# Algebra

## 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…

## 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…

## 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…

## 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…

## Choosing a Strategy

(Reposted from my blog for my students) I will often tell my students to select strategies that work best for them to solve a problem, rather than focusing on a single specific strategy. What do I mean by this? I…

## Filling in Blanks: The General Case

Discussion about the puzzle behind the previous post led to this question: Is it possible to construct a similar question, where there are three of five values and a mean given, and where the median is half one of the…

Algebra

## Filling in Blanks

Here’s an interesting mean/median question. Assume your lab assistant took five readings, but only recorded three of them: -3, 5, and 7. The assistant also recorded that the mean was 6, and (for reasons lost on you) that the median…