Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

Saturday, December 27, 2008

Good Bye Craps

Alternate title: Probability, Why Have You Forsaken Me?

Throughout my adulthood I have been an infrequent patron of casinos. As in, maybe once a year. It made me feel a little dirty, but I found a game that was fun, social, and that I could reliably win, or at least not lose much.

I had many justifications! I had math, for one. I didn't mindlessly feed my quarters into the loud, blinky machines with the terrible house advantage. I didn't sit down at a blackjack table, barely understanding the game, begging the dealer to relieve me of my chips, the sooner the better.

I played probability, not possibility. I played a game that had nothing to do with luck! Every event an independent event! The lowest house advantage in the house! Still more likely to lose my money than not, of course, but the least likely! Barely likely!

For around six years, this worked out fine. Sometimes I would lose a small amount, say less than $30, but I rationalized that a couple hours of entertainment was a fair trade. More often, I would come away ahead by $50 or $70. Soooo smug I was.

Well 2 winters ago, disaster. I walked away from Turning Stone $200 poorer. That stung. So much that I didn't go back last year.

This year I was feeling confident enough to give it another go...but...pffft. At first I wasn't going to play because the lowest table minimum was $10. I prefer $ least if you lose money there, you do it slowly. But, since I was there, I felt silly for not playing at all...and 45 minutes later had donated $112 to the Seneca Nation.

You win, craps. I give up.

I hope they at least support some decent schools with all that profit.

Wednesday, December 24, 2008

T^3: Regents Review Done Right

(Note: This post is part of Teaching Tips Tuesday at Notes from the School Psychologist.)

The documents:
Practice Test

Part I Answer Sheet & Scoring Sheet from June 2008 Math B

Powerpoint of Solutions and Scoring Rubric

In New York they are called Regents Exams, other names in other states, but I'm just going to assume these things are the same everywhere: stressful, inevitable (at least in today's edu-accountability climate), and sharing particular features.

A major goal this year is to spend time preparing for the test itself: format, question type, environment, timing, scoring, calculator use as well as of course the content.

In the three days before winter break my Algebra 2 classes had finished Transformations, and I didn't want to delve into Trig until after break. So, I made my first foray into this explicit Regents Review. I created a practice test identical in structure to a Math B Exam: 20 multiple choice, 12 4-pointers, and 2 6-pointers. I used questions from old exams, except I only chose questions the kids would already know how to do.

Last Friday, they took a whole class period and they just completed the multiple choice. I provided each a copy of the scoring sheet from the June 2008 exam, all official-like.

Over the weekend, they were to complete the 14 extended response questions as homework. On Monday and Tuesday, we reviewed the solutions together and, and I think this is important and everyone should be doing it, I showed them the scoring rubric for each question that we use to grade the exam.

They could see that out of the 88 available points, they would lose a whole point for not reducing (8 + 6i)/2. They could also see that if they could translate a problem into a quadratic equation, they would get 2/4 points even if they couldn't solve it. If they wrote the WRONG equation and solved THAT, they would STILL get 2/4 points. If they could prove a given quadrilateral was a parallelogram, even if they couldn't do what the question asked (prove it's a rhombus but not a square), they would get 2/6 points. They could see that a correct answer with no work shown would only earn them 1/4 points.

I think this little exercise made the point far more effectively than me saying it over and over again: write down anything you can about the problem, even if you don't think you can solve it, or do so correctly.

At the end, they tallied up their points and finally, I showed them the conversion chart from last June (it's the last page in the powerpoint) - what their score would have been had they taken this exam. Students vaguely know that the Regents are curved, but they could see exactly what their 62/88 translated to, and see how many points they would need to pass.

I hope it works! I figure anything I can do to make the details of the test mechanics automatic will help them focus on just getting the math right. And also arm them with a little test-taking savvy so they can eke out a couple extra points here and there. We'll see.

Tuesday, December 23, 2008

oooooOOOoo Baaaaybeeee......

My lovely and talented colleague is playing "All I Want for Christmas is You" out her door during passing time. The Mariah Carey version. On repeat. All day.

Related: Kids get ridiculously giggly about a dancing teacher in the hallway.

