Geometry, getting our groove back

It was a fabulous break, two full weeks plus three weekends. The snow came on strong right before break started and continued right into the first week, until we had a couple feet of beautiful snowpack. Then the sun came out. Bliss! So now we are back at school and trying to get back into the swing. Geometry meets first thing, and everyone is tired.

I meet them at the door, hand everyone a slip of paper as I welcome them back. Projected are instructions to write on the slip:

1) Your name

2) Something really fun you did over break

3) Name for a quadrilateral with two parallel sides

4) Definition of a kite.

They work on it while I take roll; I collect the slips and toss them on my desk. “OK, people, it was an awesome break and now we are back. I am glad to see you all. You have amazing Geometry knowledge in your heads but it may be kind of hard to access it after all the fun time off. You may not have thought about Geometry once over break and that’s fine; in fact, maybe I didn’t either. But now we are going to blow out the cobwebs so get ready.”

We are most of the way through a unit on special quadrilaterals. I grab the deck of cards (with a kid’s name written on each card) and shuffle, and fire up the slides. We play “Name it” where I project a general quadrilateral with some extra information, draw a card, and a kid has to try to specify what kind of figure it is. They have the right to pass, but are encouraged to make educated guesses, or give a partial answer.

I do everything I can to avoid revealing whether the kid is right or not. Clever Hans is always on my mind these days. “What makes you say that?” “How sure are you?” “Who agrees or disagrees?” I encourage dissent and try to make it OK, really OK, to be wrong. I’m trying to avoid the correct answer, said once. I notice several kids are making sketches to help figure these out, which is awesome but I wish more kids were. It can take a long time for the class to come to a consensus about a difficult slide. Once they do, next slide and next card.

By the time we get through all the slides (about 20) they are worked. I have everyone get up, stretch. Go to the bathroom if you need to or get a drink, but hurry. We are about halfway through an 80-minute block.

New topic: midsegment of a trapezoid. I explain what it is, show some examples and non-examples, make sure everyone gets the definition. “Okay, what can you conjecture about these things? Consider angles and segment lengths, go!” Usually I would make small groups with the cards but today I just go with who’s already close, waving them into clumps of four.

After 10-15 minutes I call them back to order and start asking groups to share. The things they need to know about the midsegment, they’ve pretty much figured out. Every group gets parallel to the bases and corresponding angles. Most get midsegment length average of the two base lengths. One group has figured out a way to prove it by chopping the trapezoid up to make a rectangle. Nice.

Okay, what if we take this trapezoid and shrink one base down to a point? We now have the midsegment of a triangle. What can you conjecture? They get it and some kids make pretty clear, convincing arguments. I think (hope) we have dodged the bullet of the right answer, said once.

Ten minutes left. Do the homework now and you won’t have to do it at home. Work with a friend, go! I grab the stack of entry slips and thumb through. I cruise around and connect with several kids about things they did over break. “You sledded down that? Epic!” “Glad you got to see your cousins. Where were they visiting from?” “You saw the King Tut exhibit. I’m so jealous!” I also help with the homework, a little. Then the bell rings. We’re back.

A project in Algebra 2

Given two points in the plane there is exactly one line through them. Of course. But there is also exactly one exponential function, and one power function. Being a math nerd, I think this is cool. I also thought it could form the basis for a project wrapping up our unit on exponent and log functions. The assignment is to begin with two “personal points” formed from some numbers about you (like birthday, jersey number, etc.) and provide the graph and equation of the linear, exponential and power functions through those points. I assigned this as a portfolio, meaning it will be a summative grade (like a test), and gave them the scoring rubric in advance. The assignment sheet is here.

As a lead-in to the project, we did a lot of graphing on log scales this week. We also graphed on semi-log and log-log paper, and saw how they linearize an exponential/power function (respectively).  This helped us discover and understand why the function through two points must be unique. It was nice to do some graphing by hand, both for the tactile experience and to see all the problem-solving opportunities that arose. Choosing an appropriate log scale was especially difficult for many kids.

To do the project, kids will need to use several mathematical tools:

  • point-slope form of a linear equation (can’t hurt to review it yet again)
  • point-ratio form of an exponential
  • “logged” form of power function, with the slope as log ratio
  • various exponent and log solving techniques

There is obviously no “real-world meaning” to the personal numbers students use during the project. During our log graphing week we’ve looked at several different data sets. I’m hoping the curve-fitting skill from this project can carry forward to future data analysis projects. At the least, it will give them some good algebra practice and might help make an interesting connection between these parent functions.

Introducing differentials

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Beginning proofs as groupwork

Today in Geometry students were taking their first crack at making up proofs. I have modelled this for them, walked them over the pons asinorum and tried to show them how we can use a deductive argument to see how our compass-straightedge constructions work. They have done a bunch of the classic SSS etc. problems and should know the triangle congruence theorems. They were ready, but I knew that it would be hard for them, so I wanted to have them work in groups.

I break my classes into groups nearly every class, if only for a short investigation. But I’ve been feeling unsatisfied, and recently returned to the book Designing Groupwork (Cohen) for some deeper insight. One thing the author stresses is the importance of assigning individual jobs to help break down students’ prejudged status structures. In an effort to bring the groupwork up a notch, and to be really clear and formal about what makes a mathematical proof “right,” I created both a scoring rubric for student proofs and a corresponding set of group member roles. I made up a list of simple proofs I thought the students could handle. The whole thing is a one-page (front-back) sheet, found here.

