## 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, December 22, 2009

### I Finally Used the Cell Phones for Something

Sixty students! Two days! Two buildings! Twelve clues! One map! I went a step beyond the classroom locus treasure hunt this year and made a school-wide locus treasure hunt.

In class the students worked with large (11"x17") maps of the school, compasses, rulers, and locus-type pirate map style clues (100 feet from Miss Nowak's room, and also equidistant from the math hallway and the resource hallway) to narrow down the location of 12 stars I stuck around the school. Then they had the whole next day to look for the stars and send a picture of them standing next to it to a picasaweb drop box for a bonus point.

I'm not going to bother posting documents because obviously they would only work for my school. But, it was fun, and I was proud of myself for thinking to use the phone cameras like that.

And, it seemed much, much easier for the kids to get a hold of the locus concepts when it was tied to a concrete map of the school. For some reason saying "equidistant from room 2122 and 2126" has magical super powers that "equidistant from two points" does not. Weird.

Update: Here are the files, for whatever they are worth.

## Friday, December 18, 2009

### Introducing Logs

(Adapted from James Tanton's monthly St. Mark's Math Institute newsletter.)

And then let them in on the dirty little trick that in math we insist on calling it a logarithm instead of a power when we write it like that.

To answer the inevitable "

*What the hell?*", I go into a little history of how John Napier invented logarithms to make multi-digit multiplication easier for Renaissance astronomers.

Then we mention that it's an inverse of an exponential equation, play a little more with shifting between exponential equations and log equations, and we are done for the day.

Here are my filled in notes:

Here is the smart notebook file. Here is a google doc with text of a task for easy copy and pasting.

## Wednesday, December 16, 2009

### Row Games Galore

**Update**: I moved all the files that were shared on Box into a Google Drive folder. You can upload, download, and add the folder to your Drive if you wish. Please use responsibly.

I love a good Row Game but they are sure a pain to make. I just made a new one for reviewing surface area and volume, and in a minute I'm going to go lay down with a hot compress and a glass of wine.

I know some others are out there making them too, and I'd love it if we could all put them in one place, and save each other some work. So here is that place! It's a

## Friday, December 11, 2009

### If You are a Fan

I'm not sure what the acceptable social norms are for promoting oneself (this is new to me), but I wanted to let readers know the nominations were up, and voting is open until Wednesday the 16th. I'm stunned and humbled just being nominated.

## Thursday, December 10, 2009

### A Very Special Episode of f(t)

And then, at the beginning of one particular class, I begged whoever swiped one of my loaner graphing calculators, that was missing for over a week, to please return it. I told them I provided them so people who forgot theirs would be able to borrow one. Because it seemed like a considerate thing to do, to make math class less stressful. I knew if someone had it, it didn't mean they were bad, but that I really needed it back, and it was within their power to make a more honorable choice. I didn't care who it was, they could put it in my mailbox in the office so I didn't have to know. I wouldn't be able to loan out calculators anymore if I didn't get it back. And it was a shame that the 120 people who came through my room each day would have to suffer for one person's actions. That people without a calculator would have to make arrangements before Monday's quiz, because they won't be able to borrow one from me.

So we had a half-period lesson, then a half-period practice / peer tutoring kind of activity. I was all over the place, the kids were up and around, everyone was working hard.

And then after the bell rang, I walked to the back of the room, and OH SNAP, SITTING THERE, IN THE MIDDLE OF MY DESK, THE MISSING CALCULATOR! It was a CHRISTMAS MIRACLE! I couldn't believe it. I still can't. Little speeches like that never work out for me. I felt like a movie teacher. I wish I knew who it was. I would hug the little bastard.

## Tuesday, December 8, 2009

### Edublog Award Nominations 09

Best individual blog : dy/dan. Even though Dan went half corporate this year, he keeps feeding me lessons that are made of adolescent catnip. And if you hadn't noticed, is constructing a new framework for math ed.

Best individual tweeter : @msgregson I don't know when this girl sleeps, but she's the most reliably helpful tweeter around. She's going to be a technology coordinator to be reckoned with.

Best group blog : 360. I spend a little time trying to hack my way through some intimidating math blogs, so I enjoy the way these college professors can make sophisticated concepts relatable.

Best new blog : Questions? Dave Cox came out swinging this year, asking profound and demanding questions of his not that new practice. Take a moment to appreciate how rare that is. And he can carry a tune, too.

Best elearning / corporate education blog: Not sure what category works best for Colleen K, but http://www.mathapprentice.com/ and http://www.mathplayground.com/ both blow me away. I know I didn't do anything with my computer degree, but if I had, I hope it would look something like that.

Best resource sharing blog The Exponential Curve Blog posts are sporadic, but Dan Greene figured out how to share all of his stuff, all of the time, and keeps adding to it.

Best teacher blog Continuous Everywhere but Differentiable Nowhere There should be a preservice course about how to improve your practice through written reflection, and a large part of it should be reading Sam's blog.

## Monday, December 7, 2009

### School and Other Miscellany

2. I was asked to appear with a crowd of other veterans in the last scene of the school's production of White Christmas. Even though it's a neat idea, I don't really want to, because I don't like people looking at me. Well, more than 30 people. Who are over the age of 16. And they want us to not wear a whole uniform but just like a cover (that's a hat, civilians). Which I was taught is Wrong. I haven't decided yet.

3. I've seen some Edublog award nominations, and they are very cool and flattering. Thank you. I don't expect f(t) to be competitive; I think it's a little too niche. I might have a distant shot if they made a category best math teacher blog that is not dy/dan.

4. I need some more music for running. I'm in week 3 of couch to 5k and my playlist is played out. I need help. Here is my current playlist. You should post yours so I can pick through it. Also, you only get to judge mine if I get to see yours. No deleting the embarrassing tracks first.

