              ### Course 1 Unit 7 - Simulation Models 1st Edition

Simulation Models is the seventh unit in Course 1 of the Contemporary Mathematics in Context program. By the time students begin this unit, they will have developed the ability to make sense out of real-world data through the use of graphical displays and summary statistics. They will be able to recognize important patterns of change among related variables. They will have developed skill in using linear and exponential equations to model real-world problems. Students will also have developed skills in algorithmic problem solving and learned how to model a variety of situations with vertex-edge graphs. They will have enhanced their visualization skills and developed a deeper understanding of two- and three-dimensional shapes and their properties. (See the descriptions of Course 1 Units.)

#### Unit Overview

The Simulation Models unit introduces students to simulation and to the idea of probability distributions. Simulation is a powerful, easily understood method that also has instructional advantages. Setting up a simulation helps students clarify their assumptions about such things as whether trials are independent. Simulation helps develop students' intuition about probabilistic events. And, perhaps most importantly, students who have been introduced to simulation have a feeling of control over probability.

Important probabilistic concepts explored in this unit include simulation, frequency tables and their histograms, the idea of independent events, the Law of Large Numbers, and expected number of successes in a series of binomial trials.

 Unit Objectives To design and carry out simulations in order to estimate answers to questions about probability To use the Law of Large Numbers to understand situations involving chance To use tables of random digits in order to perform simulations and to understand some properties of random digits To understand the concept of a probability distribution and how an approximate probability distribution can be constructed using simulation in order to understand situations involving chance

#### Sample Overview

In Lesson 1 (a portion of which is included on this web site - see below), students examine various plans to limit population growth, simulating situations in which the probability of each outcome is 0.5. Students construct frequency tables and display corresponding histograms for geometric or waiting-time distributions.

#### 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.

#### View Sample Material

We have provided for your perusal a file containing the Simulation Models Table of Contents and another containing sample material from Lesson 1.

#### How the Statistics and Probability Strand Continues

In Course 2, students study the appropriate use of correlation and regression to describe bivariate association. Students continue to develop their ability to understand situations involving chance by using simulations and mathematical analysis to construct probability distributions and study the following topics: multiplication rule for independent events, geometric distributions, expected value, rare events, summation notation, and an introduction to binomial distributions.

In Course 3, students study survey methods, sampling, sampling distributions, relationships between a sample and a population, confidence intervals, and margin of error. In Unit 5, Patterns in Variation, students extend their understanding of measurement of variation, use the normal distribution as a model of variation, and are introduced to probability and statistical inference involved in the control charts used in industry for statistical process control.

In Course 4, students extend their understanding of the binomial distribution and how the binomial distribution is used in statistical inference to test a single proportion and to compare two treatments in an experiment.

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