Pick’s Theorem

Happy 2018!  Here I am posting on this infrequently-updated blog, and who knows, maybe I’ll keep it up.

In the past few years, I’ve seen Pick’s Theorem alluded to in various places. This theorem gives a surprisingly simple way to calculate the area of a polygon drawn on a lattice (most people imagine a Geoboard) based on the number of points on the polygon’s boundary and the number of points on the polygon’s interior. I won’t post the formula here in case you’d like to discover it for yourself, but here is one of many websites illustrating the theorem.

For example, the polygons below each have 5 boundary points and 3 interior points. Despite their different shapes, Pick’s Theorem predicts that each will have an area of 4.5 units.

I wanted to explore Pick’s Theorem with our Math Circle, a group of about 8-14 middle schoolers (mostly 6th graders). After examining lots of other math-circle Pick’s Theorem explorations, I handed the students the following much simpler version:

Math Circle – Picks Theorem

The handout ends right about where you would want to start making conjectures, with students being asked to find polygons with a given number of boundary points, number of interior points, and area. The last question is deliberately impossible.

The ideal math circle as described by its founders, the Kaplans, is a relaxed session in which students are presented with an interesting mathematical situation, then ask questions and make conjectures and discoveries together. I find that my group is a little young for that kind of Socratic seminar style: they really want to be doing things. So my aim last time was to given them lots to do, but eventually put them in a position where they would begin to generate some questions and ideas of their own.

Last time, most students got to various places on the last page. For a few, their interest waned before the end of the session. Next time will be tougher to manage! I’d rather not make a handout for the actual discovery of Pick’s theorem because I’d like the investigation to begin with student questions/observations. This means more “talking together” time, which will tax their attention spans.

My plan is to ask students to continue working on the last page, and then whenever enough people grumble about the problems being hard or impossible pool all of our data and see if we can find a pattern that might help us determine which polygons are possible. Depending on what they come up with, I might ask everyone to figure out (B,I,A) for rectangles with dimensions of their choosing, and try to generalize. Or, I might ask them to generate a bunch of polygons all with the same number of boundary points, but to vary the number of interior points, and see what happens.

I’ll bet we can come up with the formula, and at that point some students might lose interest (and I can give them a 1to9puzzle), those who are still with me can think about why the theorem might be plausible. We can come up with a good explanation for rectangles. I’m not sure if we can get beyond that, but the students may surprise me!

We’re hiring!!!

Hello, virtual world! It’s been a long time since we’ve updated this blog, but in case you’re still reading, we are once again looking for a thoughtful, enthusiastic math teacher to join our community. If you like our curriculum, maybe you’d like to try teaching it with us!

Feedback!

At Park, we write about our students at the end of each quarter. This feedback is meant to be global feedback that goes beyond performance on any one assignment.  This almost always takes the form of a paragraph or two about each student, and possibly some sort of rubric.  Parents love this evidence that teachers understand their students’ personalities and learning styles.  However, teachers wonder if it really makes sense to do this four times a year, and if it’s really worth all that work.  By the fourth time you are writing about a student, it can be hard to think of new things to say.   So we’re now looking at mixing it up and maybe doing something different for some of those four times. 

The Mathematics of Fountain Design

Park teacher Marshall Gordon has an upcoming article in Teaching Mathematics and its Applications: An International Journal of the Institute of Mathematics and its Applications.  The article details the experience he had teaching a project-based unit on the mathematics of fountain design.  In the process of designing their fountains, students were naturally motivated to explore the different parameters affecting the trajectory of a parabola.

Click the link to read the article: The Mathematics of Fountain Design

Not My Best Moment

This happens more often than I’d like: we start by having a normal class discussion and end up trying to resolve something that doesn’t engage the students’ interest, or even their understanding.  It’s the exact opposite of the way I’d like my classroom to feel.

Last week in 9^th grade class we were discussing the following problem, which had been assigned for homework.

Write an equation for a rule a?b, so that the answer is odd only when both a and b are even.

Getting it right this time?

Today is the last day I can treat myself to the luxury of sitting in a coffeeshop on a weekday morning/afternoon.  I came here to think about what I wanted to do in the early days of my 9^th grade class.  In practice, this has translated into my spending most of the time solving and thinking about the “Tinker” problems.  This has worked remarkably well to help me set priorities.

