Resources

Celebrating Pi Day

Though I would not consider myself a Pi Day enthusiast, Pi Day has become a sort of Mathematical Awareness day, and so I’ve tried to find meaningful ways to observe it with my students.

One aspect of Pi that I try to get students to appreciate is its invariance. It’s not just that Pi is the ratio of circumference to diameter in a circle; it’s that Pi is the ratio of circumference to diameter in every circle. It’s an invariant of circles. And one way I try to get students to appreciate and respect that invariance is by computing “Pi” for other figures.

For example, consider the square. The circumference of a square is simply its perimeter. You could choose to consider the red segment below, which is equal in length to a side of the square, as the square’s diameter.

Thus, we can calculate “Pi” for a square to be

$Pi = \frac{Circumference}{Diameter} = \frac{Perimeter}{Side} = \frac{4s}{s}=4$

Thoughtful students may have other suggestions for the “diameter”, which can be a fun exploration in and of itself. But one way to sidestep this controversy is to simply define “Pi” in a more robust way.

Notice that, in a circle, we have

$\frac{Circumference^2}{Area} = \frac{(2 \pi r)^2}{\pi r^2} = \frac{4 \pi^2 r^2}{\pi r^2} = 4\pi$

So we can define “Pi” for any plane figure to be one-fourth the ratio of the square of its perimeter to its area.

This simplifies matters, because area and perimeter are well-defined for most figures, whereas diameter is not. And it’s nice that this new “Pi” is still 4 for a square, since we have

$\frac{1}{4} \frac{Perimeter^2}{Area} = \frac{1}{4} \frac{(4s)^2}{s^2} = \frac{1}{4} \frac{16s^2}{s^2} = 4$

Once students have generalized the notion of “Pi”, there are several interesting directions to go. First, you can explore the value of “Pi” for other regular polygons. What is “Pi” for a regular hexagon? For a regular octagon?

Of course, something wonderful happens as you look at regular polygons with more and more sides. With some elementary geometry and trigonometry to derive the formula for the area of a regular n-gon, you can numerically explore convergence to Pi. And with some knowledge of limits, you can actually prove it converges to Pi!

You could also fix n and explore values of “Pi” for irregular n-gons. For example, set n = 4 and compare and contrast “Pi” for different rectangles, rhombuses, and parallelograms. It’s interesting to investigate which kinds of figures have “Pi” values closest to the actual value of Pi. You might even use this idea to develop a metric for equilateralness.

In one of my classes. we took our discussion of “Pi” up a dimension. With help from a Pi Day grant from Math for America, we used Zometool to explore the value of Pi for solids in 3 dimensions.

We built models of cubes, dodecahedra, icosahedra, triacontahedra, and other solids. We debated which solids had “Pi” values closest to the actual value of Pi. Then, starting from the assumption that

$Pi \sim \frac{SA^3}{V^2}$

we calculated “Pi” for our various solids. Students had a great time with this hands-on activity!

And most importantly, students came away with a better understanding of, and appreciation for, this remarkable constant.

So let’s find meaningful mathematical ways to celebrate Pi Day! Make it a Pi Day resolution.

Resources Technology

An Introduction to Desmos

I’ve presented on Desmos many times to teachers, administrators, and students. So I was excited to bring that experience to the Math for America community through my workshop, An Introduction to Desmos, at the MfA offices in New York City.

Nearly 50 MfA teachers attended, and it was a very active and engaged bunch. Most attendees were familiar with Desmos, and many were using it in their classrooms. But I got the sense that everyone’s eyes were opened a bit wider to the power and possibility of this mathematical technology.

