More from James Tanton, on Twitter

Yeah, I have a Twitter account, but I seldom check it out. I don't see the point, usually. But I just discovered that James Tanton (jamestanton) is tweeting interesting problems. Hmm, can I get those delivered to my Google Reader, or something?

Republic of Math blogged about James' question regarding n plus square root of n: Can it ever round to a perfect square? (It's answered at the blog post, so you might want to play with it before clicking through.)

When I went to Twitter this morning, James' latest question was:

60houses in a row. 3 roof colors cycle in 3. 4 door colors cycle in 4. 5trim colors cycle in 5. Every house unique roof,door,trim color set?

He had to really squeeze to get that one in the 140 characters, didn't he? 

Thanks for feeding us some good math, James!

My Math Alphabet: E is for Eigenvectors and Eigenvalues

E is for Eigenvectors and Eigenvalues

This post is about fear.


Part 1. Fear.
Those words, 'eigenvector' and 'eigenvalue', sounded scary to me for years. I expected the concepts they reference to be hard. And so they were hard. These words / concepts come up in a course called linear algebra. The rest of the course was easy for me, but I struggled with these 'eigen' ideas. So when I was preparing to teach linear algebra for the first time, I got nervous again. It took a while to embrace the idea. Now that stuff doesn't seem so hard. But I remember that the word threw me off, and I know to take it easy when I teach it.

Square roots throw off my algebra students, partly because there's a weird new symbol involved. And partly because the concept that goes with it doesn't mean much to them. (I should have started more slowly than I did this semester, with finding sides of squares that have various areas. I'm kind of zipping through roots and rational expressions, because I wanted to focus more on our last book chapter, using the quadratic formula and graphing parabolas.) They know the √81 is 9, but if they're trying to simplify √162, they'll correctly write √81*2, and then proceed to change that √81 into √9, and then into a 3. It's all just moves in a game that they don't quite understand. And the game is scary.

I remember being very uncomfortable with Greek letters in my first calculus course. There were way too many of them. How was I supposed to memorize twenty-something new symbols?! So I'd read things by saying squiggle every time I saw a Greek letter. That doesn't work if there's more than one of them in the statement you're trying to read. I still slow way down when I try to read things with unfamiliar Greek letters. (Alpha, beta, gamma, delta, and theta are fine. Phi and mu are ok. I'm too lazy to go find the command to write them properly.)


If we always start with interesting problems, instead of with definitions and notations, will we intrigue people enough that fear will never come up? Or is grading going to always introduce the fear factor? What's a good problem for getting students thinking about square roots? Has fear ever come between you and your interest in math?


Part 2. Eigenstuff.
If you don't even want to think about eigenvectors and eigenvalues, stop here. But I'm going to attempt to write about them in a way that makes them feel less scary.

Eigen means something like 'innate' or 'its own' in German. That doesn't sound so bad. They're about not changing in a certain way, so what's a name I could make up that emphasizes the concept of staying the same? (I want to make up my own name for these as a way to make them my own. I think that will help me like them better.) What about home-vectors and home-values, as in 'staying at home'? It's similar to homo- which means same (homogenous, homophone, and homomorphism) and to homeo- which means similar (homeopathy, homeostasis, homeomorphism*), and that's good. It feels homey to me, and that's good too.

Linear algebra deals with vectors (think arrows) and operations on them. For the vector, let's imagine (1,2,3) in R³, our usual 3-dimensional space. (1,2,3) points 1 unit east, 2 units north, and 3 units up. An important concept in linear algebra is linear transformations, which take a vector or group of vectors, and stretch, rotate,  or reflect them.

Start with a particular linear transformation A, which is represented by a matrix. The transformation happens by multiplying the  matrix A times the vector (say x), getting Ax. Given A, are there vectors that don't change direction, but just stretch or shrink? If there are, those are called eigenvectors (home-vectors), and each one has an eigenvalue (home-value) associated with it, that describes how its length changed. The eigenvalues/home-values usually get labeled with Greek letters, lambda most commonly. I'll use s for stretch (or scalar, if you like). The definition gets us Ax=sx, which says that when the transformation A works on the eigenvector/home-vector x, you get x multiplied by a number (that's what scalar means), so that it just stretches.

Wikipedia has more, of course. (I liked seeing the applications, though most of it was beyond me. And I liked this: They pointed out that if the dimension is 5 or more, there is no method for finding exact values, and round-off error can make numerical methods problematic "because the roots of a polynomial are an extremely sensitive function of the coefficients".)

What's a good problem for getting students thinking about these ideas, before we ever say the E-word?


