#3048. Introduction to Polynomials

Mathematics, level: Senior
Posted Tue Feb 3 20:39:18 PST 2004 by Scott Krise (Scott_Krise@yahoo.com).
University of Pittsburgh at Johnstown, Johnstown, Pa
Materials Required: algebra tiles, overhead projector, transparencies, worksheets
Activity Time: 40 minutes
Concepts Taught: polynomials

Introduction (attention getter, anticipatory set, discrepant event, open-ended problem scenario, engagement):

Concept Analysis and Development:
Get the students to repeat the word polynomial.

Readiness Check:
Ask students for examples of: Constants, Variables, Exponents

What do we get if we multiply constants and variables together? Terms

Show the students that all of the examples on the board are polynomials.

Developmental Activities (Instructional components that provide opportunities for students to make progress toward intended instructional objectives):

Discuss objectives:
By the end of the class you'll know what a polynomial is and why these are all examples of polynomials. Hopefully you'll be able to pick out examples and non-examples of polynomials. We should also find time to simplify polynomials by combining like terms. And perhaps we'll even get to add and subtract some polynomials.

Define the concept:
A polynomial is a sum of terms that consist of constants and variables, where the variables must be raised to a whole number exponent.

Demonstrate the concept through several examples:
X3 + 4X +3
X2 + 1
4
X5 -- 8
4 + 8X -- 12X3
Compare examples and non-examples of the concept:
X3 + X2 -- 4X2 + 8
-6X-2

List the Noncritical Attributes (irrelevant dimensions) that are related to the concepts:
~Terms may be written in any order.
~Terms may have variables in the denominator if they have negative integer exponents.
~Expression may include differences of terms.
~Non-simplified expressions are still polynomials.

Identify and design a strategy where students practice using the concept:
~Inform the students that we will be using Algebra Tiles to reinforce the concept of polynomial structures. Distribute the tiles and instruct the students to dump out their bags and get familiar with the shapes and colors of the tiles. Ask the students what they notice about the tiles, i.e. their color patterns and shape differences. Place the three tiles of different shapes onto the overhead and relate to the students what symbols correspond to each shape. ~Blue Square = X2
~Green Rectangle = X
~Yellow Square = 12 or 1

Have the students construct the polynomials:
4
3X
2X2
2X + 10
3X2 -- 3X + 4
6 + 3 -- X2

Identify and design a strategy that evaluates student mastery of the concept: Break the students up into pairs and have them complete the short worksheet on polynomials in which they are allowed to use their algebra tiles. Collect the worksheets after the students have all finished.

Introduce simplifying like terms:
~Using the algebra tiles show the addition and subtraction of like terms.
Examples.

~2 Blue Squares + 1 Blue Square = 3 Blue Squares
~3 Green Rectangles -- 2 Green Rectangles = 1 Green Rectangle

Equate the relationships between adding and subtracting like shapes to adding and subtracting coefficients of like variables. Then experiment with adding and subtracting polynomials. Allow the students to practice using Worksheet 2 and also have them write 4 of their own sums if two of them must be sums of binomials and two must be sums of trinomials.
Collect this worksheet also.

Conclusion (Closure; a planned wrap-up for the lesson)
Wrap up the lesson with a quick review of the definition of a polynomial and ask if there are any questions.