The AMC is done! PHEW!

Today was the AMC 10/12 B contest. Another teacher and I in the department organized it … and we got 123 kids to take it!


This is a t-shirt that the math club leaders designed and ordered. The math clubbers all wore them today! SOLIDARITY!

If you recall, my school isn’t really a math contest-y school. I have been working slowly behind the scenes to make a culture change. I figure culture change of this level takes at least 3 years before it takes effect. I brought in the New York Math League two years ago. That year I also did a ton of work to push for kids to take the AMC 10/12. And from a dozen kids two years ago, we went up to 116 kids last year. And this year we had 123 kids. (It would have been higher, but there were mandatory sports and musical practices.)

More than anything, I love the fact that 123 kids thought about math for 75 minutes after school. My favorite moment from today was watching kids discuss problems after the contest ended. I mean: if they did that, then they cared.

Going through this AMC push a second year in a row also reminds me how hard it is to organize something well. I’d say that I spent between 10 and 20 hours making sure that this one contest was successful.

Flat ME!

Reminded by Mr. Knauss that I have one too, this is my equivalent of a woodshopped sign.

It’s a “flat stanley” made by a senior homeroom for a competition they have. (Each homeroom makes a different flat stanley and takes it around New York City taking pictures of it in various locations.)

Part I of a self-inflicted challenge: The Line of Best Fit

Here’s my challenge, created by me, for me. I want to explain where the line of best fit comes from. Not just the algorithm to find it, but conceptually how it is found. My intended audience: students in Algebra II. Where the derivation comes from? Multivariable calculus.

So here we go.

Let’s say we have a set of 5 points: (1,1), (3,5), (4, 5), (6, 8), (8,8)



We want a “line of best fit.” It’s tricky because we don’t exactly know what that might mean, quite yet, but we do know that we want a line that will pass near a lot of the points. We want the line to “model” the points. So the line and the points should be close together. In other words, even without knowing what exactly a “line of best fit” is, we can say pretty certainly that it is not:

Instead, we know it probably looks like one of the following lines:

LINE A: y=1.1x

or

LINE B: y=0.9x+1

Of course it doesn’t have to be either of those lines… but we can be pretty sure it will look similar to one of them. You should notice the lines are slightly different. The y-intercepts are different and the slopes are different. But both actually lie fairly close to the points. So is Line A or Line B a better model for the data? And an even more important question: might there be another line that is an even better model for the data?

In other words, our key question is now:

How are we going to be able to choose one line, out of all the possible lines I could draw, that seems like it fits the data well? (One line to rule them all…)

Another way to think of this question: is there a way to measure the “closeness” of the data to the line, so we can decide if Line A or Line B is a better fit for the data? And more importantly, is there an even better line (besides Line A or Line B) that fits the data?

(Part II to come…)

UPDATE: Part II here

Multiple Integrals! Jigga Wha?!

In Multivariable Calculus today, I let my kids loose. We are starting our chapter on multiple integrals, and I generally start out just dryly explaining what integration in higher dimensions might look like. But today, I decided to scrap that and have my kids try to see if they could generalize things themselves and come up with an idea of what integration in multivariable calculus would look like.

It was awesome. They immediately picked up on the fact that it would give you (signed) volume. That was great. They realized the xy-plane was equivalent to the x-axis. With some prompting, they understood we weren’t integrating over a 1D line (like between x=2 and x=5 on the x-axis), but now on a 2D region. (Of course, a little later, I explained that they could integrate over a line, but they’d get an area.)

Here’s the final list we generated.

It was nice, because students were coming up with some pretty complicated ideas on their own. They were motivating things we were going to be learning. Nice.

After we went through this thought exercise, still not looking at a single equation, I then threw the following up on the board:

I wanted to see if they could use our discussion to suss out some information about the notation, and the meaning behind it. They actually got that the limits 2/4 correspond with y and the 0/3 correspond with the x. And that the region we’re integrating over is a rectangle. And the surface we’re using is 4-2xy. I mean, they got it.

I then showed them how to evaluate this double integral, briefly. I tried to get the why this works across to them, but we ran out of time and I slightly confused myself and got my explanation garbled. I promised that by the next class, I would fix things so they would totally get it.

Although not perfect (but good enough for me, for now), I whipped up this worksheet which I think attempts to make clear what is going on mathematically.

I strongly believe, however, that this will drive home the concept way better than I ever have done before. If you teach double integrals, this might come in handy.

PS. I, a la Silvanus P. Thompson in Calculus Made Easy, talk about dx and dy as “a little bit of x” and “a little bit of y.” So if you’re wondering what I’m looking for question 2 on p.2, I want students to say dy. Then the answer to A is (\int_{0}^{1} x^2 e^y dx)*dy. That’s the volume of one infinitely thin slice. Now for B, we have to add an infinity of these slices up, all the way from y=0 to y=2. Well, we know an integral sign is simply a fancy sign for summation, we so just have \int_{0}^{2} (\int_{0}^{1} x^2 e^y dx)dy

Calculus Mottos

There is one motto I like to bring up in calculus:

Take what you don’t know, and turn it into what you do know.

We also have a few extra mottos floating around:

Calculus above all.

Don’t be a hero.

(The last one refers to doing all the calculation in a problem at once, in one line of work. Too much room for horrible horrible error, they aren’t communicating their ideas, and a million other problems.)

On their midterm, I gave two easy bonus questions. I do this as a way to give a little “error correction” to my grading… With midterms, I have to grade a lot of long tests with a lot of questions, and I suspect that I probably make a small error on one or two of the tests. I figure giving a little bonus boost to everyone helps me feel better about any inadvertent errors.

This year the questions were:

A. Make up a new calculus motto and give examples of using it in this course.
B. Draw a picture of Mr. Shah. Make it snazzy.

I loved these bonuses. Reading them after grading the midterm (which kids did awesome on, b.t.dubs) was hilarious.

I am going to share the mottos with you:

  • Calculus is not just math, it’s a way of life.
  • Use yourself, then some others. After that, calculus won’t be a bother.
  • Practice makes perfect!
  • Take what your friends know, and turn it into what you know!
  • “Crazy for Calculus!”
  • Calculus: I find its commentary insightful and would like to subscribe to its newsletter.
  • Calculus: I promise it’s not as hard as you think it is
  • D.A.N.C.E. — Derive Awesome Nifty Calculus Equations
  • Build the basement before you shingle the roof.
  • Don’t forget the old skills from 1st semester or you won’t be able to register.
  • Calculus is life, life is calculus.
  • History repeats itself: Even if you won’t be tested on a skill, the concepts will come up later on.
  • Don’t freak out.
  • Break it down to its most basic form, to understand it.
  • Go to your limits, reach for infinity, and always aim for the other side of the curve!
  • Write it right and you’ll get it right.
  • “Zero is okay when you’re at the top, but terrible when you are at the bottom.” [re: dividing by zero]
  • (Almost) anything can be simplified in calculus.
  • Calculus above all, and even above senioritis!

Loves me some mottos. Their descriptions of them using it were priceless and spot on. Wondering if I should be ordering some new buttons soon? #methinkso!

A class of diversions

Yesterday in my Algebra II class I went a bit off the deep end in terms of tangents. We were studying complex numbers, and the day before, I had shown them Schrodinger’s Equation (which has i in it). One of my students wanted to know what it was, so I naturally told them.

NOT.

I had this student research it and come to class and explain what he could suss out. He did, in a really funny way, and it was totes good times.

When he got to Schrod’s cat, however, his explanation fell a bit short, so I had him go back an re-research it. We’ll see how that goes.

That was DIVERSION 1.

Then when we were discussing how all our plethora of numbers fit together, we produced

and of course, of course, a student raises his hand and asks “where does infinity go?”

