NCTM Comes to Music City

The National Council of Teachers of Mathematics held a regional conference in Nashville, TN last week. Since I live in Music City, this was too good an opportunity to pass up! I’ve attended several NCTM conferences, both national and regional, and this was by far the best and most technologically savvy one.

To begin, I installed the conference app on my phone, which made planning and organizing my experience a breeze. It synced the sessions I wanted to see with my calendar, so I received reminders and updates for each one. The app also promoted interaction among attendees through its Activity Feed. We could also keep up with what was happening via Twitter using the hashtag #NCTMregionals.

The first session I went to was “Hands On Trigonometry”, by Elizabeth Petty. She led us through a lesson on the unit using paper plates and twizzlers to demonstrate what a radian is. We then folded the paper plates in half to create an x-axis, then thirds to mark the 30 and 60-degree points on the circle. It was an excellent example of using a hands-on activity to reinforce a difficult math concept.

The next session, “Keeping It Real: Authentic Real-World Math Lessons”, was presented by Ginny Stuckey of Mathalicious. Their mission is to develop lessons that use real-world data and situations to spur critical thinking in students. The lesson Ms. Stuckey demonstrated involved how municipal fines can quickly become impossible to pay off for low-income offenders.

Next up was one of the best sessions of the conference, “Using Manipulatives and Investigations to Teach Geometry”, by Christine Mikles. Ms. Mikles uses the CPM Geometry text, which is full of hands-on activities. Here’s an example of using two mirrors and a protractor to learn about central angles in regular polygons:

Polygon with mirrors

The last session I attended on Thursday was “Slices of Calculus”, by Nina Otterson. As a veteran calculus teacher, I was very excited to learn a new way to teach how to calculate the volumes of solids of known cross-sections. Ms. Otterson’s approach makes a lot of sense: she has her students learn what ratio of a square’s area is an equilateral triangle, an isosceles right triangle, and a semicircle. Then, they find the volume of the solid using a square, which is easy, and apply the appropriate ratio. Her students build models of each type of solid. Here’s one I built using equilateral triangles whose base is a pair of intersecting parabolas:

Solid with Triangle cross-sections

Friday’s first session was “The Math Department I’ve Always Wanted: Twitter As My PLC”, by Michael Felton. It was an excellent presentation on how math teachers can use Twitter to ask questions of other teachers, get great ideas for lessons, and get feedback on their own lessons. Michael is part of the “Math Twitter Blogosphere” (#MTBoS on Twitter), where some of the most innovative teaching is being developed today. If a teacher needs some ideas on how to teach practically any math topic in a creative way, he or she can find it at MTBoS.

Next, I went to a session sponsored by CPM (College Prep Mathematics): “CCSS Math Practices? Trust CPM’s 25 Years of Writing Experience”. This company is a nonprofit textbook publisher run by math teachers. Their series of books stresses the importance of hands-on explorations to teach math concepts. I was very impressed with the passion and excitement of the teachers who use these books.

Finally, I went to another session by Michael Felton, “Desmos and Modeling”. I’ve used Desmos‘ online grapher for several years now (I even did a presentation on it at a TAIS conference last year). They have moved far beyond a simple function grapher, though. Their site now includes all kinds of activities that teachers can use in the classroom. The one Michael demonstrated involved matching transformed sinusoidal curves, and it was a lot of fun. Go to https://teacher.desmos.com/ to see all the fantastic lessons they offer. Teachers can also keep track of students’ progress as they work through the activity.

All in all, an excellent conference. It is fascinating to see how social media is transforming teaching, and how much teachers are trying to incorporate active learning into their lessons. I figure a conference is worthwhile if I can take home at least three good ideas/lessons/activities from it. After this one, I have more than dozen to try out with my students!

 

Math Is Everywhere (cont.)

Being a math teacher is both a blessing and a curse (to paraphrase Adrian Monk). I’ve just returned from an overnight trip to Decatur, AL with my cross country team (where our varsity crushed all competition with an incredible team score of 20, and our JV earned third with an excellent 69 points). As I got off the elevator last night, I noticed the carpet in the hallway had a very interesting pattern. I’d be willing to bet the carpet company has an undercover mathematician working in its art department! Parts of it looked like Riemann Sums:

Carpet1 Carpet2

and the overall pattern looked like a sum of trigonometric functions:

Carpet3 Carpet4

I’ve been playing around on desmos, trying to match it (and not being very successful). Does anyone have any suggestions?

Desmos

A Simple Way To Teach Polar Coordinates

The courtyard outside my building is laid in a circular pattern, which makes it a perfect setting to teach polar coordinates: Sr Patio   I took my Precalculus class outside and stood on the polar axis. I assigned an r and theta-value to each student. They then had to start at the pole and figure out where their given polar coordinates would position them. It was a great way to incorporate physical movement with math. They even understood what happens when is negative! To wrap things up, we worked out what the equations theta = pi/4 and r = 6 would look like, and we compared them to their rectangular counterparts. Then, back inside to apply our new knowledge to some problems!

Sunrise, Sunset; a Sinusoidal Story

Today in precalculus, we looked at some data that follow a sinusoidal pattern and calculated a regression function to model it. Here’s how we did it:

The hours of daylight over a year increase, then decrease in a regular, periodic fashion – just the kind of data that result in a nice sine curve. The US Navy maintains a website where you can enter any location and see the sunrise and sunset times over an entire year:

http://aa.usno.navy.mil/data/docs/RS_OneYear.php

We entered our location (Nashville, TN), and used the sunrise and sunset times for the middle of the months:

Raw Data

The times are given in hours and minutes, so we converted them to decimal hours. We then subtracted sunrise time from sunset time to get the number of daylight hours for each month:

Daylight Hours

Now that we had our data, we entered it into our graphing calculators:

Stat

After setting our window parameters, we plotted the data:

StatPlot

Nice! Next, we calculated the regression function that best fits the data (This might be a good time to discuss pros and cons of different regression functions: quadratic, cubic, quartic, etc.):

We then plotted the regression function with the data points. Nice fit!

Plot

Because we did this in mid-February (x = 2.5), we used our model to estimate how many hours of daylight we should expect to have today: 11.4 hours.

If you’re ready to move beyond graphing calculators (and I certainly am!), then you can do this activity with the Desmos online function grapher. What’s nice about that approach is the ability to set up sliders, and let students fit the regression function to the data manually.

Here are a couple of screenshots:

Desmos Data

Desmos Regression Function

 

Update: Apparently this is the source of the Chinese Yin/Yang symbol.