The Law of Cosines is presented as a geometric result that relates the parts of a triangle:
While true, there’s a deeper principle at work.
The Law of Interactions: The whole is based on the parts and the interaction between them.... Read article
The Law Of Sines is something I memorized in a class once, but didn’t internalize:
Ok, that’s a neat connection, and maybe we can prove it by drawing some right triangles (of course) and re-arranging terms.
But what does it mean?... Read article
After a decade of writing explanations, I’ve simplified the strategy I use to get new concepts to click.
Make explanations ADEPT: Use an Analogy, Diagram, Example, Plain-English description, and then a Technical description.
Here’s how to teach yourself a difficult idea, or explain one to others.... Read article
What’s the essential skill of a cartoonist? Drawing ability? Humor? A deep well of childhood trauma?
I’d say it’s an eye for simplification, capturing the essence of an idea.
For example, let’s say we want to understand Ed O’Neill:
A literal-minded artist might portray him like this:
While the technical skill is impressive, does it really capture the essence of the man?... Read article
Here’s a trick for thinking through problems involving exponents and logs. Just ask two questions:
Are we talking about inputs (cause of the change) or outputs (the actual change that happened?)
Are we talking about the grower’s perspective, or an observer’s?... Read article
Ratios summarize a scenario with a number, such as “income per day”. Unfortunately, this hides the explanation for how the result came about.
For example, look at two businesses:
- Annie’s Art Gallery sells a single, $1000 piece every day
- Frank’s Fish Emporium sells 250 trout at $4/each every day
By the numbers, they’re identical $1000/day operations, right?... Read article
Trig mnemonics like SOH-CAH-TOA focus on computations, not concepts:
TOA explains the tangent about as well as x2 + y2 = r2 describes a circle. Sure, if you’re a math robot, an equation is enough. The rest of us, with organic brains half-dedicated to vision processing, seem to enjoy imagery.... Read article
After months of work with the help of Neil, a great designer, and my Excel-blogging friend Andrew, I’m happy to launch a brand-new design.
My goals were to be friendly, readable, and easy-to-navigate. Here’s a quick before-and-after:
Neil did a fantastic job here — I’d been looking for a way to convey a welcoming, conversational tone.... Read article
Algebra is really about relationships. How are things connected? Do they move together, or apart, or maybe they’re completely independent?
Normal equations assume an “input to output” connection. That is, we take an input (x=3), plug it into the relationship (y=x2), and observe the result (y=9).... Read article
Hi all! I’m happy to announce a public availability of the BetterExplained Guide To Calculus. You can read it online:
And here’s a peek at the first lesson:
(Like the new look? I’ve been working with a great designer and will be refreshing the main site too.)
The goal is an intuition-first look at a notoriously gnarly subject.... Read article
Summary: I’m building a calculus course from the ground-up focused on permanent intuition, not the cram-test-forget cycle we’ve come to expect.
Update: The course is now live at http://betterexplained.com/calculus
The Problem: We Never Internalized Calculus
First off: what’s wrong with how calculus is taught today?... Read article
I’m thrilled to announce the print edition of Math, Better Explained is available on Amazon:
With the magic of print-on-demand, you can order the book with overnight shipping (Amazon Prime!), and be reading full-color insights tomorrow. Yowza.
I’ve often been asked if a print version can be made, and I’m beaming to say it’s now a reality:
- 12 chapters (~100 pages) of full-color explanations
- Professional-quality typesetting & layout
- Gorgeous, high-resolution text and diagrams
- Compact, easy-to-carry size with comfortable margins (7″ x 10″)
The best part?... Read article