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Advanced Placement Statistics

Dover Public SchoolsCurriculum Guide

Mathematics Department

STATEMENT OF ASSURANCE 1

MATHEMATICS DEPARTMENT

AP Statistics

Statement of Assurance

I, (Tonalee Frame), District Supervisor of Mathematics certify that the AP Statistics curriculum contains the following required components:

The curriculum

1. is aligned to the latest New Jersey Core Curriculum Content Standards (NJCCCS) and/or Common Core Standards (CCSS) for the department stated above.

X

2. contains a pacing guide X 3. contains formative and summative assessment strategies

consistent with the District’s vision of increased academic achievement.

X

4. promotes understanding of content material at levels concomitant with 21st century skills and themes.

X 5. was developed using a standards assessment or similar

mapping strategy to insure that it is part of a district curriculum that is aligned vertically and horizontally.

X

6. addresses standards from Technology and Career Education, Consumer, Family, Life Skills, Arts, Health and Physical Education.

X

7. details available instructional materials and resources concomitant with indicated objectives and standards. X

Signature: _Tonalee Frame Date: September 11, 2013

Board of Education Approval Date:

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AP Statistics Curriculum (Models Audit Syllabus)

COURSE DESCRIPTION: AP Statistics is the high school equivalent of a one semester, introductory college statistics course. In this course, students develop strategies for collecting, organizing, analyzing, and drawing conclusions from data. Students design, administer, and tabulate results from surveys and experiments. Probability and simulations aid students in constructing models for chance phenomena. Sampling distributions provide the logical structure for confidence intervals and hypothesis tests. Students use a TI-83/84 graphing calculator, Fathom and/or Minitab statistical software, as well as Web-based java applets to investigate statistical concepts. To develop effective statistical communication skills, students are required to prepare frequent written and oral analyses of real data. COURSE GOALS: In AP Statistics, students are expected to learn

Skills • To produce convincing oral and written statistical arguments, using appropriate terminology, in a

variety of applied settings. • When and how to use technology to aid them in solving statistical problems

Knowledge • Essential techniques for producing data (surveys, experiments, observational studies, simulations),

analyzing data (graphical & numerical summaries), modeling data (probability, random variables, sampling distributions), and drawing conclusions from data (inference procedures – confidence intervals and significance tests)

Habits of mind • To become critical consumers of published statistical results by heightening their awareness of ways

in which statistics can be improperly used to mislead, confuse, or distort the truth. COMMON CORE STANDARDS: Since AP Statistics is the high school equivalent of a one semester, introductory college statistics course, much of the course content is beyond the common core standards. For reference purposes, the most appropriate standard available for the content is listed. Deep into the course, after the process standard, a default content standard of HSS-MD.B.7 is used where no other comparable content standard exists. Much of the AP Statistics curriculum involves “analyzing decisions and strategies using probability concepts.” RESOURCES: Text: The Practice of Statistics (3rd edition), by Yates, Moore, and Starnes, W. H. Freeman & Co., 2008. (referred to below as TPS) ISBN: 0-7167-7309-0 Supplement: Prep for the AP Exam Guide for Yates, Moore, and Starnes’s: The Practice of Statictics by Michale Legacy, W. H. Freeman & Co., 2008. ISBN-10: 0-7167-7709 – 6 Student CD to accompany The Practice of Statistics. ISBN: 0-7167-7712 - 6 Various Internet Applets and resources found at sites such as:

o The Rice Virtual Statistics Lab http://onlinestatbook.com/rvls.html o The Marzano Research Lab http://www.marzanoresearch.com/resources/proficiency-scale-bank o ITCConline

http://www.ltcconline.net/greenl/java/Statistics/catStatProb/categorizingStatProblems12.html o AP Stats Monkey http://www.apstatsmonkey.com/StatsMonkey/Descriptive_Statistics.html o The Data and Story Library http://lib.stat.cmu.edu/DASL/

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COURSE OUTLINE:

Overview: What is Statistics? (6 days) Course Content Assignments Standards

Activity: Water, water everywhere Can students tell bottled water from tap water? This activity models the components of the statistical problem solving process: research question, data production, data analysis, probability model, and inference

Handout: Getting to know your textbook

MP3 HSS-IC.A.1

Water, water activity analysis and simulation. Journal: Compare and contrast the simulation methods we used.

MP3 HSS-IC.A.2

Activity: Dolphin Therapy (see attachments) Are Yawns Contagious? MP3 HSS-IC.A.2

Data production methods: Surveys, experiments, and observational studies Activity: Snapple Capples from Statistics Through Applications

1. Read preliminary chapter section on data production and do P1-P5 2. Administer survey to 4 other people and yourself during your next lunch period.

MP6 HSS-IC.B.3

Data analysis fundamentals: Key questions; Individuals, variables: quantitative vs. categorical; basic graphical displays—dotplot, bar graph Activity: Analyze data from infamous survey

P13 – P17 MP6 HSS-ID.A.1

CASE CLOSED: Group activity and discussion P19, P21, P22, P24, P27, P28 MP3 HSS-ID.A.1

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Unit 1: Producing Data Surveys, Experiments, Observational Studies, and Simulation (10 days)

Course Content Assignments Standards Sampling: good and bad methods: Voluntary response; convenience samples; Simple random sample (SRS); stratified sampling; cluster sampling, systematic sampling, multi-stage sampling

TPS 5.2, 5.6, 5.7, 5.9, 5.11, 5.24, 5.26, 5.32

MP6 HSS-IC.B.3 HSS-IC.B.6

Designing polls and surveys: Undercoverage, nonresponse, question wording, potential bias; Skill: Choosing samples with technology

TPS 5.15, 5.16, 5.18, 5.19, 5.20, 5.25, 5.27

MP6 HSS-IC.B.3

Long-term Project: Students work in teams of 3-4 to design and carry out a survey project on a topic of their selection, write a summary report, and give a 10-15 minute oral synopsis to their classmates.

MP1, MP3 HSS-IC.B.3 HSS-IC.B.6

Basics of experimental design: Subjects, factors, treatments, explanatory & response variables; completely randomized design

TPS 5.33, 5.35, 5.37, 5.39, 5.40, 5.43


Principles of experimental design: control, random assignment, replication; placebo effect; blinding and double-blinding; multi-factor experiments

1. TPS 5.45, 5.46, 5.67 2. AP Exam Free Response (surveys)


More advanced experimental designs: Block designs (RCB); why block?; blocking vs. stratifying Video: Against All Odds: blocking

1. TPS 5.47, 5.55, 5.57 2. Begin Multiple Choice practice packet


Matched pairs designs: A special form of blocking! cross-over designs Activity: Standing vs. sitting pulse rate

1. TPS 5.48, 5.49, 5.62, 5.68 2. Continue practice packet


More Matched pairs designs: Continue Practice Packet MP1, MP3 HSS-IC.B.3 HSS-IC.B.6

REVIEW OF PRODUCING DATA Finish practice packet MP1, MP3 HSS-IC.B.3 HSS-IC.B.6

TEST ON PRODUCING DATA Case study: magnets and pain MP1, 2, 3, HSS-IC.B.3 HSS-IC.B.6

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Unit 2: Analyzing Univariate Data (11 days) Course Content Assignments Standards

Activity: memorizing digits (see attachments) Hair cut prices MP3 HSS-ID.A.1 HSS-ID.A.1

Basic graphical displays: categorical variables—bar graphs and pie charts; quantitative variables—dotplots and stemplots

TPS 1.1, 1.2, 1.4, 1.5, 1.6 MP1 HSS-ID.A.1

Displaying quantitative variables: histograms; constructing and interpreting; histograms vs. bar graphs Skill: Histograms on the calculator

TPS 1.7, 1.8, 1.11, 1.12, 1.26 MP1,5 HSS-ID.A.1

Ogives and timeplots: Using ogives to determine percentiles from scores or scores from percentiles; seasonal variation, trends, cycles

