Archive Page 2

Differentiation in Algebra 2

Where I teach, Algebra 2 is where differentiation gets kinda extreme. There are two sections in the school – I teach both – and each is a mix of kids. The reason there’s no Honors/Regular distinction is that, in a school this size, a split like that tracks out the whole schedule; you end up with a “high” and a “low” class in English, History, and science. So in each Algebra 2 class there are kids who totally love math and are excited for Calculus and science/math university studies, and kids who hate/fear math, barely passed Algebra 1 and Geometry, and are only taking the course because they’re hoping for a spot at a four-year college. And, of course, lots of kids in between.

Other courses are like this too, I know. But something about the advanced algebra content of A2, and its reliance on prior high school level math couses, makes it stand out. Here’s an example: We’ve been studying arithmetic and geometric sequences, recursive and closed-form versions. Modelling growth and decay, and some financy loan/investment type stuff. We’re headed for linear models of data and linear systems. I give a short pre-assessment to see what kids know and remember from Algebra 1 (and junior high); they have to find some slopes and solve a couple easy systems.

I get the pre-assessments back and look them over. In each class, four students have completely aced it. A couple of them even solved the systems by both substitution and elimination, and demonstrated (unasked) which name went with which solution method. Aside from these eight, everyone else pretty much bombed. But, within the bomb group there are several kids who I suspect have just forgotten how to do these problems, and some who probably never really got it the first time.

We do a one-week intensive unit on lines, slope, and solving systems, with a lot of skill practice. Kids really liked the “Arithmetic Simultaneous Linear Equations” activity found here. During this time, the students who killed on the pre-assessment work as a small group learning about matrices and operations on them. At the end of the week, I give a test (to everyone). Now I have all students meeting standard on solving basic systems (albeit with some algebra errors) except two kids in each class, whose algebra skills are seriously sketchy. I mean, they are consistently making some wild and shocking algebra blunders.

That brings us to today. After looking over the tests, we break into groups. At this point three different things are happening in class:

  • The fast group is moving forward with matrices. They know that there’s a way to use the calculator to solve any linear system, but not exactly how. They are still looking for inverse matrices by hand, and I think their overall understanding is getting to where  I will feel good about handing them the tech option soon.
  • The bulk of the class is working in small groups on linear systems application problems, with each group presenting their solutions to the other groups. Sometimes I throw in the mistake game to keep things interesting. I am seeing signs of increasing confidence and facility from almost all of them.
  • The kids who really struggled are working directly with me. We are hitting the basic algebra moves and trying to build understanding, and doing a ton of practice. Because everyone else was sort of cruising, I could work with these kids one-on-two for most of the class today (with some quick circulation to the other groups). The kids responded to this attention; I just hope they follow up with my advice to keep practicing outside class.

By Friday I’m hoping to see everyone meet standard on basic systems, so we can move on with confidence to more complex material.

A joke for hard times

So, in Geometry we hit the ground running with a Geogebra lab which I was pretty excited about. The kids have to make some different polygons, and I get to try to “break” them with the move tool. So, an isosceles triangle should stay isosceles even though we stretch and drag it around. This requires building it in one of several different ways (using a line of symmetry, or fixing two points to a circle to preserve a common distance, etc) which kids have to be creative to find. If you haven’t played around with Geogebra yet, this is a great way to get going with it. One of my goals is to have my students proficient and comfortable with the program, starting in the geometry year, and this lesson was supposed to serve that goal whle helping kids get prepared for the upcoming test on vocabulary, polygon classification, etc.

I teach two sections of Geometry and they meet on alternating days. Monday, the lab went pretty darn great. I did a launch using the classroom projector, and we headed out. In the computer lab, kids were engaged, there was a good mix of sharing and independent work, several kids had breakthroughs, and the lesson differentiated itself fairly well. I was happy. One item on the list is a triangle with a segment from each vertex to the midpoint of the opposite side. We had sketched this during the bellwork and wondered whether those segments always had to meet in a single point. Opinions differed. Now they were seeing that the medians always did meet, no matter how they morphed the triangle. I intended this as a setup for introducing inductive and deductive reasoning, and it seemed to be working. We had a a little discussion about how you can really know something. Joy!

Tuesday was a whole different story. The class is a little larger and has a lot more attitude. Still, I felt ready, emboldened by my success on the day before. Well. First off, I got a glitchy launch of GGB in the classroom, it kept freezing, and I had to relaunch twice. Kids were like, “um, this is lame.” I soldiered on. We got through it and headed to the lab.

