One of the most useful strategies I have in life is using timers. Timers allow me to create a temporary container for focused attention. A time to put away all my other concerns and focus on my practice. You may have heard of the Pomodoro Technique. It’s worth checking out if you’re not already familiarContinue reading “Using Timers”

# Author Archives: Paul Gafni

## Conjectures and Counter-Examples

Without question, introducing the terms “Conjectures” and “Counter-examples” into my classrooms has been one of the most impactful changes to student engagement in my classes over the past few years. And just as exciting–it’s super easy for you to put it in your classroom too! Although I’ve used these terms day-in-and-day-out while I was studyingContinue reading “Conjectures and Counter-Examples”

## [Laser Cutting] Intro to Rhino and Grasshopper

I’m excited to share a brief, hands-on introduction to Rhino and Grasshopper–powerful software for 2D and 3D design work. Rhino is perfect for designing all sorts of objects–art, tools, toys, prototypes, etc. Rhino creates files that are ready-to-go on laser cutters and 3D printers. Rhino has a 90 day trial, and you’ll find that anyContinue reading “[Laser Cutting] Intro to Rhino and Grasshopper”

## What is an Algebra? (Part III)

In our next few Algebras, we broaden our sense of elements. Until now, we’ve looked at different types of numbers as our elements. Now, consider matrices as elements. Given any two 2×2 matrices, and , we can certainly computer and . And crucially, this sum and this product are still 2×2 matrices. So, we see that the set ofContinue reading “What is an Algebra? (Part III)”

## What is an Algebra? (Part II)

(This is Part II of a multi-part series. See Part I) Consider the following statements about integer arithmetic. even + even = even even + odd = odd odd + even = odd odd + odd = even even * even = even even * odd = even odd * even = even odd * odd =Continue reading “What is an Algebra? (Part II)”

## What is an Algebra? (Part I)

We’ll answer this question in this series of posts by giving a hands-on introduction to a variety of different Algebras (yes, there are lots!). Loosely speaking, an Algebra is a system that consists of elements and one or more operations. For now, you can think of the elements as numbers and the operations as the basic arithmetic you’re familiar with: addition or multiplication,Continue reading “What is an Algebra? (Part I)”

## A Novel Take on Graphing

Here’s the pitch: Draw a horizontal axis and label it with the numbers (What numbers should go to the left of 1?) Draw a vertical axis, labelled similarly. Draw a downward sloping line at a 45 degree angle through the “origin” (which seems to now be the point . The equation no longer seems appropriate. CanContinue reading “A Novel Take on Graphing”

## A Fractal Inducing Problem about Polygons

Here’s a question I like: take an n by n grid (start small), and draw straight lines connecting dots. If you end where you started and don’t cross lines, you get a polygon. What’s the most number of sides the polygon can have? I received those words in an e-mail from Dan Finkel in AprilContinue reading “A Fractal Inducing Problem about Polygons”

## Projects

Jewelry My newest artistic endeavor is jewelry–earrings and necklaces. See my Etsy shop for more. Pentaflower Since the start of 2015, I’ve been interested in a simple mathematical flower structure, made of 52 regular pentagons. I’ve made this shape in a couple different sizes and media–the one pictured hereis 6 feet in diameter. Couch Armada IContinue reading “Projects”

## Donald Duck in Mathemagic Land

I watched this video a few times growing up and was really happy to rediscover it recently. I hope you enjoy it as much as I do!