Monthly Archives: January 2009

Yesterday I ahem-ed and winked to my Algebra II class about them needing to know how to complete the square for class today. Teaching this topic last year was a nightmare. A total trainwreck. Students were having difficulty all over the place — they couldn’t simplify radicals, they didn’t get why the procedure worked, they were wondering how imaginary numbers came into play here, they confused the steps, they didn’t *get* it. And it was my fault.

Part of the problem was that we were doing too much, too fast. We had brought in graphing quadratics early on, and we were emphasizing the relationship between the equations and the graphs from the start. We also — in the middle of the quadratic unit — taught complex numbers. That’s too many huge things to deal with. Quadratics bring too much together, and we needed to keep the ideas and skills organized so they make sense.

So the other Algebra II teacher and I decided we’d try something different. First, this year, we introduced complex numbers without talking about quadratics. We motivated these numbers, and then we had students practice working with them, getting really comfortable with them. Second, when we started quadratics, we did so without any graphing. Period. We were doing all algebraic work.

Here’s how we progressed.

PART I: Review
Regular, very simple equations with solutions involving square roots, imaginary numbers, and real numbers:

Quick review of factoring:

A brief discussion of solving equations with perfect square terms — with imaginary and real solutions:

Part II: Completing the Square

Perfect Squares:

We talked about what a perfect square is and noticed a relationship between the four terms — when you FOIL. Importantly, students are going to see that the second and third terms are the same.

***

Creating Perfect Squares!:

The next step of creating perfect squares really has them grapple with the fact that the missing constant term is simply half of the coefficient of the term squared.

***

Completing the Square:

For me, it was this step, just a short distance from the last step, which made the entire unit a success. Because now my students had seen the relationship between all the terms in a perfect square and actually seemed to understand them. My favorite part was that most students were getting problem 10 right — and it involves fractions! We also talked about how important signs are for this process.

***

The End Game: Completing the Square

Before we actually “completed the square” I had students look at the last section of the review sheet.

We talked about how if you can write a problem in this form, that you can ALWAYS solve it. And what we were going to be doing is finding a way to write any quadratic equation in that form, so we can solve it.

Then, I went through an example — step by step — to get a problem to that form:

Then I had them solve it, like they had done previously. Most of them had no trouble solving it.

They practiced doing a few problems on their own — some which gave “nice” answers, some which gave answers with radicals, and some which gave complex answers.

Part III: Reinforcement

I made them practice a few more times, with some harder problems, and then I threw them a curve ball — a coefficient in front of the term. We conquered that, although there were same difficulties with fractions. Then I put some terms on one side and some terms on the other side (e.g. ).

Overall, they really rocked it. How do I know they got it?

I started off this post by saying that I ahem-ed about giving a pop quiz in my class today. Well, I followed through on that. gave a pop quiz to my class on completing the square. I gave them one easy problem and one much more difficult problem — with fractions and radicals — on completing the square.

I got a whole bunch of perfect scores.

If you want, the worksheets I created are below:

Factoring Quadratics
Completing the Square, Part I
Completing the Square, Part II
Completing the Square, Pop Quiz

Today a student in my calculus class asked why L’Hopital’s Rule works. I paused, and failed to think of an easy way to explain it. But now I’ve found a really easy way to explain it — at least for the 0/0 case. (Thanks Rogawski!)

We want to show that .

At least in the 0/0 case, we know that and . Great! If that be true, we can say that:

Of course that has to be true, because we’re subtracting 0 from the top and the bottom! Now we can say:

(We are dividing the top and bottom by the same number.)

Finally, we take the limit as approaches of both sides, to get:

By basic limit rules, we can rewrite the right hand side of the equation to be the limit of the top and bottom separately. But the limit of the top and bottom separately are just the derivatives! (See the definition of the derivative there?)

Q.E.D.

Today was exhausting. It’s been a while since I really, really planned for classes. The past week was this weird interim week where I didn’t work full-force. Midterms were canceled and classes continued, but in this half-hearted way. So it’s been a bit of a shock to sit down and plan 3 lessons, and do a giant pile of grading.

I counted. Today I graded 216 papers, which consisted of 600 problems (not counting parts a, b, and c as separate problems). I created 57 SmartBoard slides (well, some I cribbed from last year, or from previous classes). I entered 24 grades in my gradebook. I modified and printed one worksheet on the quadratic formula. I created my next problem set for my Multivariable Calculus class [1]. I set up my calendar for the next two weeks. Go me!