Things didn’t exactly soar. Several kids were way not into it, and let everyone know. The toughest thing about this group is the intelligent, high status kids with shitty attitudes. One of them, H., was clearly feeling way too cool for this. She has the potential to be a great math student but has already let me know all she cares about is the grade. Finally I got them going and most kids were making progress.

We get pretty far along and I notice several kids have made the triangle with medians. I get a discussion going about how they always seem to meet in one point, no matter how the triangle changes. “So we can see how that always seems true, but is that the same as knowing that it’s true? As we go forward we will gain the tools to prove that it’s true, which is the same as understanding why it is true.” Pretty nice lead-in to reasoning and proof, right? Lockhart-esque, even? Wrong. Says A-student H.: “Things just are. I don’t really care why.” General agreement from her peers in 5th block Geometry.

Which brings me to my joke.

Jesus was out walking when he came upon a man crying. “What is wrong, my son?” “Once I was a great violinist, Lord. But last week I crushed my hand, and now I am unable to play.” And Jesus laid his hand upon the man, and Lo, he was healed and able to make music once again.

Later Jesus came upon a woman who sat crying. “What is wrong, my child?” “Once I was a painter, Lord, but now I am blind and all art is lost to me.” And Jesus laid his hand upon the woman, and Lo, she was healed and went on to paint once again.

Still later Jesus came upon yet another man who was crying. “What is the matter, my son?” “Lord, I am a high school math teacher.” And Lo, Jesus sat down and wept with him.

Wrestling with homework (Algebra 1, Geometry)

I started this year with a homework policy that came out of tempered idealism. I believe that in high school kids should gradually take ownership of their learning, and that includes practice. I think they are capable of doing a small amount of practice for no reason other than it will help them learn and retain new skills. No points! No homework grading!

On the other hand, I knew that a certain percentage of kids would work the system however they could, including to their own detriment. No homework grading? Cool! No homework!

Then there are some kids who probably will not do the homework whether you grade it or not. Some of them have chaotic home lives. Homework inherently discriminates against these kids. Many of these kids are the ones we will work ourselves to the bone to prevent from failing.

So, what to do? I feel I need to encourage practice, and responsibility. I do not want to grade 135 kids’ homework. I don’t want to “check in” homework, unable to tell whether it is correct, or copied. I want everyone focused on learning, as much as possible.

I told kids the first week that I’d be giving a weekly “Homework Quiz.” These would be designed so that, if you did the homework, you should ace the quiz. Quizzes are given a full class later than the homework is due, so we have a chance to go over the work together and find any misconceptions. I’ve been making the quizzes short (half a page) and only allowing about 15 minutes of classtime. They are quick and easy to grade, and have the tested skills listed at the top. They get entered into the gradebook, divided by skill, under Formative Assessment, which altogether will be 20% of their grade (schoolwide SBG-hybrid policy… long story).

This week, in an attempt to see how things were working (and to keep everyone on their toes) I checked homework. Students were, needless to say, shocked. Three weeks in, and they had already decided I would never check the homework! “I thought you didn’t do that!!” I love teenagers.

Does it work? Yes and no. Here are some gross generalizations based on the first few weeks:

  • Kids who are quick learners either did or didn’t do the homework, and did fine on the quizzes. Honestly, if they could have nailed the assessment just from paying attention in class, do they need homework?
  • Compliant kids – who probably would have done the homework without the quizes – did the homework, but seemed a little clearer on what they should be getting out of it.
  • System-working kids who would do anything to avoid the homework did a minimal amount of the work and often did poorly on the quizzes. I am still hoping that as the weeks go by they will really swallow the connection between practice and success.
  • Kids who lack home support have mostly not done the homework. I suspect that would also be the case in a graded-homework setting. They have done poorly on many parts of the quizes, but not all. The only advantage of this system, for them, is that we have some data on what they know, and what they don’t, so I can get some targeted help headed their way.

There is one other thing about this system that I like: it’s a low-stakes test. Kids get more practice in a testing environment, and another opportunity to recall and build memory.


The lesson inside the lesson

I love my calculus class. They are mostly seniors, and seniors are busy people. I know that my predecessor’s greatest peeve was kids missing class and not taking responsibility for catching up, and I know that in the past many seniors have had a less than rigorous final academic year. This week I confronted these issues head-on.

We are on the block, and meet Tuesday, Thursday. Friday (today) is a professional day, so no class. Tuesday, five students were gone to help with an elementary school campout. This is a neat opportunity to mentor young kids, and I encourage it. They came to me the week before with a planned absence form. I let them know that makeup work would be available on the website, and when they mentioned they’d be out camping Tuesday I promised to post it a day early. The lesson they missed was about sums, products, and composites of functions, and finding the domains. It went pretty well; kids were getting it and I assigned some additional practice. Thursday, folks came to class and promptly took a quiz on the subject. And? those who had attended the previous class did well; those who did not attend… Their faces told the story. I let everyone know this was a formative assessment, and not high stakes. Still, these are the school’s high-achieving kids and they are not used to bombing quizzes.

