Tuesday, March 25, 2014

Archimedes' Twin Circles

So the Futility Closet, a constant source of neat results, puzzles, quotes and more, posted this pretty result:

As with many visual theorems, my first impulse was to make a dynamic visualization. Off to GeoGebra!

But I quickly hit a snag... I didn't know how to construct the tangent circles. I made the basic set up and then starting monkeying around. Eventually I thought about how to just make a circle tangent to one of the interior circles and the line.  I made a tracing point with the distance to the interior circle and the dividing line and traced along where it would be equidistant - a requirement for a circle tangent to both.

Of course - a parabola. To be tangent to a circle and a line is like the  definition of a parabola as all the points equidistant from the directrix and focus. The point D had to be on the parabola, which helped me to find the directrix. The focus had to be the center of the circle.

So then the other parabolas weren't too hard to find. The center of one of the twins had to be both on a parabola of tangents to an interior circle and the line, and a parabola for the enclosing circle and the line.

And now we can see that Archimedes result was true in general.

This is a special case of the Apollonian Circle problem (finding a circle tangent to three non-concentric circles &/or lines), and I feel like it was helpful in deepening my understanding of that. To be specific, a special case of the Circle-Circle-Line special case. But it was fun.

Now that we can construct them, how would you prove the twin-ness of these circles?

The sketch is, as usual, up on GeoGebraTube for you to play with yourself.

Postscript: the always educational Alexander Bogomolny (proprietor of Cut-the-Knot) had this to add on Google+:
They are no longer twins. E.g., Circle Triplets.

But there are more, Arbelos, and even more: C.W. Dodge, T. Schoch, P.Y. Woo, and P. Yiu, Those ubiquitous Archimedean circles, Mathematics Magazine, Vol. 72 (1999), 202-213. (JSTOR) and Some More Archimedean Circles in the Arbelos, Frank Power, Forum Geometricorum, 2005. (postscript).

Gaussian Skirmish

I have a particular fondness for Gauss, as he was my entry into math history.

John Hocking, whom I usually refer to as my undergraduate math guru, was, by great fortune, my math teacher for the first two years at Michigan State. A good researcher himself, he brought math alive with deep connections, intriguing applications, lively stories of discovery and some history of the great mathematicians. It had never really occurred to me to wonder who invented the stuff we were studying. Unthinkable in most other subjects.

But when Karl Friedrich Gauss came up, the usually passionate Gibb kicked it into another gear. "This is a man whose name should be as famous as Beethoven's!" And yet none of us had heard of him! "Like the magnetic unit?" one person asked.

In that pre-Google world of 1982, it took some effort to find out more.

One day in our capstone class is devoted to Gauss, then. Given an hour to talk about him, what should you cover?

Excuse the Google docs, but here's what I share:

One interesting thing in a math history class is that the closer we get to being relevant to today, the harder the math is to understand. And class is diverse in the sense that despite all being math majors, there are future elementary teachers, secondary teachers, grad students and applied mathematicians. For Gauss, he loved number theory so, and modular arithmetic was so crucial to his number theory, I came up with a game to increase experience with modular addition. It seems to me that in a half hour students go from hesitant to fluent, and the discussions at the start of playing are powerful.

These directions are in particularly good shape, as Sue Van Hattum helped edit them for her upcoming book.

Hope you give it a try and enjoy it as much as we do!

P.S. Looking for more Gaussian fun? Here's today's longer than usual list of To Do choices.

To do:

Daily: lots to choose from, of course...

  • Make sure you understand the Fundamental Theorem of Algebra.
  • One neat result of Gauss was proving Fermat’s conjecture that every counting number can be expressed as the sum of three triangular numbers. Good thing to play with, and fun that it connects to Gauss’ 7 year old result.
  •  work on a classic Gauss problem: how many ways are there to write a whole number as a sum of perfect squares? For example, 5 has 8 ways, not using zeroes but yes using negatives. So (-1)^2+2^2=5 is one way. 
  • Follow Mike’s process through explaining that problem, Keep notes and make sense yourself of what he and his kids share.

