BERKELEY HEIGHTS PUBLIC SCHOOLS BERKELEY HEIGHTS, NEW JERSEY

GOVERNOR LIVINGSTON HIGH SCHOOL MATHEMATICS DEPARTMENT

AP AND DISCRETE MATHEMATICS #MAY1230

Curriculum Guide

September 2010

Mrs. Judith Rattner, Superintendent Mrs. Patricia Qualshie, Assistant Superintendent Mrs. Susan Rembetsy, District Supervisor

Developed by: Steve Hess

This curriculum may be modified through varying techniques, strategies, and materials, as per an individual student’s Individualized Plan (IEP)

Approved by the Berkeley Heights Board of Education at the regular meeting held on 10/28/10 .

TABLE OF CONTENTS

Page

Vision Statement ...... 1

Mission Statement ...... 2

Course Proficiencies ...... 3

Course Objectives ...... 3

Student Proficiencies ...... 6

Methods of Evaluation ...... 8

Course Outline/Student Objectives ...... 9

Vocabulary for AP Statistics/Discrete Math ...... 14

Resources/Activities Guide ...... 15

Suggested Materials ...... 17

Resources for Students ...... 17

Resources for Teacher ...... 17

VISION STATEMENT

AP Statistics and Discrete Math is a college level course that includes the content covered by the AP Statistics Exam and selected topics from discrete mathematics. The text being used was written for high school students who are at an advanced reading level. The objective of the course is to provide students with a non‐ introduction to statistical literacy, by presenting the tools for collecting, analyzing, and drawing conclusions from data. The themes of the course are exploring data, by observing patterns and departures from patterns, planning a study and deciding how to collect and measure data, anticipating patterns by using theory and simulation, and analysis to confirm models.

The course is practical. It does not deal in‐depth with numerous mathematical proofs, although some formality will exist. Most of what is covered is done through an intuitive and logical approach. This offering will provide an added opportunity for students to strengthen their quantitative reasoning abilities. The AP Statistics portion of the course is designed around the use of technology. Graphing calculators are used extensively. The computer will be used to handle larger data sets. The course will develop marketable skills through the use of and interpretation. Students will also frequently work in groups, which will help develop their communications and teamwork skills.

Effective communication, using the language of mathematics, is essential for all math courses and will be emphasized throughout this course. The development of mathematical definitions, notations, terminology, syntax, and logic will be imbedded in the course studies. Students will also be required to interpret statistical measures in the context of the problem and be able to clearly discuss the reasonableness of conclusions that can be formed from specific data sets.

Five (5) credits are given for passing this full year course. Standards from the New Jersey Core Curriculum Content Standards for Mathematics have been integrated throughout the curriculum.

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MISSION STATEMENT

This course provides an opportunity for students to learn the principles of statistical analysis and how to apply them to a wide variety of real world situations, from the social , business, biology, and other fields. The students will also explore a variety of discrete math topics, which will expand their problem‐solving skills and expose them to a number of math fields, which have not been covered in‐depth, in previous courses.

Through both the statistics and discrete math topics, the students will become familiar with a variety of problem‐solving strategies and develop an understanding of the applications of math to many other fields. The students will also learn to use mathematical language to communicate ideas through the use of model, diagrams, and symbols. The students will be able to build mathematics models to represent specific information provided in real world situations. Technology will be integrated throughout the instruction of the course, and will be used to expand the problem‐solving capabilities of the students.

The students will be held responsible for individual work while also being expected to function productively in a cooperative learning environment. The prerequisites for AP Statistics and Discrete Math are a B+ or better average in Math Analysis, or a B average or better in Math Analysis Honors with a teacher recommendation. Five (5.0) credits will be earned for successful completion of this full year course. National and state standards are integrated throughout the curriculum.

