f(t)

This Logic Game Needs a Name

2012-01-25T13:18:00.000-05:00

This is a game to give Geometry students practice evaluating the truth value of conjunction, disjunction and conditional statements.

This is what one game set looks like, for use by two students:

Each set has: 36 statement cards, 18 T/F cards, a cube with logic operations on it, and 4 negation chips. (All the cards are one-sided.)

The sides of the cube look like this:

I bought unfinished wooden cubes at a craft store and wrote on them with a Sharpie (hence the bleeding.) I'm sure someone clever will comment with a better way to make these. You could also just use regular 6-sided dice and provide a decoder (rolling a 1 or 2 means "or", etc), but I was in overachiever mode yesterday.
The kids are very much beginners in the Logic unit, so we gradually dialed up on the cognitive load by playing two easier warm-up games before the real game. I also had them set their notes from yesterday out on their desk for reference.

Warmup Game 1: Easy Mode

1) Use only the True/False cards and the cube.

2) Distribute half the cards to each player.

3) The person whose birthday is next wins when the outcome is True. The other person wins when the outcome is False.

4) Game play is like “War.” On each turn, each player flips over one card, and the cube is rolled. The players work together to determine whether the resulting compound statement is True or False. The winner keeps both cards.

5) The game is over when time is up or one person gets all the cards.

So if this happened, "True" would win the round:

But if this happened, "False" would win the round:

We only played Game 1 for a couple minutes, because it's pretty lame. Not very challenging, no strategy, also the "True" player has an advantage, because more of the possible statements come out True. On to...

Warmup Game 2: Like Game 1 but Harder

1) Use only the Statement cards and the cube.

2) Shuffle and randomly distribute 10 cards to each player. I gave them a few minutes to look through them to familiarize themselves with what the statements looked like, and think about whether they were true or false.

3) The person whose birthday is next wins when the outcome is True. The other person wins when the outcome is False.

4) Game play is like “War.” On each turn, each player flips over one card, and the cube is rolled. The players work together to determine whether the resulting compound statement is True or False. The winner keeps both cards.

5) The game is over when time is up, or when one person gets all the cards.

6) VARIATION: Each player also gets two negation chips. A negation chip can be played at ANY TIME, but can only be used once and must be discarded after use.

So if this happened, False would win the round:

But if this happened, True would win the round:

But if the False player still has a negation chip, she could opt to throw it down, and take the round:

So that was all a warm up to familiarize ourselves with the materials, and remember the stuff we learned about yesterday. Still kind of lame because there's not really any strategy. Finally, we get to play the very fun...

The Real Game (which still needs a good name)

1) Each player gets: 10 Statement Cards, 5 True/False Cards and, 2 Negation Chips. You will be choosing cards to play on each turn, so it’s ok to look at all the cards in your hand. (You could deal out more or less cards if you want the game to take more or less time. This number was manageable for about a ten-minute game. Most groups were able to play two games.)
2) The goal is to get rid of all your cards by making statements that work.

3) Game play is turn-based. On your turn, you select three cards and place them in the field of play: two statement cards and a True/False card.

4) Then, roll the cube.

5) Both players should agree on whether the resulting compound statement works or not. If the statement works, you discard the three cards used in the turn, and go again. If the statement doesn’t work, you keep the cards in your hand, and lose your turn.

6) A negation chip can be played at any time. Even after the cube is rolled. However, once a negation chip is used, it must be discarded, whether the resulting statement worked or not.

7) The player that gets rid of all her cards first, wins.

So if a player selected these three cards, and rolled OR, the statement works:

They discard those three cards from their hand and take another turn.

But if a player selected these three cards, and rolled IF THEN, the statement does not work:

And they return the cards to their hand, and lose their turn.

HOWEVER, if they still have negation chips, they could play one now:

and now the statement works, so they can discard these cards and go again.

All the kids were engaged in playing for the whole period. Some of them asked "Can we play this again?" which blew my mind. I intended to do an exit assessment but didn't, so I'll give it to them at the beginning of class tomorrow and see what they retained. If I made new games, I would make the Statement Cards a different color from the T/F cards for easier sorting. It also needs an awesome, catchy name! But I haven't thought of anything worthy yet.

The final version owes a big debt to my colleague Dina Kushnir who talked through the game play with me, and came up with some of the basic mechanics. I'd also like to thank Maria Andersen for all her writing and insights about what makes a good math game - I don't think this would exist without her.

Here are some resources so you can make your own games. Enjoy!

Good News Postcard

2012-01-20T09:16:00.002-05:00

Here is a picture of the postcards we use that I talked about here. Unfortunately I don't have a digital file used to produce them, but at least you can see what they look like. I find the hand-holding-pen clipart very helpful, don't you? (Oh! That's what I'm supposed to do. Write on it. Thank you, clip art.)

Project Spiky Door

2012-01-19T13:06:00.000-05:00

Ever since Riley and I had this exchange, I've been dissatisfied with my classroom's plain door that is not covered in spikes. Well, like, lo:

This took three days of class time (129 minutes) for a non-honors Geometry class of almost all tenth graders. Many students also needed some additional time outside of class. The goal was for students to get hands-on grokking for the meanings of height, slant height, surface area, and volume. They designed the net of a pyramid and labeled dimensions, calculated the total surface area and volume of their design, and constructed and decorated the model.

I offered a bonus for a base that was not a square. I also suggested making a cone instead, but none of them chose to make a cone, so next time I should make the cone worth a bonus as well. I specified a target range for the area of the base, and a target range for the volume, mostly so that the door would look cool.

Here is the project description and scoring sheet.

In order to make grading less ultimately-annoying, since they were given lots of flexibility in choosing measurements, I programmed my TI to do calculations for me (link goes to TI-Nspire .tns file.) The vast majority made a square pyramid, and just about every student first decided on the length of a base edge and the slant height, and went from there, so the program takes these measurements as inputs, and returns all the other values they were supposed to calculate. This way, if their numbers match my numbers, I can just move on, and only have to spend more time inspecting their work if there is a discrepancy.

Here is an example of what a student turned in. Since they had to organize their work, I had to say things like "Label EVERYTHING! Make it REALLY EASY for me to grade! I am VERY LAZY and I DON'T CARE if you get a good grade!" That sounds awful but it's funny only because they know neither of those things are true. It would be easier to grade these if you gave them like a template where they had to fill in stuff, but I think there's a ton of value in them deciding how to present their work sometimes. This is hard for me to explain, but when they are in filling-in-blanks mode it's different from figuring-stuff-out mode.

This was pretty fun and a nice break from the usual. I believe the goal of reliably distinguishing both slant height from height, and volume from surface area was achieved. And my door is looking pretty badass. I'd do this again.

Shoulders this project stands on:
Mimi
Riley

Good News from FM High School

2012-01-11T20:27:00.000-05:00

Realizing that "make positive phone calls home" and "reinforce positive behavior" are on every teacher's to-do list, and that some of you regularly do these things, and that I'm not patting myself on the back too much because many of you are more virtuous than I...

But also recognizing that these behaviors being on my good intentions slash nice if I have time list and being on my checked-off list are two different things...

And also not judging myself too harshly for six years of failures in virtuosity...

I'm happy to report that the school providing me with a fresh, stiff pack of yellow postcards that have address blocks on one side and "Good News from FM High School..." on the other side has prompted me to send at least three of these every Friday afternoon to children's homes for...

acknowledging determination to not drop out of a class that is a challenge for them
or he is a cool kid and he volunteered to partner with the kid everyone thinks is a little weird
or because she had a rough start this year but she is killing it this quarter
or something I halfway made up but I know they have been having conflict with their parents and could use a boost

has been, I wouldn't call it transformative, and I may be imagining it, but I have noticed an additional warm fuzziness, a we're all in this together, a patience in my room with each other and with me...

and I'm going to keep doing the postcards.

A More Auspicious Beginning to Surface Area and Volume

2012-01-03T12:41:00.000-05:00

For a few years, by the time December rolls around, I am so beat up by daily grading, making copies, paperwork, being a good example for the youth of America, etc etc, that my Surface Area/Volume unit has always been just fantastically awful. Although it's been a long December, I was determined to make this at least a little better.

I start with surface areas, because we just spent tons of time working with areas of composite figures and shaded regions, so it makes sense to extend that.

So to start out, instead of this (try not to barf:)

I sneakily ask them to calculate some composite areas.

I am not super-thrilled with these. In the future I would make them bigger and use more integral values, and not use such a tall, skinny cone.

As they were working, I asked them to write how they calculated the area in the middle of the shape. So in that last one they might write "Area = 1 rectangle and 2 circles."

The cylinder was VERY interesting, since there's really nothing given about the height of the rectangle, but by now they tend to assume that if they need a dimension, there should be a way to figure it out. Some kids realized we were going to turn it into a cylinder and used the circumference of the circles, but some kids estimated that the height of the rectangle was about three circles, which I didn't anticipate but hey, pi is about 3, and that's kind of awesome.

Anyway, so now we are going to cut these out and hold on to them for a few days.

Hopefully avoiding the phenomenon of having no idea which dimensions we need or randomly plugging given numbers into given formulas.

I don't really know where we're going to go from here. I might freak out and go back to a traditional approach, but at least more kids might have a better idea of what those formulas are all about.

Math Lesson Formula

2011-12-29T14:38:00.001-05:00

Okay so, seven years in, and I feel I am finally cracking this nut: how do you make any math lesson work for most kids under most circumstances? Throughout the year I have been tweaking most of my lessons to follow the same basic formula. Not that we do the same boring thing every day - there are infinity variations to make it work for me or a particular group of kids. Not that I'm saying teaching doesn't require a whole mess of skills besides knowing how to set up a lesson. Anyway.

I will illustrate with the most frustrating of topics : log laws. I can't think of a topic that seems more boring and pointless to most math teachers and students. I know their virtues as well as you, but let's be honest, 99% of your kids don't really need to know them for anything they are likely to do for the rest of their lives. I posted about it last year, but there was a piece missing, and now it really sings. To believe this works, you have to believe that the one doing the work is the one learning. Nobody gets much out of Miss Nowak doing dramatic performances of math problems and proofs other than Miss Nowak learning how to do dramatic performances of math problems and proofs under the withering attention of 24 bored and irritated teenagers. I don't want to give the impression that I'm giving them a worksheet and being all like, "You're on your own, kids! Time for me to kick back and drink coffee." Because I'm running around, scanning for common questions or points of confusion or missed connections, re-capping with the whole group every five to ten minutes, encouraging and validating, etc. But if you believe "teaching" = "lecturing" then you are not going to see the validity of this approach, and I can't help you.

Phase 1: Productive Struggle
Hook the new thing to something they already know or know how to do. Then make them do it. A few times. Let them discuss and work together. No reason this has to be done in silence. Whether calculators are allowed depends on whether the calculator will let them avoid the things you want them to remember and see. (This particular lesson is no-calculators.)

Phase 2: Generalize. Make them write whatever they have been doing with letters. This is harder for most kids than you'd probably expect, especially if they've never been asked to do it before.

Phase 3: Use it. Presumably this new thing you've discovered is good for something. Even if that something is obviated by ready access to a shmancy calculator.

Phase 4: Prove it. The hardest part for kids, and the hardest part for me to figure out how to get them to be the ones doing the work. I have had some success with this approach of setting up an organizer and basically telling them what to write. But they still need lots of hand-holding. But at least they are doing more than watching/copying a dramatic performance.

Phase 5: Lots and lots of practice. I want them to understand, but I also want mental automation of relationships and procedures. Because later they are going to use this stuff to learn something new.

I would like to say Phase 6 is apply it to a novel and interesting problem, but I'll be real, I'm not there yet with log laws. Though I am there with good projects on some other topics that lend themselves to applications. Give me another seven years.

All I Really Need to Know about Teaching

2011-12-16T16:48:00.002-05:00

Dave started it.Here's mine: It's been on a bulletin board at eye level at my desk for seven years. It's a little embarrassing to share because it's more than a tad hubristic. But I think it helps me be better. I don't really need to have it hanging up because I can recite it from memory. Like a mantra. But it's comforting to have it there. Like a talisman.

“I have come to the frightening conclusion that I am the decisive element. It is my personal approach that creates the climate. It is my daily mood that makes the weather. I possess tremendous power to make life miserable or joyous. I can be a tool of torture or an instrument of inspiration, I can humiliate or humor, hurt or heal. In all situations, it is my response that decides whether a crisis is escalated or de-escalated, and a person is humanized or de-humanized. If we treat people as they are, we make them worse. If we treat people as they ought to be, we help them become what they are capable of becoming.” ― Johann Wolfgang von Goethe

(There is a teacherified version floating around by Haim Ginott. I like the original better.)

I don't take away from this "it's all about me." The take away is more like, "There is a ton that is in my control, and that makes all that happens here my responsibility." Which is maybe a little oppressive and maybe a little "duh." But I still like reading it every day.

In Which Ben Articulates My Reasoning Better Than I Could

2011-12-12T19:12:00.000-05:00

...and then some.

Read.

#takethetest

#passiton

I'll add, since I didn't provide much in the way of explanation. The NY Algebra 2/Trig test is a horror show. It tests many things. Notation. Vocabulary. Procedures. Graphing calculator button sequences. It is not a test of mathematical understanding. I am pretty sure any reasonably mathematically-literate adult would sit down to take it, and within twenty minutes be all like, "What the HELL is all this CRAP? And WHY are we inflicting it on our young people? Get me the Governor! Oh wait, I am the Governor!"

I just want the guy to know what his organization says is important for college-bound kids to know. Thats all. I'm not even totally anti-test. I'm anti horrible, very-bad, no-good test.

"Favorite No"

2011-12-11T08:30:00.001-05:00

Just a quick share - I have tried this a few times this year, because I was looking for ways to more frequently but still quickly assess a whole class. It works really nicely. I don't have anything to add - Lea covers it all. Just watch.

This Journey to Wherever...

2011-12-10T13:14:00.001-05:00

...is about to get slathered in chimichurri. If you keep up with me on Twitter this is old news, but everyone else: I accepted a position at an International School in Buenos Aires for 2012-2014. Reactions fall into two camps: 1) Awesome! 2) WHY IN THE WORLD WOULD YOU DO THAT? so, here goes. While a young pup Navy officer I spent great chunks of time overseas, predominantly Italy and Bahrain (which seems like an odd pairing only if you've never been a Med/Gulf Sailor,) and hearted it. I like being a stranger in a strange land. I like navigating mysterious cultural waters. I like spending twenty minutes of gesticulating to communicate the idea "I think this thing is awesome but I am not willing to pay 20 dinars for it." "Okay fine you can have it for 15." "10." "12." "Fine."

So when I started teaching in 2005 the idea of an International School really appealed, but at the time, the reputable schools wouldn't consider teachers with no experience. I hear that's not always the case these days, but at the time, it was off the table for a few years anyway. Cut to seven years later, and you'd be right to wonder what took so long. I wonder that, too.

It took me about the past three years to Get Serious about making this happen. There was comfort with the known and fear of the unknown. There were two boys who captured my attention for a time but things just didn't go that way. I had to improve my math teacher fu. I had to gain and lose fifty pounds. Would that it had all not been so painful but it was all probably necessary. There were excuses but once I Got Serious I realized my excuses were really no thing.

For instance, I was all concerned about What Would Happen with the Cat, but here is what my point of contact at my new school said when I asked if it would be reasonable for her to join me:

YES!!!! Bring Kitty! (what is his/her name....that is very important for me to know!) I came here many years ago with 2 young German Shepherds and an old grumpy cat! So you KNOW I understand bringing your little furry friend!

So yeah, excuses loom large in your mind but sometimes go poof when exposed to daylight. And they are just that, excuses, i.e. not the real reason you are hesitating. I think my real reasons (mostly fear) were alleviated by meeting and talking to and reading the blogs of teachers currently working at International Schools. These are real people not that different from you and this is their life. I have to especially thank Mimi who spent lots of time patiently answering my questions at PCMI and afterward, and offered lots of good advice.

So when I finally Got Serious I was looking for three things in a placement: 1. a non English speaking country so I could become fluent in another language 2. a stable, reputable school committed to supporting and developing their faculty and 3. a major urban center, for all the reasons people like living in cities. Beyond that I wasn't even particularly concerned about what continent I would end up on.

I joined Search Associates and joyjobs. Search Associates turned out to mostly be useful for demonstrating to potential employers that I was "serious" - since I interviewed and accepted an offer quickly, and never even had to go to a job fair. Joyjobs is a source of lots of good information, and frequently updated vacancy postings, and worth the small fee in my opinion.

In conclusion.... I'm ridiculously delighted with the way things turned out, and I can't wait to get there. This placement has lots to be excited about - the probability of teaching IB or AP, a block schedule, much smaller classes, no fire drills, no Regents nonsense...not to mention I won't have to scrape ice off my car any more. Considering that my new school hasn't hired a math teacher in five years, maybe all the delays make sense in the grand scheme. Now for a long six months of trying to learn Rioplatense Spanish and find new homes for all my stuff. (But not the cat. She's coming.)

f(t) is having a moment

2011-12-06T20:08:00.001-05:00

Hello, faithful readers.

I need you all to do something for me.

I never ask you for anything.

I need everyone.

EVERYONE.

Just go toot my buddy Governor Cuomo about how he should take the exam that decides whether my cherubs are college-ready. That's all.Here's a template you can follow. Heck you can just re-tweet me.

If you really, REALLY, TRULY want to have an impact? Repost this on your Facebook and your Twitter and etc. We are about to get exponential on his ass.

It'll just take a sec. Then you can go back to watching My Drunk Kitchen. Promise.

xoxoxk8

Still Relating Those Rates

2011-12-03T10:49:00.001-05:00

First I have to express an obscene amount of gratitude to Bowman Dickson for illuminating what will be challenging for students learning related rates, and sharing how he deals with it. I basically just took his post and reorganized it into a lesson that will work for me. This post will probably make more sense if you read his first.

Second, I have been thinking this morning about what this lesson has to do with the recent discussion at dy/dan. There's probably a way to turn these into a problem we could pose without words through the cunning use of video production skills I don't have. It's really fun to think about.

Here's what I'm giving the kids. Here are relevant documents: handout for the kids, smart notebook file, ggb's that I made.

Update: Mimi Yang, a.k.a. thebomb.com, revised the cone tank ggb to reflect a constantly changing volume. That file is in there too.

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SUPA Calc Lesson 4-6 : Fold this paper in half to hide the bottom half. Please don’t look at the example problem while we are doing the investigation. It will just get in the way of your learning.

1. Blow up a balloon!

2. Go here: http://bit.ly/calcballoon

3. Figure out everything you can about rates with the balloons. Record your observations below… (there is no one right way to do this. Make it make sense to you.)

4. Which one is more like inflating a real balloon and why? Write about it.

5. What is going on with the other one? Write about it.

Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?

(Here, we'll set up a solution with a diagram, givens, equation, etc, in a very structured way.)

6. Go here: http://bit.ly/calclad

7. The model depicts a 10-foot ladder leaning against a wall. If the bottom of the ladder slides along the floor at a constant rate, what happens at the top of the ladder? Why? How did you figure it out? Write about it below.

A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?

8. Go here: http://bit.ly/calccone

9. A conical tank is filling with water. Use the slider to change the height of the water in the tank. How are the height and radius related? How are the height, radius, and volume related? Write about it.

10. Imagine you are standing in a municipal pumping station, watching this tank being filled with water. What do you think is more likely: (a) the height of the water is changing at a constant rate, (b) the radius of the water is changing at a constant rate, or (c) the volume of the water is changing at a constant rate? Why?

A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep.

