Group member(s): Chastity Colbert, Theresa Hogan, Danielle Ortega

 

Author of Lesson: Chastity Colbert

 

Title of lesson:  QUADRATIC Equations: Solving by quadratic formula    

 

Date of lesson: N/A

 

Length of lesson:   50 minute class period

 

Description of the class:

                     Name of course: Algebra II

                     Grade level: 10th or 11th grade

                     Honors or regular: N/A

 

Source of the lesson:

            http://distance-ed.math.tamu.edu/peic/lesson_plans/intro_quadratics2.pdf

 

Academic Standards:

Algebra 2 TEKS:

 

 (1) The student understands that quadratic functions can be represented in different

ways and translates among their various representations. Following are performance

descriptions.

(A) For given contexts, the student determines the reasonable domain and

range values of quadratic functions, as well as interprets and determines the

reasonableness of solutions to quadratic equations and inequalities.

(3) The student formulates equations and inequalities based on quadratic functions,

uses a variety of methods to solve them, and analyzes the solutions in terms of the

situation. Following are performance descriptions.

(B) The student analyzes and interprets the solutions of quadratic equations

using discriminants and solves quadratic equations using the quadratic

formula.

(D) The student solves quadratic equations and inequalities.

 

I. Overview

            Students will continue in their learning of solving quadratic equations in the lesson by focusing on the quadratic formula.  Students will build on their knowledge of completing the square, which was taught in the previous class period, to derive the quadratic formula.  Also, in the lesson, students get introduced to the determinant and its relationship to existent and non-existent solutions.

 

 

 

 

II.  Performance or learner outcomes

Students will be able to:

á      derive the quadratic formula

á       apply to various quadratic equations.

 

III. Resources, materials and supplies needed

            dry erase board markers or overhead markers, Paper and Pencil, and warm-up         worksheet

 

 

 

Engagement (Warm-up) and Exploration

Teacher Does

Students Do

As a warm-up, give students three quadratics to solve: one that could be solved by taking a square root (ex: ) one that could be solved by factoring, and one that could be solved by using completing the square (make sure this example has no leading coefficient and that the middle coefficient is even)..

 

Discuss why each problem is best solved with the given method.

 

In their project groups, students work on worksheet.

ÒOops, I left off one-more problem that I need for you to solve.Ó Then, give students a quadratic that needs to be solved using completing the square, but is not Òas niceÓ as the first. (For example, ). Allow students a little more time to solve this problem.

Students try to solve last problem. 

 

 

 

 

 

 

Explain

Teacher Does

Students Do

Ask about 2 groups to come up and show their work to the class.

ÒHow did you most of you solve the last equation?  What method did you use?Ó

                  

 

 

 

ÒWho found it a little difficult and cumbersome to keep track of all those coefficients?Ó

ÒWell, today we are going to invent a new method of solving quadratic equations using the complete the square method.Ó

 

 

Many students will have probably tried to complete the square considering that they learned it the previous day and that the equation is hard to factor.

Then, by guiding the students, start deriving the quadratic formula.

Start with the standard form of a quadratic equation with arbitrary coefficients, a, b, c, and using completing the square to derive the quadratic formula.

 

 

Finally, show how to use the formula to solve any quadratic by using the ÒextraÓ question that was given at the beginning of class.  Then, have students in class check that the formula works by using the quadratic formula for the problems used in the warm-up.

 

 

 

Extend/Elaborate

Show various graphs of parabolas with the solutions from using the formula.  One should show about 2 that have true roots and 2 in which the roots do not exist.

 

ÒWhat do you notice about the roots in which the graphs cross the x-axis? What about those that donÕt cross?Ó Give students about 5 minutes to discuss within their groups

 

 

 

 

 

ÒWhen it crosses the x-axis, the number in the square root is positive; when it doesnÕt cross, the number in the square root is negative.Ó

 

ÒCorrect, we call this number the  determinant.  When the determinant < 0, the solution/root does not exist.  When the determinant greater than or equal to zero, the solution/root exists.

 

 

 

Evaluation

Students will perform have an oral evaluation at the end of class, briefly reviewing the quadratic formula and other methods they have learned. They will discuss when it may be useful to use the various methods to solve different quadratics. Sometimes one method is faster than the other (i.e. factoring is often faster than using the quadratic formula if you can factor a quadratic).  Students will also be given homework out of the textbook.