My laptop is broken. I hope to have it fixed soon. Meanwhile, I can't do email during week days.

- - - - -

Update: We solved the problem - broken monitor - by connecting my laptop to a very old monitor at work. So now it stays at work. I have a computer at home, too, but neither is completely satisfactory. (My desktop computer at home doesn't have very comfortable seating. Time to upgrade my chair.)

## Monday, March 28, 2011

## Saturday, March 26, 2011

### Blogaversary

I started this blog in 2007, and promptly abandoned it for a few years. On March 26, 2009 I came back to it and began writing a few posts a week. In the two years since then, I've gotten over 47,000 page views. I'm so honored that people are reading what I write.

I remember how hard it was to write papers back in college. Part of the problem, for me, was knowing that only one person would read my paper, and that person wasn't reading it for pleasure, but just to give me a grade. (Yuck!) I failed one course because I just couldn't write for that teacher.

I just went back and looked at my first week of posts. I hadn't started linking to other people's goodies yet, and I included a few pieces I'd written before I started blogging. Four of my first five posts are pretty meaty:

I am so grateful to be a part of a community of math enthusiasts and passionate teachers. As I mentioned a few weeks ago, the time I spend playing with math has at least tripled, because I get a chance to discuss it with peers. I feel like you all have helped me to become a mathematician.

Thank you for playing with math with me!

I remember how hard it was to write papers back in college. Part of the problem, for me, was knowing that only one person would read my paper, and that person wasn't reading it for pleasure, but just to give me a grade. (Yuck!) I failed one course because I just couldn't write for that teacher.

I just went back and looked at my first week of posts. I hadn't started linking to other people's goodies yet, and I included a few pieces I'd written before I started blogging. Four of my first five posts are pretty meaty:

- A poem about the number i,
- A children's story about aliens who count in base eight,
- A post on the joy of learning,
- A poem about my ideal school.

I am so grateful to be a part of a community of math enthusiasts and passionate teachers. As I mentioned a few weeks ago, the time I spend playing with math has at least tripled, because I get a chance to discuss it with peers. I feel like you all have helped me to become a mathematician.

Thank you for playing with math with me!

## Monday, March 21, 2011

### Tesselation Maker

**Tesselate**: to cover the plane with a repeating pattern of identical shapes. (See wikipedia for lots more.)

You've seen it in Escher's artwork*, except that he usually had the shapes evolve. My favorite may be the one that starts with fish in a pond at the bottom, which somehow become birds in the air.

When I taught a math for elementary teachers course, I let my students do an extra credit project where they created their own tesselation. Lots of them tried it out, and some of them came up with some great ideas. I never did it myself, and I didn't realize how hard it is to get the shape to keep coming out the same. When I tried it with kids, it was too hard for them. (Maybe if I'd had more experience with it, I would have seen how to help them.)

But this is exactly the sort of thing computers are good at - repetition. Creating a tesselation on the computer is easy. You can get started here, where I made the tesselation above. You can start with any basic shape that will tesselate: triangles, rectangles, and hexagons all work. If you're doing it by hand, you can cut a piece (pretty much any shape you want) out of one side and attach it to the opposite side in the same place for rectangle or hexagon bases. For triangles, you need to rotate your cutout around the midpoint of the line it's on, and attach it to the other half of that line. On the site where I made this, any change you make will make a corresponding change where it's needed.

If you're comfortable with a drawing program (I'm not), here's a tutorial that looks good. Hmm, maybe I can do something in Geogebra...

_______

*To see the tesselation prints at the Escher site, go to Picture Gallery, and then Symmetry.

### Why Teachers Like Me Support Unions

I've always supported unions. Perhaps it has to do with being from Michigan. When I was young, regular working people could afford to buy a house, and maybe even have a cottage on a lake. Not any more. The auto industry in Michigan is decimated. And the unions, too.

The percentage of working people who are unionized has shrunk dramatically in recent years. (About.com says, "While more than one-third of employed people belonged to unions in 1945, union membership fell to 24.1 percent of the U.S. work force in 1979 and to 13.9 percent in 1998." That number is now below 12%.)

It seems to me that this country has gotten more and more conservative, and that many who would benefit from working people having more power are against it. I don't understand the big picture here, but I know that teachers' unions are big, and anti-union forces would be thrilled to weaken us (since those forces support Republican candidates and unions support Democratic candidates). I haven't been active in my union, and I don't see how our small union can help us when the whole state of California is in such huge financial trouble. But maybe...

