Marion Walter’s Theorem Via Mass Points

I recently had the good fortune of learning Marion Walter’s theorem from Marion Walter herself:

blog_071214_00

Marion Walter’s Theorem: In a triangle, draw line segments from each vertex to the trisection points on the opposite side. The six drawn line segments will form the edges of a central hexagon. The ratio of the area of the hexagon to that of the whole triangle is 1/10.

An efficient way to prove Marion Walter’s theorem is to use mass points.

In this post, I’ll give details because the proof is a model example of the mass points technique. If you’re having difficulty learning the technique, I hope this post will help it all come together for you. As always, try to use mass points to prove the theorem yourself, and, only after you have tried, read on. If you’ve never heard of mass points before, google it or check out Volume 7, Number 3 of the Girls’ Angle Bulletin.

Spoiler Alert! Proof Below!

Continue reading

Posted in math | Tagged , | Leave a comment

Girls’ Angle Bulletin, Volume 7, Number 5

blog_063014_01The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

A paraboloid of revolution adorns the cover. Anna investigates cross sections of paraboloids in this issue’s Anna’s Math Journal. It sure feels like Anna is embarking on an interesting mathematical journey with this new topic. We hope you’ll be inspired to follow-up on her work.

However, first up is the concluding half of our interview with University of Oregon Professor Emerita Marie Vitulli. Read some of the ways she thinks gender bias in mathematics can be countered.

Next, Akamai Technologies computer scientist Kate Jenkins discusses algorithms that find the “maximal subsequence” of a sequence.  Her first part closes with an interesting challenge. Can you find a solution to her challenge before she gives it in the next issue?

blog_063014_02Brit Valeria Golosov presents a fictional account of how she imagines that Brahmagupta derived his famous formula for the area of a cyclic quadrilateral. Valeria is entering her final year of secondary school in London.

This issue’s Math In Your World was specifically requested by Vida John. We love receiving content requests from members and subscribers. This Bulletin is written for members and subscribers and members and subscribers are welcome to control Bulletin content by emailing us comments and suggestions.  Please don’t be shy about emailing us about anything to do with math! We also welcome and encourage all members and subscribers to send in solutions to the Summer Fun problem sets. We might even publish your solution in the Bulletin.

blog_063014_03This summer’s batch of Summer Fun problem sets address magic squares, mass points, quadratic reciprocity, and permutations. Contributors include Johnny Tang and Cailan Li, both recent high school graduates who will be heading to college this coming fall. The central theme of Volume 7, Number 3 of the Bulletin was the concept of center of mass which underlies the technique of mass points. In that issue, we didn’t have enough room to include many problems to practice the technique. So that’s one reason why we included a problem set on mass points. The problems range from introductory level to some that will hopefully entertain those experienced in the technique. Cailan’s problem set takes readers from the rudiments of modular arithmetic all the way through a proof of Gauss’s Law of Quadratic Reciprocity, following a proof by D. H. Lehmer. The set on permutations culminates with a result of Zolotarev that links signs of certain permutations to the Legendre symbol introduced in Cailan’s problem set.

To whet your appetite, suppose AB, and C are the angle of a triangle. Can you prove that

9 > 3 + 2(\cos 2A + \cos 2B + \cos 2C) \ge 0

with equality if and only if the triangle is equilateral? For a spiffy way to prove this, check out the Summer Fun problem sets!

We conclude with a brief account of a wonderful field trip we took to MIT’s Department of Aeronautics and Astronautics, which was generously organized by Professor Karen Willcox,

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!

Posted in math, Math Education | Tagged , , , , , , , , , , , , , , | Leave a comment

Cubes in one-point perspective

This post is a reply to Jamie’s comment on Drawn to Math:

Would anyone know how to construct a cube in one point perspective?

This is an excellent question because it isolates an important, simplified situation that enables one to study key aspects of perspective drawing.

(If you are completely new to perspective drawing, I’d suggest working through the Summer Fun problem set on pages 21-22 of the June, 2013 issue of the Girls’ Angle Bulletin.)

Continue reading

Posted in applied math, math | Tagged , , , , | Leave a comment

Girls’ Angle Bulletin, Volume 7, Number 4

Cover of Volume 7, Number 4The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

The cover features a planar configuration by Leah Berman, associate professor of mathematics at the University of Alaska Fairbanks.  There are 240 lines and 240 points arranged so that each line contains 6 of the 240 points and each point sits on 6 of the 240 lines. More images of planar configurations by Leah and Nadine Alise can be found in Mathematical Buffet.

This issue’s interview is with University of Oregon Professor Emerita Marie Vitulli.  In this first part of a 2 part interview, we learn about Prof. Vitulli’s field and how she got into mathematics.

GABv07n04_TMoore_01Special thanks to Professor Emeritus Thomas Moore for contributing a problem about Pythagorean triples.  If you haven’t heard of Pythagorean triples, Prof. Moore gives a brief introduction and more challenges in Pythagorean Triples Challenge.

 

GABv07n04_angledglass_12Angles pervade much of this issue.  In Learn By Doing, Addie Summer covers the basic of angle measure.  Then, Lightning Factorial follows Emily and Jasmine as they use angles to design a stained glass window.  Finally, in this issue’s Math In Your World, I write about one of John and Jane Kostick’s latest creations, which they dub the Quintetra Assembly. blog_043014_02 To compute the necessary angles needed to create a wood sculpture for this amazing polyhedron, several angles must be computed.  In this article, I sketch how to determine these angles and include a net, courtesy of the Kosticks, for Jane’s Quintetra block, 30 of which can be used to build a model of the Quintetra Assembly.

Finally, Anna continues her investigation of x to the x.

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!

Posted in math, Math Education | Tagged , , , , , , , , , , | Leave a comment

Girls’ Angle Bulletin, Volume 7, Number 3

Cover of the Girls' Angle Bulletin, Volume 7, Number 3The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

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.”

GABv07n03_MIYW_g01To 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.

Graph of x to the x.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!

Posted in math, Math Education | Tagged , , , , , , , , | Leave a comment

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

Posted in math | Tagged , | Leave a comment

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!

Posted in math, Math Education, WIM videos | Tagged , , , , | Leave a comment