# Author Archive: Paul Hartzer

## Geometry for multiplication, division, and roots

Contemporary plane geometry of the sort taught in the standard American high school is most heavily informed by two books and a third mathematician. The first of these is Euclid’s Elements, which is so conceptually tied to planar geometry that…

## Factoring and long division

This morning, I’ve been watching YouTube videos. I started with Tarleen Kaur’s video on Middle Term Splitting. What I find interesting about Kaur’s Chapter to Chapter videos is that, because she’s a student in India, her methods are often different…

## Positive numbers and absolute value

They say that when you’re a hammer, everything looks like a nail. Since I’m currently thinking about conceptual vs procedural teaching, I’m noticing examples. Here’s a good definition of absolute value: “the magnitude of a real number without regard to…

## The smallest angle

I have been thinking about procedural vs conceptual thinking, which Skemp’s seminal article refers to as relational vs instructional. One of the questions on this year’s geometry final asks: Given a triangle ABC with sides AB = 5, BC = 6,…

## Concepts vs procedures

A persistent topic in mathematics education is whether to focus on conceptual or procedural knowledge. After reading Kris Boulton’s recent post that argues, “It depends,” I found myself thinking about the disconnect between arithmetic and algebra. What is needed to understand…

## Graphing and the coordinate plane

Dan Meyer’s latest post is on an exercise involving using a gridless coordinate plane to place fruit along two dimensions. The goal is a worthy one: To give students the opportunity to explore what the coordinate plane is without getting tied…

## Writing Rationals with the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic says that all integers greater than one can be written uniquely as the product of prime numbers. Another way of stating this is that, if $$P = (p_1, p_2, p_3, …)$$ is the (infinite) set…

## A trio of math limericks

Inspired by Math with Bad Drawings, here a trio of my own limerick creations: Two circles surrounding a square Was more than the poor thing could bear. It made itself fetal ‘Til planar was hedral. Cylindrical nets are a snare!…

## Transformations as Functions

Most high school geometry textbooks will proclaim that there are four basic transformations. Three of these (translations, reflections, and rotations) are rigid transformations; the resulting copy (image) is congruent to the original version (pre-image). Here are examples: The first example…

## Multiplication Table Slide Rule

Using Publisher, I’ve created a slide rule for multiplication tables (up to 10×10). To use it: — Print it out and cut along the dotted line. — Move the 1 on the bottom part to any single digit on the…