7.8 - Completing the Square and the Quadratic Formula

Introduction:

This is a set of guiding questions and materials for creating your own lesson plan on completing the square, which leads to a natural derivation of the quadratic formula.

Common Core State Standards:

A-SSE.1
A-SSE.2
A-SSE.3
F-IF.8

Learning Objectives:

To construct and understand the quadratic formula through a geometric interpretation.
To understand the technique of completing the square.
To solve quadratic equations either by completing the square or applying the quadratic formula.
To understand the connections between completing the square and the quadratic formula.

Guiding Questions:

Suppose you have a square of side 5cm and a rectangle with dimensions 4cm and 5cm. Can you arrange the shapes so as to form a square?
If cutting is allowed, can you do it?
Is anything missing?
Can we review what we just did?

What did we start with?
What did we do with the square and rectangle?
What did we end up with?

What if our square and rectangles have dimensions $x^2$ and $4x$ ?
What if our square and rectangles have dimensions $x^2$ and $6x$ ?
Can we use this technique to solve a quadratic equation?
Does it work if instead of adding we subtract the linear term, like in $x^2-8x$ ?
Can we use this technique to solve any quadratic equation?

Case 1: $x^2+bx+c=0$
Case 2: $ax^2+bx+c=0$

Notes for Teachers:

I demonstrate how to use some of the most difficult guiding questions in the videos below. Also, it is important to construct the quadratic formula from Case 2, above. A lesson to cover these topics might take between 90 and 120 minutes.