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COURSE PROFICIENCIES

COURSE OBJECTIVES

1. Display univariate data with graphs, including , stemplots, and box and whisker plots, and describe data through center, shape, and spread. (4.4/12A5)

2. Apply and analyze the and use the normal probability plot to assess whether a dataset is approximately normal. (4.4/12A5)

3. Analyze form, directions, and strength of relationships in bivariate scatterplots. (4.4/12A5; 4.5/12B4)

4. Understand and apply correlation and discuss cause‐and‐effect relationships versus causation. (4.4/12A2,5; 4.5/12B4)

5. Use regression, including ordinary least squares, nonlinear models, and multi‐variate regression to analyze bivariate relationships. (4.4/12A4; 4.5/12F3)

6. Explore categorical data through frequency tables, bar charts, and two‐way tables, including marginal, joint, and conditional frequencies. (4.4/12A2)

7. Distinguish between census, survey, observational study, and and understand most appropriate applications for each. (4.4/12A6)

8. Understand the role of randomization, simple random sample, stratified random sample, cluster , and multi‐stage sampling. (4.4/12A1)

9. Recognize and anticipate sources of bias, including selection bias, response bias, non‐ response bias, convenience sampling, voluntary response bias, undercoverage, lurking variables, confounding, and common response relationships. (4.4/12A1)

10. Use three principles of experimental design – comparison of treatments, randomization, and replication. (4.4/12A2‐3)

11. Investigate random phenomena through simulation using a random number table and a graphing calculator (4.5/12A1‐3, F3‐4)

12. Estimate and predict outcomes from real world investigation, using Venn diagrams, decision trees, and conditional probability. (4.4/12B1,4‐6; 4.5/12A2‐6)

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COURSE PROFICIENCIES (continued)

13. Create and interpret discrete and continuous probability distributions and understand their application. (4.4/12B5; 4.5/12C4)

14. Use and interpret the mean and standard deviation for sums and differences of independent random variables. (4.4/12A5, B1)

15. Create sampling distributions and calculate their statistics for proportions and means, determine their variability based upon same size, and apply the Central Limit Theorem. (4.4/B6, C6)

16. Estimate confidence intervals for sample means and proportions, and calculate the necessary sample size to achieve a desired margin or error. (4.4/12A2,5)

17. Design and interpret hypothesis tests for statistical significance for proportions and means, choose an appropriate significance level, and explore the trade‐off between Type 1, Type II error and Power of test. (4.4/12A2,5; 4.5/12B2, C4)

18. Use Chi‐Square Goodness of Fit Test and inference for two‐way tables. (4.4/12A2,5; 4.5/12B2)

19. Create confidence intervals and hypothesis tests for the slope of a least‐squares regression model. (4.4/12A2,4,5)

20. Formulate a statistical hypothesis, design an appropriate statistical experiment, conduct the experiment, and interpret and communicate the results. (4.4/12A3; 4.5/12B1‐2,4)

21. Use formal symbolic logic notation to analyze statements, to build truth tables, and to compare expressions for logical equivalence. (4.2/12A4; 4.5/12B4, D1‐6)

22. Explore problems related to scheduling and conflict, network design, and management using graph theory including vertex coloring, Euler paths and circuits, Hamilton circuits, and critical path analysis. (4.4/12D1; 4.5/12A1‐6, C3‐4, D2,4, E1,3)

23. Apply the tools of game theory, including payoff matrices, extensive forms, and the concept of Nash Equilibrium to explore how several agents may act strategically and predictably in a competitive or cooperative situation. (4.5/12A1‐5, B1‐4, C2‐6, D2,4,6, E1,3)

24. Examine issues of social choice, including methods of voting and of fairly dividing discrete sets of goods, such as in an estate settlement. (4.4/12D2; 4.5/12A1‐5, C3‐4)

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COURSE PROFICIENCIES (continued)

25. Identify and create self‐similar fractal images through iteration, to find patterns in the area and perimeter of these images, and explore the concept and applications of fractal dimension. (4.2/12B4, E2; 4.3/12A1‐3; 4.5/12A1‐5, C3, E3)

Note: Instructor may choose a set of discrete topics from #’s21‐25. It is understood that not all topics will be covered every year.