1. TPS 1.13, 1.18, 1.25 2. AP exam free response (study design)

MP4 HSS-ID.A.1

Numerical measures of center and spread/variability: Mean, median, mode; Range, IQR; boxplots and the 1.5xIQR criterion for outliers Skill: Numerical summaries on the calculator

TPS 1.27, 1.29, 1.31, 1.33, 1.35, 1.36

MP5 HSS-ID.A.2 HSS-ID.A.3

Numerical measures of center and spread/variability: standard deviation; determining which summary statistics to use when; changing units of measurement

TPS 1.39, 1.40, 1.42, 1.43, 1.45, 1.46

MP5 HSS-ID.A.2 HSS-ID.A.3

Comparing distributions: Side-by-side or segmented bar graphs; back-to-back stemplots; parallel boxplots

TPS 1.47, 1.49, 1.50, 1.53 MP5 HSS-ID.A.1 HSS-ID.A.2 HSS-ID.A.3

REVIEW OF ANALYZING UNIVARIATE DATA Activity: Matching boxplots, histograms, summary statistics

1. Multiple choice practice packet 2. Begin TPS 1.52, 1.55, 1.60, 1.61, 1.64, 1.66, 1.67, 1.70

MP5 HSS-ID.A.1 HSS-ID.A.2 HSS-ID.A.3

SURVEY PROJECT WORK DAY Finish practice packet and review questions

MP1,3 HSS-ID.A.1 HSS-ID.A.2 HSS-ID.A.3

TEST ON ANALYZING UNIVARIATE DATA Case study: Nielsen ratings MP1,2,5 HSS-ID.A.1 HSS-ID.A.2 HSS-ID.A.3

Short-term project: Critical statistical analysis – each student collects data and analyzes it using the techniques learned in this unit, prepares a written analysis. Evaluation using a four-point rubric.

MP1, HSS-ID.A.2 HSS-ID.A.3

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Unit 3: Describing location in a distribution (10 days) Course Content Assignments Standards

Activity: Do people have psychic abilities? (see attachments)

AP Sample from prior tests MP5 HSS-ID.A.2 HSS-ID.A.3

Measures of relative standing: percentiles and z-scores; Chebyshev’s inequality

TPS 2.2, 2.3, 2.4, 2.7, 2.8 MP5 HSS-ID.A.3

Density curves: Normal distributions and the 68-95-99.7 rule

TPS 2.9, 2.10, 2.12, 2.23, 2.24, 2.25

MP5 HSS-ID.A.4

Computer Lab Lab summary and description Standard Normal curve and table: Nonstandard Normal curves and calcuations

1. TPS 2.29, 2.32, 2.33, 2.35 2. Work on critical statistical analysis

MP1,5 HSS-ID.A.4

Assessing normality: Normal probability plots; other graphical and numerical methods

TPS 2.36, 2.37, 2.38, 2.39, 2.50 MP1,5 HSS-ID.A.4

PRACTICE PROBLEMS WITH DENSITY CURVES

TPS 2.43, 2.44, 2.45, 2.48, 2.54, 2.58, 2.59

MP4 HSS-ID.A.4

PRACTICE PROBLEMS WITH DENSITY CURVES

AP free response samples from past tests.

MP4 HSS-ID.A.4

SURVEY PROJECT WORK DAY Prepare for Quiz QUIZ ON UNIT 3 Finish Critical statistical analysis MP1,2,5,6

HSS-ID.A.1 HSS-ID.A.2 HSS-ID.A.3 HSS-ID.A4

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Unit 4: Analyzing bivariate data (9 days)

Course Content Assignments Standards Scatterplots: constructing and interpreting, direction, shape, strength (and outliers) Skill: Making scatterplots on the calculator

TPS 3.1, 3.4, 3.5, 3.7, 3.9 MP2, MP3 HSS-ID.B.6

Correlation: calculations & properties, defining correlation; what affects correlation? Activity: Guess the correlation game (java applet)

TPS 3.13, 3.16, 3.19, 3.20, 3.23, 3.24

MP2, MP5 HSS-ID.B.6

Introduction to linear regression: interpreting the slope and y-intercept in context; prediction vs. extrapolation Skill: Finding the LSRL on the calculator Skill: Interpreting computer regression output

TPS 3.29, 3.32, 3.33, 3.36, 3.38 MP2, MP5 HSS-ID.B.6

More linear regression: the least-squares principle and properties xy ssrb ⋅= ; ( )yx, on LSRL Activity: Java applet: minimizing sum of squared error Activity: Calculator discovery of LSRL properties

TPS 3.6, 3.34, 3.35, 3.37 MP2, MP5 HSS-ID.B.6

Analyzing model quality: residuals & 2r residual plots – constructing & interpreting; 2r – calculation & interpretation

Skill: residual plots on the calculator

TPS 3.39, 3.41, 3.43, 3.47 MP6 HSS-ID.B.6a HSS-ID.B.6b

Unusual points in regression: outliers, influential points

1. TPS 3.60, 3.61, 3.62 2. Begin case study on new SAT scores

MP1, MP5 HSS-ID.C.7 HSS-ID.C.8

Cautions about correlation & regression: 1. TPS 3.46, 3.55, 3.70, 3.71 2. Begin Multiple choice practice packet

MP5, MP6 HSS-ID.C.9

REVIEW OF ANALYZING BIVARIATE DATA 1. Finish multiple choice practice packet 2. TPS 3.77, 3.80, 3.83, 3.84, 3.85

MP1-6 HSS.ID.B.5,6 HSS.ID.C.7-9

TEST ON ANALYZING BIVARIATE DATA Case study: Are baseballs juiced? MP1- 7

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Unit 5: More on Relationships between Two Variables (9 days) Course Content Assignments Standards

Transforming to achieve linearity: powers and logs Skill: Transformations and regression models on the calculator

TPS 4.2, 4.4 MP1, MP5 HSS.ID.B.6a

Exponential models: Exponential growth; log y transformation

TPS 4.5, 4.7, 4.9 MP1, MP5 HSS.ID.B.6a

Power models: log x, log y transformation TPS 4.11, 4.12 MP1, MP5 HSS.ID.B.6a

Choosing the best model with technology: Skill: PwrReg and ExpReg on the calculator

AP Exam review assignment (from ARTIST website)

MP1, MP5 HSS.ID.B.6a

Relationships between categorical variables: marginal and conditional distributions

TPS 4.23, 4.24, 4.25 MP2

Relationships between categorical variables: Simpson’s paradox

TPS 4.29, 4.31 through 4.35 MP2, MP3

Establishing causation: Lurking variables; causation, common response, and confounding

TPS 4.41, 4.45, 4.50, 4.51 MP6 HSS.ID.C.9

REVIEW OF UNIT 5 TPS 4.37, 4.53, 4.54, 4.57 MP1 HSS.ID.B.6a HSS.ID.C.9

QUIZ ON UNIT 5 Case study: insurance MP1 HSS.ID.B.6a HSS.ID.C.9

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Unit 6: Probability (12 days) Course Content Assignments Standards

Simulations: Basic process and examples—one where labels represent individuals; one where labels represent outcomes of chance phenomenon

TPS 6.1, 6.3, 6.13 MP4 HSS.IC.A.2

Basic probability concepts: Probability as long-run relative frequency; randomness; legitimate probability models; sample spaces, outcomes, events Activity: Spin 123

TPS 6.23, 6.24, 6.27, 6.28, 6.29, 6.33, 6.36

MP3 HSS.CP.A.1

Basic probability rules: Addition rule for disjoint events; complement rule; Venn diagrams – union and intersection; equally likely outcomes

1. TPS 6.37, 6.39, 6.43, 6.44 2. Begin AP Exam review assignment (from ARTIST website)

MP7 HSS.CP.A.1 HSS.CP.B.7

Independence & the multiplication rule; general addition rule Definition of independent; multiplication rule for independent events

