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Course 1 Unit 1 - Patterns of Change
©2008
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
|
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.
You will need the free Adobe
Acrobat Reader software to view and print the sample material.
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.)