Participants began by working through a document I’ve put together that functions as a guided tour of Desmos. I’ve used this document many times with both teachers and students: it provides a quick overview of the power and breadth of the functionality of Desmos, and it allows me to circulate and answer, and ask, questions. [You can find the document here: Introduction to Desmos]

The second part of the workshop had participants working on a series of content-specific challenges. The goal was to use get teachers using Desmos to build mathematical objects. For example, some teachers worked through these parabola challenges:

Construct an arbitrary parabola
(a) with vertex (2,3)
(b) with vertex $(x_1, y_1)$
(c) with roots 2 and 3
(d) with roots $r_1$ and $r_2$
(f) with focus $(a,b)$ and directrix $y = c$

There were similarly structured challenges for Lines, Transformations, Regions, and several other areas. Participants could choose what to work on based on what they taught or what they were interested in.

As I circulated the room, I answered lots of good questions. And I listened in as teachers talked about how they were already using Desmos in their classrooms. I was especially gratified to hear several teachers tell me that they learned something in the workshop that would have made yesterday’s lesson better. It felt good to deliver immediate impact to my colleagues, and I’m excited to know that many teachers have already integrated Desmos into their instruction.

Throughout the workshop I emphasized that the real power of Desmos is not as a presentation tool, but as a creative tool. I often describe Desmos as a mathematical makerspace: a place where we can design and build using the tools and techniques of mathematics. As teachers, it’s tempting to see Desmos primarily as a tool for demonstrating mathematics to our students, but it’s true power lies in how it can help us all, teachers and students alike, make mathematics.

You can find more of my work with Desmos here. And you can see pictures of the workshop here.

By MrHonner, 10 yearsFebruary 25, 2016 ago

Resources Teaching

TCM at NCSSM

I’m excited to be presenting at the upcoming Teaching Contemporary Mathematics (TCM) conference at the North Carolina School of Science and Mathematics (NCSSM).

NCSSM is an internationally known leader in providing advanced math, science, and engineering education to public school students. They have a residential program that serves 11th and 12th graders and distance programs that serve students all across the state of North Carolina.

NCSSM hosts the annual TCM conference to bring together teachers to talk about innovations in teaching modeling, technology, and problem-solving in advanced high school courses. My talk, Mathematical Simulation in Scratch, details some of the work I and my students have done in our mathematical computing course.

TCM runs January 29-30 on the campus of NCSSM in Durham, North Carolina. You can find out more information about the TCM conference and see the schedule of talks here.

By MrHonner, 10 yearsJanuary 11, 2016 ago

Resources Teaching

MT^2 2015

I am excited to be hosting Math for America’s Master Teachers on Teaching (MT^2) event this December 10th at the Gerald D. Fischbaum auditorium.

MT^2 is an evening of short talks from MfA Master Teachers that are meant to inspire and challenge the MfA community. And it is a showcase of the passion and talent of that community, which now numbers nearly 1,000 teachers of math and science in New York City.

This will be the 4th annual MT^2, and the theme is Equality / Inequality. The evening’s lineup features eight talks from middle and high school math and science teachers offering a variety of different interpretations on the theme.

I have given talks in each of the first three MT^2’s: first, on the bad habits students learn from standardized tests; then on my relationship with change; and last year, about how to turn technology’s failures into teachable moments. While I enjoy the challenge of presenting, I am honored to be hosting this year’s event, and I look forward to an evening of great ideas.

And Math for America hopes to live-stream the event, so stay tuned!

By MrHonner, 10 yearsNovember 29, 2015 ago

Resources Student Work Teaching Technology

Student Desmos Projects

Desmos, the free, browser-based graphing utility, has quickly become an indispensable tool in the mathematics classroom. It provides easy, intuitive access to graphs of functions and relations, and creates unique opportunities to understand mathematical relationships dynamically.

But to me, its greatest virtue may be that Desmos provides opportunities to use mathematics to create. I like to think of Desmos as a mathematical makerspace, where the tools at our disposal are exactly the tools of mathematics.

To that end, when I introduce students to Desmos, we always work toward the creation of something mathematical. Below are some beautiful examples of student work from our latest round of Desmos projects.