Curriculum Writing for the Reluctant

2011-11-30T12:11:00.001-05:00

I am really trying to beef up my Area, Surface Area and Volume unit for Geometry this year. It gets the job done regents-exam-wise, but it is so dissatisfying and I feel it could be so much better. Overall it basically boils down to plugging things into formula-sheet-provided formulas, and isolating variables in formula-sheet-provided formulas. There are some good things in there... we find composite areas and perimeters using aerial and other images, for example. Finding the areas of regular polygons is a good application of right triangle trig. There is an investigation of how areas change when dimensions change, which is serviceable but I suspect kids don't really see the big picture. We "do" volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres. My students tend to do very well on questions from this unit on the Regents exam, and I don't want to mess that up, but in this case I don't believe that the exam is valid for measuring understanding.

These are the kinds of things I want them to understand and/or be able to do:

what physical property you are actually calculating when you calculate a volume or a surface area
why the formulas are what they are
how changing a 2D or 3D figure's dimensions affects its area or volume. for example, I think they understand that if you order a pizza that has twice the diameter, you get way more than double the amount of pizza. But I don't think that intuition has any ties to math class.
isolate a variable in a formula. for example, solve S = lw + wh + lh for w.

I have a bunch of great resources and problems and tasks that I have collected in my Evernote over the past few years that could potentially work very nicely here.

1. Design a new label for a given tennis ball canister, oatmeal canister, or soda can. (a) Create a prototype label so that it covers the entire lateral surface of the canister with little to no overlapping paper. (b) Congratulations! The company chose your design and wants to produce 100,000 labels. Calculate how much material (paper, aluminum, whatever) you will need to order.

2. This game at NLVM is quite nice for challenging your intuition about how volumes are related to dimensions.

3. This video features people with charming accents complaining about how the volume of their chocolate bar decreased even though it appears that the surface area stayed the same or possibly increased. I've shown this in the past and found that students are unable to articulate what these people are upset about using the word "volume" (much less intelligently discuss surface area.) The word "volume" from math class is not connected in their brains to "how much stuff inside."

4. Starting with a piece of copier paper, roll it into a cylinder both the long way and the short way. Will it contain the same amount either way? If not, which way holds more? Mathematically justify your response.

5. Starting with a sheet of copier paper, cut four congruent squares out of the corners and fold up the sides to make a box. Who can make the box that holds the most? Kristen Fouss did something like this but in pre-calculus. Geometry probably doesn't need to get into deriving and optimizing a polynomial equation.

6. Starting with a sheet of copier paper, design, cut out, and assemble a right pyramid with a square base. First pass: any pyramid will do. Second pass: make the area of the square exactly ___. Third pass: make the overall height of the pyramid a specified length. Present your best-looking pyramid, including the area of its base, its overall height, its lateral surface area, its total surface area, and its volume.

7. Investigate what happens to area when dimensions change. What happens to volume when dimensions change. (Somehow.)

8. The car talk fuel-tank problem.

9. Some version of the PCMI volume/surface area problems. (If you know the perimeter and area of a rectangle, can you determine its dimensions? Are there any rectangles whose perimeter = area? If you know the surface area and volume of a rectangular prism, can you determine its dimensions? Are there any rectangular prisms whose volume = surface area?)

10. Derive the formula for the volume of a sphere without calculus. From Exeter Book 3. Would pose quite a challenge for my students. They would not be able to do it on their own. In fact, as it is written, it would completely mystify them.

What I am struggling with and probably will be for the next week or so is, how do I take any of these things and fit them into a logical, coherent unit of study of surface area and volume? NY/my district/my school does not provide us with a curriculum. We have : a list of standards, a collection of previous exams, a pacing calendar, and a kind-of crappy textbook, which are all useful in their own limited ways, but none of them tells you what to do in class. I have lessons already written that get it done, so there is no incentive to bother, other than it bothers me when I feel I could be doing a better job. Part of the dilemma is, I feel that any of this would have to be added to what I already do, not replace it. I still need them to be able to, for example, identify that the bases of a prism are the parallel sides, even if they are not on the top and the bottom. And I'm already about two weeks behind in this course.

How do you take a compelling resource and turn it into an effective lesson?

IVT the sensible way

2011-11-21T20:22:00.001-05:00

I heard of a lovely activity for pseudo-discovering the Intermediate Value Theorem at a recent compulsory workshop for the calculus course I teach. It has everything i like in a thing. I do not have a record of the name of the teacher who presented it (and even if I did, I don't know if he wants to be famous on the Internet) so if you are he, please email me if you want credit.

(begin basic text of student handout/activity:)

The intermediate value theorem states: If a function y = f(x) is continuous on a closed interval [a,b], then f(x) takes on every value between f(a) and f(b).

Think about what you remember of conditional statements (from your geometry course:)

1) State the hypothesis of the IVT.

2) State the conclusion of the IVT.

3) In the following, be sure to use the endpoints (a, f(a)) and (b, f(b)).

A. Sketch a diagram where both the hypothesis and the conclusion hold true.

B. Sketch a diagram where the hypothesis is false, but the conclusion is true.

C. Sketch a diagram where the hypothesis and the conclusion are false.

D. Sketch a diagram where the hypothesis is true, but the conclusion is false.

(on to the back of the page)

4) Which one is impossible to do? Explain why.

5) Compare your diagrams with a partner. How are they similar? Different? If they are different, are they both valid?

6) Is any real number exactly 1 less than its cube?

A. Create a function whose roots satisfy the equation.

B. Find f(1) and f(2). How do you know there is a point (c, 0)? What do you know about c?

When the Problem Does the Teaching

2011-11-11T12:53:00.001-05:00

This recent Geometry lesson is a good example of setting the kids in pursuit of a problem, where they have to learn the thing you want them to learn anyway in the process. (That wasn't that eloquent, sorry, I will illustrate.) On Tuesday, we developed the rule for the sum of the angles in a polygon by the chopping-into-triangles technique that many of you are probably familiar with. The next day I wanted them to be able to find the degree measure of one angle in any regular polygon, so I set them this task, which I stole from a PCMI problem set:

I did not include that first question when I did this in class, and many students stumbled over restricting their search to regular polygons. So I added it after the fact for next time I give this problem.

There are lots of these triplets to find, so all the kids met with some success pretty quickly. It is also a little like finding a pearl in an oyster, so they were rewarded and motivated to keep looking. Regular polygons are hard to draw, so with a little reminding and prodding, they started to find the degree measure of one angle in a regular pentagon, hexagon, octagon, etc (the whole, covert point of the activity, anyway! Yay!) I had them add their finds to a whiteboard everyone could see as they were discovered. They also wanted to verify by using the Smartboard to render regular polygons perfectly, and fit them together like puzzle pieces, which I was happy to allow them to do. This was actually a pretty great class - some kids conjecturing likely candidates, some kids armed with calculators cranking out angle measures, some kids organizing all their finds, some kids going up to the smartboard in groups of two or three for visual/spatial verification. And when I assessed them the next day, no one had any trouble understanding the question or coming up with correct angle measures. This problem is a keeper.

This Is a Fun Question to Ask Your Calculus Students

2011-11-07T20:24:00.001-05:00

"What is 1 Radian?"Try it. Dare ya.They'll do a little better with: "What is 1 Degree?"

Completing the Square

2011-11-02T16:33:00.000-04:00

I made some final tweaks to Completing the Square in Algebra 2, and I find it just amazing the difference between this year and previous years, in that so much more often now, I just know what to do.

It doesn't feel like I changed all that much, but the kids just get it. I don't think it's a difference in delivery or anything. Here are the important bits.

First, I took two days instead of one. Go to hell, pacing calendar. The first day is just to see the pattern and get the idea with easy easy problems. a = 1 and b is even. The second day we work with a != 1 and odd values of b (fractions. eep. but the kids are even dealing with fractions okay.)

Tee it up: why would we want to do this? It saves us time.

Look for patterns. The kids fill out this whole table all on their own. I don't say a thing. I convince them to try and focus by telling them that if they really get how this table works, their lives will be a million times easier for the next six months. It's an exaggeration but you need them to engage here.

The bottom three rows were new this year. Hardly any students needed an assist with the * rows. I was surprised. The important part - the mathematics - was the ** row. Again I was surprised that they mostly worked this out on their own. There were some kids, I had to point at numbers, and say "Look at the 10, the 25, and the 5. Look at the 14, the 49, and the 7. How are those related? How can you write that relationship but use b?"

Once we're all on board with the table, we put the pattern together with "the genius method" from before to solve a simple quadratic in standard form:

And that is basically that. We practice a bunch of easy ones. The next day, we come back and practice a bunch of really hard ones.
Here are the smartboard files: Day 1, Day 2.

I just find it stunning that you can plan out a lesson 95% correctly and it will miss most of your kids. And you can change one little thing - add three rows to a table - and now all the kids basically get completing the square, think it's easy, prefer it to other methods of solving quadratics, and tell you why they don't get why this is such a big deal. I feel a little like I have super powers.

Triangle Centers

2011-10-23T19:25:00.002-04:00

I'm attempting to incorporate triangle centers with constructions. locus, and parts of coordinate geometry, because they all go together anyway. This is one of the things I love about Geometry - there are many topics, but they all relate to each other.

Some of my questions are exact replicas of those found in Exeter Math Book 2. I'll attach all the documents I used, and I know they are imperfect. But this is the basic progression:

Incenter
Where is a point that is equidistant from all sides? Conjecture that there is such a point on Nspire.

Figure out how to construct it by pouring salt on triangles.

Circumcenter
After doing some problems to review that there is such a thing as the Pythagorean Theorem...Where is a point that is equidistant from all vertices? Use coordinate geometry to see that there is such a locus of points on the coordinate grid.

Centroid
After a detour into deriving the midpoint formula... Where is a point that is the center of gravity in a triangle? Use area and coordinate geometry to dissect a triangle into two equal areas.

Demonstrate center of gravity by demonstrating that the triangle will balance on that point.

Where is the intersection of the altitudes? This point is not interesting, so get through it as quickly as possible. But take the opportunity to teach what an altitude is.

Then there's an Nspire document with a summative review sheet so that they can keep straight the four different, confusing kinds of triangle centers.

Next we are going to do the famous locus scavenger hunt, after a day of basic locus notes, which will allow me to basically skip the whole pointless locus unit altogether.

And then I think I am going to have the children make their own locus-based scavenger hunt for the towns of Fayetteville and Manlius, for the fun, and for the backwards learning. Though I don't happen to have those documents prepared yet.

Icebreakers

2011-10-20T20:08:00.002-04:00

It is a little-known fact that I hate icebreakers. "Little-known" in the sense that I don't mention it every five minutes, but if you know me, you could have probably guessed.

Anyway. In previous years I have been frustrated that students in my classes, by the end of the school year, might not even know everyone's name in their class. So this year, I resolved to compel them into an icebreaker every time I changed the seating arrangement.

Well, first of all, what a coup if you have a lesson that depends on discussion and conversation and talking. Because, there is no teeth-pulling necessary to get people to talk about themselves for a couple minutes. And once they have broken the ice, so to speak, you don't have to worry about that component.

And second, how entertaining, and what a fun way to learn about your kids. Yesterday I changed all the seats and I asked the Geometry students (the youngest and traditionally most taciturn of my lot): if you were a Geometry vocabulary word, which one would you be and why? (with the caveat that in 5 minutes I was going to randomly select students to introduce their partner to the group.)

"A line. Because he just keeps going and going."

"A polygon. Because he has many sides."

"A circle. Because she is well-rounded."

It just went on and on. It was adorable.

You Don't Even Know What Math Is Yet

2011-10-15T23:19:00.001-04:00

Alex McFerron:

I tutor mathematics to kids and I can't tell you how many times I hear the words "I'm not good at math". This is from very intelligent kids who aren't out of high school. Honestly, I want to say, you don't even know what math is yet. You don't know the first thing about it or your ability to do it or not do it. I want to tell them that no one is good at math who doesn't work at it.

I think that what separates math people from non-math people in our culture is that math people continue doing math and don't spend anytime thinking they aren't math people. They just keep going on the journey. The older I get, the more it is obvious that a lot of really capable people quit the journey. I admit, this journey isn't for everyone. Its hard work. It takes focus. You have to want to do it. The financial rewards aren't really there in proportion to the work. But what really hurts me is that there are people who want to do it and quit. They change majors, go home, seek other work all because they have it in their heads that they lack talent and aren't naturals. The profession loses when this happens.

Alex has a unique perspective on Mathematics and I always learn from her insights. Professionally, she's a software engineer, but she is also a math enthusiast and evidently tutors kids as well. I am left wondering as a result of this post, what is our role in promoting the "not a math person" label? And what can we do to entice kids to not quit the journey? I have to think that choosing puzzling but accessible tasks is paramount, but hey I do that, and many of them still check out anyway. What else?

The Nspire Has a Complex Mode...

2011-10-13T19:31:00.000-04:00

...not just on the CAS version, but on the numerical version too.

...it's not disabled in test mode.

...it works well. (The TI-84 has an i button, but in my experience it's unreliable.)

...oh *&^%.

First, I thought, "the children must never know about this!"

Yeah. Right.

If there is a button I will find the button.

There it is.

Here are some things it can do:

So after I threw out every assessment I used to use for this unit, I settled into the place of "What the *&^% do I do now?"

But it's good. Good! Good, I say.

It made me give some super-serious thought to what the complex number system is good for.

It's good for solving equations that don't have real solutions. Hello there, quadratics! We meet again. A little earlier this year.

It stops the Fundamental Theorem of Algebra from breaking.

It's full of numbers that represent two dimensions.

I can work with that.

This Batch of Children Will Do Just Fine

2011-10-12T20:44:00.000-04:00

They didn't seem overly impressed with the origami time-lapse, but I found three (THREE!) secret-admirer origami presents on my desk throughout the day. A water bomb, a box, and a more-different box that looked kind of tulip-y. (When did they find time to make them in the midst of my dazzling instruction? It's a mystery.) (They were impressed by the ~500 views. "WHO ARE ALL THOSE VIEWS?!" "They are mostly my Mom." "Naw Miss Nowak has like 1000 Twitter followers." "WHAT?!" "Yeah she's apparently kind of a big deal." "Why?" "I have no idea.")

A child was making Vi Hart sketches of a triangle with an inscribed circle and infinity-more inscribed circles while waiting for his classmates to finish simplifying $\frac{4+\sqrt{-48}}{2}$. Me: "YOU WATCH VI HART?!?!!" Him: "Yes, I love her." Me: "ME TOO! YOU ARE MY NEW SECRET FAVORITE!"

One child keeps DOING SOMETHING to my rubik's cube in like four moves that I can't undo anywhere near that quickly and it's driving me literally insane. (Just changing the middle square color on all six faces? Wtf. I should be able to undo that THE RUBIKS CUBE HAS INVERSE FUNCTIONS FOR CRYING OUT LOUD.)

Calculus is still a very timid group, extra afraid of being wrong slash anyone knowing they are wrong, but had lots of fun looking into the "there are always two antipodal points on a great circle at the same temperature" thing today. (Which true confession I'm not totally sure is a good use of Intermediate Value Theorem day but I kind of suspect is it, so I went with it.) There was a distinct, entrenched camp arguing "no," and a diverse, poorly-organized camp valiantly arguing "yes," and the "no's" were very gracious when they realized their position was untenable.

They will do just fine indeed thank you.

This all makes up for how I got observed today during quite possibly the worst-ever Geometry class of the year. (Observation the first day back from a four-day weekend? Who does that? We spent 45 minutes recalling the mathematical differences between their assholes and a hole in the ground. Not really but it took ten minutes for someone to summon a vague recollection of having heard of the pythagorean theorem before. I wish I was kidding. Needless to say we did not even get to the dazzling lesson that was planned. Why didn't she come last Thursday? We were pouring salt on triangles. It was epic.)

2011-10-09T21:31:00.000-04:00

I purposely didn't bring any school work home after I spent all day Saturday in school, planning. (We can't get into school on Sundays. Well, we can, but you have to disarm the alarm system, which I am too afraid to try.) I've had trouble this year Turning It Off, and I was trying to force myself into some down time.

Here's what my down time looks like: (or will, until it gets pulled down for copyright violations)

Also, if you're an Nspire person, here's a file with the area proof of the pythagorean theroem.

Days off are exhausting.

Destroy My Problem Set, Please.

2011-09-24T10:09:00.003-04:00

Students should be able to complete this in groups without too much assistance from me. We already had a lesson on what the cube root means and simplifying cube roots to simplest form, which was also a refresher on how to simplify square roots. When it says "check on a calculator" they will have access to a CAS calculator for this lesson. I realize that if the roots don't come out rational, the calculator displays the answer with a fractional exponent. I don't know what to do about this yet. Maybe I will just put the answers on cards they can check instead of futzing with the CAS's.

The goal is for them to practice multiplying and simplifying, and investigate multiplying conjugate pairs to set us up for rationalizing denominators, both monomial and binomial.

I know the two questions at the end are kind of weird but it seems like a shame to waste the opportunity.

So...

okay?
Crappy?
Suggestions for decrapifying?
Am I missing any relevant opportunities to make connections?
Or show multiple ways of seeing something?
How are the kids going to noob this up in ways I'm not anticipating?

Also, I am a Latex beginner so no making fun of my typesetting. It took me four hours to make this.

Everything is a Parabola

2011-09-19T17:52:00.000-04:00

Lined up the kids at the board along a big number line. Everybody picked a number : their "x." I said, "Shawn's x is at 2. However far away you are from Shawn, move that many floor tiles into the room, perpendicular to the board." They move into a lovely approximation of the graph of f(x) = |x - 2|. "Hey what kind of a shape are you guys making?" "A PARABOLA!" Practically in unison. FACEPALM.

Geometry Project Follow-Up

2011-09-17T18:21:00.000-04:00

So I think I know what I'm going to do with all these Powerpoint files submitted for the Geometry project. I'll grade them as promised, but when the kids come to class Monday, I'll...

First, show them an uber-presentation of the best possible answer to each statement. This way they will get their feathers fluffed up when they recognize their work on display, but we will also hopefully resolve any lingering doubts. I did not necessarily confirm or deny all their questions while working on the project, because it was more important they keep thinking about it, and the teacher giving the answer from on high shuts down thinking. Most often this sounded something like, "Miss Nowak, will you check my ten Always/Sometimes/Nevers and tell me if they're right?" Me: "No...but I'm happy to discuss with you any specific questions you have about what words mean or what would make a statement true or false..."

Second, each group will get 4 copies of a printout of someone's presentation slide that exhibited a misconception. The task will be, what could you do to help clear up this group's misconception? What could you say/draw/show to convince them that they misunderstand, and also help them understand?" Then, we will jigsaw to mix up the groups, and everyone will share what they learned with their new group.

I always felt weird about the flow of : Work on Project, Submit Project, Teacher Grades Project and Hands it Back, and that is The End. I think there's much to be gained from re-visiting this work and catching all those lingering misunderstandings.

Geometry: Points, Lines and Planes

2011-09-16T19:11:00.003-04:00

I was going to write up a description of this project we just wrapped up in Geometry, but luckily Allison already did it! (Maybe go read that post if the rest of this doesn't make any sense.) I liked it because it gave students an opportunity to get messy with points, lines and planes. There was a whole lot of productive struggle going on in my Geometry classes.

Here are some nice examples of student work:

Here are some examples that show misconceptions! I'm going to do something with these. Not sure what precisely yet.

Some kids really did seem to be enjoying themselves while learning, but there was also an awful lot of complaining going on. Managing all of their digital photos and getting them from their phones and cameras into their accounts was a bit of a hassle, so I can appreciate the frustration there. But it was good! There were lots of conversations, which if they were blog posts or magazine articles would have titles like:

Why You Are Making Us Do This
It Is No Fair Making Us Think
We Would Prefer to Just Fill Out Worksheets That Ask the Same Questions Over and Over
Why Miss Nowak or Anyone Would Like This Job
If You Give Miss Nowak Your Phone She Will Change Your Wallpaper to a Math Picture

However, there were also hopeful conversations like:

I Would Rather Be in This Class Where I Actually Learn Something Even Though It's Harder
My Friend is Jealous She Doesn't Get to Do This Project
Holy Crap, a Three-Legged Table Can Never Be Wobbly

Algebra 2: Graphing Absolute Value Functions

2011-09-16T11:33:00.003-04:00

My goal with this was for students to understand "why the V shape" for absolute value functions. I think it will take three days. The TI-Nspire is used. This leads up to something very much like this, but with much more scaffolding ahead of time.

Phase 1: Learn lists and spreadsheets, data and stats skills on nspire:

1. Students watch and follow along on their handhelds "Data and Statistics: Adding and Rotating a Movable Line" tutorial.
2. Close file and do not save.
3. Send students document Squares.tns containing the case vs gray squares table from the beginning of the No Sleep Til Brookline problem set. Display these directions:

open file
add a Data & Statistics page
create a scatter plot of case vs gray squares
add a movable line, and try to get it through all the points
remove the movable line
menu, analyze, plot function
type in the equation you know fits the line

Phase 2: Understand why absolute value graph has a V shape and what it means
1. Write a number line on the whiteboard from -8 to +8 or even longer if possible.
2. Students line up against wall (however many will fit - maybe a subset of the class). Students note their position along the number line written on the whiteboard. This is your "x."
3. Choose whoever is at 2 to be the vertex. Let's call him Jake.
4. Give Jake something to hold up like a flyswatter.
5. When I say go, you are going to move away from the board. The rule is, however many floor tiles you are away from Jake on the number line, you are going to step that many floor tiles into the room away from the board. You'll move perpendicular to the wall. Take a moment to decide how many tiles you will move....Go.
6. Students move into a V shape.
7. Display in succession on projector:

jump up and down if you are a solution to |x - 2| = 5

jump up and down if you are a solution to |x - 2| > 5
jump up and down if you are a solution to |x - 2| < 5
jump up and down if you are a solution to |x - 2| = 3
jump up and down if you are a solution to |x - 2| = -3
8. Let's call the distance you stepped into the room y. What is the equation of x vs y?

9. Reset students back to line up against the board. (Or get a new group up there.)

10. Get the flyswatter away from Jake.
11. Our new function is |x + 3| = y.
12. Who gets the flyswatter? (Let's call her Jill.)
13. Your position is still your x. Decide what your y is and move there.
14. Display a few "jump up and down" questions.
15. Note your position! Come to the smartboard and enter your name and position in the Lists & Spreadsheet.

Phase 3: Create Absolute Value Scatter Plot on TI-Nspire
16. Send everyone the lists&spreadsheet with name, position on NL
17. column 3 call it distJill
18. In formula cell, how can we calculate everyone's distance from Jill? Try difference...note problem with negatives.
19. Show how to enter absolute value function: template or abs(position - -3)
20. Show how to sort the whole spreadsheet from closest to farthest
21. Add a DataStats page
22. Can you get the dots to arrange themselves into the V-shaped graph?
Can you add the function that goes through all the dots?
Can you add a horizontal line function that represents "4 away from Jill"?
Can you shade the region that includes people that were within 4 spaces of Jill?
Can you add a vertical line (Plot Value) that represents the average distance from Jill?

Phase 4: Apply skills to novel problem
For the past week, I left this sitting out on a desk in my room:

Something like 100 students entered their guess. I was able to copy the column containing their guesses from Google Docs into an Nspire lists and spreadsheets page. They'll get this Nspire file and these directions, and have the period to do what they can with it.

Phase 5: Transformations on the Absolute Value Function
Students will spend time playing with this Nspire file with this investigation, to understand how changes to the parent function transform the graph. To assess, they will try to match the pictures with a function.

Algebra 2: Solving Absolute Value Equations

2011-09-15T17:44:00.000-04:00

You know how you can show them this way, all justified and with lots of practice untilblueintheface:

But then a couple days later half of them will do this

and the other half will do this:

So, I stopped teaching it that way. I'm starting with something much like what most of us probably do:

Allison lives at 15 Sycamore Drive, and Sarah lives 8 houses away. Where does Sarah live?

But then, I'm sticking with that model for all kinds of problems.

and

Earlier in the lesson I made them write it out in words, i.e., "the distance from 200 to 3x is 896."

It was more of a pain initially, and not the most effervescent lesson I have ever delivered, but MAN, it did the trick. No more of that autopilot, forget to write two equations, forget that absolute value can't equal a negative nonsense.

This is an idea I stole wholesale from the article "A Conceptual Approach to Absolute Value Equations and Inequalities" by Mark W. Ellis and Janet L. Bryson, Mathematics Teacher April 2011, Volume 104, Issue 8, Page 592.

Here's my smart notebook file for equations.

Inequalities are a natural extension of this concept. Where on the number line are all the values that are more than 896 away? That are less than 896 away?

Here's my smart notebook file for inequalities.

My First Days of School

2011-09-12T20:53:00.000-04:00

My first day schtick has changed quite a bit over the years. The first year or possibly two I drank the Wong kool-aid because I was terrified and I had no idea what else to do, and also my school gave us all the Wong book for free. That didn't work. It probably works for the Wongs but it didn't work for me. I felt like a pretend drill instructor. (Wong. Wong-wah-wah-wong-wong.)

Then for a while I got kids to fill out a sheet about themselves that I stole from Dan Meyer. I had hopes these pieces of paper would capture the essence of each child, and had every intention of perusing them leisurely with loving tenderness. In reality, I just scanned the "Anything going on you want me to know about?" section so I'd have a heads up about all the imminent divorces and cancers and then I threw them in a drawer. Since that only took about ten minutes of day 1, I would just start in on a lesson for the rest of the period. Giddy up!

This year is the first one I feel I started out semi-competently. The cluebird is circling and coming in for a landing. My only measure for this is the proportion of students that will make eye contact when they talk to me. I considered what I actually wanted to achieve from the interaction (imagine that): I want them to know they can be successful. I do want to start getting to know them, but just as importantly, I want them to know that I want to know them. I want to know who has the tech in their pocket that we can exploit for the learning. I want them to think about what they want to improve about themselves as students, and I want it to dawn on them that it's in their power to change whatever that is. I want them to feel comfortable in my room, to not feel trapped and helpless, at least know some of their classmates' names, know that I expect them to work hard but I'm pretty darn reasonable, and to tell me where funny things are on the Internet. Okay I'm going to stop with this paragraph now because it's getting out of control and was obviously was too ambitious for 43 minutes with 121 strange kids. Ahem.

Phase 1: Snowball Icebreaker
(I saw this on somebody's blog but I can't find where. Speak up and I'll happily give you credit.) Everyone gets 1/2 sheet of paper and writes three distinctive (you will have to give examples of distinctive vs non-distinctive) characteristics about themselves, but not their name. Everyone crumples up the paper, and we toss them around the room (suggested verbiage: "My biggest rule is that you show respect for each other and for me. If you can do that we will get along fine. So there will please be no whipping your snowball at anyone's face. Now, all together, let's pick a direction and gently fling our snowballs.") Everyone picks up a snowball and uncrumples it. Their job is to find its owner and write his/her name on it, and be prepared to introduce this person to the class. Their job is also to facilitate being found. As soon as they do both things, they sit down in the closest empty seat. (Suggested verbiage/modeling: "This is what I don't want to see: (hold up sheet in someone's face) 'IS THIS YER SHEET?!' (pause for giggling.) This is what I do want to see: 'Hi! My name is Kate. What's yours? Audrey? Nice to meet you! Tell me, Audrey, are you allergic to wheat? No? That's too bad. Although fortunate for you, I suppose. Ok, is there anything you'd like to ask me?")

Once the dust settles, start asking kids to introduce each other. This was great fun, because I kept asking for more details that would let them show off a little and/or amuse us all.
"This is James. He works at Wegmans and plays guitar."
Me: "Cool! I wish I could play guitar. What is your best song?"

"This is Angelina. Her brother is a pilot who lives in London."
Me: "How old is your brother?"
Angelina: "30"
Me: "Is he single?"

So that all took 15-20 minutes. I liked it because it fit one of my goals for this year: everything we are doing is for a reason, and we will follow up on things and not drop them without processing them and assessing you and making sure we all got the point (more on my lofty '11-12 goals in a later post.) It also hit several of my goals for day one: some kids learned some other kids' names, it was low-stress and all the chatter made the room feel inviting, I got to learn a little about them and they saw me being interested.

Phase 2: Distribute Books and Collect Data
Then I passed out my heavily modified Who I Am sheets, and the kids worked on them while I passed out textbooks and made smalltalk. This exercise felt different than previous years, because they weren't filling them out all scared in stony silence, but there was productive chatter and informal sharing and it just felt nicer.

Since I asked them questions I actually cared about, it was no drudgery to take all their filled in sheets and read them thoroughly and enter them in a spreadsheet. I sort of had the idea that analysis would yield some interesting things but I don't know what I was thinking. There are six days a year where three of my students have a birthday at the same time...that's kind of neat. I got a roughly even mix of math-likers and -haters:

And we have mixed opinions on whether Mathematics is invented or discovered:

And I suspect that some of the kids who claim they like math, don't know what it is, and like it for all the wrong reasons:

But we will see what we can do to change that.

Phase 3: Some Blah Blah
I spent the last 5-10 minutes telling them about supplies they need and what to do if they have to use the bathroom, that sort of thing. We didn't do any math. I don't feel bad about it. I feel better about the way this year started than I ever did. I'm excited about how we're learning, too. More soon.

Pre School Year Jitters

2011-08-31T20:18:00.002-04:00

I don't think I've ever posted at this point in the school year before. To recap, I'm coming up on year seven of this delightful and terrifying profession.

I appreciate that I get the opportunity to totally start over every year. And I don't even have to go to the trouble to press RESET and pay a new quarter.

Today I looked around my classroom and realized it is TOTALLY TRICKED OUT. Smartboard. Document camera. TI Navigator. (Really, Nowak, you have no excuses. None.) Gigantic Darth Vader poster. (Fun Miss Nowak fact: my childhood dog was an all-black ChouChou/shephard mix, with a black tongue. Named Darth Vader. We just called him Darth. He was great.)

I have things to complain about. My Regents classes that are supposed to be a healthy mix of accelerated and not kids are pretty much totally NOT. Out of 54 Geometry kids, I have one 9th grader. ONE. Out of 42 Trig kids, I have four 10th graders. FOUR. I don't know if this is some kind of conspiracy or scheduling fluke or what, but this year is not going to be a walk in the park from the classroom management or instructional perspective. NOT. But they're in my computer scheduling thingie and I can see their names and pictures. And I love them a little already. And I can't bring myself to object although I realize Guidance is probably trying to see how far they can push me. Now, when I can only speculate about them. By reputation. Much like they're looking at my name on their schedule, and making predictions about me, by reputation. She's hard. She's easy. She's a bitch. She's awesome. Just make her laugh. Just be yourself. You're doomed. There is hope.

I posed an inservice class to my department that was basically "us hanging out and working collaboratively on difficult math with maybe food" and they were totally on board. I wrote up a proposal and sent it in. That class is going to be amazing if it gets approved. And maybe if it works and I'm a little bit lucky it will change the way we teach and change the nature of what the children learn.

Since something like 90% of our faculty are new in the last five years, our new-ish principal wants to open discussions of just about everything, including scheduling and grading. We talked about it in small cross-discipline faculty groups this morning, and that experience surprisingly did not leave me in abject despair. I gingerly broached my lunatic-academic-fringe stance on grading and they did not treat me as if I were radioactive.

Shocking!

You're doomed!

There is hope!

It's Amazing What a Good Night Sleep will Get You

2011-08-24T09:37:00.003-04:00

Linear review lesson for Algebra 2 - version 2 - I like this much more better.
Here it is as a word document if you want to steal parts.

Linear Equations Review Lesson for Algebra 2

2011-08-23T20:42:00.002-04:00

So, I just spent two days making one lesson for one class! Yeah, this does not bode well for this year. This pays serious homage to the PCMI problem sets by Bowen Kerins and Darryl Yong, who I already know are way funnier than I will ever be.

The goals for the lesson are the students remembering and being able to... (NB, they should already "know" all this from their previous Algebra 1 and Geometry courses)

Explain the meaning of all the terms in slope-intercept form
Write equations of horizontal and vertical lines and know how their slopes work
Sketch the graph of a line given various kinds of information about the line
Interpret point-slope form
Write the equation of a line in point-slope form given its slope and a point on it
Find the slope of a line given two points on the line, or its graph, or its equation in either form
Know how slopes of parallel and perpendicular lines work

Open questions

Is this too ambitious and going to scare the bejeezus out of the poor summer-addled adolescent brains?
How am I going to assess who knows what as the students are working?
What's the best way to organize the kidlets so that they might benefit from some cooperation? I'm thinking groups of three or four with minimal guidance about how they "should" work together.
Aside from the lame jokes in the marginal notes, how can I bring more joy into this exercise?
Are there better ways to ask any of these questions that make them more tangible?

As always, I welcome your thoughts.

So...I had the first version here? But because of box.net's helpful versioning, it's no longer available. The latest version is posted here.

New Blogs You Should Read

2011-08-15T20:17:00.002-04:00

The authors in this list have one thing in common - I have met them all in person! I know, weird, right? So I feel utterly qualified to endorse them as smart, interesting, nice people. I will try to tell you something about them that is compelling and not readily apparent. Their blogs are relatively new, but all shaping up nicely. Check it out:

Tina (not sure if she wants her last name used) was at PCMI '11. She is SMIZZ-ART, yo, and one of those earnest, wholesome, authentic people who you suspect might not own a television and might spend her weekends hiking and canning seasonal produce. She could also fit in your pocket.

Bill Thill is one of the most thoughtful educators I have ever met. He will push back against all your assumptions and you can count on him to ask the most laser-like, insightful questions. Seriously, your bullshit is not safe within 50 yards of him. Also does a mean Chloe Sevigny impression.

Allison Krasnow, in the first conversation I had with her, gave me a brilliant way to manage homework to make it much more useful as a self-checking tool for the kiddos, but no more work for me. She's warm, genuine, and wears very cool earrings. Her new blog has four posts so far and I want to hug every one of them.

I met Paul Salomon at a School of Math session where we worked on a super-fun problem together. Paul teaches at Saint Ann's School, where they have no grades and the loosest of a math curriculum a.k.a. heaven. He writes a lot about the way math should be taught but he has a bit of authority in this arena, as he gets to teach math the way it should be taught. He's also a demon on Twitter and has been stirring the pot lately on the "how much paper/pencil computation is too much" front.

Chris Luzniak has really just dipped his toe into blogging about teaching math and running his school's speech and debate team, and I am hoping he sticks with it and starts writing some more. But this pattern fits with his persona - he mostly keeps is own counsel when it comes to teaching math and how to do it, but when he does weigh in, it knocks you over, and you wonder just what is going on in there the rest of the time. A real tour de force.

Good Problems: Follow That Diagonal

2011-08-12T22:01:00.001-04:00

This is a sweet little problem because it is simple to state and understand. It seems like anybody should be able to make progress investigating it, but it won't be obvious to your smartypants kids.

Draw a 9 by 3 rectangle on a square grid. Draw one diagonal. How many squares does the diagonal pass through? Draw some non-similar rectangles with one diagonal. How many squares does the diagonal pass through? Develop a rule to determine the number of squares a diagonal passes through for any rectangle of any size.

I think I'm going to keep it in my back pocket for a day when I need to kill half a period. It might be nice for the first day of school if you like that sort of thing. I don't think I've seen it before. It was sent to me by Øistein Gjøvik - he has a post about it that includes access to a Geogebra file. (One benefit of blogging I would have never predicted: a cool Norwegian sends awesome math problems to my inbox.)

I have been on a bit of a Sketchpad bender since we used it at PCMI, so here's a sketch I made.

I am torn about giving guidance about posting solutions in the comments. I have one way to think about it that works, but I'm sure there are more and I really want to hear them. On the other hand, I don't want to spoil anyone's fun. So maybe if you want to work on it, resist looking at comments?

Another thing I'd like to hear about is, do you see this fitting into a curriculum? Or is it just a nice problem that doesn't have a home in a unit of study?

Summer Learning, PCMI Edition: Odds and Ends

2011-08-01T14:26:00.000-04:00

This is a series of posts that are reflections from the Park City Mathematics Institute Secondary School Teachers Program.

This is a catch-all for things I want to remember and post that aren't big enough for their own post.

Livescribe
In a 5-minute short, Cal Armstrong presented his use of Livescribe smart pens. I had a little "holy cow" moment during his presentation, because I've long dreamed of kids' recording their problem-solving process, but there's only one smartboard in the room, and writing with a mouse is hard. Enter the Livescribe pen which records both your writing as you write, and audio along with it. And they are only like $100 a pop. I could ask kids to record a livescribe of them solving a problem as their reassessment, or record a tutoring session of them teaching it to someone else. We could put them on blackboard and build up a little library of these, or upload them to voicethread for feedback.

Google Forms for Recording Small-Group Discussion
I am pretty good at incorporating small-group or partner discussion, but I don't often have an efficient way for groups to share their thinking. One technique I noticed frequently deployed at PCMI was to give groups a link to a google form, so that each group could send in a summary of their discussion or response to a prompt. We aren't a 1:1 school, but it would be sufficient for each group to have one laptop for this purpose, and I'm pretty sure I could secure 5-6 laptops to keep in my room. Then again, I am supposed to have a TI-navigator system next year, so maybe I could just use it for this purpose.

Other Kinds of Tasks
Do you ever get stuck in a problem-writing rut? I do. Throughout, I was keeping track of all the tasks I saw that were something other than "find the missing value:"

write an equivalent expression
give an example
show that two expressions are equivalent
interpret expressions/equations in writing
interpret a graph in writing

Metacognition: See How I Think
We spent a few days talking about what is metacognition, and ways for students to "do" metacognition. We participated in an exercise that I think could be adapted for students to use. In a group of three, students take on three roles: problem solver, listener, notetaker. The listener is NOT HELPING solve the problem, just asking the problem-solver to clarify their process and state it out loud. Meanwhile, the notetaker is writing down any evidence of metacognition or "thinking about thinking" that she hears. I think this could be very beneficial in helping students see how the same thought processes (making use of structure, considering extreme cases, organizing data, etc) cut across mathematical content, but I wonder at designing it in such a way that they can see the point. I need to spend some more time thinking about this.

The Vampire Animations
I worked on a lesson as part of our working group, and I don't think I'm supposed to disclose all the inner-workings of the lesson because it may be reviewed for publication as part of a larger project, but I do want to share this super-fun simulation we made. If you can use it, steal away.

Here is a "question" video of an infection spreading up to 64 victims:

And here is an "answer" video up to 512 victims:

Arts and Crafts
Finally, what is camp without crafts?

Summer Learning, PCMI Edition: Formative Assessment

2011-08-01T13:13:00.000-04:00

This is a series of posts that are reflections from the Park City Mathematics Institute Secondary School Teachers Program.

Our middle session every day was called Reflecting on Practice, and it was basically a mini ed-school class. The focus this year was on formative assessment or assessment for learning. These types of classes are not usually my favorite (make a fake assignment! watch a video of someone teaching! talk about your feelings!), but in this case they were exceptionally well planned and executed so I didn't have much time to feel sorry for myself.

Biggest takeaway - there needs to be deliberate feedback, not attached to a numerical grade, built in to classes. Because when there's a number there, kids don't pay attention to anything else. (On the flip side, in the absence of a grade you run into kids not taking the work seriously, so giving feedback on their marginal efforts feels like a waste of time.) At least some of the time, the attitude toward assessment should be less "judgment day" than "a conversation about learning and understanding." I was influenced especially by two articles we read: Classroom Assessment: Minute by Minute, Day by Day and Working Inside the Black Box: Assessment for Learning in the Classroom (which does not appear to be available online.)

These simple ideas lead me to rethink the whole process for "level 1" quizzes - the kids' first stab at a concept on an SBG quiz. I spent way too much time re-designing what a quiz paper should look like:

That's super-helpful for you, right? Sorry. After I try this in class I'll have more to say about it with a nicely-typed up version. But the idea is, there's half a page for the student to do his work, and predict his score. The bottom half of the page is set up for structured teacher, self, and peer feedback. I want the message for level 1 to be "I want to help you figure out what you still don't understand" instead of "Fear my red pen!"

