I really like both of these slides:

## Alert!

**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!**## Tuesday, March 31, 2009

## Monday, March 30, 2009

### An Interesting Development: E-Texts Now Available

People keep saying that textbooks need to get off paper, online, be interactive, customizable, and easy to change. That ideal vision is a long way off. But it's getting closer.

Last year the math teachers from the Arlington Central School District wrote and published online, for free, an Algebra 1 e-text that is aligned with the New York Integrated Algebra standards. They began using it last year and were happy with their results. Each lesson includes work to complete in class and a 1-page homework. You could, theoretically, print the whole thing out and make copies for your students and use it to drive your entire course. I imagine homeschoolers could make great use of it, too.

Today I learned that one of the architects of the Algebra project, Kirk Weiler, has published a similar e-textbook for Algebra 2. This time, it's the first offering of his company emathinstruction.com. The book is still free, but he plans to charge for things like an answer key.

I have mixed feelings here. I'm very impressed by the amount of work and thought that went into the Algebra 1 book. It's well paced and sequenced and the examples they chose to illustrate the concepts are top-notch. The integration with graphing calculator technology is seamless. I'm sure the Algebra 2 book is similarly impressive though I haven't had time to look through it yet. I had the opportunity to see Kirk speak around a year and a half ago when they were wrapping up the Algebra 1 book and he is a smart, smart, enthusiastic, engaging guy. (I said "smart" twice on purpose. He's really smart.) It's amazing to have free books that hit all the state standards, no more no less. And yet...pdfs? That I can't change? To download and print out? One lesson at a time? On paper? It feels a bit web 1.0. However, within the paper textbook paradigm, I don't think you could do any better.

Last year the math teachers from the Arlington Central School District wrote and published online, for free, an Algebra 1 e-text that is aligned with the New York Integrated Algebra standards. They began using it last year and were happy with their results. Each lesson includes work to complete in class and a 1-page homework. You could, theoretically, print the whole thing out and make copies for your students and use it to drive your entire course. I imagine homeschoolers could make great use of it, too.

Today I learned that one of the architects of the Algebra project, Kirk Weiler, has published a similar e-textbook for Algebra 2. This time, it's the first offering of his company emathinstruction.com. The book is still free, but he plans to charge for things like an answer key.

I have mixed feelings here. I'm very impressed by the amount of work and thought that went into the Algebra 1 book. It's well paced and sequenced and the examples they chose to illustrate the concepts are top-notch. The integration with graphing calculator technology is seamless. I'm sure the Algebra 2 book is similarly impressive though I haven't had time to look through it yet. I had the opportunity to see Kirk speak around a year and a half ago when they were wrapping up the Algebra 1 book and he is a smart, smart, enthusiastic, engaging guy. (I said "smart" twice on purpose. He's really smart.) It's amazing to have free books that hit all the state standards, no more no less. And yet...pdfs? That I can't change? To download and print out? One lesson at a time? On paper? It feels a bit web 1.0. However, within the paper textbook paradigm, I don't think you could do any better.

### The Perfect Challenge

I am not a huge fan of making kids memorize things for its own sake. But memorization does have its place. For example, for a student to be able to efficiently simplify a radical, it really, really helps if she recognizes a perfect square when she sees one. For example, $\sqrt{338}$ looks like it would be complicated to simplify, but by the time I'm done with my freshmen, they can do this without a calculator. They notice it's even, so they write it as $\sqrt{169}\sqrt{2}$, and then they recognize that 169 is the square of 13, so they write $13\sqrt{2}$. Done. But recognizing the 169 is the lynchpin.

I actually have them memorize all the perfect squares from 1 to 400. The first 10 they kind of know already, so it's really a matter of learning the square of 11 through the square of 20. How do I accomplish this feat? Bribes. This little exercise is a testment to the lengths a 15 year old will go for a Lemonhead. (Also lemme just say, I'm pretty sure I stole this idea from someone, but I have no idea who or when. I'm not trying to claim credit, just get the word out.)

Once the Perfect Challenge starts, we begin every class period with a short, timed quiz, and they know exactly what will be on it. I just jumble up the questions. If a student can complete it perfectly in the allotted time (60 - 90 seconds, depending on my mood), she gets a piece of candy. We do it every day until every student can do it perfectly. Some years one or two students can do it the first day. It has never taken more than 7 tries.

Here is my Excel Document with 7 quizzes, 2 per page.

## Saturday, March 28, 2009

### Meet the Math Mama

I am very excited that my friend Sue VanHattum has started blogging at Math Mama Writes. She teaches community college and runs a math salon in the California bay area. Sue loves math and learning and has a very warm personality. She is another Math Circle Summer Institute alum. And she is starting off big by compiling an anthology. I am looking forward to good things from her blog.