TPS 6.45, 6.47, 6.49, 6.61, 6.66, 6.67

MP1, MP3 HSS.CP.A.2 HSS.CP.B.8

QUIZ ON PROBABILITY MP1 HSS.CP.A.2

Conditional probability: General multiplication rule & tree diagrams

TPS 6.70, 6.72, 6.73, 6.78, 6.86(a)-(d)

MP6 HSS.CP.A.3

MORE Conditional probability: use and abuse of tree diagrams

Sample AP questions MP6 HSS.CP.A.4

Independence & Bayes’ theorem: Proving independence; disjoint vs. independent

TPS 6.71, 6.81, 6.82, 6.87, 6.90, 6.91

MP6 HSS.CP.A.4

Activity: No dice! Begin practice packet MP3, MP4 HSS.CP.A.4 HSS.CP.A.5

PRACTICE PROBLEMS WITH PROBABILITY Continue practice packet MP1 REVIEW OF PROBABILITY Finish practice packet MP1, MP3 TEST ON PROBABILITY MP 1-7

HSS.CP.A.* HSS.CP.B.*

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Unit 7: Random Variables (6 days) Course Content Assignments Standards

Introduction to random variables: Discrete vs. continuous; probability distributions; notation

TPS 7.2, 7.3, 7.4, 7.5, 7.7, 7.9 MP1, MP6 HSS.MD.A.1

Mean and variance of a random variable; law of large numbers

TPS 7.25, 7.30, 7.32, 7.33, 7.43 MP1, MP6 HSS.MD.A.2

Rules for means & variances: linear transformations; linear combinations of random variables; independence

TPS 7.38, 7.39, 7.41, 7.47, 7.51 MP1, MP6 HSS.MD.A.2

Combining Normal random variables: Activity: Simulation approach

TPS 7.44, 7.45, 7.46, 7.50 MP1, MP6 HSS.MD.B.6

PRACTICE PROBLEMS WITH RANDOM VARIABLES

TPS 7.55 through 7.60 MP1, MP2 HSS.MD.A.4

QUIZ ON RANDOM VARIABLES AP Exam review assignment (old free response questions)

MP1, MP2 HSS.MD.A.1 HSS.MD.A.2 HSS.MD.A.3 HSS.MD.A.4

Unit 8: Binomial & Geometric Random Variables (6 days)

Course Content Assignments Standards Binomial settings & the binomial random variable :BINS; X = # of successes; introduction to calculating binomial probability

TPS 8.1, 8.3, 8.4, 8.5, 8.8, 8.11, 8.12

MP1, MP6 HSS.MD.A.1

Binomial distributions: mean and variance Using the calculator; Binomial pdf vs. binomial cdf Skill: Binomial distributions on the calculator

TPS 8.13, 8.14, 8.16, 8.23 MP1, MP6 HSS.MD.A.2

Normal approximation to the binomial distribution; binomial simulations Estimating binomial probabilities with Normal calculations

TPS 8.19, 8.24, 8.27, 8.29, 8.30 MP1, MP6 HSS.MD.A.2

Geometric distributions: BITS; Y = # of trials up to and including 1st success; calculating geometric probabilities Activity: Mr. Nerdly & the Birth Day Game

TPS 8.36, 8.41, 8.43, 8.44, MP1, MP6 HSS.MD.B.6

PRACTICE PROBLEMS WITH BINOMIAL & GEOMETRIC RV'S

TPS 8.50, 8.51, 8.52, 8.59, 8.60, 8.63, 8.65, 8.66, 8.67, 8.68

MP1, MP2 HSS.MD.A.4

QUIZ ON BINOMIAL & GEOMETRIC RV'S Case study: ESP MP1, MP2 HSS.MD.A.1 HSS.MD.A.2 HSS.MD.A.3 HSS.MD.A.4

EXAM REVIEW: 3 DAYS SEMESTER 1 EXAM: Simulated AP format Including Multiple Choice, Short Answer, and Free Response Questions.

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Unit 9: Sampling distributions (7 days)

Course Content Assignments Standards What is a sampling distribution? Moving towards inference; bias and variability

TPS 9.1, 9.2, 9.3(a)(b), 9.5(a)(b), 9.6

MP1, MP4 HSS-IC.A.2

Sampling distributions of p̂ Mean and standard deviation of sampling distribution; normal approximation and rules of thumb Activity: Reese's Pieces Java Applet

1. TPS 9.8, 9.10, 9.19 2. Begin AP Exam review assignment

MP1, MP4 HSS-IC.B.2

Sampling distributions of proportions: calculations and conditions

1. TPS 9.25, 9.27, 9.30 2. Continue AP Exam review assignment


Sampling distributions of x Mean and standard deviation of sampling distribution; Central Limit Theorem (CLT) Activity: Rice University Java applet

TPS 9.24, 9.31, 9.33 MP5, MP6 HSS-IC.B.5 HSS-IC.B.6

Calculations involving x : Normal population distribution vs. CLT

TPS 9.35, 9.37, 9.38, 9.47 MP5, MP6 HSS-IC.B.5 HSS-IC.B.6

PRACTICE PROBLEMS WITH SAMPLING DISTRIBUTIONS

TPS 9.49, 9.50, 9.51, 9.58 MP1 HSS-IC.B.5 HSS-IC.B-6

QUIZ ON SAMPLING DISTRIBUTIONS Finish AP Exam review assignment MP1-7

Unit 10: Estimating an unknown parameter (9 days) Course Content Assignments Standards

Idea of a confidence interval; connect with sampling distributions Activity: Confidence interval capture simulation on calculator and computer

TPS 10.1, 10.2, 10.5, 10.6 MP3 HSS-MD.B.7

Confidence interval for µ when σ known: Inference toolbox introduced

TPS 10.7, 10.9, 10.11, 10.12 MP3 HSS-MD.B.7

Confidence interval considerations: Changing confidence level; interpreting CI vs. interpreting confidence level; determining sample size

TPS 10.15 through 10.18 MP3 HSS-MD.B.7

Confidence interval for µ when σ is unknown: t-distributions and the one sample t interval

TPS 10.13, 10.27, 10.28, 10.31 MP3 HSS-MD.B.7

Paired t procedures & Robustness of t procedures: Skill: Performing t procedures on the calculator

TPS 10.35, 10.36, 10.42 MP3 HSS-MD.B.7

Estimating an unknown population proportion: CI's for p with the inference toolbox

TPS 10.45, 10.46, 10.47, 10.49 MP3 HSS-MD.B.7

Determining sample size: for proportion intervals 1. TPS 10.52, 10.54, 10.55 2. Begin practice packet

MP3 HSS-MD.B.7

PRACTICE PROBLEMS WITH CI’s for a single population parameter

1. TPS 10.66, 10.68, 10.72, 10.73 2. Finish practice packet

MP3 HSS-MD.B.7

TEST ON ONE-SAMPLE CI’S Case study: Need help? Give us a call!