Decimal Games: Burger Time

 My last three games for Mr. Schiller's 5th graders have all been about decimals. They worked pretty well, and I definitely tried new things for myself as an educational game designer.

The first game came when they were first digging into decimal multiplication, just doing whole number times the decimals.  I went through a large number of gyrations about a good context, but I kept coming back to measurement. I thought about a race game, where students measured out multiple decimal portions of a meter or centimeter - and I still think that has some potential. If it was warm, especially, I'd love to see them out with meter sticks on a course they made around the school. Of course, we were in the midst of the harshest winter in 70 years. No one even remembered what the sidewalks looked like. I also thought about physically stacking things, but I wasn't sure about what materials we had to stack. I'd like to see more of what game designers call dexterity games in math.

I finally went with burgers - I suppose with Robert Kaplinsky's In-n-Out burger activity rattling around my skull. The game itself is almost more of an activity. Some students built their burgers and didn't care about the game layer on top of that. Choosing ingredients, making the picture - that was very engaging for most of them.

Build a Burger
Who makes the best burger in town?
Materials: 5 dice (or 5 dice per team/player)

Idea: Roll 5 dice to get ingredients for your burger. The numbers correspond to how many mm tall each part is. Three 5s means 3 all beef patties. Two 6s means two layers of bun. You get to reroll one time. Pick the dice you want to reroll.

1 – sauce, .1 cm. Choose from ketchup, mustard, mayo, hot sauce, , barbecue sauce, secret sauce.
2 – cheese, .2 cm.
3 – bacon OR onions (Mix if you have 2 or more)
4 – tomato OR lettuce, .4 cm. (Mix if you have 2 or more)
5 – patty, .5 cm. Maximum: 3 patties, UNLESS you have 3 buns, then you can have 4 patties.
6 – bun layer, .6 cm. Maximum: 3 layers. Every burger needs at least 1 layer of bun!

The choice on the different rolls seemed crucial - their customization increased variety and student ownership. The rerolling mechanic is the child of Yahtzee, of course, but a great way to add choice and chance for redemption.

We launched the game with stories of great burgers with excessive description. I rolled up a burger with help from the class and demonstrated the multiplication. As the students played, they ignored some of the rules - which I see as making the game their own. In the end of class discussion, that gave them plenty of fodder for suggestions for the game. Add another ingredient, let people go meatless or bunless, and - certainly - we should actually make the burgers.

The game was an undeniable hit, and seemed to provide some good experience for multiplication of decimals as groups of decimal quantities. As you can see in the student work below, there was lots of material generated for discussion between student and teacher and students with each other.

And finally, a student who clearly has a future in fast food marketing:
"See, it has meat instead of a bun, but still the regular meat, with the bacon and cheese in the middle."

Here's the form as a pdf if you're interested. Email me for the Word doc if you want to modify.

Sunday, March 9, 2014

Carnival of Mathematics 108

Welcome to this months Carnival of Mathematics!

I thought it was auspicious that this is 108, a number closely connected with the Golden Ratio, because of its appearance in the regular pentagon. (I am also Golden, if you can't see the auspices.)

108 is pretty interesting in other ways, too. I got wondering if many other numbers are multiples of the same number with zeroes removed. (What about other digits?) New to me was the idea of a refactorable number: a number divisible by the count of its divisors. They are rarer than I would have first expected; also called the tau numbers. 108 is the 18th tau number. Can you find all 17 prior?

Let's get to the submissions! Where I could find them, I linked the author's twitter account as well.

There are many states with a Highway 108, but Kansas gets the picture, since they use a sunflower (their state flower), which is also - of course - associated with the Golden Ratio. 

Patrick Honner asks who has done a billion dollars worth of work? Wages, worth and gender bias all figure in.