## Thursday, March 26, 2009

### Nailing the Warmup

This is the warmup slide from my trig class today. I wanted them to remember about inverse functions before talking about graphing logarithmic functions.

These kids are starting to rebel if I pose a problem that doesn't come with a cartoon/picture/and or require a solution they actually care about. I wonder if I am ruining them for other teachers.

Blogger is being dumb about letting me make the slide any bigger: xkcd comic is here.

## Tuesday, March 24, 2009

### This Game Really Is Worth 1000 Worksheets

I learned about this from Denise at Let's Play Math - turn evaluating expressions or comparing quantities into a game of War. For some reason, every kid loves War. I used it today for the trig classes to mentally evaluate the values of logs. It was a big hit. I made decks of 30 by printing out these (word doc) and introducing them to Mr. Paper Cutter. Here is the pdf (Thank you, Kenneth.)

## Sunday, March 22, 2009

### If Next Year is Your First Year

I tried to comment on MissCalQL8's post, here, but her commenting really doesn't like me, or my browser, or something, I couldn't make it stick. But I know there are other preservice teachers who read me from time to time, so anyway. Not that I'm some great expert, I'm just talking to you from the other side of the first few years.

If next year is your first year...

There are lots of things you will only learn with experience. That's ok. Be prepared to make mistakes, learn from them, and make adjustments. It doesn't matter if this is your first career or your fifth. You are probably not going to be great right away. The Wongs wrote a helpful book, but their way is not the only way. You will find what works for you. It will take a few years.

Join NCTM, but also join any local teachers' associations you can find. Your state almost certainly has one, and your county might, too. Visit their websites, forums, and join their listserves, if they exist. Go to any conferences that interest you and you are able to get to. Take every opportunity to meet teachers in the districts where you want to work.

I wouldn't try to buy supplies for your classroom. You will likely be able to place a supply order with your school, or they will give you an allowance to spend. You also don't know what will be already available for you in the classroom you inherit or as castoffs/extras from other teachers. You could certainly start building a professional wardrobe, if you need to. It will be less painful if you get a little at a time. Now is a good time to find winter clothes on clearance. Don't forget about shoes. Check out discount places like Marshall's.

If you don't already, start taking good care of yourself. Get plenty of exercise, rest, water, and vegetables. Make it a habit and a routine now so that you don't have to spend mental energy on it when you are busy teaching.

Those are the most helpful things I can think of. Veterans? Anything I missed?

## Saturday, March 21, 2009

### My Favorite Moments of the Week

1. Me: "See how (5+3)/(1+3) = 2? It does not = 5, right? So no one is going to break my heart and cancel terms that are not factors, right?"

Kid: "Why are you showing us this? This is like teaching someone to use a computer by opening it up and looking at the insides."

Me: "Let's get one thing straight. Opening it up and showing you the insides IS MY JOB."

2. Kid: "You know I just realized something really cool - if you showed me this problem 20 minutes ago I would have looked at you like you were crazy, but now I can do it! And it's easy!"

Other Kid (deadpan): "The miracle of learning."

3. Me: "The Smartboard has this neat tool I just found, look." (I freehand the graph of an ellipse by connecting some plotted points, and when I lift the pen, the shape snaps to a perfect curve.)

Kid: "But can she make a Kessel run in 12 parsecs?"

Kid: "Why are you showing us this? This is like teaching someone to use a computer by opening it up and looking at the insides."

Me: "Let's get one thing straight. Opening it up and showing you the insides IS MY JOB."

2. Kid: "You know I just realized something really cool - if you showed me this problem 20 minutes ago I would have looked at you like you were crazy, but now I can do it! And it's easy!"

Other Kid (deadpan): "The miracle of learning."

3. Me: "The Smartboard has this neat tool I just found, look." (I freehand the graph of an ellipse by connecting some plotted points, and when I lift the pen, the shape snaps to a perfect curve.)

Kid: "But can she make a Kessel run in 12 parsecs?"

## Friday, March 20, 2009

### Three

So....

I called Math Teachers at Play #3 "Three is a Magic Number"

which

made me sing that little phrase all day every time I said "three" in class

and

planted the earworm

and

reminded me that it was, in fact, my #2 (not #3, unfortunately) favorite school house rocks after "Electricity, Electricity". Well Electricity was my favorite original SHR but it lost some charm in the remake which TISMN retained, so maybe it's a tie

and

found me playing the video for a delighted eighth period Algebra 1 class

and

lead me to post a stinking link in my blog in hopes of eradicating the earworm.

