Solution below the break.
Thursday, March 31, 2016
See if you can integrate some Calculus tricks into this week's Advanced Knowledge Problem of the Week!
Solution below the break.
Solution below the break.
Tuesday, March 29, 2016
Thursday, March 24, 2016
This week's Advanced Knowledge Problem of the Week diverges a bit from previous problems, as it covers topics from multivariable Calculus. See how you fare!
Solution below the break.
Wednesday, March 23, 2016
Women in STEM: Bridging the Gap
With Women's History Month winding to a close, will the conversation about STEM education for females also dwindle to a mere whisper? We have already written about the accomplished women in the STEM fields, but what about the young mathematicians and scientists that are trying to break into the male-dominated area of study? While the disparity between males and females in STEM is easy to spot, the root of the problem is not as obvious. For years, researchers have made efforts to determine what exactly causes females to shy away from careers in math and science. Pinpointing specific problems would allow for an effective solution to take shape. This post is aiming to inform people about the obstacles that stand in the way for women in STEM, as well the work that has been done to bridge the gender gap.
So, what's the problem? And why does it matter?
In the general workforce, females are only slightly outnumbered by men. Men encompass 52% of the workforce, while females make up the other 48%. However, in the fields of science, technology, engineering, and mathematics (STEM), woman make up only 24% of the workforce. This statistic shows that as most industries are closing the gender gap, the STEM field is making much less progress. This is significant to women, because in the STEM field the wage gap between men and women is actually much smaller. In STEM, the wage gap is only 14%, whereas is all other fields the gap is 21%. In addition to this, STEM careers have higher salaries in general, averaging over $10 more an hour. There is also a greater chance for career advancements. More importantly, in an era of global competition and technology, shouldn't we be utilizing all possible talent in the important fields of math and science? The gender gap in STEM shows that a significant supply of intelligent and capable female workers are not being used efficiently.
Disparity in Higher Education
New studies show that females are actually more likely to earn a college degree than males are. However, this trend does not translate into the fields of math and science. In fact, the opposite seems to be true, as a smaller share of STEM degrees are going to females. Between 2004 and 2014, the percent of STEM degrees earned by women decreased in each discipline area. On average, only 35% of STEM degrees are earned by females, and 65% are earned by males. The fields with the greatest discrepancy between men and women are computer science and engineering. Each only confer about 18% of degrees to females. Interestingly enough, the gap grows throughout college. In other words, there is a lower retention rate for females in STEM. This is particularly true in the fields dominated by men, such as engineering. Out of 100 female students, only 12 will graduate with a stem degree.
Perception and Stereotypes
Picture it. A stereotypically "nerdy" male with a female friend that struggles with basic math–a twist on the classic, "Beauty and the Geek". How many shows follow this plot line? I can think of a few off the top of my head. Even in the modern era, our culture is pervaded with the idea that young boys should play with legos, while girls should play with Barbies. It may seem trivial, but even simple toy or show choices can impact the decisions young children make, and set into motion their likes and dislikes. According to the Organization for Economic Co-operation and Development, at age 9 the gender gap in science and math starts to widen. About 2/3 of girls under the age of 12 claim to like science, yet by the time they reach high school, opt out of advanced level math and science courses. What causes them to diverge from their original interests? Around age 12, girls start to lose confidence in their mathematical abilities. In a survey, a 15 year-old females were more likely to"get nervous when doing a math problem", or "worry that they will get low marks" than their male counterparts. This does not necessarily reflect their grades, but instead shows the mindset that young girls assume when it comes to learning in the STEM fields.
Unfortunately, some teachers fail to reassure these students. In a NBER study, a researcher found that when grading math, teachers who knew the gender of the test taker were more likely to show a bias favoring male students. When they were unaware of gender, the exams were scored more evenly. This information seems to show that students and teachers alike are likely to adhere to the gender stereotypes set forth in the past. In fact, students in entry level college biology classes were asked to guess which students in their class had the highest grade. The answers showed that male students overwhelmingly selected other males, while females selected males and females equally. Could this be a reason why more females don't continue taking STEM classes after beginner courses?
What's being done to help?