The next week is going to be brutal — the start of a new quarter, along with the task of entering and finalizing all the grades from the previous quarter.

I’m actually nervous about this upcoming week — but not only because of the work I’m going to have to do. Because of the student death at my school, two weeks ago, everything has been in disarray. Midterms were canceled, tests were modified to be open note or take home, movies were shown in classes (including some of mine), and students were given lots of flexibility. (“You were unable to focus last night? Of course you can take the quiz in study hall tomorrow.”) In other words, for the past two weeks, I unclenched my fist. My expectations were blurred. They were also greatly lowered. Starting tomorrow, I have to go through the process of re-clenching my fist. It’s going to be hard. I’m going to be clear with everyone that even though we’re not going to be the same, life has to go on in my classes. We’re going to go full force, and I am going to have the same high expectations for everyone that I had earlier in the year.

Here’s to hoping that the transition will go easily. I believe that as long as I’m clear with them, they’ll rise to the occasion. They have in the past.

[1] Actually, this problem set is going to be different from the others, and I’m hoping pretty interesting. I am assigning them one problem on using multivariable calculus to find the line of best fit (some of them are concurrently taking statistics), and then they are asked to create their own problem set for the material we’ve covered. Three problems. One which they have to make up themselves, but they can be inspired from any resources. The other two can come from other resources. Heck, you can read the problem set here. I’m hoping that having them dig for problems that seem interesting to them will keep them excited about the material. I’ll let you know how that goes.

In fact, here are all my problem sets so far.

multivariable-calculus-problem-grading-rubric
problem_set_1a
problem_set_1b
problem_set_2a
problem_set_2b
problem_set_3a
problem_set_3b

The images below are thumbnails. Click on the image to see it properly.

ALL 4 SLIDES TOGETHER:

To see the slides individually (click on them to see bigger versions) and read my reflection, jump below the fold.

Read the rest of this entry →

Just a quick link passing you along to Dan Meyer’s Annual Report contest.

Design information in four ways to represent 2008 as you experienced it. This can mean:

 

• four separate PowerPoint slides with one design apiece,
• one JPEG with four designs gridded onto it,
• an Excel spreadsheet inset with four charts,
• etc.

I had a blast working on it last year. To get a sense of what we’re talking about, check out examples from last year.

The memorial service for the student who passed away last weekend was held today. I went with a group of teachers so we could support each other and students. The church was packed and hushed, and the service was emotional. Students — his friends — spoke about their memories of the deceased, and about his best qualities, and about the love that exists between friends. The pain in the tremors in their words was hard to hear. That was the hardest part of being there for me. But hearing all the wonderful memories, how much this person touched those around him with his large heart and humor-filled personality, was also the best part about being there too. I almost cried when someone read “Goodnight Moon,” which was one of his favorite books when he was a child. 

A few of us went out for a late lunch afterwards and just being around each other was helpful. Not that I’ll ever forget this incident, and surely there is still going to be an aftermath for weeks to come, but with this, I think I’m done talking about it on this blog. There’s probably just going to be one more post about how this relates to teaching at some point in the future.

Goodnight stars
Goodnight air
Goodnight noises everywhere

Today in Multivariable Calculus I was supposed to teach my students how to find the plane tangent to a surface at a point.

The book, however, was not clear how to do this. They had an equation involving the gradient of a function, but the equation was derived via local linear approximations. Fine and dandy, but I didn’t like it. I didn’t “see” it or grasp what was going on.

What’s clear is that to find the equation for the plane — for any plane — we need a point and a vector pointed in the direction normal to the plane. We are given the point, but we need to find the direction normal to the plane. That’s the same as the direction normal to the surface!

So I set my class up with the task of doing this on their own. They’re still working on it.

But honestly, I’m not quite there yet. I don’t want to just give them the equation and method on how to apply it, but I don’t think I can explain it in any good way. I’m almost there, at a conceptual tipping point, but I need one last shove over the edge. Anyone out there ready to help?

First of all, I decided that working with surfaces is silly and I’d reduce the problem to curves. So let’s start simple.

Let’s say we have the graph of and we want to find vectors normal to the curve at and (the blue and green dots).

Well, traditionally, we’d be crazy and parametrize the parabola by creating the vector-valued function and then calculate the unit tangent vector () and then from that calculate the unit normal vector (). [1] Then we’d calculate and to find the vectors.

