math

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How to ace the SAT math sections—by doing less math
On a timed exam like the SAT, when you have barely a minute to solve each math question, there’s nothing worse than getting bogged down in a problem. We’ve all been there: the small square of space in the test booklet becoming crammed with writing, the pile of eraser shavings growing, the clock ticking, and…wait, what was the question even asking ...
How to solve (almost) any math problem
Math is all about problems -- questions for which you don’t currently know the answer -- and problems can be really frustrating. That feeling of being stuck, for me, goes from a scattered confusion to a mind-numbing blankness. It’s really easy to shut down and give up, so the first step to solving any math problem is to persist! Don’t let the ...
How to ethically use WolframAlpha, Mathway, and Photomath
As a middle school and high school math teacher, I’ve seen my students try to get out of doing work in all sorts of ways. If you haven’t heard of WolframAlpha, Mathway, or Photomath, you may want to stop reading this article now - the temptation may ruin your hard-work ethic.
Einstein’s proof of E = mc^2
In this post, we’re going to prove the most famous formula in all of science, E = mc^2! We’ll do this using a simplified version of Einstein’s original 1905 proof. In this post I will assume that you are familiar with special relativity and Lorentz transformations.
The 3 most important GMAT math formulas
To solve many of the quantitative questions on the GMAT, it is essential to understand a couple key equations. This article will clearly lay out 3 very important formulas.
Can you “hear” the Fourier Series on a guitar?
As a musician, I had always wondered why different instruments sound dis tinct from one another, despite being in-tune and playing the same note. Why is it so easy to distinguish someone singing a C major scale versus someone playing the same scale on the piano? Timbre, tone color, or tone quality of a sound are those characteristics separate from ...
Introduction to Functions
You’ve made it through algebra: now it is time to start talking about functions. While functions are often used to make upper-level mathematics easier to understand, they can be confusing at first. So – what is a function? How do functions relate to the algebraic equations we have used before? And how do they help us with mathematics and computer ...
Slide rules, logarithms, and analog computers
Growing up, one of my favorite films was Studio Ghibli’s The Wind Rises—an animated historical drama about a 20th-century Japanese engineer named Jiro Horikoshi. Each time I rewatched it, I was always intrigued by a device that Jiro used for performing calculations. It consisted of two wooden rulers, with the top one able to slide freely. Somehow, ...
Think quickly: can you ace the world's shortest intelligence test?
Below is the world's shortest intelligence test. See how many questions you can answer!
Proof by contradiction: how to be so wrong you end up being right
Mathematical proofs are what make math objective: while you could find a few examples that "prove" a mathematical statement, it is often more important to write a rigorous proof that holds true in all cases. Mathematicians have a few methods in their toolkit to tackle different proofs. In this post, we will learn how to write a proof by ...
Solving algebraic equations with variables on both sides
If you are reading this, I can tell you’ve mastered solving simple linear equations. You’ve mastered the art of balance. You know that whatever you do to one side of the equal sign, you must do to the other. You can perform inverse operations until the cows come home, and you are a pro at isolating the variable. I bet you even check your work by ...
How to solve linear algebraic equations
Today, we are going to learn how to solve linear algebraic equations like 3y + 3 = 18 or 5x - 4 = 16. If these equations make you feel a bit queasy, have no fear! I am going to break the process down into five simple steps.
Calculating flood risk using probability and statistics
If at some point you ever want to buy property near water, a variation of this question will undoubtedly pass through your head: what are the chances that my {insert name of your expensive piece of property close to water} floods?
The Intermediate Value Theorem explained by everyday life
Calculus can be tough stuff. Calc AB was the first AP class I ever took in high school, and though I love the subject now, I certainly didn’t love it when I was first struggling with limits or with the chain rule for derivatives.
How to approach initial value word problems
We’ve all been there: on a homework set or in an exam, you turn to the final page and, to your dismay, it’s a wall of text. The dreaded Word Problem. Some of the words are useful, but some of them are meant to distract. Let’s look at a strategy for answering initial value word problems.
The power of prime factorization for the GMAT quantitative section
We have all encountered factor trees at some point during grade school. When I first encountered them as a kid, the whole exercise seemed unnecessary and silly. I thought to myself, “Great. I can list all the prime factors of 48. But, to what end?” It was not until much later that I realized the utility of prime factorizations. On an exam like the ...
Introductory statistics: are my data normal?
Statistics is fun, I promise! But before we can start having all the fun, it is important to describe the distribution of our data. We will need to handle problems differently depending on the distribution.
Gaining intuition for Taylor series
I’m writing this blog post because when I first came across Taylor series I found that a lot of my previous intuitions for mathematics were suddenly inadequate. It took time for me to build intuition not only for how these things work but why they are important. I hope this post will be illuminating for those just beginning to learn about Taylor ...
