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 This lesson provided by: Author: Cliff Allred System: Mobile County School: Pillans Middle School
Lesson Plan ID: 27005
Title:

Now, where did THAT come from? Deriving the Quadratic Formula

Overview/Annotation:

Generally, teachers expect students to memorize the quadratic formula and to know that you use it after exhausting all other means of solving a quadratic equation, i.e. as a last resort. This technology-based lesson is designed to assist students with deriving the formula on their own. Students must first be familiar with complex numbers and the process of “completing the square.”

This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI project.

Content Standard(s):
 MA2013(9-12) Algebra 18. Solve quadratic equations in one variable. [A-REI4] MA2013(9-12) Algebra 21. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. [A-REI7] MA2013(9-12) Algebraic Connections 6. Use the extreme value of a given quadratic function to solve applied problems. (Alabama) MA2013(9-12) Algebra II 4. Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]
Local/National Standards:

NCTM Standards

Understand numbers, ways of representing numbers, relationships among numbers, and number systems

• compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions
Primary Learning Objective(s):

Students will learn to derive the quadratic formula using a quadratic equation in standard form.

Additional Learning Objective(s):

Students can use the discriminant within the quadratic formula to determine the type and number of solutions for a quadratic equation.

Students can use the quadratic formula to find the solutions of quadratic equations.

Approximate Duration of the Lesson: 31 to 60 Minutes
Materials and Equipment:

Pencil

Paper

NO calculators!

Technology Resources Needed:

Lesson can easily be modified to be conducted using either or both of these setups:

• Internet equipped student computers

• Teacher computer with Internet access, Interactive board, and LCD projector
Background/Preparation:

Before attempting this lesson, students should be familiar with factoring quadratics and completing the square.

Procedures/Activities:

ENGAGE

By now your students have likely struggled through the process of solving quadratic equations by means of factoring, graphing, and/or completing the square. Use this quadratic equation solver to “blow their minds” as it instantly provides the quadratic equation, solutions, discriminant, number and type of solutions. (This activity can be particularly meaningful if you collect some of the more difficult problems with which your students have struggled during this unit of study.)

EXPLORE

Divide the class into smaller groups and ask them to brainstorm how and why the quadratic equation solver works. Depending on the level of students you have in class (or even within groups) you may want to ask them to begin by examining the “quadratics in disguise” at the bottom of the website with the quadratic equation solver on it. Hopefully, student discussions will lead to the “conclusion” that the way the “solver” works must have something to do with the coefficients of the quadratic, linear, and constant terms of the equation in standard form. Use various questioning techniques related to the methods used thus far for solving quadratics : factoring, graphing, and completing the square to assist as needed. Caution: do not allow those students who remember the quadratic formula from Algebra I to get away with just providing that as the answer. Question them regarding why the formula works. Don’t worry about those students who do not remember the quadratic formula, that’s part of the purpose of this lesson.

EXPLAIN

Using the student ideas expressed in EXPLORE, guide students toward considering solutions for the general quadratic equation in standard form (ax2 + bx + c = 0). Have them work within their groups to see whether any of the techniques they’ve used so far with quadratics can be applied to this general formula. For those who recall the quadratic formula, do not let them get away with just writing it down. Insist that they “do the math” to derive the formula within their group. Even those who remember the formula must be able to explain the process of getting to the solution(s) from the general equation.

A fun way to help students check their work Deriving the Quadratic Formula

EXTEND

Have students conduct research on the use of completing the square to solve quadratic equations by the Babylonians, Arabs, Greeks, and Chinese. Students should prepare a presentation of their findings on how real-world problems led to the discovery of using general equations, particularly quadratics, cubics, and quartics.

EVALUATE

Teacher will closely observe and monitor groups as students brainstorm and struggle to derive the quadratic formula from the general form.

Teacher will use open-ended questioning to help guide students discussions and to attempt to ascertain depth of student understanding.

Teacher will use the attached rubric for grading student presentations.

Attachments:**Some files will display in a new window. Others will prompt you to download. DerivingQuadraticFormulaProjectRubric.htm
Assessment Strategies:

Teacher will closely observe and monitor groups as students brainstorm and struggle to derive the quadratic formula from the general form.

Teacher will use open-ended questioning to help guide students discussions and to attempt to ascertain depth of student understanding.

Teacher will use the attached rubric for grading student presentations.

Extension:

Students who excel with the primary objective of this lesson may be introduced to the discriminant and how it can be used to determine the type and number of solutions for a quadratic equation.

Remediation:

Students who struggle with this concept might find assistance by means of an Internet search related to "deriving the quadratic formula." A number of step-by-step instructional videos are available.

Each area below is a direct link to general teaching strategies/classroom accommodations for students with identified learning and/or behavior problems such as: reading or math performance below grade level; test or classroom assignments/quizzes at a failing level; failure to complete assignments independently; difficulty with short-term memory, abstract concepts, staying on task, or following directions; poor peer interaction or temper tantrums, and other learning or behavior problems.

Presentation of Material Environment
Time Demands Materials
Attention Using Groups and Peers
Assisting the Reluctant Starter Dealing with Inappropriate Behavior

Be sure to check the student's IEP for specific accommodations.
Variations Submitted by ALEX Users: