The Girls’ Angle Support Network is a crucial component of Girls’ Angle’s math educational strategy. The Support Network consists of professional women who use math in their work in some vital way and have agreed to visit the club to show our members how and for what they use math.  Each visitor provides yet another reason to study math and a role model. Over 30 such women have visited so far and it has been a dream of ours to create a video series that features these extraordinary women.

Thanks to Jan Rimmel and the AboutFace Media team, people from all over can now meet Girls’ Angle Support Network member and Advisor Dr. Elissa Ozanne, a Decision Scientist who works at the Helen Diller Family Comprehensive Cancer Center at the UCSF Medical Center. It’s hard to think of a more vital application of mathematics than hers.

We hope to raise additional funding so that we can produce more such videos. You can help!

Find the positive integer such that

.

She was able to solve this using the sum of arctangents identity

but was wondering if there was another way less algebraically involved.  Also, what do you do if you can’t recall this identity?

The problem certainly suggests using the sum of arctangents formula and using that formula certainly works. If you forget the formula, you could derive it during the contest, and I’ll say more on this point further on. But, often, contest problems can be cracked by going back to basics and applying a little clear-headed thinking and a few strategic calculations.

The left hand side of the equation in the problem is a sum of four angles. Let’s recall that by definition, in a right triangle with legs of length 1 and x, the measure of the angle opposite the leg of length x is equal to the arctangent of x.  For example:

This suggests the following approach to the AIME problem (the following figures are not drawn to scale):

We start with an angle that measures the arctangent of 1/3:

We then supplement that by an angle of measure arctan 1/4 by “going over 4 and up 1″:

And then we add an angle of measure arctan 1/5 by “going over 5 and up 1″:

And now, we have to figure out so that if we “go over and up 1″, we end up on the line . This amounts to computing for which value of the point  has equal coordinates. That is, we seek so that:

.

This is a linear equation in one variable whose solution is . Done!

Explaining things tends to use much more space than what is needed to actually carry out the computation, so to give an even better sense of just how short and quick this solution is, here is my actual scratch work for this problem:

It looks like the arctangent function just disappears entirely!

Often, on competitions, a problem that strongly suggests the use of some advanced identity lends itself to a simple solution if one goes back to basics. That’s because constraints on contest problems sometimes force problem designers to use specific situations whose numbers work out so conveniently that it actually saves some effort to not use the general formulas associated with the problem content. This doesn’t always happen, of course, but in any case, it’s still useful to see if you can find a solution of this elemental nature.  Doing so increases your understanding of the basics.

By the way, the solution also shows how one can recover the sum of arctangents formula (because, of course, the solution does add up angles that are specific arctangents!):

Thus 

This issue kicks off with the concluding half of Winding Numbers by Stanford professor Søren Galatius.  The cover design is inspired by his article…the drawing is one long loop with seven-fold symmetry. Points with the same winding number are painted the same color. The central highlight consists of points where the winding number is 4.

Do you remember this result from high school geometry?

Also inside, we give a small illustration that shows how mathematics is a magical substance…one where the more you try to put into your mind, the more your mind can hold!

If you made a gumdrop & pretzel structure this holiday season, then Katherine Sanden’s Math In Your World column is for you. And if you are prone to algebraic computation errors when you expand things like , check out Cammie Smith Barnes’ Errorbusters! column. Coach Barb gives us more fraction satisfaction in another whimsical interview with . And, if you enjoyed last issue’s scrambled proof, this issue has another quite challenging one: a scrambled version of Chvatal’s Art Gallery Theorem.

You’ll note that we didn’t include the second half of our interview with Harvard professor Sophie Morel in the electronic version. Some day, we will put it in, but for now, we are hoping to entice some into becoming Bulletin Sponsors. The full interview does appear in the printed version.

Happy New Year!

For those of us born in or before 1991, we’ve lived through two palindromic years: 1991 and 2002, and there won’t be another palindromic year for another century.

But did you ever happen to notice that if you take 2011, which isn’t a palindrome, and multiply it by its reversal, 1102, you get a palindrome? .

That didn’t happen in 2010: , nor the year before that: .