_____
*Homomorphisms and homeomorphisms are two different things, both mathematical. See wikipedia.

My Math Alphabet: D is for Dance

D is for Dance

People say music and math are closely related, but deep down, I don't really get it. I'm not so good with music. Let's say I'm a slow learner. I can sing pretty well, if it's a song I've sung lots and lots of times. I can play the penny whistle well enough to get compliments now and again. (Penny whistle is probably the easiest instrument there is.) I tried to learn guitar for years, and was always mediocre (at best). I was terrible at the timing.

Then there's dance. I love both music and dance, but I'm slow at both. My favorite sort of dance is contra dance. If you go to a contra dance, you're welcomed in, and taught how to do it. You don't have to bring a partner, and no one scowls when you mess up (well, almost no one). Good thing folks there are patient, because ... I'm a very slow learner.

So I never would have thought, on my own, about looking at the connections between math and music or dance. But I'd seen a number of questions about this on the Living Math (Yahoo group) and Natural Math (Google group) lists, so when Malke Rosenfeld emailed me about her Math In Your Feet program, I paid attention.

I sent a message to both those lists, with a link to her blog, and figured she'd help a few people with their questions.

But when I read her blog, I got excited myself. It turns out she works in public schools, and is trying to help the schools see how important movement is for kids' brain function. Yes! (My son, who never has to sit at a desk, would wither if he were stuck sitting in a classroom for hours on end.) She also addresses some cool mathematical questions in her work with kids. Things like: How many times do I have to repeat this 'jump and turn' to get back to where I started?

I think this post is my favorite for getting a sense of what she does. If you're like me, this blog will stretch your notion of how to approach math. You might also like this collection of math and dance links. (It's a wiki, please add to it if you're moved to.)

One more story about me and dance... At contra dances, when the band (there's always a live band) takes a break, waltz music goes on the sound system, and it's time for waltzing, which I. Could. Not. Do. I'd try to count - 1 23, 1 23, 1 23 - but it never helped. Part of my problem is that I like to take B.I.G steps when I try on my own to dance to that music. But there's another problem. The proper way to waltz involves 3 counts forward and 3 counts back. Two years ago, at the Queer Contra Dance Camp, a marvelous dancer showed me a different way to count, 1 23, 4 56. Somehow, that made all the difference. Now I can waltz - more or less.

My Math Alphabet: C is for Calculus

C is for Calculus, Which Gets Me Jobs

Calculus helped me get both of my full-time community college positions, and it was calculus that got me back to teaching in the first place. I love calculus.


The Slope of a Curvy Line
Back in the early nineties I was doing user support for a progressive internet provider, known as PeaceNet, EcoNet, Institute for Global Communications, and a few other xNets. We worked out of a small office in San Francisco, and I did phone support all day, helping people all over the country get online, while also trying to write manuals and learn more in between calls.

In January of 1995 I flew to Seattle to visit some friends, and on the way back home I sat next to a man who was a Native American AIDS activist. We talked about many things, and at one point I explained to him what calculus is:

You know how in algebra, you graph lines, and find the slope? The slope tells the rate of change, which is important in lots of real-life applications, but most of those don't make straight lines. Finding the slope for a curvy line is what calculus does.


If we draw a tangent line to a curve, we can define the slope of the curve to be the same as the slope of that line. The problem is that a tangent line to a curve only touches it in one point, and you need two points to find the slope. So we cheat. We use the point of tangency, and then for a second point we look at secants  (lines that touch the curve in two places) through that point, with the second point sliding closer and closer to that first one, so that the secant line is twisting closer and closer to the tangent line.

I drew lots of diagrams on our napkins, something like this (though of course I couldn't animate mine). And he told me I should be teaching. He suggested I get a job at an Indian college. I missed teaching, so I looked up the Indian colleges, and thought about it.  I was too tired of moving to new places to go for it, but that got me started on looking into community colleges, and that summer I applied at a number of colleges back in Michigan, where my family is.


What's It Good For?
One of my interviews was at Muskegon Community College, only an hour away from my family in Grand Rapids. During my teaching presentation, JB, who was on the committee interviewing me (they were all pretending to be my students) asked, "So what's this good for? Anything?" I asked what he was interested in.

JB: "Rocks." (He's a geology teacher.)

SV: "Hmm, well, would it be possible to know the shape of a layer of rock underground, and want to know its volume?"

JB: "Yeah!"

SV: "And would it be likely for the shape to have a circular shape, so it would be the same in any direction from a central point?"

JB: "Oh yeah, that's common."

SV: "OK. What if it were shaped like this..." And I drew a hypothetical rock layer formed by two parabolas crossing. We imagined it spinning around the y-axis. I then explained how to use something the textbooks call the 'shell method' and I call the 'tin can method', to figure out the volume.They were impressed, and I got the job.


Related Rates
I worked there for six years, and was happy with my work. But I wanted to adopt a child, and it became clear I wasn't going to be able to adopt in Muskegon. (As a single pagan lesbian, I just wasn't the sort of family the social workers were looking for there.) I decided to try to move back to Ann Arbor, or back to the Bay Area. During my interview at Contra Costa College, I had to do an unplanned lesson. I got to choose between linear algebra and calculus, and then they would give me a topic. I considered doing the more advanced linear algebra topic, in order to impress them, but decided it might backfire if I couldn't explain it well enough. I stuck with calculus, and they asked me to explain related rates.

That's one of my favorite topics (I know, lots of pseudo-context in those problems, but I think they're neat!), and I enjoyed getting to play with it. I had thought my planned lesson went well, but later found out they weren't particularly impressed with that, and it was my impromptu talk on related rates that got me the job. They must have seen my eyes light up as I started to explain how one rate of change was related to another, and how we could solve our problem using those relationships.

I adopted my son a year and a half after I moved back out here; I'm now the happy mama of an 8-year-old. I've been at CCC for nine years now, and I love the diversity of my college. Thank you, calculus!

Deriving the Quadratic Formula: James Tanton's Twist

I've always enjoyed showing students how to derive the quadratic formula. I don't test them on it, so the stress level is lower. And it's late in the term, so they appreciate a break from the pressure, and most really do try to get it. I get a few making those appreciative sounds that happen when the lightbulb goes on, and that makes it especially fun.

But it's hard slogging through some of the weird steps. Here's the standard derivation, if you haven't done it in a while. Check it out, and imagine trying to explain it to people who are pretty fragile around math.

So the math education gods were smiling on me last week, and the day before I brought this topic to my students, I interviewed James Tanton, who (out of the blue) showed us his twist on this. (Thanks, James, for helping me with my lesson plans!)

If you just can't find the time to watch the video, it goes something like this:

ax²+bx+c=0 

We want a perfect square in the first term, 

so we multiply both sides by a:                  a²x²+abx+ac=0

We want the second term to have a factor of 2, and to keep the first term a perfect square, 

so we multiply both sides by 4:            4a²x²+4abx+4ac=0

We almost have what we see in the box above, but we want b² and not 4ac, 

so we do a little adding and subtracting:   4a²x²+4abx+b² = b²-4ac

Now factor the left side:

(2ax+b)² = b²-4ac

Taking the square root of both sides steals away a solution, so we include a plus or minus:

2ax+b = ±√ b2-4ac  

Subtract b from both sides and divide both sides by 2a, and you've got it. Much prettier than the standard derivation, I think.

I used this in class last Thursday, and I think it went much more smoothly than the standard version. I always do it twice, so I did it again on Monday. Students said they got it, and some liked it. I haven't spent enough time on completing the square (our days together are numbered, at this point in the term), so I don't expect their understanding goes very deep, but it's a start.

I'll do this again next semester in my intermediate algebra course, with a better grounding in completing the square. I look forward to it. Maybe all our conics problems will be easier, with James Tanton's brilliant help.

Number Tricks Show the Power of Algebra: Subtract by Adding

I gave a mastery test last Tuesday, and my first class had worked hard to get ready for it. I wanted to show them something fun the next day, instead of diving into our next unit: roots, completing the square, and the quadratic formula. I really enjoy the buildup of all that, but it takes some serious work on their part. I wanted to start with something more .. light-hearted.

So I showed them a number trick I learned recently. (Was it at AMATYC? Was it online? Or was it somewhere else? I haven't a clue.)


The Trick

If you don't want to do a subtraction, say one with lots of borrowing, you can add instead.

Let's try an example:
Lots of borrowing needed here, and two borrows in a row can get confusing. So instead we'll do this...

1. Find the "9's complement"
2. Add
3. Subtract one from the highest-value digit (leftmost position) of the result, and add that one in the ones place.

To find the nines complement, subtract each digit from 9, and write the result in place of the digit.

Now we add, and do a little fussing...




Why it Works
The real fun, with Helen on the plane and with my students, was showing why it works. (One student wanted to know why they hadn't been taught this before. I said I thought it was too much of a trick, and you might forget just how to do it, and math shouldn't be that way. But maybe it would improve some people's subtraction enough to be worth it for them.)