Getting to the root of the problem, however, involves a change in mindset from educators, students, and STEM professionals alike. In an article by Tech Crunch, Erin Sawyer explains the importance of incorporating STEm education at an early age and continuing to foster a connection to math and science. This involves showing young girls practical uses of STEM. For an example, Kids' Vision is an after-school program that brings girls into Silicon Valley and exposes them to the tech industry. Likewise, Technovation is placing leadership in young girls' hands, holding competitions to build confidence and familiarity with the STEM field. These programs can help initiate an interest in STEM, and broaden the career opportunities that young women see.
So what can you do?
Tuesday, March 22, 2016
Friday, March 18, 2016
Game Time: Math Style
We'll start by taking a look at our inspiration- March Madness.
Before the tournament tips off, it is estimated that more than 40 million Americans flock to ESPN or other sports-based websites to fill out their own tournament bracket. In all, it is estimated that about 70 million brackets will be completed. Bracket-filling strategy is a personal preference. Some stay loyal to their favorite college, selecting them to win-it-all, even though it isn't probable. Others hope to shock their competitors and pick underdogs, crossing their fingers that a rare bracket will do the trick. On the other hand, some people pick top seeded teams, leaning towards favorites.
According to USA Today, the odds that you randomly select a perfect bracket– that's guessing every game correctly– are 1 to 9.2 quintillion.
Busy swinging the bat and keeping their eye on the ball, baseball players aren't consciously thinking about the mathematics that make their sp ort possible. However, hitting a home run or even making it to first base is centered heavily on angles, velocity, and energy. The speed of a pitch or of a ball after it's hit can be found using a specific equation. Ho is the height from which the ball is thrown, α is the angle at which the ball is thrown, vo is the speed at which the ball is thrown, and x is the distance that the ball travels. From the graph below, it can be seen that hitting the ball at a 45 degree angle will cause the ball to travel the farthest.
|Distance baseball will travel|
|Graph showing range with different α's.|
Black graph α = 30o, blue graph when α = 45o, red graph when α = 60o
|Projectile motion of a baseball|
It can be argued that any sport involves similar physics. So why was baseball one of the sports chosen to examine with a mathematical lens? Mathematical approaches to managing baseball teams have surfaced throughout the sport. In fact, the term sabermetrics specifically describes the way in which statistical analysis is applied to baseball records. The term was coined from the acronym SABR (Society for American Baseball Research), and is used to evaluate and compare baseball players. Sabermetrics takes an emotionless, objective approach to baseball. It aims to answer only questions that can be proven with facts. Perhaps one of the most celebrated proponent of sabermetrics is Billy Beane, who inspired the movie Moneyball. Beane was the General Manager of the Oakland Athletics and used statistical analysis to lead the A's to a winning season. He looked at the risk factor of each player he brought into the program, and examined who was worth the money. He looked for players who may not have carried a famous name, but could still contribute to the team. In other words, he stretched the dollar and looked for economically smart choices. This is one example of how sabermetrics and analyzing stats is prominent in the sport of baseball.
Soccer is the world's most popular sport, and every day millions of people around the world take the field– from professionals to small children. However, I doubt any of these players take the time to examine the geometry that makes a soccer team successful. The basic shape of soccer is a triangle. the players on the field are connected by imaginary triangles, that build upon each other to diamonds and other shapes. Why a triangle? It allows for the most passing lanes and provides an option in every direction. The best teams in the world are known for being able to move the ball around the field in these triangles.
Free kicks provide another use for the application of geometry. Defenders line up in a wall, in hopes to impeded a direct path to the goal. The wall is set up at a specific angle and the person kicking the ball tries to bend the ball behind the ball. However, the amount of people that stand in the wall is dependent on where around the goal the ball lies. While the goalie isn't thinking about math when he/she sets up the wall, the logic behind it is definitely mathematical. The chart below shows that as the ball moves away from the center of the goal, less people are needed to defend the wall. When the ball is in the middle of the goal, the shooter has a larger angle, so the width of the wall needs to be greater.
Thursday, March 17, 2016
Legend has it that Saint Patrick's day involves the shamrock, or a three leaf clover. After St. Patrick trained as a priest and bishop he traveled Ireland in 432AD with a goal to convert the pagan Celts who inhabited the Island. Having previously lived there, he was aware that the number three held a special significance in Celtic tradition, and pagan beliefs. Essentially as a technique to convert the pagan Celts, he used the shamrock to explain the Christian concept of the Holy Trinity.
