Tuesday, December 19, 2006

The name's Overun. Rise Overun. But you can call me Slope.

The year ended with a great assembly. The dance classes performed their jazz dance final, complete with white gloves and top hats. The group of students working with local "guerilla-style theater" group headRush performed their one act play that they put together; the play focused on Mexican immigrants coming to the US for a better life, and the difficulties they encounter, along with the next generation's difficulties in dealing with gangs. The student body loved the play and quite a few of the teachers ended up with wet eyes (as one of the "actors" told us later, the DCP staff is too sensitive). The students working with headRush loved their experience and have decided to keep the project going, even though headRush's fellowship with Montalvo (where they are currently artists-in-residence) is coming to a close. The new theater company will be known as "Teatro Lobos Unidos" (I think I may be their club advisor). Finally, the world-famous Numeracy Project made our yearly debut (Rise Overun is my stage name) with Back in SSR, Equis Cuadrada, and Honor Roll-a.

On the last day of school, students were totally high on the promise of sweet freedom that lay just around the corner (so was the staff, to be quite honest). The day went pretty well, though students got more and more restless throughout the day. For me, the kids I taught during the last period before break was my honors Algebra 2 class. I was interested to see how these students would behave. They actually did spend most of the period getting their work done. I knew lecture would be a waste, so I had them work on a packet to practice graphing parabolas; they had to graph the absolute value of quadratic functions; graph quadratic inequalities; and solve quadratic inequalities graphically. They did pretty well, but some of them looked like they were about to explode out of their skin. Toward the end, when I was trying to get their attention to explain the homework and what would happen when they got back from break, one of the students wouldn't sit down. I asked the class what would happen to him if he made me keep them past the bell. It didn't take long for him to sit down after that!

One other thing: presents. On the last day of school, there are always students who bring in Christmas presents for their teachers. Only a few students in high school still do this (I assume many more do in elementary school). The spirit behind this is really touching. Our families don't have to get us anything, and many don't have the budget for extra expenses. The presents are usually little trinkets like a box of candy or a bottle of hand lotion. This year, I received a few boxes of candy (the staff room is always piled high with open candy boxes during this time of year). One student gave me a mug filled with marshmallows (other staff later had to explain to me that it was a hot-cocoa making kit!). But my most interesting gift was from the student who gave me a plaid golf scarf and a set of XL men's thermal long underwear. A bit odd, but very sweet.

The year is over, but not the semester. When we get back from break, there will be a week of classes for review, and then final exams. I didn't get to finish my current unit, so I think I will actually have to give the unit test in the second semester, after the final exam. That will be a bit odd, but I haven't gotten the chance to teach students completing the square yet, and I think I can't chop that out.

I'm out of town now, visiting family in Cleveland and then St. Louis, and I probably won't be posting much more until the beginning of January. Happy holidays!

Tuesday, December 12, 2006


I forgot to post about the last lesson - I gave students a quiz on radicals and complex numbers, and I also taught them how to factor trinomials in the form ax^2 + bx + c when the gcf = 1. I graded the quizzes over the weekend, and they weren't that great, so I decided to spend today reviewing with them.

So after reviewing the homework carefully, we spent the rest of the class doing an activity called "Showdown". I got this from another teacher, and I'm not sure where he got it from. When managed properly, I find that this activity works extremely well for review. Students always want to know if it's a competition, and it's not - it's a collaborative activity. Though competitions are fun and have their place, they often lead to one student in the group doing all the work so that the team can get the highest score. This activity helps students communicate with each other and lets them go at their own pace. I made a set of 16 problem cards (with answers on the back) for each group, which I'll post on ILoveMath. Here is the overhead of the rules I made for the class:

Showdown Setup:
  • Each group selects a team captain who will set the pace of work.

  • Each student needs a handful of answer slips (there are 16 problems in total).

  • Each group needs a team answer sheet.


  1. The captain shows problem 1. All team members work individually and silently on the problem. Write answers on answer slips, turning them upside-down when done (or totally stuck).

  2. When the captain sees all slips are upside-down, he/she calls “Showdown”. Group members show and compare their answers, explain their work, and come up with a group answer. The captain writes the group answer on the team answer sheet.

  3. The captain turns the problem card over and compares the given answer to the team’s answer. If the answers are different, discuss. At this point, you can ask me for help if you can’t figure out how to get the answer.

  4. Move on to the next problem and repeat.

The individual time gives students a chance to access their own knowledge. The group time gives students a structured way to collaborate, and it also prevents them from asking me for help before they have thoroughly discussed a problem. I am freed up to be able to walk around and see what's happening, listen to their discussions, and point things out to them when needed. I've done this activity in the past where it did not work exactly as it's supposed to (i.e. students start collaborating right away, or they look at the answer right away), but my class today did an outstanding job with it.

Monday, December 11, 2006

The Numeracy Project

To complete the dorky math teacher persona, I am in a band with some other teachers (and a couple of students). We are called The Numeracy Project, and we so rock! The other teachers in the band are actually quite good; I just strum rhythm guitar and look goofy. Sometimes we write our own songs (Wednesday, we are debuting an A. Malizia original: "Equis Cuadrada"), but mostly we take existing songs and rewrite the words. Old favorites: Sweet Home Alameda (Sweet Home Alabama); Honor Roll-a (My Sharona); My Graduation (My Generation); you get the picture. I have finally put pen to paper to try my own skills as a composer. What do you think? (The tortoise and hare is a reference to an activity done at the beginning of the slope unit.)

Slope is Rise Over Run (sung to House of the Rising Sun, by The Animals)

There is a ratio in my math class,
(they call it Algebra 1),
and it’s been the ruin of many a freshman.
For reals, it’s hella not fun!

My mind is always wandering,
the hare and tortoise run.
The teacher points, and then declares,
slope is rise over run.

The change in Y, the change in X,
the stupid race is won.
And the only answer that’s required is
slope is rise over run.

Organ Solo

Oh seniors, help the freshmen
not to do what I have done.
I wasted my time in office hours,
pretending to get homework done.

Well, I open up my textbook,
I wipe away the drool.
I’m ready to do whatever it takes
to not go to summer school.

There is a ratio in my math class,
(they call it Algebra 1).
Though it’s been the ruin of many a freshman,
now, I’m no longer one.

Saturday, December 09, 2006

Financial Literacy Project (or, screwing over the freshmen for fun and profit!)

I have the beginnings of an idea for a project to do later in Spring, for the exponential functions unit, in conjunction with our freshman College Readiness classes.

I was thinking about when I was a freshman in college, and how there were always tables set up by credit card companies who would attract crowds of freshmen with such irresistable items as Citibank t-shirts and Bank of America frisbees. They would give a credit card to just about anyone. There have been lots of reports about how so many college students get into incredible credit card debts because they don't know how to manage a credit card, and they are preyed on by these vultures.

I would like to run a simulation, where my Algebra 2 students run their own credit card companies. We will examine what interest means, and the types of predatory practices credit card companies engage in (i.e. a low introductory rate that suddenly changes for various reasons, allowing very low monthly payments, etc.). Then, the students will design their own cards and draw up a credit card offer. They will then set up booths in the school one day, and the freshman College Readiness students will be asked to sign up for at least one card (my students will need to come up with ways to attract their business with their own exciting promotional items... I'm predicting free bags of hot cheetos and the like).

Then, for the next few weeks, in the College Readiness classes (who will be in their Financial Literacy unit), the freshmen will have a given job and salary, mandatory expenses, and optional expenses. They can use their credit card at will, and pay as much of their balance as they choose. To make this more exciting for the students, I'd like to think about how to tie their decisions to real-life consequences (i.e. if you buy an iPod with your credit card, you earn the privilege to listen to one during tutorial time, or if you go on a shopping spree, you earn a free-dress day). Students will be responsible for tracking their budgets and payments.

Then, after a few weeks of the simulation, my students will collect the data and calculate how much the freshmen owe. Whichever credit card company made the most money will get some sort of prize (maybe I'll take them out to lunch or something). Though this may seem counter to the spirit of the project, I'd like to do this for two reasons. One, by thinking of ways to be as deceptive as possible, students will be more attuned to potential deception later when it really counts. Two, they'll enjoy "being bad" (which will increase engagement) and it's a safe environment in which to play the bad guy.

My students will use this simulation as a springboard for the exponential functions unit, compounded interest, and so on. To wrap it up, they will put together a presentation to make at our weekly school assembly where they report on the results of the simulation, explain why we did it, and include a note of caution to all students about the dangers of using credit cards.

The logistics are not thought out yet, but I think this has the potential to be both fun and very educational. Any thoughts?

Thursday, December 07, 2006

Next Lesson: Powers of i; Basic Trinomial Factoring

Today in class, we started with a mini-lesson on finding the powers of i, using the "multiplying by i = rotating by 90 degrees about the origin" metaphor. I misjudged - I made it a self-paced worksheet, but it was too complex (no pun intended) for them to handle on their own. When I went over it with them as a class, they started to understand it a lot better, but it ended up taking much more time than I had planned. The circle help students see why the pattern repeats in 4s. If you start at 1 on the real axis, and make a full circle, you will have multiplied by i 4 times. Therefore, any multiple of 4 in the exponent will cause an integral number of circles, landing you back at 1. Then, the remainder indicates how many more rotations you need and where to end up on the complex plane.

I'm at the point in the year where finals are looming and I feel pressed to rush through the curriculum to finish the unit. But what's the point, if they're not going to retain information as I charge ahead? I may need to scale back my learning objectives for this unit, or for the upcoming units.

After the powers of i, I wanted to do a quick review of factoring out the greatest common monomial factor from a polynomial, as well as factoring quadratic trinomials where a = 1. This is basic stuff that they did in algebra 1, but they had forgotten so much of it. By the end of class, most of them were starting to remember having done this before... so I go back to my earlier question about retention. I feel like this is a fundamental algebra 1 concept that should be retained from spring of this year (when they learned it in algebra 1) until now. Maybe going through it again now will help them get it into longer term memory... but how many times do they need to review the same topic until it sinks in?

