Mr Honner

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

Math Quiz — NYT Learning Network

Through Math for America, I am part of an ongoing collaboration with the New York Times Learning Network. My latest contribution, a Test Yourself quiz-question, can be found here

Test Yourself — Math, April 2, 2014

This question refers to an article about how dynamic ticket pricing has put the musical The Lion King back on top of the Broadway box office. Approximately how many people see the musical each week?

By MrHonner, 12 yearsApril 2, 2014 ago

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, 12 yearsMarch 30, 2014 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.

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, 12 yearsMarch 23, 2014 ago

Balancing Act

Math Quiz — NYT Learning Network

Math Photo: Escalating Ellipses

This is Not an Exponential Function

Math Photo: Pi/4 Radians

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