### New Blogger: Launch the Alert-5 Subscribe Buttons

## Sunday, December 6, 2009

### My Favorite Theorem

I immediately chose Cantor's proof of the nondenumerability of the reals, for its counterintuitive-ness and yet easiness to understand if you can hang with a pretty simple argument. It's the best way to show the uninitiated that math is beautiful that I've ever found. And not because you drew a pretty picture of a golden rectangle, but because the ABSTRACT ARGUMENT all by itself is BEAUTIFUL and once you get it makes you go ooh and aah. I've explained this to middle school kids, roommates, people in elevators, my mom, and one time (disastrously) a boy I had a crush on.

The big, impressive, revolutionary idea embedded in here is that there is more than one size of infinity. It takes most people a while to stretch their brains around that. It drove Georg Cantor to an asylum, so no need for anyone to feel bad if it takes them a while.

Children's first encounter with infinity is (usually, I think) when they realize that the counting numbers never stop. There's no last one. You can keep counting incrementing by one forever and you will never...reach...the end.

What if you count just the even numbers? Nope. By fives? Nope. Tens? Nope. Hundreds? Millions? Brazillions? You get the idea.

What if we want to compare how many counting numbers there are to how many even numbers? We obviously can't count them and see which set has more. It makes intuitive sense to conjecture that there are more counting numbers than even numbers. Seems like there should be twice as many.

Cantor's genius move was to invent a way to compare the 'sizes' of these infinite sets, which he called 'cardinality.' Instead of trying to count the number of elements in each set, which is impossible, he said let's look at it another way. For example, when I take attendance in my class, I don't count the number of students, and count the number of chairs, and compare the two counts. I just see if every chair is matched with a butt. If there are no chairs left over, I know the sets are the same size, and I mark All Present.

Cantor applied the same logic to infinite sets. He said if we can match every element in one set to an element in the other, then they are the same size. Since both sets go forever, we don't have to worry about leftovers. They both keep going forever, so every element will match something in the other set, and they are the same size. Sets that can be matched this way are said to be in a one to one correspondence.

Looked at this way, there are the same number of elements in the counting numbers as there are in the even numbers. We can match 1-2, 2-4, 3-6, 4-8, 5-10, and so on, forever. The elements of the sets can be placed in a one to one correspondence, so they are the same size.

You can make a matching to show that the cardinality of the counting numbers = the integers. The counting numbers = the multiples of ten. And even, with a little clever organization, the counting numbers = the rationals.

Intuition resists this conclusion. Sometimes with students, I have to stop here. They just aren't ready to concede the point, and I'm not about to make them push the "I believe" button on something we're doing for fun. I think it's healthier to let them be bothered by it. And if you don't believe that part, you definitely won't come along with me for the rest of the ride.

So you believe that all these infinite sets that can be matched with the counting numbers are the same size. Great! Then what? There's just that one size of infinity?

Nope.

There's an infinity bigger than that. Infinitely bigger than that. And that infinity is located, for an example, on a number line between zero and one.

It takes a little meditation to absorb that statement, too. There are infinitely more numbers between zero and one than there are counting numbers.

Here's where my favorite proof starts. :-) Quite the setup, I know.

Let's say that that conjecture is NOT true. Let's say that there are the same number of elements in the set between zero and one as there are counting numbers.

If that's true, then I can make a list of all of them. It will be infinitely long, but a list can be made. Much like you can list the counting numbers.

So here's the beginning of my list:

0.124598560

0.7463728642893

0.00033033845788

0.1111111111111....

0.7465564748383

0.123123123123123...

0.141592653589...

0.799999999999...

.

.

You get the idea. The decimal expression of every rational and irrational number between zero and one will be in my list.

Now, I'm going to write a new number. But I'm going to write it in a very particular way. The first digit will not be a 1. The second digit will not be a 4. The third digit will not be a 0. The fourth digit will not be a 1. The fifth digit will not be a 5. etc*.

By doing this, my new number must be different from any number in my list, because for the

*n*th list item, it differs in the

*n*th digit.

But wait! We said we were going to list ALL the numbers between 0 and 1! What gives?!

What gives, is that the only thing that could be wrong here is our original supposition. When we said, "

*Let's say that there are the same number of elements in the set between zero and one as there are counting numbers.*" that must have been wrong.

Which means there must be MORE elements in the set between zero and one, than there are in the counting numbers.

Another size of infinity. Cool, right? That's why it's my fave.

*A little caveat needs to go here about not choosing 0's or 9's for your new number, since for example 0.50000.... = 0.499999....

## Wednesday, December 2, 2009

### Reporting from the EduTech Front

I tried a little polleverywhere experiment earlier this year. I am in love with the idea of this technology. My school has a few sets of clickers, and they are a total pain. All hail clicker functionality using the tiny computer the kids already have in their pockets! The kids were amped, too. When I started talking about how I wanted us to try out polleverywhere, and giving them instructions like "find the slope, and text your answer to this number!" there was a palpable "this is so cool" energy running through the room.

Except! 1. Not every kid has a cell phone. My students are predominantly middle class, but we are a large public high school and serve

*plenty*of families who consider cell phones for their kids a luxury. Also, ironically I suspect some of these non-cell-phone-having kids are from families trying to adhere to what the school is telling them, that cell phones are technologia non grata up in here. Even if you bring them, you are expected to keep them turned off and out of sight during the school day (unless a teacher gives you explicit permission to use them for a classroom activity.)

*Not every kid has a cell phone.*Not every kid with a cell phone has texting enabled. How to employ them as a learning tool when not every kid has one is a major teacher training challenge that needs to be addressed. It basically limits us to opt-in kind of use, like a project where students can choose from several options, and one of them involves a cell phone. Or small group work, where only one group member needs access to unlimited texting. I can not think of a good solution for kids without a phone if you are trying to implement it as a frequent whole-group feature.