Apart from the religious aspects of this day, many in Ireland and other Western nations use St. Patrick's as an excuse to drink and attend their city's parade. With so many websites focusing on the science of beer on St. Patricks day, the Center would like to reference Matthew Francis science blog, Galileo's Pendulum. The article discusses one of Ireland's greatest mathematicians, William Rowan Hamilton, and one of his great discoveries.
In 1843, he was walking along the Royal Canal in Dublin in deep thought regarding complex numbers and if they could be extended to higher dimensions. Once he realized they could be he carved his solution on the Brougham Bridge. The Plaque shown on the right commemorates his discovery.
The equation he carved on the Brougham Bridge is:
For physics enthusiasts: Hamilton is also the man who discovered Hamiltonian dynamic, which in turn underlies quantum mechanics and several aspects of chaos theory.
To better understand this solution, remember that an imaginary number is the square root of a negative real number. A real number, includes all counting numbers, negative numbers, ratios and irrational numbers, like π that can't be expressed as fractions. The imaginary unit is the square root of -1. Complex numbers are the sum of a real number and an imaginary number.
For more information about Hamilton's solution please refer to this article: http://bit.ly/1PcxDEt
Q: What is your favorite St. Patricks day memory in Ireland?
A: As a kid each village had a bag pipe band. I loved watching them perform on the streets with my friends. Also, St. Paddy's day was during lent, so we were allowed to eat some of the candy we were saving up for Easter Sunday.
Q: Whats the main difference between how the holiday is celebrated in the United States and in Ireland?
A: Since St. Paddy's day is a national holiday in Ireland, no one goes to work or school. Also, the older community in Ireland sees it as more of a religious holiday and will attend church.
Q: What is the legend of the leprechaun?
A: No idea - it's an American tourist thing!
Q: Are there any St. Patrick's day themed problems the Center has been working on?
A: Yes this is one for the American audience!
For the solution video click here!
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In lieu of our normally scheduled Advanced Knowledge Problem of the Week, today's problem is St. Patrick's day-themed. See if you can help Paddy escape from his captors!
Stumped? Check out the solution below!
Stumped? Check out the solution below!
Tuesday, March 15, 2016
Give this Problem of the Week a tri-angle, and let us know what you came up with! There are a lot of great possibilities for this week's problem.
Solution(s) below the break!
Solution(s) below the break!
Monday, March 14, 2016
Its March 14th (3/14) and The Center of Math would like to wish everyone a happy Pi day! Weather today is an excuse to make pie, eat pie or throw it in someones face, we hope its a good one! Taking a trip back to basic mathematics, Pi (π) or 3.14, is the ratio of a circles circumference, its distance around, to its diameter. As Pi is a constant number, it will remain the same for all circles of any size.
HistoryThe History of Pi dates back to the Old Testament of the Bible. The mathematician Archimedes used polygons with several sides to measure circles and concluded that Pi was approximately 22/7. The adoption of "π" was first used in 1706 by William Jones, symbolizing "p" for "perimeter." In more recent years, Pi has been calculated to more than one trillion digits past its decimal. However, only 39 digits past the decimal are required to calculate the spherical volume of our entire universe. Many find it an entertaining challenge to memorize as many digits as they can, especially today, when several Pi memorizing competitions take place.
To date, the longest recitation of Pi is held by Akira Haraguchi. On March 14, 2015 (last year's Pi day), the 60 year old man needed more than 16 hours to recite the number to 100,000 decimals, breaking his 1995 personal best of 83,431. Haraguchi developed a system where he assigns kana symbols, or Japanese scripts, to memorize Pi as a collection of stories. The same system was developed by C.S. Lewis, who assigned letters from the alphabet to numbers, creating stories to memorize numbers.
However, the Guinness Book of World Records has not yet accepted any of his records set, making them unofficial. The Guinness-recognized number of π is held by Rajveer, 21, reaching 70,000 digits. Rajveer wore a blindfold throughout the entire recall, which took nearly 10 hours.
Tau VS Pi
Currently, the "Tau VS Pi" debate lies under the premise that,"pi is a confusing and unnatural choice for the circle constant. Far more relevant, according to the algebraic apostates, is 2π, aka tau." Tausupporters argue that mathematicians don't necessarily care about pi's ratio, a circles circumference with its diameter. Almost every mathematical equation about circles in written in terms of r for radius, and tau is the number that connects a circumference to that quantity.