Enjoy. Here's hoping you are not as susceptible as I.

## Thursday, March 19, 2009

### Math Teachers at Play #3

Welcome to this installment of Math Teachers at Play! Three: It's a Magic Number!

Secondary Grades

Pat takes midpoint all the way through three dimensions for his precalculus class with Just an Average Point posted at Pat'sBlog.

Mr. D of I Want to Teach Forever shares a quality Bingo and a Matching Game for an Algebra 1 class in Two Review Games: Multiplying Polynomials and FOIL.

The Number Warrior, Jason Dyer ignores his lack of a handy lake, mountain, or orbiting satellite and has his students take measurements anyway in Imaginary Mountains (Laws of Sines and Cosines). He has lots of posts worth your time recently, check him out.

John D. Cook's When does the sum of three numbers equal their product? has a surprising result that I confess has me a little confused because I have to set down and hash out a proof for myself.

"Because acquiring this knowledge is difficult. Because you will feel triumphant when it no longer confuses you. Because you will enjoy what you can do with it. Because in learning it you may discover new perspectives on life, new ways of thinking. Because its possession will make you more alive than its alternative, which is ignorance."

- Banner & Cannon, The Elements of Teaching

Primary Grades

Denise explains how Math Facts Are Like Learning To Type posted at Let's play math!. Also click the link at the bottom of the story for the Game Worth 1000 Worksheets. I play these variations with my cousins every chance we get.

Cassandra Turner describes a nice mental math warmup in Number Strings as a Math Warm-Up.

William Wallace, when he's not leading Scottish revolts, re-invents that old standby binary trick in base three, with graphics.

Lists

Larry Ferlazzo provides a helpful list of lists: The Best Places To Find Theatrical Movies On Science, Math, & History.

Maria H. Andersen shares her Top 10 Technology Tools for Math - lots of good stuff here, and a few that were new to me.

In my search for distinctive features of various ages, someone (forget who, sorry!) turned me on to this cool list: What's Special About This Number?

Pi Day Roundup

Some math teachers were surely having some fun during this 2-week period, which of course contained the annual nerdvana, Pi Day. The value of this observance is in debate, as some celebrated enthusiastically and some remain dubious. This year congress made it official and Rachel Maddow let her geek flag fly. 360 told us about some things that equal pi and gave us a special edition pi day sudoku.

Unclassifiable

Here's a newfangled Rubik's Cube that will make your head explode.

Ian at Logic Nest reviewed JAME, fractal exploring software that looks amazing, and will hopefully be amazing once I actually have time to play with it.

Pagetutor uses Google Sketchup to give us some perspective on What A Trillion Dollars Looks Like. Like, in cash money. Stacked on pallets.

Jon analyzes Nim with some crazy directed graphs.

The TED site shows us Hans Rosling, using stats and infographs that will make you ooh and ahh, while oh by the way exploding myths about the developing world.

Have you done something fun lately that you want to share? Use the handy submission tool to get it in the next Math Teachers at Play.

And please indulge my favorite thing from the web in recent memory:

## Wednesday, March 18, 2009

### Pi Day 2009

For the third year my math department put on what can only be called an extravaganza for Pi Day. Why? It's fun. It generates authentic interest in math, of the "what's the big deal?" variety, and opens opportunities for us to discuss and teach about pi and math history in our classes.

The centerpiece of the festivities are three options assemblies held during the school day. ("Options", meaning, no one is required to go, but students are free to go if they have a coincident lunch, study hall, or their teacher during that period takes them.) For the past two years we have filled the auditorium to capacity. For three periods. For an optional math assembly.

The assemblies all follow a similar script. After the audience is seated and latecomers are finding seats, a band played this year's song called "All the Round Things" (a parody of Blink 182's "All the Small Things"). Both students and teachers wrote and performed the song this year.

Next, we conduct a game show styled after TV's "1 vs. 100", wherein a faculty member (not a Math teacher - this year they were all English and Social Studies) faces off against 30 students armed with clickers. We write the questions to be challenging but do-able for the students, and select faculty to 1. be funny and trash-talk the students and 2. with secret math/science backgrounds. The teacher usually wins.

Toward the end of the assembly we hold our digits of pi memorizing contest. Students sign up ahead of time who have memorized a minimum of 50 digits. During the first assembly, 5 students write as many digits on whiteboards on stage as they can in 3 minutes 14 seconds (while music blares and the crowd cheers). During the second assembly, a different 5 students do the same thing. All of the assembly participants get a certificate for a free pizza from a local place that donates them.