Cav has a post trying to make sense of maths testing. Maybe separating it out into two different subjects... On a more personal note, he looks at amusing, infuriating and worrying answers on his own students' exams.

Jennifer Silverman shares her visual approach to the quadratic formula at her tumblr. College math majors often are missing this connection between completing the square and the QF.

Malke Rosenfeld writes about Beautiful Objects at Math in Your Feet. The objects in consideration are lovely group symmetries from Christopher Danielson.

Mike Lawler got great reasoning out of Fawn Nguyen's bridge problem. Inspired me to try it with freshmen and grad students to great effect. And now on to the grad students classrooms, too!

Sue Van Hattum makes an argument that optimization is the best application topic in standard math classes, and uses her recent calc lesson as an example.

This graphic designer, Jack Hagley, made a neat logo based on his research at Wolfram-Alpha. He thought 108=1x2x2x3x3x3 was beautiful in its own right. (I agree!) These are hyperfactorial numbers.

Edmund Harriss has several threads converging in his post at Maxwell's Demons about Rational Parameterisation of the Circle. Very novel idea, connections with stackexchange, and beautiful images from Lissajous variations. He also has some very mathy fun with generating "huge worlds of potential logos" for Twitter Math Camp 14. (Full, but there is a wait list.)

John D. Cook accesses Category Theory to determine whether the stars go up or down at his blog, the Endeavor. Hint: yes.

Dan McQuillan writes about induction in the Unreasonable Efficiency of Mathematical Writing. I love his advice: "focus on the beautiful idea."

At the By Way of Contradiction blog, there is a probability post on how to pick odds in the most favorable way. Includes this sentence: "Satisfied with your math, you share your probability, he puts 13.28 on the table, and you put 2.72 on the table." This is, I think, the pub for me.

In some Hindu and Buddhist practice, a mantra is recited 108 times. This leads to malas (prayer beads) often having 18, 27, 54 or 108 beads. This may be connected to the Sri Yantra, which has "9 interlocking triangles forming 43 smaller triangles."

Cool geometry, regardless.

Edward Frenkel, author of the recent hit Love and Math, had a much talked about editorial in the LA Times: How our 1000 year old math curriculum cheats America's kids. (That's what we get for buying an 800 year old curriculum when we started, I say.)

Askhat Rathi at the Conversation has my favorite recent math in the news story, the discovery of a new class of polyhedra. Yeah!

108 figured prominently on LOST. (Here's some of the connections from the show.) Most significantly, it was the sum of the numbers 4 8 15 16 23 42. Never sufficiently explained, I think, and I'm still mad about the ending. So there.

Evelyn Lamb digs into the history of the Parallel Postulate at her Scientific American blog Roots of Unity. Triangles with angle sums of 0 degrees, rectangles held hostage - exciting stuff. Evelyn also has a post at the AMS (her Blog on Math Blogs) looking at Michael Pershan's Math Mistakes site and contemplating SBG. (Go for it, Evelyn!)

Fiona Keates, who blogs in The Repository at the Royal Society, has a piece on a mathematician in the movies. Yes, Ramanujan is coming to a theatre near you!

Shecky Riemann interviews the  fascinating Cathy O'Neil, author of the Mathbabe blog, at Math Tango. (Cathy's on Twitter, too.)

The oddest 108 connection I found was 108 Rock Guitars. Since Math Rocks!, they get a link. (Maybe I meant Math Rock?)

Antonio Chinchón has a post on the sound of the Mandelbrot Set. A neat combination of pretty, computation and sonification of data. He also has a post about Warholing data (after the pop artist), which he does to Grace Kelly.

Mike Croucher at Walking Randomly has several different mathematizations of the heart that go far beyond the standard cardioid.

Sam Shah derived the curve found in string art as he did Doodling in Math Class.

