 |
 |
 |
 |
 |
 |
 |
 |
 |
Course 3 Unit 2 - Inequalities and Linear Programming
©2009
Solving systems of linear inequalities was informally addressed in Course 1 Unit 3, Linear Functions. In Course 2 Unit 1, Functions, Equations, and Systems, and Unit 2, Matrix Methods, students solved linear systems of two equations in two variables by graphing, substitution, and linear combinations, and by using matrix methods. This unit reviews and extends students' understanding of and their ability to solve inequalities in one and two variables.
Unit Overview
This unit develops
student ability to reason both algebraically and graphically to solve
inequalities in one and two variables, introduces systems of inequalities
in two variables, and develops a strategy for optimizing a linear function
in two variables within a system of linear constraints on those variables.
Topics included are inequalities in one and two variables, number line
graphs, interval notation, systems of linear inequalities, and linear
programming.
Objectives
of the Unit
|
Sample Overview
The sample material
provided below is Investigation 3, "Complex Inequalities," from
Lesson 1, "Inequalities in One Variable." The goal of
this investigation is to generalize students' graphic and algebraic
understanding and technique for solving inequalities (developed in
the first two investigations of this lesson) to somewhat more complex
situations and algebraic expressions. Students use algebraic methods
to locate intersection points of graphs and use the visual images provided
by graphs to inform the solution of the related inequalities. Interval
notation is introduced in this investigation.
Selected
homework tasks from the Connections and Reflections sections are provided.
See Implementing
Core-Plus Mathematics page 13 for more detail on the design
features of the On Your Own homework section.
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 are two more algebra and functions strand units, Unit 5, Polynomial
and Rational Functions, and Unit 8, Inverse Functions.
Unit 5
extends student ability to represent and draw inferences about polynomial
and rational functions using symbolic expressions and manipulations.
Topics included are the definition and properties of polynomial and
rational expressions, operations on these types of expressions, completing
the square, proof of the quadratic formula, solving quadratic equations
(including complex number solutions), and the vertex form of quadratic
functions.
Unit 8
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.)