And 2. Turns out my classroom is a Verizon dead zone. AT&T and T-Mobile work fine, but Verizon is the dominant carrier around here. More than half the kids were not getting enough bars to send a text.

So much for my grand polleverywhere plans.

My other jaw dropping technology moments this year have come from the class blogs. Or rather, from panicked and frustrated kids the morning after they tried to access the class blogs. (This goes double if Geogebra was embedded - I've basically given up on that.) Here are some choice quotes: "I am not good at logging into things." "Our computer at home runs Windows 98." "It wouldn't load the page. Something about cookies." And my personal favorite, "Whenever something is on the Internet, I can't take it seriously." By this point, I've been able to plan with these kids a way for them to use a school computer during the school day. But this was not nearly as easy as the "They're All Digital Natives! They Will Teach US!" propaganda would have us believe.

Maybe it's just a matter of time before every single student, indeed, carries a cell phone. Maybe Verizon will install another cell tower near my school. But for what it's worth, these are my realities on the ground.

### Tales from the Google Forms

I think this means they couldn't find the sides of the square with the integral diagonal. We shall find out tomorrow.

## Monday, November 23, 2009

### The Answer is Not the Answer

...which did wonders to relieve their anxiety, and took the emphasis off the answer, and put it on the process.

## Sunday, November 22, 2009

### Exponential Growth and Credit Cards

(Note: After some revisions a new, improved document was uploaded, and the link changed, at 11:55AM EST on 11/23/09.)

## Thursday, November 12, 2009

### All Worksheets Are Not Created Equal

Yes and no.

*This is not so hot:*Disjointed. Pointless. Teaches nothing. Brain need not engage. Follow some procedure whether you know what you are doing or not. Kids either refuse to participate in the charade or tolerate it. They might be tolerating it with sweet, deceiving smiles on their faces, but they are not relishing it as a learning opportunity.

*This is much better:*Believe it or don't, but you get a much different reaction from a kid, who has the appropriate background, to a worksheet like this. A narrowing of the gaze. Quiet focus. Murmuring and grunting. Meaningful questions.

I'll leave it as an exercise for the reader to think about why it's better. That's not really the point I want to make right now. Propogating the myth that All Worksheets are Bad isn't just unhelpful, it's harmful. It has led to a situation where we think Good Teachers Don't Use Worksheets and Bad Teachers Use Worksheets. Textbook companies are putting out crap. Grad programs aren't showing new teachers how to make good ones.

Argue all you want about what public schools should look like, but in the schools we have in late 2009, I am looking at 30 kids for 45 minutes a day, sitting in the available chairs at the available desks. Some of them have a cell phone, a subset of those get reception in my room, some number could bring a laptop every day if we asked them to. But they all have a pen or a pencil, and none of them want to waste their time. This reality means I must, inevitably, find or make paper materials to put in their hands. If I believed all worksheets were bad, and using them made me a bad teacher, I would eventually print out some garbage provided by the textbook, close the door, and keep my head down. Thankfully I spent my first couple years, before they retired, around some very wise mentors.

Here is that exponent worksheet if you liked it.

## Wednesday, November 11, 2009

### Transformation Golf

I know I know...in other news, water is wet.

With that in mind, I bring you...Transformation Golf. The directions are "Get the blue shape to match the green shape." Shots are taken by commanding your dynamic geometry software to perform line reflections, transformations, rotations, and dilations on the blue shape. The early holes start with one-offs, and later holes require compositions of transformations. Accountability and notation practice come from having the students write down the specific transformation they used for each shot. There are deeper conversations to be had about how to decide par for particular holes, and for any hole. But, still, the beauty is in the undo button*.

I wrote this a few years ago, before I knew about Geogebra. So, the golf game file and the associated instructions use Geometer's Sketchpad. Perhaps some industrious soul will adapt it to Geogebra and share it with the rest of us.

**This paragraph was brought to you by The Passive Voice.*

## Monday, November 9, 2009

### Remembering Function Definitions

## Saturday, November 7, 2009

### Calculator Customization Using Transformations (Post #2)

**Part 3: Plot Transformations of the Drawing**

Now we want to take our picture and transform it using the lists. For example, let's say I want to reflect the picture over the y-axis. Instead of graphing (x, y), I need to graph (-x, y). Go back to the lists in STAT, Edit. Put the negation of L1, the x-coordinates, into L3. You don't have to type them all separately. With the cursor, highlight the list title L3. Then type the negative sign (-) and L1 with 2nd, 1. Then Enter. It should automatically populate L3 with the negation of every value in L1.

We still need to graph (L3, L2) to graph the reflection. Go to 2nd, Y= (STAT PLOT). Turn on Plot 2. Now use L3 for the XList and L2 for the YList. Press GRAPH. You should see both your original picture, and its reflection over the Y-axis.

You can plot any transformation you like now by using L4, L5, and L6 in the same fashion. Plot (-L2, L1) for a Rotation of 90 degree counterclockwise. Plot (-L1, -L2) for a point reflection through the origin. Plot (L1, L2 - 10) to translate the picture down by 10 units. Etc etc.

**Part 4: Add Text**

This part is optional, but it is nice to add some text, like your name, if you are going to use the drawing as the start-up screen for the calculator.

- If you are not already there, go to the GRAPH screen.
- Press 2nd, DRAW, and arrow down to "Text". Press ENTER. You will see a flashing cursor on the graph screen along with your drawing.
- Use the arrow keys to move the cursor to where you would like to start typing.
- Hit 2nd ALPHALOCK to put the calculator in alphabetic mode.
- Start typing your text. If you need a space, it's above the zero key. Press ENTER when done.
- If you mess up, you can't delete mistakes. Unfortunately, you have to start over again. To start over, go to 2nd DRAW and select ClrDraw. Go back to step 1 in this section.

**Part 5: Save Your Picture**

- Once the picture is complete, press 2nd DRAW. Arrow right to STO and select StorePic.
- You should see StorePic appear on the home screen with a flashing cursor. You must enter a number between 0 to 9. I suggest you use a number other than 1, because many programs use 1 to temporarily store graphics. If that happens, your beautiful drawing will be overwritten. For this project, let's enter a 5.
- Once you type in a number, press ENTER. Your design is now saved in PIC5.

**Part 6: Use the picture as the calculator start-up screen**

- You need the application in the APPS menu called Start-Up. If you don't have it, you can get it from someone else with a link cable. Or, you can download it from ti.com to a computer, and download it to your calculator with the computer link cable.
- If you have Start-Up, launch it from the APPS menu.
- Under "Choose Settings" turn Display ON, Type PIC, Name PIC5, and whatever Time you like.
- Press FINISH when done.

## Friday, November 6, 2009

### Calculator Customization Using Transformations (Post #1)

This first post describes how to get the picture into the calculator. The next post will talk about transforming the picture, and using the final drawing as a start-up screen in a TI-83 or 84.

Overview: Put a drawing in the calculator, do some transformations on it, add your name, and use it as the start up screen in your calculator, like this:

Part 1: Create a Design:

- On graph paper, create a simple design that can be entirely drawn without picking up your pencil.
- You may want to trace a cookie cutter or a stencil.
- Draw axes on the paper in a sensible place, so that all your points are in Quadrant I. (x and y both positive)
- Label points on the drawing with their coordinates. Eventually, the calculator will be connecting these points with straight line segments, so be smart about it. Curvy sections will need more points if you want them to look smooth. Here is an example:

Part 2: Calculator

- STAT, Edit. Clear all old data in any list. (Highlight the title of the list, and press CLEAR, Enter.)
- Enter the x-coordinates of all points in L1 and the corresponding y-coordinates in L2. Enter them in the order you want them connected.
- If the design is a closed figure, you have to enter the coordinates of the starting point again as the last point.
- 2nd, Y= (STAT PLOT). Press Enter on Plot 1, and set it up as below:

## Friday, October 23, 2009

### The Row Game

*das uber-class*that now has 31 kids in it.

*I don't even have 31 desks. I just hope at least one person is absent every day. I am totally serious about this.*

Make a worksheet of problems organized in two columns. Column A and column B. The tricky part is the pair of problems in each row has to have the same answer. Obviously some topics are more suited to this than others. (Solving linear systems, easy. SOHCAHTOA, easy. Graphing inequalities, hard.)

Pair up the kids. Decide who is A and who is B. Tell the kids to only do the problems in their column. When done, compare answers to each question number with their partner. And if they don't get the same answer, work together to find the error. That last step is where the magic happens. I know how well I taught the topic by how busy I am while they are row gaming it up. (Sipping coffee: go, me. Running around like lettuce with its head cut off: self-recrimination time.)

I'll also do this by projecting 2-3 pairs of problems for 5-10 minutes of practice at the end of a lesson. Row Game Lite.

Here is an Operations on Radicals and a Permutations and Combinations worksheet to get you started.

## Friday, October 9, 2009

### How to Embed a Geogebra Sketch into a Blog Post

## Triangle Centers

David Griswold/Kate Nowak, Created with GeoGebra

How to Do This

- Make your sketch in Geogebra. Save it as a ggb file, like normal
- In Geogebra, File -> Export -> Dynamic Worksheet as Webpage (html)
- Go to Geogebra Upload Manager and login or create an account and login
- Follow their instructions for uploading materials. Basically they want you to put it in a folder with your name on it.
- You'll want to upload the *.ggb and the *.html both for your sketch.
- Open the html file in a browser. Select View -> Source.
- Copy all the code between (but not including) the TABLE tags to the clipboard.
- Paste it into the HTML editor of your new blog post.
- In the line that stars with "param name =," change the filename from yourfilename.ggb to the whole URL copied from the geogebra upload site. For example, I had to change
*centers.ggb*to*http://www.geogebra.org/en/upload/files/kate_nowak/centers.ggb* - That should do it.
- Has anyone else noticed that "Geogebra" sounds like an undergarment for a woman made out of rocks?

## Tuesday, October 6, 2009

### Speed Dating

To prepare, you will need cards or slips of paper with problems on one side and the answers on the other. (Here are 16 Rational Expressions cards with answers and here are 15 Probability & Permutations cards with answers you can cut out, fold, and use.)

It's important to use problems that will take all your kids about the same amount of time to complete. To differentiate, use a mix of difficulties. If you will have the whole class working together, you need as many problems as students. If you are breaking the class into two distinct groups, you need half as many problems as students, but two copies.

Arrange your desks in two rows facing each other, like this:

Each student gets a problem. Here's where you can differentiate, by giving quick workers more difficult problems. You can mark the problems "easy" "medium" "hard" and let the kids pick their challenge level - it works surprisingly well.

They have several minutes to solve and become the expert on that problem for the day. After a few minutes have passed, tell them the answer is on the back so they can check if they did it right.

When ready, the students trade problems with the person across from them and work it. If they have a question, they are

*looking at the expert*on that problem. If someone raises their hand to ask me a question, I first ask the expert student, "What is his question?" If she says "I don't know," I tell them I'll be back around in a few minutes.

When ready, students

**get their original problem back**(you will have to remind them to get their original problem back before shifting seats for a while, until they get used to the structure.) One row stands up and shifts in the same direction. The student on the end that gets bumped off circles around to the other end. Now everyone should have a new partner and trade problems.

Repeat until all possible partners are exhausted or you run out of time.

This was christened "Speed Dating" by my third period trig class last year. Another class I had called it "The Math Train." If we haven't done it in a while, kids start asking for it. The social component makes it fun, whatever you call it.

## Thursday, September 10, 2009

### Mo' Bandwidth Mo' Problems

I set it up so that they needed to be logged into both to complete a homework assignment the first week. I gave them super explicit written instructions for everything they were expected to do, several days to do it, and practically continuous email support.

I'm not handing out books. Not yet. After a few weeks, after we get over the resistance to something new, then if they still want a book they can have a book.

Apparently this makes me THE MEANEST TEACHER IN THE WORLD. (She won't even give us a book!)

Digital natives, my butt.

I'm embarrassed at the number of times I've had to suppress the "chuck it all and go back to business as usual" reaction. And at its severity.

This week has also been riddled with maddening technical issues with the Smartboard and the laptop it's attached to. My grand plans to use polleverywhere in class were foiled by the lack of Verizon coverage in my classroom. (Verizon is supposed to be the good one, around here!)

It's only been 3 days.

Did anyone get the number of that truck driving school? Truck Masters, I think it was called?

## Friday, August 28, 2009

### WCYDWT: Maxine

The obvious question is "How far does Maxine run?" but a more critical thinking question would be "What minimum information do you need to determine how far Maxine runs?" What do you all think?

## Tuesday, August 25, 2009

### Resolutions

I copped out in the comments with a link to a google document full of ideas I didn't want to forget when the exhilarating panic is followed by the depressing "time to make the donuts" of the first few weeks of school. But those are just ideas. I can't do them all. And they're mostly technical. They would be minor improvements to the housing, when I really want to increase the payload.

I hereby resolve to:

1. Rewrite Boring Lessons. I am good at making my lessons effective in the mode of a few well-chosen examples coupled with ample and varied opportunities for practice. This year I want to rework lessons with the priority of making them not boring. I am not good at this. Yet. I'm afraid they will necessarily have to be less absolutely clear and thorough. Trying not to freak out. Deep breaths.

2. Stay Positive. I won't complain about a student to my colleagues or family. It makes me feel better for half a second, and then makes me feel like crap, and is otherwise totally unproductive. In the past four years I have gotten a million times better at lengthening the pause between think and speak, and installed some industrial-grade filters. One more shouldn't be that hard.

## Saturday, August 22, 2009

### Is This Your Life's Work?

I can't speak for anyone else. But speaking for myself...this is not so. I decided to publish part of an email I received recently, and most of my response, to clarify my very non-super human reality.

From my inbox, excerpted:

I am curious if .... this is a common phase of the dedicated teacher's career: the year you give your life over to making a curriculum you can be proud of.

I have spent much of my summer planning out my curricula so that, at the start of the school year, I will be prepared to begin my own year of designing smart lessons. However, as September approaches, I am becoming anxious. The daily routine I've been looking forward to seems grueling. How can anyone balance making fresh lesson plans and worksheets and homework assignments and quizzes while keeping up with grading and the hundred or so other chores that become part of a teacher's job? Coming up with challenging work is challenging itself, and I have found no way to speed up the process.

How do teachers, who are just getting into the brave new world of curriculum development, manage to get it all done? Is a 5am-7pm schedule, like I'm anticipating, required? Does every teacher who makes their own materials spend at least a year in monk-like isolation working continuously?

(From Alison Blank, who it turns out is a blogging natural.)

And my response...

The year you teach a course for the first time is definitely going to take more time than one you have already taught and refined. I have found it goes like this : first year of a new course, I'm keeping my head above water and trying to stay a unit ahead of the kids. (Sometimes, I'm only a day ahead of the kids.) In the second year, I am heavily revising my materials from last year. And in the third year, I'm polishing it.So I hope that's helpful to someone up late, panicking. Getting good at this will take time. No one who has spent more than ten minutes in front of a classroom expects you to be an expert teacher tomorrow. Maybe you plan to teach for a while and then move on to something else. In which case, it might make sense to light the other end of the candle. But are you in this for a couple years? Or is teaching school your life's work?

I plan a unit at a time. I like to get the unit calendar, activities, smartboard files, assessments, and keys all done during unit planning, so that when I'm delivering it I can focus on the kids. Unit planning has to be a big chunk of uninterrupted time, like 3PM-7 or 8PM or come in on a Saturday. But that's not every day. That's maybe once every one to two weeks for each course.

I don't make all my own stuff. I have colleagues who are very generous about sharing their materials on a share drive on our school network. I rarely make anything from scratch. I usually start with someone else's unit and revise it to my liking. It's still a significant amount of work, but it's way easier than starting from scratch.

When I am planning something from scratch, I don't try to do it all. I use some selected materials (assignments, activities, etc) provided by the textbook. Yes much of the textbook materials are crap, but if you look through everything they provide for a particular chapter, you can usually find two or three things you can use, maybe with modifications. And I find stuff online to use.

Or, I will decide to use something from the book or that I find online that is not perfect, but it's good enough for now. (My goodness, I might be kicked out of the blogging community for that.) Then when I refine the course next year, I will focus on designing something better for that particular lesson. You don't have to achieve perfection in one year. Shoot for perfection in three years.

I guess what I'm saying is, for me, trying to invent everything out of whole cloth would be unrealistic. I can't work twelve hours a day every day. I wouldn't be a very good teacher if I tried to work that way. I need a few hours in the evening to go to yoga, go for a walk, cook a healthy dinner, watch tv, or honestly sometimes I just sit on my couch and stare off into space.

## Friday, August 14, 2009

### Dr. Strangeblog, or How I Learned to Stop Worrying and Love Scott McLeod

For

I'm not alone - just look at the comments on that post. My impulse is to write all kinds of snotty retorts.

"This

is the time of calculators ; any educator who spends public money on paper and pencil should be fired!" "Of course! Clearly the best way to teach a child about a sphere is with a

flat computer monitor! I have been such an idiot!" "Take out your laptops, kids! Oh wait, you're not allowed to bring your laptop to school! Now what?! I guess I should quit, and do a different job until conditions are perfect for 21st century learning!"

