# ALEX Lesson Plan

## Discover the Roots of a Polynomial Function

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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.

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: