Algebra II

From whence this came…

UPDATE: Here is my Advice from Algebra 2 Students Past, in full glory.

Today I was compiling and typing up my “advice from Algebra 2 students past” to hand out to my kids this year. And I forgot about a striking passage from one of my students:

It will behoove you to understand going into class what exactly Mr. Shah wants from you, which is an attentive, honest, and interested student. It has taken me six months to realize what Mr. Shah really wants from you; he wants you to ultimately be a good person in the world.

Where that came from, I have no idea. None of my other students wrote anything like it. It rang so sincere and specific in the way it was written that I know it wasn’t just a casual, flippant remark. Of course I glowed when I read it — because the student hit on something even I hadn’t been able to articulate. My primary goal is to get my students to understand and be able to do mathematics, but these kids are in the process of learning how to be adults. And whether my kids remember completing the square years down the road is ultimately less important than the person they forged through trying to learn how to complete the square. Not that I ever say that to them, really.

Which is why I’m  darn curious from whence this remark came from, though. Cuz I have no freakin’ clue.


My 2009-2010 Course Expectations

Below are working drafts of my course expectations for next year. Most things — in terms of wording and text — haven’t changed, although the grading breakdowns have. In case it wasn’t glaringly obvious, I’m all about having super clear expectations for my students. Anyway, you can see that aspect of my teaching come through in these.

Algebra II Course Expectations, 2009-2010

Calculus, Course Expectations, 2009-2010

Multivariable Calculus, Course Expectations, 2009-2010

Feel free to steal anything, if you like anything.

Factoring, Schmactoring

So factoring is super useful, yes. But at the Exeter Conference, one of the keynote speakers was making an impassioned, clarion call for CAS in the classroom and threw up an image. It was of which quadratics are actually factorable, and which aren’t. I tried to make my own 15 minute version of that to show you below (where b and c are non-negative, just because I got lazy). Apologies if there are any mistakes.

Picture 3

This image struck me so hard I can’t even tell you. Because although we teach the quadratic formula, in reality, most of our assessments which come after the quadratics unit give factorable quadratics. But in one powerful image, we are reminded that most quadratics are not factorable (at least, over the rationals). And we all know why we give factorable quadratics all the time — and it’s nothing to be ashamed of. We don’t want to have students spend all their time using the quadratic formula (and possibly generating incorrect answers) when we’re trying to teach an unrelated skill.

Still, the implicit lesson we’ve taught our students, by always giving nice, factorable quadratics is that most things are factorable. I mean, how many times have you been asked “is there a mistake in this question?” when you’ve given students a non-factorable quadratic on a test not on the quadratic unit? I thought so.

So next year I vow to show my students this chart, and remind them that most things in this amazing universe are NOT factorable. Heck, most quadratics that come up in engineering won’t even have integer coefficients, I will say, while showing ’em a picture of a falling cow and the equation governing its vertical motion in metric units. And that I tend to give more factorable quadratics than unfactorable quadratics because I want to save them computing time, not for any other reason.

The Summer is Winding Down; Alg II is Winding Up

On Twitter, in the last day or two, teachers are returning to their schools for the requisite start-of-the-year-teacher-meetings. I have three weeks before that happens (phew), and I’m now ready to buckle down and set my sights on the start of the year. I found out officially what I’m teaching for next year, and it’s the same as what I taught this year.

1 section of Algebra II; 2 sections of non-AP Calculus; 1 section of Multivariable Calculus

My goals for Algebra II and Calculus are ambitious [1], so tonight I’ll briefly outline just one of them: Algebra II.

Algebra II: The History

For those who have followed my blog from the beginning, you’ll know two years ago when I started teaching Algebra II, the curriculum was kind of insane. We were doing so much — rational functions and the rational root theorem, a heck of a lot of trigonometry, and who knows what else — that the kids were simply following the motions. It was too much. Plus the textbook was written at way too high a level for where our kids were at. With the entrance of a new department head, we reviewed the curriculum and recognized that were were duplicating half of what students covered in precalculus.

So last year we took a hacksaw to the curriculum and asked what our kids needed to know, what they were going to see in precalculus, what was crucial and what was extra ballast. There was blood, lots and lots of squirting blood from every section of the curriculum. Nothing was safe! But out of the massacre, we came up with what I think was a tight curriculum — one that was paced well, one that allowed our kids to really understand ideas instead of procedures. The only regret I have from this year’s curriculum is that we required our kids to buy the old textbook, since we barely referred or worked out of it. (I work at an independent school, where the students purchase their own textbook.) I created and/or provided most everything we did.

Algebra II: The Future

Next year, we’re going to take this course to the next level. There are three ways we’re going to try to do this.

