## Girls’ Angle Bulletin, Volume 7, Number 3

Much of this issue is about the center of mass, including Julia Zimmerman’s cover drawing which features a mobile fantasy. In Math In Your World, we cover the basics involved in ensuring that a mobile will balance properly. You could use this knowledge to get your pet hamster to lift your car. In Center of Mass, we explain the “piecemeal” property of the center of mass: that you can compute the center of mass part by part.  This property forms the basis of the problem solving technique known as “mass points.”

To understand the center of mass and torque, it helps to learn about vectors.  So Robert Donley, a.k.a. Math Doctor Bob, introduces vectors in his second installment of Learn by Doing.  If you work through the problems in his column, you’ll learn about vector addition, scalar multiplication, and the dot and cross products.  Because Bob wanted to lay the groundwork for understanding physical vectors such as force and torque, he confined himself to discussing vectors over the real numbers.

In our interview, meet University of Michigan Professor of Mathematics Karen Smith.   Prof. Smith discusses a wide range of topics, including math that interests her, how she goes about solving math problems, and how she got into math.  Her route to math is rather exceptional in that after graduating from Princeton with a Bachelor’s degree, she taught in the school system before going to math graduate school.  She is a recent Clay Scholar and she contributed an autobiographical essay to the book Complexities: Women In Mathematics, edited by Bettye Anne Case & Anne M. Leggett.  In this interview, she even describes an unsolved problem which middle school students could begin to explore.

In Anna’s Math Journal, Anna launches into an exploration of the function $f(x) = x^x$ after being asked the following question: For what number x other than 1/2 is $x^x = (\frac{1}{2})^{1/2}$?

“You” helps 3/7 find a nifty solution to a curious dilemma in Coach Barb’s Corner…involving fractions, of course!

Finally, if you like dissection problems, see if you can find a nice way to tile a regular hexagon into pieces that can be rearranged to form an equilateral triangle. Henri, a student at the Buckingham, Browne, and Nichols middle school in Cambridge, was presented with just this challenge, and we exhibit his beautiful solution inside.  This problem is an instance of the Wallace-Bolyai-Gerwien theorem which says that any polygon can be dissected into parts that can be rearranged to form any other polygon of the same area. Even though there is a general construction to create such tilings, there’s still the challenge of finding nice ones, and Henri’s is quite elegant.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

## Intuiting the Chain Rule

If you’ve just learned the chain rule but feel that you have no intuition for it, this post might help.

Recall that the chain rule tells us how to compute the derivative of a composite function. Specifically, suppose $f : \Bbb{R} \to \Bbb{R}$ and $g: \Bbb{R} \to \Bbb{R}$ are differentiable functions.  Let $f'(x)$ be the derivative of $f(x)$ with respect to $x$ and let $g'(x)$ be the derivative of $g(x)$ with respect to $x$.

Then $\frac{d}{dx} f(g(x)) = f'(g(x))g'(x)$.

One way to gain an intuitive feel for the chain rule is to think of an example of a composite function that you understand well and then see that what you think about the example is consistent with what the chain rule says. Here, I’ll describe a scenario that I often use to explain the chain rule and seems to help. Continue reading

## WIM Video: Pick’s Theorem

In our latest WIM Video, Depaul University Associate Professor of Mathematics Bridget Tenner discusses Pick’s theorem.

The Girls’ Angle WIM Video series is the brainchild of Girls’ Angle director Elisenda Grigsby. WIM Videos are often shot on the spur of the moment and feature women in mathematics explaining some piece of math that excited them when they were students. We hope to create a large gallery that will showcase the diverse group of women working in mathematics today. If you support this concept, please consider making a charitable donation to Girls’ Angle. Girls’ Angle is a 501(c)3 and every amount helps!

## Do you believe this?

Here’s a triangle:

I’ll use the vertex labels to also denote the corresponding angle measures.

Take the sines of all 3 angles in the triangle: $\sin A$, $\sin B$, and $\sin C$.

Would you believe it if I told you that these three sines satisfy the triangle inequality?

Even more, would you believe it if I told you that a triangle with sides of length $\sin A$, $\sin B$, and $\sin C$  is similar to the original triangle?

## Happy New Year!

How many dots are in the base?

## Girls’ Angle Bulletin, Volume 7, Number 2

In Bisection Envelopes, a paper to appear in the journal Involve, Noah Fechtor-Pradines proves many interesting properties of the envelope of the set of lines that split a planar shape into two halves of equal area.  The cover shows the bisection envelope of the regular pentagon and in Math Buffet, you can see more images of bisection envelopes that illustrate just a few of the properties Noah discovered.  Try to sketch the bisection envelope of a semicircle, then check out the one in the Bulletin.

In this issue’s interview, meet Associate Professor of Mathematics at Loyola Marymount University, Alissa Crans.  She gives a lot of great advice and also tells us how she got into math. She explains some of her goals as the Associate Director of Diversity and Education at the Mathematical Sciences Research Institute and her work at Pathways.

This fall, Akamai mathematician Kate Jenkins visited the Girls’ Angle club and explained Dijkstra’s algorithm for finding optimal paths in graphs.  You can try your hand at applying Dijkstra’s’ algorithm in Mole Map, USA. She also described what it is like to work at Akamai, which creates products that make the internet run more efficiently.

Robert Donley, a.k.a. Math Doctor Bob, launches our new column, Learn by Doing, with a problem concerning dice.  In Learn by Doing, we are going to present mathematics by posing problems. We hope you make a point of learning math actively, with pencil and paper in hand. If you particularly enjoy learning by solving problems, check out Combinatorial Problems and Exercises, by László Lovász.

Two hurdles to overcome in mathematics are gaining facility with variables and thinking in higher dimensions. For variables, Tim Chow, a mathematician who works at the Center for Communications Research, brings us It is a Variable! And for the fourth and higher dimensions, we offer The Fourth Dimension. It’s truly remarkable that we are able to think about objects that cannot be realized in the physical world but only exist in our minds. In fact, we define mathematical objects with such precision and detail, that, in some sense, they can even seem more real than physical objects!

A bisection envelope.

In our regular columns, Anna continues looking at products of consecutive integers with respect to whether or not they can be perfect squares and Lightning Factorial discusses Fermat’s Principle of Least Time in Math In Your World.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

## 2013 AIME 2 Problem 10

I’ve been asked about the following problem from the 2013 AIME 2 a few times, so I decided to blog a couple of solutions for it:

Given a circle of radius $\sqrt{13}$, let A be a point at a distance $4 + \sqrt{13}$ from the center C of the circle. Let B be the point on the circle nearest to point A. A line passing through the point A intersects the circle at points K and L. What is the maximum possible area for $\Delta BKL$?