MP1 - 7 HSS-MD.B.7

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Unit 11: Testing a Claim (8 days)

Course Content Assignments Standards Introduction to significance testing; Stating hypotheses Activity: Pick a card

TPS 11.1, 11.3(a), 11.5, 11.6 MP3 HSS-MD.B.7

Components of a significance test: Conditions, calculations, interpretation; one-sided vs. two-sided tests; statistical significance and P-value

TPS 11.7, 11.8, 11.11, 11.12, 11.13, 11.14

MP3 HSS-MD.B.7

Inference Toolbox & Tests from CI’s: duality TPS 11.27, 11.29, 11.31 to 11.33 MP3 HSS-MD.B.7

Uses and abuses of tests: Statistical significance vs. practical importance;

TPS 11.43 to 11.48 MP3 HSS-MD.B.7

Type I & II errors, Power: Type I and II error in context; connection between power and Type II error Activity: Calculator program that connects these three concepts

TPS 11.49, 11.51, 11.53, 11.55, 11.56, 11.57

MP3 HSS-MD.B.7

REVIEW OF SIGNIFICANCE TESTS 11.36, 11.65, 11.66, 11.71, 11.72, 11.73

MP1 - 7 HSS-MD.B.7

AP EXAM PRACTICE DAY Complete practice packet MP1 - 7 HSS-MD.B.7

TEST ON SIGNIFICANCE TESTS Case study: I’m getting a headache! MP 1 - 7 HSS-MD.B.7

Unit 12: Significance Tests in Practice (6 days)

Course Content Assignments Standards Testing a claim about µ : the one-sample t test TPS 12.1, 12.3, 12.6, 12.20 MP3

HSS-MD.B.7 Paired t tests Skill: t tests on the calculator and computer

TPS 12.9, 12.10, 12.12, 12.16 MP3 HSS-MD.B.7

Testing a claim about p: Significance tests with the inference toolbox Skill: Proportion inference on the calculator

TPS 12.23, 12.24, 12.25, 12.30 MP3 HSS-MD.B.7

What if the conditions aren’t met? A brief look at some nonparametric testing options

TPS 12.31, 12.33, 12.34, 12.37, 12.38

MP3 HSS-MD.B.7

REVIEW OF ONE-SAMPLE TESTS Practice packet MP3 HSS-MD.B.7

QUIZ ON ONE-SAMPLE TESTS Case study: Do you have a fever? MP3 HSS-MD.B.7

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Unit 13: Comparing Two Population Parameters (8 days) Course Content Assignments Standards

Comparing two population parameters: paired data vs. independent samples; estimating 1 2µ µ−

TPS 13.1 to 13.4, 13.11 MP3 HSS-MD.B.7

Two-sample t tests: and assorted df possibilities TPS 13.5, 13.7, 13.8, 13.9 MP3 HSS-MD.B.7

LAB: two-sample t TPS 13.13, 13.14, 13.15, 13.17 MP3 HSS-MD.B.7

Estimating 1 2p p− : the two-proportion z interval TPS 13.25, 13.27, 13.23 MP3 HSS-MD.B.7

Significance test for comparing two population proportions

TPS 13.29, 13.32, 13.33, 13.39 MP3 HSS-MD.B.7

AP EXAM REVIEW DAY Begin TPS 13.40, 13.41, 13.44, 13.45, 13.46, 13.47

MP3 HSS-MD.B.7

REVIEW OF TWO-SAMPLE INFERENCE Finish practice problems MP1 - 7 HSS-MD.B.7

QUIZ ON TWO-SAMPLE INFERENCE Case study: Fast Food Frenzy! MP1 - 7 HSS-MD.B.7

Unit 14: Inference about Distributions of Population Proportions (6 days)

Course Content Assignments Standards Chi-square goodness of fit test : The chi-square family of distributions Activity: M&M color distributions

TPS 14.1, 14.5, 14.8 MP3 HSS-MD.B.7

Chi-square test of homogeneity: Independent SRS's or randomized experiments

TPS 14.11, 14.15, 14.16, 14.18 MP3 HSS-MD.B.7

Chi-square test of association/independence: Distinguishing between homogeneity and association/independence questions Skill: Chi-square tests on the calculator

TPS 14.22, 14.24, 14.25, 14.29 MP3 HSS-MD.B.7

PRACTICE PROBLEMS WITH CHI-SQUARE TPS 14.35, 14.36, 14.39, 14.41 MP3 HSS-MD.B.7

AP EXAM REVIEW DAY Practice for Quiz on Chi-square MP1 - 7 HSS-MD.B.7

QUIZ ON CHI-SQUARE Case study: Does acupuncture promote pregnancy?

MP1 - 7 HSS-MD.B.7

Unit 15: Inference about Linear Regression (2 days) Course Content Assignments Standards

The linear regression model: Population vs. sample regression lines; CI for slope Activity: Investigating Old Faithful eruption data

TPS 15.2, 15.3, 15.9 MP3 HSS-ID.C.7

Significance tests about β : Nasty formulas; computer output; abbreviated inference toolbox Skill: Regression inference on the calculator

TPS 15.8, 15.15, 15.16 MP3 HSS-ID.C.7

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AP EXAM REVIEW (6 days)

• TPS Part Review Exercises • Practice AP Free Response Questions • Mock Grading Sessions • Rubric development by student teams • Practice Multiple Choice Questions

AP STATISTICS EXAM (1 DAY)

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AFTER THE AP EXAM: Students complete a final project, alone or in teams, on a topic of their choosing. The instructor has final say on team composition and topic approval Both a written analysis and a brief oral presentation are required for this project. Evaluation (Grading): Your grade in this course will be determined by your performance on tests, quizzes, homework, graded assignments, projects, and exams. Late work is penalized 10% per day.

• Tests Tests will be given about once every 3 weeks. Corrections with reflections may be made on any test for up to half-credit. I will provide more information following our 1st test.

• Quizzes There will be occasional announced or unannounced quizzes on course content. Students may retake up to two quizzes per marking period. Retakes require one session of extra-help and a second session to complete the retake. The higher of the two quizzes will count.

• Homework Homework assignments will be inspected and/or collected regularly. For each assignment, a þwill be awarded for a satisfactory effort to complete all assigned questions according to directions provided in class within a one hour time limit. A þ+ may be awarded for exceptional work, and a þ- may be awarded for incomplete work or for failure to follow prescribed format. A þ+ earns an extra 2 points above the normal 4 point value of each homework assignment, a þ results in 4 points, while a þ- earns only 2 points. Failure to submit an a reasonable portion of the assignment earns zero points. You will receive one HOMEWORK PASS per quarter that you may submit in lieu of an assignment. You may also "redeem" an unused pass at the end of a quarter for a 4 point increase in your homework point total.

“Flashback” problems, which will often be graded, should be written up separately. • Graded assignments Computer assignments, labs, CSA’s, and cumulative reviews will be scored

on their statistical accuracy, organization, appearance, and communication quality. Point values will vary based on the complexity of the assignment.

• Projects I will distribute a grading rubric with each project. Remember that each member of your group will earn the same grade, and that I expect you to do an equal amount of work.

• Exams There will be a mid-term exam during the scheduled exam week. In addition, there will be a final practice AP exam that counts for 2 test grades. Students with low grades and/or effort marks may be required to take a final exam second semester.

Obtaining your grade: Here is how you can determine your course average: Your quarter grade will be

determined by computing 100possible Points

earned Points× . Your semester grade will be determined by computing 80%

of your pre-exam average plus 20% of your exam score (on a 100 point scale). On the pages that follow, you will find descriptions of a typical case study and a CSA, as well as details of our survey project (1st semester) and our final project (2nd semester).

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Chapter 1 Case Study Nielsen Ratings What does it mean to say that a TV show was ranked #1? The Nielsen Media Research company randomly samples about 5100 households and 13,000 individuals each week. The TV viewing habits of this sample are captured by metering equipment, and data is sent automatically in the middle of the night to Nielsen. Broadcasters and companies that want to air commercials on TV use the data on who is watching TV and what they are watching. The results of this data gathering appear as Ratings on a weekly basis. For more information on the Nielsen TV Ratings, go to www.nielsenmedia.com, and click on “About Us.” Then under “Related,” select “What Are TV Ratings?” Here are the top primetime shows for viewers ages 18 to 49 during the week of November 22-28, 2004. SHOW NETWORK Viewers (millions) 1. Desperate Housewives ABC 16.2 2. CSI CBS 10.9 3. CSI: Miami CBS 10.5 4. Extreme Makeover: Home Edition ABC 9.7 5. Two and a Half Men CBS 8.8 6. Without a Trace CBS 8.2 7. Raymond CBS 8.0 8. Law & Order: SVU NBC 7.8 Monday Night Football ABC 7.8 Survivor: Vanuatu CBS 7.8 11. Seinfeld Story NBC 7.6 12. Boston Legal ABC 7.4 13. Apprentice NBC 7.1 14. Fear Factor NBC 6.5 15. Amazing Race CBS 6.1 CSI: NY CBS 6.1 17. NFL Monday Showcase ABC 5.7 18. According to Jim ABC 5.5 19. 60 Minutes CBS 5.4 Biggest Loser NBC 5.4 Source: USA Today, December 2, 2004 Which network is winning the ratings battle? Give appropriate statistical evidence to support your answer.