During the third assembly, the winners from the first two assemblies face off head to head. They each get a microphone, and alternate saying digits of pi until someone makes a mistake and can't go on. The competition this year took a dramatic turn, as the two finalists from last year both won their writing portion, so the final round was a rematch. After reciting over 180 digits perfectly, last year's champ made a mistake, and a new champion was crowned!

So that's what we do. It's a fair amount of work for several teachers, but we delegate. Examples: one person ran a t-shirt design contest and got the t-shirts made, 2 people wrote game show questions and set up the clicker software, one person liaised with the AV crew and custodians, and a million other things. It all got done and we had a great time.

What does your school do for pi day? We are always looking for ideas to make ours better!

## Tuesday, March 17, 2009

### Reminder

Teachers, homeschoolers, or anyone who enjoys playing around with math: Tomorrow night is the deadline to submit an article to this week’s

You can use the link above or email them directly to me. But I suggest you use the link and check out all the other "carnivals". I think my favorite is the "living with food allergies" carnival. Fun!

**Math Teachers at Play**carnival, which will be posted Friday right here! Posts must be relevant to students or teachers of preK-12th grade mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.You can use the link above or email them directly to me. But I suggest you use the link and check out all the other "carnivals". I think my favorite is the "living with food allergies" carnival. Fun!

## Saturday, March 14, 2009

### Funny Story

In Germany, I was informed by an exchange student, they don't have Pi Day, because they write the day before the month. So Pi Day would fall on 31 April. Oh, Germans. You should celebrate on 14 March with us, anyway.

## Friday, March 13, 2009

### How I Spent Pi Day

I firmly believe that my school puts on one of the truly great pi day observances. More on that later, when I have some pictures.

In my classes on Pi Day, I have found that kids, understandably, want to know "how they came up with all those digits". (My school has a digit memorizing contest.)

So I spend half the period telling them about the life and death of Archimedes. My material comes from Chapter 4 of Journey Through Genius by William Dunham (a fantastic read). I actually spend a fair amount of time reading to them from the book - I skip some parts, emphasize the cool stuff - they seem to really like it. It is a very well-written book.

I tell them about how he was born in Syracuse. Not the one we live in, but the one our city is named after.

View Larger Map

I tell them about how he was both a mathematician and an engineer. I show them a picture of the Archimedean screw, an invention for moving water uphill, and tell them that it is still in use today.

I tell them about how he was an eccentric genius, who would focus such single-minded attention on a problem that his friends and family had to force him to eat and bathe. About how King Hieron set him to the task of figuring out if his new laurel wreath crown was real gold, or if his goldsmith was cheating him.

And Archimedes puzzled over how to find the volume of the crown, so that he could calculate the density, and finally had a breakthrough while getting into one of his infrequent baths (water displacement = volume), and was so excited that he jumped from the tub and ran through the streets of Syracuse shouting "Eureka!", sans toga.

I tell them about how Hieron was so impressed with this, and with Archimedes' amazing work with levers and pulleys, that he set Archimedes to defending Syracuse from the invading Roman General Marcellus. Archimedes rigged a wall to shoot 1000 arrows simultaneously. Archimedes dropped boulders on their heads. Archimedes picked up their ships by the prow and dumped them into the sea. Archimedes may have used a mirror or lens to turn the sun into a death ray and set their ships on fire. Dunham calls him a "one man military-industrial complex".

Archimedes made the Roman army turn back. Marcellus retreated and instead laid seige to Syracuse. The seige continued until the Syracusans, after too much partying during a Feast honoring the goddess Diana, left a portion of their city wall too lightly defended. The Romans overcame them and entered the city triumphant. It is said that Marcellus wept when he saw the beautiful Syracuse, knowing the devestation his army would wreak.

Marcellus had come to respect Archimedes so much that he ordered him to be captured unharmed. However during the invasion, Archimedes' mind was so lost and absorbed in an interesting problem that he didn't even notice the Roman invasion. When a Roman soldier found him, Archimedes (instead of identifying himself and surrendering), begged the soldier not disturb his work. The soldier, annoyed, ran him through.

Then I start to tell them about the technique Archimedes used to calculate Pi to three decimal places. How it used to drive people like Euclid, Plato, and Pythagoras nuts that they didn't know the exact length of the circumference of a circle, given the diameter. People knew it was a little more than three times as long as the diameter, but hadn't done much better than that. Three generations after Euclid, Archimedes was the first to take a scientific approach to a more precise value.

I pass out patty paper and bullseye compasses (a compass/stright edge hybrid). I have the students construct a circle and inscribe and circumscribe a hexagon. We take the diameter of the circle to = 1 unit. Everyone notices that the perimeter of the inscribed hexagon is 3, and it has to be shorter than the circumference. The older students calculate the perimeter of the circumscribed hexagon. I then have them double the number of sides of both polygons, so they have an inscribed and circumscribed 12-gon. We notice that the perimeter of the inside one got longer, but is still less than the circumference. And the outside one got shorter, but is still greater than the circumference.