It seems to me that teachers are the most likely people to speak up for students. Unions don't just push for higher pay. More importantly they push for smaller class sizes, better working conditions (which will usually mean better learning conditions), and fair procedures. If a teacher isn't paid well, she's likely to take on another job in the summer. A well-paid teacher can spend her summer dreaming and planning a better classroom and better lessons for the next year.

When I first started teaching, I thought I wanted to teach kids. I got elementary-certified, and then got hired to teach junior high math. I was no good at classroom discipline, and got fired halfway through my second year by a principal who was no good at helping new teachers improve. (I had a half-year contract, so they claimed they weren't 'firing' me, just letting me go. But I was teaching year-long classes.) All of the other teachers I talked to told me that no one is good at classroom management in their first year or two. I filed a grievance, the union fought for me, and I won.

Ironically, during the months over which this played out, I had gotten work with the community college that paid more. It was decided that the school district owed me any difference between the pay I lost and the pay I earned. I earned more, so they didn't owe me money. They also owed me a job in the fall, but I declined. I had come to realize that teaching at college level was going to be much easier for me. So I got no direct benefits from winning my grievance, but I may have set a precedent that helped other teachers.

I've read much more eloquent posts on why teachers need unions. Here's a complete list for this campaign. And here are a few I especially like:

- Kate Nowak (on women's pay)
- Julia Tsygan (in Sweden)
- John Golden (with a bit of history)
- Jason Buell (with some serious humor)
- Jonathan Halabi (with some perspective)
- Mark Bousquet (at college level)

## Sunday, March 20, 2011

### Playing Around...

I use sitemeter to keep track of how many people visit my blog. You'll see it at the very bottom of the page. It keeps track of a few other things, and I was exploring some of those this morning. 'Recent visitors by referrals' showed me how people get here. I found a few blogs that have me on a blogroll. (Thank you!) And I also found a few google searches.

I was glad to see someone searching for 'murder mystery math', which got them a project I use when teaching about exponential functions and logarithms. I also saw that someone had been searching for 'pythagorean triples for kids'. They got this post, which isn't for kids, because Math With My Kids is on my blogroll. I looked at the other results of that google search, and found this cool bunch of pythagorean puzzles, on Dr. Mike's Math Games for Kids Blog. That looks like the best answer for the person's query. There seem to be lots more goodies on his blog - I'm subscribing.

I was glad to see someone searching for 'murder mystery math', which got them a project I use when teaching about exponential functions and logarithms. I also saw that someone had been searching for 'pythagorean triples for kids'. They got this post, which isn't for kids, because Math With My Kids is on my blogroll. I looked at the other results of that google search, and found this cool bunch of pythagorean puzzles, on Dr. Mike's Math Games for Kids Blog. That looks like the best answer for the person's query. There seem to be lots more goodies on his blog - I'm subscribing.

## Saturday, March 19, 2011

### Contest: Find Math in Your City

"If you know a good location that tells a fun mathematical story — a piece of interesting architecture, a mathematical sculpture or the maths behind something more mundane, such as traffic lights — then enter our competition and tell us all about it." [from the Maths in the City site]

Sounds a lot like Math Trails (examples from South Bend and National Gallery of Canada in Ottawa) or Math Trek. Some day I'd like to help set one of these up around here.

Sounds a lot like Math Trails (examples from South Bend and National Gallery of Canada in Ottawa) or Math Trek. Some day I'd like to help set one of these up around here.

## Wednesday, March 16, 2011

### Links (at Lots of Levels)

I have 12 tabs open in Firefox right now, all things I want to remember to follow up. Maybe if I put a few links here, I can close some of those tabs...

**Pre-school.**- I think I posted before about this. Gwen Dewar writes:

This preschool math game was designed by researchers who wanted to know if a board game could help kids develop their number sense (Ramani and Siegler 2008). The premise? That a game featuring sequentially-numbered spaces would help preschoolers learn about the number line and about the relative magnitude of numbers. The game was very effective. After only 4 game sessions totaling less than 80 minutes, kids made substantial, lasting improvements in the areas of mathematical knowledge mentioned above.

She describes how you can make the same game yourself. Instead of making a spinner (as she suggests), you could modify a die to have 3 ones and 3 twos on it. I found this older article when I was reading her current article on good educational toys. It hadn't occurred to me how cool digital cameras might be for kids.

**Elementary.**- Denise has posted a cool game, Times Tac Toe.
- Division of fractions, done well.

**High School.**- The New York Times has an intriguing article about 8 high school students who were allowed to form their own mini-school within the school, which they called the Independent Project.

