Advanced Knowledge Problem of the Week

See if you are up to the challenge of this week's Advanced Knowledge Problem of the Week!

Solution below the break.

Women History Month

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!

Hypatia

Despite there not being records of the first female mathematician, Hypatia is certainly recognized as one of the earliest. Following in her fathers footsteps, she invested her time int he study of math and astronomy. Her father, the last known member of the famed library of Alexandria, collaborated with Hypatia on various commentaries of classical mathematical works. This process involved translating mathematical methods and translating them, while incorporating explanatory notes. The partnership between her and her father inspired Hypatia to create her own commentaries and including them in her practice as a teacher. Her passion of philosophy and her commitment to the Neoplatonism belief system, a system where everything emanates from "The One", lead her to become a convenient scapegoat in various political outbreaks. Unfortunately, she was killed by a mob of Christian zealots.

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.

Ada Lovelace 

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. 

Sofia Kovalevskaya 

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." 

Sources: http://www.smithsonianmag.com/science-nature/five-historic-female-mathematicians-you-should-know-100731927/?no-ist
http://www.bbc.com/news/science-environment-28739373

Problem of the Week

See if you can factor in your Number Theory knowledge to help you solve this week's Problem of the Week!

See the solution below the break.

The SAT Gets a Makeover

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.

Google Images

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".

Google Images

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!

Good Luck!

Sources:

https://collegereadiness.collegeboard.org/sat/inside-the-test/math
http://www.boston.com/news/education/2016/03/03/what-expect-from-the-new-sat/TV4MttWNBl6PPHKPHUPq6O/story.html?p1=feature_pri_hp

Advanced Knowledge Problem of the Week

This week's Advanced Knowledge Problem of the Week asks you to prove a statement about a sum of trigonometric functions.

Solution below the break!

Everyday Math: Mathematics in Art

Everyday Math: The Art of Mathematics

Students and teachers alike often place art and mathematics on opposite sides of the academic spectrum. They consider there to be little overlap between the logic that blooms in a math class and the creativity that flourishes is an art room. Rather than running beside each other as parallel lines, mathematics and art intertwine in a way that creates some of the world's greatest masterpieces. The mathematical techniques used by renowned artists proof that math is everywhere, seeping into every aspect of academia and entertainment. 

The Golden Ratio

The Golden Ratio is explained algebraically as .
This ratio, also known as the Divine Proportion, is used by artists on a geometric scale. For an example, the Golden Rectangle is a specific rectangle that follows the Golden Ratio in its side lengths. The Golden Ratio also manifests itself in nature and the human body, making it useful for artists.



Da Vinci

Leonardo Da Vinci is credited with some of the world's most famous paintings, and also is one of the prominent artists to utilize the Golden Ratio in his work. The most acclaimed painting showing clear examples of the Golden Ratio is Da Vinci's "The Last Supper". From the architecture of the background to the minuscule details on the shield, implications of the Golden Ratio are evident throughout. Other, more controversial examples of the Golden Ratio is Da Vinci's work can be found in the "Mona Lisa" and "The Annunciation". Da Vinci's fascination with ratios can be seen in "The Vitruvian Man". Here, Da Vinci measured the human body using specific ratios drawing ideal proportions according to the work of Vitruvian.


Raphael

Working in the shadow of Da Vinci, Raphael also used the Golden Ratio to create visual beauty and harmony in his art. In his piece, "The School of Athens", a perfect Golden Rectangle can be seen painted in the forefront. Many scholars say that this was Raphael's way of noting his recognition of the Golden Ratio, and prompting others to seek them in his work. Like Da Vinci, the placement of subjects as well as the architecture in the background is painted with the Golden ratio in mind.

Geometric Patterns

Many artists create pieces that involve only simple shapes, yet are situated such that a beautiful design is created. Using only circles, triangles, and squares, many artists have created masterpieces that are still viewed today.

Flower of Life

The Flower of Life can be traced back 5,000 years to Egypt. It is made by overlapping circles at specific points. By starting the next row of circles on the circumference of the row before, a certain pattern in created that stays constant throughout. In the religion of Islam, animals and people are not typically depicted on Mosques. Because of this, geometric shapes like the Flower of Life are painted on Mosques around the world. It also serves as a symbol of the divine order of the Universe. 

Modern Tessellations

Originally, tessellations were large images created by placing smaller tiles in a certain pattern. However, a modern interpretation of a tessellation is a design that incorporates non-square tiles and completely fills a space without gaps or overlaps. Tessellations can be found in nature, but are commonly created in art. M.C Escher is known as the father of modern tessellations and is infamous for his passion for creating complex tiled designs.