## Every Third Triangular Number

This is a quick proof based on an observation inspired by “Mathematical Lens” in the May 2016 Mathematics Teacher (“Fence Posts and Rails” by Roger Turton). A triangular number is the sum of all integers from 1 to n. The…

## Pizza Time!

The Internet is in a tizzy yet again about the evils of mathematics education. At least Common Core isn’t being demonized quite as front-and-center as in the recent past, but still. This time it’s about pizza. Which means every mathematics…

## Inscribed Right Triangle

Here’s a fun puzzle (via Brilliant.org): What is the area of the square $$ABCD$$? There may be a simpler approach; my solution wound up being more complicated than I expected. Since $$\Delta AEF$$ is a right triangle, $$AE = 5$$…

Geometry

## al-Jabr: Integer Parameters

I was thinking about the third scenario described in al-Khowarizmi’s al-Jabr: $$x^2 = 3x + 4$$. I was curious about the integer solutions of the general pattern, $$x^2 = ax + b$$. It’s easy enough to demonstrate that this will…

## Naming Variables

First of all, let me get this out of the way: “Hey, you kids! Get off my lawn!” In this post, I comment on the notational shifts from what I was trained in back in the 1980s and what textbooks…

## Right Triangle Similarity

Today’s lesson in my Geometry class was on the use of the geometric mean when finding missing values of right triangles. For every right triangle, two of its altitudes are the legs and the third is perpendicular to the hypotenuse….

## Al-Jabr (continued)

In my previous post, I looked at the first two detailed examples provided by al-Khowarizmi in his compendium, the title of which gives us the word “algebra”. Al-Khowarizmi discussed three types of mathematical objects: Numbers (N, constants), roots (R, unknowns),…

## Al-Jabr

The word “algebra” comes to us from the title of a book, Hidab al-jabr wal-muqubala written by Abu Abdallah Mohammed ben Musa al-Khowarizmi (there are variations in the transliterations of both the title and the author) around AD 825. He was…

## Proof: Isosceles Triangles in a Quadrilateral

In my last post, I noted that it’s possible to create an isosceles trapezoid from four isosceles triangles, but I wasn’t sure if there was a way to construct a quadrilateral from isosceles triangles such that the quadrilateral was neither…

Geometry

## Isosceles Triangles in a Quadrilateral

In this post, I’ll discuss two issues. First, I’ll look at a problem taken from a major textbook, and explain why the solution is wrong. Then, I’ll discuss why this particular problem bothers me in the greater context of mathematics…

## Carole and the Common Core

Here’s another meme criticizing the Common Core: The criticism is that the student has provided a fully correct answer and gotten dinged for not providing an estimate. This is, of course, taken as yet another illustration of why Common Core…

## Complex Numbers making Real Numbers

As a point of curiosity, I found myself wondering when a complex number to an integer power creates a real number. For the sake of completeness, I also looked at when the result is a fully imaginary number. More rigorously,…

Mathematics