1. We have a new textbook (Holt, Algebra 2), which serves all my anticipated needs. The students can buy a hardcopy or an e-book, which is a nice option for them. (The e-book is much cheaper too! I think $15.) The best part of the book, though, is the online homework help. Check, for a moment, the homework help for Chapter 3, Lesson 1 (click the image below):


Wow – right? The videos! The text! Clear, amazing. And the problems aren’t the exact problems from the book, but almost the same problems. So students are truly getting guided practice, and not simply given the answers.

I’m going to assign only around 10-15 homework problems a night, but I’m going to expect absolute perfection, because of this additional resource which students can use to target their own misunderstandings.

(2) To emphasize mathematical communication, we’re going to institute a class blog. Mathematical communication was one of my goals last year, and I tried to include at least one “explain…” on every assessment. However, I think we need to practice more frequently. Inspired by the likes of David Cox and Darren Kuropatwa, and blog posts like Jackie Ballarini’s, I’ve convinced myself that this could be the solution.

To be clear, I’m not envisioning this really expansive web-hub for the class. It’s going to be very limited. I want student scribes (individual or in pairs) to record what we did in class each day. Record and explain. That’s all. I don’t expect and won’t require the rest of the class to read it — though they may want to as a quick review before an assessment. Honestly, I could very well ask for the daily notes to be written on paper and turned in. The online aspect is simply to make it easier for me to keep track of all these notes. Also, I want to teach my students that they can write equations and create graphs with their computer! (We’re a 1-1 laptop school, and as of last year, I learned that even my multivariable calculus kids were typing x^3 in MS Word instead of learning they had a built-in equation editor!) Details on how this is going to be rolled out will be forthcoming… like, um, when I come up with them.

(3) Homework… homework, homework, homework. I talked with the other Algebra 2 teacher, and we’ve decided — after reading over the awesome and extensive homework survey results — to really choose our homework carefully and limit the number of problems when they are coming out of the book. We had a long-ish talk about creating different “levels” of homework like some of the respondants recommended, but when push came to shove, we decided that it made a lot more sense in an accelerated class, which our class is not. So, for now, we’re assigning easy and medium level problems and expecting homework perfection on them.

However, one thing we’ve noticed is the absolute MESS that our students make of their class notes and homework. Although they are sophomores and juniors, their ability to take class notes, show clear and organized work on problems, and keep all their work together in one place, is virtually nonexistant.

Okay, okay, not for all of ’em, but a lot of students have never learned the skills of being organize in math. So the other math teacher and I are going to require each student to have a 3″ binder (kept in their lockers, or in the classroom) and a folder. When we’re done with a unit, students will — in class — place and organize their homework and notes and assessments in this binder.

Here’s the kicker. The student’s homework grade is going to be divided into two parts: aperiodic daily check of homework (walking around the classroom and giving students a 0-3 score on their homework) and a binder check. The binder check will happen twice in the first quarter (to get students used to the expectations), and then at the end of the second, third, and fourth quarters. On the day of the binder check, students will bring in their binders and be given a list of 15 things they have to find in their binders:

Homework assigned 9/21/2009; Section 2.5, #32

And students will have to circle this homework problem and their solution in red, and then put a mini post it tab on that page so I know to look there when I collect the binders. Then the problem will be graded on correctness and work shown.

And two added bonuses of these binders? Students will have all their assessments in one place when studying for the midterm and final. And I can have these binders on hand for parent-teacher conferences and for comment writing — both of which will be a much more powerful source of information than my gradebook and scattered notes.

So that’s the plan. Now the real question is if I can pull all of this out of a magic hat before school starts.


[1] MV Calculus was such a success last year — by any metric I want to assign — that I plan on doing the same format and let the course grow organically out of the personalities of the kids in it. Last year I had 4 students in that class. This year, I only will have 2.

UPDATE: I’ve uploaded a draft of the new curriculum here. We added a few more things from what we did last year, but it is largely the same.

My Exponential Function Unit

My Exponential Function Unit for Algebra II

Basic Context: This unit is coming right on the heels of function transformations. Students are familiar with translating functions up, down, left, and right; reflecting functions over the x- and y-axes; and vertically and horizontally stretching and shrinking functions.

Structure: The work on exponential functions is broken into four parts.

Part 0: Preliminary Diversion into Inverse Functions
Part I: Graphing exponential functions
Part II: Solving basic exponential function equations
Part III: Applications of exponential functions (carbon dating and compound interest)

Time: This took a total of 13 days — including an introductory activity day, a review day, a day where we did an exponential decay simulation as an entre to carbon dating, and two assessment days.

Nature of Class: I teach 15 students in a non-accelerated Algebra II class. The ability level of the students range the gamut. Many have a hard time thinking abstractly. All have graphing calculators and know how to use them at the basic to intermediate level. We meet 4 days a week for 50 minutes each day.

Broad Goal: The goal for this unit was to really drive home the concept of exponential functions.

