Course 1 Unit 7 - Quadratic Functions

This is the fourth unit from the algebra and functions strand in Course 1. This unit begins the study of quadratic functions that will be continued in each of Core-Plus Mathematics Courses 2-4 and in distributed Review tasks in many units.
     The unit was written assuming that most students entering this curriculum will have had modest prior experience with quadratic functions and equations. However, most of that experience may have focused on "finding the unknown x," not on how a function rule like y = ax2 + bx + c relates all values of x to values of y. The emphasis in this unit is on quadratic functions, and quadratic equations arise as a way of expressing questions about quadratic functions. The approach in the unit assumes that the development of an understanding of quadratic functions in realistic situations will make manipulating symbols much more meaningful.

Unit Overview
To be proficient in the use of quadratic functions for problem solving, students must have a clear and connected understanding of the numeric, graphic, verbal, and symbolic representations of quadratic functions and the ways that those representations can be applied to patterns in real data. The lessons of this unit are planned to develop each student's intuitive understanding of quadratic patterns of change and technical skills for reasoning with the various representations of those patterns. Understanding and skill in working with quadratic functions is developed in three lessons.
The first lesson develops students' understanding of the characteristics of quadratic functions and the connections among the various representations. Special attention is given to discovering the way each term of a quadratic expression influences the patterns in tables of (xy) values and the shapes of graphs for quadratics. Lesson 2 explores ways of constructing symbolic expressions for quadratic function rules by reasoning and by data modeling, and strategies for rewriting quadratic expressions in equivalent forms by expanding products of two binomials and/or one monomial and one binomial factor. Lesson 3 explores strategies for solving quadratic equations algebraically by reducing to x2 = n, by factoring, or by using the quadratic formula. The final lesson takes a look back and reviews the key concepts and skills of the unit.

Objectives of the Unit
  • Recognize patterns in tables of sample values, in problem conditions, and in data plots that can be described by quadratic functions
  • Write quadratic function rules to describe quadratic, or approximately quadratic, patterns in graphs or numerical data
  • Use table, graph, or symbolic representations of quadratic functions to answer questions about the situations they represent: (1) Calculate y for a given x (i.e., evaluate functions); (2) Find x for a given y (i.e., solve equations and inequalities); and (3) Describe the rate at which y changes as x changes
  • Rewrite simple quadratic expressions in equivalent forms by expanding or factoring given expressions and/or by combining like terms

Sample Overview
The sample material is Investigation 1 from Lesson 3. In Lessons 1 and 2, students developed a connected understanding of the first three objectives above. With that understanding in place, they develop effective methods for solving quadratic equations algebraically. In Investigation 2 (not provided here), students use the quadratic formula to solve quadratic equations. They also connect the formula to the information it provides on the x-intercepts and maximum or minimum of the quadratic function that corresponds to the equation.

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.

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How the Algebra and Functions Strand Continues
In Course 2, students review and extend their ability to recognize, describe, and use functional relationships among quantitative variables, with special emphasis on relationships that involve two or more independent variables. They also develop matrix and linear combination methods for solving systems of two linear equations. They are introduced to function notation, review and extend their ability to construct and reason with functions that model parabolic shapes and other quadratic relationships in science and economics, with special emphasis on formal symbolic reasoning methods, and are introduced to common logarithms and algebraic methods for solving exponential equations.
In Course 3, students extend their understanding of formal reasoning in contexts, study linear inequalities and linear programming, polynomial (including the vertex form of quadratic functions) and rational functions, sequences and series, and inverse functions.
     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.)

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