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STUDENT PROFICIENCIES

The student will be able to:

1. Explore data and carefully describe the characteristics of a dataset. (4.4/12A5)

2. Use the graphing calculator to analyze data. (4.4/12F3,4)

3. Analyze normal distributions. (4.5/12B4, C3,4)

4. Obtain and use regression equations for both near‐linear and non‐linear data. (4.4/12A4; 4.5/12F3)

5. Produce data using samples and . (4.4/12A1)

6. Use probability theory to predict outcomes. (4.4/12B5)

7. Make inferences about the population, from samples. (4.4/12B5‐6)

8. Apply various significance tests using proportions and means within one sample and across two samples. (4.4/12A1‐5, B5; 4.5/12A2)

9. Use chi‐square procedures and regression to test the relationship between variables and to test the goodness of it. (4.4/12A2‐5, B5; 4.5/12A2)

10. Use formal symbolic logic to link statements, form conclusions, and recognize logical equivalence. (4.4/12A4; 4.5/12B4, D1‐6)

11. Solve problems of conflict and scheduling through the use of graphing theory tools (4.4/12D1; 4.5/12A1‐6, C3‐4, D2,4,5, E1‐3)

12. Use tools of Game Theory, including Nash Equilibrium, to analyze situations of conflict and bargaining under various scenarios of information, and both rational and irrational motivation. (4.5/12A1‐5, B1‐4, C2‐6, D2,4,6, E1‐3)

13. Evaluate numerous voting methods against several carefully defined fairness criteria. (4.4/12D2; 4.5/12A1‐5, C3‐4)

14. Examine various methods to fairly divide continuous and discrete bundles of goods and then identify the circumstances that produce equitable, envy free, and Pareto Optimal divisions. (4.4/12D2; 4.5/12A1‐5, C3‐4)

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STUDENT PROFICIENCIES (continued)

15. Identify and construct fractal images and explore applications of fractal dimension. (4.2/12B4, E2; 4.3/12A1‐3, C3, E3)

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METHODS OF EVALUATION

1. Tests and quizzes.

2. Homework and class work.

3. Projects.

4. Cooperative learning assignments.

5. Mid‐term and final examinations.

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SCOPE AND SEQUENCE COURSE OUTLINE/STUDENT OBJECTIVE

The student will be able to: NJ Core Curriculum Strands & Standards/ Indicators Course Outline/Student Objectives Grade 4.4/12 A5 I. Exploring Data Distributions (2 Weeks) 4.5/12 B1‐4 A. Display Univariate Data with Graphs Including C3‐4 Histograms, Stemplots, Dotplots, Dumulative Frequency Plots, Box and Whisker Plots, and Modified Box and Whisker Plots B. Describe Univariate Data Through Center, Shape, and Spread 1. Measure center: median, mean 2. Measure spread: range, IQR, standard deviation, quartiles, deciles, and five number summary 3. Identify outliers and measure which are resistant/non‐resistant to outliers 4. Describe shape: skewness, symmetry 5. Understand the effect of changing units on summary measures C. Apply and Analyze Normal Distribution 1. Interpret standard deviation and apply 68‐59‐99.7 rule 2. Use standardized normal curves 3. Use normal curve distribution tables and TI 83/84 to estimate outcomes that follow a roughly normal distribution D. Create and Interpret Normal Probability Plot 4.4/12 A2,4,6 II. Exploring Bivariate Data (4‐5 Weeks) 4.5/12 A1‐6 A. Analyze Form, Direction, and Strength of Relationship in F1‐4 a Scatterplot B. Understand and Apply Correlation C. Discuss Cause‐and‐Effect Relationships vs. Causation D. Use Regression to Analyze Divariate Relationships 1. Develop models using Ordinary Least Squares regression, using graphing calculator and Excel