Sunday, December 21, 2008

The Flyswatter Game for Identifying Conics

I heard about this game from Rebecca Bell and immediately thought to use it for identifying the equations of conics*. We spent about 10 minutes lecturing/notetaking and then played the game.

The Preparation:
1. Buy 4 flyswatters. (At the hardware store. They are cheap.)
2. Create equations to identify and swat. I used 2 of each of the main types of conic sections.
3. Print out 2 copies and separate them with a paper cutter.
4. Tape them randomly to the whiteboard. I used 2 identical boards, playing with 4 teams, for a large class.
5. Create several rounds of things to identify. In earlier rounds I made it easy, like "a parabola" and later made it more difficult like "a parabola that opens to the left".

Setting up the Class:
1. Divide into 4 teams.
2. 2 teams play on each identical whiteboard.
3. Each team stands in a line facing the board.
4. The first person in line holds a flyswatter and stands behind a line on the floor. (I made "lanes" with the desks - careful to place the front of the lines so that the 2 teams are the same distance from the center of the board.)

The Rules:
1. Stand behind the line until I call out what you are swatting.
2. The first person to swat the correct equation gets a point for their team. Sometimes there is more than one right answer on the board, so you can both get points.
3. If 2 people go for the same equation, the swatter on the bottom wins.
4. If any team members yell out help like "top right!", the round is over and no one gets any points.
5. If you win a point, you pick up a marker and give your team a point on the whiteboard.

I found the best place for me to stand was on top of a desk behind all the lines.

The kids were engaged, had fun, and didn't want to stop playing. Success! A few got so competitive (pushing, jumping to front of line out of turn) I had to put them in time out. That's right. 16 year olds in time out.

I wish I had some pictures but my camera had an inexplicably dead battery that day. Also, if anyone knows how to get a cell phone picture onto the Internets, please enlighten me.

*because I am a big nerd.

What I Am Doing Today Instead of Playing Bioshock*

1. Writing the Level 3 OCMTA exam.
2. Filing half the papers at my desk and throwing the rest away.
3. Cleaning up the house.
4. Shoveling (after it stops snowing).

Maybe I will get to play Bioshock* after dinner, if I'm good.

*Laugh all you want, you wouldn't believe the mileage this gets me with your average 15 year old boy.

Tuesday, November 25, 2008

Choose Your Own Adventure

I can't believe it took so long for Choose Your Own Adventure to show up on Youtube. There has to be some justification for showing it in a math classroom. Every film has two options? Exponential growth? I'll think of something.

Wednesday, November 19, 2008

Pi Whats?

Edit: Here is the file you can download and edit.

(Alternate Title: What is a Radian?)

What I wanted them to grok:

Discussed before they started: what is pi? Could you cut a string exactly pi inches long? Most of them said "no". They thought it was theoretically possible, but couldn't be measured precisely enough. I promised them that they'd walk out of class with a string of length pi in their pocket.

The Activity:

Materials: Circular object (collected from around my room: a CD, a water bottle, paper plates, cardboard mailing tube, etc etc)
Metric ruler
Strip of Paper (I purchased adding machine paper at big box office supply store)

1. Measure the diameter of
your circular object. Calculate the radius to the nearest 10th of a centimeter and record: r = _______ cm

2. Cut a strip of paper at least 8 times longer than your radius, and tape it to a flat surface.

3. With a marker, make a mark on the edge of your circular object. Make a corresponding mark near the end of your strip of paper. Line up the two marks, then roll your circular object one complete revolution. Make a second mark on the paper at exactly one complete revolution. What does the distance between the two
marks on the paper represent? Verify this value by using the radius to calculate

4. Cut a length of string the same length as the distance between the marks. Set it aside.

5. Using one of the marks on your strip of paper as "zero", use your ruler and a pencil to mark off intervals along the paper of the same length as your radius. About how many radii before you get to the one-revolution mark? This value should be the same for all of the various circles. Why?

6. Take your length of string, fold it in half, and cut it exactly in half. This string is precisely pi long. You and your partner can walk around all day with pi in your pocket! The question is….

