MrHonner

4/9/16 — Happy Square Day!

Happy Square Day! It’s quite rare for the numbers of the date to all be perfect squares, and even rarer for them to be consecutive!

April, 2016 has been a fun month for mathematical dates! So far this month we’ve had 4/2/16 (an Exponent Day), 4/4/16, and 4/8/16 (a Geometric Mean Day, or alternately, a Powers of 2 Day). And there are more fun mathematical dates to come!

By MrHonner, 10 years ago

Regents Recap — January 2016: No, It Wasn’t

“The graph below was created by an employee at a gas station.“

No, it wasn’t.This problem from the January, 2016 Common Core Algebra Regents exam is just the latest in a long list of examples of absurdly contrived contexts on high-stakes exams. There seems to be a school of thought that believes we should go to great lengths to humanize test questions; I honestly can’t imagine why.

Not only does this fabricated context add nothing of value of this problem, it sends the message to students that applications of math are pointless and nonsensical. And as I argue in my talk g = 4, and Other Lies the Test Told Me, I fear that these messages add up over time.

Yet we know at least one good thing that came from this absurd test question: statistician Thomas Lumley was creatively inspired by this graph to imagine an amusing back-story! You can read it at his blog.

And you can find more of my critiques of New York State mathematics exams here.

Appreciation Photography

The Most Mathematical Subway Stop in the World?

Next stop, Perpendicular?

By MrHonner, 10 yearsMarch 20, 2016 ago

Resources Student Work Teaching

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

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