Teaching

Balancing Act

We’ve been discussing center of mass in class recently. While this powerful idea extends well beyond its physical interpretation, it’s good to frame the conversation around the idea of balancing objects on their centroids.

One student was inspired by the claim that all physical objects have centroids, and that they can be found fairly easily. So he took a small plate of aluminum, drilled a bunch of random holes in it, and tried to find its center of mass.

He claimed he had done it. But how could we test his hypothesis? It seemed like there was only one way!

It took us a while, and a few attempts, but we were able to do it! It created a nice little challenge for us, and a nice physical experience with the center of mass.

By MrHonner, 12 yearsApril 7, 2014 ago

This is Not an Exponential Function

A question from the January 2014 Integrated Algebra Regents exam asks students to identify a graph showing exponential decay. This graph was the correct choice.

We regularly see terrible graphs on these exams: non-trigonmetric trig functions, functions intersecting their vertical asymptotes, and unscaled coordinate systems. So it comes as no surprise that this graph is not actually the graph of an exponential function!

First, note that all exponential functions have the form $y = C a ^{kx}$ . Since this graph passes through the point (0,4), we immediately see that $C = 4$ .

Note also that the graph passes through the point (2,1). Thus, $y(2) = 4 a ^{2k} = 1$ . We can now use this information to compute $y(1)$ .

Since $4 a ^{2k} = 1$ , we see $a^{2k} = \frac{1}{4}$ . But $a^{2k} = (a^{k})^{2}$ , and so $(a^{k} )^{2} = \frac{1}{4}$ .

Taking the square root of both sides, we see that $a^{k} = \pm \frac{1}{2}$ . Assuming $a > 0$ , we have $a^{k} = \frac{1}{2}$ , and so $4a^{k} = 2$ . But $y(1) = 4 a ^{k}$ , so we now know that $y(1) = 2$ .

Notice, however, that the graph does not pass through the point (1,2)! Thus, this is not the graph of an exponential function.

NYT Probability Resources Teaching

Teaching Math Through March Madness

My latest piece for the New York Times Learning Network leverages March Madness to explore some basic ideas in counting and probability.

Begin by having students explore how to count the number of possible brackets. Start by analyzing a four-team bracket, say, with Team A playing Team B and Team C playing Team D in the first round. Have the students directly list the eight possible tournament outcomes: For example, A beats B, D beats C, and then D beats A is one such outcome. The use of tree diagrams may be helpful in representing the possible brackets.

Then ask students to predict and explore how many brackets are possible with an eight-team tournament. There are 2 raised to the 7th power, or 128, such brackets. One way to see this is first by noting that eight teams in a single-elimination tournament will end up playing seven total games: Seven of the eight teams must be eliminated, which requires that they lose a game.

I’m glad I could make a small contribution to the Math Madness surrounding March Madness! You can find the entire lesson here.

By MrHonner, 12 yearsMarch 20, 2014 ago

Teaching

Speaking at NYSMATYC

I’m excited to be speaking at the upcoming conference of the New York State Mathematics Association of Two-Year Colleges (NYSMATYC) .

The NYSMATYC is the country’s oldest professional organization devoted to mathematics and two-year colleges. The NYSMATYC focuses on supporting mathematics teachers and students at two-year colleges across New York state through publications, awards, scholarships, and regional conferences.

The theme of this year’s conference is “The Career Journey” and runs April 4th-6th. I’ll be giving the banquet keynote address on Saturday night, where I’ll be talking about my own career journey and offering my thoughts on the rapidly evolving landscape of of mathematics education.

You can find out more about the organization and the conference at the NYSMATYC website.

By MrHonner, 12 yearsMarch 18, 2014 ago

Appreciation Resources Teaching

Pi Day 2014

I don’t usually celebrate Pi Day, but a variety of inspirations intersected at a fun project idea this time around.

Thanks to Math for America, I participated in a terrific workshop on Zometool earlier this year, led by George Hart. We built, explored, conjectured, proved, and collaborated around a lot of rich mathematical ideas. And this semester, Steven Strogatz is teaching a History of Math course at Cornell, and he has been generously sharing thoughts and resources online. As a result, I have been reading up on the derivations of the volume and surface area formulas for Platonic and Archimedean solids.

So when I recently re-watched James Tanton’s brilliant video “What is Pi for a Square?“, the idea hit me: for Pi Day, students could explore the value of “Pi” for various Zometool-constructible solids!

What makes James Tanton’s exploration so wonderful is that it highlights the invariance of Pi in circles (as circumference by diameter) while inviting students to play around with the notions of “diameter” “radius”, and “Pi” in regular polygons. We’ll be taking the question “What is Pi?” up a dimension, and thanks to Zometool, we have a tangible context for our conversations and calculations.

We’ll be exploring properties of polyhedra, calculating areas, volumes and ratios, arguing about definitions, and comparing the sphereness of various things. I’m looking forward to a great day of mathematics! And perhaps discovering what the value of “Pi” is for a rhombic triacontahedron.

By MrHonner, 12 yearsMarch 14, 2014 ago

Balancing Act

This is Not an Exponential Function

Teaching Math Through March Madness

Speaking at NYSMATYC

Pi Day 2014

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