I asked the cherubs to weigh in on Facebook and got mixed responses.

So anyway, the idea for a process will look like this for Level 1 questions:

Students take the quiz, and predict a score.
I collect the quizzes and write feedback in sentence form, like "right idea but computational errors" or "a more careful and accurate diagram would be helpful"
Next day, students give feedback to each other. One idea, so that they are working to understand and not to just get the right thing on the paper without understanding, is to not let them use pens or pencils, but communicate with mini-whiteboards. While they are doing this, I can be assessing/rewarding/publicizing helpful dialog that I hear.
Then, students have an opportunity to re-work the problem, or possibly a new but similar problem.
Then...what, grade the quality of their feedback to each other? Or never grade this part of the process? I am thinking that if I want the focus to be on the learning and not a numerical grade, I can't give a grade to this part ever.

Obviously there are questions here that will take time to sort out. But this seems like a step in the right direction.

Summer Learning, PCMI Edition: Deeper Criteria

2011-07-30T06:43:00.000-04:00

This is a series of posts that are reflections from the Park City Mathematics Institute Secondary School Teachers Program.

One afternoon we listened to a lecture/powerpoint by Douglas Corey of Brigham Young University about comparisons of effective teachers at home and abroad. Toward the end of his talk, he seemed to have partaken of a generous serving of the edureformer kool aid and came across as anti-teacher or at least teacher-concern-dismissive, which obviously turned many people off. However I took away some notes about his research that struck me as important.

Based on classroom-level comparisons between different countries from the TIMSS video study, researchers found

there is no single effective teaching method
all high-achieving countries teach quite differently
we can not judge a lesson's effectiveness by methods used, but rather
effective lessons have deeper criteria in common he called "instructional principles"

He asked us to predict the instructional features which must be present for students to learn with understanding. These were the guesses that my group brainstormed. If you want to play the home version of the game, take a moment to jot down what you think they are too.

teacher content knowledge
problem-solving
students have to be working and thinking
deeper explorations
students making connections
continuous assessment that informs instruction
deliberate metacognition is part of instruction
teacher believes all students can learn rigorous, conceptual mathematics
students need to spend time thinking about math outside of class

However researchers only found two:

"intellectual engagement" - the teacher has to get the kids thinking about a problem. Students have to struggle. "Struggle" means students expend effort to make sense of math, to figure something out that is not immediately apparent. It does not mean needless frustration.

"connection-making" - the focus of the teaching has to be on making connections. Connections don't come by accident but must be an explicit focus of planned instruction.

The struggle thing rang true for me. At some level I internalized that idea long ago. I'm still coming to terms with the connection-making point. The same concept was approached earlier by Gail Burrill with respect to the Common Core standards. She pointed out that in American classrooms, teachers can plan and ask connection-making questions and activities, but students mostly still end up doing procedures. A big question I am still grappling with is how to design and deliver instruction so that the students are doing connections. I have only vague notions about what that would even look like. I don't really know what to do with this yet beyond hang a sign on the bulletin board next to my desk at school that says "make the students do connections."

Summer Learning, PCMI Edition: Keep Learning

2011-07-29T20:25:00.001-04:00

This is (Hopefully...it's a goal but I am distractable...ooh! shiny!) a series of posts that are reflections from the Park City Mathematics Institute Secondary School Teachers Program - the best professional learning out there, in my opinion, except it sounds like next year it's basically canceled or at least way diminished (thanks, dysfunctional government!) We wrote personal reflections on a regular basis, and I spent 20-30 minutes before bed every day writing down things I didn't want to forget. The reflections were supposed to be private, but you all know I'm an exhibitionist about my learning.Without further ado,

Lesson 1: The most important thing I can do to keep improving as a teacher is to keep placing myself in the position of learning new things. The discomfort, confusion, coping - I have to keep coming back to it and back to it. The goal is perceiving myself, even when facing thirty teenagers, as an "accomplished novice" instead of an "answer-filled expert." How do teachers move from expert to novice? That is hard to do and even hard to think about - a desirable pathway but a difficult one to find. But the "accomplished novice" attitude is one I respect in the best teachers I know. If I don't keep placing myself in the learner role, I forget, forget, forget. "Answer-filled expert" is the pattern of habits I fall into when I forget.

Virtual Conference on Core Values: Mistakes are Made

2011-07-02T06:00:00.012-04:00

To sum up the center of my classroom in a phrase: We Make Mistakes. Sometimes deliberately and sometimes not, but we celebrate both kinds.

Non-educators are maybe puzzled at this statement, but all the teachers in the audience have already started nodding along with me.

See, lots of people think that learning happens like this:

and that that sparkly rainbow is the maaaagical majesty that separates good teachers from bad.

But those who have spent some time, you know, actually responsible for other people's learning know that it really happens more like this:

We could just call it "Piaget for Dummies." Other authors have likened the confusion to the conflict in a story. That little star is a place a teacher's skill is really important, because that's the point where kids might feel stupid and check out and also probably hate you for making them feel that way.

So, I'm very deliberate about centering lessons around mistake-making in non-threatening ways. If all my lessons have one thing in common, that would be it. Here are two specific strategies.

GoodQuestions

The first one became a staple in all my classes late last year and is really just Cornell University's GoodQuestions project, by way of Helen Doerr at Syracuse University, who sums it up this way: "A good question is divisive."

Pose your question. Here is one I plan to use next year in Calculus:

Compel the students to register a choice. You could use clickers, poor-man clickers, or polleverywhere. But it is important that they commit to a choice. And it's best if they can't see what other people selected until everyone's vote is registered. Ideally the responses will be more-or-less evenly distributed, or at least there will be no clear winner. Which is awesome! Because even if it ends up that you're wrong, you can't feel too bad about it if a third of the class agreed with you.

The point is to instigate an argument that can be settled with mathematical justification. If you're not sure how to get the justification ball rolling, try "Would anyone care to defend choice A?" If you're desperate, pass out slips they can write their name on and hand in for participation credit whenever they present an argument. Whatever you do, for goodness sake, don't tell them the right answer. (Like, ever. Let them come to consensus. Learn how to ask helpful questions without giving away the store.) Unless for some reason you want to completely shut down discussion. And thinking. You know you've got a really good question when you don't have to pull teeth to get kids to talk because they are so compelled to explain their reasoning that it overcomes their fear of everybody looking at them.

The Tyranny of Randomness

My second strategy is, I imagine, common in most math classrooms, and that's getting kids to the board to present their work. When I get them at the beginning of the year, they are freaked out by this. Of course. Standing in front of a room of your peers, potentially exposing your ignorance, is super intimidating until you do it a few times and everybody is doing it and you realize it's not such a big deal.

"Will you check it first?"
"We'll all check it together!" (This is not optional!)
"What if it's wrong?"
"That would be awesome! I'm kind of hoping it's wrong actually!" (If anyone points and laughs, I'll kill them!)
"I didn't get an answer..."
"We'd still like to see the progress you made!" (Get up there, kid!)

I've always done this but sometimes have used volunteers, and sometimes have played with ways to randomize selecting students. For example, use the random integer function on the TI and match it to a numbered list of names. I also tried the popsicle sticks, but I am not that organized and kept losing them. Neither of these methods was very satisfying, because they couldn't see the process and had to take my word for it being random. But random selection really helps, here - when any of them could be selected at any time, they become more likely to give a problem some honest effort. Also, if they see a machine pick, they don't blame/hate me for picking on them.

Then last year I found the best thing - SMART Notebook has a flash random word chooser in the gallery (if you can't find it you might need to update the Lesson Activity Toolkit) - so I just have a file saved with a different page for every class with the students' names. When it's time to select a student, I just fire it up and we can all see it ping around before it settles on a name. It's very suspenseful.

Whatever you do, you can't let them off the hook. When selected, they're going to stand in the front of the room and pick up a marker and write something. (Alternatively, going to put their work under the document camera, or whatever process you've worked out.) Or you might as well go back to asking for volunteers, and it will be the same three kids who are the only ones doing work and going to the board. You'll probably have to be annoyingly insistent for a little while until they become convinced of the inevitability of the situation.

There is a free random word chooser here that seems to work pretty well, but I have not tried it in a class. (And you'll want to turn the sound down/off because, ugh.)

The moral of the story: confusion and mistakes are necessary for learning. So much so, in my opinion, that I center my classes around them! Thanks for attending! Check out the other presentations!

2010-2011 Wrap Up

2011-06-28T17:52:00.000-04:00

There are just some things I want to get down right now. There was less blogging this year (I know...the spirit just didn't move me, as much, maybe I should set up some kind of schedule.)

Things to Consider about Grading
-go back to 5 point scale
-first assessment - feedback only?
-require that homework be done, checked, discussed, etc before reassessment?
-"show me what you practiced" - you must provide evidence of practice before reassessment. at a minimum, corrections on the two in-class assessments - and the practice problems from that day on the calendar (?)
-summative assessments - all MC questions - every 6-10 weeks
-pare down standards list somehow
-write reassessment questions ahead of time and have them ready
-alternative assessments/reassessments? record themselves solving a problem?

Add to Lessons
-notebook checks and grades for notebook organization
-Discovery Education videos? (we have a subscription, apparently)
-more structure for vocab in geometry
-open notes for quizzes?
-allow single-page crib sheet for quizzes?
-compelling the kids to write more. Don't just tell me the answer, describe how you got it. Describe how you know. Save samples of good and bad writing.

Things I Don't Yet Know the Answer to:
Ways for kids to share their thinking in the middle of class without interrupting the flow too much. Options right now are: 1. Throw whatever they have in front of them under the document camera. Quick but sloppy. They aren't necessarily writing for an audience in their notes. 2. Haul their butts up to the whiteboard to copy what they've already done so everyone can see it. Takes time. That we don't have to spare. 3. Some elegant tech-y solution I haven't thought of yet. (Idea: Poll them early to figure out who has a camera phone on them, and who can text/email an image for free. Go with that.)

Big Overall Thing I Notice about Myself this Year: I used to be the annoying asshole in the group who was always trying to show she was smarter than everyone else. I've turned into the slightly less annoying asshole who's paying attention to actively supporting whatever good is transpiring around her slash shutting up and listening. I like new me a whole lot more than old me.

I Can't Do Both

2011-06-11T09:47:00.000-04:00

This comment got way out of hand, and Dave's question really gets to the heart of the matter. I feel like I've said all this before, but it bears repeating, in case any non-educator policymakers are actually listening and actually give a crap.

David Cox said:

"The question, as I see it, is this: Can we do both? Can we teach students math—real, interesting, thought provoking, man-this-stuff-is-cool, math—and still have them show growth on whatever assessment is put in front of them?"

That's an awesome question, Dave. There are three problems with our current state exams in NY anyway that complicate the issue.

#1 There is way too much content tested for a course that is supposed to be done in one year. I think most of us would agree that authentic learning takes time, but if you take the time to do it right, you can only cover maybe 2/3 of the stuff tested. We have to make the choice of exposing students to all the content by frog-marching them through it, or teaching it in a way they really learn it and conceding there will be stuff on the test they've never seen before.

#2 Many of the questions strictly test knowledge of notation and vocabulary. A kid could know the math inside and out but still miss these.

#3 Many of the questions have a gotcha nature that are clearly not intended to assess understanding. For example, this week one of my students couldn't understand why she picked the wrong answer on a multiple choice trig question - she had done everything right, but her calculator was in radian mode instead of degree mode. Those weasels had made the radian-mode answer one of the distractors, AND, this wrong answer was reasonable in the context of the question.

These are my frustrations around NY exams that make me feel like I can't both teach for understanding and teach such that the exams show progress. It remains to be seen how the common core assessments address these issues.

The Immovable Object vs. the Unstoppable Force

2011-06-10T18:29:00.001-04:00

The narrative seems to be boiling down to a showdown.

The Immovable Object: Mathematics
see also:
Paul Lockhart
Ellen Kaplan
George Polya
Andrei Toom
Michael Serra
Dan Meyer
Vi Hart
Jo Baoler
Euclid
m-f'ing etcetera

The Unstoppable Force: Poorly-Written Purpose-Confused Tests with Non-Existent Validity
see also:
The College Board
Pearson/Prentice Hall
Bill Gates
The New York State Board of Regents
Governor Cuomo
The U.S. Department of Education
President Obama
The US Governor's Association
The Common Core common computer-based assessments : coming soon to a misery factory I mean school near you!

Until now, reasonable, effective, educated, brain-in-their-head math teachers have been able to shut their doors and concentrate on the immovable object.

But now, New York gets APPR (crammed down our throats, lawsuit-pending)? (40% of teacher evaluations based on student-progress-no-we-don't-really-know-what-that-means-yet-and-stop-asking!) Which I think means if we keep ignoring the unstoppable force, we get to lose our jobs.

Anyone get the number to that truck driving school? TruckMasters, I think it was called?

The Description of the Wondrous Canon of Logarithms

2011-05-07T12:29:00.001-04:00

Sent in by fellow math history nerd Elizabeth (aka @cheesemonkeysf):

"I was reading Napier's Preface to The Wondrous Canon of Logarithms (in Latin, so sue me) in preparation for using ... Log War cards this coming week, and I just totally fell in love with the sweet and completely nerdy generosity of his introduction.

So I'm providing this translation.

I agree with the better translators on *most* of the translation, but they leave out some of the little endearments that make it so charming -- and that make me forgive him (somewhat) his rather cumbersome ideas (thank the gods for calculators in our day).

I especially love his commiseration with his "dearest fellow mathematicians" and with "most diligent students of mathematics." The guy had class. :-)"

The Description of the Wondrous Canon of Logarithms, and the use of that which, not only in Trigonometry, but also in all mathematical logistics, is most fully, most easily, and most expeditiously explained.

By the author and inventor John Napier, ___

On the Wondrous Canon of Logarithms
Preface

Since nothing in mathematical practice, my dearest fellow mathematicians, is more tiresome than the great delays suffered in the tedium of lengthy multiplications and divisions, the finding of ratios, and in the extraction of square and cube roots– and in which not only is there the time delay to be considered, but also the annoyance of the many slippery errors that can arise: I had therefore been turning over in my mind, by what sure and expeditious art, I might be able to improve upon these said difficulties. In the end after much thought, finally I have found an amazing way of shortening the proceedings, and perhaps the manner in which the method arose will be set out elsewhere: truly, concerning all these matters, there could be nothing more useful than the method that I have found. For all the numbers associated with the multiplications, and divisions of numbers, and with the long arduous tasks of extracting square and cube roots are themselves rejected from the work, and in their place other numbers are substituted, which perform the tasks of these rejected by means of addition, subtraction, and division by two or three only. Since indeed the secret is best made common to all, as all good things are, then it is a pleasant task to set out the method for the public use of mathematicians. Thus, most diligent students of mathematics,
please accept and freely enjoy this work that has been produced through my good will.

ON LOGARITHMS.

By which all the sines, tangents, and secants,
are set out for you by means of great labour and prolixity;
And which this little table of Logarithms, gentle reader,
Gives to you all at once, without great labour.

Farewell.

f(t)'s Greatest Hits

2011-04-28T10:16:00.000-04:00

f(t) has its very own Best Of page! My little blog is all grown up. Next thing you know it will be getting its own top level domain and growing facial hair.

Those who have been around for a while will probably not think it's too exciting, but I thought it might be useful for newbs. I spent all of 20 minutes on it, so let me know if I missed one of your faves. Enjoy!

QBert

2011-04-12T21:23:00.000-04:00

This year I look forward to and dread at the same time broaching binomial expansion/bernoulli trials/pascal's triangle in Algebra 2. (They are so stickily intertwined, their little tentacles all wrapped up around each other.)

I'm looking forward to it because I have a problem to pose that the kids will dig. That I am shamelessly stealing from Jason Dyer.

See QBert.

See him hop. Hop, QBert, hop. 8-bit music is rad. Kids will be amazed that I have an Atari. And brought it in. And let them play it. (Side note: saw a kid in Learning Support today playing Google Pac Man. Way to get around the filters, kid! Well played.)

I'm sure you see where this is going. QBert starts at the top. How many ways to get to any block? I used Geometer's Sketchpad and made them a beautiful blank QBert board to scribble on. I figured out how to do it with iterated transformations. It was exhilarating.

In an ideal world the curious cherubs will count pathways and write them down and notice some patterns! And think it's cool and try to explain mathematically where those patterns came from and everything will be lovely and we will break into song like we're in a Disney TV film about a high school where everyone is happy all the time and even when they are not happy they only have problems which you might describe as adorable.

I dread it because this is what will happen in my world: they will get confused looks on their faces for 5 seconds. Then they will start talking about unrelated topics. I will say "How can I help you?" They will want me to show them what to do. I will resist. I will say "How many paths can he take to this close block right here?" They will say 2. They will write it down. They will get confused looks on their faces for 5 seconds. They will start talking about unrelated topics. I will try to prod them along to listing out the rights and lefts. I will say the words "right" and "left" approximately 900,000 times. Each.

We will waste 20 minutes.

I will feel like the world's crappiest math teacher.

I will whine about it on my blog.

I will try something similar next week.

Force Vectors and Youtube for Educational Purposes

2011-03-29T06:16:00.001-04:00

Vectors are fair game in Alg 2/Trig as Law of Sines/Cosines application problems, and we get one measly day to spend on it. Not an issue for kids who are taking Physics concurrently, but we don't get many of those since Alg2/Trig is sophomores and juniors. With 42 minutes I couldn't come up with a manageable way to use those static spring loaded pull scale things the physics teachers have.

Without some kind of physical demo, at least, this lesson is beyond meaningless. You want me to draw arrows? Why? So we push desks around. Pushing desks around is pretty good.

But what else is good is watching an airplane flying in a direction other than the one it's pointing. To crazily incongruent music. This one, I think, is the best one, but it wouldn't let me embed it. So here is another decent one:

Do not at all remember if I thought up this idea or got it from somewhere, so if it was yours, email me and I'll happily give you credit. Best guess, this prompted me to go searching around youtube.

The Wheat from the Chaff

2011-03-23T12:38:00.000-04:00

I am frequently poo-pooing TI's offerings on their website around here, so I thought I'd balance the karma by advertising a pretty decent activity we did today: Midpoint Quadrilaterals.

It has the right amount of intellectual rigor and things for the kids to actually do, and the documents that suggest a progression of inquiry did not require much modification. There are four parts, and for each part I made sure they knew what they were expected to do and look for and write down, gave them time to work, then regrouped and had them share their conclusions and reasoning.

I also inserted a task before proving that the inscribed quadrilateral was a parallelogram (after Page 1.3) to remind them of triangle midsegments which looked like this:

...which if this interests you enough to check it out, I'm sure you can figure out why.

Getting More Out of Dynamic Geometry Technology

2011-03-18T14:07:00.001-04:00

My school is ramping up use of the Ti-Nspire. This year is the first year we are using them in Geometry. Next year they will make their way to Algebra 2/Trig¹. But this post applies to the use of any dynamic geometry software (DGS) including Nspire, Geometer's Sketchpad or Geogebra. (Disclaimer: This is not a post about the pros and cons of different technology. All platforms have their benefits and drawbacks, this is what we decided to go with, and now it's my job to use them as effectively as possible. So razz me all you like, but I'll most likely ignore you.)

DGS is awesome but poses unique challenges. Kids can look at a million different examples of something (yay!), but then they think that constitutes a proof. Sometimes they don't bother looking at the million different examples - they just look at the screen as it first appears, and guess. Kids can measure anything (yay!), but they can use that to circumvent your goal of teaching them to use their understanding of a property to solve a new problem. You can set out an activity with lots for them to do, and kids like pushing buttons and it keeps them busy, but some of it is just busy work. The translating of the button pushing to the learning is where the work lies for a teacher.

