teaching Absolute Value Definitions I teach three interchangeable definitions of the absolute value function, encouraging students to choose a favorite and be able to swap between them freely. In each definition, the absolute value function is a [real function][(\(\mathbb{R}\to\mathbb{R},\) meaning that for every

math What are independent probabilities? The way they are discussed for SAT and GRE exams usually doesn't give students a good intuition. Examples Most probabilities are not independent, which means they have some sort of correlation. (Examples...)(correlated-examples) (start correlated-examples) This US presidential election year, we have Trump as

math Sums of Sequences on the SAT Monday night, my friend Jesse mentioned an SAT question he felt was actually insightful. It led us to explore a property we did not expect to find. The Initial Question The integer \(8\) may be written as the sum of \(n\) consecutive integers, for

math Math Process A number of posts I make illustrate the process of mathematics rather than the results, while still trying to appeal to those primarily interested in the results. This post serves to explain how those are written, and how to read them. Hidden Text The

math Implicit Differentiation When introducing this concept, most courses overlook the best examples: the basic ones. Implicit differentiation is what allows you to find the derivatives for so many functions and even prove some general properties of derivatives that it's ridiculous to teach the concept without emphasizing

math Derivatives of Specific Functions Every calculus student has to study the definitions of limits and derivatives (future post), the properties of derivatives of general functions (future post), the derivatives of specific classes of functions, and the applications of derivatives (future post). There will be a future post on

teaching Even and Odd Functions Most middle and high schools nowadays introduce the terms "even" and "odd" to describe functions without really explaining why they're named that way. Without that, the definitions are harder to recall, and the reason we might find them interesting is

math What is Linear Algebra? With most math courses, you can't really know what it's about until you're in it. This, hopefully, will express some of what Lin Al is about before you decide to take it. Each context assumes you've taken the corresponding course, be it Algebra I,

math Trigonometry Formula Sheet This post contains the info I pretty much reiterate verbatim for every trigonometry (or calculus, etc.) student I've tutored. There will be a future post on how to use this one. Definitions These should be the only things you have to accept without reason.

math The Blob Pythagorean Theorem The Numberphile video below is quite excellent, but some viewers might appreciate clarification on the topic. Watching the video is not required to follow this post. The Motivation We wish to prove the Pythagorean theorem using scale and similarity as our foundation, rather than