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II. Exploring Bivariate Data (continued) 2. Recognize appropriate instances to use non‐linear models and develop and apply non‐linear power and exponential models 3. Compare and contrast the impact of influential observations and outliers 4. Analyze residuals and residual plots E. Introduce Multivariate Regression with Dummy Variables and Interaction Terms F. Explore Categorical Data Through Frequency Tables, Bar Charts, Two‐Way Tables, Including Marginal, Join, and Conditional Frequencies 4.4/12 A1‐3 III. Producing Data: Samples, Experiments, And Simulation (3 4.5/12 B1‐3 Weeks) C2‐4 A. Distinguish Between Census, Survey, Observational F3,4 Study, and Experiment and Understand Most Appropriate Applications for Each B. Understand the Role of Randomization, Simple Random Sample, Stratified Random Sample, Cluster Sampling, and Multi‐Stage Sampling C. Recognize and Anticipate Sources of Bias, Including Selection Bias, Response Bias, Non‐Response Bias, Convenience Sampling, Voluntary Response Bias, Undercoverage, Lurking Variables, Confounding and Common Response Relationships D. Design Effective Experiments 1. Use three principles of experimental design – comparison of treatments, randomization, and replication 2. Identify other key experimental design techniques, including factors, levels, treatments, double blink, placebo effect, blocking, and matched pair design E. Investigate Random Phenomena Through Simulation Using a Random Number Table and the TI 83/84 4.4/12 A1‐3 IV. Anticipating Patterns By Using Random Variables (6‐7 4.5/12 B1‐3 Weeks) C2‐4 A. Estimate Probabilities and Predict Outcomes from Real F3,4 World Investigation 1. Use Venn Diagrams, Decision Trees, and Conditional Probability 2. Distinguish between independent and disjoint events 3. Law of Large Numbers

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IV. Anticipating Patterns By Using Random Variables (continued) 4. Create and interpret discrete and continuous probability distributions and understand their application 5. Use and interpret the mean and standard deviation for sums and differences of independent random variables B. Recognize Binomial and Geometric Distribution Settings and Use Their Properties to Solve Specific Probability Problems C. Create Sampling Distributions and Calculate Their Statistics for Proportions and Means D. Determine the Variability of a Statistic Based Upon Sample Size E. Understand and Apply the Central Limit Theorem 4.4/12 A1‐5 V. Statistical Inference (8‐9 Weeks) 4.5/12 B1‐4 A. Estimate Confidence Intervals for Sample Means and C3‐4,6 Proportions E1‐3 1. Compare two means (paired and unpaired) or two F1‐4 proportions 2. Calculate necessary sample size to achieve a desired margin of error and understand the trade‐off between accuracy and cost 3. Use of distribution when standard deviation is unknown B. Design and Interpret Hypothesis Tests for Statistical Significance for Proportions and Means 1. Set up formal null and alternative hypothesis 2. Choose an appropriate significance level and explore the trade‐off between Type I, Type II error, and Power of test 3. Design and implement tests for differences between two sample proportions or means, including matched pairs C. Use Chi‐Square Goodness of Fit Test and Inference for Two‐Way Tables D. Create Confidence Intervals and Hypothesis Tests for the Slope of a Least‐Squares Regression Model 4.4/12 D1 VI. Graph Theory (2‐3 Weeks) 4.5/12 A1‐6 A. Define Connected Graphs and Find Their Chromatic C3‐4 Number

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D2,4,5 VI. Graph Theory (continued) E1‐3 1. Use vertex coloring to model real world problems of conflict and scheduling 2. Draw a connected graph with a given number of nodes and a given chromatic number B. Use Graphs and the Concept of Vertex Coloring to Model lRea World Problems of Conflict and Scheduling C. Find Euler Circuits and Paths When They Exist D. Eulerize Graphs Efficiently to Create Euler Circuits and Paths E. Find Hamilton Circuits and Contrast the Types of Problems Where These are Applicable with Euler Circuits and Paths F. Use Various Algorithms, Such as Cheapest Link and Nearest Neighbor to Explore Traveling Salesperson Problem G. Use Directed Graphs to Conduct Critical Path Analysis to Solve Project Management Problems Efficiently 4.5/12 A1‐5 VII. Game Theory (4 Weeks) B1‐4 A. Explore Classic Games of Prisoner’s Dilemma and C2‐6 Chicken D2,4,6 1. Define the essential characteristics and differences E1‐3 between these two similar, two player games 2. Explain the solution of these games through elimination of dominated strategies 3. Identify and explain the concept of Nash Equilibrium 4. Adopt these models to other real world situations of conflict or competition B. Use Backward Induction to Solve Games in Extensive Form C. Understand How Mixed Strategies Can Solve Games Without Nash Equilibrium 1. Analyze classic Hawk‐Dove game and use algebra and graphical analysis to find solution 2. Explore how changes to initial conditions affect the outcome D. Explore How Game Theory Can be Used to Model Irrational Behavior Such as Altruistic, Hyper Competitive, and Vindictive Behavior 4.4/12 D2 VIII. Voting And Fair Division (2 Weeks) 4.5/12 A1‐5 A. Define Numerous Voting Methods – Plurality, Majority, C3‐4 Borda, Condorcet, Run‐Off, Sequential Run‐Off, Approval – and Evaluate Against Fairness Criteria

