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  This lesson provided by:  
Author:Lillie Coleman
System: Fairfield City
School: Fairfield High Preparatory School

  General Lesson Information  
Lesson Plan ID: 27664

Title:

Discover the Roots of a Polynomial Function

Overview/Annotation:

In this lesson, students will be re-introduced to the Four Step Problem - Solving Plan.  The plan will be used as a 'checks and balance' tool for discovering the roots (solutions) of polynomial functions.  Students will:

*Explore the problem - identify what is given and what they are asked to find.

*Develop a plan - look for a pattern, make a model, solve a simpler but related problem.                          

*Solve the problem - carry out the plan

*Examine the solution - check the results with the conditions in the problem, check the reasonableness of the solution 


 Associated Standards and Objectives 
Content Standard(s):
MA2015 (9-12) Algebra
9. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* [A-SSE3]
a. Factor a quadratic expression to reveal the zeros of the function it defines. [A-SSE3a]
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. [A-SSE3b]
c. Determine a quadratic equation when given its graph or roots. (Alabama)
d. Use the properties of exponents to transform expressions for exponential functions. [A-SSE3c]
Example: The expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
 
MA2015 (9-12) Algebra
18. Solve quadratic equations in one variable. [A-REI4]
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form. [A-REI4a]
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square and the quadratic formula, and factoring as appropriate to the initial form of the equation. [A-REI4b] (Alabama)
 
MA2015 (9-12) Algebra II
17. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. [A-APR3]
 

Local/National Standards:

This lesson addresses the following NCTM standards:

*Understand patterns, relationships and functions.

*Represent and analyze mathematical situations and structures using algebraic symbols.

*Use mathematical models to represent and understand quantitative relationships.

*Analyze change in various contexts.

*Draw reasonable conclusions about a situation being modeled.

Primary Learning Objective(s):

Students will be able to:

Find zeroes by factoring.

Find zeroes using the quadratic formula.

Find the zeroes of a cubic polynomial.

Students will utilize other problem-solving strategies such as identifying conditions, organizing data and working backwards deploying inverse methods.   

Find zeroes approximately.

Solve an equation by calculator:  an erroneous answer.

Additional Learning Objective(s):

 

 Preparation Information 

Total Duration:

61 to 90 Minutes

Materials and Resources:

Assign students as experts in groups of 3.  Have the students pull slips from a container with numbers 1 through 3.  Like numbers will be designated as experts for preselected problem-solving strategies.

Graphing calculators, graph paper, linear measuring tool, notebook and pencil.           

Technology Resources Needed:

One graphing calculator per group of 3 students.

One laptop with Internet access per group of 3 students.

Document camera, laptop with Internet access , whiteboard and projector combination for teacher demonstration and guided practice.

Background/Preparation:

Motivating the Lesson:

Teacher should remind students that we have been problem-solvers since early childhood.  Babies are presented with toys which force them to perform a task to get desired results (insert name of popular toy).  Kindergartners were introduced to worksheets with blanks representing missing numbers.  Finding solutions have evolved to finding roots of polynomial equations with large exponents.   


  Procedures/Activities: 
 

Teacher will visit each group:

Expert Group #1:

Students will use graphing strategies to find solutions to polynomial equations. 

The opening of this lesson will center around the display and review of some special functions in graph form.  Students will click on the link below as the teacher guides them through the activity below:

http://www.glencoe.com/sec/math/precalculus/amc_04/extra_examples/chapter3/amc06_0302.rtf

The students will be randomly assigned one of the sample problems and asked to make a table of values for graphing the assigned equation, then sketch the graph on grid paper.

To build the students' understanding, the following link will help students visualize the procedure for finding the roots of polynomial functions using the graphing calculator.

http://www.mathbits.com/MathBits/TISection/Algebra2/zerofunctions.htm

Activity:  Students will use the following guidelines when using any type of computer / calculator in problem-solving:

1.  Students must understand the operation of their own calculator.  Students will begin by entering expressions in a way that guarantee that their calculator is performing the operation correctly.

2.  Students will focus first on analyzing the problem.  A strategy will be developed.  The student will then use the calculator to help implement the strategy.

3.  Students will check their solutions using a calculator.  Check for reasonableness.

4.  To lesson the chance of errors, students should clear the calculator and check all settings before beginning a new problem. 

Expert Group #2:

Students will use the 'complete the square' method.

Teacher will remind students when solving a quadratic equation by completing the square, the leading coefficient must be 1.  When the leading coefficient of a quadratic equation is not 1, you must divide each side by the coefficient before completing the square.

Complete the Square:

1.  Move the constant to the right side of the equation.

2.  Take 1/2 of the middle term and square the results.

3.  Add the value in step 2 to each side of the equation.

4.  Factor the perfect square trinomial.

5.  Take the square of each side.

6.  Move the constant to the right side of the equation to reveal the roots of the equation.

Students will use the following link to work through some additional examples.    

http://patrickjmt.com/quadratic-equations-completing-the-square/

Expert Group #3:

Students will use the Quadratic Formula to find the roots of a quadratic equation.

Quadratic Formula:

-b ± √b2-4ac ⁄ 2a

Students will discover that in the quadratic formula, the radicand b2- 4ac is called the discriminant of the equation.  The discriminant tells the nature of the roots of a quadratic equation or the zeros of the related quadratic function. 

*b2 - 4ac > 0  implies two distinct roots/zeros

*b2 - 4ac = 0 implies exactly one root/zero  (The one real root  is actually a double root.)

*b2 -4ac < 0  implies no real roots/zero (two distinct imaginary roots/zeros) 

Students will use the following link for additional practice with the quadratic formula.

http://patrickjmt.com/using-the-quadratic-formula/

After each expert group has completed their assigned activities, the students will form new groups of 3 with an expert from each of the previous groups.  Strategies and methods will be shared by each expert.

Reteaching / Correcting Common Errors - Error:  Students begin to write equations for a problem before the problem is completely understood.  Prescription:  While a problem is being discussed, students will put their pencils down.  No writing is permitted until the conditions and the variables are identified.  Given problem situations, individual students will tell the steps of the solution and uncover each step.  Students will continue to work in their groups and write steps for each method.  This should eliminate mistakes made by taking short cuts or completing too many steps at one time.  



  Assessment  

Assessment Strategies

Students will present verbally and written the best method for solving a polynomial function.

Additionally, the quiz below should be taken and checked in a group setting.

http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-888482-9&chapter=6&lesson=7&&headerFile=7


Acceleration:

Real-World Application of Solving Polynomial Functions

Student will search the Internet for the following:

*How does the knowledge of polynomial functions facilitate the design of the payload area of a satellite?

*What goes up must come down!  How does the knowledge of polynomial functions prove this theory?

Kinesthetic

*How much ceramic tile does it take to cover your classroom floor? 

Intervention:

Understanding the knowledge level of the student is essential for the success of this lesson.  During the Lesson Motivation Activity, the teacher will use problem-solving examples from each level of education (i.e., early childhood, elementary, middle, etc)  Students will be called on for responses.  Teacher observations are critical during this activity.  Students needing extra preparation will be provided extra examples on their identified level.  Teacher will also schedule one-on-one time with these students to asist them.   

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.
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