Dimension?
6. Fun Stuff: Pi Day Links
7. Fun Stuff: Pi Day Videos

Round It Up to Pi?

This is what I wrote last year about Pi Day:

Let me begin the festivities with discussing my own discovery of pi. I first learned about the number pi when I was in the second grade. I remember learning about the rounded value of pi, 3.1416. Now I know that the last value has been rounded up (otherwise next year would be Pi Day of the Century), but back then, I didn't know that. I knew, though, that 3.1416 wasn't the exact value of pi. I once tried to guess the next few digits of pi -- I thought that the pattern continued, so that the digits would be something like 3.141618202224262830.... Now I know, of course, that this guess is wrong -- in fact, there is a rational number that starts out 3.141618202224262830... (it's 30791/9801), but of course, pi is irrational.

In the year since I wrote this, I notice that there's a movement to declare today to be Round It Up to Pi Day of the Century. For example, see the following link:

So somehow there are two Pi Days of the Century -- 3/14/15 for truncating the value of pi and 3/14/16 for rounding the value of pi. In some ways, this is similar to the First Day of the Century -- as there is no year 0 in the Gregorian Calendar, the First Day of the 21st Century was January 1st, 2001. On the other hand, the century of the 2000's consists of those years beginning with the digits 20, so the First Day of the Century of the 2000's was January 1st, 2000:

Leading up to the year 2000, the big century celebrations were planned for 1/1/00. Only after that date had passed did anyone think about the 21st century beginning on 1/1/01. Likewise, mathematicians were discussing 3/14/15 for several years leading up to 3/14/15. Only after that date had passed did anyone think about rounding pi to 3/14/16. In each case, the second date is the only day left to celebrate after the bigger parties a year earlier have already passed.

Lessons 8-8 and 8-9 on Pi

Recall that last month, I visited several other teacher blogs for ideas on lessons. One of these blogs has a lesson that's perfect for Pi Day:

Laura Lee is a middle school math teacher from Minnesota. Here is how she teaches her seventh graders about pi:

I teach CMP. I love discovering pi in Investigation 3.1 of Filling and Wrapping. It’s hands-on and engaging. Students love discovering that pi is a real thing not just a random number and excuse to eat pie on 3/14. If the lesson goes really well, I’ll even get a kid to ask, “Isn’t this a proportional relationship? Isn’t pi a constant of proportionality?” To which I want to use every happy emoji ever created in large, bold, capitalized letters with lots of exclamation points!!!
But then Investigation 3.2 comes along, where students discover the formula for area of a circle as pi groups of radius squares. I’ve never been able to get students to make sense of this lesson. There seem to be lots of understandings (what a radius square is, how to calculate how many times one number goes into another, understanding multiplication as groups of…) needed in order to get valid data that would allow students to conclude pi groups fit inside the circle. It just gets messy and doesn’t seem to make any sense to students, so this year I decided to take a different approach.
I decided to go with composing and decomposing. I would show them how to decompose the circle into a rectangle that is πr by r. Here’s how the lesson went (spoiler alert- a Domino’s pizza makes an appearance):
Notice that last year, I posted a lesson that actually covered area before circumference. Lee's lesson restores the order from the U of Chicago text, with circumference (Lesson 8-8) before area (8-9).

Let's just skip to the part where, as Lee writes, a pizza makes an appearance:

Method 2 (I had been talking about pizza all morning with my first 2 classes, so last hour I decided I should just order a pizza and use that to derive the area formula)
  • Order a pizza (Domino’s large cheese worked great!)
  • Reveal pizza to class, watch them go insane!
  • Have students gather around your front table
  • Slice pizza into 16 slices,
  • talk about circumference of 8 of the slices or half of the pizza: πr, record this on the pizza box
  • then start arranging pizza slices into a rectangle, listen to student “Ahas!” and “No ways!” when they see it is clearly starting to form a rectangle
  • Talk about dimensions of rectangle and then the area

The U of Chicago text does something similar in its Lesson 8-9. The difference, of course, is that the text doesn't use an actual pizza.