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AP STATISTICS Critical Statistical Analysis (CSA) #1: Exploring Quantitative Data

DUE: Locate a set of quantitative data (at least 25 data values) in a newspaper, magazine, periodical, recent book, or on the Internet. YOU must obtain this data without consultation from other students. Be certain to make a copy of the source data and to record the bibliographical information. They are part of the CSA scoring rubric. Once you have obtained your data, use appropriate graphical (dotplots, stem-and-leaf plots, boxplots, histograms) and numerical (mean, median, mode, IQR, range, standard deviation, 5 number summary) descriptive techniques to present the data. Then, write a narrative analysis of the data in context based on your graphical and numerical summaries. Comment on each of your graphical representations. Discuss which display(s) and statistics are most helpful in exposing the key features of the data set. Use appropriate terminology, and write in complete, grammatically correct sentences. DO NOT discuss your analysis with any other person – it is to be your own work. Aim for one page of narrative and one to two pages of figures. Your CSA will be evaluated on: accuracy of graphical representations and numerical summaries, quality of your written analysis, neatness, and organization. The marking guide follows. Accuracy of Graphical/Numerical Summary Techniques 4 4 The student has used all required statistical techniques correctly and appropriately. All

minor points are included. 3 3 The student has generally used each of the required statistical techniques correctly and

appropriately. There may be minor omissions or errors. 2 2 The student has used some of the required statistical techniques correctly and

appropriately. There are, however, significant errors in one or more techniques or a plethora of minor mistakes.

1 1 The student has made some attempt to use required techniques correctly and

appropriately, but the effort is flawed in some major way. 0 0 The student has used inappropriate techniques for the given set of data. Quality of Statistical Analysis 4 The student thoroughly and accurately discusses the implications of the statistical

techniques employed in the context of the data. Correct terminology is used throughout. 3 The student accurately describes the implications of the statistical techniques employed in the

context of the data and generally uses correct terminology. There are minor omissions/errors. 2 The student produces a generally accurate interpretation of the statistical techniques employed

with some use of appropriate terminology or with inadequate connection to context. A key omission or inaccurate conclusion may also have been made.

1 The student attempted to interpret the statistical techniques that were employed, but failed to

expose some key ideas. Terminology and reference to context are inadequate or missing. 0 The student interprets the statistical techniques employed incorrectly or not at all.

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Organization, Transition, Appearance 4 The CSA shows evidence of careful organization, flows naturally from statistical technique to

statistical analysis, and is neat in appearance. 3 There are minor flaws in one of the areas: organization, transition, appearance. 2 There are major flaws in one of the areas: organization, transition, appearance, OR minor flaws

in two areas. 1 Completely inadequate in two of: organization, transition, appearance. 0 Inadequate in all three areas. English Mechanics 4 The student’s writing is grammatically correct, is punctuated properly, and flows logically from

one point to the next. No spelling mistakes! 3 The student’s writing has a minor flaw in one of the areas: grammatically correct, punctuated

properly, logical flow, spelling. 2 The student has made significant errors in one of the areas: grammatically correct, punctuated

properly, logical flow, spelling, OR minor flaws in two areas.. 1 The student’s writing has major flaws in two of the areas: grammatically correct, punctuated

properly, logical flow, spelling. 0 The student’s writing has major flaws in more than two of the areas: grammatically correct,

punctuated properly, logical flow, spelling. Source Documentation 4 The student has selected a source that is appropriate for the assignment and has correctly referenced

the source. 3 The student has selected a source that is appropriate for the assignment, but has made some error or

omission in the citation. 2 The chosen source is slightly inappropriate for the assignment OR referencing is incomplete or

inaccurate. 1 The chosen source is inappropriate. 0 No source documentation is provided. Scoring Guide: Your grade on this CSA will be determined based on your total points, as follows:

21-24 A 16-20 B 11-15 C

6-10 D < 6 F

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AP Statistics Survey Project Phase I: Team members brainstorm possible survey topics on issues of school interest Phase II: Each team submits a typed proposal describing:

• Topic/question of interest • Background motivation for selecting this topic/question • Questions to be included in the survey • Methodology

Ø The type of sampling procedure do you intend to use – stratified, cluster, SRS, or systematic

Ø Precise description of your randomization, including labeling Ø When, where, and how you will administer the survey

Phase III: Select your sample and administer your survey Phase IV: Organize, summarize, and analyze your data Phase V: Prepare a written report that documents your survey. Follow these guidelines. Your written report should include each of the sections described below. The finished product will be evaluated according to the rubric on the attached page, so read it carefully.

Ø Topic/Question – should be descriptive, and eye-catching

Ø Background – Why did you decide to investigate this topic/question?

Ø Methodology – This should be clear enough so that anyone who reads your description could replicate the survey effortlessly.

ü Describe and defend your chosen sampling procedure. ü Detail your randomization process. ü Carefully explain when, where, and how you administered the survey. ü Provide a copy of your survey.

Ø Data – Organize your data in tabular form.

Ø Analysis – Include appropriate graphical and numerical summaries – bar graphs, pie charts, counts,

proportions, percents.

Ø Interpretation – Discuss what the data tells you about the topic/question you chose. What generalizations might you draw about the population from which the sample was drawn?

Ø Pitfalls and extensions – Share any difficulties you experienced during the survey project. What

might you do differently if you were to repeat the survey? Are there any possible extensions of this survey project that might prove interesting?

Phase VI: Class Presentation – a ten minute opportunity for you to share the critical aspects of your survey project with your classmates. Make it interesting!! See the attached grading guide.

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AP Statistics Survey Project Scoring Rubric Topic/question and Background 4 The topic/question selected is clearly stated, is of interest to the school community, and is

appropriately narrow in scope. The background provided gives strong motivation for the team’s choice of this topic/question and delineates its relevance to the school community.

3 The topic/question selected is clearly stated, is of interest to the school community, and is appropriately narrow in scope. The background provided gives considerable support for the team’s choice of this topic/question, and some attempt is made to show its relevance to the school community.

2 Either the topic/question is flawed in one of the areas: clearly stated, of interest to the school community, appropriately narrow in scope OR the background provided fails in either its support for the chosen topic/question or the relevance to the school community.

1 Both the topic/question and the background provided are flawed in at least one area. However, one or both satisfactorily address at least half of the areas specified.

0 Neither the topic/question nor the background provided satisfactorily address at least half of the specified areas.

Methodology – Sampling Procedure 4 The chosen sample procedure is appropriate for addressing the selected topic/question, is described

accurately, and is implemented according to the stated plan. 3 The chosen sample procedure satisfies two of the three criteria mentioned above, but is weak in the

other area. 2 The chosen sample procedure satisfies two of the three criteria completely, and does not satisfy the

third requirement OR the chosen sample procedure satisfies one of the three criteria completely and the other two partially.

1 The chosen sample procedure satisfies one of the three criteria completely, and one of the other two criteria partially.

0 The chosen sample procedure does not satisfy any of the three criteria completely. Methodology – Randomization 4 The randomization process includes a clear and correct labeling of subjects, a description of the

number selection process (random number table or calculator), and the results of that randomization (i.e. the numbers and subjects chosen). In addition, the randomization process matches the chosen sampling procedure.

3 The randomization process includes all three components listed above. However, clarity of communication would prevent easy replication of this randomization. Still, the randomization process matches the chosen sampling procedure.