I wanted to teach 9^th grade this year because I realized that I was not doing nearly as much as I could be to teach students how to be learners.  Assigning students nonroutine problems has its drawbacks: though we have great class discussions and kids learn to see math for the open book that it is, students also have the perfect excuse to say, “I just didn’t know how to do this homework problem”, and teachers have the perfect excuse to give large hints that don’t empower kids to feel that they could have solved the problems themselves. 9^th grade seems like a good opportunity to focus on changing some of my practices.  We’re all making a fresh start.

Here are the things I most want to work on with my students this year, things that we need to establish in the earliest days.

Our Own Marshall Gordon in the Journal of Curriculum Studies

Dr. Gordon argues that Habits of Mind should be the focus of mathematics instruction for students of all ages, and especially for students who will become teachers.  “Mathematical Habits of Mind: Promoting Students’ Thoughtful Considerations” appears in the Journal of Curriculum Studies, vol. 43, no. 4, pp. 457-469.  An full abstract is printed below.

Colleges in the US are being compelled to rethink what the First Year Experience or Seminar ought to be for students who have difficulty with mathematics, and what ought to be the mathematics education of teachers, K-12, given the minimal success most students are experiencing. It will be argued here that toward ensuring a more successful education for all students learning mathematics, and most especially for those who will become teachers, the inquiry process must be made explicit so that the productive practices of a mathematically-inclined mind are considered as content. That is to say, the classroom conversation needs to include discussion of the actions mathematically able thinkers use to gain insight into a problem; such as: considering a simpler problem, tinkering, taking things apart. This paper will make an argument why this is an essential consideration for promoting a robust society, and include instances of how mathematics may be presented in this framework.

If you would like to access the article online, free of charge, send an email to parkmathblog@parkschool.net.  Provided you’re one of the first 50 people to ask, we’ll send you the link.

 

The Perfect Combinatorics Problem

In this post, I’m going to extol the virtues of my favorite combinatorics problem.  You’ve probably heard it, or some version of it, before:

A pizza parlor offers ten different toppings on their pizza.  How many different types of pizza are possible to make, given that a pizza can have any number of toppings, or no toppings at all?

Just in case you aren’t familiar with this problem and want to work it out for yourself first, I’m putting most of this post after the jump.  First, a shout out: I remember doing this problem with Michigan State Professor Bruce Mitchell, who used to teach Saturday-morning math enrichment classes at my middle school, and whose enthusiasm and humor kept me coming back.  Second, some pizza:

http://www.neatorama.com

You may prefer to pretend you never saw that.

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Geometry Follow-Up: Proof in a Bag

The concept of proof-in-a-bag is simple.  Write out a two-column proof and then cut it up so that each statement or reason is by itself on a scrap of paper. Then put all the scraps in a bag (a small sandwich bag works well, though an opaque paper bag might have more of a dramatic effect) and have kids work on rearranging the scraps so that they form a coherent proof.  You can decide whether you want students to know ahead of time what it is they’re proving, or if you want them to figure it out by putting statements with “given:…”, “prove…” and a diagram in the bag as well.

Credit where credit is due: I got the idea for this from Laura Chihara while a student in her Algebraic coding class at the Carleton-St. Olaf Summer Math Program.

 It’s nice to have any activity where kids are physically doing something in a math class, of course, but I really like what kids get out of this activity.  It emphasizes the idea that you have to have enough evidence before you can conclude that triangles are congruent (otherwise, what are those “extra” statements doing in the bag?)  And it is very good for helping students understand what can be a statement vs. what can be a reason.  I often find that students want to use triangle congruence theorems like SAS when using properties of triangle congruence; the structure of this activity leads them to realize that they’ve already used SAS to justify the triangle congruence statement; they now need to use something else (CPCTC or the equivalent) to start using the congruence.

There are some times when I would definitely not use this activity.  If the proof is a particularly exciting one for kids to work out on their own, I wouldn’t rob them of the opportunity.  Proof-in-a-bag works best for simple, straightforward proofs, where the two-column proof format can be used without having to do a lot of extra explaining.  I generally use it for one day only, at a time when the class has had some practice writing proofs but has not yet reached a level of comfort with them.

Does anybody else have activities or techniques that they use to teach writing proofs?  I’d be especially interested in what people do who don’t insist on a strict two-column format all of the time.