**College.**- Keith Nabb wrote an article I like, but it's hidden in a password protected site. I'm asking if I can post it here. Meanwhile, check out these animations he has for his Algebra, Trig, and Calc courses.
- A student of mine in Beginning Algebra is struggling with negative numbers. I liked this article, and plan to send her a link to it.
- In my Intermediate Algebra class, we'll be starting roots tomorrow. This article is at a higher level than most of them will want, but I think I can share a bit of this issue with my students. How do we pick which square root is the principal root?

**Teaching.**- Research on teaching (versus pseudoteaching) and learning.
- In the 26th comment on Dan Meyer's WCYDWT: Storytelling post, Kathy Sierra wrote:

Why they don’t teach screenwriting techniques to teachers is beyond me. We used to make all the authors in our tech book series read the screenwriting book

*Save the Cat*, by Blake Snyder, and build storyboards for each topic using that simplified framework. It’s not an answer to bad teaching, but it’s a way of structuring a lesson that feels more like a hero’s journey for the learner... [I want that book.]

**Math In Use.**- How many representatives should each European Union member country get? Mathematicians studied this question. One of the criteria was that the final 'formula' be easy for everyone to understand. They settled on something pretty simple, but there are lots of little twists. (And one big hurdle: Some countries would lose representatives. Can the other countries get those countries to agree to this?) I have a story to tell about helping a friend design another formula, but that will have to wait until I have more time.

## Monday, March 14, 2011

### Happy Pi Day!

π Day is here - 3rd month, 14th day.

I'm still loving this video:

And I'm happy Pi Day is on a Monday so I can share it with all my classes.

Here's what I found online today:

- My post from last year,
- Some 'political' arguments about changing from π to τ (tau). Vi Hart says she doesn't usually blog about politics, but makes an exception for this. I guess we're sort of opposites; I'm pretty political, but haven't gotten worked up about π versus τ,
- Some history,
- The
*sound*of π, - Probability and π,
- Finding π without a circle,
- A formula for the binary digits of π and an older formula,
- A trick and some puzzles,
- And of course more songs.

What's your favorite?

## Saturday, March 12, 2011

### Richmond Math Salon Next Saturday - Math Magic

The next math salon will be on Saturday, March 19 (2 to 5, as usual). Participants will learn a few mathematically-based magic tricks, and why they work. We'll also see if we can make up variations on these tricks.

[Here's a great video of the math salon about this time last year.]

- We'll play Guess My Number, and each child will get a set of the 5 cards used. We'll explore base 2 counting to see why this works. [To do this, the 'magician' needs to be able to add numbers up to 31.]
- Anyone who really likes this game can make a more advanced version which uses base 3.
- We'll play a Card Flip game which uses any deck of cards. (Please bring your favorite deck.) [To do this one, the 'magician' needs to be able to recognize odd and even.]
- We'll play number games you can always win (if you know the secret)!

[Here's a great video of the math salon about this time last year.]

## Tuesday, March 8, 2011

### Quote on Mathematical Beauty

Where things get really interesting is when unexpected bridges emerge between parts of the mathematical world that were remote from each other in the mental picture that had been developed by previous generation of mathematicians. When this happens, one gets the feeling that a sudden wind has blown away the fog that was hiding parts of a beautiful landscape.

- Alain Connes inThe Princeton Companion to Mathematics

[Found at Mathematics Rising, which was linked to at Math-Frolic.]

## Monday, March 7, 2011

### Relief At an Easy Problem

Many of the students in my Beginning Algebra course are nervous about their abilities. And I push them to really think about what's happening. So when I asked them to solve a matching problem in the book that felt easy to them, they loved it! Relief washed though the room.

They're just learning to graph linear equations. I've begun to realize how hard it is for many people to see a point as being the equivalent of two pieces of information (the x-coordinate and the y-coordinate). One visual object turns into two numbers. So of course they struggle with graphing lines. The matching problem from the book gave 4 equations, all y = 5x+b, with 4 different numbers for b, along with their graphs. They liked doing that.

Today I'm going to start by asking them to graph 2x+3y = 6. That will take a while. Then I'm going to hand out the sheet I just made with all 8 possible equations like this (± 3y, ±6, and swapping the 2 and 3) and all 8 graphs, so they can do another matching problem. That will~~go quickly~~ take over an hour, but I think they'll really enjoy it, and will begin to think about how the same numbers can make different lines. (Embarrassing how wrong my estimates of difficulty level can be. That sheet was definitely not easy. But they worked hard on it, and I do think it worked out well.)

[How do I show a Google doc file here?]

They're just learning to graph linear equations. I've begun to realize how hard it is for many people to see a point as being the equivalent of two pieces of information (the x-coordinate and the y-coordinate). One visual object turns into two numbers. So of course they struggle with graphing lines. The matching problem from the book gave 4 equations, all y = 5x+b, with 4 different numbers for b, along with their graphs. They liked doing that.