But while chatting with this guy (and, props, some of what follows are his words, not mine), I experienced a moment of clarity about Scott McLeod. All along, I have been reading him through the wrong lens. I wouldn't call him Dangerously Irrelevant, I'd call him Necessarily Irrelevant. (Although he claims the title refers to institutions' response to new technology. I think he is just playing with us.)

Because the thing is...his writing is a fantasy. Like

But! High fashion

So that's how I choose to read Scott Mcleod. He sees the world as it could be, disconnected

## Monday, August 10, 2009

### To the New to Blogging

Here's a list of howto and general blogs.

Here's a list of subject-specific blogs.

## Sunday, August 9, 2009

### Cool Things I Want to Do with Technology in Need of Prioritization

2. Class blogs that students contribute to and comment on.

3. Send assignments and reminders as text messages.

4. Send assignments and reminders on Facebook.

5. Use polleverywhere in class more than one time as a novelty.

6. Survey students at points throughout the year to prompt them to reflect on how they learn.

Obviously I can't do it all because, my goodness, those poor children. I feel a little like I am holding a bunch of hammers, in search of a nail. Are you using any of these successfully? If so please tell me about it.

## Thursday, August 6, 2009

### Trig Reference Angle Cheat Hand

Flip down the finger that corresponds to the angle whose sine and cosine you need.

The number of fingers to the left gives you the sine, and the number of fingers to the right gives you the cosine.

So if you flip down your index finger which corresponds to 30 degrees...

$sin{(30)}=\frac{\sqrt{1}}{2}$

and there are three fingers to the right.

$cos{(30)}=\frac{\sqrt{3}}{2}$

Try it for the fingers that correspond to the other reference angles. For example, if you flip down your pinky, there are four fingers to the left $sin{(90)} = \frac{\sqrt{4}}{2} = 1$ and zero fingers to the right $cos{(90)} = \frac{\sqrt{0}}{2} = 0$. It works!

It's just another way of organizing the cofunction behavior of sine and cosine to remember the values of five reference angles, but adults and kids both flip out when I show them. Kids especially feel that they "don't have to memorize" if they know this method.

## Tuesday, August 4, 2009

### Dispatch from Wells College

I can't believe it took me until Tuesday night to post!

I delivered my workshop on Monday morning about blogging for PD. All details here. I was surprised and charmed that the people who decided to attend, with a few exceptions, had very little experience with even reading blogs! We spent the majority of the workshop allowing them to explore what's out there, and setting up Google Reader. I was also surprised that the highest barrier to participating in professional blogs is fear of privacy violation. A fear I totally respect, and want to spend more time discussing at the next session tomorrow morning.

I got to attend a number of fantastic sessions - I got some new ideas for math games (for little kids, but good ones can be extended to older kids), learned about some exciting resources to bring astronomy into a math classroom, and I might have picked up both a knitting and a quilting habit.

Today I took a stimulating trip to Watkins Glen for a leisurely walk down the Gorge Trail (it was gorge-ous), an unexpected dinner at Moosewood, and a failed attempt to hit the Cornell Dairy Barn. As a bonus, we only *almost* ran out of gas on the way home. And there was excellent ice cream when we got back anyway.

Phew! I'm tired.

## Thursday, July 30, 2009

### How Making Mistakes and Blabbing it to the Universe Improves My Teaching

These are my notes for a workshop next week. Comments/additions/criticisms? Please help, oh PLN of mine.

**Why Bother?**

- Communication tools make widespread collaboration possible where it hasn't been before.

- If you aren't thrilled by the professional development or lesson improvement opportunities offered by your school, you have other options.

- Speaking for myself, being a learner makes me a better teacher. Continue growing, learning, and improving your craft.

- Network with other like-minded professionals.

- The ability to get ideas, and to get feedback about your work, no longer has to be limited to a planning period, an annual conference, or only teachers in your school and district. As a result, your teaching will improve faster.

**Why Start a Blog?**

I always had a hard time with reflective practice as a private exercise. I was told it was important, but it wasn't rewarding in a way that led me to pursue it regularly. All my writing landed with a dull thud. Publishing those reflections, successes, failures for an audience means my work is broadcast in a community of intelligent professionals who read, use, and comment on my work. The feedback from them is the reward that keeps me doing it. (Update: Read MizT's elaboration on this point.)

**You Don't Have to Start Your Own Blog**

There are plenty of ways to develop, benefit from, and contribute to a learning network without starting a blog of your own.

*- Through the habit of reading other teacher's blogs, you will get ideas and insights in an easily accessible format, delivered to your computer. You can go to individual websites, but using an aggregator like Bloglines or Google Reader is much more convenient. Instead of bookmarking and visiting individual sites, an aggregator collects them all in one place for you. New posts in your subscribed blogs are automatically sent to the aggregator through the magic of RSS (Real Simple Syndication (you don't need to remember that.)) You only have to check the one site, the aggregator, to see if any of the blogs you read have been updated lately. For Google Reader you will need a Google account, then you can click Add a Subscription and type in an address.*

**Read Blogs**If this is all brand new to you, here are a few to get started with (I chose these to represent a spectrum of math ed bloggers out there... through linking from posts that appeal to you, you will find more and more): 360, colleenk, Dan, Elissa, Jackie, Jason, JD, John, Sarah, Sam, Sue, and me.