Let  x-y represent the subtraction problem. Let's start with supposing y is 3 digits (or less), and see if we can find a way to generalize, which is harder to write down. Find y complement, which I'll call y_c, by finding 999-y. If y_c=999-y, then y=999-y_c. And

x-y 

= x- (999-y_c) 

= x-999+y_c 

= x+y_c-999 

= x+y_c- (1000-1) 

= x+y_c-1000+1. 

Done.

I like that! And, more importantly, so did my students. I used to think number tricks were silly, but the ones I did at the start of this semester, along with this one, have convinced me to change my mind about equations versus expressions.

I used to feel like equation solving was the heart of algebra, but the algebraic explanations of number tricks start with an expression representing the numbers, and then involve simplifying until something comes out which really does explain why the trick works. Simplifying expressions suddenly seems much cooler than it used to. (I used to also think you couldn't check your answer when you simplified an expression, but all you need to do is put in a random number to the original and your answer, and make sure they match. Try 2 or 10 for an easy number, since 0 and 1 aren't good choices for catching mistakes.)

Math 2.0 Webinar Series: James Tanton

This evening at 9:30 eastern time, 6:30 California time, I'll be interviewing James Tanton as part of the ongoing Math 2.0 webinar series created by Maria Droujkova. I've reviewed a few of James' lovely math books here, and have used his videos in a few of my posts. He also sends out a monthly puzzler to people on his email list, runs math institutes at St. Mark's School where he teaches, and more.

Here's what James says about himself on his 'about' page:

James Tanton (PhD. Mathematics, Princeton University, 1994) is a research mathematician deeply interested in bridging the gap between the mathematics experienced by school students and the creative mathematics practiced and explored by mathematicians. He is now a full-time high school teacher and does all that he can to bring joy into mathematics learning and teaching.
James writes math books. He gives math talks and conducts math workshops. He teaches students and he teaches teachers. He publishes articles and papers, creating and doing new math. And he shares the mathematical experience with students of all ages, helping them publish research papers too!

We'll have fun exploring math with James. Click here to join us. Come early if it's your first webinar, so you can get situated.

AMATYC Conference in Boston

Well, there wasn't enough math to suit me. But a few of the sessions were great. I especially liked two of the Saturday workshops.

One was on the history of pi, and the presenter, Janet Teeguarden, set it up as a game of Risk (her own version).  We got a sheet with 25 multiple choice and short answer questions, and put our answers down. As she presented, she'd stop at each question and ask us to put down how many points we wanted to risk. We started with 100 points, and each question allowed us to increase that if we were right. One person was sure of his answers on ten questions and didn't try any others, so he got 102,400 points. I was only sure on a few answers, but tried a bunch. I got up over 10,000 points. The game kept me listening much more intently to a presentation that I might have mildly enjoyed otherwise. (And I'm not good at listening to presentations that don't have audience participation, so I might have drifted away, even though it was interesting.)

Although this presenter used Powerpoint for her presentation and the game (it made a cool swooshing noise as each answer was shown), I think I could do something similar with just a worksheet and the board. I think I'll try this when we're reviewing for the final exam. I could give them a practice final to do as homework, and then play Risk with them as I go over it.

[Edited to add...]  Here are the first few questions of the 'Risk Your Knowledge of Pi' game:

	Risk	Points
		100
1. What is the formal definition of pi?
2. In what book will you find the following? “Also he made a molten sea ten cubits from brim to brim, round in compass, … and a line of thirty cubits did compass it round about.”
3. What ancient Greek mathematician determined that 3 10/71 < pi < 3 1/7?
4. He used circles with inscribed and circumscribed polygons to make this estimate. How many sides did his final polygons have?

I risked 100 on question 1, got it right, and entered a total of 200.  I risked all 200 on question 2, got it right and had 400. I think I risked all 400 on question 3 and got it right, for a total of 800. I risked 50 on question 4, and got it wrong, for a total of 750.


The next session was something about improving antiquated word problems. The presenter (a textbook author) talked about how silly some problems are, and how he tweaks them to make them more relevant. He said: "Functions are the heart of intermediate algebra." I've been looking for a way to tie all the topics together, and thought that might work for me. I'm excited to look at the text again, and see whether a function focus would tie most of it together.

The problem we worked longest on started out something like:
The 10,000 seat stadium will sell out for the rock concert. The better seats are $65 and the cheaper seats are $45. How many of each ticket type must be sold to bring in revenues of $500,000?

