Course 1 Unit 1 - Patterns of Change

Patterns of Change is the first unit in Course 1 of the Core-Plus Mathematics curriculum. Other algebra and functions strand units in Course 1 are Unit 3, Linear Functions, Unit 5, Exponential Functions, and Unit 7, Quadratic Functions. (See the descriptions of Course 1 Units.) Note that there are eight units for each course. Units vary in length but are typically completed in 3 to 6 weeks.

Unit Overview
The intent of this unit is to focus student attention on the variety of types of change inherent in problem situations. This unit will provide students with a broad picture of patterns of change. Students will explore linear, quadratic, inverse variation, and exponential patterns of change. Within this unit, there is an effort to make a distinction between cause-and-effect change relationships and change-over-time relationships. In the third unit of this course, linear functions will be analyzed as a class of functions with a specific pattern of change.

Objectives of the Unit
  • Begin developing students' sensitivity to the rich variety of situations in which quantities vary in relation to each other
  • Develop students' ability to represent relations among variables in several waysusing tables of numerical data, coordinate graphs, symbolic rules, and verbal descriptions—and to interpret data presented in any one of those forms
  • Develop students' ability to recognize important patterns of change in single variables and related variables

Sample Overview
Lesson 1 begins with a bungee jump experiment in which students review the use of tables and graphs for representing relations between variables. In Investigation 2, relationships with random variation are explored. This foreshadows that aspect of Course 1 and gives an example in which the value of y is not precisely predictable from the value of x by an algebraic rule. Investigation 3 is designed to get a variety of functions on the table early. In general, it is not expected that students will be proficient at translating word problems into algebraic rules, but they might be able to use a given rule.
     The second lesson (a portion of which is included on this Web site—see below) develops students' understanding and skill in analyzing situations that change over time. In particular, the iterative perspective in which one compares the value of a variable at one point in time to the value of the variable at successive, equally spaced intervals is introduced.
     The main goals of Lesson 3 are to develop each student's ability to express problem conditions symbolically and to use symbolic representations with appropriate technology to answer questions about situations modeled by several basic patterns of change. In particular, students will learn to produce tables and graphs for functions in order to solve equations in one variable.
     Lesson 4, the "Looking Back" lesson, is intended to give students an opportunity to synthesize and pull together the main mathematical ideas of the unit. The concept of "function" is informally defined in this lesson.

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
As you examine the sample investigations and lessons, you should note that lessons often span several days and that investigations typically require more than one 50-minute class period to complete. (See Instructional Design.) The two investigations in Lesson 2, homework set, and quiz will take approximately six 50-minute class periods for heterogeneously grouped classes of ninth graders. Teachers adjust the length of time spent on an investigation for classes above or below grade level or classes that have not completed rich middle school mathematics programs. Also included here is the Practicing for Standardized Tests masters that are provided in Unit 1 Resource Masters.

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How the Algebra and Functions Strand Continues
Patterns of Change is the first of ten algebra and functions units in the first three years of the Core-Plus Mathematics curriculum. Algebraic reasoning and skill is developed in these units, but also in other strand units such as Course 2 Unit 3, Coordinate Methods, Unit 4, Regression and Correlation, Unit 7, Trigonometric Methods, Course 3 Unit 1 Lesson 3, "Algebraic Reasoning and Proof," Unit 6, Circles and Circular Functions, and Unit 7, Recursion and Iteration. In Course 4, students continuing on to college will continue to develop these skills. (See the CPMP Courses 1-4 descriptions.)

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