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District Overview:The mathematics curriculum provides sequential and comprehensive K-12 instruction in a collaborative, student-centered learning environment that fosters critical thinking, creativity, skillful problem-solving, and effective communication in order to enable all students to adapt to an ever-changing, global society and increase college and career readiness. An emphasis has been placed on conceptual understanding, higher-order thinking, and problem solving skills to prepare students for 21st century careers. This is further embedded through the integrated use of technology into daily lessons. Instruction focuses on meaningful development of mathematical ideas at each grade level where students are given the opportunity to explore, engage, and take risks with content as they build and expand their knowledge and understanding of mathematics. Students will experience mathematics as a coherent and useful subject within the context of real-life situations. In all, the curriculum aims to reach high standards while encouraging curiosity and building confidence in a collaborative atmosphere.

Introductory Statistics & Probability Course Description:The aim of this course is to provide students with a basic understanding of the fundamental concepts of statistics and probability. The early portion of the course will cover six main introductory themes: the nature of probability and statistics, frequency distributions and graphs, data descriptions, probability and counting rules, discrete probability distributions, and the normal distribution. As the course progresses, students will be exposed to a few higher-level topics used in data analysis: confidence intervals, hypothesis testing, and linear regression. The students will have several opportunities to work from a project-based perspective—making many different real-life connections to statistics and probability. Completion of this course should give students enough of a fundamental background to enroll in other statistical courses on the college level.

Introductory Statistics & Probability Units: Unit 1: The Nature of Probability and Statistics Unit 2: Frequency Distributions and Graphs Unit 3: Data Description Unit 4: Probability and Counting Rules Unit 5: Discrete Probability Distributions Unit 6: The Normal Distribution Unit 7: Confidence Intervals and Sample Size Unit 8: Hypothesis Testing Unit 9: Correlation and Regression Unit 10: Sampling and Simulation

Subject: Mathematics Course: Introductory Statistics & Probability

Suggested Timeline: 3 weeks

Unit Title: The Nature of Probability and Statistics

Unit Overview/Essential Understanding: In Unit 1, students will be introduced to many important terms and ideas in the world of statistics. They will practice identifying and classifying many different types of data, as well as discuss the main methods for gathering data. The unit finishes with having the students investigate several uses and misuses of statistics in the real world.

Unit Objectives: At the end of this unit, students will have a strong understanding of and be able to independently work with:

Descriptive and inferential statistics Variables and types of data Data collection and sampling techniques Observational and experimental studies Uses and misuses of statistic

Focus Standards Addressed in this Unit: CC.2.4.HS.B.1 – Summarize, represent, and interpret data on a single count or measurement variable

Important Standards Addressed in this Unit: N/A

Misconceptions: General confusion between discrete and continuous types of data Knowing when to use (and when not to use) different sampling techniques Being aware of common misuses of statistics in advertising

Concepts/Content: Data Statistics Probability Variable

Competencies/Skills: Demonstrate knowledge of

statistical terms Differentiate between the two

Description of Activities: Direct teacher-led instruction Large group Q & A discussion Independent student work Small group collaboration

Population Sample Descriptive Inferential Qualitative Quantitative Discrete Continuous Level of measurement Random sample Stratified sample Cluster sample Systematic sample Observational study Experimental study Treatment Control group Hawthorne effect Placebo effect Confounding variable

branches of statistics Identify types of data Identify the measurement level

for each variable Identify the four basic sampling

techniques Explain the difference between an

observational and experimental study

Explain how statistics can be used and misused

Online remedial/enrichment videos

Assessments: Informal questioning In-class formative assessments Review of homework Periodic quizzes Unit test

Interdisciplinary Connections: N/A

Additional Resources: Elementary Statistics: A Step-By-Step Approach textbook Approved supplemental resources like: USA TestPrep Approved online resources like: Khan Academy



Unit Title: Frequency Distributions and Graphs

Unit Overview/Essential Understanding: In Unit 2, students will start to learn what to do with raw data after it has been collected. They will create frequency distribution tables to being to organize data in a meaningful way. Once organized, the data can then be put into several different visual displays or graphs. The students will not only learn how to create each different type of data display, but also how to choose which one would be appropriate for given situations.


Organizing data Histograms, frequency polygons, and ogives Other types of graphs

Focus Standards Addressed in this Unit: CC.2.4.HS.B.5 – Make inferences and justify conclusions based on sample surveys, experiments, and observational studies


Misconceptions: Making miscalculations with constructing frequency distributions Using the wrong scales and/or data for the x- and y-axes on graphs Choosing the wrong type of visual display to present data

Concepts/Content: Class Frequency Raw data Categorical frequency distribution Grouped frequency distribution Class boundaries

Competencies/Skills: Organize data using a frequency

distribution Represent data in frequency

distributions graphically using histograms, frequency polygons,

Description of Activities: Direct teacher-led instruction Large group Q & A discussion Independent student work Small group collaboration Online remedial/enrichment videos Project: “Collecting & Organizing Data” –

Class limits Class midpoints Class width Cumulative frequency Histogram Frequency polygon Ogive Bar graph Pie graph Pareto chart Time series graph Stem-and-leaf plot

and ogives Represent data using bar graphs,

Pareto charts, time series graphs, and pie graphs

Draw and interpret and stem-and-leaf plot

Students will each visit the website of a popular restaurant of their choice. They will select several menu items and gather relevant nutritional information (calories, fat, ingredients, etc.). This information will then be classified according to data type, level of measurement, qualitative or quantitative, and discrete or continuous. Students will organize this data into frequency distribution tables and then construct appropriate statistical graphs to display the information visually.