But trust me, this is an awful amount of work, and is not a pretty function. We had to parametrize, take derivatives, and plug in values. And if you remember, we started out with such a simple equation . Why can’t it be easier?

And it can. And this is where I need your help.

Instead of considering the plain old boring function , we turn this into a surface by introducing a direction: .

The function is a surface. We’re only interested in one slice of the surface, when (when the height is 0). This will then reduce to our original equation . The set of level curves of the surface is below. Note that the level curve that goes through the origin is the level curve we’re interested in.

Remember that one important (perhaps the most important) property of the gradient is that the gradient of a function points in the direction of maximum of steepness on a graph of level curves.

Let’s look at the points we’re interested in!

Just looking at the graph shows we’re onto something. Look at the blue dot. Which direction is the steepest, if you were standing at the blue dot and wanted to walk in the steepest direction? Well, clearly it would be directly north. (You want to walk the shortest distance to get to the next level curve. Since the change in heights between level curves is constant, you want to minimize the distance you’ve walked to get to the next height to have the steepest slope.) What about the green dot? Clearly, northwest.

And actually calculating the gradient of gives us .

At the blue dot, we get , which is a vector pointing straight up.
At the green dot, we get which is a vector pointing northwest.

I’m plotting them below.

And without all the pesky level curves to distract us.

Clearly this method works. We take the original function and bring it into a higher dimension (). We use the fact that the gradient gives us the direction which is “steepest” on this surface, if we were trapped at a particular point. (In this case, or . Notice these points lie on the level curve we care about, the level curve which actually is the equation we were initially concerned about (). Then we recognize — somehow — that the gradient of the higher dimension equation somehow gives us the normal vector of the original equation we were concerned with.

The questions I have after doing this:

(1) Why did we have to change our nice curve into a surface to solve this problem? And why this surface?

(2) How can we understand that the vector normal to the curve somehow is “magically” the gradient of the surface we created — one of whose level curves is the curve we’re interested in.

(3) Extending this analysis to problems where we want to find the normal vector to a surface like an ellipsoid (like ) at a particular point, we’re going to be using the function — whose level curves will be surfaces, stacked one on top of another. To find the normal vector, we take the point on the “level surface” which describes our ellispoid, and find the quickest way to get to the next “level surface”? Is that right? I think that seems right. Strange, but right.

(For a picture of some level surfaces, check it out here.)

Anyway, this is just my musings, my way of thinking through this. I’m not quite there. Any help you can give, great. If not, that’s cool too.

[1] I guess to make things simpler, we could simply calculate the direction of the normal vector and not worry about making it a unit normal vector, so we could simply calculate $\vec{T}’(t)$ only. We’re not concerned about the magnitude of the normal vector, only the fact that it’s normal.

Dan Meyer of dy/dan fame has a series of posts titled “What can you do with this?” — where he shows a picture or video with some sort of math connection, and asks teachers how they might use it in class. (Others are jumping on the bandwagon.) I figured why not. I love the idea. What I’ve noticed is that many of the pictures deal with ratios and proportions. I wonder if we can get pictures that deal with other things — like radical equations and limits. 

Michael Lugo at God Plays Dice directed me to the following picture:

Fantastic, isn’t it! What you could do in a math class isn’t obvious at first glance. But let’s see what you come up with! For a spoiler (do NOT check it out until you’ve come up with an idea yourself), see below the jump.

Read the rest of this entry →

Today, besides waking up tired, I went through a whole range of other emotions. Frustration being the most prominent. Now that day 1 and day 2 are over, the school is slowly starting to go into full swing again. Classes are starting to put more content in them, students are becoming more engaged, jokes are being made, and I can finally smile again. Which is all really wonderful.

Still, the logistical aftermath of the tragedy is starting to unfold. We had midterms scheduled for next week, and the powers that be decided instead of changing the nature of the midterms (have teachers make them take home, or one hour instead of two, or have group midterms, or any number of other things), we are canceling them entirely. Which I can support, and will support. Sometimes top-down decisions have to be made. However, the result of this is that teachers are expected to continue teaching next week.

But no teacher, at least none of the teachers I’ve talked to, is in any way ready to start teaching our new material next week. We have a bunch of work we planned on doing during the lull of midterm week. At least, I have a ton of work — in addition to prepping for next semester. 

So things are stressful. And the fact that this is all coming out today, Wednesday, makes it hard to manage.

