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Course 3 Unit 5 - Polynomial and Rational Functions
©2009
In units of the algebra and functions strand in Courses 1 and 2, students built a repertoire of function families that model linear, exponential, quadratic, and power (direct and inverse) relationships. They also developed skill in using algebraic expressions to represent those patterns of change, skill in manipulating the expressions to solve equations and inequalities and to gain insight into relationships, and skill in solving a wide variety of authentic quantitative problems. In this unit, polynomial and rational functions are added to students' toolkit of families of functions.
Unit Overview
Topics
studied in this unit include the definition and properties of polynomials,
operations on polynomials, completing the square, proof of the quadratic
formula, solving quadratic equations (including complex number solutions),
the vertex form of quadratic functions, definition and properties of
rational functions, and operations on rational expressions.
This
unit is an introduction to polynomial and rational functions and expressions
and should be accessible to most students. Further work toward developing
proficiency with symbol manipulation occurs in subsequent Review tasks
and in Course 4.
Objectives
of the Unit
|
Sample Overview
The sample investigation
is Investigation 2 of Lesson 1. In the first investigation,
students learned how to model complex graphical patterns with polynomial
functions of degree 3
and 4. They explored a variety of polynomial functions to discover
the type of graph possibilities. They began to see the relationship
between the degree of a polynomial function and other important features
of its graph, especially the number of local maximum and local minimum
points.
In the sample
investigation provided below, students work with polynomial expressions
and functions, learning how to combine them by addition and subtraction.
They look for a pattern relating the degrees of components to the degree
of the sum or difference of two polynomials and a pattern relating the
degree of a polynomial expression to the number of zeroes of the corresponding
function.
This
investigation makes use of the public-domain CPMP-Tools computer
software. Select "Course 3" from the menu bar,
then choose Algebra and CAS. Enter a polynomial equation with parameters a, b,
and c (such as the one on Course 3 page 325) in the
Y= tab. Then select the Graph tab to view.
Instructional
Design
Throughout the curriculum,
interesting problem contexts serve as the foundation for instruction.
As lessons unfold around these problem situations, classroom instruction
tends to follow a four-phase cycle of classroom activities—Launch,
Explore, Share and Summarize, and Apply.
This instructional model is elaborated under Instructional
Design.
View the
Unit Table of Contents and Sample Lesson Material
You will need the
free Adobe
Acrobat Reader software to view and print the sample material.
How the Algebra
and Functions Strand Continues
In Course 3,
there is one more algebra unit, Unit 8, Inverse Functions.
This unit develops student understanding of inverses of functions with
a focus on logarithmic functions and their use in modeling and analyzing
problem situations and data patterns. Topics studied in this unit include
inverses of functions, logarithmic functions and their relation to
exponential functions, properties of logarithms, equation solving with
logarithms, and inverse trigonometric functions and their applications
to solving trigonometric equations.
Course 3 Unit 7, Recursion
and Iteration, is technically a discrete mathematics unit,
but working with sequences and series helps students strengthen their
symbolic skills.
Course 4:
Preparation for Calculus extends student algebraic skills and understandings
in equations and functions in algebra units but also in geometry units
such as Unit 2, Vectors and Motion, and Unit 6, Surfaces
and Cross Sections. (See
the CPMP Courses 1-4
descriptions.)