Parabolas and Quadratics | www.MathEdPage.org

Mostly Algebra 1

A manipulatives-based complement to this work (or to any work on quadratics) can be found in my book Lab Gear Activities for Algebra 1. (Send me e-mail if you need help locating a copy.) The manipulatives make factoring and completing the square accessible to the vast majority of Algebra 2 students, while giving them a geometric perspective which is unfortunately missing from many curricula. That approach is all the more important if you teach this material to students in grades 7-9.: Sample activity from the end of the book, which makes a connection between the Lab Gear and graphical representations of certain quadratics (PDF).; Applet that presents a geometric view of completing the square.

An apparently unrelated unit: Constant Sums, Constant Products: Images that reveal many connections.

An application of quadratic functions, from Algebra: Themes, Tools, Concepts:: Maximizing Area | Condensed version.

The lessons below (in PDF format) involve the sparing use of the graphing calculator. Unlike some other calculator-based approaches ("graph this, graph that, what do you notice?") this is a rather theoretical unit, which helps develop students' symbol sense and goes for a lot more depth of understanding than is customarily expected at this level.: Factored Form of Quadratic Functions | Applet; From Factored to Standard Form; Moving Parabolas Around; Quadratic Forms (a condensed version of the above three lessons, probably better suited for teachers); Make These Parabolas: TI-83/84, TI-89; Quadratic Parameters

A derivation of the quadratic formula using graphs:: A Graphical Approach to the Quadratic Formula
It is especially useful if you eschew the manipulatives / geometric approach to completing the square, or emphasize transformations of graphs, but you will find it interesting no matter what.

A student-created visual proof: Integrating y=x², by Jacob Regenstein.

Late in our Algebra 1 course at The Urban School, we ask students to create a short, illustrated report explaining the connections illustrated here, using their own example. One year, while grading this assignment, it hit me that there must be a new proof of the quadratic formula based on this, and sure enough, there is. You can read it in my article in the February 2008 issue of The Mathematics Teacher, under the title "A New Path to the Quadratic Formula". The proof involves neither parabolas nor completing the square!: Worksheet (for teachers and perhaps some precalculus students); (background: Constant Sums, Constant Products); A related Cabri applet
See an online presentation of the proof. [Let me know if it doesn't work on your computer.] For better results, including animations, download the file (3.2 MB), in .mov (Quicktime) format. Either way, grab a pencil to follow the argument, and click your mouse to advance through the slides.

Parabolas are a major topic in secondary school, and yet in the US, students and teachers do not often think about them as geometric objects. This may be a consequence of the strict division between algebra and geometry courses. As a result, many basic properties of parabolas are not well understood. These pages present some key concepts in this domain:: Geometry of the Parabola: 2D | 3D