Completing the Square

AUTHOR: Amber Blakley

DATE LESSON TO BE TAUGHT: ?? Also note this lesson will probably take 2 days

GRADE LEVEL: Algebra I; 8^th 9^th or 10^th

CONCEPT(S): Students will be taught completing the square through an applet and from exploring an equation with the teacher. During the exploring phase students will play with the equation to try and get what they want.

OBJECTIVES:

Performance Objectives

1. Students will be able to complete the square of a quadratic equation.
2. Students will be able to solve a quadratic equation.
3. Students will be able to manipulate an algebraic equation.

TEKS:

(A.10)  Quadratic and other nonlinear functions. The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods.

The student is expected to:

(A)  solve quadratic equations using concrete models, tables, graphs, and algebraic methods; and

(A.11)  Quadratic and other nonlinear functions. The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations.

The student is expected to:

(A)  use patterns to generate the laws of exponents and apply them in problem-solving situations;

MATERIALS LIST and ADVANCED PREPARATONS:

The Cow for laptops

Know what homework will be assigned (out of book or worksheet)

ENGAGEMENT:

Teacher will do:	Student will do:	Probing Questions:
Draw the equation y=x^2+x-6	Look at graph and gives many suggestion: things they know about it.	What can you tell me about the graph? What about where it crosses the axis? What values does it have there?
Circle where x=3,-2	Answer where the graph crosses the x axis. At 3 and -2. Where y equals zero.	What about this part of the graph specific? What does it represent?
The part of the graph we identified is where x=3, -2 Today we will be learning a method to find this values.	May have ideas, may not.	Without looking at the graph does anyone know how to solve for these values?

EXPLORATION:

Teacher will do:	Student :	Probing Questions:
Write problem on board: x^2 + 6x – 7 = 0	Take out scratch paper and write down the problem. Students will get the equation to x^2 + 6x =7.	Now what might we do with this equation to solve for the x’s?
Write what students have told them: x^2 + 6x =7.	Silence or maybe some ideas. (x+3)(x+3), but that gives us x^2 + 6x +9.	What else might we do to the problem? What might we do to the x^2 + 6x part? Can we represent this part differently? Can we maybe represent the data with (x__)*(x___)? If it’s not exactly x^2 + 6x , can we get close to it?
Writes down on board x^2 + 6x +9 =7and any addition ideas students have.	We added nine. We need to add nine to both sides to get x^2 + 6x +9 =16 (x+3)(x+3),=16 (x+3)^2=16	So what did we do to the left side that we did not do to the right side? Then what can we make the left side look like? Can we break that down?
Writes down the steps the students are leading them through. Ask a student to write it down on the board if someone feels comfortable. Yeah, we just solve for x by a method called completing the square.	X+3=plus or minus 4; then x equals -7,1. Students may not be sure about the plus or minus, be sure to make this clear or clarify if needed.	Now can we solve this for x? How do we do that ?

EXPLANATION

Teacher will do:	Student will do:	Probing Questions:
Tell students that there is a shortcut to this method.	Possibly will see it, but some won’t.	Can anyone see the short cut?
Well let’s look at the original equation we started with.	Plus nine is 6 /2 and then squared. Can get three by just dividing 6 by 2.	What might we do to get our plus nine? What might we do to get our three?
Write down a completing the square work with already knowing the shortcut. Do with general formula. Assume a=1. Think about the problem before this and try to relate it.	ax^2 + bx +c=0 then ax^2 + bx=-c then add (b/2) Assume a=1. squared to both sides and (x+b/2)^2=-c+(b/2)^2 If a equals another number besides one.	What might we start off with? Does it still work with the symbols? What might cause problems for this method?

ELABORATION:

Teacher will do:	Student will do:	Probing Questions:
Assign student to computers and write down website on board.	Get computers and go to http://www.ltcconline.net/greenl/java /BasicAlgebra/CompleteTheSquare AndSolve1/CompleteTheSquareA ndSolve/classes/Complete TheSquareAndSolve.html
Has the student explore applet and work on completing the square.	Students will start the applet and work various problems with the applet.
Instruct students to do five problems and write down their work. Monitor students and answer any questions they may have.	Will do five problems with applet and write down the problems and steps they used to solve the problem.

EXTENSION:

Teacher will do:	Students will do:	Probing Questions:
Have student try and come up with a visual proof that this works. Possible have them work on this in groups of 3-4.	Explore and think about how they might represent this. May come up with squares and rectangles. Could be another figure.	What do we need to represent the equation? What might we use to represent it? Can we represent this will blocks ect?
Have some students come up to the board. Have them draw their visual proof and explain why it makes sense to them.

EVALUATION:

Teacher will do:	Student will do:	Probing Questions:
Hand out homework.	Work on the homework with the remaining time in class and take it home if needed.
Take up homework the next day and look for common misconceptions to see if students get the idea.	Will need to do the work and turn it in.