I tell them that this is the exact technique Archimedes used. He honed in on the value of pi from both sides. He went up to 96 sides, and found

$3\frac{1}{7}<\pi<3\frac{10}{71}$ And he didn't have the place value decimal system. Or any kind of calculator. The really neat thing is that until the invention of the calculus nearly 2000 years later, people used the exact same technique to get more and more digits' accuracy, they just had better tools. And that's the classroom portion of Pi Day. A lotta history, a little math. Gotta love it. Here are some files. The Smart Notebook of the pictures related to Archimedes, a collection of potential projects students could do around pi day, and a short classroom activity about how pi is related to hat size.

In my classes on Pi Day, I have found that kids, understandably, want to know "how they came up with all those digits". (My school has a digit memorizing contest.)

So I spend half the period telling them about the life and death of Archimedes. My material comes from Chapter 4 of Journey Through Genius by William Dunham (a fantastic read). I actually spend a fair amount of time reading to them from the book - I skip some parts, emphasize the cool stuff - they seem to really like it. It is a very well-written book.

I tell them about how he was born in Syracuse. Not the one we live in, but the one our city is named after.

View Larger Map

I tell them about how he was both a mathematician and an engineer. I show them a picture of the Archimedean screw, an invention for moving water uphill, and tell them that it is still in use today.

I tell them about how he was an eccentric genius, who would focus such single-minded attention on a problem that his friends and family had to force him to eat and bathe. About how King Hieron set him to the task of figuring out if his new laurel wreath crown was real gold, or if his goldsmith was cheating him.

And Archimedes puzzled over how to find the volume of the crown, so that he could calculate the density, and finally had a breakthrough while getting into one of his infrequent baths (water displacement = volume), and was so excited that he jumped from the tub and ran through the streets of Syracuse shouting "Eureka!", sans toga.

I tell them about how Hieron was so impressed with this, and with Archimedes' amazing work with levers and pulleys, that he set Archimedes to defending Syracuse from the invading Roman General Marcellus. Archimedes rigged a wall to shoot 1000 arrows simultaneously. Archimedes dropped boulders on their heads. Archimedes picked up their ships by the prow and dumped them into the sea. Archimedes may have used a mirror or lens to turn the sun into a death ray and set their ships on fire. Dunham calls him a "one man military-industrial complex".

Archimedes made the Roman army turn back. Marcellus retreated and instead laid seige to Syracuse. The seige continued until the Syracusans, after too much partying during a Feast honoring the goddess Diana, left a portion of their city wall too lightly defended. The Romans overcame them and entered the city triumphant. It is said that Marcellus wept when he saw the beautiful Syracuse, knowing the devestation his army would wreak.

Marcellus had come to respect Archimedes so much that he ordered him to be captured unharmed. However during the invasion, Archimedes' mind was so lost and absorbed in an interesting problem that he didn't even notice the Roman invasion. When a Roman soldier found him, Archimedes (instead of identifying himself and surrendering), begged the soldier not disturb his work. The soldier, annoyed, ran him through.

Then I start to tell them about the technique Archimedes used to calculate Pi to three decimal places. How it used to drive people like Euclid, Plato, and Pythagoras nuts that they didn't know the exact length of the circumference of a circle, given the diameter. People knew it was a little more than three times as long as the diameter, but hadn't done much better than that. Three generations after Euclid, Archimedes was the first to take a scientific approach to a more precise value.

I pass out patty paper and bullseye compasses (a compass/stright edge hybrid). I have the students construct a circle and inscribe and circumscribe a hexagon. We take the diameter of the circle to = 1 unit. Everyone notices that the perimeter of the inscribed hexagon is 3, and it has to be shorter than the circumference. The older students calculate the perimeter of the circumscribed hexagon. I then have them double the number of sides of both polygons, so they have an inscribed and circumscribed 12-gon. We notice that the perimeter of the inside one got longer, but is still less than the circumference. And the outside one got shorter, but is still greater than the circumference.

I tell them that this is the exact technique Archimedes used. He honed in on the value of pi from both sides. He went up to 96 sides, and found

$3\frac{1}{7}<\pi<3\frac{10}{71}$ And he didn't have the place value decimal system. Or any kind of calculator. The really neat thing is that until the invention of the calculus nearly 2000 years later, people used the exact same technique to get more and more digits' accuracy, they just had better tools. And that's the classroom portion of Pi Day. A lotta history, a little math. Gotta love it. Here are some files. The Smart Notebook of the pictures related to Archimedes, a collection of potential projects students could do around pi day, and a short classroom activity about how pi is related to hat size.