Pi whats?
At the appropriate time we discussed this, and came around to "pi radii of the circle". Many of them still didn't like this and insisted it "wasn't really pi", so we had to have the discussion about operating in the perfectly abstract realm...some of them noticed that if their circle had a radius of exactly one centimeter, then the string would be pi centimeters...that seemed to help. Then I had them convert a bunch of degree measures to radians and back, and come up with formulas.

The scary thing was, most of them didn't recognize it from last year until they started writing down a formula. But I'm hoping that more of them actually know what the heck a radian is, and this makes the unit circle a slightly easier mountain to climb. We shall see.

Tuesday, November 18, 2008

Flashes of Brilliance

I'm sure I'm not the first one to think of this, but this lesson was pleasingly effective. In past years graphing inequalities in two variables was something that seemed easy for the kids in class, but they could never remember what was up when we reviewed it later.

This year, to start off, I had the kids pick the coordinates of three different points (x, y) on the plane, test them in the given inequality, and mark the point with a closed circle if it came out to be true, and an open circle if it came out to be false. After a few minutes I had to say, "If you only got true points, find a point where it comes out to be false." We ended up with a screen that looked like this:

We discussed the significance of the boundary between the blue and white spots, and also tried a few non-integral points. Then I had them try a different one on their own. They developed a very serviceable method for indicating the "true region". One class wanted to write the words "True" and "False" instead of shading, which isn't bad, but I warned them that eventually we'd need to know where two of them overlapped, so they shaded.

This seems like a stunningly obvious approach now that I've tried it, but I thought I'd share since it took me 4 years to think of it. Does anyone do anything similar, or have a better way?

Tuesday, November 11, 2008

Thanks, Vets

I know lots of other teachers are veterans, too. I am spending today writing marking period 1 comments, going to see madagascar 2, and grumbling about civilians. So just a quick thanks. And one of my favorite videos from 116, showing the lengths people will go to to keep each other sane in close quarters. They are having so much fun, they almost make me nostalgic for being at sea. Almost.

Sunday, November 9, 2008

My Word Wall

Well it looks cool, but I honestly don't know how much good its doing. I haven't carved enough time out of the curriculum to really make the vocab a part of the course. I feel like we're just slapping copied definitions on the wall. Anyway:

And just for fun, here are some other pictures of my room:

Tuesday, October 28, 2008

Nova Nerdplosion

Check out "Hunting the Hidden Dimension", with a basic explanation of fractals, the Mandelbrot set, and a large collection of applications (blood vessels! forest health! textiles! eyeballs! heartbeats!). The website has neat stuff now, and you can watch the whole thing online starting Wednesday, October 29.

I love Nova. Especially Nova episodes about math. I show "The Proof", about Andrew Wiles' quest to prove Fermat's Last Theorem, to my classes when we have a down day. It gives them a taste of what professional mathematicians actually do all day. Also when they complain that a problem takes too long, I can remind them that Wiles kept at the same problem for seven years.

Monday, October 27, 2008

I <3 Green Globs

If you have access to this game by Sunburst software called "Green Globs", use it. One of my more smartass freshmen exclaimed today "I'm learning! I'm learning!", but they were mostly having fun. It's a game that gives you a bunch of globs on a cartesian plane, and you knock out globs by typing in the equations whose graphs intersect them. The more you hit with one graph the higher your score.

I like it because they *need* to know how to write the equations for graphs they want for some other purpose (an artificial purpose, but one to which they are accustomed to submitting)...I haven't run across anything better for clarifying for them positive vs. negative slopes, rise vs. run, how to slide the graph up and down, how to adjust the steepness. I drag it out again when they graph quadratics.

Anyone know of a free version of something like it? I'd love to hear about it.

Wednesday, October 22, 2008

The Professional Development I Need

Secret message to School District Administrators responsible for Staff Development:

I have many good ideas, and shamelessly adopt other people's good ideas, but I'm not great at executing them. Some people seem to be born with a gift for it, but I'm not. I need practice, feedback from people who know what they are talking about, and chances to observe people who know what they are doing.

