The Web Sketchpad model below shows the function *f(θ)* = 1 – cos 2*θ* in both Cartesian and polar form. For each graph, the independent variable appears as a red bar that corresponds to a particular value of *x* (for Cartesian) or *θ* (for polar). The red bar has tick marks that … Continue Reading ››

# Category Archives: Advanced Mathematics

# The Swimming Pool Problem

In a prior blog post, I presented an uncommon method for solving the well-known Burning Tent problem. My solution, modeled on the approach in the *Connected Geometry* curriculum, used a dynamic ellipse to pinpoint the optimal solution. Now, I'd like to offer a related problem from *Connected Geometry* where … Continue Reading ››

# An Interactive Approach to Finding nth Roots

When students find the nth roots of a complex number, they use de Moivre's Theorem and a fair bit of calculation and trigonometry. In this blog post, I'm going to approach the topic from a more visual perspective and make use of the following geometric way to think about complex number multiplication: To multiply two complex … Continue Reading ››

# A Mathematical Mystery Story with Web Sketchpad

Several years ago, I wrote a blog post about the value that students derive from writing mathematics with Sketchpad. The post included an example of a simple Logo iteration, easily implemented in Sketchpad, that produces some very complex and interesting shapes depending on the values of several input parameters. In the … Continue Reading ››

# The Art of Parametric Equations

Can mathematical curves be beautiful? Most certainly! Precalculus students glimpse the connection between mathematics and art when they graph roses, cardioids, limaçons, and lemniscates. But these curves give just a taste of the beauty that can be achieved when graphing equations.

In a recent article from the online science magazine Quanta, Pradeep Mutalik reviews a gorgeous new math book, Creating … Continue Reading ››

# The Broken Stick Puzzle

Several weeks ago, my friend Martin shared the following probability puzzle with me: *Two points are chosen independently and at a random on a stick. The stick is broken at those points to form three smaller sticks. What is the probability these three sticks can form a triangle?*

This is a classic problem, dating back to … Continue Reading ››

# Pi Day 2015: Pieces of Pi

For this year’s Pi Day post, I thought I’d continue our Web Sketchpad (WSP) construction theme. But rather than adapting the visualizations from last year’s Pi Day post to the new construction capabilities, I decided to take a different approach. Some time ago, I built a set of custom tools … Continue Reading ››

# WCYDWT: A Ball, a Trash Can, and Web Sketchpad

Dan Meyer has posted a number of "What Can You Do With This?" activities on his blog. (Activities is probably too prescriptive a word; they're more in the nature of prompts for student thinking, noticing, and wondering.) One of the first was the image below, which he made by superimposing frames from a … Continue Reading ››

# Pentaflake Chaos

Dan Anderson commented on my Pentaflake post to observe that the pentaflake can also be created by a random process, sometimes called the Chaos Game. In this game you start with an arbitrary point and dilate it toward a target point that's randomly chosen from some set … Continue Reading ››

# The Brouwer Fixed Point Theorem

According to Wikipedia, the Brouwer Fixed Point Theorem, named after mathematician and philosopher Luitzen Brouwer, states that "for any continuous function *f* mapping a compact convex set into itself, there is a point *x*_{0} such that *f*(*x*_{0}) = *x*_{0}.

This is a deep theorem, but one aspect of it is lovely, surprising, and entirely approachable by high-school geometry … Continue Reading ››