4 - Completing the Square through the Millennia

Summary

Introduction

Solving quadratic equations is a topic relevant to modern mathematics instruction, as it has been for thousands of years. As we start the 21st century, more often than not students will use calculators and computer algebra systems to solve quadratics. Today, we associate solving quadratics with curves (parabolas) rather than rectangles and squares (even though the word quadratic is from the Latin quadratum, a four-sided figure). A centuries old method which hopefully will survive in classrooms in this millennium is the method of completing the square. Understanding the process of completing the square is important for our students, for a wide range of reasons including that it provides arguably the best approach to deriving the quadratic formula. In the examples below, we outline the use of completing the square as it was done in four previous millennia.

Over the years, the method has had various representations. Understanding the historical, geometric representation may help students internalize the method when algorithmic or algebraic representations alone may not. Multipleways of learning and knowing are offered by including the historical perspective. The examples given in this capsule are actual problems solved in the past, and your students are invited to solve them today using the methods of antiquity as well as current techniques. In my courses, I present the information as an interactive lecture that extensively involves students, as described below.

Historical preliminaries

About 4000 years ago, Mesopotamian scribes pressed the method of completing the square into clay tablets, the technology used to record information in that time.