Teaching

Appreciation Teaching

The Perils of Teacher Blogging

I find my blog to be a very useful teaching tool. It’s full of resources for students to explore, and after doing so they often follow up in class with interesting questions and comments. The blog helps extend our mathematical conversations beyond the classroom. Sometimes, it works too well.

In a recent discussion on probability, we considered the following question:

Suppose you randomly choose a positive integer. What is the probability that the number you choose is divisible by five?

The students thought about the question and discussed their ideas. I asked for their thoughts. The usual good answers came out ( $0, \frac{1}{5}, \frac{1}{2}, 1)$ as well as at least one bad one (infinity!?). I asked students to explain their reasoning, and an eloquent student sitting in the back volunteered to defend his answer.

“The answer is zero,” he said. “Although there are infinitely many multiples of five and infinitely many total integers, the probability is zero because a small infinity divided by a big infinity is zero.”

“A small infinity divided by a big infinity is zero?” I responded, trying to appear as perplexed as possible. “That sounds kind of crazy to me. What does that even mean?” I tried to stir up the anti-zero sentiment in the room.

The student persisted. “According to you, a small infinity divided by a big infinity is zero.”

“That doesn’t sound like something I’d say,” I said, which is what I say when students remember something I wish they hadn’t. I usually get away with it. Not this time.

“You didn’t say it,” replied the student. “You wrote it on your blog.”

In a rare moment, I had no response. What could I say? I did write it on my blog. I had nowhere to hide.

The class celebrated this clear and decisive victory.

By patrick honner, 14 years9 years ago

NYT Resources Teaching

Math Lesson: Charting Population Growth

My latest contribution to the New York Times Learning Network is a math lesson designed to get students thinking quantitatively about the increase in population growth around the world. Here is an excerpt.

A typical feature of population growth is that the rate of increase itself increases over time. Visually, this means that the line segments get steeper from left to right. When the slopes of each line segment are computed for each 10-year interval, students can look for a pattern in how the slopes, i.e. the rates of population growth, change. For example, students might notice that, every ten years, the slope of the line segments increase by 0.5 million people per year: this means that the rate of change of population increases by 0.5 million people per year.

Once a pattern is identified, students can then extend their graph beyond 2010 by drawing a line segment from the 2010 data point whose slope fits this pattern. By extending this new line segment so that it covers 10 years on the horizontal axis, this will create a population projection for the year 2020. By repeating the process, students can create population projections for 2030, 2040 and beyond.

Using a recent revision on world population growth by the United Nations population bureau as a starting point, students choose a country to profile. By using available population data, students create piece-wise linear graphs to model that countries population growth, and look for trends in order to make population projections.

You can find the full article here.

By patrick honner, 14 years8 years ago

Challenge Teaching

Fun With Folding: Centroid

Here’s another entry from my Fun With Folding series: folding the centroid of a triangle! The centroid of a triangle has a lot of interesting properties, most notably serving as the center of mass of the triangle.

To fold your way to the centroid, use the midpoint fold three times to construct midpoints of each side of the triangle. Then, fold the line through each vertex and opposite midpoint. (Click here to find instructions for these basic folds.)

Like magic, all the medians intersect at the centroid!

Be sure to try some other fun mathematical activities with folding!

Have more Fun With Folding!

By patrick honner, 14 years6 years ago

Challenge Teaching

Fun With Folding: Circumcenter

Here’s another entry from my Fun With Folding series: folding the circumcenter of a triangle! The circumcenter of a triangle is the center of the triangle’s circumscribing circle.

To fold your way to the circumcircle, use the perpendicular biscctor fold three times to construct the perpendicular bisector of each side of the triangle. (Click here to find instructions for these basic folds.)

Like magic, all the perpendicular bisectors intersect at the circumcenter!

Be sure to try some other fun mathematical activities with folding!

Have more Fun With Folding!

By patrick honner, 14 years6 years ago

Challenge Teaching

Fun With Folding: Incenter

Here’s another entry from my Fun With Folding series: folding the incenter of a triangle! The incenter of a triangle has a lot of interesting properties, most of which are related to the fact that it is the center of the triangle’s unique inscribed circle.

To fold your way to the incenter, start with an arbitrary triangle and use the angle bisector fold on each angle. (Click here to find instructions for the basic folds.)