The scene: 8th period. My miraculously small, at 11 students, Algebra 1 class. I decide that I would rather stab myself in the eye than do the notes-practice-repeat routine about the slope formula.

I close the smartboard file and have them arrange their desks in a horseshoe. I lie and say that last period, before they were even aware, they were in pitched competition with 7th period. A competition to see which class can do "the wave" fastest. We talk about how we can compare which class is fastest, since we have about half the students they do.

And when I say "talk", I mean it's me and about 3 kids, the rest look listless, like they are waiting for me to just tell them what they need to know. We "decide" we need to measure how many kids can cycle through doing the wave at various timed intervals.

They have fun with this. There is open but good-natured rebellion when I won't let them do the wave for a full 5 minutes.

I decide how to record and organize everything on the board. I don't even know how I'd get them to do that part. I sketch a hasty graph.

There is lively debate about whether it would be better to add up all the people and all the seconds from all our trials and divide, or rather divide people by seconds for each trial and average the rates. These calculations come out differently, but they don't see why. I do a crappy job of explaining why the first option is better.

Then I fumble through a discussion of how "somethings per one something" is called "slope". They remembered the word, they remembered "rise over run". Briefly talk about extrapolation and interpolation, and an equation for the graph. A good teacher would have had just the right sequence of questioning prepared.

5 minutes before the bell, I panic and have them write m = (y2 - y1)/x2 - x1) in their notes and copy a few examples.

It's not easy for me to write that I'm not better at this by now. I hate sucking at stuff. I've seen how they need concrete models to hang their learning on. I don't know how to give it to them.

On Friday, all the high school teachers will come to school for a Staff Development Day. I am leading a "Smartboard Work/Share" session in my room. Not that I won't get some good work done, but honestly I am already excellent at using a computer. I need some one or ones to help me improve the execution of this lesson.

Tuesday, October 21, 2008

Leading a Math Circle Proves at Least as Difficult as Expected

I finally had a first math circle meeting today. I invited any and all Math League students to come. Around 25 showed up at the beginning...8 or 10 had to leave in the middle and 2 more stopped in toward the end. I didn't know most of them but I think they were mostly juniors and seniors (in precalc and calculus). 6 or so students completely dominated the conversation.

I started with "what's the biggest number?" which quickly morphed into "are there different sizes of infinity?" and stalled at "is infinity a number?". They never agreed on whether or not infinity was a number, and by the end decided that they didn't even know what a number was. That was kind of neat. They also kept looking at me imploringly and saying "IS THERE AN ANSWER?!" and you all would have been proud of my aikido-like deflections.

For my part, I tried to steer them to focusing on the countable sets first, and broached the idea of one to one correspondence, but no one picked up that particular ball and ran with it.

So I guess I need help with -

Managing such a big group? More than 20 seems too big to guarantee everyone gets involved. I have 2 other teachers who are interested in "helping" but one of them could only come for about half the time and the other just stopped in said hi and left.

Encouraging dominant students to be more inclusive and collaborative?

Suggestions for steering them toward Cantorian set theory and the proofs I have in mind? I want to get them at least through the diagonal proof of the nondenumerability of the continuum, so that they can see that there is more than one distinct cardinality. At the end I postulated that there were as many positive even integers as there were positive integers but I didn't want to do too much. How should I start next week? I tried to pick a topic where I already knew "how it went", maybe that was a mistake, maybe I should have picked something unfamiliar.

Monday, October 20, 2008

When They Are Going to Use This

A graduate of my school, currently a freshman studying fine arts, sent these pictures to my colleague. She had to make a sculpture from a single piece of poster board. Check out all the geometry she used!

Buried Treasure

In Out of the Labyrinth: Setting Mathematics Free
, a book I have turned to a dozen times, I found this passage about "teaching by indirection" that informs all of my lesson planning. I don't remember ever seeing it before, though I suspect I must have:
Means to an end seem to be more readily absorbed than the end itself...So if you want your students to learn some topic A, find another topic B, which A entails - and have them work on B. You'll see that A is effortlessly taken in stride (although B itself may escape).