But…it happens again next year: .

So, we’re about to live through two consecutive non-palindromic years whose product with their reversals is palindromic…something that won’t happen again until…any guesses?

It won’t happen again until the years 10011 and 10012! That’s a long time to wait.

This palindrome madness is all a bit silly anyway…after all, if you switch to, say, base 16, the palindromes go poof! (And if you play this game with the coefficients of a polynomial, the product will always be palindromic.) So let’s not attach undue significance to these pieces of mathematical trivia…it’s all in the lightness of the holiday spirit.

p.s. If you’d like a math problem to think about, here’s one: Suppose you pick a random whole number between 1 and 1,000,000 (inclusive) with all choices equally likely. What is the probability that the number is a palindrome? More generally, if the whole number is selected from between 1 and N, roughly how does the probability depend on N?

But I wanted to mention a few things that caught my attention these last couple weeks.

First, congratulations to the members of Girls’ Angle for persevering so valiantly at the latest end-of-session math treasure hunt at Girls’ Angle last week. I love that several of the solutions the girls came up with were different from the ones I had in mind. I’ll write more about this in the next Bulletin.

Next, there’s Math or Mess’ latest post about telescoping sums. I just wanted to point out that telescoping sums can be used to find formulas for things like the sum of the first squares. Often such formulas are used as exercises in mathematical induction. But these leave the student wondering, how were the formulas deduced in the first place?

To use telescoping sums to find a nice expression, say, for the sum of the first squares, one observes that . If one sums over from 1 to , the left hand side telescopes and we get:

,

where is the sum of the first squares. One can then simplify to obtain the well-known formula . (Another way to deduce the formula is to observe that because is a quadratic, the formula for must be a cubic in . One can then determine which cubic passes through the first 4 values of .)

And then, I love this line from Mathspig’s Zombie’s series ”Another way of calculating a zombie’s velocity or speed is to use direct proportion.” Somehow, it really captures the essence of zombie movement! I don’t know if this makes me frightened of direct proportions now or if this just makes zombies seem simple and tame…but direct proportions are one of those basic math concepts that pervade all kinds of phenomena, such as brownie cookie time (which is proportional to brownie thickness), earnings (if you’re paid on an hourly rate or something similar)…even Einstein’s famous formula is an example of direct proportion (assuming that the speed of light in a vacuum is, in fact, a constant).

Finally, there’s a new article by Jonathan Kane and Janet Mertz in the Notices of the American Mathematical Society: Debunking Myths About Gender and Mathematics Performance. I confess to not having had a chance to read it yet, but I still wanted to mention it because the contents seem interesting and several people have pointed it out to me.

When Girls’ Angle’s first semester of existence was drawing to a close, I made a small treasure hunt for the girls. I brought in identical gifts for each member, but to find their gift, each girl had to locate precisely where her gift was “buried” on a big map of Cambridge. Each girl was given a math problem. Solve the problem and find the location! When a girl thought she had the solution, she’d go to the map (which was on a computer) and erase away the area where she thought her treasure was hidden. If she was right, the word “Yes!” would appear in that spot on the map. You can read more about this mini-treasure hunt and see a “Yes!” in Volume 1, Number 2 of the Girls’ Angle Bulletin.

The next year, I brought in a big box of gifts. But when the girls went to open the box, they found it tightly wound with ribbon that was held together with two combination locks! When they protested, I handed them a stack of math problems. To get their gift, they had to solve a bunch of math problems whose solutions clued in the combinations of the locks. (See Volume 2, Number 2 of the Girls’ Angle Bulletin.)

That event proved so fun, we turned it into a tradition.

At the end of every semester, these “math treasure hunts” became more and more elaborate, and I started getting pretty good at timing it so that the members would sometimes open the last lock with just minutes to spare.

Then, about two years ago, it suddenly dawned on me that these math treasure hunts were the non-competitive, math intensive event I was looking for! After all, there is no incentive for any girl to withhold information from another because they either all succeed together in cracking the locks or they all fail together. There’s no competition of any sort, neither at the individual nor at the team level. In fact, girls have every incentive to help each other with the math so that they can all succeed!