Clearly, you can see from this that I like order, stability, and clear expectations. When things are messed up, I get messed up.

As a last side note, I found out that I’m not even going to be in the classroom next week. I was told this after trying to figure out what I’m doing in each class, and when I’m doing it. The school is hiring me a substitute for every one of my classes, because I’m a 10th grade adviser and we have to put on gradewise community service project that was planned for midterm week. So apparently I have to leave all 4 of my classes in the hands of substitutes for a week. Which is such a waste of time for the kids. 

I hate to say this, but I think showing math movies is on the table in some of these classes.

Yesterday, the key word was exhaustion. The fact that it was Monday, and when the day was over and it was still Monday, was unthinkable. Wednesday, Thursday, surely it was much later in the week. 7:45-3:10 felt much longer than 7 hours and 25 minutes. Waves of tiredness hit all of us, at various points of the day, and the thought of going home and preparing for the next day unfathomable.

I ended up having 3 cups of coffee and a can of Coke to get through the day.

As with many bad things, one of the most striking things that I noticed about yesterday was that I still had moments when I was laughing. Not the awkward, nervous laughter that comes out of not knowing what to say or do. It was true laughter at funny things. I recognized that I was going through a range of emotions, and something the school psychologist told us stuck with me. “You may, for a second or minute or a few minutes, forget what happened. You may be happy. That’s okay.” When he said it, I understood it but I did not comprehend it. I went home exhausted at 8, passed out at 10.

This morning I woke up feeling… refreshed. Like my battery had been replaced and I had a fresh supply of energy. I had rested away the sheer exhaustion that yesterday brought on. I wasn’t sad or happy or anything — except functioning. And this, my friends, was nice.

As I approached the school, the heaviness started to set in again. But at least today I had the restedness to deal with everything. Today people were figuring out not how to cope, comfort, and deal with the shock and tiredness, but how to move forward. This wasn’t all people — but I saw smiles and laughter, overheard some normal conversations instead of yesterday’s hushed whispering and muffled whimpering. I saw hustle and bustle instead of trudging and robotic-front-stare-walking. Not that it was all roses, or roses at all. A sadness and heaviness blanketed the school, but underneath this weighty shrowd, stirrings. Stirrings. That’s when the school psychologist’s statement really was highlighted:

“You may, for a second or minute or a few minutes, forget what happened. You may be happy. That’s okay.”

And I did forget, too. Not all the time, not even a majority of the time, but there were moments when I realized I hadn’t thought about the death of our student for the past three minutes. It was initially scary — because I wondered if that made me unfeeling. I don’t think that’s the case. I think that I’m in the initial stages of moving on. Not forgetting — this students’ death will be with me for years. But moving forward. Being able to function.

Not all my students are at that point, nor should they be. Some seem to be able to function. One of my classes carried on fine. Another one was quieter but students weren’t incapacitated. The last was the hardest, because it is populated with students the grade of the student who passed away.

I knew they were not sleeping — a student told me that no one in that grade has been able to sleep well. We spent only 15 or 20 minutes learning new material, and it was very basic stuff. Almost no one raised their hands to answer my questions. When I gave them the remaining 25 minutes to work on a worksheet covering some of the same material, I told them: you can work on this if you feel you can focus, you can work alone or with friends, you can not work on this if you don’t feel you can focus, whatever. I expected students to get with their friends and maybe work on the worksheet or maybe talk. Or if they were sick of talking about the death of the student, they would at least talk about math.

But the level at which they were affected was so enormous that literally, for those 25 minutes, no one said anything to anyone. I encouraged students to work with each other, but no one moved. Teenagers, not wanting to talk. Let me say that again: Teenagers, not wanting to talk?! The only voice in the room was mine. All students worked alone on the sheet, and they worked pretty assiduously. And quietly.

In the last four minutes of class, I had to ask them if they thought us doing a small bit of lesson and some worksheet work (and no homework) was helpful. Again, silence. Quiet. One mumbled something about it being fine, and then another two said it was good.

I could have asked them if the moon was made of cheese and gotten a similar response.

It’s hard to see this class so deeply affected — because all teachers want to protect their students. But we can’t. And we’re all grieving in different ways and at different paces. I wish I could neatly tie today up in some summative way. But every hour was different, every class was different, every interaction was different.

So with that I’ll stop. I’m exhausted again. Maybe more tomorrow.