              Course 4 Unit 7 - Concepts of Calculus ©2010

This final algebra and functions unit of Course 4 builds on the prior units that developed student understanding of functions in general to focus on the key concepts of calculus—the derivative and the definite integral. (See the CPMP Courses 1-4 descriptions.)

Unit Overview
This unit consists of two main lessons. The goal of Lesson 1 is to draw on students' many prior experiences with functions and rates of change to address the question of instantaneous rate of change. It highlights the difference between average rate of change over an interval and rate of change at a point.
The goal of Lesson 2 is to develop understanding of ideas for studying accumulation of quantities that are changing at variable rates and for which the function that describes rate of change is what one knows. It focuses on the interpretation of area beneath a rate graph as accumulated change and develops the basic idea of the Riemann sum that defines the integral. The lesson does not develop the fundamental theorem of calculus linking anti-derivatives and integrals.

 Objectives of the Unit Develop the concept of instantaneous rate of change in a continuous variable and strategies for estimating those rates of change Define the derivative of a function at a point in its domain Connect the derivative of a function to local approximation of slope of its graph Develop derivative formulas for linear and quadratic functions Develop the connection between area under a rate function graph and accumulation of change Define the definite integral of a function and its application to problems

Sample Overview
The sample investigation below is the first investigation of the unit. In this investigation, students are asked to think about the problem of estimating the rate of change at a specific point in time for three continuously changing phenomena—height of a bungee jumper, motion of a person walking toward and away from a motion detector, and profit of a business. In each case, they are asked to use graphs and tables of sample values for each variable to estimate instantaneous rates of change.

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.