Critics don't mean anything as been miscalculated, Pi still equals the same infinite string of never-repeating digits, however professor Robert Palais started the "pi is wrong" outburst with an article to match. The article delves into the various ways using tau in beneficial when learning and solving fundamental concepts in trigonometry. For example, with pi-based thinking, one third of a circle equates to two thirds pi radians, and three quarters around is one and a half pie radians. Sounds complicated for a beginner, right? By contract, a third of a circle is a third of tau, and three quarters of a circle is three quarters tau. Palais remarks, "the opportunity to impress students with a beautiful and natural simplification is turned into an absurd exercise in memorization and dogma."
Another major plus side of a universe where Pi is replaced by Tau, you get to eat twice as much pie on Tau Day!
On a lighter, more fun note, check out some Pi Day inspired youtube videos!
To accompany our new Pre-Calculus Blueprint, I have created a short series of problems highlighting concepts that students often struggle with while going through a Pre-Calculus course. These problems stand alone from the Blueprint resource, so feel free to work through them on your own.
Following are the problem and solution transcripts:
Solution videos on this playlist:
Following are the problem and solution transcripts:
Solution videos on this playlist:
Friday, March 11, 2016
Coming soon to theaters near you: Mathematics
You may not run to the nearest theater to watch a mathematician prove a theorem, but math may be starring in your favorite movie without your recognition. This installation of Everyday Math features big screen hits that would not be possible without mathematics. Usually a math story line is intertwined with a romance, mystery, or comedy, which can make it easier to miss. Don't worry! We're here to point out our favorite films that carry a heavy dose of real math information. Grab your popcorn and make your next movie night a math-themed experience!
The Classic: A Beautiful Mind
A Beautiful Mind is based loosely on the real life story of John Nash, a Nobel Prize winner. Russell Crowe plays John Nash, a mathematical genius that specialized in game theory, differential geometry, and differential equations. Game theory can be utilized in fields such as economics and political science. In fact, Nash won his Nobel Prize in economics. In a famous scene, the film dramatizes Nash's discovery of the Nash equilibrium, a term used in economics and game theory. The film is said to take artistic interpretation of Nash's real life, but the mathematics in the movie are based on real theorems and theories. The director of A Beautiful Mind enlisted the help of a mathematics consultant, Dave Bayer of Columbia University, to ensure the mathematics were correct throughout the film. The movie contains some math jokes and facts that may only be clear to those well versed in mathematics. For an example, at the end of the film, a student wants to show Nash a proof exploring the idea that "finite Galois extensions are the same as covering spaces", which is actually a true statement. Along with his prowess in mathematics, A Beautiful Mind also demonstrates Nash's story of mental illness and schizophrenia. His schizophrenia initially impacts his career, but he is able to recover and take his place as one of the leading mathematical and economic minds of his time.
|Watch the dramatized discovery of Nash's Equilibrium here!|
The Romantic Drama: Proof
Proof, starring Gwenyth Paltrow and Anthony Hopkins, ties in universal themes with the backdrop of mathematics. The movie tells the story of 2 mathematicians, father and daughter, fighting mental illness and attempting to prove various mathematical theorems. Before he dies, Robert (the father) makes note of the interesting characteristics of the number 1729. The daughter eventually takes the place of her ailing father and dedicates herself to mathematics, even though she lacks formal training. Critics explain that the film realistically expresses the field of mathematics. It shows the nuances of proving a theorem, as well as the work and studying that are involved. Paltrow's character, Catherine, describes how she feels when she is attempting to solve a difficult problem, comparing "elegant proofs" to music. Proof also makes reference to other real mathematicians that may or may not be known to the public, such as Sophie Germain and Carl Friedrich Gauss. The director of the film consulted heavily with Timothy Gowers, a Fields medalist from Cambridge University.