Hopefully you got a chance to see Carnival 107 at White Group Mathematics. Next month's 109 is at Tony's Maths Blog. Be sure to check out Sue Van Hattum's hosting of the Math Teachers at Play carnival 71 (with 71 links). You can submit a post at the Carnival's homepage at the Aperiodical. Katie Steckles makes these happen, and submitted several interesting items above; Thony Christie also made some nice picks. Gracias!
108 Eyes by playful_geometer
There were some posts I wasn't sure what to make of... John Gabriel arguing that no valid construction of the reals exists. Katie submitted a Windows 8 math game app called Equations. I haven't been able to test it out, though. The reviewer dislikes it for pretty valid sounding reasons.

Friday, February 28, 2014

Math and Inequality

Got this email, and I'm wondering how a math educator should respond:
Dear Colleague,

A teach-in is being planned for the purpose of mutual education among students, faculty and staff of the GVSU community.  The teach-in is designed to address topics related to inequality and systems of oppression, as well as social justice and liberation.  Recognizing the multi-faceted dimensions of these topics, we are planning this teach-in as a day-long event.

The details are as follows:
Date:                     Wednesday, March 26th
Time:                     8am – 10pm
Location:              Exhibition Hall, Mary Idema Pew Library  (Room 040)

The learning objectives of the teach-in are to raise awareness, share knowledge, and create dialogue.  Therefore, in an effort to involve as many students, faculty and staff as possible, we will be adhering to the MWF class schedule, with the possibility of fourteen 50-minute sessions, each starting on the hour. 

We hope that you will attend as many sessions as your schedule allows and encourage your students to attend.

If you’re interested in taking a more active role, you’re encouraged to work with other colleagues around campus to propose a 50 minute session. Please keep in mind that a teach-in is practical, participatory and action oriented. We especially encourage contributions with an intersectional framework (race, class, gender, sexuality, religion, ability, etc.).  Sessions may include student leaders as co-presenters or panelists.
I love that the university is doing this. Will you help me brainstorm how to participate?
I have an intro to math class, and a grad class on secondary math education issues that meet during this time.

Thanks for any ideas!

Lucky Duck is a recurring character
in Tom the Dancing Bug by Ruben Bolling

Tuesday, February 25, 2014

Capstone Book Club

One of the courses I'm teaching this semester is a capstone course for our majors. Rather than me picking the book, I let them choose from a list that the Math-Twitter-Blog-o-Sphere helped me put together. Last summer, The Mαth Book and Joy of X were the big hits, this semester, they made some very different choices. (Here's the list and their choices.) Having them choose makes me feel a little less like this:

My notes of the discussion, with links to some of their blogposts. Since they're notes of a discussion, pretty terse. Also, not my point of view, this is the students'. The people who got the most new students interested in their book were those reading The Mathematician's Lament and e: the Story of a Number.

Euler: Master of Us All. History at HS level, proofs at graduate school. Felt outclassed reading it. Simplify or explaining the proofs would have helped. Averaged a paper per week. Very dense.
Review: Alex

Visions of Infinity. Each chapter is a different idea, then has history and the author explains the important proofs about the idea. How mathematicians think vs what they thought about. Eg. squaring the circle, explained why you couldn't do it. Was advanced, but not frustrating. Recommend it to someone who knows about math. Last two chapters… where math is going. Strongly the author's point of view.
Review: Kristine, Emily

The Math Book: one discovery per page. Good if you like quick little synopsis of many topics. Different big discoveries. Learned a lot of little things, but so rudimentary in some places. There were some over your head, too. I would recommend it. Gives sources to dig deeper. Easy to make connections. Biased towards white Christian males.
Pro reviews: Kenton, Erin. Con review: Brittney
Journey through Genius: history, picks important results and proves them. No non-western mathematics. Tries to justify why no islamic mathematicians but comes across as a cop out. The Greeks asked why, but the Egyptians were satisfied with just working. Did give a personal view of the mathematicians. Recommended to people who want to get to know the mathematicians well.
Review: Biz looks at it from a multicultural perspective.

