# Arenas of mathematical mastery

## What Do Digits Mean, Anyway?

Puzzle I found this puzzle in the G+ Mathematics community, courtesy of Paul Cooper. Solve the final addition: 50 + 60 + 90 = 380 30 + 40 + 60 = 330 90 + 60 + 70 = 350 50…

## All Lines are Congruent

A standard high school geometry textbook talks about congruence in terms of three types of objects: Line segments, angles, and polygons. Congruence is then defined in terms of measurable parameters: “Two figures are congruent if they have the same size…

## Solving Simultaneous Equations: Multiple Methods

Introduction and Terms Recently, a post on the G+ Mathematics community involved how to determine $$x$$ and $$y$$ when: $3x + 5y = 12 \\ x + y = 2$ This is generally referred to as simultaneous equations or a…

## Russian peasants, number sense, and bases

Russian peasants do too much work There is a method of multiplication called the Russian peasant method. I’ve seen it mentioned here and there, but I was not explicitly educated in the process; it struck me as being more trouble…

## Factorials and the meaning of “is”

In a YouTube video, James Grime of NumberPhile makes the claim that the meaning of the factorial is $n! = \prod_{i=1}^n i$ for n > 0, and proceeds to explain why 0! = 1 using a recursive proof. This echoes…

## Negative numbers squared

Background Mathematical conventions represent the linguistic aspect of mathematics. One of the strengths of modern mathematics is the way in which we can represent some fairly complex ideas in a shortened, rigorous symbol set. However, as a result of these…

## 10101 and 11011 are never prime

One particularly tricky aspect of number sense is being able to separate the abstract notion of value from more concrete visual representations of numbers, and the even more concrete notion of countability. For instance, some people get caught up on…

## Factoring quadratics and linear equations

Factoring a quadratic equation involves finding two linear equations whose product is the quadratic equation. This is an example where mathematics teachers often act as if (a) there is one method of solving and (b) there is one solution. The…

## =

When I was a lad studying mathematics, the equality sign seemed particularly simple: The stuff on the left is equal to the stuff on the right. However, I have since been developing a much more sophisticated perception of the simple…

## Let’s Make a Deal

The Problem In the misnomered “Monty Hall” problem, the rules are set out as follows: You as the contestant are faced with the choice of three doors, behind exactly one of which is money or something else of significant value….

## Expressions as Names

One basic concept in mathematics that I see students struggle with, and with which I struggled myself, is the notion of expressions. However, when we remove the mathematical component, we can see that expressions behave much like a concept that…

## The Difference of Squares

In our Mathematical Reasoning class tonight, we discussed this problem: Can your age in years be written in terms of the difference of two square numbers? If so, what two numbers? There are at least three mathematical problems contained here:…