For the Sport's Enthusiast: Moneyball
|Watch a mental math scene here|
The Teen Rom-Com: Mean Girls
This may be an outlier of the group, as the movie is geared towards teenage girls and features all the facets of your typical high-school movie. Mean Girls star Lindsey Lohan plays a homeschooled girl who tries her luck at navigating high school for the first time. She comes across some new friends, and they attempt to steer her in the right direction. Cady Herring, Lohan's character, has an aptitude for math and even joins the math club. Her math teacher, played by Tina Fey, is featured several times in the movie explaining various high-school math concepts. Late in the movie, Cady attends a mathlete competition and is faced with a limits problem. There are also scenes where she is being tutored in calculus, and the math errors showed are typical errors a high school student would make. This movie is perhaps the epitome of teen-movies circa 2004 but the mathematics represented, although correct, definitely correspond with the high school setting.
|Click here to watch the scene!|
The Story of An Underdog: Good Will Hunting
Known as a classic math movie, Good Will Hunting is a must-see. This underdog tale has a romantic twist and stars both Matt Damon and Robin Williams. Matt Damon's character, Will Hunting is a troubled young adult who's life path weaved in and out of foster homes and trouble with the law. While working as a janitor at MIT, he is able to solve two math problems that were created for graduate students. A professor at MIT took interest in Will, noticing his affinity towards mathematics and genius-level ability. Along his journey, Will faces his inner struggles with the help of a therapist (Robin Williams), and meets a romantic interest who helps to shape his life. The math problem that Will faces on the board is actually a real problem, although not as difficult as it is made out to be. The problem involves a feature of graph theory, homeomorphically irreducible trees. Pictured below is the problem that sent Will into the realm of academia. Interestingly enough, the math brains behind the movie actually appeared on screen as well. Patrick O'Donnell, who had a minor roll in the bar scene, actually ran the math department at University of Toronto at the time. O'Donnell and John Mighton, who plays the professor's assistant, chose the equations and theorems used in the movie.
|See the solution here!|
Wednesday, March 9, 2016
Monday, March 7, 2016
With Hilary Clinton leading the democratic polls - potentially being our first female president, and National Women's History being one of March's claims to fame, the Center of Math found it appropriate to shed light on historic female mathematicians!
Sophie Germain (1776-1831)
Her love for the study of mathematics was sparked by her isolation during the French Revolution. She thought herself Latin and Greek to understand classic mathematical works. Being female, and unable to study at the École Polytechnique, she obtained lecture notes and submitted papers to professor Joseph Lagrange, under a false name. Once he discovered she was a female, he became her mentor. His many networks lead Sophie to make her own connections with other prominent mathematicians. At the time, her work was not up to par with other male mathematicians due to her informal training and lack of resources. Despite the odds against her, she became the first woman to win a prize from the French Academy of Sciences on the theory of elasticity and her proof of Fermat's Last Theorem, though unsuccessful, was used as a foundation for work on the subject.
Her overprotective mother, encouraged Ada to stray from her father, Lord Byron's path. The poet left Ada and her mother in England after a scandal shortly after his daughters birth. Ada's mother pushed her toward the study of science and mathematics, later collaborating with the inventor and mathematician Charles Babbage. He gave Ada the responsibility of translating and Italians mathematician's memoir, analyzing his Analytical Engine - some would consider this one of the first computers. Beyond translating the memoir, Ada included her own set of notes and a method for calculating a sequence of Bernoulli number, now acknowledged as the first computer program.
Like Sophie Germain, Sofia was not allowed to attend university due to her sex. She contracted a marriage with Vladimir Kovalevsky and moved to Germany. There, she was tutored and after writing treatises on partial differential equations, Abelian integrals and Saturn's rings, received a doctorate degree. She was later appointed lecturer in mathematics at the University of Stockholm and later became the first woman in that region to become a full time professor. In 1988 she won the Prix Bordin from the French Academy of Sciences and in 1989 won a prize from the Swedish Academy of Sciences the next year.
Emmy Noether (1882-1935)
Albert Einstein, in 1935 wrote a letter to the New York Times describing deceased Emmy Noether as "the most significant creative mathematical genius thus far produced since the higher education of women began." After receiving her PhD, for a dissertation on abstract algebra she had a hard time obtaining a position a university position. She overcame this hurdle by moving to America and becoming a lecture and researcher at Bryn Mawr College and the Institute for Advanced Study in Princeton, New Jersey. There, she developed many mathematical foundations for Einstein's general theory of relativity while making advances in the field of algebra.