How it worked: I pitched the task to kids as their first real attempt at mathematical proof. They were actually sort of excited, and nervous. I told them their work would be scored and entered in the gradebook. ( I don’t usually do this, striving for a more righteous SGB, but sometimes it helps to raise the stakes just a tiny bit.) I provided one last model so they could see the level of clarity and justification I expected. Then I shuffled the cards and made groups of four. Each group got a sheet with one of the proof tasks highlighted. I put their names by the roles. And they got after it with the whiteboards.

All the groups were able to write a correct and clear proof, working together. One group in one section took a lot longer than the others, so kids had to do some other problems while they waited. The presentations were pretty good, though lacking in the class feedback department (which I encouraged, but didn’t require or formalize).

The whole lesson, with launch, group work, and sharing, fit into an 80-minute block just right. There are a lot of things left to prove on the list, and I plan to do the lesson again after Thanksgiving. I anticipate that they will be much faster, and more confident. Then, eventually, I’ll be asking them to come up with proofs without the group support, and I plan to use the same rubric (without the participation component) to assess them.

Calculus: Big Ideas, and a visit from Mr. Feynman

We’ve been cruising along through various applications of derivatives. As an interesting optimization application, we talked about the refraction of light by a change of media (like a pencil appearing bent in a glass of water) and derived Snell’s law, using Fermat’s principle, that light takes the path requiring the least time (hence optimization). And a student asked, “How does the light know which path will take the least time?” My response: “That is an awesome question! I promise to do right by it, but I am going to need to do a little homework, and we are out of time for today anyway. But thank you so much for asking that!”

So I had to dust off the modern physics books and really get to thinking how to answer that question. My training is in math, not physics, but I took this one amazing course in college that was an intensive physics class based on the Feynman lectures. I had a vague memory that the path of least time was most probable, but I wasn’t prepared to give a full answer, by any means.

My homework led me to the videos of Feynman speaking in New Zealand on the topic of QED, which are truly marvelous. I recommend every teacher watch them, just to see how he communicates with the audience. One interesting thing I discovered is that Fermat invented the principle of least time to fit the experimental findings of Snell; of course it makes sense that the model was created to fit the data, but we obscure this when we use Fermat’s principle to derive Snell’s law!

In the end, I decided to talk with the class about the history of our understanding of light, from Newton’s particle model to Maxwell’s wave theory, and explain why neither model completely worked. I introduced the twentieth-century quantum model where “waves” of probabilities rule particles. I tried to tell them just enough to let them know there’s a big, amazing world of knowledge out there waiting for them. Then I played a clip of Feynman from the first NZ lecture (“Corpuscles of Light”); on the YouTube version it’s from about 20:00 to 26:00. In this part of the lecture he speaks eloquently about not understanding. He describes the laws of nature as hard to accept and calls them “screwy,” then announces, “If you don’t like it, go somewhere else! Another universe, where the laws are simpler!” The kids were losing it; it was so fun to watch their reactions. Then I told them that links to all the videos were posted on the class website, and invited them to explore more on their own.

Function transformations: Desmos rocks it

In Algeba 2, we’ve got our growing list of parent functions, and we know how to slide, flip, and stretch them. Some students have developed a pretty deep understanding of the transformation formulas – that is, they know why they work – and I believe they will be able to recall and use them far into the future. Others are able to use the formulas correctly (some more often than others) but maybe don’t really get exactly how they do what they do. These kids may memorize the formulas for the test, and then forget them.

Today I projected a bunch of  graphs of transformed functions, and kids sought to write an equation for each. It’s a classic Algebra 2 lesson, developing more fluency between the algebraic and graphical contexts. When I handed out the iPads and sent kids to Desmos.com, things really got going.

Desmos is so much better than a graphing calculator for this purpose. It provides instant feedback, and you see the equation and graph at the same time. The graph changes while you edit the equation. Or, you can make a copy of an equation and edit that, then compare. I got some great “What would happen if…” and “Why” questions, which I love. In both sections we had a detailed discussion of why translating then dilating a half-circle would yield a different result than dilating then translating.

The lower-achieving kids seemed to do well with this lesson. It was nice to be able to try something, see what happens, then adjust. And because everyone was engaged, I got to spend some more time working directly with them.

Downsides of Desmos: On an iPad it’s pretty easy to lose your work by deleting or navigating away. Also, the parentheses take some getting used to. The downsides were totally outweighed by how immersed the kids were in the algebra-graph environment, and how much they liked it.

Compass-edge constructions, curiosity

I’m introducing the basic compass-straightedge constructions in Geometry this week. For the most part lessons seem OK; it’s helpful that everyone has something to do. Busy hands are less the devil’s playground than bored hands. My biggest disappointment is that only one or two kids in each section seem to wonder – like, at all wonder – why something like the angle bisector construction works.

This is a big contrast from last year, when I taught at an alternative high school. There, the students were historically low achievers in math (and most other subjects) and in many cases had done some construction or other labor. Hands-on geometry was the best thing I did with those kids. For them, finding the balance point of a cardboard triangle was, like, very cool math. And what I loved was they always wanted to know why something like that worked. It was a very natural, in-the-world kind of curiosity, that felt to me like a true mathematical instinct.

My geometry classes this year are mostly composed of what we would classify as better students. They will undoubtedly be more successful on average on the state end-of-course math test than the cohort I had last year. But I’m waiting – impatiently – for their curiosity to emerge. And I’m wondering what things I can do to inspire and nurture that quality in them.


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


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