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

Tuesday, December 21, 2010

Log Laws

(Update! After this post and all the comments, I modified the student handout to include a little proof section after each part. You can find it here.)

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.)

Tuesday, December 14, 2010

Review and Practice: Add Em Up

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!

Thursday, November 4, 2010

Special Right Triangles

 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.")


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.

Wednesday, October 20, 2010

The People's Blog

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

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.

Monday, October 18, 2010

Counterexamples in Geometry

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!

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

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.


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.


All trapezoids have 2 congruent sides.

All triangles have a line of symmetry.

Quadrilaterals can't have two obtuse angles.


A circle's circumference is equal to its area.

Every rectangle is a square.

Every scalene triangle contains a right angle.


2 complementary angles are adjacent.

Adjacent angles are supplementary.

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.

All similar shapes are congruent.

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

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

All squares are kites.

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

The diagonals of a quadrilateral always intersect.

Diagonals of parallelograms are also angle bisectors.

corresponding angles for two lines cut by a transversal are congruent

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

No squares have greater area than circles.

All rectangles are squares.

All equilateral triangles are right (or obtuse.)

Any figure with four equal sides is a square.

All pentagons are concave.

All parallelograms are rectangles.

Thursday, September 30, 2010

A Whole New Kind of Number

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.

Monday, September 20, 2010

Radical Comes from Radix Which Means Root in Latin*

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.

Friday, August 20, 2010

The NSpire Lowdown

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.

Thursday, August 5, 2010

Put Your Money Where Your Math-Loving Mouth Is

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.

Wednesday, July 21, 2010

Who else is sensing a theme here?

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.

Friday, July 2, 2010


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.

Thursday, June 24, 2010

Absolute Value Both Rigorous and in Context

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:

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.

Wednesday, June 16, 2010

Where Mah Physics Peeps At

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.

Thursday, June 10, 2010

I Kind of Hate the Stupidly Ubiquitous Video Cameras

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.

Tuesday, June 8, 2010

The Envelope Please

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.

Monday, June 7, 2010

The Personal Invitation

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.

Monday, May 31, 2010

Do Not Be Discouraged

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.

Sunday, May 30, 2010

Love, Your Fairy Blogmother

[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

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?

Saturday, May 29, 2010

Get Your Hot Fresh SBG Checklists

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
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.

Thursday, May 27, 2010

Just How Long Is That Quarter Mile Track Anyway?

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?