Course 3 Unit 7 - Discrete Models of Change
1st Edition

Discrete Models of Change is the final unit from the discrete mathematics strand in Courses 1-3 of the Contemporary Mathematics in Context program. (See the descriptions of Course 3 Units.)

Unit Overview

Discrete Models of Change extends student ability to represent, analyze, and solve problems in situations involving sequential and recursive change. It summarizes and formalizes much of the work with recursion across Courses 1-3, while at the same time reviewing linear, exponential, and power functions.

Unit Objectives
  • To use iteration and recursion as tools to represent, analyze, and solve problems involving sequential change
  • To formalize and consolidate previous study of NOW-NEXT equations, particularly through the use of subscript notation and the introduction of recursive formulas
  • To understand and apply arithmetic and geometric sequences and series
  • To understand and apply the method of finite differences
  • To explore function iteration and, in the process, informally introduce function composition
  • To understand and apply recursive formulas, particularly combined recursive formulas of the form An = rAn - 1 + b
  • To review linear, exponential, and polynomial models from a recursive perspective

Sample Overview

The sample material from Discrete Models of Change is the second lesson of the unit. In the first lesson, recursive formulas were used to model situations, and the effect of parameter changes on long-term behavior in these situations was examined. Lesson 2, "A Discrete View of Function Models," develops student understanding of arithmetic and geometric sequences and their connections to linear and exponential functions. Finite differences tables are used to find power function formulas for selected recursive formulas.

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

In Course 4, the Counting Models unit extends student ability to count systematically and solve enumeration problems, and develops understanding of, and ability to do, proof by mathematical induction.

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