So let's look at an example of a typical lesson from my Geometry curriculum: G.G.40: "Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals." Surely someone somewhere has posted a decent lesson about trapezoids using DGS. I head over to TI's repository of Nspire lessons, and look under Geometry/Quadrilaterals. It has a lesson called "Rhombi, Kites, and Trapezoids." Oh, goody. This is what I find about trapezoids in the student document:

...and that's it. Hm. Thanks, TI.

So we head over to NCTM Illuminations, search 9-12 Geometry for "trapezoid," and the only result is a lesson making an origami pinwheel:

Which is made out of trapezoids and looks kind of fun, but is not going to help us with "Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals."

How about Geogebra Wiki? Maybe there's something I can adapt there. But a search on the High School Geometry page for "trapezoid" returns nothing, as does the same search on the Quadrilaterals page.

So, yeah. I guess I'm stuck making a new thing. Awesome.

My file ends up being three pages, with questions to explore about each, with the goal of discovering the properties of trapezoids and isosceles trapezoids.

First a plain old trap:

Then an isosceles:

And then some "use the properties to find the missing thing" problems. I am not super into demonstrating examples and then making them do the same exact thing. Then they're exercises, not problems.

Then an isosceles with the diagonals constructed, which they are supposed to measure and change the shape of and conjecture about:

Then they're asked to prove the conjecture about the diagonals:

So, this lesson gets the job done, but it's not terribly satisfying. The kids completed the tasks gamely enough, and they'll remember the properties better for having seen a bunch of examples using the DGS. But it was pretty boring. (Although not as boring as listening to me wah wah wah my way through telling them the properties and doing example problems for them to copy down.) You got a better idea? Let me hear it. And I swear if anybody says do the same thing except put a picture of a bridge in the background, I'm going to throw up.

Here are the files for the trapezoid investigation, and also a similar lesson for kites.

¹With reciprocal trig functions and logs to any base, features which will be disabled in Test Mode for state exams. Deity help us.

We Have a Winner

2011-02-02T13:17:00.000-05:00

Alex, on Files for Riley's Intro to Trig, asking the question backwards and displaying the kind of lesson-design genius that I wish came easily to me:

Challenge the class to figure out sin57 with just a ruler and protractor - no calculator. Hopefully some bright spark will put the equipment together with last night's homework, and draw a right-angled triangle with a diagonal of 1.

So, same challenge - cos23. This time, draw the triangle yourself, putting the 'angle' in the same place.

Finally, draw the unit circle in. Pick a third point on the circle, in the same quadrant. Draw an 'x' there.

"What's special about this point?"

Hopefully someone (perhaps someone who's done the homework) will now tell you that you should draw the triangle, and that the two straight sides will give you sin and cos of the angle.

Files for Riley's Intro to Trig

2011-02-01T14:54:00.000-05:00

I assigned this to be done outside class today, since tomorrow's lesson is "Trig on the Unit Circle." You should go read it because otherwise I'd just be copying and pasting the whole thing.

I'm a little flail-y, still, about where to take it tomorrow. I have never had success getting the cherubs to see the connection between the coordinates on the unit circle and sine and cosine. Or when I have, it's been fleeting. I was thinking we'd discuss the patterns and shortcuts, and then I'd pose "Let's come up with a way to find precise coordinates for 57.4 degrees without having to draw it." But I'd love to hear better ideas.

Anyway, I thought it would be helpful to make the files I'm using available. They are here. One's a GSP and the other is an Excel.

An Omen of Sorts

2011-01-20T19:29:00.000-05:00

Me to Kid 1: "I had a terrible nightmare last night, and you were in it."

Kid 1: "Really? What was it about?"

Me: "I was giving a bunch of you a ride home and we got into a horrible accident!"

Kid 2: "You know what it means when a math teacher has a dream about you, right?"

Us: "????"

Kid 2: "You're going to give birth to the antichrist."

Megan Golding is My Hero

2011-01-16T14:13:00.000-05:00

Not just for making this crazy-fun project overlaying triangle concurrencies on a school map so that they could be clues for a treasure hunt, but blogging it up so other teachers can hit the ground running. Fantastic use of sketchpad, great game mechanics - she and her colleagues really thought of everything and knocked it out of the park. This is why teachers blog, so the next time someone asks, send them there. :-)

e is a Slippery Little Devil

2011-01-05T15:25:00.008-05:00

Can someone please explain this to me in a few sentences that make sense, without calculus?

If we start with P, increase it at a rate of r, compounded n times per time period for t time periods, we end up with

$P(1+\frac{r}{n})^{nt}$

If we compound it more and more frequently to the point where we're compounding it all the time, we're basically doing... (I know this is not rigorous. I'm sorry. I'm trying to make this graspable to any old 16 year old. Not just the future engineers etc.)

$P(1+\frac{r}{\infty})^{\infty t}$

And since

$e=(1+\frac{1}{\infty})^\infty$

(Again, I realize this would make a real mathematician bleed from the eyeballs. Sorry sorry sorry.)

That means we can replace part of our percent change equation with e, and calculate continuous growth with

$Pe^{rt}$

To recap:

My question is, why is r in the exponent now? I don't get that.

Log Laws

2010-12-21T13:06:00.002-05:00

Hate em. Can't teach em. Kids are confused by em. Kids never, ever remember em when they need em.

This year, like last year, we took the definition of a logarithm and knocked it out of the park. We were all feeling good after Day 1. "This is pretty easy." "Why does everyone say logs are such a big deal." Etc, etc. Right on, kids. Right on.

Then on Day 2, boom, the fit hits the shan. We left class looking dazed, bewildered, scared. Well this is my third year teaching Algebra 2 and I decided that This Would Not Stand. (I guess I only have one good, new lesson per unit in me every year. But like ten years from now, LOOK OUT.) So we re-did Day 2, differently, and even though losing a day in Algebra 2 gives me an ulcer, it was worth it.

What was I doing that was so awful? This:

Barfity barf barf. It's like I forgot how to teach math. I think I thought this topic was sort of hopeless and useless so I gave up on it for a little bit.

This is what we did that was better. Go get it here if you like. (Note: It uses the mathtype plugin for Word.)

Review and Practice: Add Em Up

2010-12-14T12:48:00.000-05:00

This worked really nice as a practice activity today, by my criteria of : the kids talk to each other, have ways to figure out if they're correct, and have ways to find their mistakes if they're not. I like when I can spend my time helping kids who need it and asking and answering meaningful questions, and don't have to hear "Is this right?" over and over.

The students got in groups of four, and each group got a total of 16 problems. Four pieces of paper with four problems each. The papers were different colors.

The students completed one problem on each page. So they all worked on one, rotated papers, worked the next one, etc. These guys even coordinated their calculators.

The paper color corresponded to the difficulty of the problems, which I let them know.

When all four problems were complete on a page, they added up their four answers. I posted the sums on colored index cards.

So if they check their sum and it's correct, great. But the best dialog started if it wasn't correct. Because first, they had to figure out which out of the four answers was wrong. And is it smart to start over and re-do the problem? Not in this case, since they were solving equations. It made much more sense to plug the answers in and see which one didn't work. Then they could start error-checking their work, which is great practice in itself.

The topic was solving exponential equations by changing the base, though this could work for anything. The document with the problems is available here. Enjoy!

Special Right Triangles

2010-11-04T21:17:00.002-04:00

You know the ones...

Convenient, nice to know going into trig, time-saving...especially for the SATs and GREs. It's a little hard to believe just how much the College Board hearts it some special right triangles.

But still, hard to motivate.

Enter...the humble dollar bill. It's a rectangle. What sort of rectangle? Not all rectangles are the same shape, of course. Some are squares! Their sides are in a 1:1 ratio. Some are square-like. Some are long and skinny. A long, skinny one's sides might be more like 10:1 or even more severe.

What about our paper currency in the US? All denominations are the same size and shape. This is not true in every country. But ours are all this very familiar rectangle.

What ratio do you think its sides are in? If you use two bills to measure, to see how many short sides make up the long side, you can see that it's skinnier than 2:1 (If it were 2:1, it would make a square when you folded it in half, and it doesn't. Credit for that observation: Rachel B., class of '13), but not quite 5:2 (or 2.5:1, if you prefer). (Cue annoying kid shouting "TWO AND A THIRD! IT'S TWO AND A THIRD" employing a technique commonly known as "proof by intimidation.")

Hm.

Now we do a little folding. I handed out photocopies of dollar bills. They were black and white, and one-sided, so I don't think it was a felony.

It's an isosceles trapezoid!

It's a rhombus!

It's an equilateral triangle!

At this point, you can also gently unfold it and coax it into a tetrahedron.

Unfold it all the way, and behold...

Don't see it? Look at the creases.

What does this mean? In my classroom it means I can launch the derivation of side relationships in 30-60-90 triangle with a tiny amount of Davinci-code-type intrigue which is totally worth navigating the murky waters of incommensurate lengths. But in a nutshell, it means our familiar dollar bill is made of a rectangle in a supremely weird ratio of $4:\sqrt{3}$

and come on, that was no accident. I bet the Freemasons were involved.

The People's Blog

2010-10-20T06:15:00.001-04:00

Teachers : Help me out here. Jessica can benefit so much more from the collective than from just me.

Kate,

I am a follower of your blog and others who use SBG in their classrooms. This year I am implementing it with my Algebra classes this year for the first time with the hopes that next year I will be able to do it with my Pre-Algebra classes as well. I teach 8th grade in a middle school.

I love SBG. It is working very well, minus a few small issues that I am trying to address. Since I do not have a blog of my own (school district policy), I am resorted to emailing those that use SBG to find a solution to my issues. I hope that you are able to help, even though you are busy being a teacher yourself. If you are not, feel free to pass this along to anyone you think could provide me with some assistance.

Being a math teacher, I love lists:

1. Any suggestions for how to make creating reassessments easier? Even though I do not currently use it to help me, I have Examview available to me, but it is only organized by Chapter in the book, not skill and in order to find the skill I need I have to view ALL problems available for the chapter and weed out what isn't necessary. I haven't been able to find an easier way, which is why I don't use it. So then I am finding that my reassessments are mostly hand written and given to the students when they come in. I want a better way to organize my problems into skill electronically to use from year to year and I can't figure it out.

2. Since I teach middle school, specifically 8th grade, I found that by only keeping track of the highest score (in my hard copy and electronic gradebook), I have NO idea how many times a student has reassessed. Which causes three problems, one I don't know what specific reassessements per skill they have taken so they don't receive repeats, two I have no way to gauge their continual struggles if I don't know how many times they have taken a reassessment, which also prevents me from sharing that with a parent. With that information I feel I could have a great discussion with the student and parent about their comprehension of said skill. And three, I am finding that my students are just coming in to reassess over and over and not coming in to receive help BEFORE reassessing. I want to find a way to almost force them to come in for help, for example, before their fourth reassessment of a skill, they have to come in for additional help in that skill first. But then I have to keep track of that information somehow and I don't have enough spots in my gradebook. :)

3. Actual grading, I use the 5, 4, 3.5, 3, 2, 1, 0 scale with the 5 being two 4's. I sometimes struggle with what would receive a 3 or a 2. I only use 3.5 if a smaller mathematical and non-conceptual error is made, but the difference between a 3 and a 2 seems to be my gray area...I don't know how to decifer between the two.

Okay, I think that is it. Thank you for reading and if you can't address my questions, please forward them to someone you think could help!

How about 987 someones I think could help? Comment away.

Counterexamples in Geometry

2010-10-18T20:41:00.049-04:00

Yesterday on Twitter I asked for false Geometry statements for which it's easy to draw a counterexample. Twitter is brilliant for this - everybody can come up with a-couple-a-three no problem, but it would be a pain to sit and think of a dozen. And even when you did, they might not be the best dozen for your purposes. After waiting a day, I got to pick from lots.

I'm consciously trying to do a better job of motivating proof this year. Many times this will take the form of asking the students to provide a conjecture before we try and prove anything. For example, I have this planned for next week: Start with an isosceles triangle. Make an exterior angle at the vertex. Bisect the exterior angle. What appears to be true? Now prove it. I'm looking forward to using the Nspire for this purpose - they'll quickly be able to look dynamically at tons of examples and bring the inductive reasoning to bear towards a conjecture.

Anyway, I'm getting ahead of myself. On Wednesday we'll be learning about the Triangle Sum Theorem, which they already "know." But a proof is very accessible and uses what we just learned about angles made by parallel lines and a transversal. So I was challenged with how to motivate proving something they already "know."

Hence, my need for false statements.

True or false? If false, draw a counterexample.

1. All right triangles are isosceles.

2. All rectangles are similar figures.

3. All pentagons are regular polygons.

4. Altitudes are always inside a triangle.

5. All quadrilaterals with four congruent sides are squares.

6. For any two lines cut by a transversal, corresponding angles are congruent.

7. All quadrilaterials have congruent diagonals.

8. Diagonals of a quadrilateral always intersect.

9. The three angles in the interior of a triangle sum to 180 degrees.

10. The acute angles in a right triangle are always complementary.

Kids draw counterexamples. Miss Nowak sticks them up on the document camera. Everyone gets in counterexample drawing mode, remembering that vocabulary, thinking, drawing, etc. Then they get to #9.

"But wait, the angles in a triangle do add up to 180."

"How do you know? Did you draw every possible triangle? And measure their angles? And add them up?"

"No. We learned it in eighth grade."

"But how do you KNOW?"

"My teacher told me."

"Okay, I believe you, but that's not math. That's like maybe some weird sort of religion."

I LOVE THIS. I can't wait for Wednesday. I'm going to leap out of bed and run to school to teach it.

Here are other false statements offered by Twitterzens. There are some great ones here but I had to pick an appropriate mix for my crowd. (Also, if you're wondering who actually answers the questions you throw out into the Twitter-void, follow these rock stars.) Enjoy!

@Stelladuma
angle 1 is supplementary to angle 2. Angle 2 is supplementary to angle 3. Therefore, angle 1 is supp to angle 3.

@untilnextstop
all equilateral triangles are congruent to each other.

if it has two pairs of congruent sides, it's a rectangle.

3 coplanar points/lines always form a triangle.

all isosceles right triangles are congruent.

all coplanar points are collinear.

@j_lanier

The perimeter of a rectangle is larger than its area.

An altitude of a triangle is also a median.

The center of a triangle's circumcircle is inside of the triangle.

Pairs of the following kinds are similar: isosceles triangles, scalene triangles, rhombi, isotraps, equiangular hexagons.

If two circles are the same size, then lines tangent to both of them are parallel.

If two circles intersect, then their two common tangents are parallel.

@mathhombre

All trapezoids have 2 congruent sides.

All triangles have a line of symmetry.

Quadrilaterals can't have two obtuse angles.

@misscalcul8

A circle's circumference is equal to its area.

Every rectangle is a square.

Every scalene triangle contains a right angle.

@AmberDCaldwell

2 complementary angles are adjacent.

Adjacent angles are supplementary.

@Stelladuma
if 3 angles of one triangle are congruent to the corr angles of another triangle, then those two triangles are congruent.

if diagonals of a quadrilateral are perpendicular, then the quad is a kite.

@Mrs_Fuller
All similar shapes are congruent.

All corresponding angles formed by a line transversing 2 other lines are congruent

@fpumathguy
the sum of the interior angles of a triangle is 180 degrees (see earlier post about non-Euclidean space)

All squares are kites.

@daveinstpaul
Two lines that are perpendicular to a third line are parallel (not true in 3D.)

The diagonals of a quadrilateral always intersect.

@PamLPatterson
Diagonals of parallelograms are also angle bisectors.

@restonkid
corresponding angles for two lines cut by a transversal are congruent

@calcdave
if a triangle does not have all equal angles, it is not an isosceles triangle.

No squares have greater area than circles.

@Mrs_LHenry
All rectangles are squares.

All equilateral triangles are right (or obtuse.)

@thescamdog
Any figure with four equal sides is a square.

All pentagons are concave.

All parallelograms are rectangles.

Circumcenters

2010-10-05T09:58:00.002-04:00

A Whole New Kind of Number

2010-09-30T15:12:00.000-04:00

I think I've finally got my introduction to imaginary numbers and the complex plane to a point where, let's say, the students can make room in their brain for the idea. They still don't like it, but they leave with some sense of what the heck we mean when we say i.

I start by asking them to place some real numbers on a number line.

Then I ask them to think about the lengths of sides of different squares. We try several fractions and terminating decimals to try to find one that we can square and get a 2, and we are unable to find one.

So that's why people needed to invent irrational numbers: to solve this problem. We just define radical 2 to be the number that gets you a 2 when you multiply it by itself.

Then we read through this story. I have them read the slides popcorn style (reader of this slide chooses reader of the next slide.)

They enjoy the story, except sometimes they make comments about how John and Betty are freakishly precocious, and sometimes they wonder what is up with Betty's hair. We don't read the whole thing. Just up to where they have to invent i.

The story gets them up to: i is the number people had to invent because there aren't any real numbers we can square and get -1. And if i² = -1, it stands to reason that we can define i as the square root of -1.

This is the best, most grabby part of the lesson: I put the number line back up, and say

So if i is a number...where do we put it?

Stop and wait and let the room be silent for a little while. They're considering things, and deciding against them. They sometimes suggest putting it at both 1 and -1, but of course they don't really know. So I say:

i isn't on the line. But it is on the board.

Then I carefully measure with the thumb and finger of one hand the distance between 0 and 1, turn my hand, and put i the same distance above 0. Then they can tell me where 2i and -i are located, and they can pretty much figure out where we should put complex numbers like 3+2i and -1 - 3i.

This lesson goes on to consider what we might mean by things like 5i + 6i, 2(4i), and 25i/5. Having the graph to refer to really helps. It sets us up nicely for powers of i tomorrow, too.

With all three groups today, there was a moment of "ick." "I don't like this." "This is weird." I tried hard to acknowledge and legitimize that feeling. I told them that feeling of discomfort is normal when you're making room in your mind for a brand new idea. I likened it to that saying "Pain is weakness leaving the body."

Except I said that weird feeling is ignorance leaving the brain. They seemed to like that.

Radical Comes from Radix Which Means Root in Latin*

2010-09-20T15:29:00.001-04:00

Factoid from the title care of Justin Lanier (which, Justin factoid, is pronounced the opposite of "La-Far.") It is unclear if radix has anything to do with radishes. Radishes are root vegetables, right? I'm thinking yeah.

So anyway I thought I'd throw this out there and see if anyone's got anything better. My irrational and complex numbers unit is pretty anemic. The way we have it calendared, we also kind of have to race through it. This intro lesson is my effort to give them something to grab onto. I'm open to suggestions.

The first three are new. The last two are what I used last year (I tried it as a puzzle - see if you can figure out the patterns and determine the missing values - some of them really liked it, and some of them sat and freaked out for 20 minutes because they couldn't find the cube root function in their calculator.) I think I'm going to give them all of it this year, and instruction will be a combo of encouraging them to think/play/struggle and good old d.i.

*I said Greek first. Oops.

The NSpire Lowdown

2010-08-20T15:07:00.004-04:00

I just completed a 3-day TI-Nspire training that my school contracted TI to come in and do. I haven't tried to teach with it yet, but of course, I have opinions. Let me say right off I don't necessarily think this technology is the right solution for every school, and I'm not trying to sell it to you. I'm just going to talk about what I like and don't like about it, because I know some people are curious, because they've been asking me - hopefully this will be a useful source for people making their own decisions.

Implementation: My school started phasing in the TI-Nspire with last year's freshman class. New students will now be expected to purchase a TI-Nspire handheld ($150 retail) as opposed to a TI-84 ($139 retail). For families who aren't able to purchase their own, we check them out a calculator for the year like a textbook. There are two versions of Nspire: a CAS and a numerical. The students get the numerical. We have a few classroom sets of CAS to share, too. It has a non-qwerty keypad - the keys are in alpha order - this is annoying but makes it more standardized test-compliant.

The Enthusing

Functionally/mathematically, as far as I can tell, the technology doesn't do any more for you than say Wolfram Alpha plus Geogebra. It is however very sleek and every bit as good.

