Course 4 Unit 4 - Counting Models
1st Edition

Students have learned concepts and methods for solving counting problems in previous courses of the Contemporary Mathematics in Context program, but not as explicit instruction. This unit from the discrete mathematics strand pulls together and formalizes this work. (See the descriptions of Course 4 Units.)

Unit Overview

Counting Models extends student ability to count systematically and solve enumeration problems. The unit also develops student understanding of and ability to do proof by mathematical induction.

Unit Objectives
  • To develop the skill of careful counting in a variety of contexts
  • To understand and apply a variety of counting techniques, including the Multiplication Principle of Counting, tree diagrams, systematic lists, and combinatorial reasoning
  • To identify, understand, and solve counting problems involving combinations and permutations
  • To understand and apply the General Multiplication Rule for probability
  • To understand and apply the Binomial Theorem and Pascal's triangle
  • To develop the ability to prove statements using combinatorial reasoning and the Principle of Mathematical Induction

Sample Overview

This sample material consists of the two investigations of Lesson 2, "Counting Throughout Mathematics." In the first investigation, students use counting methods developed in Lesson 1 to solve probability problems. In the second investigation, the focus shifts to algebra as students learn about the connections among combinations, the Binomial Theorem, and Pascal's triangle.

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 Discrete Mathematics Strand Continues

The counting and reasoning skills developed in this unit will be applied in future units. For example, combinations are required in Unit 5, Binomial Distributions and Statistical Inference, for the binomial probability formula, and counting arguments related to binary strings are used in Unit 9, Informatics (the final discrete mathematics unit). Pascal's triangle is revisited in Unit 10, Problem Solving, Algorithms, and Spreadsheets.

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