Lee writes that for her, the key is proportionality. This fits perfectly with the Common Core:

Prove that all circles are similar.

Then again, notice that Common Core seems to expect a proof here. How does Common Core expect students to prove the similarity of all circles without Calculus?

Unfortunately, none of our sources actually prove that all circles are similar. What I'm expecting is something like this -- to prove that two circles are similar, we prove that there exists a dilation mapping one to the other. For simplicity, let's assume the circles are concentric, and the radii of the two circles are r and s. So we let D be the dilation of scale factor s/r whose center is -- where else -- the common center O of the two circles. If R is a point on the circle of radius r, then OR = r, and so its image R' must be a point whose distance from O is r * s/r = s, and so it must lie on the other circle of radius s. Likewise, if R' is a point on the circle of radius s, its preimage must be a point whose distance from O is s / (s/r) = r, and so it must like on the circle of radius r. Therefore the image of the circle of radius r is exactly the circle of radius s.

Of course, this only works if the circles are concentric. If the circles aren't concentric, then it's probably easiest just to compose the dilation with an isometry -- here a translation is easiest -- mapping the center of one circle to that of the other. Therefore there exists a similarity transformation mapping any circle to any other circle. Therefore all circles are similar. QED

To get from the area of the unit disk (pi) to the area of any disk (pi * r^2), we are basically using the Fundamental Theorem of Similarity from Section 12-6 of the U of Chicago. This time, though, we are using part (b) of that theorem:

Fundamental Theorem of Similarity:
If G ~ G' and k is the ratio of similitude [the scale factor -- dw], then
(b) Area(G') = k^2 * Area(G) or Area(G') / Area(G) = k^2.

We skipped this formula back when we covered Lesson 12-6 because at the time, we hadn't learned about area yet. Although Wu attempts to prove a special case of the Fundamental Theorem of Similarity using triangles, it's much easier to do it using squares, as the U of Chicago does. If G can be divided into A unit squares, then G' can be divided into A squares each of length k. And the area of a square of length k is clearly k^2, so the area of G' must be Ak^2. For the circle problem k is the radius r, and A is the area of the unit circle or pi, so the area of a circle is pi * r^2. We can do this right on the same worksheet -- there's already a circle drawn of radius 10 times the length of a square, so instead of the length of each square being 1/10, let it be 1 instead. Then the area of the circle of radius 10 is equal to the number of shaded boxes, or 314, since the old unit square has been divided into 100 unit squares.

Subbing on Pi Day

Here's one more thing I wrote last year about today's lesson:

If it's possible, I wouldn't mind bringing in some pies (either fruit or pizza, like Lee) and having the students calculate arc length and possibly even the area of a sector -- to my surprise, the U of Chicago doesn't give the formula for area of a sector either -- by using a slice of pie. The arc length and sector area would have to be given before the student is allowed to eat the slice. Of course, it will be expensive to have enough pie for every student in every class -- but it may be worth it to have a Pi Day worth celebrating.

Well, since I wrote this last year, I actually have the opportunity to bring pie to school today! This is because today, I am subbing at that continuation school where I often work (you know, the one that often plays Jeopardy on Fridays). This particular program has only a few students who stay in two classrooms the entire day (only six were present today), so I can easily afford to buy enough pie for everyone today.

I asked the students to calculate the circumference of an apple pie with diameter eight inches (typical for a medium-size pie). Then I had them calculate the length of an arc formed by dividing the pie into ten pieces (one for each student, the rest for the adults). At the end of the day I served the students the amount of apple pie that they had calculated. (I know -- I actually prefer cherry pie on Pi Day, but there's already a cherry pie at home so I ate apple pie at work.)