2 The randomization process includes all three components listed above. However, the randomization process does not match the chosen sampling procedure OR the randomization process matches the chosen sampling procedure, but the clarity of communication would prevent easy replication of this randomization, in spite of it being correctly designed and implemented.

1 There is some flaw in the randomization procedure itself. Some aspect of the randomization – labeling, number selection, or results is completely correct.

0 The randomization is flawed in all three areas – labeling, number selection, and results.

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Methodology – Administration of Survey 4 The survey is administered according to the stated plan. All those selected in the randomization

process actually complete the survey successfully. No evidence of bias is present. 3 The survey is administered almost entirely according to the stated plan. Nearly all of those selected

in the randomization process complete the survey successfully. No evidence of bias. 2 The survey is administered almost entirely according to the stated plan, and nearly all of those

selected actually complete the survey. Some evidence of bias is present. OR The survey administration deviates from the stated plan in some way that does not introduce bias, but that might impact who completes the survey.

1 Bias has impacted the survey administration to a great extent, but the stated plan was generally followed OR the administration procedure deviated markedly from the stated plan, with some bias.

0 The administration process deviates markedly from the stated plan and bias is noticeable. Methodology – The Survey 4 Survey questions have all been pre-tested and refined. They are clear and unbiased. 3 Survey questions have all been pre-tested and refined. They are unbiased, but somewhat unclear. 2 Survey questions have been pre-tested, but not completely refined or show some bias. 1 Survey questions have not been pre-tested, but are somewhat clear and relatively unbiased. 0 Survey questions are unclear and show distinct bias. Data Recording and Summarization 4 Original data provided and summarized in appropriate tabular form. Neat and accurate. 3 Original data provided and summarized in tabular form, but with a minor error in tabulation or

sloppy presentation. 2 Either original data is omitted, but the data is summarized neatly and accurately in tabular form OR

the original data is presented, and there is a major flaw in the presentation of the data (but not in the accuracy of the tally).

1 Original data is provided, but is not appropriately tabulated OR the original data is omitted but the data is summarized partially correctly in tabular form.

0 Original data is not provided and the data is not tabulated appropriately. Interpretation 4 The student thoroughly and accurately interprets the meaning of the graphical and numerical

summaries in the context of the data. In addition, the student identifies any generalizations that may be drawn about the population from which the sample was drawn.

3 The student interprets the meaning of the graphical and numerical summaries in the context of the data correctly, except for minor errors or omissions. In addition, the student identifies any generalizations that may be drawn about the population from which the sample was drawn.

2 The student accurately interprets the meaning of either the graphical or numerical summaries in the context of the data, but makes serious errors/omissions in interpreting the other. In addition, the student identifies any generalizations that may be drawn about the target population.

1 The student makes a genuine attempt to interpret both the numerical and graphical summaries, but fails to completely or correctly address either one. The student might also omit generalizations to the target population.

0 The student’s interpretation of both the numerical and graphical summaries is inadequate.

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Graphical and Numerical Summaries 4 The student has correctly summarized the data using bar graphs/pie charts and

counts/proportions/percentages. Graphs and calculations are neat and accurate. 3 The student has correctly summarized the data using bar graphs/pie charts and

counts/proportions/percentages, but has made a minor error in either computation or graphing. 2 The student has correctly summarized the data either graphically or numerically, but has made a

major error in the other component. 1 The student has used appropriate techniques to summarize the data either graphically or numerically,

but has not executed the techniques correctly. 0 Neither the graphical nor the numerical summary is appropriate. Pitfalls and Extensions 4 The student articulates all pitfalls encountered, and clearly explains how (s)he dealt with each of

these obstacles. In addition, the student shares at least one plausible extension of the survey project. 3 The student articulates all pitfalls encountered, and explains how (s)he dealt with each obstacle,

though not in a clear manner. In addition, the student shares at least one plausible extension of the survey project.

2 The student articulates some of the pitfalls encountered, and explains how (s)he dealt with some of these obstacles. The student also shares at least one plausible extension of the survey project.

1 The student articulates some pitfalls encountered, but does not explain how (s)he dealt with these obstacles or does not share at least one plausible extension of the survey project.

0 The student does not articulate the obstacles (s)he encountered. English Mechanics 4 The student’s writing is grammatically correct, is punctuated properly, and flows logically from one

point to the next. No spelling mistakes!! 3 The student’s writing is grammatically correct, is punctuated properly, and flows logically from one

point to the next, except for minor errors in one of these categories. No spelling mistakes!! 2 The student has made significant errors in one of the areas: grammar, punctuation, spelling, flow OR

minor errors in two areas. 1 The student’s writing is flawed in two or three of the areas: grammar, punctuation, spelling, flow. 0 The student’s writing is deficient in all four areas: grammar, punctuation, spelling, flow. Scoring Guide: 35 – 40 A 26 – 34 B 17 – 25 C 10 – 16 D < 10 F

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AP Statistics Final Project Now that you have demonstrated your understanding of statistics content on the AP exam, you get a chance to apply what you have learned in designing, carrying out, and presenting the results of a statistical study on a topic of your choosing. Choose a research question that you can answer using statistical methods that we have studied this year. Your project must involve data production, data analysis, probability models, and inference. This project will consume most of our class time and homework time from now until the end of the year. A detailed timeline and grading rubric follow. Project Timeline First Monday Post AP Exam Project overview; topic selection begun; groups/individuals First Tuesday Post AP Exam Topic and study design proposal due First Wednesday Post AP Exam Project work First Thursday Post AP Exam Project work First Friday Post AP Exam AP Exam discussed; course evaluation Second Monday Post AP Exam Interim report due Second Tuesday Post AP Exam Project work; data collection completed Second Wednesday Post AP Exam Project work; data analysis Second Thursday Post AP Exam Project work; data analysis Second Friday Post AP Exam Project work; data analysis Third Tuesday Post AP Exam Rough draft of written report due Third Wednesday Post AP Exam Prepare for presentations Third Thursday Post AP Exam Written Reports due; Oral Presentations Third Friday Post AP Exam Oral Presentations Grading Rubric This project is worth 250 points, to be allocated as follows: Topic/study design proposal 25 due: Tuesday, May __ One-page, typed, double-spaced proposal detailing research question(s), rationale, proposed study design (very thorough), and anticipated method of data analysis Research question(s) – clear and achievable 5 Study design – thorough; proper use of terminology 10 Method of analysis – correct for the design 5 Communication quality – grammar, flow, organization, spelling 5 Interim report 25 due: Monday, May __ One to two-page, typewritten, double-spaced summary of progress made thus far, with specific reference to the individual contributions of all group members. Clear evidence of progress toward answering initial question(s) 10 Individual contributions significant and balanced 10 Communication quality – grammar, flow, organization, spelling 5 Performance assessment 50 Daily observations of work; individual conferencing with project groups – 5 points per day

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Rough draft 25 due Tuesday, May ___ Three to five-page, typewritten, double-spaced skeleton of the written report. Use the same major section headings as for the written report. Written Report 75 due Thursday, May ___ Final, typewritten, double-spaced report.