Major Failures: I see two major failures. One is not seriously talking about how fast exponential functions grow. This would have been a really fun day, working on a problem like: “Would you rather have (A) $1,000,000 a day for the month of May, or (B) $1 on the first day, $2 on the second day, $4 on the third day, $8 on the fourth day, etc.” The second is just not having a lot of fun with this. The exponential decay simulation we did could have been so much more powerful, and changed in so many fun and really great ways. We could also have done an activity for exponential growth, using real data — population growth, Moore’s Law, or something to do with the Supreme Court. It would have been nice to finish off with a nice 2 day research activity. If for nothing else, to let my students produce something they could be proud of.

Major Strengths: In terms of getting students to understand exponential functions conceptually, I think I’ve done a pretty good job. My students can relate tables, graphs, and equations. They understand why the functions look the way they look. By the time we finished the exponential application days, students were coming up with the formula for the depreciated value of an object without any help.

Materials [NOTE: If you are opening these docs on a Mac, “Select All” and change the font to “Gill Sans.”]

Part 0: Preliminary Excursion into Inverse Functions
PDFs of My Smartboards before class: 1, 2, 3.

Part I: Graphing Exponential Functions
1. Introductory exercise introducing students to exponential growth and decay (.doc)
2. Introduction to exponential functions, and graphing basic exponential functions (.doc); HW (.doc)

Part II: Solving Basic Exponential Equations
1. PDF of My Smartboard before class: 1

Review Sheet on Part 0, Part I, and Part II to prepare students for the assessment (.doc)

Part III: Applications of Exponential Functions
1. Coin Drop Simulation for Exponential Decay (.doc); HW (.doc)
2. Carbon Dating (.doc); HW (.doc)
3. Compound Interest (.doc)

What I Want You To Know: Looking at just the stark documents, this whole unit seems like it might be a bit formulaic. However, particular moments of the guided notes, or the SmartBoards, or during the activities, were actually designed to be places where we have classroom discussions. For example, when one of the worksheets reads:

we actually had a great 5-7 minutes talking about the answers! So I’m afraid these resources make it seem like we might not have really interrogated exponential functions. But we did.

You can really see what I mean because… during this unit, my friend came to observe my class. (It was an assignment for a class she was taking for her Masters.) It happened to be the class where we first talked about exponential decay. While I was teaching, she decided to make a (partial) transcript of the entire class. The transcript is very rough and partial, and you can’t really tell what’s going on exactly, but you can get a sense of what the class was like:

Transcription (with student names redacted) after the Jump


Thrills & Frustrations

I was frustrated and thrilled today.

I will not whine… too much… about the frustration. For those of you out there who want to know some details, let’s put it this way: when asked to do something, or when I volunteer to do something, I tend to go at it heart and soul, full force. When that effort doesn’t get reciprocal respect (note I’m not talking about reciprocal effort here… just respect)… when I feel like my time and efforts are seen as expendable… I get upset. I put on a grin and I bear it, but not happily.

With that rant over, I can now go to the thrill, which came from having a couple really great classes. My favorite class was my Algebra II class where we were doing exponential functions, and I used a self-created guided worksheet on carbon dating.

I was really pleased with the kinds of questions the students were asking, the mistakes they were making (and correcting), and the aaaah haaaa! moments they had. I could have “taught” carbon dating in 20 minutes: here’s the equation, this is why it should make sense, this is how you apply it. But I loved getting them to the point where they are on the cusp of figuring out the equation for themselves. My students aren’t experts at solving the problems yet. I know if I had taught it traditionally, in 20 minutes, they would be able to solve these problems much better than they can at this moment. But they wouldn’t get things nearly as deeply as they are now. It’s a trade off. And you know what? The time spent was worth it.

I really want to really debrief this lesson soon on this blog. Partly because I thought it went so well and thought it might be useful for others. But partly because it has raised some questions for me about Dan Meyer’s methodology — and its implementation. But for now, I’m going to be content.

Function Transformations

I just wanted a quick post to share the documents I created to teach function transformations. All documents are in .doc format. They aren’t flashy, but they really got students thinking about everything. (This is a regular Algebra II class.) They nailed the final assessment, and are now doing amazingly on transformations of exponential functions. In other words, I see my work as a success.

Function Transformations 1 BASIC INTRODUCTION (here): HW (here)
Function Transformations 2 UP! DOWN! LEFT! RIGHT! (here): HW1 (here) , HW2 (here & here)
Function Transformations 3 VERTICAL STRETCHING (here): HW (here)/ Solutions (here)
Function Transformations 3.5 PRACTICING THINGS STEP BY STEP (here)
Function Transformations 4 HORIZONTAL STRETCHING (here): HW (here) / Solution (here)

Also I handed this practice sheet out to all students to practice their 8 base functions (here).

Just so you know, I don’t always teach via handouts. But with all this graphing, I decided it made good sense.
I’m happy if you want to critique them, or make suggestions on how to improve them.