## Modeling in GeoGebra

Introduction In this entry, I’m going to demonstrate the use of GeoGebra to estimate a value for a fairly tricky trigonometry problem, then illustrate how to find the value using trigonometry and an appeal to WolframAlpha. In so doing, I…

## Angles and Congruence

Congruence As I discussed in an earlier post, there are two basic definitions of geometric congruence that are presented to students. The first is based on measurement: Definition 1. Two objects are congruent if all of their measurements are the…

Mathematics

## Programming, Mathematics, and Language

I’ve been struggling for a while now to find a way to frame and articulate the answer to what seems like a simple question: “What is mathematics?” At the same time, I’ve been seeking to layout the similarities and differences…

## The Golden Ratio and Generalizing Quadratics

A poster on the Google Plus Mathematics community commented that one feature of the Golden Ratio ϕ is that adding one to ϕ yields the same value as squaring ϕ does. That is, $\phi^2 = \phi + 1$ He was surprised that there…

## Negative Bases

And now, for something silly. In general, number bases are expected to be positive integers greater than one. The most widely used are decimal (because we have ten fingers and ten toes), binary (how computer data is stored), hexadecimal (a…

Mathematics

## 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…

## A pair of probability problems

I’ve recently come upon two probability problems with counterintuitive solutions. One I’d seen before and dismissed because I didn’t understand the write-up (mea culpa); the other is new to me. Born on a Sunday Puzzle: You are introduced to a…

Mathematics

## 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…

## GeoGebra Tutorial: Golden Ratio / Power of a Point

Introduction In my previous post, I included this image, which I’d made in GeoGebra. The image satisfies the conditions of the problem: $$AD$$ is tangent to $$\odot P$$ and $$\overline{BC} \cong \overline{AD}$$. In order to create this image, I…

## The Golden Ratio and the Power of a Point Theorem

The Golden Ratio By definition, the Golden Ratio is a ratio involving overlapping line segments. Given collinear points A, B, and C, such that B is between A and C, if the ratio between the two subsegments is the same…

Mathematics

## Schrödinger’s Brat and 3-Door Monte

The Monty Hall problem persists in Internet mathematics discussions, as if its results are somehow spectacularly unique or mystifying. Here is the problem: You are on a game show and are presented with three doors. Behind one door is some…

Mathematics

## Proof of the Power of a Point Theorem

I had to dig for a bit to find a complete proof for each part of the Power of a Point Theorem, so I thought it would be useful to compile my own proof. The Power of a Point Theorem…

Mathematics