Research question(s) and rationale 5 Study design, including any modifications made 10 Raw data summarized 5 Exploratory data analysis – graphical, numerical, verbal 15 Inference 15 Obstacles encountered and how addressed 5 Conclusion, including possible extensions 5 Communication quality – neatness, grammar, flow, organization, spelling 15 Oral presentation 50 Thurs, May ___ or Fri, May ___ 10 to 15-minute class presentation of major aspects of your study. Meets time guidelines 5 Content accuracy – technical vocabulary used correctly 20 Presentation dynamics: Organization, flow, transitions, Voice quality 10 Visual aids 10

(including something for posting in the classroom; required!) Equitable participation of all group members 5 SAMPLE DOCUMENTS FOLLOW:

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In Class Support Program For Classified Students

Secondary Level

In order to have the teaching/learning process take place in the least restrictive environment, classified students may be placed in this course in accordance with their IEPs. Differentiated instruction, provided by the teacher for the course, and based on the course curriculum, addresses the needs of both general and speical education students. In addition, classified students may be assisted by either a special education teacher or a teacher assistant, again in accordance with their IEPs. Together the general education teacher and the special education teacher or teacher assistant insure that accomodations and modifications prescribed in each student’s IEP are implemented in the general education program. Accomodations for Classified Students (IEP Dependent) Accomodations are made to more fully engage the Special Education student. Accomodations provide the basis for differentiated instruction/assessment which, in turn, allows the teaching/learning process to take place in the least restrictive environment. Each individual student’s IEP includes specific accomodations which must be provided to meet the child’s special needs. In some cases, accomodations needed by individual classified students can be used on a whole class basis to support all students in the class. Instructional Accomodations:

• Graphic organizers such as the following: cause/effect, compare/contrast, time order, simple listing, problem/solution, organize by categroy, concept webs, semantic maps, notetaking format organizers, data charts

• Study guides • Think sheets • Laptop computers for notetaking, homework, research projects, and class

assignments • Power Point presentations as visual aids (instruction) or alternative way to

demonstrate learning (assessment) • Blackboard games such as “Jeopardy” for review • Interactive white boards to provide visual aid and promote active student

engagement • Preteaching of vocabulary for understanding of concepts • Wordsearch or crossword puzzles for vocabulary reinforcement • Movies, music and artwork • Videos, posters, articles, charts, graphs, illustrations, maps, and other

visual aids • Skill building activities in small groups or pairs

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• Postreading discussion points; critical thinking topics • Projects and presentations using a variety of modes of expression • Cooperative learning activities in small groups or pairs • Authentic products: greeting cards, brochures, menus, surveys, etc. • Role playing • Journal and portfolio entries • Limit notetaking to key points • Provide scaffolding for notetaking • Emphasize oral class participation

Testing Accomodations:

• Word banks • Timeline charts • Underlining key words or phrases • Modifying time allotted or length of test • Giving more frequent, short tests • Giving only one page of a multipage test at a time • Pretesting • Open book tests • Take home tests • Use of a preferred response (e.g. written, dictated, oral, illustrated) • Use of an interpreter • Administering test orally to student • Administering test in a distraction-free environment • Administering test one on one or in a small group • Clarifying test directions • Using other alternatives to written tests such as oral reports, class

discussion, projects, interviews, and portfolios

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In Class Support Program For Classified Students

Secondary Level In class support is currently provided in the following academic courses at Dover High School and Dover Middle School. The list is modified each year, as needed, based on students’ IEPs. Dover High School English English IB English IIB English IIIB English IVB Mathematics Math Foundations I, Math Foundations II Math Foundations III Pre-Algebra Geometry Algebra II Science General Science Introduction to Biology General Chemistry Environmental Science Social Studies World History & Cultures U.S. History I U.S. History II

Dover Middle School Language Arts Literature Grade 7 Literature Grade 8 Communications Grade 7 Communications Grade 8 Mathematics Mathematics Grade 7 Mathematics Grade 8 Science Science Grade 7 Science Grade 8 Social Studies Social Studies Grade 7 Social Studies Grade 8

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INSTRUCTIONAL ADAPTATIONS FOR ENGLISH LANGUAGE LEARNERS Algebra I operates within the framework of the NJCCCS for Mathematics. In addition, the WIDA (World-Class Instructional Design and Assessment) English Language Proficiency Standards are incorporated to insure that ELLs are accessing the academic language necessary for classroom understanding or communication. The WIDA CAN DO descriptors provide a range of expectations for student performance within an ELL’s proficiency level. A variety of accommodations and instructional strategies are used to assist ELLs (sensory, graphic, interactive and native language support). Mathematics Examples of Sensory, Graphic and Interactive Supports for ELLs

Sensory Supports Graphic Supports Interactive Supports

Supports related to the language of Mathematics

• Real-life objects (realia)

• Manipulatives • Pictures &

photographs • Illustrations,

diagrams & drawings

• Magazines & newspapers

• Physical activities • Videos & Films • Broadcasts • Models & figures

• Charts • Graphic

organizers • Tables • Graphs • Timelines • Number lines

• In pairs or partners

• In triads or small groups

• In a whole group

• Using cooperative group Structures

• In the native language (L1)

• With mentors • With the

Internet (websites) or software programs

Blocks/Cubes Number lines Calculators Protractors Geoboards Counters Compasses Calendars Coins Rulers, yard/meter sticks Models of geometric figurds Clocks, sundials and other timekeepers

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Technology

The Dover Public School District recognizes the use of technology as an integral

component of every student’s education. As the district acknowledges the

importance of providing the student with experiences that model “real life” use of

technology in society and the workplace, student use of technology on a regular

basis in all disciplines and at all grade levels is encouraged and supported.

Technology skills are not taught in isolation, but rather infused into the

mathematics curriculum providing students with authentic interdisciplinary

projects that address the New Jersey Core Curriculum Content Standards and the

Common Core Standards in Mathematics, 21st Century Skills and Themes,

technology and other disciplines. Student achievement and growth are measured

using performance-‐based assessments. In addition to increasing higher-‐order

thinking skills and creativity, these technology-‐infused projects provide students

with the opportunity to work in collaborative settings. The ethical use of technology

and on-‐line safety is promoted in numerous ways and is constantly modeled by the

classroom teacher.

Students have ample access to technology via the use of laptop computer carts,

computer labs, and classroom computers and interactive boards. Every student has

an eChalk account, an individual login and a home directory that provides individual

storage space. Students work in a wireless environment with secure Internet

access.

Technology resources used include, but are not limited to:

• NETTRECKKER – Secure educational Internet search engine

• STUDY ISLAND– Online study program for standard based assessments

• UNITEDSTREAMING – Digital video-‐on-‐demand, aligned to NJCCCS and the

CCSS

• GIZMOS

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• PROMETHEAN – Interactive classroom whiteboards for teachers and

students to actively interact with the NJCCCS and the CCSS

• ACTI-‐VOTES

• ONCOURSE – Online lesson plans, provides students and parents with

continuous interface to class assignments and downloads

• GENESIS – Online student data system, provides teachers with an online

grade book

• GRAPHING AND SCIENTIFIC CALCULATORS – Provides students with the

ability to “see” mathematics graphically

o TI83PLUS

o TI89 – Calculus

o Ti-‐nspire/TI-‐84

§ TI-‐SMARTVIEW Emulator software for the TI-‐84 Plus family

o Casio

o Scientific calculators

o GEOMETER’S SKETCHPAD – Dynamic construction, demonstration and

exploration tool for the study of mathematics; provides students with

the opportunity to build and investigate mathematical models,

objects, figures, diagrams and graphs

• GEOMETER’S SKETCHPAD CURRICULUM MODULES

§ EXPLORING ALGEBRA I

§ EXPLORING ALGEBRA II

§ EXPLORING GEOMETRY

§ EXPLORING PRE CALCULUS

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KINGSTON:Math Stds ChecklistCCCHonorsREV#!G.xls9/12/13 1/28

Standard/Course Algebra IH Geometry H Algebra II H PreCalc H NJIT

Calculus

Number and QuantityDOMAIN: The Real Number System N-RN

Extend the properties of exponents to rational exponents.

1. Explain how the definition of the meaning of rational exponents follows from extending the pro;erties of integer exponents to those values, allowing for a notation for radicals in terms of rationalexponents.For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

X X2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

X XUse properties of rational and irrational numbers.3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational nuber is irrational; and that they product of a nonzero rational number and an irrational number is irrational.

XDOMAIN: Quantities N-QReason quantitatively and use units to solve problems. (Foundation work with expressions, equations and functions.)1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistenlty in formulas; choose and interpret the scale and the origin in graphs and data displays.

