Course 2 Unit 4 - Power Models
1st Edition

Power Models is the fourth major unit from the algebra and functions strand of the Contemporary Mathematics in Context program. As students begin this unit, they are able to recognize a variety of patterns of change among related variables, and are able to use linear and exponential equations to model and solve real-world problems. (See the descriptions of Course 2 Units.)

Unit Overview

Power Models develops student ability to recognize data patterns and problem conditions that involve direct or inverse power variation and quadratic relationships. Students analyze these models and equations from the tabular, graphic, and symbolic viewpoints and apply them to a variety of problems.

Unit Objectives
  • To develop understanding of the types of situations that could be represented by power models, inverse power models, and quadratic models
  • To model situations involving direct variation, inverse variation, and quadratic relationships
  • To compare patterns found in tables and graphs of models of the forms y = ax2, y = ax3, y = a/x2, y = a/x3, and y = ax+ bx + c.
  • To develop understanding of the different rates of change in various models
  • To find good estimates of solutions to quadratic equations and to be able to check those solutions
  • To simplify expressions by applying properties of exponents and rewriting radical expressions

Sample Overview

In the first two investigations of Lesson 3, "Quadratic Models," students build quadratic models for motion, income, and profit. They investigate the contextual meaning and impact of each term of the quadratic expression and use the models to solve problems. In Investigations 3 and 4 of this lesson (not included in this sample material), students analyze the variety of patterns that can be modeled by the full quadratic expression ax2 + bx + c and the ways that questions corresponding to the quadratic equations can be solved by numerical and graphical methods.

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 common pattern as elaborated under Instructional Design.

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How the Algebra and Functions Strand Continues

Course 3 contains three units devoted to extending student ability to represent and solve problems using algebraic models and methods. Students develop the ability to construct, reason with, and solve equations involving several variables and constraints in Unit 1, Multiple-Variable Models. Unit 3, Symbol Sense and Algebraic Reasoning, formalizes the function concept, introduces polynomial and rational functions, extends the solution of equations and inequalities by methods including factoring and the quadratic formula, and develops student ability in algebraic proof. The final algebra and functions unit in Course 3, Families of Functions, reviews and extends student understanding of the basic function families and develops student ability to adjust function rules to match patterns in tables, graphs, and problem conditions.

Four units in Course 4 extend student understanding of algebra and function concepts in preparation for post-secondary education. Students develop understanding of the fundamental concepts underlying calculus, of inverse functions, and of logarithmic functions and their use in modeling and analyzing problem situations. Students also extend their ability to use polynomial and rational functions to solve problems and to manipulate symbolic representations of exponential, logarithmic, and trigonometric functions.

A unit that develops understanding and skill in the use of standard spreadsheet operations while reviewing and extending many of the basic algebra topics from Courses 1-3 is included for students intending to pursue programs in social, management, and some of the health sciences or humanities.

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