Let's say for some reason or other you thought it important for your young students to learn their times tables. Set them instead to puzzling over which of the first 100 or so integers are prime. This is always a heady pursuit for our pattern-hungry minds, and in the course of it such "facts" as that 51 is (disappointingly) 17 x 3 are hit on -- and stick. Once they dope out how the times tables are made ("once you know 3 x 4 your get 4 x 3 for free"), a very few significant encounters tamp down the actual values. You could also have had them figure out how to add fractions: finding a common denominator is tense and rewarding, and puts times tables in their right (ancillary) place. Or if you want them to master deductive reasoning, put in front of them some intriguing topic in Euclid. If the subject is calculus, embody it in a particular physical problem. It takes some thought to find a good B for each A - fortunately no system of facile entailments is ready at hand.

(Emphasis mine.) It's that "fortunately" that undoes me. That the Kaplans expect me to know my subject with such intimacy that I can set my kids on a path where they will necessarily overcome the thing I want them to learn anyway, without even being aware of it. And I can't look it up which B goes with my A. And I shouldn't even try to compile a list. And the idea we can hand the uninitiated a script, or a binder, and expect real mathematics to be learned, would be laughable if it weren't so sad and terrifying.

Friday, October 17, 2008

A fun little limit problem

You drive the first half of a car trip at 30 mph. How fast would you have to travel during the second half of your trip so that the average speed of the entire trip was 60 mph?

Wednesday, October 15, 2008

Direct and Inverse Variation

Planning a lesson about direct and inverse variation, I thought I'd share the hook I use to introduce the concept. This is the script I use in class, but it was heavily cribbed from

If 5 cats can catch 5 mice in 5 days, how many days does it take 3 cats to catch 3 mice?

The "obvious" answer is three. Three cats catch three mice in three days.

However, this answer is not correct!

As the number of cats increases the number of mice they can catch increases. "Direct variation" means that as one quantity increases, the other quantity will also increase.

On the other hand, as the number of cats increases, the number of hours required to catch a certain number of mice decreases. "Inverse variation" means that as one quantity increases, the other quantity decreases.

If 5 cats can catch 5 mice in 5 days, how many mice can they catch in one day? (discuss)

Now, if 5 cats catch 1 mouse in 1 day, How many days would it take 1 cat to catch 1 mouse? (discuss)

And finally, since 1 cat catches 1 mouse in 5 days, if you have three times as cats, they'll catch three times as many mice in the same amount of time. So 3 cats catch ____ mice in 5 days.

photo credit: Denis Defreyne via flickr

Monday, October 13, 2008

Lesson Reflection: Measuring Error

New to our Integrated Algebra standards last year was percent error in measurements. Last year I did a miserable job of it. "Copy notes followed by practice problems" miserable. This year was better.

Things to keep in mind:
  1. They already have learned how to calculate "percent error" in science labs.
  2. This concept is a hybrid of "greatest possible error" and "percent error".
I started this year with a slide and a discussion:

They all said "5.4 cm". To which I replied, "Are you sure it's not really 5.38 cm? 5.41 cm? 5.40236 cm? We discussed the mechanics of measuring something, how you end up visually rounding to the nearest marking on the measuring device. They talked about their digital scales in science class, and how you can set the readout to the nearest tenth or hundredth, and we discussed that the little onboard computer is doing the rounding, even though you can't see it. (For next year: at this point, have them measure several different items to the nearest _______th, to ensure everyone groks the "visual rounding" idea.) Then, I got them to conclude verbally, stated several different ways, that any measurement could really be up to half a unit smaller or larger than the stated value.

Then a few problems:

Then brought in area:

The next day, we tie greatest possible error to percent error, and calculate measuring error (greatest possible area divided by measurement).

On the assessment, 83% of the students scored a 4 or 5 on the question compared to 26% last year. I think the difference was in the introduction.

For next year: take 5 minutes and have them measure some stuff to the nearest whatever. Measure your desk/textbook/etc to the nearest inch, nearest centimeter, nearest 16th of an inch, nearest millimeter, etc. Perhaps borrow a few of those digital scales from science.

Also, it's still pretty dry. Any suggestions to make it more engaging? Other NY teachers (and anywhere else you cover this topic), what are you doing?