Another cool thing about this math treasure hunt concept was that because the girls all work together, the problems could include ones that were deeper and richer than the typical math contest problem.

But creating such an event for the general public is a huge undertaking. There are a lot of logistics to work out and a boatload of math problems to construct. I began discussing the idea with others, but the discussions always petered out. Some would think it a good idea, but too much labor. Interestingly, some thought the idea terrible…because there was no competitive element! And some of those would proffer that without a competitive element, there wouldn’t be any motivation.

This past summer, Anne Juan, an undergraduate at MIT contacted me on behalf of MIT’s Undergraduate Society of Women in Mathematics (USWIM). She suggested we do a joint project and wondered if Girls’ Angle had any ideas for one. Well…

So discussions began between Anne, USWIM President Hilary Monaco, and myself. Instead of petering out, things began inching forward, and so today, USWIM and Girls’ Angle are collaborating to bring the traditional end-of-semester math treasure hunt out of the club and to the general public in the form of SUMiT 2012, a fun-filled, fully collaborative, math-intensive event for girls.

SUMiT 2012 will be limited to 30 girls in grades 6-10 and take place on January 21 at MIT. SUMiT 2012 promises to be the first of many SUMiTs which we hope will grow into a major weekend extravaganza with girls from all over the country participating in an exciting event where participants immerse themselves in mathematics while making fast and lasting friendships as they combine their thoughts and solve the problems.

Although SUMiT 2012 grows out of the in-club math treasure hunts, it is going to take place on a grander scale and have a richer story. To pull this off, USWIM and Girls’ Angle have been working closely together for months sharing ideas and divvying up tasks. For example, the beautiful SUMiT website was designed primarily by USWIM member Yuxin Xie who also made an awesome contribution to the story line, which we’re keeping a secret!

All in all, things are really coming together and if you’re a girl in 6th-10th grade who enjoys math and would like to meet other girls who like math and make new friends as you work together on math problems toward a common goal, I think you’d love SUMiT. Registration opens December 5. Also, because we are limiting SUMiT 2012 to 30 girls, and because we want to create an inclusive event, we decided to accept girls on a first-come-first-served basis. That way, if you don’t get in, you’ll know that it has nothing to do with your math abilities. It just means that 30 other girls registered before you. Hopefully, some day SUMiT will have grown large enough that we’ll be able to accommodate every girl who wants to take part.

For more information, please visit the SUMiT page.

I suppose one could theoretically imagine an individual without language skills inventing a cellular phone, but I think such an event is even less likely to occur than witnessing a chimpanzee produce Hamlet on a typewriter. People who came before us left us a legacy of symbolic representations. This post itself contains 1070 of them, some quite modern like the word “blog” (at least, in the modern sense). Symbolic representations help us organize our ideas and enable us to reach higher levels of sophistication in our thoughts.

In mathematics,  the act of creating new symbolic representations happens routinely. Look at any random mathematical post by Professor Terrence Tao, and you’ll frequently find the construct “Let X be…,” where X is some symbol. This symbolic naming skill is important at all levels of mathematics, not just at the professional level where Professor Tao operates. One early, math-specific, instance of this skill is the concept of the variable. Students who haven’t mastered variables typically get stuck solving “word problems.” Here’s a common example:

Heather routinely put her pocket change in a big jar. Because she found quarters useful for parking meters and laundry, the jar contained only pennies, nickels, and dimes. After several years, the jar became filled to the brim and she decided to cash the coins in. When she stacked them, the coins formed a column 3,593.4 mm high. When she cashed them in, she learned that there were 2,340 coins with a total value of $141.26. How many pennies, nickels, and dimes were there? (According to the U.S. Mint, the thicknesses of pennies, nickels, and dimes are 1.55 mm, 1.95 mm, and 1.35 mm, respectively.)