So, we moved on to the lesson. Their task was to extend from our current understanding – derivative of power functions, and function composition – to differentiating a general polynomial. They did the work in groups; there was a worksheet to provide scaffolding; I circulated. They struggled mightily, because of the abstract level we were working at, but ultimately everyone got it. It was a great hour.

I thanked everyone for their hard work. I mentioned that three students were gone today (for peer mediation training; again, a worthy cause!) And I think that it hit them. This lesson had been good, and hard. It would be really difficult to make it up outside class. I let them know that my goal was to make every class worthwhile; I asked them to please think carefully about their absences, to take responsibility, and to work very hard to make up for each one. This was the lesson. It felt like a step in the right direction.

Test prep, whiteboards

My most difficult class to manage is an end-of-the-day block of Algebra 2. I’ve been feeling frustrated by their lack of focus, and amazed that even though the class is quite a bit smaller than the other section of Algebra 2, and comes later (allowing me to be better prepared), it consistently feels less successful. Partially it’s the 2:00 start time; also, there is one posse of junior and senior boys in there who love to chat, have not-that-great basic algebra skills, and are big and loud. I hate having to repeatedly ask/demand/holler for attention, but that’s what I find myself doing.

We’re coming to the end of our first unit, with a test next class. So yesterday I split them into groups, and gave each group a different problem, and a whiteboard to work on and present with. When they were ready, I told them that the questions were very similar to what they would encounter on the upcoming test. The presentations went pretty well, with all groups showing complete solutions, but what I was particularly pleased with was how well they did as an audience. They listened and asked clarifying questions, and stayed attentive for longer than I’ve seen so far this year.

I feel a very small amount of guilt for using the threat of a test as a lever to encourage better listening. But I like that they listened to each other.

Working our way towards the derivative…

First, we played with pullback cars and calculated average velocity, and realized that a smaller interval could yield a more accurate estimate of instantaneous velocity.

Then we went to boot camp and got really familiar with the polynomial functions and their qualities. (Also reviewed slope, for good measure.)

Next we talked briefly about reflection (with a detour to pool tables) and scientists’ quest to understand curved mirrors and lenses.

Then it was Geogebra in the lab. Given a polynomial curve, place a point on it where you would like to know the “slope.” It is cool that if you zoom in far enough, the curve looks linear. Place another point on the curve, make a line that connects them, and apply the slope tool. Now, zoom in, and move the second point ever closer to the first. Watch the slope measure change. You will be able to estimate the slope right at that point. This exercise felt like both a good introduction to some new GGB tools, and a really visceral experience of calculating a one-point difference quotient.

Next, we did the algebra to find the slope of the tangent to y=x^2 at (1,1). (We are working without the formal definition of limits. We have a variable h for the interval, which “gets really small” and we are left with a numerical value for slope. It is great how they get this, but think it is fishy. It is fishy. Once they get more comfortable with it, I will pull the rug out with a rant about how messed up everything gets if you allow division by zero. And so, hopefully, motivate limits. I don’t really know how else to motivate limits.) Homework: Choose three more points on the same curve; use this technique to calculate the slope of the tangent at each of those points.

So, having done all this, I believe hope they will be ready for the real difference quotient definition on Thursday. The big hurdle remaining is understanding that we are deriving one function from another – that is, that the derivative gives us that numerical slope for any x value we input.

As a side note: I have three exchange students from Beijing in my class, who all took calculus previously, in China. They seem fascinated; I think their previous experience of learning math was very different. The hometown kids are fascinated, too, by the algebra prowess of the exchange students. They can factor quadratics just like that! One of the host parents approached me this week, and said his guest student is really enjoying math. “He told me he learned something new on the very first day!” That was the day we played with cars.  🙂

Classroom layout





Thanks to another teacher in the building, who was also looking for a change, I replaced the single desks (which had always been in straight rows) with some two-person tables. I’m trying to encourage student discourse, both in pairs and in inclusive class discussions. 

It is easy for me to walk through the U-shape and look at the work students are doing, and I think there may be more sharing of ideas. On the downside, kids who get distracted are that much harder to keep on task, and may distract others.

We are all born into this world, and at some point we will die and that will be that. In the meantime, let’s enjoy our minds and the wonderful and ridiculous things we can do with them. I don’t know about you, but I’m here to have FUN.
-Paul Lockhart

In theory, theory and practice are the same. In practice, they are not.

-Yogi Berra