Where do Taylor series come from and why do we learn about them?
Taylor series can often seem a bit mysterious the first time that we learn about them. The formula for the Taylor series of a function f(x) around a point x=a is given by
What is implicit differentiation and how does it work?
One topic that seemed a bit mysterious and magic to me when I first learned calculus was implicit differentiation. In this post, we’ll start by reviewing some examples of implicit differentiation and then discuss why implicit differentiation works.
But what is “dx” really? Calculus terms explained
The symbol “dx” comes up everywhere in calculus. For example:
The Idea of Taylor Approximation
If you type √5 into your calculator, it’ll output something like 2.2360679775. But how did your calculator find that answer? Is there any way you could have found it by hand?
Tackling unfamiliar problems on the SAT math sections
Let's face it: there's no way to control exactly what math questions will pop up on test day. Questions at the beginning of each section tend to be simple and straightforward--you might be asked to isolate a variable, determine the slope between two points, or solve a system of equations--but later questions can often feel like they've come out of ...
How to survive a proof-based math class
Probably the most common challenge that I see my students struggle with is understanding and writing out mathematical proofs. Although most higher-level college math and computer science courses rely heavily on proofs, there aren’t many courses that really prepare students before they’re thrown off the deep end. I wanted to discuss some tips and ...
An application of calculus: finding optimal road networks
Suppose that we have many towns spread across the country and we are trying to connect them with a network of roads. If we would like to do so by laying as little road as possible, how do we do it? In this blog post, we will use Calculus to tackle a special case of this optimization problem.
Which is bigger?: Set cardinality, injective functions, and bijections
Comparing finite set sizes, or cardinalities, is one of the first things we learn how to do in math. From a young age, we can answer questions like “Do you see more dogs or cats?” Your reasoning might sound like this: There are four dogs and two cats, and four is more than two, so there are more dogs than cats. In other words, the set of dogs is ...
An Insider's Tip to Prepping for the SAT Math Section: Plug in Numbers
Preparing for both mathematics sections on the SAT can be a bit intimidating. You can’t expect yourself to know every topic that might come up, and the time limit adds to the stress. Much more efficient than trying to learn everything you might come across is to start with what you have already learned in high school and use examples to apply it ...
The key to mastering mathematics? Quit memorizing.
There are many misconceptions when it comes to the subject of mathematics. One of the most common myths I encounter is related to the way one approaches learning math.
How to get started on the GRE Math section
How do you feel about math? Let’s take a minute to think about math and how we feel about it. Good memories or bad? Or both? If I were to draw an analogy about math and anything else, it would be a foreign language. You can’t forget what you learned in arithmetic and expect to do well in algebra, just like you can’t forget how to conjugate French ...
How To Multiply Matrices Quickly and Correctly in Six Easy Steps
Step 0 – Make Sure the product makes sense! Say we’re given two matrices A and B, where
The top strategy for the math section of the SAT and ACT
Complicated algebra is the last thing many students want to deal with on a high-stakes test like the SAT or ACT. Yet it seems like there is no way around it, with the alphabet soup of variables scattered throughout the exam. Thankfully, there is a strategy for those problems where your algebraic manipulations are leading nowhere. It’s called ...
How to Study for a Math Test: 4 Essential  Tips
It’s a Tuesday night, and you’ve just remembered that you have a huge algebra 2 test tomorrow. Panic starts to sweep over you as you think back on all you’ve learned this semester. How will you ever succeed? You call up your friend to ask how you can study, and he calmly tells you he has just spent about an hour reviewing the main concepts from ...
How Majoring In Math Will Change The Way You See The World
There is a humorous, misguided stigma associated with math. Just the mere utterance of the word “math” conjures up, for many, the image of intimidating, arcane equations strewn about blackboards and calculators gone on the fritz. For many, math was a painful experience in grade school (and beyond) -- myself included! In fact, I did not become ...
Tips for getting a perfect score on a standardized math test
SAT, ACT, SSAT, ISEE, GRE. What do these acronyms all have in common? Well, they’re all standardized tests, but more importantly, they all have multiple-choice math test sections. Despite whether or not they’re accurate indicators of student performance in the classroom, lab, or office, they are all essential for entry into some educational career ...
An introduction to proofs: the structure of induction
Induction. It's a mathematical concept that is no doubt familiar to any student taking an introductory proof class. It is also a concept that can bring complex feelings---the excitement of learning a new cool proof technique, the fear of being asked to prove something "obvious", or the confusion of where to start.
Solving the “I’m not good at math” problem
You’ve heard it before. Or you’ve said it. I’m not good at math. I hear it from seventh graders struggling with fractions, high school students preparing to take the SAT, friends at a restaurant when splitting a check, and even from parents assuring me that their child’s own difficulties are in fact genetic. And while I’ve heard it countless ...
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