If I were teaching the lesson to several full-size classes, I might bring a pizza to just one of the classes, just as Lee does. But I'd have the classes earn the pizza. For example, the class who earns the highest grade on the previous test would get the pizza party -- all other classes would have to use Lee's "Method 1" (the one without the pizza) to find the area. This is one reason that I posted a Chapter 8 Test last week, even though we hadn't finished the chapter yet -- the results from the test for Lessons 8-1 through 8-6 can be used to determine which class wins the pizza party.

A Final Comment on Area

Yesterday I watched a magic show -- the 60th annual performance of It's Magic. I actually attended this show last year, but didn't mention it on the blog because it has nothing to do with Geometry. But as it turns out, this year, one of the magic tricks is actually relevant to this blog!

The magician Kyle Knight begins with a puzzle, consisting of nine pieces that fit together inside of a rectangular frame. Then his wife Mistie starts telling a story about life as Kyle removes the pieces from the frame. Then Kyle takes a tenth piece -- a small red square -- and then starts to put the pieces back together. Amazingly, the ten pieces appear to form the same rectangle as the original nine, even though there is an extra piece! Then Kyle takes the puzzle apart again and adds an eleventh piece -- a red rectangle. If we take the length of each side of the red square to be one unit, the new red rectangle appears to measure one unit by two units. Finally, the magician puts the eleven pieces back together to form -- you guessed it! -- the original rectangle. He replaces the original frame to show that the pieces fit inside the rectangle perfectly, even though we apparently added three square units of area!

Actually, I somewhat know how the Knights performed the trick -- it's not magic, it's Geometry! Or perhaps I should say that it's mathemagic.

And the reason that I know the trick is because David Kung mentions it in his Mind-Bending Math -- in Lecture 17, "Bending Space and Time." Recall that I spent all of January discussing Kung's lectures, and we reached Lecture 17 back on January 30th. But I didn't mention this trick that day because I'd subbed for math that day and spent most of that post writing about that class. It makes sense to mention this trick today since we're still in Chapter 8 on area -- though I wish that I'd mentioned it last week when my posts were short instead of the middle of this super-long Pi Day post.

So instead, I post a link to the Missing Square Problem:

At the above link, Dr. Presh Talwalkar uses a right triangle with legs of length 5 and 13. The Knights, meanwhile, use a rectangle. Talwalkar explains that this problem is related to Fibonacci numbers, so I wouldn't be surprised that the Knights' rectangle is nearly a Golden rectangle, with side lengths of consecutive Fibonacci numbers.

The Missing Square Problem is interesting enough for the Knights -- two professional magicians -- to include it in their performance, so it might be nice to include a simple version of this in class.

The Third -- and Fourth Dimension?

Today's lesson marks the completion of Chapter 8, so now we move on to Chapters 9 and 10, which cover three-dimensional solids. We know that the important lessons here are on surface area and volume, which appear mostly in Chapter 10, so we'll mostly skip Chapter 9.

Meanwhile, I haven't discussed any books or DVD's on the side since Kung's lectures. Well, it's time for me to announce my latest side-discussion book.

We know from my last two side discussion works that Mandelbrot discovered the dimensions that lie between the well-known 0, 1, 2, and 3. But this leads to another question -- can there exist any integer dimensions beyond 3? Is there possibly a fourth dimension?

Kung alludes to the fourth dimension in some of his lectures. But about ten years ago, I first read a book called The Fourth Dimension: Toward a Geometry of Higher Reality by Rudy Rucker. As it turns out, Rucker also wrote a book called Infinity and the Mind, where he discusses infinity (which also appears in some of Kung's lectures).

And so I decided that while our Geometry students are learning about the third dimension in Chapter 10, you readers can learn about the fourth dimension here on the blog as we revisit Rucker's book, starting tomorrow.