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VIII. Voting And Fair Division (continued) B. Explore Weighted Voting Systems and Banzhaf Power Index C. Evaluate Various Divide and Choose Methods of Splitting a Continuously Divisible Set of Goods with Two and Three Players D. Evaluate Methods of Splitting a Set of Discrete Goods 1. Compare dan contrast sealed bid vs. adjusted winner 2. Use method of markers for large batches of items E. Consider the Impact of Asymmetric Information and Strategic Play on Voting and Fair Division 4.2/12 B4 IX. Fractals (1 Week) E2 A. Identify and Explain the Characteristics of Self‐Similarity 4.3/12 A1‐3 B. Construct Fractals Through Iteration Involving Removal, C3 Addition, and Rotation E3 C. Derive the Seed and Rules Set for Various Fractal Images D. Derive Formulas in Terms of the Iteration Step for Area and Perimeter of Various Fractals E. Calculate the Fractal Dimension of Man‐Made and Natural Fractal Images and Explore Applications 4.2/12 A4 X. Logic (1 Week) 4.5/12 B4 A. Translate Written Statements Into Symbolic Logic and D1‐6 Assess the Logical Equivalence of Various Written Statements B. Link Chains of Logical Statement to Form Conclusions Using Syllogism and Law of Detachment C. Use Truth Tables to Test Logical Equivalence Between Various Statements to Prove de Morgan’s Laws Note: The New Jersey Core Curriculum Content Standards can be accessed at www.state.nj.us

Topics VI‐X provide a set of choices for the instructor. Typically the instructor will cover three to four of the Discrete Math topics in a school year.

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VOCABULARY FOR AP STATISTICS/DISCRETE MATH

This is a list of essential vocabulary:

1. Sample means 2. Measure of center 3. Measure of spread 4. Outlier 5. Skewness 6. Univariate data 7. Symmetry 8. Standard normal curve 9. Bivariate data 10. Correlation 11. Regression 12. Standard deviation 13. Normal distribution 14. Randomization 15. Sampling 16. Sources of bias 17. Statistical inference 18. Confidence intervals 19. Game theory 20. Euler circuits 21. Hamilton circuits 22. Backward induction 23. Voting 24. Fair division 25. Fractals 26. Symbolic logic 27. Truth tables

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RESOURCES/ACTIVITIES GUIDE

Yates, Moor, and Starnes. The Practice of Statistics: TI‐83/84/89 Graphing Calculator Enhanced. 3rd ed. New York: W. H. Freeman, 2008.

Gonick, Larry and Woollcott Smith. The Cartoon Guide to Statistics. New York: Harper/Collins, 1993.

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SUGGESTED AUDIO VISUAL/COMPUTER AIDS

1. Text Instruments TI‐83 and TI‐84 graphing calculators.

2. Microsoft Excel.

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SUGGESTED MATERIALS

Resources for Students

College Board AP Statistics Website: http://www.collegeboard.com/student/testing/ap/sub_stats.html

Resources for Teacher

College Board AP Statistics Website: http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2151.html

Peck, Olsen, and Devore. Introduction to Statistics and Data Analysis. 2nd ed. Brooks/Cole‐ Thomson Learning, 2005.

COMAP. For All Practical Purposes: Mathematical Literacy in Today’s World. 6th ed. W. H. Freeman & Company, 2003.

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