For people who lack facility with variables, the idea of writing “Let P be the number of pennies. Let N be the number of nickels. Let D be the number of dimes,” is beyond their ken. For this particular problem, though it may not be absolutely necessary to introduce variables to solve the problem, it helps enormously. The best way to judge the value of variables for solving this problem is to try to solve the problem yourself without using variables. (If you’re well-versed in this kind of algebra, you’ll probably find it hard to think in a way that isn’t influenced by your facility with variables.) The use of variables to solve this problem helps to isolate the mathematics from obfuscating details and organize your mathematical manipulations.

In fact, students who cannot introduce symbolic representations for the numbers of pennies, nickels, and dimes often find themselves completely stalled. They stare at the problem with no idea how to proceed. (I should state that the ability to introduce the variables is not a panacea. Problem solving requires additional skills.) Yet, these students can succeed in solving more transparent versions of the same problem. For example, though they lack facility with variables, they can usually solve the following:

Heather finds some coins in her pocket. She sees only pennies and nickels. There are 7 coins total and they amount to 15 cents. How many pennies and nickels are there?

For this problem, one can avoid variables and reason like this: “If there were 7 pennies, that would be 7 cents, so that can’t be right. If there were 6 pennies and 1 nickel, that would be 6 plus 1 times 5 or 11 cents, so that can’t be right. If there were 5 pennies and 2 nickels, that would be 5 plus 2 times 5 or 15 cents…and that’s consistent with the given information, so the answer is 5 pennies and 2 nickels.” Perhaps, such students would attempt this kind of procedure with the first problem, but they would surely give up before finding the answer. (If you’re skeptical, try this method on the first problem yourself!)

So far, my point has been that possessing the ability to make new symbolic representations (such as the act of introducing a variable) is extremely important if one wishes to be able to solve more challenging problems and understand more advanced mathematics. Now I’ll tie this in with the MCAS.

Take a look at the 2011 MCAS Grade 10 Mathematics Test. See if you can solve all 42 problems without ever using the skill discussed in this blog. You will find that it’s no problem to do so. For none of the problems is there any great benefit to introducing a new variable or making a new symbolic representation. For example, here’s problem #39:

Ken and Jerome went to the same electronics store.

• Ken bought 2 video games and 1 DVD for a total of $105.
• Jerome bought 1 video game and 4 DVDs for a total of $105.

Each video game cost v dollars and each DVD cost d dollars.

Which system of equations can be used to find the cost, in dollars, of each video game and each DVD at the store?

In this problem, the test question itself introduces variables so the student doesn’t have to.

A person can get a perfect score on the 2011 MCAS Grade 10 Mathematics Test without having the extremely important skill of being able to make a new symbolic representation.

In fact, a person can lack this skill and get good grades in math classes and high scores on math standardized tests throughout K-12. Turn it around and we see that students who present good math grades and test scores may still be lacking at least one extremely useful problem solving tool that they will need if they wish to study more advanced mathematics. I’ll dare say that this skill is critical in all research arenas. (I’m reminded of when Professor Eric Lander introduced the term “chromosome” at about 44:35 of this videotaped lecture on genetics at MIT: ”What is the appropriate scientific procedure when you have no clue what something is?… You need to give it a name…”)

I’ve discussed only one critical skill, but problem solving involves many such skills that are similarly ignored by standardized tests. Our society seems obsessed with simple, “scalable” solutions to all our problems. Some day, artificial intelligence may improve to a point where it will provide a scalable solution to quality math education for all, but until then, we must continue to support excellent math mentors. At this time, it is only skilled mentors that can diagnose what standardized tests are blind to and help students equip themselves with these important problem solving tools.

It’s fun to try to do that without watching the video first, but even after seeing the video, it can still be a bit of a challenge to figure out.

Another good exercise to try is to close your eyes and imagine the whole process. I’m always one to point out another chance to exercise the mind’s eye!

If you enjoy this sort of thing, you’ll probably enjoy disentanglement puzzles of which this is a human-sized example.

For more mathematical tricks of this nature, check out Martin Gardner’s book Mathematics Magic and Mystery. In fact, he describes vest removal there on page 87 although he begins: “First unbutton the vest.” Notice that Ryan succeeds without un-zipping hers! Also, notice how Ryan never pulls her fists into her sleeves (although there are times when she does put one of her fists up the other sleeve).