Silly question. Why do we want that revenue in particular? If we change it to a function question, it becomes more interesting: Create a function that takes number of $65 $45 seats as input, and produces revenue as output. There are all sorts of things that can be done with this:

• Graph this function.
• What does the y-intercept represent?
• What does the slope represent? (Hey, the slope is negative. More tickets means less revenue? How's that possible?)
• What x and y values make sense?

On Thursday (backing up), I went to a fun workshop on using the abacus. Now I'd like to spend some time trying it out with kids.


I also visited a math circle on Sunday morning and had a great talk with one of the presenters afterward, just before I caught my plane home. The woman I sat next to on the plane liked to talk, and that made the time go faster for the first half. After 3 hours, she'd run out of stories and neither of us was looking forward to the 3 hours left to go. I offered to show her some math, and she said sure. I showed her binary numbers by starting with a magic trick. (Hey, I think I have to revise my post on binary. I left that out!) You normally start with 5 cards, each showing some of the numbers from 1 to 31. I just wrote them on my paper. She got into it, and I showed her a way to subtract by adding (decimal first, then binary). Our flight wasn't over yet, and she said she liked algebra, so I showed her some of the Pythagorean triple patterns. We were landing as we finished that up. Thank you, Helen, for being an eager student - you made my flight so much more fun!

It's good to be back home, and I'm pumped up for the last few weeks of the semester.

Blogger Ethics: Book Reviews

Some ethical considerations for my blog are obvious to me:

• I'm always honest, and do my best to be fair. 
• I choose not to take any ads, because that's what I prefer in my internet experience. 
• I don't even read most of the email solicitations I get, from people requesting to do guest posts, or asking me to link to their '10 Best X' sites. I have asked a colleague to do a guest post because I loved what she wrote on a small population email list, and I considered letting another blogger do a guest post here because I liked his blog (we just never worked out the details). To me a guest post must be from someone whose work I know and respect.
• If I'm reviewing anything I got for free, I mention that fact.

But I've had a question in my mind for a while about book reviews, and haven't answered it yet.

Once in a while, when I see an interesting new math title, I request a review copy from the publisher. If it's good, I read it and review it, mentioning in the review that I got the book for free. (There's one book at least that I liked but haven't yet reviewed. I want to find time to look it over again before I review it.)

But if I don't like the book, my feeling is that the best thing to do is just not write a review. I think other people might like the books that I don't like; our reactions to books can be as varied as our reactions to any art.  If I say that I don't like a book, that requires evidence to back up my displeasure. But I don't want to bother with all that if I'm not fond of the book. I think I'd like to pass those books on to another blogger. Perhaps they'll find a reader who likes them better.

Is this the right thing to do, or is there some reason I ought to review the books I'm not fond of?

AMATYC Conference in Boston, Next Week

I'm headed to Boston on Wednesday for the annual conference of the American mathematical Association of Two Year Colleges. Let me know if you'll be there too.

My Math Alphabet: B is for Binary

On, On, Off, Off, On = 16+8+1=25

B is for Binary

I loved teaching about binary numbers when I used to teach basic computer courses. What's going on inside that box? Lots of little electronic switches are going on and off, and that's enough to represent any number (and anything else you put in a translation table). The switches are not the mechanical sort you see to the right; they're microscopic.

On=1, Off=0. Start on the right with the ones position (2⁰), double each time for the twos (2¹) place, fours (2²), eights (2³), sixteens (2⁴), and further. One byte in a computer is 8 bits (bit is short for binary digit), so the highest-valued binary digit in a byte is worth 2⁷ or 128, and a byte can take on any value from 0 to 255. For instance, I'm 54 - that's 32+16+4+2, or 00110110.

Rick Regan really digs binary, and his blog, Exploring Binary, says it all. Much of it is too technical for me, but I've enjoyed a number of his articles. How I Taught My Mother Binary Numbers looks like a good starting point, along with One Hundred Cheerios in Binary. His post on the look and sound of binary numbers is delightful, as is The Binary Marble Adding Machine.

Here's one more fun site for binary numbers: Computer Science Unplugged.

Outside of math, I think we too often think in binary terms - good or bad, left or right, boy or girl, etc. I like it when I can break out of those boxes and find a third path. But in math, binary is fun.

One-Way Functions: What Are Some Simple Examples?

Tanya Khovanova gives a great simple example of a one-way function, which is becoming dated. She's looking for other examples. Can you help her out?

A one-way function can be used to encode a message. To decode the message, we'd want to go backwards, using the inverse of the function. It's very hard to compute the inverse of a good one-way function. Anything involving financial transactions online should use a secure encoding.