Book contents
- Frontmatter
- Preface
- Introduction
- Contents
- Learning from the Medieval Master Masons: A Geometric Journey through the Labyrinth
- Dem Bones Ain't Dead: Napier's Bones in the Classroom
- The Towers of Hanoi
- Rectangular Protractors and the Mathematics Classroom
- Was Pythagoras Chinese?
- Geometric String Models of Descriptive Geometry
- The French Curve
- Area Without Integration: Make Your Own Planimeter
- Historical Mechanisms for Drawing Curves
- Learning from the Roman Land Surveyors: A Mathematical Field Exercise
- Equating the Sun: Geometry, Models, and Practical Computing in Greek Astronomy
- Sundials: An Introduction to Their History, Design, and Construction
- Why is a Square Square and a Cube Cubical?
- The Cycloid Pendulum Clock of Christiaan Huygens
- Build a Brachistochrone and Captivate Your Class
- Exhibiting Mathematical Objects: Making Sense of your Department's Material Culture
- About the Authors
Why is a Square Square and a Cube Cubical?
- Frontmatter
- Preface
- Introduction
- Contents
- Learning from the Medieval Master Masons: A Geometric Journey through the Labyrinth
- Dem Bones Ain't Dead: Napier's Bones in the Classroom
- The Towers of Hanoi
- Rectangular Protractors and the Mathematics Classroom
- Was Pythagoras Chinese?
- Geometric String Models of Descriptive Geometry
- The French Curve
- Area Without Integration: Make Your Own Planimeter
- Historical Mechanisms for Drawing Curves
- Learning from the Roman Land Surveyors: A Mathematical Field Exercise
- Equating the Sun: Geometry, Models, and Practical Computing in Greek Astronomy
- Sundials: An Introduction to Their History, Design, and Construction
- Why is a Square Square and a Cube Cubical?
- The Cycloid Pendulum Clock of Christiaan Huygens
- Build a Brachistochrone and Captivate Your Class
- Exhibiting Mathematical Objects: Making Sense of your Department's Material Culture
- About the Authors
Summary
Introduction
Why is the algebraic process of forming a perfect quadratic expression referred to as completing the square? With simple cutouts for an overhead projector, the geometric underpinnings as well as many aspects of quadratic equations can be exhibited quickly and effectively. This can be expanded to the cubic with the use of five wooden blocks. This chapter will show how to use these items in the classroom to give students a geometrically intuitive as well as an historical understanding of quadratic and cubic expressions.
Creating squares
At several points in the undergraduate curriculum, as well as in some secondary courses, we find ourselves “completing the square.” I found myself in just such a position a short while back while teaching a sophomore level calculus course. On the spur of the moment, I asked the class if they knew why it was called “completing the square”. The response ranged from blank stares to shrugged shoulders. Looking at the clock, I decided I could sidetrack for a few minutes.
I started by drawing the Figure 1 on the board to represent the statement x2 + bx. (In general, you will have some expression of the form ax2 + bx + c. But the actual process of completing the square is done on an expression of the form x2 + bx, after division by a and “removal” of c.) I start with the comment that for the ancient Greeks as well as the Babylonians, geometry was physical; that x (in modern notation) represented a length, x2 an area, and so on.
- Type
- Chapter
- Information
- Hands on HistoryA Resource for Teaching Mathematics, pp. 139 - 144Publisher: Mathematical Association of AmericaPrint publication year: 2007