X2. Define appropriate quantities for the purpose of descriptive modeling. X3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

X

Mathematics Standards Assessment 2011-2012 Honors Sequence

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CalculusDOMAIN: The Complex Number System N-CNPerform arithmetic operations with complex numbers. 1. Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real. X2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

X3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. X XRepresent complex numbers and their operations on the complex plane.

4. (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. X

5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (–1 + √3 i)3 = 8 because (–1 + √3 i) has modulus 2 and argument 120°.

X6. (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

X

Use complex numbers in polynomial identities and equations. (polynomials with real coefficients.)

7. Solve quadratic equations with real coefficients that have complex solutions.

X8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).

X X

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Calculus

9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

X DOMAIN: Vector Quantities and Matrices N-VM1. (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes e.g., v, |v|, ||v||, v).

X X2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

X X3. (+) Solve problems involving velocity and other quantities that can be represented by vectors.

X X

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CalculusPerform operations on vectors. 4. (+) Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

X X5. (+) Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v x, vy) = (cvx, cvy). b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). X XDOVER

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CalculusPerform operations on matrices and use matrices in applications. 6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. X X7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. X X8. (+) Add, subtract, and multiply matrices of appropriate dimensions. X X9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

X X 10. (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. X X11. (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

X X12. (+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. X X

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Calculus

ALGEBRASeeing Structure in Expressions A-SSE 1. Interpret expressions that represent a quantity in terms of its context.★ a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

X linear,

exponential, quadratic

X(a) polynomial and rational

expresssions2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

X XWrite expressions in equivalent forms to solve problems 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★ a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

X quadratic and exponential X(b&c)

4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.★ X

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CalculusArithmetic with Polynomials and Rational Expressions A - APR

Perform arithmetic operations on polynomials 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

X Linear and quadratic

X beyond

quadraticUnderstand the relationship between zeros and factors of polynomials.2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

X3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. XUse polynomial identities to solve problems 4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

X 5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.1

X XRewrite rational expressions 6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. X

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Calculus7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

X

X linear and quadratic

demoninators

Creating equations A-CEDCreate equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. X X

2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

X linear,

quadratic, exponential

(integer inputs only) X X

3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. X

linear only

X Equations using all available types of

expressions, including

simple root functions

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

X XReasoning with Equations and Inequalities A-REIUnderstand solving equations as a process of reasoning and explain the reasoning.

1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. X

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Calculus2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. X

X simple radical and rational

Solve equations and inequalities in one variable 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. X4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. X XSolve systems of equations

5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. X6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. X7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.

X Linear-linear,

linear-quadratic X

8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. X X9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

X X

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CalculusRepresent and solve equations and inequalities graphically 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

X11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

X

X Combine

polynomial, rational, radical,

absolute value and

exponential functions

12. Graph the solutions to a linear inequality in two variables as a half- plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. X

FUNCTIONS

Interpreting functions F-IFUnderstand the concept of a function and use function notation1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

X2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

X

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Calculus3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

X XInterpret functions that arise in applications in terms of the context 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

X Linear,

quadratic, and exponential

X Empahsize selection of appropriate

models

5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★ X X

6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

X X

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Calculus7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. X(a&b)

X Focus on using key features to guide selection of appropriate type of model

functions.

9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

X XBuilding functions F-BF

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CalculusBuild a function that models a relationship between two quantities 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

X(a&b) lLnear,

exponential and quadratic

X(b) Include all types of functions studied X(c)

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★

X XBuild new functions from existing functions

3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

X

X Include simple

radical, rational, and exponential functions emplying

common effect of each

transformation across function

types

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Calculus4. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain.

X(a) Linear only X(a)

X (b,c,d)

5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

X X

Linear, Quadratic, and exponential models F-LEConstruct and compare linear, quadratic, and exponential models and solve problems

1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. X

Functions and Relationships1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs. X

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Calculus2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). X3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. X4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

X Logarithms as solutions for exponentials

Interpret expressions for functions in terms of the situation they model

5. Interpret the parameters in a linear or exponential function in terms of a context.

X Linear ind

exponential of form

f(x) = bx + k X

Trigonometric Functions F-TFExtend the domain of trigonometric functions using the unit circle1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. X X2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. X X3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number. X4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

X

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CalculusModel periodic phenomena with trigonometric functions 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★ X X6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

X7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.★

XProve and apply trigonometric identities 8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

X X9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. X

GEOMETRYCongruence G-CoExperiment with transformations in the plane

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. X

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CalculusUnderstand congruence in terms of rigid motions

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. X7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. X8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. XProve geometric theorems 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

X10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

X11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

X Make geometric constructions

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Calculus

12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

X 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. XSimilarity, right triangles, and trigonometry G-SRTUnderstand similarity in terms of similarity transformations 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

X2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles X3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. X

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CalculusProve theorems involving similarity 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

X5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. XDefine trigonometric ratios and solve problems involving right triangles

6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. X 7. Explain and use the relationship between the sine and cosine of complementary angles. X8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

XApply trigonometry to general triangles 9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. X X10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. X X11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). X XCircles G-C

Understand and apply theorems about circles

1. Prove that all circles are similar. X2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and X

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Calculus 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. X4. (+) Construct a tangent line from a point outside a given circle to the circle. XFind arc lengths and areas of sectors of circles

5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. XExpressing Geometric Properties with Equations G-GPETranslate between the geometric description and the equation for a conic section

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

X X2. Derive the equation of a parabola given a focus and directrix. X X3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

XUse coordinates to prove simple geometric theorems algebraically 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). X

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Calculus5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

X6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. X7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. XGeometric measurement and dimension G-GMDExplain volume formulas and use them to solve problems 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. X2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. X X3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

XVisualize relationships between two-dimensional and three- dimensional objects

4. Identify the shapes of two-dimensional cross-sections of three- dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. XModeling with Geometry G-MG

Apply geometric concepts in modeling situations

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso X

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Calculus

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★

X3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★ X

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CalculusPROBABILITY AND STATISTICS

Interpreting Categorical and Quantitative data S-IDSummarize, represent, and interpret data on a single count or measurement variable1. Represent data with plots on the real number line (dot plots, histograms, and box plots). X

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Calculus2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. X3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). X4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. XSummarize, represent, and interpret data on two categorical and quantitative variables 1) and using the quadratic formula5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. X X6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. XInterpret linear models 7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. X8. Compute (using technology) and interpret the correlation coefficient of a linear fit. X

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Calculus9. Distinguish between correlation and causation. Xmaking Inferences and Justifying Conclusions S-ICUnderstand and evaluate random processes underlying statistical experiments 1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

X X2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? X X Make inferences and justify conclusions from sample surveys, experiments, and observational studies

3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. X4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. X5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. X6. Evaluate reports based on data. XConditional Probability and the rules of Probability S-CPUnderstand independence and conditional probability and use them to interpret data 1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). X X

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Calculus2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. X X3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. X X4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

X X5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. X XUse the rules of probability to compute probabilities of compound events in a uniform probability model 6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

X Introduction

to apply counting

rules X

7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

X Introduction

to apply counting

rules X

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Calculus 8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

X X9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. X XUsing Probability to make decisions S-MD

Calculate expected values and use them to solve problems

1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. X X 2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. X X3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

X X4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? X XUse probability to evaluate outcomes of decisions

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Calculus5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast- food restaurant. b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. X(a) X(b) X(a&b)6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). X X7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). X X

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Honors Sequence

Mathematics Department Dover Public Schools

Algebra I Grade 8

Geometry Grade 9

Algebra II Grade 10

Pre Calculus Grade 11

NJIT Calculus/AP Statistics Grade 12

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Advanced Placement Statistics - dover-nj.orgdover-nj.org/wp-content/uploads/sites/2/2016/07/Mathematics-AP... · Advanced Placement Statistics ... analyzing data (graphical & numerical