## Thursday, March 12, 2009

### More Locus - Finding Buried Treasure

I briefly mentioned in the previous locus post that a treasure map is a fun application of the concepts in locus. Think about pirate movies..."5 paces from the tree, and 3 paces from the fence!" (There could be up to four possible locations for that treasure...)

I first tried to design a treasure hunt using the whole school, a la geocaching, but the impracticality of that became quickly apparent. So, I designed a smaller scale treasure hunt within my classroom.

It's designed for use with student desks in a 6 x 5 grid. Before class, you hide treasures (I used colored index cards with a funny quote or comic) by taping them under certain desks. You provide the students with a map of the desks they can write on, and a sheet of compound locus descriptions that tell them how to locate the treasure.

I've used this several times and it was successful and fun. I hope you find it useful!

## Wednesday, March 11, 2009

### Not Everyone Can Be in the Top Quartile

Dina wrote a very engaging dialog explaining how she might go about calculating her "teacher percentile", and why it is impossible with the data available.

And I realized what was the most bothersome thing about Bill Gates' recent bloviating about teacher quality. (Not, by a long shot, the only bothersome thing.)

He made this particularly asinine and ill-informed observation:

So simple!

Aside from all the other reasons it's not simple, Bill, your math is wrong.

You don't understand what a quartile is.

Take a group of people and measure something numerical about them. Their height, let's say. Take those heights and make a list of them from greatest to least. Cut that list into four equal sized chunks. The top chunk is the top quartile.

Surely everyone with half a brain notices the problem with his statement. We can't replace all the teachers with top quartile teachers. You only have a top quartile when you compare and rank the entire group. Only 25% of teachers can EVER be "top quartile". (Not to mention, say you buy this line of crap and go ahead and fire 75% of the teaching force. Where, exactly, is this magical pool of top teachers to replace them?)

You have to wonder what would happen if Bill got his wish. If 75% of working teachers were replaced with all these awesome, amaaaaazing teachers who are currently for some mysteeeerious reason not teaching. Do you think that after the first year he would realize that only 25% of the new group of teachers were top quartile? I picture him at his big Surface(tm) desk, pulling his hair, rubbing his temples, lamenting "How is this POSSIBLE?"

I thought he was supposed to be so smart...he should come sit in my 9th grade algebra class in a few weeks. That's when our Stats unit is. I will enjoy the moment when the kids realize they know more than Bill Gates.

And I realized what was the most bothersome thing about Bill Gates' recent bloviating about teacher quality. (Not, by a long shot, the only bothersome thing.)

He made this particularly asinine and ill-informed observation:

A top quartile teacher will increase the performance of their class -- based on test scores -- by over 10 percent in a single year. What does that mean? That means that if the entire U.S., for two years, had top quartile teachers, the entire difference between us and Asia would go away. Within four years we would be blowing everyone in the world away. So, it's simple. All you need are those top quartile teachers.

So simple!

Aside from all the other reasons it's not simple, Bill, your math is wrong.

You don't understand what a quartile is.

Take a group of people and measure something numerical about them. Their height, let's say. Take those heights and make a list of them from greatest to least. Cut that list into four equal sized chunks. The top chunk is the top quartile.

Surely everyone with half a brain notices the problem with his statement. We can't replace all the teachers with top quartile teachers. You only have a top quartile when you compare and rank the entire group. Only 25% of teachers can EVER be "top quartile". (Not to mention, say you buy this line of crap and go ahead and fire 75% of the teaching force. Where, exactly, is this magical pool of top teachers to replace them?)

You have to wonder what would happen if Bill got his wish. If 75% of working teachers were replaced with all these awesome, amaaaaazing teachers who are currently for some mysteeeerious reason not teaching. Do you think that after the first year he would realize that only 25% of the new group of teachers were top quartile? I picture him at his big Surface(tm) desk, pulling his hair, rubbing his temples, lamenting "How is this POSSIBLE?"

I thought he was supposed to be so smart...he should come sit in my 9th grade algebra class in a few weeks. That's when our Stats unit is. I will enjoy the moment when the kids realize they know more than Bill Gates.

## Tuesday, March 10, 2009

### Locus

Do other states even have to teach Locus? I have a feeling it might be one of those annoying NY topics that they like to cram in the curriculum for no good reason.

Anyway, Locus is fun for me to teach since I cracked the code.

Just in case anyone is not on the same page yet, "Locus" comes from the Latin for "place", and we can think of it as a location in space. More specifically, a collection of points that fit some description.