It has a learning curve, but no worse than Geogebra or Sketchpad. If you already know either of those applications, you can pretty much hit the ground running.

If you go to one of these trainings (also may require a good relationship with your regional trainer-rep-salesman guy), they bring you lots of swag.

As part of the workshop fee my school paid, I received: a TI-Nspire CAS handheld, a canvas case with extra pockets (for cables, I guess? or maybe candy), a copy of the teacher edition software (emulator and connecting to your class functions), a Belkin hub and four cables for linking up and sending documents out to student handhelds, and a sweet t-shirt.

This is what I see as the big trade-off: Because of the proprietary nature of the software, we're not going to experience the wide-open, awesome universal sharing and embedding we get with Geogebra. This is crappy. However, the trade-off is access - every student should have their own handheld - in their hands in class every day, in their backpack, at home with them at night. I can't say the same for Geogebra. We're not a 1:1 school and won't be any time soon (for many reasons, and cost is only one of them). Getting computers into kids' hands in class every day is rather a hassle at my school. And not every kid has a computer at home (despite my history of getting snippy about it).

The Complaining

TI releases new software versions and hardware updates at a pace that makes Apple look downright sensitive to early adopters. This coming year in Geometry I am going to have kids with both generations of keypads, and it's going to be a nightmare.

The trainers will use TI-Navigator, which is a wireless document transfer/communication between your computer and all the student handhelds. It's seriously impressive. You feel like you're on Star Wars. Or Star Trek. Or some other show with "star" in the title. You probably don't have Navigator. It's $2500.

The instructional materials they make available, while plentiful, well-organized, and technically very nicely designed, are a pedagogical train wreck. Several times during the training, I heard about how a particular investigation + pre-made file required very little thought or work on the part of the teacher or student! As if this were desirable. Mostly, they have the student opening a pre-made file and dragging a point or slider around and looking for what changes or stays the same, which: 1. yawn, and 2. how is that supposed to stick in a brain?

Which begs the question of how to best exploit its capabilities for learning some math better than you can without it. Which I feel is an open question wrt classroom tech in general.

Off the bat I see three really nice uses.

1. Getting a handle on a problem and its constraints. Here is a sweet little problem I just swiped from What's Math Got to Do with It?, because it happened to be laying around.

If this were not just a screenshot, you could drag the skateboarder, represented by the red point, around the merry-go-round and get a good feel for what the problem is stating and asking.

2. Checking a solution by measuring. Ideally I know we love to pose problems where we can see how close we got by, like, playing the end of the video. But until that library reaches critical mass, we have:

3. Dynamic Notes : A document with a page for each key concept in an instructional unit, grabbable and playwithable.

No doubt people have questions... see you in the comments.

Put Your Money Where Your Math-Loving Mouth Is

2010-08-05T17:27:00.002-04:00

In my humble estimation, this project is worth your support. You can send as little as $1! But you get some cool stuff if you can send a little more. If you're a fan of the Math for Primates podcast, as I am, it's the same Tom! Check it out! I am particularly looking forward to the six weeks of having my mind bent. More than it is already, natch.

Who else is sensing a theme here?

2010-07-21T07:51:00.000-04:00

Exhibit A : I don't know the answers, but it turns out the experts in the field don't either. Not because they haven't tried, but because it's that complicated and messy.

Exhibit B : Just don’t make this about some magic set of rules that are going to make your classroom perfect. Guess what? That will never happen. Stop looking. Education is always going to be ugly.

Exhibit C : Let it be clear that there is nothing magical that I am doing. There is no algorithm. I don’t woo them in with some charm and they are all of a sudden amazing students.

Union

2010-07-02T12:29:00.000-04:00

This summer I am very purposefully avoiding thinking about work. A break seems crucial and necessary. I can't really explain why. I can't think about work. I don't want to think about work. My brain and body rebel at the notion of thinking about work. When people on Twitter start talking about the minutiae of their grading systems, I have to close the laptop and leave the room. This is the first time I've begun summer feeling like this. Maybe I will understand it better when I come out the other side.

One of the things I've been doing instead is attending Bikram yoga practice. The executive summary of Bikram is: very hot room, an hour and a half, same difficult 26 poses in the same order every time, lots of sweat. I've gone every day. I am shooting for thirty days in a row. My everything hurts.

But as you've probably guessed, despite my determination to think about other things for six weeks, aphorisms from Bikram apply without editing to a teaching practice. This is just me writing some of them down.

Everything matters. At yoga, which direction your fingers are pointing and where your eyes are focused matter. At school, where you stand, how long you pause, and the numbers you choose for every problem matter.

...but don't be too serious about it. Wink at yourself in the mirror.

Many teachers is better than one teacher. At yoga, I haven't had the same instructor twice. They lead you through the same poses, but the individuals are all different and equally awesome. This one told me to point my tailbone at the floor so that I really felt my spine lengthen, that one told me to press my chin into my chest. At school you can and should engage all the students in helping teach the course. This goes to deep, philosophical methods by which you approach instruction with collaborative problem solving, and the surface of how you structure practice activities.

Push yourself, try your best, and aim for perfection. At yoga, you can move a half inch deeper into the pose on the next breath. You can inhale another sip of air when your lungs are full up. At school, you can pick one student in each class you haven't talked to this week and ask them about their sport/hobby/pet.

...but be gentle and forgiving, and kinder to yourself than you think you deserve.

...and then let it go. Did you fall out of standing bow? Twice? It's over now. Let it go. One of the instructors says this and it's awesome: "Exhale...set you free." Did something go down at school you could have handled better? Acknowledge, learn, let it go.

If you are doing something mentally and physically demanding, don't forget to eat and drink enough water. Or you will feel like crap. At school, sometimes I am feeling cranky in the late afternoon and realize I haven't had any water all day. The consequences are a bit more extreme at Bikram: dizziness, nausea, feeling faint.

Finally, I took a picture of their poster, which might make a cute WCYDWT. What are they trying to maximize or promote with this pricing scheme? You'd probably want to hide that bottom part at first.

Absolute Value Both Rigorous and in Context

2010-06-24T13:09:00.004-04:00

I know I said I was done for the year. SORRY. I am literally sitting around school twiddling my thumbs today. I am ripping this idea off from Dan, but trying to extend it to be appropriate for Algebra 2. Absolute value is one of the first lessons of the year, and in the past my students neither understand it conceptually nor remember an algorithm for solving equations and inequalities with anything like reliability. This feels more like an Algebra 1 lesson to me, but I think it will be necessary.

This is my version... peanut M&Ms were the cheapest/most voluminous things I could find. There are about 230 in a large bag, by the way. Yesterday I polled 50 faculty and staff. In the fall I am going to have to get my butt into overdrive within a day or two to collect at least as many data from students.

I have yet to nail down the details, but the flow will go something like this:

Preliminaries
Put up a picture like this.

Ask how far away the houses are from school. Get a few volunteers to describe the mental procedure they used to determine distance from school. Point out that everyone naturally used a difference and absolute value to express distance. And that further, if we can represent distance as absolute value with an equation, we will be able to use it to ask and answer more interesting and difficult problems than our intuition can handle alone. Graph by hand y = |x| by making a table of values. Note the characteristic V shape.

Questions to Answer
Bust out laptops and distribute excel file. As per Dan's original plan, kids will have some choices about what questions to explore and time to flail.

- Who won?
- Rank everybody.
- Top 10 Guessers.
- Any ties?
- Worst guesser?
- Which grade guessed best?
- Which job guessed best?
- Calculate percent error.
(Maybe some/all kids can present aspects of the results on posters we can display?)

Once that's all squared away, I want everyone to explore:
- On average, how good were the guesses?
- Create the scatterplot that displays the characteristic V shape.
- What is the equation of the connected graph of that plot and what do the variables represent?

(This popping up on my screen should not have been, but was, the best part of my day yesterday:)

Follow-on problems once equation is achieved. Solutions using both the graph and the equation.

- What guess corresponds to the average distance from the correct guess?
- What did the worst guesser guess? The best?
- In what range did the better-than-average guessers guess?
- In what range did the worse-than-average guessers guess?

New problems and generalization:
Write an equation/inequality that models the scenario. Make sure to define your variables.
- Today’s temperature is 10 degrees off from the usual temperature.
- Today’s temperature will be within 10 degrees of the usual temperature.
- Today’s temperature will be more than 10 degrees off from the usual temperature.
- If the usual temperature is 68, find values for the three forecasts above using algebra. Show all work at every step.
- Graph the scenario. Indicate the three different forecasts on the graph.

- Write a general expression for the distance between a changing value and a known value. Define your variables.

- Put this equation into words: |x – 10| = 3
- Solve it, showing all work at every step.
- Write down/discuss a procedure for solving any absolute value equation.

- Put this inequality into words: |x – 10| < 3
- Solve it, showing all work at every step.
- Put this inequality into words: |x – 10| > 3
- Solve it, showing all work at every step.
- Write down/discuss a procedure for solving any absolute value inequality.

Feel free to poke holes in this or let me know how you would implement it differently. Also I need to get them solving and graphing more complicated equations and inequalities like say 10 = 2 |3x - 4| + 7, so I'd love to hear if you see any natural ways to make that happen. I haven't been able to think of any yet.

Where Mah Physics Peeps At

2010-06-16T18:09:00.001-04:00

I usually talk about vector forces by pushing desks around. You know like 2 people push in the same direction, 2 people push in opposite directions, then they push on adjacent sides and the desk moves diagonally. But then we just make up forces and do practice problems.

Would it work to get 3 bathroom scales to measure the two component forces and the resultant force at the same time? Is there an easier way to do it than trying to balance a scale against a corner of a desk?

I would ask the Physics teachers at my school but I'd have to walk all the way down two hallways and I am very, very lazy.

I Kind of Hate the Stupidly Ubiquitous Video Cameras

2010-06-10T17:10:00.000-04:00

So in the last ten minutes of fifth period, awards had been given out, instructions about what to bring and not bring to the Regents exam had been given, and D plugged his ipod into my computer speakers to entertain us...fine, cute. He started dancing, so did another student, they were very talented, well-practiced, adorable, etc. So then D asks if I want to learn The Stanky Leg and I'm like "Sure! I hope I don't hurt myself! I am comically uncoordinated!" He starts trying to show me, and I start trying to imitate him, and I'm sure it was hilariously awful. But after like a minute, I look up and there are at least three cameras pointed at us. And I stopped. I couldn't make myself continue. I don't know how to feel about that. It would have been fun to continue, and I was fine embarrassing myself in front of these 20 people I've spent so much time with this year, but I wasn't fine embarrassing myself in front of the universe.

The Envelope Please

2010-06-08T20:51:00.001-04:00

Riley thinks we should inject a little ceremony and gravitas into the last day of school. And I agree! He was lamenting the departure of his kids before he was able to properly see them off, but "luckily", my last day of classes is Thursday. I thought it would be cute to think of an award for everyone.

Awards I might give out if I can think of a few more:

The Heat Seeking Missile - for the Best Pattern-Noticer
The Bulldog - for the Most Tenacious Problem Solver
The Honorary TA - for the most prolific/effective peer tutors
The Librarian - "Sshhh! You guys! I'm trying to LEARN!"
The Scion of Pythagoras - for the most beautiful compass and straight-edge constructions
The Helpdesk - for the angels I can put on tech support duty on computer days
The Up Up Down Down Left Right Left Right B A - for the nerdiest t-shirts

I am still working on it. I have two whole days. I'll get there. But I had lots of other ideas.

Awards it would be impolitic to give but amuse me nonetheless:

The Your Tutor Gets an A+ Award - for the highest discrepancy between grades on in-class and take-home assignments
The Draco Malfoy - for s/he with the parent that made the most threats
This is Not the Phone You're Looking For - for the stealthiest texter
The Deanna Troi - for the Cleverest Hans
You Make Me Die Inside A Little - for the kid who tells me things like 0 is a solution to (x+3)/2x = (x-2)/x because undefined = undefined.
Are You Taking a Class Called 'Advanced Field Trip'? - for the most excused absences
The I Love You Man - For the girl who writes something like MISSNOWEEZIE IS THE BOMBDIGGITY on the whiteboard every freaking day.

The Personal Invitation

2010-06-07T17:54:00.001-04:00

I have a trick for recruiting students for voluntary activities. For example, an enriching day of mathematics, or to contribute writing for community outreach, or to mentor some freshmen. And it's not to make announcements to whole classes to say, come talk to me if you are interested. That doesn't work.

I know people might object to this, because maybe it seems unfair, like opportunities are being limited. (Even though, as I said, the open invitation never works anyway.) But I think of which students would be good candidates. Who has appropriate talents and who will benefit. Then I ask them to take a lap with me around the building (the corridors make a giant rectangle), and I explain what I want them to do, why I think they're the right person to do it, how it will benefit them. And I ask if they would be interested.

It always works. Nobody has ever said no. They usually say something like "I am totally into that." And, they follow through. They jump headlong into these projects with enthusiasm and grace.

It's powerful, the personal invitation. To know your teacher sees something special in you, despite your maybe not being an academic superstar, despite whatever flaws your fears tell you are evident. It's hard to talk back to that.

But, I've been thinking, wow. I need to invite them to learn some math. Frequently. Not as a group - charismatic lecturing is not my forte. Not necessarily every kid every lesson every day. In a way that appeals to their individual talents. Because once it's given a chance, this stuff is startling, beautiful, descriptive. Once they know they bring something to it, and it can benefit them. I have no idea how to pull it off. But I need to find a way.

Do Not Be Discouraged

2010-05-31T11:36:00.001-04:00

f(t) didn't have many readers for a long time, either.

What happened in March 2009? I think it was Math Teachers at Play #3 and Logarithm War.

Love, Your Fairy Blogmother

2010-05-30T21:27:00.007-04:00

[update: Sam Shah disagrees with half of what I say here, but in very compelling ways. Highly recommended reading there, too.] [Also, Elissa has awesome and high-larious wisdom on the matter.]

So you want to start a blog.

What you should not do: Email me. I mean, I don't mind, but I'm just going to be generically encouraging. I've been getting deluged lately, so this post is a public service. "Start a blog" must be a popular summer project for teachers this year. I'm sure commenters will add many helpful suggestions I didn't think of, so be sure to check them out, too. Also note: I violate these pointers all the time! They're just suggestions.

What you should do before you start:

Start reading. Set up Google Reader, subscribe to every blog you can find that's like the one you want to start. Check your reader at least once a week. You don't have to read every word. Read the posts that interest you.
Start commenting. Don't just use your first name if it's common. Especially if it's "Dan" "Dave" or "Matt." Use your last name or make up a hilarious or distinctive nom de plume. Also, don't just comment to comment. Say intelligent things. Add to the conversation. Point to good resources or tell a helpful anecdote.
Pick a title that stands out, that people will see the title and think, "Oh yeah. That guy." Don't choose a title with "Math" in it. Don't choose a title that only differs by one word or letter than someone else's blog. It might take you a while to think of a good name. It's ok. Take your time. You're going to be stuck with it.
Write a tagline that concisely describes your purpose.
Optional: Get on Twitter. Contribute to the conversation there.

When you start writing:

Add your link to the comments you write, so people can follow it and get to your blog.
Some people publish several times a day, some a few times a week, some go months between posts. I don't think it really matters, as long as what you do publish is worth reading.
Give each post a title that makes the reader wonder what it's about and want to read on.
Don't be afraid to post about your failures as well as your successes.
Tell a story. Give it a beginning, middle, and end. Include an illustrative anecdote about how it went down in your class. Dialog helps, which you can totally make up if you need to.
Avoid posting fodder just for the sake of posting, such as : embedded videos without commentary that adds to its viewing, lists of links to other blogs, etc.
Stick to the topic. Don't badmouth anyone. Try not to complain too much.
Credit and link back to ideas you got from and references you make to other people's work.

Realizing I May Have Buried the Lede:

Be generous. This community is a gift culture - sharing is how reputations are built and respect is earned. If you have worked hard on a successful lesson, it's worth writing up. Share your presentation files, handouts, dynamic geometry sketches. There are many sites that allow you to share documents for free, and BetterLesson has a nice platform for this now.

Some suggestions for getting more readers:

Don't be too focused on hit counts or number of subscribers. Are you learning from your written reflections? Are you having worthwhile conversations? Isn't that why you're doing this?
The best way to get people to read you is to write original content worth reading.
Volunteer to curate and host a carnival like Math Teachers at Play.
Keep commenting elsewhere, and including the link back to you.

How to manage comments:

Acknowledge insightful comments and keep the conversation going for as long as it makes sense.
Respond to direct questions to you.
Learn to tell the difference between commenters who want to discuss legitimate differences in opinion and issues raised by your post, and the ones who just want to drag you into a pointless holy war. Engage the former.

I hope that helps. Good luck!

Drumroll, Please

2010-05-30T15:10:00.002-04:00

After careful consideration, my esteemed colleague Mr. Shah and I have declared a victor in the Binomial Expansion video contest: Jason Dyer! Woo! High fives! Crowd going wild!

Jason, in a moment of genius, chose a winner of a concrete basis for developing the structure of binomial expansion: the game Q-Bert. There's an outside chance he gamed the judging based on my non-secret, unabashed enthusiasm for all things with a high score and a controller, but I'm ok with that! Here was Sam's take:

Not only for the fact that students can use it to count out things, and really engage hands on with figuring out the numbers -- really get intimate with what's going on instead of just noticing patterns -- but there is something powerful about the physicality of it. I kept on imagining Qbert jumping, instead of seeing things in some abstract algebraic framework. It also, merrily, explains why the pattern we use to draw Pascal's triangle works (why the two numbers above it add to form the next). Jason too saw the power in it, because he kept on referring back to Qbert as the base structure on which he built his lesson, instead of saying "here's a cool thing" and then going into the world of abstraction never drawing the abstract back to the concrete.

All of the entries had their own strengths and deserve recognition, including:

Eric Buffington - Who produced a much better-looking, cleaner version of the type of lesson I presented in the original post.

John Scammell - I thought the mathemagic angle was effective for this topic, and well-presented to a live class in his video - a breathtaking act of bravery for a teacher.

James Tanton - Whose approach shared the appealing physical basis of Jason's Q-Bert lesson.

Any of these would make an effective lesson, and made for a competitive field and a difficult decision. Congratulations to Jason, who will be receiving a brand-new Factorial! t-shirt, courtesy me, who decided not to be a cheapskate and send him a hand-me-down, and a lovely book, courtesy of Sam - I think he has his choice of a few.

This was fun! Should we do it again sometime? Maybe better-timed than the end of spring when everyone is super busy? What tough-nut-to-crack topics are begging for the mathedublogsphere treatment? Anyone else want to host the next one?

Get Your Hot Fresh SBG Checklists

2010-05-29T08:26:00.004-04:00

For the uninitiated, SBG stands for Basing the Grades On Making Sure the Children Get What They Are Supposed to Get. People who are smarter than me have already described it extensively.

All of these were only really tried one time for real because of curricular changes and my-schedule changes. I anticipate people will object to their length. I agree that in some cases, two concepts could be pared down to one. Either I need to do that for next year, or I had a reason. Some of it is just the reality of our curricula - overloaded. Sometime when I was training to be a teacher I was told, "A good teacher is one who knows what to cut." But, I don't know, I wrestle with cutting anything. I have an overdeveloped sense of duty.

The Lists

Algebra 1

Geometry

Algebra 2 with Trigonometry

More Details
I do about one quiz a week. Sometimes it takes the whole (43 minute) period but usually it doesn't. They see a question about each concept on two separate quizzes. I score them out of five. I give them one point even if they leave a question blank. It makes it easier for me to tell later if they were absent and didn't take the quiz (zeroes) or they just did really badly. So it's like

0 - Didn't take it

1 - Left it blank or wrote absolutely nothing redeeming

2 - Wrote something correct or in the right direction but is essentially clueless

3 - The cluebird has landed, but major conceptual error

4 - Minor conceptual or major computational error

4.5 - Minor computational error

5 - Knows what's up, no kidding.