Because the answer to that takes a long time to properly address, and I wanted to move on without spending 40 minutes on that question, I talked about how infinity was, for us, not a number but rather an idea of unboundedness. The game of “the highest you can go, I can go higher.” And of course, I gave them a teaser about the levels of infinity… I didn’t explain it, but I talked them through the basic conclusion (that the real numbers are a different level of infinity than the integers), and hopefully blew their minds by saying that the positive integers are the same level of infinity than the negative integers.

That was DIVERSION 2.

Finally, I concluded by explaining the broad notion of fractals and how each point of a fractal is painted black or a color… and what being painted black or a color means. We did a few examples on our calculators (of evaluating points, and decided whether to color it black or a color).

And that was DIVERSION 3.

I didn’t get to start completing the square, as I had hoped. But you know, the kids will remember this more, and it will raise their general appreciation and wonderment of mathematics… more than completing the square anyhow.

Related Rates: See ’em in action!

A short while ago, I attended the Teaching Contemporary Mathematics conference at the North Carolina School of Science and Mathematics (NCSSM). Before leaving for the conference, I had just started related rates with my kids in calculus. I have had a problem with related rates — because the problems are so bad. It’s just one of those places where you feel: this is where calculus should come alive.

But it doesn’t.

Well, at the conference, one of the last lectures I attended was given by Maria Hernandez. The lecture was ostensibly on using video in the precalculus and calculus curriculum. But in actuality, it was a solution to my related rates problem. Maria had created a wonderful, short activity which brings to life — visually — the idea of related rates.

I stole it, slightly modified it, and used it in my classroom the day after I got back. The activity’s best quality is that students could see math in action, not only from the use of the video, but also from the graphs they produced from analyzing the video. We’d been spending months on this “take the derivative” “take the derivative” “take the derivative” that I felt like we were losing all the understanding we had built up for abstraction. It was perfect to bring us back to basics. I enjoyed doing this in the classroom, and so I asked her if I could share it here. She generously replied:

Of course you can write about the video projects on your blog and feel free to share the worksheets.  You can add my e-mail address so folks can send me a note if they have questions.

Maria’s email is: hernandez at ncssm dot edu

Things to note:

 

1. All my kids have school issued laptops with Logger Pro installed. (I’ve also heard very good things about the free program Tracker which does the same things as Logger Pro.)
2. All my kids used Logger Pro in other classes, and although most weren’t experts at it, it wasn’t completely new to them either
3. This is all Maria Hernandez’s work… I am just sharing it and my experience with it in my classroom. But if you love it, send her an email shout out!

Prelude

You teach the basics of related rates, in the same, boring way you always do. Blow up a balloon, and ask what sorts of things are changing as the balloon blows up. (Volume, surface area, circumference if you assume it’s a sphere, tension in the rubber, etc.). Then you start talking about how these things are all connected — if you have a bigger volume, you have a bigger surface area — and as one changes, the other changes too. And related rates are how these things are changing in relation to each other.

Go through some basic problems together. I use this packet of problems — where we do some together as a class, some they work with a partner, and some they do on their own. In general, I don’t do the harder related rates problems, because for my (non AP) class, I care more about them getting the fundamental ideas.

The Video

Now it’s time to show the kids The Video.

[Maria has put the video up for you to download here.]

Play them the video once, then ask them to jot down things that are changing in time while you play it again. They will come up with things like radius of the cone, the volume of the cone, the height of the cone, the surface area of the cone, the amount of water that is being poured out of the beaker, the angle of the beaker, etc.

The Question

Here’s the question you should pose: “The person who tried to pour the water into the glass tried really really really really hard to pour it at a constant rate. Watch the video again. Do you think he did a good job?”

So play the video again, and then when they’re done, pose another question:

“How does the rate of change of the volume of water being poured from the beaker relate to the rate of change of the volume of the cone?”

[Note: I’m glad I anticipated this. Interestingly, it wasn’t totally obvious for my kids why the two rates of change would be the same.]