For example, the locus of points equidistant from a given point is...a circle. The locus of points equidistant from two given points is...the perpendicular bisector of the segment connecting those two points. You can have more fun in 3D. The collection of points equidistant from a line is...a cylinder. Etc.

This leads to problems like...Two statues are 10 meters apart and connected by a fence. A tree will be planted equidistant from the statues and also 13 meters from one of the statues. How many possible locations are there for the tree? How far away from the fence will it be? Problems pinpointing the location of buried treasure are also very popular.

OK so back to cracking the code. Our success was limited while I stuck to having the kids draw pictures. We were marginally more successful when graph paper was incorporated. It's much easier to grok "4 units away" and "equidistant" when you can count distances on graph paper.

The breakthrough came from getting the concept off the paper. For a class learning the material for the first time, I acquired different-colored tape. On the floor, I used the tape to make two parallel lines, and one line intersecting them. I also used the tape to make a few small X's on the floor representing points. That took a little prior planning, $10 at the hardware store, plus, the custodians yell at you if you don't get all the tape off the floor.

In the class, we pushed all the desks aside, and as a group the kids were given vocal instructions by me like, "Everyone stand so that you are 4 floor tiles away from the green line." "Everyone stand so that you are 4 floor tiles away from the blue X." "Everyone stand so that you are equidistant from the blue X and green X." After each instruction, we discussed the shape they had arranged themselves in. I proclaimed them "locus experts!" before they had even touched pen to paper. This lesson stuck. I heard stories from other teachers and students ("Today I was a perpendicular bisector!") The rest of the locus unit was much easier for them and me to deal with and scores even improved.

This year I am sadly not teaching Geometry. No Locus unit. However, I am teaching a section of Honors Precalculus. Today I was introducing the parabola as the locus of points equidistant from the focus and directrix, and x^2 = 4py. With a gleam in my eye I told the kids to shove the desks to the side of the room, and got out my tape. After a few warm-up human loci, I threw my keys on the floor, and told them to stand so that they were equidistant from the keys and the wall.

A parabola materialized. Made of children.

Then I scooted the set of keys closer to the wall and said, "Do it again."

Anyway, Locus is fun for me to teach since I cracked the code.

Just in case anyone is not on the same page yet, "Locus" comes from the Latin for "place", and we can think of it as a location in space. More specifically, a collection of points that fit some description.

For example, the locus of points equidistant from a given point is...a circle. The locus of points equidistant from two given points is...the perpendicular bisector of the segment connecting those two points. You can have more fun in 3D. The collection of points equidistant from a line is...a cylinder. Etc.

This leads to problems like...Two statues are 10 meters apart and connected by a fence. A tree will be planted equidistant from the statues and also 13 meters from one of the statues. How many possible locations are there for the tree? How far away from the fence will it be? Problems pinpointing the location of buried treasure are also very popular.

OK so back to cracking the code. Our success was limited while I stuck to having the kids draw pictures. We were marginally more successful when graph paper was incorporated. It's much easier to grok "4 units away" and "equidistant" when you can count distances on graph paper.

The breakthrough came from getting the concept off the paper. For a class learning the material for the first time, I acquired different-colored tape. On the floor, I used the tape to make two parallel lines, and one line intersecting them. I also used the tape to make a few small X's on the floor representing points. That took a little prior planning, $10 at the hardware store, plus, the custodians yell at you if you don't get all the tape off the floor.

In the class, we pushed all the desks aside, and as a group the kids were given vocal instructions by me like, "Everyone stand so that you are 4 floor tiles away from the green line." "Everyone stand so that you are 4 floor tiles away from the blue X." "Everyone stand so that you are equidistant from the blue X and green X." After each instruction, we discussed the shape they had arranged themselves in. I proclaimed them "locus experts!" before they had even touched pen to paper. This lesson stuck. I heard stories from other teachers and students ("Today I was a perpendicular bisector!") The rest of the locus unit was much easier for them and me to deal with and scores even improved.

This year I am sadly not teaching Geometry. No Locus unit. However, I am teaching a section of Honors Precalculus. Today I was introducing the parabola as the locus of points equidistant from the focus and directrix, and x^2 = 4py. With a gleam in my eye I told the kids to shove the desks to the side of the room, and got out my tape. After a few warm-up human loci, I threw my keys on the floor, and told them to stand so that they were equidistant from the keys and the wall.

A parabola materialized. Made of children.

Then I scooted the set of keys closer to the wall and said, "Do it again."

## Saturday, March 7, 2009

### More Fun with Dates!

Birth dates, actually. Ages, to be exact.

