MrHonner

Appreciation Photography

Math Photo: Escalating Ellipses

I like the way these ellipses are climbing up their circumscribing cylinder.

And this sculpture puts me in mind of how planes and cylinders intersect: if the plane is perpendicular to the central axis of the cylinder, the intersection is a circle. But if you start to tilt the plane, the circle grows into an ellipse.

By MrHonner, 11 years11 years ago

Teaching Testing

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.

Related Posts

By MrHonner, 11 years9 years ago

Appreciation Geometry Photography

Math Photo: Pi/4 Radians

While playing around building trains, it occurred to me that one link of curved track was a nice, everyday representation of $\frac{\Pi}{4}$ radians, as each link is one-eighth of a full circle.

By MrHonner, 11 years11 years ago

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, 11 years11 years 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, 11 years11 years ago

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