The Task

So then you let them know their task. They’re going to be using related rates to check to see if the person pouring the water did a good job pouring it at a constant rate.  To do this, they’re going to use (a) a guided worksheet and (b) Logger Pro.

Set them off on the guided worksheet. Maria’s original guided worksheet (with Logger Pro instructions!) is here. My (very slightly modified) worksheet is here:

Let them at it however you want. In one class, I had them work on Section A and then we had a discussion about their results. In another, I let them move onto Section B without discussing their results until the end. You can figure out what will be best for you.

The general idea behind the worksheet is that students make predictions, and then use Logger Pro to evaluate their predictions. First, students capture data using Logger Pro…

The yellow lines are our coordinate axes (the origin being at the “bottom” of the cone). The dots give us something special. As we play the movie, frame by frame, we add these dots at various times showing where the water is at these times. Notice the wonderful thing about recording the dots on the edge of the class… the x-coordinate of these dots represents the radius of the cone, and the y-coordinate represents the height of the cone.

Once the movie has been marked up with blue dots, students can see what wonderful things Logger Pro gives them!

Not only does Logger Pro make a graph of radius v. time and height v. time (lovely!), but it also gives us a spreadsheet set of data… at each time that we made a dot, we are given the radius, the height, and for free thanks to Logger Pro, \frac{dr}{dt} and \frac{dh}{dt}.

Awesome! Well using the fact that V=\frac{\pi}{3} r^2 h, we can conclude that:
\frac{dV}{dt}=\frac{\pi}{3}(2r\frac{dr}{dt}h+\frac{dh}{dt}r^2).

Oh, how nice. Logger Pro gives us all our unknowns in that spreadsheet so we can calculate \frac{dV}{dt}. And isn’t that what we cared to find out? If \frac{dV}{dt} was changing at a constant rate?

Some things of note:

1. When discussing Section A, you can have a very nice discussion about your students’ predictions. It was cool  to discuss why the general shape of r vs. t and h vs. t should be very close to each other. In fact, if you draw the cone filled in at different heights, you can use similar triangles to argue that however fast r is changing, h must be changing at a proportional rate. Why? Similar triangles!

2. In one class, I gave my kids class time to work on the Logger Pro part of the activity. In my other class, I had them work on Logger Pro at home. It was clear to me that the class who worked on Logger Pro in class enjoyed the activity more. There was more discussion, and they had people to talk to when they had technical difficulties. The class that had to do Logger Pro at home was not super pleased by it!

The Conclusion

My kids used the guided worksheet to calculate \frac{dV}{dt} at a bunch of different times, and to graph \frac{dV}{dt} v. time. It turns out (we used Excel) we get something that looks like:

My kids all conclude that the person pouring the water doesn’t do a good job.

I’m not quite done yet, though. I ask them one last question… If the graph for \frac{dV}{dt} v. time looked like:

“this would mean that the guy was pouring at a constant rate… because the data almost fits a line.”

I tricked almost all my kids in both of my classes, when I said it like that… But one kid in each class caught on that I was faking them out. And they said that this doesn’t mean the water was being poured at a constant rate… but that more and more water was being poured out over time. The only way we would be assured that the water was being poured out at a constant rate is if our data fit a horizontal line…

Nice.

Extensions

I’ve been thinking… perhaps in the fourth quarter I am going to have my students make their own videos filling up cylinders and cones. In a group of 3, I was thinking of asking them to make 2 videos:

  • trying to pour the water in at a constant rate
  • trying to pour the water in in such a way that the graph of \frac{dV}{dt} v. time has a special shape I give them (a decreasing line? a bell curve-y shape?)

Basically it’s going to be a challenge for them, and I’ll have some sort of prize for the group that can get the water in at the most constant rate, and a prize for the group that can get the water to pour in the special shape I give them. Of course the bonus for me is that I might get some more videos to use in future years…