Today I'm going to start by asking them to graph 2x+3y = 6. That will take a while. Then I'm going to hand out the sheet I just made with all 8 possible equations like this (± 3y, ±6, and swapping the 2 and 3) and all 8 graphs, so they can do another matching problem. That will

[How do I show a Google doc file here?]

### Pseudocontext Versus Real: Using the Pythagorean Theorem

[Going through my old drafts, and deleting or finishing up... This one's just a week old.]

My textbook has a section called "Solving Equations by Factoring, and Problem Solving". In this section we get a somewhat artificial problem where an object is thrown upward (at a speed which is a multiple of 16 feet per second) from a tall building (whose height is a multiple of 16), for "an action shot" in a movie, providing the equation h(t) = -16t

In fact, after we did the book's problem, I tried to make up another gravity problem, this time on the moon. We were still using feet, which I said would never happen on the moon, where sensible scientists would always use metric measurements. My problem didn't factor though (I got to remind students about using the quadratic formula), one more bit of evidence of how careful the problem creator had to be to make up a problem that would factor. Still, gravity is a great way to start thinking about quadratics.

The rest of the 'problems' in this section of the text (which are really exercises in using the Pythagorean Theorem) are complete hooey. Please don't ask me,

a

a

Now we get a

or a

This seems to be a condition on the legs: a must be odd, so that its square minus 1 will be divisible by 2, and b can then be figured from a. Hmm, will this work for any odd number?

Isn't that a better way to practice your algebra skills?

My textbook has a section called "Solving Equations by Factoring, and Problem Solving". In this section we get a somewhat artificial problem where an object is thrown upward (at a speed which is a multiple of 16 feet per second) from a tall building (whose height is a multiple of 16), for "an action shot" in a movie, providing the equation h(t) = -16t

^{2}+ 80t + 576, which conveniently factors. That's ok with me, I love talking about gravity.In fact, after we did the book's problem, I tried to make up another gravity problem, this time on the moon. We were still using feet, which I said would never happen on the moon, where sensible scientists would always use metric measurements. My problem didn't factor though (I got to remind students about using the quadratic formula), one more bit of evidence of how careful the problem creator had to be to make up a problem that would factor. Still, gravity is a great way to start thinking about quadratics.

The rest of the 'problems' in this section of the text (which are really exercises in using the Pythagorean Theorem) are complete hooey. Please don't ask me,

*"If one leg is 7 more than the other and the hypotenuse is one more than the longer leg"*, to find the sides. How contrived can you get?! If you want students to practice their algebra skills in relation to the Pythagorean Theorem, how about getting them to think about the 4000-year-old puzzle of generating Pythagorean Triples? After seeing just a few examples, starting with 3^{2}+4^{2}=5^{2}and 5^{2}+12^{2}=13^{2}, you might wonder ... If we set the hypotenuse to be one more than the longer leg, will that tell us anything?a

^{2}+ b^{2}= c^{2}becomesa

^{2}+ b^{2}= (b+1)^{2}.Now we get a

^{2}+ b^{2}= b^{2}+2b+1,or a

^{2}= 2b+1, which we can write as b = (a^{2}-1)/2.This seems to be a condition on the legs: a must be odd, so that its square minus 1 will be divisible by 2, and b can then be figured from a. Hmm, will this work for any odd number?

Isn't that a better way to practice your algebra skills?

## Sunday, March 6, 2011

### Science help requested: Accuracy with Relativity

Years ago, the textbook I used for Intermediate Algebra mentioned relativity in the rational functions chapter. They gave an expression for observed speed (of galaxies separating). I wanted to help my students practice the algebraic steps involved in solving an equation (with lots of variables) for one variable, and made up a silly story about a galacto-cop chasing a possibly speeding pirate. I have a few questions.

The equation I used was equivalent to s = (u+v) / (1 + uv/c

What I wanted to do with my students was to solve for u. I got u = (s-v) * c^2 / (c^2 - sv). First question, is this legitimate?

My story was that the galacto-cop's 'radar' (what else should I call it?) gave her the pirate's speed of separation from her ship, but she wants to know the pirate's 'true' speed, and has to figure this version of the formula out. I claimed the galactic 'speed limit' was 1/4*c.

My understanding of relativity is weak, and I'd like to get this story down a little better before I tell it in class this semester. Help?

