Mr Honner

NCTM Annual — 2018

I’m excited to be heading to Washington, DC in April for the 2018 NCTM Annual Meeting!

NCTM’s annual meeting brings together thousands of educators from across the country to discuss mathematics, pedagogy, technology, and more. I presented at the 2017 Annual Meeting in San Antonio and had a great time, so I’m looking forward to this year in DC.

I’ll be presenting Statistics and Simulation in Scratch, a 60-minute session about using simple computer programming tools to make the study of probability and statistics more experimental and exploratory. We’ll look at ways teachers and students can use Scratch, the free, web-based programming environment designed by the MIT Media Lab, to model simple probability experiments, collect and analyze data, and create mathematically compelling projects. The technology tools we’ll be using are free and intuitive, and they open up a new pathway to probability and statistics for students and teachers. In addition, it creates opportunities to learn and apply fundamental computer programming skills in a meaningful context.

My talk is scheduled for Thursday, 4/26/18, at 3:00 pm, so if you’re planning on attending the NCTM Annual, please keep my session in mind!

Conferences like this are great opportunities for professional growth, but the logistics are often complicated for classroom teachers. I’m fortunate to have received support from Math for America, which makes attending NCTM’s Annual Meeting in Washington DC possible. And I’m proud to be one of several MfA teachers presenting at NCTM! You can find a complete list of MfA presenters here.

Appreciation

Pizza Pi Paddle

Well, Pi Day has already passed, but better late than never.

I spotted this on the wall of a college town’s pizza shop. I’ll give you 3.14 guesses which college.

Resources Teaching

MfA Workshop — Exploring Modern Discoveries in Mathematics and Science

This week I will be co-facilitating a workshop for teachers, “Exploring Modern Discoveries in Mathematics and Science”, with Thomas Lin, editor-in-chief of Quanta Magazine. We will be running the workshop for a group of Math for America math and science teachers at the MfA offices.

In our workshop we’ll look at ways to connect students and teachers with modern science research and discoveries. We’ll focus on resources from Quanta Magazine, including recent reporting on advances in mathematics, biology, and computer science, as well as some of my Quantized Academy columns.

I’m excited to be working with Tom, who in addition to being the founding editor of Quanta, is also a former teacher. Tom’s desire to make the amazing work being done by Quanta’s journalists and writers more accessible to teachers and students led to the development of my Quantized Academy column last year.

Be sure to check out Quanta Magazine, and you can find my Quantized Academy articles here.

Challenge Teaching

sin(x) + cos(x)

Here is a fun little exploration involving a simple sum of trigonometric functions.

Consider f(x) = sin(x) + cos(x), graphed below.

Surprisingly, it appears as though sin(x) + cos(x) is itself a sine function. And while its period is the same as sin(x), its amplitude has changed and it’s been phase-shifted. Figuring out the exact amplitude and phase shift is fun, and it’s also part of a deeper phenomenon to explore.

Consider the function g(x) = a sin(x) + b cos(x). Playing around with the values of a and b is a great way to explore the situation.

On the way to a complete solution, a nice challenge is to find (and characterize) the values of a and b that make the amplitude of g(x) equal to one. It’s also fun to look for values of a and b that yield integer amplitudes: for example, 5sin(x) + 12cos(x) has amplitude 13, and 4sin(x) + 3cos(x) has amplitude 5.

Ultimately, this exploration leads to a really lovely application of angle sum formulas. Recall that

$sin(A + B) = sin(A) cos (B) + sin(B) cos(A)$

If we let A = x, we get

$sin(x + B) = sin(x) cos(B) + sin(B) cos(x)$

With a little rewriting, we have

$sin(x + B) = cos(B) sin(x) + sin(B) cos(x)$

which looks similar to our original function f(x) = sin(x) + cos(x), except for what’s in front of sin(x) and cos(x). We handle that with a clever choice of B.

Let $B = \frac{\pi}{4}$ . Now we have

$sin(x + \frac{\pi}{4}) = cos(\frac{\pi}{4})sin(x) + sin(\frac{\pi}{4})cos(x)$

$sin(x + \frac{\pi}{4}) = \frac{\sqrt{2}}{2} sin(x) + \frac{\sqrt{2}}{2}cos(x)$

And a little algebra gets us

$sin(x) + cos(x) = \sqrt{2}sin(x + \frac{\pi}{4})$

And so sin(x) + cos(x) really is a sine function! Not only does this transformation explain the amplitude and phase shift of sin(x) + cos(x), it generalizes beautifully.

For example, consider 5sin(x) + 12cos(x). We can rewrite this in the following way.

$5sin(x) + 12cos(x) = 13 ( \frac{5}{13} sin(x) + \frac{12}{13} cos(x))$

$5sin(x) + 12cos(x)= 13 ( cos(\beta) sin(x) + sin(\beta) cos(x))$

$5sinx + 12cosx = 13 sin (x + \beta)$

where $\beta = arcsin(\frac{12}{13}) = arccos(\frac{5}{13})$ .

There’s quite a lot of trigonometric fun packed into this little sum. And there’s still more to do, like exploring different phase shifts and trying the cosine angle sum formula instead. Enjoy!

Geometry Media Resources

Forbes Feature

Following up on my appearance on the My Favorite Theorem podcast, co-host Kevin Knudson has an article in Forbes about Varignon’s Theorem, the topic of my episode. Kevin recaps some of the ideas we discussed, including my favorite proof of my favorite theorem.

You can read the article here, and catch the full podcast episode on Kevin’s website.

NCTM Annual — 2018

Pizza Pi Paddle

MfA Workshop — Exploring Modern Discoveries in Mathematics and Science

sin(x) + cos(x)

Forbes Feature

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