Teaching

Sum of Angles in Star Polygons

Futility Closet recently posted a nice puzzle about the sum of the angles in the “points” of a star polygon.

It’s easy to show that the five acute angles in the points of a regular star, like the one at left, total 180°.

Can you show that the sum of these angles in an irregular star, like the one at right, is also 180°? (link)

A clever proof is shown, but what I would consider the standard proof is clever, simple, and beautifully generalizable.

Consider the star pentagon below.

By the exterior angle theorem, we have

$\angle 1 = \alpha_1 + \beta$

$\angle 2 = \alpha_1 + \theta$

$\angle 1 + \angle 2 = \alpha_1 + \alpha_1 + \beta + \theta = \alpha_1 + 180 \textdegree$

where the last equality follows from the triangle angle sum formula.

Do this for each of the five “points” and sum the equations

$2( \angle 1 + \angle 2 + \angle 3 + \angle 4 + \angle 5) = \sum \alpha_i + 5 \times 180 \textdegree$

Now, the sum of the interior angles of any pentagon, regular or not, is $540 \textdegree$ , so this becomes

$2 \times 540 \textdegree = \sum \alpha_i + 5 \times 180 \textdegree$

And so, a little arithmetic gives us

$\sum \alpha_i = 180 \textdegree$

For stars of this type, where the points are formed by intersecting two sides of an n-gon that are separated by exactly one side, this method generalizes beautifully. The above equation becomes

$2 \times (n-2) \times 180 \textdegree = \sum \alpha_i + n \times 180 \textdegree$

which simplifies to

$\sum \alpha_i = (n-4) \times 180 \textdegree$

In particular, we see that when n = 5, we have that $\sum \alpha_i = 180 \textdegree$ . And one of the best things about having a formula like this is asking questions like “What happens when n = 4?” and “What happens when n = 3?”!

Now, there are other types of star polygons. For example, if you start with an octagon, extend the sides, and consider the intersections of two sides that themselves are separated by exactly two sides of the octagon, you get something that looks like this.

While the formula above doesn’t apply to this star, a similar technique does. The big difference is that, instead of the star’s points being attached an an n-gon (a pentagon, in the first example), this star’s points are attached to another star polygon! There are lots of fun directions to go with this exploration.

By MrHonner, 11 yearsMay 2, 2015 ago

2015 Rosenthal Prize

The Rosenthal Prize for Innovation in Math Teaching, presented by the Museum of Mathematics, aims to celebrate and promote the development of creative, engaging, and replicable math lessons. The winning teacher receives $25,000, and their lesson is shared freely by the Museum of Mathematics.

Applications for the 2015 Rosenthal Prize are now open. As a past awardee, I will be participating in an application workshop at the museum on Friday, May 1st. The purpose of the workshop is to help teachers with the application process by answering questions and providing feedback.

The workshop details, including registration information, are available here. For those unable to attend in person, the workshop will also be live-streamed.

I encourage interested teachers to consider applying for the Rosenthal Prize. It’s a unique and interesting challenge, and just participating in the process can be a growth experience.

By MrHonner, 11 yearsApril 25, 2015 ago

Statistics Teaching Testing

Regents Recap — January 2015: Questions with No Correct Answer

Here is another installment in my series reviewing the NY State Regents exams in mathematics.

This is question 14 from the Common Core Algebra exam.

Setting aside the excessive, and questionable, setup (do people really think about minimizing the interquartile range of daily temperatures when choosing vacation spots?), there is a serious issue with this question: it has no correct answer.

The student is asked to identify the data set that satisfies the following two conditions: median temperature over 80 and smallest interquartile range. No data set satisfies both these conditions. According to the diagram, the data associated with “Serene Shores” has the smallest interquartile range (represented by the width of the “box” in the box-and-whisker plot), but its median temperature (the vertical line segment in the box) is below 80.

The answer key says that (4) is the correct answer, but that data does not have the smallest interquartile range shown. Presumably, the intent was for students to evaluate a conditional statement, like, “Among those that satisfy condition A, which satisfies condition B?” But as written, the question asks, “which satisfies both condition A and condition B?” No set of data satisfies both.

Some may consider this nitpicking, but precision in language is an important part of doing mathematics. I focus on it in my classroom, and it is frustrating to see my work undermined by the very tests that are now being used to evaluate my job performance.

Moreover, this is by no means the only error present in these exams, nor is it the first example of errors in stating and evaluating compound sentences. If these exams don’t model exemplary mathematical practice, their credibility in evaluating the mathematical practice of students and teachers must be questioned.

By MrHonner, 11 yearsApril 22, 2015 ago

Resources Teaching

Building the Profession of Math Teachers

$msri national math festival$ This past week I traveled to Washington, D.C., to speak at a policy briefing sponsored by the Mathematical Sciences Research Institute. The briefing was part of a policy day held in advance of MSRI’s first ever National Math Festival.

The theme of the briefing was “Building the Profession of Math Teachers in America”. I was invited to give a teacher’s perspective on professional development, and to talk about what successful programs look like to teachers and the impact they have on classrooms and schools.

I spoke about the influence that programs like Math for America and PCMI have had on me, my colleagues and, in turn, our school. These programs inspire and empower teachers, and help create an environment where teachers are more willing and able to take on the challenges that schools and districts face.

Here’s an excerpt from the closing of my speech.

I’ve been teaching for a long time. I’ve seen new curricula, new standards, and new tests come, go, and come again. And I know that the reality of policy-making is inextricably tied to these things.

But these things don’t really lead to the kind of authentic, sustainable change that comes from empowering teachers. I have experienced this personally, I am witnessing it right now in my school, and through digital communities, I see it happening all over the country.

I was honored to represent teachers and talk about the positive impact these programs have had on me and my school. But I was definitely nervous following speakers like Nancy Pelosi, Harry Reid, Al Franken, and many other politicians and policy makers! Hopefully, my message resonated with those in attendance, and I’m thankful to the MSRI for giving me the opportunity to contribute to the conversation.

By MrHonner, 11 yearsApril 19, 2015 ago

Teaching Testing Uncategorized

Regents Recap — January 2015: Admitting Mistakes

Here is another installment in my series reviewing the NY State Regents exams in mathematics.

This is question 27 from the Geometry exam.

This question has two correct answers: it is possible to map AB onto A’B’ using a glide reflection or a rotation. The original answer key indicated that (4) glide reflection was the correct answer. After the exam was administered, the state Department of Education issued a correction and told scorers to award full credit for both (2) and (4).

Mistakes happen, even on important exams that many people work hard to produce. But when mistakes are made, those responsible should accept responsibility, not equivocate.

Here’s the official correction from the state.

It’s hard to accept that the issue here was a lack of specificity in the wording of the question. The issue is that someone wrote a question without fully thinking through the mathematics, and then those tasked with checking the problem also failed to fully think through the mathematics. This isn’t a failure in communication; this is a failure in management and oversight.

And it has happened before. This example is particularly troubling, in which those responsible for producing these exams try to pretend that an egregious mathematical error is really just a lack of agreement about notation. Sometimes errors are just erased from the record with little or no explanation, and then, of course, there are the many mistakes that are never even acknowledged.

Mistakes are bound to happen. But by pretending that substantial errors are just misunderstandings, differences of opinion, or typos, the credibility of those responsible for these high-stakes exams suffers even further damage.

By MrHonner, 11 yearsMarch 27, 2015 ago

Sum of Angles in Star Polygons

2015 Rosenthal Prize

Regents Recap — January 2015: Questions with No Correct Answer

Building the Profession of Math Teachers

Regents Recap — January 2015: Admitting Mistakes

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