Accessible Mathematics: 10 instructional shifts. Works with field experience, works with math ed classes, what happens when you don't use these. Multiple representations, number sense. Good to examples and non-examples. Recommended to anyone going in to teaching or to teachers who need to change.
Review: Danielle, Becky, Keegan

Concepts of Modern Mathematics: overview of the math classes you have to take here. Like 310, statistics, history on those areas. People who have made significant work in that area. Some new to us, like topology. Pretty dry and boring. First chapter was the peak, then funniness ran out. Long lines of equations. Not a bad book for someone in 210, to know what's coming, but then it would be hard to understand. Examples are pretty simple - maybe even dumbed down. Pretty dull read. Good at explaining ideas like 1 to 1 and onto. Large scale review.
Review: Bryce. Killer line: "Mr. Stewart slowly became less of my friend who I wanted to hang with and more of a dreaded professor whose class you have to sit through every week."

Godel Escher Bach. It's about cognitive science, psychology, computer science, math. The Godel part is the most on math. You think because you are, but you have to be to think. Need a lot of time to dig in. Hurts to read, in a good way. Challenges the way you think. Connections with taoism and Buddhism. Isomorphisms and symbols… uses all these metaphors. Kind of like A Brief History of Time by Hawking or The Elegant Universe by Brian Green.
The Joy of X. Goes from things people have been told is true to why they're true. Negative times a negative, infinity… goes further at the end.
Short review but long notes: Jennifer
Sensible Mathematics, sister to Accessible Mathematics. Written more to administrators than to math teachers. Why it's good to have students explore more. Convinces teachers and parents to go with modern views of learning and teaching mathematics.
Review: Kerry
Love and Math. Hard to understand because it's so into physics. His life story is interesting, discrimination against Jews in Russia as a backdrop.
Kate's review says that you should like physics and math to pick up this book.
The Mathematician's Lament. Talked about how math is an art. It's about discovery, and we need a big do over in education. We're just handed formulas, not told where they're from or who made them. Can't teach teaching. Recommend to any future teacher.We take away the art of math.
Powerful review from Sara.
e: the Story of a Number. I didn't know where e came from. That's why… the math amazes me more. Everything I learn unites math more. This book did that and combined it with physics, through the number e. Pascal's triangle, derivative of e^x is itself. My optics class is about that. The challenging thing the author had to face. Pi could fit in a few pages, but e, there's no place where it came from. Napier and his tables, and how logarithms require e; when you extend to continuous cases… Book does a good job of balancing history, different contributions. His jumps are really strange and don't make sense until you finish reading. Balances history with math, saves all the proofs for the appendix. Bernoulli's conversation with Bach about the space between notes, a logarithmic increase... that was amazing.
Review: Duncan
There wasn't a whole lot of discussion in the whole class setting, but the individual discussions among students who read the same books were deep and intense. Even the group of people who all had read unique (in our class) books.

What do you think of their reactions? Any disagreements? Is there a new book in which you're interested?

Wednesday, February 19, 2014

Followers Beware

Smarter Balance has released several items for the fast approaching Common Core assessments. One really caught my eye as a dynamic context: safe following distance. I can't find a way to link to the specific item, but it is #43060 at http://sampleitems.smarterbalanced.org (CCSS: F-BF.1a, F-LE.1b). The sketch is on GeoGebraTube, as well.

How far should you drive behind the car in front of you?

GeoGebra notes: With the spirit of Jennifer Silverman hanging over my shoulder, I got real car images to use. The trickiest thing was sizing the images to look right since the scale was not 1:1. I like the flexibility of looking for feet (sigh, USA) or for car lengths or for time separating the vehicles.

The actual assessment question:
 Pretty complex problem, as illustrated by the rubric:
Just a quick post to encourage you to think about using some dynamic visualization with your math work. Many of the released items have a little gif like animation instead of text.