Our grades are calculated by 1/5s: four marking periods and a regents exam. I tend to cut off remediation opportunities at the end of the marking period for all the concepts up to that point. It just makes sense for us.

The best modification this year was: require that if you are staying after school with me, you are either there to get help, or you are there to re-test. Never both. If you want my help, great, but you have to come back to re-test. Retesting is a no kidding, materials put away, sitting at a desk by yourself with a pencil and a calculator situation. It was a good change because: they are more likely to at least try to do some preparation on their own, and their grade is a better reflection of what they've learned as opposed to what they just stored in their short term memory.

Plans, Big Plans
Be more proactive about insisting students come for remediation. Like, the instant their average goes below 70%, assign them detention if need be. Most kids, once they come in once, they realize how much it helps their grade, and then they take it upon themselves.

Be more insistent that everyone have a place to keep their checklist and graded quizzes from the current marking period. How to do this, I don't know. The worst ones keep all their subjects in a huge spiral notebook, and stuff handouts in the insubstantial pockets in the dividers. What a terrible solution. Pretty soon they can't find their checklist, and they have quizzes everywhere. One step up, but still pretty bad, is a sturdy pocket folder. It's impossible to keep stuff in order. The best would be a binder but I think that's a pipe dream. Kids hate binders. They are that awkward triangle shape, and you can't quickly deal with paper that's not hole-punched. In Geometry this year, I tried individual file folders, kept in the classroom. Organization was better, but they didn't have their old quizzes with them for studying. Major flaw.

That's all for now... I will probably add more to this as I think of it.

Just How Long Is That Quarter Mile Track Anyway?

2010-05-27T12:31:00.000-04:00

This morning I got up early and came to school to run around the track. (I know, right? Nobody was chasing me or anything.) And I know it's a quarter-mile track, but of course the inside lane is much shorter than the outside lane. So I started wondering, what is a quarter mile? The inside, the outside, somewhere in the middle? I'm getting this down now so I remember to revisit it in Geometry next year when we do all those composite perimeters and areas.

Here is our track from Google Earth:

I'll ask the kids to decide what to measure.

If you try to measure in miles, the diameters of the semicircular ends only differ by a hundredth of a mile. So I measured in feet for more precision.

I'd love to give the kids a printout and have them do the measuring, but measuring in centimeters or inches and scaling it up might be more trouble than its worth. Maybe if I find a few more interesting questions like this, we can have a computer day and they can use the GE measuring tools. Times like these I envy you 1:1 schools.

For the outside track, rounding to the nearest foot I get 1444 feet or 0.273 miles and for the inside track, I get 1320 feet which is exactly 0.25 miles. Can anyone verify that? Track stars?

Haxx

2010-05-21T10:31:00.003-04:00

Straight-edge hack:

Paper-saving hack (Thanks to Sarah Cannon):

The document camera doesn't do video hack (Thanks to Maria Anderson):

Floppy compass hack (Or just order these. They are the bomb.):

End-of-May Motivation Hack: Make something cool

Teach Us Something

2010-05-15T11:37:00.000-04:00

I continue to be frustrated with kids afraid to try something because they're afraid of being wrong or admitting they are confused - you know, the nodders, the silent ones... They hate taking a risk in front of peers like a terrorist hates freedom. I know it's totally normal and human. I'm just frustrated by my "What do you think? What ideas do you have?" being met with "Can you check if it's right first? What if I'm wrong?" I often give a friendly "So what if you're wrong?" and then some crack about how I won't let M.I.A. take them away on a prison bus even though they have red hair.

Does anyone else do something like this in the first few days of school? Get in a group of 2-4 and tomorrow, your group is going to teach the rest of us something that you are good at. It just has to be something that we can do here in school, that we will be able to try a few times within about 10 minutes. If you need any special equipment, you'll need to bring it.

There are plenty of things that these kids like to do that I'm sure I would be comically awful at. And maybe it will show that it's safe to take a risk and/or admit that you're clueless - that it's really sort of essential if you want to learn from another person. But I also wonder if this is a terrible idea, because I want them to feel safe with me, and throwing them in front of the class within the first few days of school will freak them out.

Has anyone tried something like this? Did it work? Was it a waste of time? Was it awesome or horrible in unexpected ways? Please let me know.

Djibouti

2010-05-06T22:29:00.000-04:00

is the country with the most amusing name.

But The Booty, on The Challenge, has just gotten better! So if you're thinking maybe you got the goods that will bring the kids along for the ride, well, you know what I'm hoping you'll do. (Bring it.)

It's About That Time

2010-05-03T22:23:00.001-04:00

At lunch today (provided by our fantastically supportive STPA - it's such a nice thing they do for us every year) we were talking about how soon the year is going to end. It's only a few weeks until Memorial Day. After that, we just have two four-day weeks of classes, a week of exams, and then a week of sitting around and twiddling our thumbs because we already took all of our personal days. Ok that's maybe just me.

I am not quite sure yet about this year, but in other years I have used this Survivor-themed review game to help review for final exams. I was going to write up a whole big thing about it, but then I realized that the original description was pretty darn close to what I ended up doing. So I am just formally endorsing it. Check it out.

I made a few minor modifications:

I score each challenge out of 10, but there are only 5 points available on it. So their score is never lower than 50%. This prevents all these scores from really killing their quarter grade if they bomb a few. Or more than a few. Reviewing a whole year can be rough.
I can't exempt anyone from Regents exams, so I work "immunity" thusly (I've been watching a bunch of Alton Brown, can you tell?): The winning tribe for each challenge automatically gets a perfect score on the next challenge. However, they still take it and try hard on it, because they could always get immunity a second time!
To up the joy quotient, I gave some incentive for the tribes that made beautiful signage of their tribe name to hang above their work area.
One year I gave $5 Amazon gift cards to each winner of the winning tribes, and another year I had t-shirts made. The t-shirts were the far more popular prize. They wore them to the exam and took pictures. You want to see a bunch of regents kids act like math nerds? Give em a free t-shirt.
The first few days of the game, and more if you feel like it, you MUST blare the survivor theme song in your room as class starts, and you MUST wear a safari jacket and bush hat. NO EXCEPTIONS, buddy.

Negative Numbers are Hard

2010-05-01T22:02:00.001-04:00

Go read, especially if you teach math in grades 6-9, but really, go read it anyway. Ben Blum-Smith is performing a public service, reporting on primary sources you should read but are too-hard-a-slog and when-do-you-have-the-time and teasing out recommendations you can use in any classroom. The post is lengthy but well worth your time if you've ever watched a student struggle with negative numbers, and who hasn't.

I won't give away the punchline because Ben lays it out much more elegantly than I can summarize, but here's as good an analogy for a typical clumsy, contrived introduction of algebra as any:

The way I learned it was the opposite of “natural” or “meaningful.” Very cool, but only in an after-the-fact sort of a way. Like you came to the show at the very end and only saw the climactic scene, and everybody in the audience gasped and shrieked, except you because you didn’t care about any of the people in the show because you just walked in one second ago. And then after it was over your friend explained to you what had been going on and you understood why the other audience members cared, and were kind of mad you hadn’t gotten to watch the rest of the play first.

Why Teaching Probability Messes with My Mind

2010-04-21T21:04:00.002-04:00

It's hard to measure things that haven't happened yet. It's quite a challenge to keep front and center the question: Just how sure do we need to be right now?

Remember this?

Hurricane Ike made U.S. landfall in Galveston, Texas.

WCYDWT: Fiestaware

2010-04-21T19:32:00.008-04:00

I know darn well this isn't going to knock any adolescent socks off...and honestly it's barely a WCYDWT. But, it's the best I have so far, and maybe if I put it out there, someone has something better.

Traditional Lesson:

(Complete the example. Practice practice practice. Quiz. Quiz. Done until end-of-year review time.)

That formula, with the combination and the n's and p's and r's and q's, is like a bouillon cube. Very useful, but not intended to be consumed raw. (To rip off an Ellen Kaplan analogy.) As awful as that slide is, this is what is going on in classrooms across America, because the sobering volume and ill-conceived spiraling of content means we are rushed beyond reason. I know Algebra 2 teachers whose ulcers live in fear of a snow day for the loss of instruction time.

So here's my idea and how I intend to deploy it in class. I was indeed fortuitously inspired. I didn't think of this just because Bernoulli trials are coming up.

Please bear in mind this anticipated dialog is boiled down, like bouillon, and there will be much waiting, hemming, hawing, frowning, clarifying questions, etc etc. And, it will go differently in all three classes.

"When I go visit my mom, it's my job to set the table for dinner."

"I was visiting for five days last week, and on Thursday, this happened:"

"My mom said, 'Hey, did you do that on purpose?'

'No' I said. 'That's weird.'

Why do you think we were surprised?"

(The bowls and plates match.)

"Now, my mom thought this was a miracle. A once in a lifetime Fiestaware occurrence. So, what do you think is my question?"

(What are the chances you choose matching plates and bowls.)

Here, someone will assert that I subconsciously tried to make them match, skewing the chances of a matched set. I will acknowledge this but ask if we can, for the sake of learning, pretend I was choosing plates and bowls randomly.

"So, listen. I wasn't so sure this was so improbable, because remember, I set the table for five days."

(It would be more likely the more times you did it.)

"Right. So in order to convince my mom this wasn't so unlikely, what would I need to calculate?"

(The probability of choosing two matched sets one time in five days.)

Write this on the board.

"I'd like you to discuss possible approaches with your partner. You may not be able to solve the problem yet. That's ok. I'm going to give you four minutes to discuss. I want you to make progress. What would we need to know to make progress? Write it down. And if you can't, what questions do you need answered that would help you make progress? Write them down. Is there an easier problem, that you can solve? Write it down. Go."

I don't think I can script any more after this. It depends on what they come up with. They are likely to be able to, in order of increasing difficulty/decreasing likelihood:

calculate the probability of choosing one matched set
calculate the probability of choosing two matched sets, performing the experiment once
calculate the probability of choosing two matched sets on a particular day during the five days

After the four minutes, I would ask each pair what was their finding, or what is their question, and write all these on the board. And then I would go from there. With the intention of deriving the formula for Bernoulli trials that day or the next day, with copies of some practice problems at the ready to hand out, once we got this one.

Suggestions? Better ideas? I readily admit this question kind of sucks, because there's no intrinsic buy-in. Nobody really cares about the dishes at my mom's house. Or maybe it is the best I can do in my classroom, because I thought of it, so I can do it justice. What do you think?

Regression Project Debrief

2010-04-16T17:29:00.001-04:00

These are intended to be notes for me for next time. Though if they help you to deliver this experience better than I did, awesome.

Here is the project description. The students' work is published to the class blog.

Percent of students who were able to:

Transcribe the data into an accurate, useful table of values: 100%
Model the scatterplot with an appropriate regression equation: 77%
Correctly use the equation to make predictions about future values: 60%
Interpret the meaning of the variables in the regression equation (or would-be equation): 94%
Interpret the meaning of the constants in the regression equation in the context of effect on the graph: 31%
Interpret the meaning of the constants in the regression equation in the context of the data: 20%

Those last two kind of bug me. I understand that it was a difficult thing to ask them to do, but I can't decide how important it is. If they know what x and y represent, they can use the equation to make predictions and solve useful problems. Is it really important that they know that in y = (0.44)(1.1)^x, that the 1.1 means 10% growth? I don't have a good answer for that.

Changes for next time to the project description:

Write more explicit instructions for uploading images and making and editing blog posts.
Add a template for each problem that they can copy and paste.
Add more implicit instructions to try several types of regressions and compare r-values to find the best one.
Delete 'A Piece of Cake' or consider curmudgeon's comment below to just delete the 'cake' part and add a known data point that allows them to check their model.
Delete interpreting the equation constants from the quadratics problems.
Started out requiring four problems - one from each group - amended that to two problems. Four class periods were only enough time for them to reasonably complete two.
Make the easier tasks (with high percentage of success based on this year) worth most of the points, so you won't have to fudge the scoring such that everyone wouldn't fail.

Examples of Excellent Work:

Jarrod and Stephen: Sinusoidal Regression

Harry and Matt: Exponential Regression

And...I'm on goin' on break.

Regression Project Day n

2010-04-06T17:47:00.025-04:00

Day 3

Oy. So the file-save errors from yesterday were caused by running out of room on our network servers. Luckily I had a couple flash drives lying around we could save to. I learned that kids don't as a rule carry around flash drives.

A student said "This is too hard. You should teach honors, not regents." Apparently using actual data and not made-up crap = "honors."

There are a number of changes I need to make if I ever undertake this insanity again:

"Shiver in My Bones" needs to all be graphed on the same axes.

"Piece of Cake" just needs to be cut from this project because it is the one thing that is not like the other.

"Walking after Midnight" needs to be made at least as attractive as "Harder Faster Stronger" because it was not nearly as popular as it should have been. Also specify that it must be recorded in miles vs hours and not the other way round and not in seconds or...non-miles.

"Will the Ball Hit the Can" needs to de-emphasize the meaning of the coefficients and cut it down to 1 to 2 throws, and emphasize "figure out whether or not will the ball hit the can?"

I need to throughout the year ask more often what numbers in an equation mean. Like what the a and the b mean in y=a*b^x. They were universally, alarmingly, totally stumped by these questions.

(more to come soon. this post is a hydra of a post.)

Day 2

Goods:

Had a student teacher helping today who was really smart on tech so, wonderful.
More victories from refusing to tell them what to do.
Asking pointed, leading questions is just as bad as telling them what to do. I wasn't playing that game, either.

Others:

Google docs wouldn't let us insert an image. It was giving a weird, stupid error. I think it was a google problem and not an us problem.
As a result, we gave them the alternative of writing up their analysis as a Blogger post. Dear Google: people don't realize that Blogger is Google and they can use the same login. Just call it GBlogs already.
Blogger is not as good a solution for this as Gdocs was. No tables, no equations. And no I'm not teaching tenth graders html and latex in 2 days so don't bother.
The student accounts are not able to use TI-Connect to upload a screenshot with the USB cable, because they don't have administrator privileges. Super annoying because I was expecting this to work.
Yesterday we had a problem saving the TI-Smartview emulator state because of a "cannot save to that location" error. (I hate that. Why the hell not?) Super frustrating when they did all that work, and it wouldn't let them save. I was going to have kids who were able to save, just email their files to the kids who couldn't.
But, some of the kids who did save their emulator state weren't able to open them. Unreadable file something something.

Summary: The tech is very much getting in the way of the learning. Is this still worth doing? I think so. Real data imposes a priceless logic and inevitability that I will continue to walk over broken glass to get to.

Day 1

Kids get a laptop and instructions to head to class blog and start reading instructions. Many dire warnings about reading all instructions before starting, reading carefully, blah blah blah. Still get kids going "Can I be in Group A?" Not until you learn how to read, babe.
Only a few need to create or recover a Google account, and it is once again surprising to me how hard this is for them.
Some don't know how to make a new Google document (add instructions for that somewhere)
17% successfully invite me as an editor as instructed.
Pretty sure everyone has at least created the document to edit and started tabulating data.
Lots of complaints along the lines "You mean we have to use all this data?!" Only if you want a good model. And have fun in SUPA Stats next year because you thought it was going to be easier than Precalc.
Some IEP-ers have trouble with written instructions and breaking large tasks down into smaller ones. Sent lunchtime email to Resource teacher, pleading for help. There are a lot of instructions. I can't give them all verbally.

Group A: In Facebook World Domination, I am patting myself on the back for not telling them how to tabulate months. They kept asking for validation. I kept saying "Does that make sense on a number line? Are data that are 6 months apart going to be 6 apart?" etc etc. They all figured this out and got a little shot of whatever brain chemical you get when you do something intellectually satisfying and I'm glad I didn't rob them of that.

In Blink, lots of kids graphed Transistors vs Years instead of Years vs Transistors. Let it get to the scatter plot, when they realize they don't know a function that will fit that shape? Or nip it in the bud first thing tomorrow? They also used years like 79, 84, 93 but then when they got to 2002 they called it 02. Headsmack. I'm letting that one go until they get confused by their scatter plot.

Group B: Everybody, I think, who got that far, picked A Piece of Cake because it looked easier than measuring and timing the pendulum. They have been miserable at generating data for a Piece of Cake, though. Three cuts only gets you 6 pieces! What do you mean, maximum pieces?! Nobody cuts a cake like that!

Miscellaneous Notes:

Plug in all the computers during lunch or they will start dying during 7th period.
Scrounge as many TI link cables as you can before tomorrow.
This is suffering as much from the "Get it done while learning as little as possible" mentality as anything that ever happened in a school, even though the whole point was for it to be interesting. How in the world do I tackle that 900 pound banana, I have no idea. (Why don't I let them analyze whatever data is interesting to them? Because it was hard enough for me to find manageable, relatively easily-tabulated data that would fit a function appropriate to the course content. And they're supposed to think of something interesting and go find bivariate data about it? Puh-lease.)

Day 0 and less than 0
Collected ideas for fun data to regress. Wrote project description. Got some helpful Twitter feedback and made a few last minute changes.

Formulas? What Formulas?

2010-04-01T22:46:00.001-04:00

I banished the textbook slope, distance, and midpoint formulas from Geometry class this year. I feel like such a rebel.

First of all, kids have trouble with subscripts. They write them like exponents and then madness ensues. Second, that big long radical sign freaks them out. Third, if they learn and remember it conceptually, they won't try to memorize a formula they don't understand and dink it up later.

So here's what they did learn, and even though I'd like a more organic way for them to get there, the outcome was not bad at all.

Distance

Really Just the Pythagorean Theorem. Distance² = (subtract the x's)² + (subtract the y's)². (To be followed next week by Standard Form of a Circle: Really Just the Pythagorean Theorem.) This is awesome because they heart them some pythagorean theorem.

First I asked how far away are these places as the crow flies. (We had to discuss what "as the crow flies" means.) (Thanks Visalia, CA, for laying out your streets on a square grid.)

Then we practiced a little treating the segment whose length we wanted as the hypotenuse of a right triangle, drawing the right triangle, using PT:

Then...dun dun duuuuuun:

We go back to the ones we can graph, and figure out how we can get the legs of the right triangle from the coordinates:

Then we are basically home free:

Midpoint

(average the x's, average the y's)

Same basic plan. Practice a few times, throw one in that doesn't fit on the graph, backtrack.

Slope

Subtract the y's, subtract the x's, divide. Yep we played the song. (We sang the song. We claimed we wanted to write our own song about the distance formula and film it and put it on youtube. We are probably not that ambitious.)

Which part of the roof would you rather stand on in a flood?

How do we compare their steepnesses numerically? They've seen slope before.

Do the y's or the x's go on top? Well...which of these slopes do you think should come out to be the bigger number?

Put it together:

I was a little uneasy about not making them copy down and memorize the formulas as they appear in books...am I setting low expectations? Will they get to college and fail because they are unfamiliar with subscripts? But then we were settling in for a quiz this afternoon, and someone said all panic-stricken "What's the midpoint formula?!" And someone else said, "Calm down, you just average the x's and average the y's." And I think that's all right.

Greasing the Skids of Group Work

2010-03-30T16:45:00.002-04:00

I've pretty much had it up to *here* with, not all the kids, but the 10% that make my life annoying. I won't go into the gory details about why they are annoying. Nobody wants to hear that.

Out of frustration, though, sometimes good stuff pops out. Today in Algebra 2 the children were working in groups of four exploring transformations of functions (BL)*. This wasn't the kind of exercise with a group component grafted on to individual work - it was essential that they compare the four different graphs to draw meaningful conclusions. My classes had five or six groups each, and most of them were functioning brilliantly - helping each other troubleshoot, refining each other's wording - it was music. Really.