Maryam Mirzakhani - Fields Medallist
More recently, in 2014, the first woman was awarded the Fields Medal. Awarded by a committee from the International Mathematical Union (IMU), the Fields Medal is almost synonymous with a Nobel Prize in the world of mathematics. The medal, usually awarded to between two and four researchers, must be younger than 40, motivating the winner to strive for "further achievement" as well as recognizing their success. Maryam Mirzakhani broke the male dominated winning streak, dating back to 1936. As a moment hailed as "long overdue", Mirzakhani was recognized for her work on complex geometry. Professor Dame Frances Kirwan, a member of the medal selection committee, expresses her contempt toward the math field and how it is viewed as "a male preserve", while women have contributed to mathematics for centuries. She describes Mirzakhani's win: "I hope that this award will inspire lots more girls and young women, in this country and around the world, to believe in their own abilities and aim to be the Fields Medallists of the future."
Thursday, March 3, 2016
Stressed About Tests? The SAT is getting a makeover and we're here to help you put your best foot forward!
When college decision day rolls around, every high school senior hopes to see a “Congratulations!” envelope stuffed into their mailbox. However, as competition amongst students increases, it is becoming increasingly difficult to get accepted into your college of choice. While extracurricular activities and GPAs are important in adding substance to your college application, there is one factor that may just stand above the rest— the SAT. No high school student wakes up on a Saturday morning yearning to take a 6 hour exam, but it may expected that students who have success in high school math classes would have a similar fate on the math portion of the SAT. Surprisingly, this often isn’t the case. Students who excel in high level math courses are often discouraged and frustrated when their SAT scores fail to align with their school performance. Why does this disparity occur so rampantly? The focus of math classes is typically centered on methodical thinking and importance is placed on the correct steps rather than just the correct solution. This focus on rationale and problem solving is great for grasping complex concepts, but does not match up with the format of the SAT. The multiple choice questions of the SAT do not allow for partial credit, and only the correct answer gives the student points. In math classes, shortcuts are discouraged, but those same shortcuts can prove to be incredibly advantageous during the SAT. The new SAT attempts to help ease the frustration and dread that surrounds the test. The focus shifted to mirror the Common Core and material being covered in classes. A "real-world" f
Know Your Enemy: Understanding the new SAT
In order to combat the trend of more students choosing the ACT over the SAT, the College Board decided to give the SAT a makeover. It is important to understand the alterations made in order to best prepare to take the exam.
Here are some prominent changes that will appear on the new SAT:
1. Eliminating the Guessing Penalty
Historically, the SAT subtracted points for guessing an incorrect answer. This led students to score higher if they left certain difficult questions blank. However, now that the guessing penalty is eliminated, students should make sure to fill in each question. Eliminate answers until you have narrowed down the options, then make your best guess!
2. Fewer Answer Choices
This change goes hand in hand with number 1. Rather than 5 answer choices, there will only be 4 possible options. This raises the percentage of guessing correctly from 20% to 25%. This will also save time, as test takers only have to read and debate 4 options rather than 5.
3. Change in Vocabulary Questions
That's right.. put your flashcards away! Rather than extra challenging and even obscure vocabulary words, the College Board is filling the language sections with words likely to be heard in college classrooms. Words like "empirical" or "synthesis" are more closely aligned with the new exam than old SAT words like "impecunious" or "legerdemain".
What about the Math section?
The new SAT also made changes to the math concentrations on the exam. The new exam will focus on linear equations, complex equations/functions, and percentages/proportional reasoning. Like the vocabulary words, the math portion is more closely linked to math you will use in the 'real world'. There will still be both a calculator and non-calculator section, as well as a grid-in portion that accompanies the multiple choice questions. According to the College Board, the new exam is intended to "mirror the problem solving and modeling you'll do in college math and science courses, the jobs you hold, and your personal life".
The new version of the exam will premier on March 5th. While some students welcome the changes, others are apprehensive because there is less practice material available for the new test. College Board also explains that the grading period will be longer than in previous times, due to the rollout of the new version. The old exam and new exam can not be super-scored together, so those students who don't perform as well as they hoped, may have to retake the new exam again. Our advice from the Center of Math is not to over-stress! Study as much as possible then relax, sharpen your #2 pencil and try your best!