## Inscribing an Equilateral Triangle

I was reminded of the cylindrical wedge that casts shadows of a triangle, a square, and a circle, and it got me wondering: What if I wanted to create such a shape with an equilateral triangle as one of its…

## Multiplying Negative Numbers

Students often struggle with the concept of multiplying negative numbers, particularly with the notion that multiplying two negative numbers results in a positive. I’ve seen numerous attempts by teachers and teacher educators to explain why, conceptually, it is that the…

## The Game of Set

The game of Set consists of 81 cards. Each card has one, two, or three identical symbols of one of three shapes (oval, diamond, or squiggle), in one of three colors (red, green, or purple) and one of three textures…

Mathematics

## Units: “How many days…”

This is an example of a common sort of story problem encountered in standardized tests: “1. A team of five professionals can do a certain job in nineteen days; a team of nine apprentices can do the same job in…

## The Free Throws Problem Part 2

Here’s an extension to the problem in my previous post. Time has run out, and a player is at the free throw line. If he makes the first shot, he gets a second try. If he makes both shots, his…

## The Free Throws Problem

At a recent workshop on collaboration, the other participants and I were presented with a version of this problem: Adam hits 60% of his free throws. He gets fouled just before the buzzer, and his team is down by one…

## Length of a Tangent

I’ve seen variations of this one a few times, so I thought I’d give it a quick write-up. The simpler version is: Given two circles that are tangent and a line that is cotangent to them, what is the length…

## Divisibility Tests

Most people are aware of two or three basic tests for divisibility by a prime number: A number n is divisible by 2 iff it ends in an even number (0, 2, 4, 6, or 8) by 5 iff it ends in a…

## 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…

## Indefinite vs infinite

I have borrowed from a colleague a copy of G. A. Wentworth’s Plane and Solid Geometry, copyright 1899 and published 1902 by The Athenæum Press of Boston. I enjoy reading old textbooks because they either reinforce or give lie to certain…

## Rationals except Zero

Here’s a quick one: All rational numbers except 0 can be expressed as $(-1)^s \Pi p_i^{n_i}$ where $$s \in \{0, 1\}$$, $$p_i$$ is a prime number, and $$n_i$$ is an integer. This reminds me of the restriction on the definition…

Mathematics

## 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…