But in fourth period, I had a group that was sitting there, staring at their calculators in silence, waiting for, what, I don't know. My fed-up-ed-ness came over me and I towered over the group, intimidatingly, saying, "Do you know her name? No? Why don't you introduce yourself, like a normal person? Shake hands. Good. Now what's her name? What's his name?" I did this several times. Until the group was bonded in their feelings of persecution and shared opinion of my weirdness. After that, they started talking. The same thing happened again seventh period! I did it again. It worked.

The moral: sometimes it's my job to be an overbearing jerk. And sometimes I need to remind myself that the children are children.

*Credit for original lesson I shamelessly repurposed goes to my colleague, Dina.

Subversive Mathematics

2010-03-24T09:34:00.003-04:00

There have been two students coming to eat lunch in my room. They're both in my homeroom. Juniors now. Lanky. Maddening combination of wise and clueless. Don't know what to do with themselves half the time.

R is in the just-enough-to-graduate math classes. He is convinced math is a stupid and pointless torture device of adults who are out to make his life miserable.

I had G when he was a freshman in Algebra 1. If he doesn't see the point, he's out. He's not going to disrupt class; he'll just quietly work on something else. But, he came into homeroom super excited a few weeks ago. He likes to mix, in GarageBand, for fun, and he realized he didn't have to count beats for a whole minute if he set up and solved a proportion.

Me and G and R are very slowly working our way through finding the Ramsey number for three. They don't know that's what we are doing. They are having a great time. I'm having a great time.

G's Trig teacher saw him in there and told me it's great he's getting help with math. Uhhh. Is that what we are doing? A student teacher in my room this week got really confused. She wanted to categorize what she was seeing, real bad. Is this extra credit? Test prep? What?

I'm kind of used to other teachers thinking I'm nuts, but I could use a little validation right now. Somebody tell me I'm not wasting my time and theirs. I just need one comment from a sentient being that says "You are not crazy."

Thank You

2010-02-23T21:19:00.001-05:00

OK I wasn't going to post this story, because I thought it was a tad salacious for a teacher blog, but I'm feeling extra grateful right now for the great suggestions for data to regress (keep them coming!) I don’t respond to comments or feature comments in new posts as much as I would like or probably should. But, I assure you, no word you take the time to send my way goes unread or unappreciated. I'm extremely honored to be a part of this community. To summarize: you all kick serious ass. Thank you.

And, enjoy:

Today I was walking around the room, and a kid was telling a very suspenseful and dramatic story about driver ed or something, and gesticulating, and as I was walking up behind him, his gesticulating hand crashed right onto my boob. Not like a minor brush, either. He got an unignorable handful. Thank goodness I was wearing a bra today.

He snatches his hand away as if he was burned. I'm making a wide-eyed, silent-scream "!!!!" face at him; he's making the same face back at me.

In my misguided attempt to use humor to diffuse the situation, I say probably the worst possible thing of all things I could say at that moment,

"Geesh. The last guy who did that bought me dinner first."

Child: mortified. Classmates: in hysterics. Somehow, we got on with a math lesson.

So THEN, after school, I look up from my desk, and his mom is standing at my door. I think, "Oh, nooo" and brace for impact.

But, no, she just happened to be at school, and wanted to say hi and check in.

Sentence. Deferred.

A Regression Project : Help

2010-02-23T18:22:00.001-05:00

I'm planning a unit for Algebra 2 with Trig called "Function Transformations and Regressions." I'm all set with the transformations part, but I am trying to plan something interesting for the 3 to 4 days of regression analysis. My idea is to give the children a choice of relationships for which they will collect data, graph scatterplots, decide the best model, perform a regression. Then use the function they get to gain insight into the type of relationship and make preditions. Ideally they will present their findings to the group but I don't know if we will have time for that. Also, I'm in the annoying situation of needing to teach them how to do regressions on a TI, but that makes sharing their results very difficult.

I'd like to provide opportunities to explore data that kids might actually be interested in. I'm looking for any more brilliant ideas out there. Ideally where the data is not too difficult to obtain within a 43-minute class period, either by measuring or because it's available online and you can provide a link. (What I'm not looking for, please, pretty please, I beg you, is "SWINE FLU!". The growth and decline of influenza, while interesting, is not appropriate for my purposes. I mean what is that - quadratic growth, followed by linear decline, followed by exponential decay? Piecewise functions are beyond the scope, here.)

This is the best of what I have so far:

period vs length of a pendulum (quadratic)
square footage vs price of residential real estate in Manlius (messy, but linear)
the growth of the population of Facebook (exponential)
time vs distance and cost vs distance of various airline flights (linear)

So, come on. Who's holding?

Mathtype Challenge

2010-02-22T10:10:00.001-05:00

I got all the way to number 12 out of 15 on Maria Anderson's Mathtype/Latex challenge before I ran out of my 5 minute Jing time. Fun! Like Typeracer! But math! I used to hate Mathtype/Equation Editor until I learned some shortcuts. Now I'm a keyboard ninja. Here's my video.

Update, if you're curious about shortcuts: If you hover your mouse over any button in MathType, it will tell you the keyboard shortcut in the lower left corner of the window. The most time-saving ones for me are Ctrl-f (fraction), Ctrl-h (superscript), Ctrl-r (radical), Ctrl-g p (pi), Ctrl-g q (theta), Ctrl-e (plain text), Ctrl-+ (back to math), and Ctrl-F4 (save and go back to document.) To invoke MT from Word, I can type Ctrl-Alt-q, but that might not work for everybody.

KQ

2010-02-18T18:56:00.000-05:00

"I wouldn't even go to class if I had another teacher. I just come because you're awesome."

KQ likes me because I like gaming, and I'll talk about it with him for a minute before the bell rings, and I refer to people as noobs and things as epic and items as loot, and that's a language he not only understands but viscerally appreciates.

But it makes me wonder. KQ is getting A's in Trig because he likes me, so he'll make an effort. He didn't do very well in math last year, nor the year before. He's not exactly a model student. I believe him when he says he wouldn't do as well with a different teacher.

So if some kids are doing better because they have me, there must be some who are doing worse because they have me. And would be doing better if they had someone else. Someone who was way into SU basketball, or improvisational comedy, or spelunking.

There are teachers at my school who are universally adored - but - they are either 1. extraordinarily gifted, funny storytellers or 2. conduct easy or widely enjoyable classes*.

Is there a way for mortals to achieve that? In a discipline where we're rigidly held to account for mastery of difficult skills? Is the answer to find a basis to forge a deep affinity with all 125 of them? Or is there some other secret recipe, besides natural charm? I know it's my responsibility to foster a comfortable, cooperative learning environment, and I think I do that well. I've just never been overly worried about being well-liked.

But I would love for them all to do as well as KQ is unexpectedly doing this year.

*I'm not being negative here - just because everybody gets A's if they show up, doesn't mean it's not vitally important. And I know math can be enjoyable, but the mandates of much of our curriculum make that awfully difficult to deliver. Don't you love how I can anticipate comments and get pre-emptively defensive? Blogging hax.

Error-Finding in Geometry Proofs

2010-02-10T16:28:00.001-05:00

This got kids asking some really good questions and they seemed to like the activity. I'm required to teach two-column proofs, but wanted to elevate the drudgery with some higher order thinking.

I asked them to choose two proofs from recent assignments, write them out, except make one major error.

Their ideas of what constitutes a "major error" in a 2-column proof were enlightening. They wrote a list on the board including missing a whole step, using an invalid reason, using parallel lines to conclude the wrong angles are congruent, and assuming something from the way the diagram looks that wasn't given. They decided not to include things like omitting the horizontal bar from a line segment or the congruency squiggle.

Many of them wanted to discuss their idea for messing up their proof. They kept their voices low and hid behind a folder to talk to me. Some of them are very devious. I will be contacting them when I plan my next bank heist.

Before they handed them in, I asked them to describe the error on an attached post-it. They turned in something like this:

I told them I would copy and redistribute the proofs the next day for them to find the errors.

Notes for next time:

Many of the proofs actually had more than one error. I left this alone. When I passed out the error-ridden handouts, I just told them that lots of the proofs had more than one error, and their job was to find them all.

I had them write on blank copier paper. Next time I should probably hand out a template instead. I think a standard format would be easier for everyone to correct, and also to ensure they leave some room between each line.

The post-its were unnecessary. I don't know why I thought that was a good idea.

I have 60 students in Geometry. Checking 120 proofs would be unreasonable, and there were plenty that would have turned out repetitive. I ended up printing four per page, and giving the kids one page to check for errors.

The error checking takes a while, but they didn't need much help with that part. Finding the errors would work fine as a take-home assignment. Then in class we could project a select few and have the kids describe what corrections they would make.

The arguments over when and how to decide a proof was okay were amazing, and worth ten days of plain-old Geometry class. Consider this format for other things, too.

A way to do this same exercise with less paper and more tech? I don't know. They need to be able to create the proofs and then write on the proofs to correct them. Proof-writing kind of defies computer use because of all the symbols and formatting and drawing diagrams. Unless you have like tablets with stylii for all the students, which we don't. Now I'm thinking of maybe like a smart notebook template with all the weird symbols they would need, infinitely cloned, sitting around the edge of the page so they could just drag them into place. But, still. That would definitely take more time and probably be a total disaster.

The students liked that none of the tasks felt super-difficult, and they loved creating something they knew was going to be shared with their peers. A few seized the opportunity to write ASS as a triangle congruence theorem. Whatever. It could have been worse. Fifth period thinks it's hilarious to say "schlongbongler" and I don't know what that is, but it doesn't sound good.

Phew. I'm glad I got all that out. I think I enjoyed writing f(t) more when I mostly used it to reflect on my way-imperfect practice. I need to get back to that.

Periodic Functions

2010-02-03T21:34:00.001-05:00

For injecting some why-you-might-ever-care-about-this juice into your trigonometric graphing:

In this one the good part starts around 0:08, then again in the final seconds -

Plan - show one clip or both. What are we looking at here? What are the features of the motion? What words would you use to describe it? How might you compare one swimmer's kick with another's? What attributes might you try changing to improve a kick? How does the sun's path differ at other times of the year? At other latitudes? How is it the same? Generally keep poking at it until we describe it as periodic, and mention synonyms for amplitude and frequency. Proceed as usual.

Later in the unit, project a still of the kick trail, and write an equation for it.

If anyone's got other clips of periodic motion bookmarked or favorited, I'd love to see them.

Triangle Congruence Theorems

2010-01-27T09:31:00.000-05:00

are so boring, and there is no nice way to teach them. A google search turns up a hojillion versions of "state the theorem and show an example." Here's what I did this year. It sucks and I'm looking for better ideas.

What makes triangles congruent? They're the same size and shape. This means all the sides and all the angles are congruent. The kids had to sit through one proof like this. What a pain, having to write 6 statements to show that two triangles are congruent.

In class the next day, I had them draw three segments with a ruler on scrap paper: one the length of their index finger, one the length of their ear, and one the width of their palm. I told them to construct a triangle with sides of these three lengths. I passed out compasses but didn't give them any instruction on how to do it. Some of them figured it out. A few of these guys showed everyone what they did with the giant compass and yardstick on the whiteboard.

We had a "discussion" about how there was only one triangle you could make with the three lengths. And when I say "discussion" I mean I said - did you notice that those three lengths would only make one triangle? That the angles were sort of 'locked in'? You couldn't just draw the other sides at any old angle? Then I did a lame little demo of how with 4 sticks, the angles are all wobbly and you can make a ton of different quadrilaterals, but with three sticks, you can only get them to make one triangle. I said something about bridges. Then I modeled and they practiced a bunch of SSS proofs.

Today was SAS. In previous years I passed out protractors, and asked them to draw a triangle with two specific side lengths and a specific included angle, then look around and notice how everyone ended up with the same triangle. I didn't feel this really sent the message that the angle has to be included. This year, I asked them to solve these two problems:

I don't honestly know how effective this was. Yeah "we" eventually made the point, but it took FOREVER, and I'm sure some kids got the point, but I'm also sure that half the class was just sitting politely waiting for the torture to be over. And some kids, of course, get downright indignant when you ask them to do something that turns out to be impossible. Which, whatever, but it's much harder to teach somebody who has concluded you are a crap teacher.

Then we practiced determining which other pair of corresponding things you would need in order to use SAS, then we did one example proof.

I know this sucks but I don't know anyone doing anything better. So let me hear it, hot shots.

Why Being Able to Look It Up is Insufficient

2010-01-25T22:42:00.004-05:00

Eddie* came after school because he wasn't happy with his 6/10 on graphing log functions. I asked him to, without a calculator, sketch a graph of y = 2^x and y = log_2(x), then state the domain and range of each.

He did an admirable job sketching the graphs. He made a table of x vs 2^x, inverted them to make a table for the log function, and plotted points appropriately, connecting them with a smooth curve. An all around serviceable graphing effort.

Then for Domain and Range for both, he wrote "All real numbers." (Dun Dun Duuuuuun.)

Eddie, I say, your graphs are beautiful, but tell me how you decided on the domain and range. Oh, well, he says, I didn't really know. I just guessed, because sometimes it's all real numbers. OK Eddie, Domain refers to possible values of x. Look at your graph of the log function. What x's are not on the graph? Easy, he says, the negative ones.

Why should that be? Why can't you take the log of a negative number? Right now I'll give you an 8/10. You can have a 10/10 when you can tell me, in your own words, why the log of a negative is undefined. Invite its inverse function to the party. I hear he's a big help. This is not that difficult, you will just need to focus on the question for a little while. Maybe 10 minutes of concentrated effort. You are capable of figuring this out.

Eddie takes off down the hall and returns, breathless, 5 minutes later, bursting into my room Kramer-style, "Miss Nowak! You can't take the log of a negative because it's impossible! You just can't! Because it's not in the domain." Did he talk to someone in Precalc? Another teacher? I wonder. "I'm aware of that Eddie. I want you to tell me why it's not in the domain."

Eddie comes back a few minutes later, iphone in hand, and starts reading wikipedia at me. It was a drive-by Walesing. "EDDIE! No points for reading crap that you don't understand verbatim off wikipedia! Here's a secret, Ed: I ALREADY KNOW THE ANSWER! I'M NOT LOOKING FOR ENLIGHTENMENT HERE! I WANT TO KNOW THAT YOU KNOW! You have just spent 20 minutes searching for an easy way out, when in 20 minutes you could have figured it out for yourself! You can do better!"

I really hope that I go into school tomorrow and can write a happy ending to this story. Everyone send calming, logical brain waves Eddie's way.

Confrontations like this make me wonder if the "They can just look it up on their iphones" crew really know what they are saying. Knowing the domain of log(x) is not tops on anyone's list of crucial life skills...that's not my point. But insisting Eddie understand the why will make him re-investigate the behavior of the exponential function that is its inverse. And exponential functions are slapping you in the face every time you turn around, whether you realize it or not. Open your credit card statement? Slap. Open up a 401k? Slap. Nuclear waste will never totally go away! Slap. Your repairables are depreciating? Slap. The effects of one-child policy in China? Modeling the adoption of disruptive technologies? (Yeah, Disrupting Class includes graphs on a log scale.) The Richter Scale - Slap. The point is we spend our lives (incoming: barely coherent analogy) in an intellectual ocean - facts float in and float out - sometimes we can grab onto a mechanized vehicle that will move us around faster, sometimes we are provided with a guide, more often, not. We can increasingly get our hands on any discrete bit of information we want. But to do anything with all this information, we kind of need either direct experience with the ocean floor or very good maps. I'm in the underwater expedition and map-making business.

*Eddie is not his real name, of course.

Marginal Improvements

2010-01-22T10:34:00.000-05:00

Is something that would flow like this any better? I don't know. It's better for ME, but I don't know how many kids are going to come along for the ride.

And now I'm enamored with Sue's comment on the pyramid-slide post:

Yeah, we've gotta build this crazy thing. Pharoah's given us this diagram, and wants it 432 Pharoah-feet from the center to the edge. You know it's gonna be coming in at a 53 degree angle, the way we're gonna build it. He wants us to tell him how tall it's gonna be - I think he wants to see whether it will beat the last pyramid we built for him.

What about Students Who Don't Have a Computer at Home?

2010-01-21T22:22:00.008-05:00

This is a faithful transcription of a conversation I had today. I am not making this up. And this is pretty much how these always go.

"Miss Nowak. The speakers on my computer at home broke. It is impossible for me to watch the videos without speakers. I mean...I've tried! But it's hard!"

(Color commentary: This kid is hoping and expecting I am going to say, that's ok, you poor unfortunate broken-speaker child, you don't have to do your assignments.)

"I agree that does pose a problem. Let's think of some other options you have for using a computer. Do you have a lunch period?"

"No. I don't have a lunch in my schedule."

"Do you have a study hall?"

"No. I don't have any free periods."

"Hm."

Walk nonchalantly over to my computer. Look up student's schedule. (Pro tip: Kids think they can get away with lying about their schedule even when everybody knows you can look it up. See, they think you would never suspect they're lying, because you know they know you can look it up. What they don't know you know, is that they are capable of some impressive psychological jujitsu.)

"It says here you have a Study Hall every day 7th period."

"Oh...well...I always go to the computer lab to...type essays! So, see, I don't think of it as a Study Hall."

(Color commentary: He goes to the computer lab to play Linerider and read the Blizzard forums.)

"So...you go to the computer lab for 7th period every day."

"Yeah."

"Does your typing take you 43 minutes every day? Or can you spare 10 minutes for math?"

"No...Yeah...I can do it then."

"You will need headphones. Do you usually carry headphones with you?"

Other kid who is not involved in the conversation but is obviously enjoying himself: "They have huge boxes of headphones in the computer lab to borrow! You can just take them!"

And, scene.

If you want to find excuses for why you can't possibly teach class differently than you have for the past n years, you will find excuses. If you want to find solutions, you will find solutions.

Edit, because I got called out, and fairly: Look...I'm just weary of that question I used to title my post. If what you're doing is working for you, and your kids are learning well what you think they need to be learning, great. Awesome. You should say this: "It's not broke. No reason to fix it." And I'm not trying to convince you to. If you feel overwhelmed, you should say "This is too much to take in all at once. Can you suggest something smaller in scale to try out?" If you think this is an ineffective way for kids to learn, argue with me on the merits. I want to have that conversation. But if you live in a place like where I live, where I have taken surveys of classes that indicate home computer and internet penetration approaches 100%, and while at school, kids are tripping over a computer every time they turn around, "What about students who don't have a computer?" is not the rhetorical trump card people seem to think it is.

Yer Doin' It Wrong

2010-01-21T18:32:00.005-05:00

Getting ready to introduce trig functions I opened the first lesson from last year and found this horror show.

Many thanks to Dan Meyer for the upgrades to my supposedly-educational crapola radar. (Don't you love how the textbook labels this "Egyptology" for us? That adds so much awesomeness to this problem. It really enhances the pixellated image.)

Now I don't really want to spend 30 minutes rehashing right triangle trig, just remind them about it before I peg a vertex to the origin and one side to the x-axis and start spinning the hypotenuse around. As of now I'm at the filtering-signal-from-noise stage of devising something better.

Conversations with the Deficient in Attention

2010-01-16T14:26:00.003-05:00

"Hi Miss Nowak."

"Oh! Hi ___. What's up?"

"What are you doing?"

"Getting ready to teach this class. How are you? How's your year going?"

"Oh fine. Hey is that a quiz? Can I try it?"

"Um, sure...it's a Geometry quiz. Probably a little boring for you. Aren't you going to be late for 8th period?"

"Yeah maybe. Hey I like your nail polish."

"Thanks."

"The answer to the first one is 80 degrees."

"I know."

"Remember that time in detention, when you let me sit at the big table with you to do my homework, and I started flipping through your notepad (pause for laughter) and found your (pause to laugh and catch breath) grocery list, and it said... tampons? That was hilarious."

"Good times."

"So I just stopped in to tell you I got into RIT."

"....What?! Wow! That's great! High five!"

"But if I want a hug, I should go to the English wing, right Miss Nowak?"

"That's right."

"Yeah I'm majoring in physics." (bell starts ringing) "Whoa, I better go."