So, here's the thing. When people have a birthday I often tell them why their new age is a cool number. (I know, I know, the nerditude. It consumes.) I'm already planning a 33 1/3 birthday party. For myself. Also, when kids ask me how old I am, I just give them a mathematical clue. They find this very annoying.

My favorite is students turning 16, because we sing them a special little song (originally written in honor of my friend Denisse) -

Happy birthday to you!

You're a power of two!

The next time that happens,

You'll be 32!

I've lately been trying to compile a list. I left out all the primes - they're more interesting as people get older. Can you fill in any holes? And no cheating - everyone older than 3 is the sum of two primes!

If your age is _____, you are a(n) ________

6 - a perfect number

7 - the only single digit number spoken with two syllables

8 - a perfect cube

9 - a perfect square

10 - a triangular number

11 - the first three-syllable counting number

12 - an abundant number

15 - a triangular number

16 - a power of two, and a perfect square

24 - a factorial

25 - a perfect square

27 - a perfect cube

28 - a perfect number

32 - a power of two

Anything else I come up with is either "prime" or a big stretch! I could really use good ones for 17 and 18.

update: 13, 19, 23 and 31 are happy numbers.

17 is a hungry number.

17 is also a Fermat Prime.

65 is the least number that can be written as the sum of two squares in more than one way.

Thanks for all the comments!

So, here's the thing. When people have a birthday I often tell them why their new age is a cool number. (I know, I know, the nerditude. It consumes.) I'm already planning a 33 1/3 birthday party. For myself. Also, when kids ask me how old I am, I just give them a mathematical clue. They find this very annoying.

My favorite is students turning 16, because we sing them a special little song (originally written in honor of my friend Denisse) -

Happy birthday to you!

You're a power of two!

The next time that happens,

You'll be 32!

I've lately been trying to compile a list. I left out all the primes - they're more interesting as people get older. Can you fill in any holes? And no cheating - everyone older than 3 is the sum of two primes!

If your age is _____, you are a(n) ________

6 - a perfect number

7 - the only single digit number spoken with two syllables

8 - a perfect cube

9 - a perfect square

10 - a triangular number

11 - the first three-syllable counting number

12 - an abundant number

15 - a triangular number

16 - a power of two, and a perfect square

24 - a factorial

25 - a perfect square

27 - a perfect cube

28 - a perfect number

32 - a power of two

Anything else I come up with is either "prime" or a big stretch! I could really use good ones for 17 and 18.

update: 13, 19, 23 and 31 are happy numbers.

17 is a hungry number.

17 is also a Fermat Prime.

65 is the least number that can be written as the sum of two squares in more than one way.

Thanks for all the comments!

## Tuesday, March 3, 2009

### 3/3/09

This only happens nine times a century! Like a comet! We have to wait 7 (not 5) years for the next one! (5 years since the last one) (That will teach me to blog before coffee.) What are you doing to celebrate?

## Monday, March 2, 2009

### The Potential Snow Day Survival Guide

This was originally written as a gentle ribbing of my downstate colleagues, who let out a collective "wtf" last week when a school day was called

Potential Snow Day Survival Guide

*way*too late. I wasn't happy with the potential for hurt feelings. I have toned down the rhetoric but for my own amusement would still like to humbly offer....Potential Snow Day Survival Guide

- If a snow day is suspected, watch the 11pm news. This way you can tell if there's anything to even get excited about. The newscast gives you the best picture about the inches of snow that will fall at what time. Weather.com is not going to cut it. You are looking for heavy snowfall starting around 3AM and continuing throughout the morning. Something the plows won't be able to keep up with, at just the right time.
- If you drive to work, park your car facing out. Preferably at the top of a hill. That way you will be able to just brush it off and barrel out with minimal digging.
- Calculate the last possible minute you can stay in bed and still make it to work when you absolutely have to be there. Not a half hour early, like you normally do. The last possible minute. (Take into account that the commute might take longer in bad weather.) For me this is 6:22.
- Set your clock radio to go off 10 minutes before that time. Tune it to a news/weather station.
- When your alarm goes off, don't hit the snooze. Listen to the radio. Drape your hand over it, so you can turn it off as soon as your school is called. With a little practice, you will not have to open your eyes during this part.
- Either your school is called... (turn off radio, roll over, go back to sleep. optional: point and laugh at anyone living with you who has to go to work.)
- Or it's not called by your absolute last-possible minute. In which case...
- Take a radio into the bathroom with you. Play it loud enough so you can hear it in the shower. Take it into the kitchen while you make and eat breakfast.
- Sometimes even though you take all these precautions, they still call it after you are out the door. That is very annoying.
- The day is yours! Go back to bed if you want, or make pancakes, or a snowman. Enjoy!

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