Fun Stuff: Pi Day Links

Now let's move on to some fun stuff. Here are some Pi Day links. Let's start with a new link for this year's Pi Day:

According to this link, Pizza Hut is having a Pi Day math contest today where the top prize is pizza for approximately pi years:

Calling all math experts and Pizza Hut fans alike! National Pi Day is here and this is your chance to win free “pie,” that’s 3.14 years of Pizza Hut pizza (awarded in Pizza Hut® gift cards)! Take a look at the math problems below and provide your answer to Option A, B, or C in the comments section. Please be sure to note which you are trying to solve. Answers will be time stamped to determine the potential winner and participants can only win once.
Best of luck!
– Pizza Hut & John H. Conway

Now John Conway is one of my favorite math professors -- I've mentioned him several times on the blog before. I might try answering some of the questions myself. Since this is a contest, I can't post any answers I figure out today. Maybe I'll post the answers in tomorrow's post -- if I can figure any of them out.

Here's another link discussing the Pizza Hut contest. I only post this one because Common Core is mentioned in one of the comments.

"The number pi is a mathematical constant consisting of billions of digits"
Really? How does the writer get something so basic wrong?

@MikeLemon <--- Common core! What else?!

Well, I already listed the Common Core Standard for pi ("Prove that all circles are similar.") In fact, the number pi isn't directly mentioned in the standards -- except for pi^2 as an example of an irrational number:

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

But nowhere in that standard does it state that irrational numbers have billions of digits.

As other commenters point out, MikeLemon criticizes the article because it states that pi consists of billions of digits, when it really contains infinitely many digits. In this subthread, one commenter argued that "infinitely many digits" includes "billions of digits," but another countered that no one would say that pi contains "tens of digits."

What does Common Core have to do with this? Most likely, StarfishPrime is an opponent of the Common Core. He/she makes a logical fallacy here -- Common Core is bad and lack of understanding is bad, so all lack of understanding is attributable to the Common Core.

I myself have pointed out that Common Core has its good features and bad features. The Core isn't as uniformly terrible as StarfishPrime makes it to be.

This is the Exploratorium, a science museum located right here in California. I live in Southern California and the Exploratorium is up in San Francisco, so I've never been there. But it is the museum where Pi Day was invented 28 years ago.

This is the National Pi Day site. National Pi Day was, believe it or not, declared an actual holiday by Congress back in 2009.

Fun Stuff: Pi Day Videos

Now here are some Pi Day videos:

This new video, "Pi Must Die," comes from Mathologer. I see that Mathologer has created videos for several math topics I've mentioned on the blog, including the Mandelbrot set.

"These Are the Droids You're Looking For": The Massachusetts Institute of Technology, or MIT, has a tradition of sending students their admission letters on Pi Day.

"Pi Skyline": Here young students create a Pi Skyline, where the first building has 3 floors, the second building has 1 floor, the fourth building has 4 floors, and so on.

"Just for Fun": This video introduces Buffon's Needle Project, which is a method where students can compute the value of pi via an experiment.

Plus here are a few obligatory music videos involving pi, some of which I posted last year:

"Pi Day": This is a parody of Rebecca Black's "Friday."

"Lose Yourself in the Digits": This is a parody of Eminem's "Lose Yourself."

"Happy Pi Day": This song, by Paradox, is not a parody of anything (as far as I know).

This last link is not a video -- instead it links to the old Sailor Pi site that I mentioned last year. This is a parody of Kermit the Frog's "Rainbow Connection," called "The Digit Connection." (An old MIDI version of the song still plays in the background in Internet Explorer.)

We notice the middle lines:

All of us under its spell,
We know this must be math-e-magic...

And of course, mathemagic is the theme of today's Pi Day celebration.

These are the worksheets for today, which I obtained directly from Lee's blog. Lee includes a QR code on one of her worksheets -- otherwise it is similar to what I posted last year. Since I don't have access to QR technology, I'd just ignore the code.

If I timed this correctly, this is being posted at 3/14 1:59 PM Pacific Time.

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