*- Don't be afraid to add to the conversation on a blog by commenting. Include your first and last name, unless you are cultivating a pseudonym (see below). If you start a blog later, you will have built a reputation you can use to draw people to read your posts. When it's available, I always check the box for "email me follow-up comments," so I don't have to go back to the post later to read the latest comments.*

**Comment on Blogs***- It's more than celebrities and talking cats! Many educators are coming on board, and I'm finding more and more useful links and discussions on there. It can be a good way to get started. Set up an account and describe yourself in a few well-chosen words. If you are looking for ideas about who to follow, start by checking out who I and other math teachers are following. Once you follow someone, they will often follow you back, if they see that you are a teacher. Congrats, you have readers! (Of your tweets. Limited to 140 characters.)***Twitter**

*Delicious***- Save bookmarks that are available anywhere there's Internet. I use this as the place to save all those links that I want to remember later. Add tags so you can search them easily. It also has a social component, whereby you can follow people and see what they are saving.**

**, Considerations, and Potential Pitfalls**

How to Start a Blog

How to Start a Blog

There are gazillions of articles about the technical aspects. Spend some time googling and reading articles. I use Blogger and like it fine. Lots of people like Wordpress.

Decide why you want to start a blog. To share your great lessons? To comment on education policy? To relate funny stories about your day? To post pictures of your lunch? It's good for a blog to have a focus. If you don't want to write original content and just want to share links or pictures, consider another solution like Twitter, Tumblr, Flickr, or Delicious.

If you and some friends all want to start, consider a group-authored blog (example). You will immediately have each other as readers and commenters.

Decide whether you want to post with your real name or anonymously/pseudonymously. The decision to use my real name was a personal one and works best for me. Plenty of awesome blogs are written by teachers who would rather not identify themselves. If privacy concerns are keeping you from starting a blog, consider a nom de plume. Either way, I advise against publishing anything you wouldn't want a student, parent, or administrator to read.

Have several ideas for posts before you start. When you begin, you will want to update at least once a week. Otherwise people will lose interest and you won't generate a following.

When you write a new post, don't publish it right away. Several commenters made this suggestion, and it's a good one. Instead of "Publish", the blogging software should let you "Save as Draft." No matter how much you like your post, you will think of a crucial edit 5 minutes later. You will realize that the way you stated something could be taken the wrong way. You will think of more elegant phrasing. Of course, you wouldn't want blogging to put you in a bad light, or give you a reputation for being unprofessional. Write the first draft and let it marinate for 24 hours before publishing.

Stick with it. It will take time to generate a following. You can get readers faster by leaving comments on other blogs (and including the URL of yours,) linking to your new posts on Twitter, or submitting a post to a blog carnival.

It's generally considered bad form, not to mention lawsuit-inducing, to post identifiable pictures or videos of students' faces. If you want to do this, you should get guardians' consent in writing.

**Further Reading/Watching**

This post wants to convince you of the value of educational blogging.

In this TED talk, Seth Godin says we all have the potential to "change everything" by leading people just waiting to be led. Not specifically about education, but relevant.

This episode of Ze Frank talks about the meaning of creativity, and what it takes to go from 0 to 1.

This post at the Fischbowl describes the value of a Personal Learning Network for both teachers and students.

Darren has his students blogging to expand the walls of his classroom.

Sitmo is a relatively easy way to include mathematical equations in blog posts.

### Be Brave

Math Be Brave

## Thursday, July 23, 2009

### A Lesson Plan Using a Virtual Manipulative

Pretty cool, right? A way to play with volume that avoids water fights. Love it. I used this in a remedial geometry class several years ago. It was fun for the kids for about five minutes. We merely used it to poke at the edges of our intuition. I didn't really know how to exploit it.

It raises a compelling question for teachers: there are some really good digital resources out there, but how do you best use them in a classroom to enhance learning? I'd like to use it this year when I teach Geometry, but I need to write an effective lesson around it.

The barest outline of a plan:

1. Playtime. Let kids slide the height thing and push the buttons, or be teacherbot and do what they instruct me to do. Solicit guesses for heights. Have kids verbalize why they think their guess is correct. Test to see how close they are.

2. Start talking about how you would calculate the new height. Go back to universal problem solving techniques that you should be hitting over and over again. What is the given information? What do you want to find? What stays the same? Encourage/coach them to do this with the rectangular prisms. They should be able to find numerical solutions easily. Develop and write on the board an equation involving equal volumes with an unknown height and solve it. Test to see if it works in the virtual manipulative. Have them calculate a few more.

3. Go through the same procedure with cylinders, then cones. It's going to look different depending on if they already know formulas, what age the kids are, what level, etc.

It wants for structure. I could develop a worksheet and break out the laptops. I'm not a huge fan of many worksheets, because I think they shift the focus from the problem-at-hand to "guess what to write in the blank." (I'm also not a fan of the laptops.) I could try to keep it as a large group discussion, but that could easily turn into me talking to 2-3 kids while everyone else zones out.

What would you do with this?

## Wednesday, July 15, 2009

### Math around the House

Ideally I think parents should make understanding organic to daily occurrences. If they treat math like spinach (we have to get through your multiplication flash cards before you can have fun, because it's good for you) their child will grow to resent it. Gaining real understanding of logic, structure, measurement, relationships, and numbers, done well, should be a creative, joyful process.

These are some things I've noticed while puttering around the house. I hope they are helpful. For all of these ideas, I would:

- Know at least a little about the concept/solution before posing it (I put very broad categories after each one to possibly make research a little easier.)
- Respond to the child's curiosity - don't force her to do anything.
- Don't try to turn everything into a textbook math problem. It's ok to start with undefined terms and not enough information. Learning how to ask clarifying questions is part of learning how to solve problems.
- It's ok to start with too-complicated a problem. Learning how to pose a simpler problem is a part of learning how to solve problems.
- Don't be afraid to leave questions unanswered for another day - give the ideas time to marinate. Cultivate patience with irresolution.

