# Mathematics

## Polygons as Functions

A recent comment from a colleague got me thinking about describing polygons using functions. His intent was that polygons (and all closed shapes) can be described as sets of functions; for instance, a triangle could be described by three linear…

## Indefinite vs infinite

I have borrowed from a colleague a copy of G. A. Wentworth’s Plane and Solid Geometry, copyright 1899 and published 1902 by The Athenæum Press of Boston. I enjoy reading old textbooks because they either reinforce or give lie to certain…

## Rationals except Zero

Here’s a quick one: All rational numbers except 0 can be expressed as $(-1)^s \Pi p_i^{n_i}$ where $$s \in \{0, 1\}$$, $$p_i$$ is a prime number, and $$n_i$$ is an integer. This reminds me of the restriction on the definition…

Mathematics

## Defining a Line

The version of Geometry most widely taught in high schools in the United States is an amalgam of the two most basic fields of geometry: Synthetic and analytic. The mixing of these two is done in such a way as…

## But is it math?

It is a persistently popular thing to do on social media to post challenges like this one. I used to be of a mind to be outraged at the abuse of the equal sign: Clearly these are not addition problems!…

## 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…

## 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…

## 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

## What is Multiplication, Anyway?

Yet again, the internet has seized upon an elementary student’s math homework and has decided to argue about Common Core. This time, it’s about a test question. The student was asked to “Use the repeated addition strategy to solve :…