Special thanks to Claudia Meyer and Clara Chan for recording Scott Joplin’s The Entertainer for this video!

Mathematician’s Guide to Vest Removal
Starring Ryan Heffrin.
Music: Scott Joplin’s The Entertainer performed by Claudia Meyer and Clara Chan
Filmed and edited by Elisenda Grigsby and Ken Fan.
Corporate funding provided by Big George Ventures.
Produced by Girls’ Angle.

Many thanks to all those who participated!

To solve, one identifies the symbols down the left side as stock ticker symbols for various Fortune 500 companies. Next, one notes the CEO of the corresponding company as of the summer of 2011 and writes their first and last names next to their company ticker symbol. Reading down the marked squares produces the name URSULA BURNS, the CEO of Xerox. The final answer is the stock ticker symbol for Xerox: XRX.

When the puzzle was constructed, exactly 12 CEOs of Fortune 500 companies were women, and by a stroke of good fortune, 11 of these CEO’s names could be arranged to extract the 11 letters in the name of the 12th CEO! That fact formed the basis of the puzzle. Today, there are a few more women heading Fortune 500 companies, but they still comprise less than 5% of all Fortune 500 CEOs. The gender disparity among full professors in the upper echelons of mathematics is similarly lopsided.

Here are the 11 CEOs (as of the summer of 2011) whose names appear in the Puzzle of Fortune (the extracted letter is in boldface):

LAURA SEN (BJ’s Wholesale Club BJ)
CAROL BARTZ (Yahoo YHOO)
LYNN EISENHANS (Sunoco SUN)
ELLEN KULLMAN (DuPont DD)
CAROL MEYROWITZ (TJX TJX)
PATRICIA WOERTZ (Archer Daniels Midland ADM)
ANGELA BRALY (WellPoint WLP)
ANDREA JUNG (Avon Products AVP)
INDRA NOOYI (PepsiCo PEP)
BETH MOONEY (KeyCorp KEY)
IRENE ROSENFELD (Kraft Foods KFT)

According to Wikipedia, Ursula Burns is not only the first African-American woman to head a Fortune 500 company, she is also the first woman to succeed a woman as CEO of a Fortune 500 company.

One reason for this unacceptably high attrition rate is that schools do not, generally, prepare students appropriately for the demands of the college STEM (Science, Technology, Engineering, Math) majors. One student he interviewed left his STEM major despite having had strong SAT scores and multiple AP courses under his belt. This was not an isolated incident. When I taught at MIT and Harvard, I witnessed many students abandoning their mathematical aspiration even though they had taken everything their high schools had to offer in mathematics.

At Girls’ Angle we have designed a program where girls can get the kind of preparation they need in order to thrive in a STEM major at the best universities in the country. Our program begins with a careful selection of our mentors. We recruit mentors who are at various stages of the STEM career path, including undergraduates, graduate students, and at least one mentor with a doctoral degree in math at every meet, and we select only those who show an aptitude for nurturing the girls to develop their thinking skills and confidence. These mentors know what it takes to surmount their respective rung of the ladder. Collectively, the Girls’ Angle community is like a human chain from Middle School all the way up to the highest levels of the mathematics profession.

Preparing students for a STEM career is a long and gradual process that continues beyond school through college into graduate school and even at the postdoctoral level. The process cannot be forced lest it becomes an unpleasant experience. So we take a very long view on our member’s education. That’s why Girls’ Angle is open to girls all the way through High School. (It’s also why one cannot get any sense of the possibilities at Girls’ Angle by attending just a few meets.)

Of course, not everybody will enjoy the STEM subjects, and if you discover that about yourself there’s no use in forcing yourself to pursue one. But for those students who do enjoy STEM subjects it would be unfortunate if they were forced to abandon a STEM career because of inappropriate preparation. At Girls’ Angle, our goal is to provide girls who have the STEM dream with the necessary ability, skills, confidence, and mindset so that the STEM career becomes a choice that they make for themselves. It is this long term goal of empowerment to make one’s own choices that guides our every move at Girls’ Angle.

Girls, learn to think well, think for yourself, and make your own choice!