Instead of giving a worksheet for practice, I wanted to show students how I create factorable trinomials, and for them to create their own and quiz each other. I think that this will be an effective way to help give them a deeper understanding. Unfortunately, we ran out of time because of the earlier problems. I am moving this into tomorrow's lesson, so hopefully it will work out.

Monday, December 04, 2006

Next Lesson: Operations on Complex Numbers

Tomorrow, the Do Now will introduce the idea of finding the distance to the origin on a coordinate plane by creating a right triangle and using the Pythagorean Theorem. Students must calculate the shortest distance from a given point back to the origin, where they left their sweeties, and therefore want to hurry back to.

Following this, I will do direct instruction on addition, subtraction, multiplication, and division. Multiplying by the complex conjugate for division will be a nice follow up to simplifying fractions like 2/(3 + root2) from the previous lessons. Then, we will recall what was learned during the Do Now to understand how to calculate the absolute value of a complex number. Students will plot complex numbers in the complex plane, and then draw a right triangle and calculate the length of the hypotenuse. I will ask students to figure out other complex numbers that have an equal absolute value, and to determine, given a set of complex numbers, which one is farthest from the origin.

Finally, students will have about 20 minutes to work on these types of problems in pairs and ask for help. Not a very exciting lesson, to be sure, but, for some reason, complex numbers and their operations seem to be hit pretty heavily on the STAR test, so it needs a lot of class time. I don't think students will be asked to find the absolute value of a complex number, but it is good scaffolding for the distance formula, which is an indispensable tool in any high school mathematician's bag of tricks.

The lesson will be posted on ILoveMath.

Friday, December 01, 2006

Next Lesson: Intro to Complex Numbers

Still catching up - this was today's lesson...

This lesson's scaffolding is based off of some ideas in the book Where Mathematics Comes From, by George Lakoff. I found this book to be fascinating, as it speaks directly to the question of whether math is a human discovery or a human invention. The book is firmly on the side of invention, explaining how a set of basic "grounding metaphors" create a foundation on which we've built the rest of mathematics. If you're interested in these ideas, I highly recommend reading it. Whether or not you agree with the conclusions, it'll really get you thinking.

In the book, they discuss the metaphor that is used to understand where the imaginary number i comes from. Essentially, you can think of multiplying by -1 on the real number line as a 180 degree rotation. Once you accept this metaphor, then it opens the question: multiplying by what number gives you a 90 degree rotation? If you multiply by that number twice, that means you've rotated 90 degrees twice. Since this is equal to 180 degrees, multiplying by the mystery number twice gives the same result as multiplying by -1. Therefore, the mystery number must be the square root of -1. Whoa. I thought that was an interesting way of looking at it. It allows you to then see why the imaginary axis is perpendicular to the real axis. It also allows you to see why the powers of i follow that 4 number pattern which students need to know. You can go around in a circle, stopping every 90 degrees at an axis. If you start from 1 on the real axis, you'll pass through 1, i, -1, -i, then back to 1. Also, this had an added benefit: one student asked what would happen if you rotate 180 degrees twice. He realized that this would be the same as multiplying by -1 twice. When I pointed out that this rotation would get you back to where you started from, I think his jaw dropped a bit - he may have finally understood a reason that explains why a "negative times a negative is a positive"!

So I designed a handout for students to work through these ideas, we discussed them, and I finished with about 20 minutes of lecture to tie it all together and do some exaxmples. I love this lesson, because I feel like it allows for some deep learning, and the mathematics are not that tough, so almost all students can follow it pretty easily.

This lesson will be posted on ILoveMath.

Next Lesson: Properties of Square Roots

I'm catching up a bit - this was Thursday's lesson...

The homework due for this lesson was a review/exploration of the multiplication and division properties of square roots. During the Do Now, students practiced this. I gave them problems that couldn't be simplified also (i.e. root 3 + 2root 5) because students often confuse the properties and apply them to addition and subtraction as well. They seem to need to hear about it quite a few times before it sinks in.

After this, I did some direct instruction. I started with some historical background on radicals which I just read about recently. The checkmark part of the radical sign is thought to be a manuscript form of the letter "r", which of course stands for "radix", Latin for root. It was first used in the early 1500s. What I didn't know was that the extended top bar is the last remaining holdover of the old notation of grouping, called the vinculum. I decided to explain this to students so they would understand why "you do what's in the radical first", and it actually did seem to click for some of them.

Following this, I showed them how to reduce square roots by finding the prime factorization and pulling out pairs of like factors. Or, as one student recalled from his 8th grade teacher, it's like two convicts trying to escape from jail, where one of them gets out but the other gets shot. I tried to go for a more mathematical explanation... We also looked at rationalizing the denominator, which was new to them. This is a holdover from the days before calculators, when dividing by a radical was quite a chore. But, the concept of rationalizing is very useful in all those fun limit problems in calculus, so I thought we should still go ahead and learn it.

After this, there was pair work practice, with lots of different types of problems combined together, the hardest being simplifying fractions like (5 + root3)/(4 - root6).

I'll post this lesson on ILoveMath.