Have you ever wondered where the formulas for volumes that you studied way back in geometry come from? It’s not too surprising that the volume of a cube is , but why is the volume of a cone ?Read More
Sitting at the cross-roads of mathematics, statistics, and computer science, the emerging field of data science (ranked by many as the top career in the US) seems daunting to those still developing strong technical skills. At the same time, a host of dynamic and highly-efficient libraries give coders the power to treat complex areas like machine learning as a black box.
There is, however, middle ground: an intuitive understanding of mathematics that makesRead More
Learning about logarithms is one of those times in math class where you wonder if this will ever be useful in any way. I see lots of students struggle with topics like logs, since they can seem abstract and they aren’t obviously useful. But I’m here to explain why they are actually incredibly important and describe so much of the world we live in! Let’s take a look at an example from chemistry and physics that shows just how powerful logs can be - the Arrhenius Equation.Read More
If polar equations have you second-guessing your future as a nuclear physicist, fret not! Almost every Pre-Calculus student I have tutored has struggled here, and it isn’t surprising at all. Remember the first time you saw an equation and were introduced to these strange x and y variables? It may seem like second nature now, but you were learning about a whole new way to communicate about points and curves.Read More
Triangles can be very different!
The SAT math section is full of triangles. They’re the most basic 2D shape that you can create and they can be found everywhere, so the SAT wants to make sure that you have the basic facts down pat. So let’s look at some tricks and facts about triangles. We’ll start by looking at the triangle below (not drawn to scale).Read More
In math, art, or life, it's not hard for things to get ambiguous... [image source: Magritte]
Trigonometry should be simple-- you’re just using the given information to solve for only one answer, right? Well, with the Law of Sines, sometimes there is more than one right answer. This situation is also known as the Ambiguous Case. Before we dive into the Ambiguous Case, let’s review the Law of Sines and Congruence.Read More
Dungeons and Dragons is complicated. Mastering probability doesn’t have to be.
Probability is one of those topics that haunt children from grade school days of the determining the likelihood of picking out red marbles from a box. Even my most advanced math tutoring students in Cambridge sometimes feel bamboozled by it. Why? Because probability and statistics can quickly become overwhelming with the many different distributions and definitions. The key is to find the underlying patterns and to stay organized.Read More
If you’re taking a Geometry class, chances are that you’ve spent the last couple of months learning about proofs (and if you haven’t gotten there yet, you will soon). Proofs are something students either love or hate, but mostly they love to hate them. Why, they often ask me, are we learning how to do this? How is this relevant to anything?
I’m here to tell you that Geometry proofs are one of the best lessons you’ll ever have—about writing. Many of my advanced math tutoring students in New York think that these two subjects are totally separate, that they are either a math or an English person and never the twain shall meet, but the truth is that there is a lot of overlap. The proof is in, well, the Proof.
What Geometry Proofs Can Teach About Writing an Essay OutlineRead More
It’s a shame that so many people can go through college as math majors and minors without ever learning the history of mathematics. Who were Euler and Gauss? Newton and Leibnitz? Euclid? We all know their theorems and mathematical contributions, but rarely do most of us think of the people —with their messy lives, quirks, and stories— behind these brilliant minds. Understanding these lives can provide context to learning math at a whole new level.
Even as an advanced math tutor in Boston and a student at MIT, I am guilty of this myself. But since it’s the time for making New Years resolutions, I’ll make one of mine to learn more about the people behind the incredible mathematical discoveries. Today, I’ll share three of my favorite mathematical anecdotes:Read More
Merry Christmas Eve! Why yes, I do believe volume rotation integration is as easy as apple pie!
Every once in a while something you learn in class really just knocks you off your feet. Those of you who have taken AP physics will know that position is the integral of velocity. About a month ago, on a fairly dreary Boston morning, I was sitting in my neurobiology class, semi-interested, when the professor informed us off something truly shocking: our brain determines and maintains our eye position by literally integrating velocity information into position information.
It took humanity all the way until Newton and Leibnitz to figure out calculus, and it takes the education system about twelve years to teach calculus. All the while you are subconsciously integrating every second that you are awake. So while it is completely understandable that AP Calculus can be intimidating, it’s at least somewhat comforting to know that you—or at least your eyeballs—have been doing it your entire life!
Unfortunately, as a math tutor near MIT, I’ve found that even for ambitious students, doing calculus is much more difficult. So here are 4 easy steps to remember when doing those pesky volume rotation integration problems.Read More