Notice two cylinders side by side and wonder aloud which uses more material, and which holds more. (Canisters for oatmeal, tennis balls, and Pringles come to mind.) Follow up: roll up a 8.5 x 11 piece of copier paper both the tall way and the short way. Wonder which way holds more. (Volume/surface area)

Do you frequent a particular ice cream place? I bet they offer well-defined options for ice cream flavors and toppings. Wonder how many possible different sundaes you could order. (Or the related questions, such as how long would it take you to eat every possible sundae, if you had one a week?) If the initial problem is too complicated, solve a simpler one. (Combinatorics)

Does your family ever clink glasses at the dinner table? Next time you do that, wonder how many clinks. (They don't have to be clinks. Do they smooch all their relatives at family gatherings? High five them?) If you only have 4 people, this is easy and boring, but if you can get them to wondering at the pattern for 5 people, 6 people... (Combinatorics)

When you are cutting a cake, or pancakes, wonder how you can get the most pieces out of the fewest cuts. (Quadratic relationships)

While traveling, set them to figuring out "How much longer?", using the data from highway mileage signs and the speedometer. (Distance/Rate/Time)

If you are playing a game with two dice, wonder what outcome is most likely. What implications does that have in the game? (Probability/sample space)

"I need 1 and 3/4 cups of butter. Can you get them for me?" When she does this successfully, talk about how she divided by a fraction to arrive at 3 and 1/2 sticks. Draw a picture, write it out with symbols. Notice how weird it is that a division can yield a result larger than the thing you were dividing. Be on the lookout, together, of other examples around the house where one might divide by a fraction. (Division/fractions) (Thanks for this one, Jackie.)

Find number-y fun in your child's favorite sport. These examples are from baseball because I know more than nothing about it: How many outs in a perfect game? Minimum/maximum at-bats in a scoreless game? (Multiplication/division)

Take advantage of anything a child asks about. Sometimes they will hear an older friend or sibling refer to something exotic: prime numbers, or square roots, and wonder about it. I have seen parents give a definition and not exploit the opportunity! Don't stop at "A prime number is only divisible by 1 and itself." Say something tantalizing like, "I wonder if we could find all the prime numbers between 1 and 100? That seems impossible!"

And finally, I can't recommend highly enough a collaborative math circle (as opposed to the type geared toward training for competitions.) If I had kids, I'd be googling "my city math circle" to see what's out there, and starting one up if it didn't already exist.

I'd love to hear about other things you and your kids have come up with! Happy math-ing!

## Tuesday, July 14, 2009

### Nine

## Thursday, June 25, 2009

### How Math Must Assess: a Post-Mortem

For the uninitiated, read this first. Dan lays down his system.

**What I Kept**

Each skill considered, numbered, recorded as a separate grade, and tested twice. Unlimited opportunities for remediation, but outside of class. I would typically re-do the test items with the student and troubleshoot her misunderstandings or more trivial errors. Then if she could prove that she knew what she was doing on a different problem without assistance, I changed her grade to 10/10. I still gave assessments at times most teachers would be giving a quiz or a test: typically the middle of a unit and the end of a unit.

**What I Changed**

I made each question a maximum of 5 points. Sometimes I made half-point deductions, like for rounding errors. After the first assessment of a skill, it was recorded as a grade out of 5 points in my gradebook. After the second assessment, I changed the entry to be out of 10 points. If the student scored a 5 the second time, I made her grade 10/10, no matter what she scored the first time. If not, I recorded the sum of the scores from the two attempts.

I didn't use a stamp. I put a small sticker on each question earning a 5 on the second attempt, and it was up to the student to transfer the sticker to his checklist. Major instructional time saver.

**What Worked**

I tried this with both Algebra 1 (9th grade) and Algebra 2/Trig (10th and 11th grade). It worked really well for the older students. Every time I was available after school, I could count on anywhere from 2 to 15 kids wanting to remediate something. I have no doubt that spending more time on difficult topics benefitted these kids' enduring retention; the results of their state regents exam (administered almost 2 weeks after classes ended) bear it out. I received an email this afternoon which pretty much made it all worth it. This is the complete text:

I PASSED???????????????????!????!????? with a seventy five?? not just a

64.5?????? i see the numbers on my gradebook but seeing is not always believing.

That girl worked her tail off all year because she had a tangible incentive. If she did well the first time, she wouldn't have to stay after school. And if she didn't, she could change her grade.

**What Didn't Work**

Dan promises students excited about tests... I didn't see that. They were happy that they weren't stuck with bad grades and felt empowered to do something about it, but I wasn't feeling a big "Yippee!" on test days.

Since we are required to publish our grades on mygradebook.com, which students and parents have access to, the concept checklists were mostly a flop. A handful of hyper-conscientious-types made good use of them, but students were more likely to go print out their scores before coming to remediation. I may do away with the checklists altogether next year. Not sure. I am not good at compelling students to keep a good notebook in general. I need to work on that.

Sometimes my grades felt a little inflated. I'm ok with that.

I think it worked better for older students because they have the maturity to take ownership of their grades and learning, they are invested in their performance because they are starting to worry about college applications, and they follow through by monitoring mygradebook.com and coming to remediation. In practice, for me, the system relies at least partially on student initiative and how much he is motivated by grades. I did not have as much success with the freshmen, as much as I tried to sell what a good deal this was. They are much less likely to stay after school in general, and as far as grades go, they just want to do well enough to keep their parents off their backs. I'm not sure what changes I could make to do better there.

**Conclusion**

It's not perfect yet, but I'm sold on the idea. Much gratitude to Dan for coming up with and publicizing a better way. I thank you and my kids do too.

**Documents**

Algebra 2/Trig Concept List 08-09

(I can add to this, just ask.)

## Thursday, June 18, 2009

### Evolving Rational Expressions and Equations Unit

To recap from comments in previous post, I have this for day 1, reviewing operations on polynomials and monomials.

For Day 1 homework, they will research a way to explain why dividing a number by zero is undefined and be prepared to explain it in class the next day.

For Day 2, we will start by explaining to partners our reasoning for a/0 is undefined. Then a few students will explain it to the class.

Then we get into this... I'm looking for feedback to make it better if you have any. Thanks Dan Greene for the idea and not minding me ripping you off.

Classwork 8 2