The equation I used was equivalent to s = (u+v) / (1 + uv/c

^{2}), where u and v are the observed velocities of two objects moving in opposite directions. s is the speed at which they're separating. At earthly speeds, s would equal u+v. Simple. But near the speed of light things get complicated. If u and v are each over half the speed of light, u+v would give us an s value over the speed of light. That's apparently not possible. Einstein (and others?) came up with the equation above, which describes relativistic effects on velocity. I think. (Please correct my statements here if they're inaccurate, misleading, or confusing.)What I wanted to do with my students was to solve for u. I got u = (s-v) * c^2 / (c^2 - sv). First question, is this legitimate?

My story was that the galacto-cop's 'radar' (what else should I call it?) gave her the pirate's speed of separation from her ship, but she wants to know the pirate's 'true' speed, and has to figure this version of the formula out. I claimed the galactic 'speed limit' was 1/4*c.

My understanding of relativity is weak, and I'd like to get this story down a little better before I tell it in class this semester. Help?

### Sexism Is Still With Us

[I wrote this last summer and never posted it then. I think it's worth saying, though the article I'm referencing is from June 2010.]

The assessment of beliefs about math and learning I posted last summer included this zinger:

I'd rather be working on math problems, or dreaming up ways to encourage my students to work on math. But Tanya Khovanova pointed to an article in the New York Times by John Tierney, which once again claims that the evidence is on the side of more men at the very top of math and science ability, and I had to respond. This article is shockingly unprofessional. I've known for a long time that the New York Times isn't always accurate, but I guess I was still naive enough to think their science section would be well-done.

Tierney starts his article with a reference to enforced anti-sexism training, to be sure to mix up different issues. He leaves out the most convincing parts of the Hyde and Mertz research (here's one article to check out). And he has crazy links that do nothing to help the reader understand the issues, which look to me like a sophomoric attempt to inject humor into his ugly opinion piece.

Someone at the New York Times should do a better job making sure the 'science' articles they print are more than someone's biased, damaging opinions.

The assessment of beliefs about math and learning I posted last summer included this zinger:

It's genetic, men are better at math than women.It's a belief that cannot be disproved as long as we live in a world that values gentleness and kindness more in women, and competition and strength more in men, a world in which math ability in girls is looked upon as odd. (Gentleness and strength are both good, so why should we try so hard to divvy them up between the sexes?) There is a clear cultural component to this equation. (And not having any examples of cultures in which women's intelligence is valued equally with men's, there's no control group to compare to.)

I'd rather be working on math problems, or dreaming up ways to encourage my students to work on math. But Tanya Khovanova pointed to an article in the New York Times by John Tierney, which once again claims that the evidence is on the side of more men at the very top of math and science ability, and I had to respond. This article is shockingly unprofessional. I've known for a long time that the New York Times isn't always accurate, but I guess I was still naive enough to think their science section would be well-done.

Tierney starts his article with a reference to enforced anti-sexism training, to be sure to mix up different issues. He leaves out the most convincing parts of the Hyde and Mertz research (here's one article to check out). And he has crazy links that do nothing to help the reader understand the issues, which look to me like a sophomoric attempt to inject humor into his ugly opinion piece.

Someone at the New York Times should do a better job making sure the 'science' articles they print are more than someone's biased, damaging opinions.

## Wednesday, March 2, 2011

### Tutoring & Number Theory

I've been tutoring 'Artemis' for over a year and a half now*. This fall he picked up the book

Number Theory includes so many cool ideas. I especially like playing with modular arithmetic, and seeing the power it can have. I've never taken a number theory course myself, so I'm consolidating my thinking, and enjoying the stretching.

Here's a problem from the book (modified a bit):

[Please don't give complete solutions in the comments. But I'd love to know if you've played with it, and what approach you took.]

Enjoy!

_____

*Search on Artemis to find all the posts.

*Introduction to Number Theory*, by Mathew Crawford, from Art of Problem Solving, and we've spent most of our sessions since then working out of that. I like not having to worry about what we're going to explore, and he seems to like the problems we encounter each week. He and I work through each problem together. Sometimes I hold back more, and get him to do one on his own, and sometimes I do more (with him correcting my mistakes). We skipped most of the first 3 chapters, because he knew most of that.Number Theory includes so many cool ideas. I especially like playing with modular arithmetic, and seeing the power it can have. I've never taken a number theory course myself, so I'm consolidating my thinking, and enjoying the stretching.

Here's a problem from the book (modified a bit):

I've got one part solved, and am still thinking about another part...Consider n! and the sum 1+2+...+n.

When is the sum a factor of the factorial?(problem 6.31)

[Please don't give complete solutions in the comments. But I'd love to know if you've played with it, and what approach you took.]

Enjoy!

_____

*Search on Artemis to find all the posts.

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