Subject: Mathematics Course: Introductory Suggested Timeline: 4 weeks

Statistics & ProbabilityUnit Title: Data Descriptions

Unit Overview/Essential Understanding: Unit 3 will have students summarizing the central tendency of data. Examples of this will involve mean, median, and mode, among other methods. The students will also use range, variance, and standard deviation to summarize the variation of data. Percentiles, quartiles, and z-scores will help students to measure the position of data values in the set.


Measures of central tendency Measures of variation Measures of position Exploratory data analysis

Focus Standards Addressed in this Unit: CC.2.4.HS.B.3 – Analyze linear models to make interpretations based on data


Misconceptions: Arithmetic errors when calculating measures of central tendency Mistakes when calculating variance and standard deviation on a calculator Misinterpreting how to handle outliers in a data set

Concepts/Content: Mean Median Mode Midrange Parameter Statistic

Competencies/Skills: Summarize data using measures

of central tendency such as the mean, median, mode, and midrange

Describe data using measures of variation such as the range,

Description of Activities: Direct teacher-led instruction Large group Q & A discussion Independent student work Small group collaboration Online remedial/enrichment videos

Unimodal Bimodal Multimodal Symmetric Skewed Modal class Weighted mean Range Variance Standard deviation Quartile Percentile Boxplot Interquartile range (IQR) Outlier Coefficient of variation Five-number summary z-score Empirical rule

variance, and standard deviation Identify the position of a data

value in a data set using various measures of position such as z-scores, percentiles, and quartiles

Use the techniques of exploratory data analysis including boxplots and five-number summaries to discover various aspects of data



Additional Resources: Elementary Statistics: A Step-By-Step Approach textbook Approved supplemental resources like: USA TestPrep

Approved online resources like: Khan Academy



Unit Title: Probability and Counting Rules

Unit Overview/Essential Understanding: In Unit 4, the students will learn about the basics of probability—in particular, the three types: classical, empirical, and subjective. They will work with situations involving mutually exclusive (and non-mutually exclusive) events and the addition rules. There will also be work with independent (and dependent) events and the multiplication rules. The unit finishes with the students working with several of the counting rules and how they relate to probability situations.


Sample spaces and probability The addition rules for probability The multiplication rules and conditional probability Counting rules Probability and counting rules

Focus Standards Addressed in this Unit: CC.2.4.HS.B.4 – Recognize and evaluate random processes underlying statistical experiments


Misconceptions: Misunderstandings involving the concepts of mutually exclusive events and independent events Confusion over whether a situation involves permutations or combinations Difficulty turning a real-life situation into a probability problem

Concepts/Content: Probability Sample space Outcome Event Tree diagram Venn diagram Classical

Competencies/Skills: Determine sample spaces and find

the probability of an event, using classical probability or empirical probability

Find the probability of compound events using the addition rules

Find the probability of compound

Description of Activities: Direct teacher-led instruction Large group Q & A discussion Independent student work Small group collaboration Online remedial/enrichment videos Project: “Games of Chance” – Students will

select and research different types of games of

Empirical Subjective Mutually exclusive Independent Law of large numbers Complement Conditional probability Fundamental counting rule Factorial Permutation Combination

events using the multiplication rules

Find the conditional probability of an event

Find the total number of outcomes in a sequence of events using the fundamental counting rule

Find the number of ways that r objects can be selected from n objects using the permutation rule

Find the number of ways that r objects can be selected from n objects using the combination rule

Find the probability of an event using the counting rules

chance. They will investigate and share the important details of each game relating to winning, losing, betting, strategies, variations, etc. A main area of focus for the students will be to make direct connections to several of the probability topics that have been covered in this unit. The students will be expected to use a combination of words, numbers, pictures, charts, and/or graphs as they communicate the descriptions of their games with each other.






Unit Title: Discrete Probability Distributions

Unit Overview/Essential Understanding:

Unit 5 will have students working with discrete probability distributions, in particular the values that random variables can assume and the corresponding probabilities of these values. The students will also work to find the mean, variance, standard deviation, and expectation of these discrete random variables. Several other types of distributions will be discussed as well in this unit: binomial, multinomial, Poisson, and hypergeometric.


Probability distributions Mean, variance, standard deviation, and expectation The binomial distribution Other types of distributions

Focus Standards Addressed in this Unit: CC.2.4.HS.B.6 – Use the concepts of independence and conditional probability to interpret data


Misconceptions: Difficulty creating a probability distribution from real-life information Arithmetic errors when calculating mean, variance, standard deviation, and expectation Confusion over which type of probability distribution is relevant to a certain situation

Concepts/Content: Random variable Discrete probability distribution Expected value Binomial distribution Binomial experiment Multinomial distribution Poisson distribution Hypergeometric distribution

Competencies/Skills: Construct a probability

distribution for a random variable Find the mean, variance, standard

deviation, and expected value for a discrete random variable

Find the exact probability for X successes in n trials of a binomial experiment

Find the mean, variance, and


standard deviation for the variable of a binomial distribution

Find probabilities for outcomes of variables using the multinomial, Poisson, and hypergeometric distributions






Unit Title: The Normal Distribution

Unit Overview/Essential Understanding: In Unit 6 the students will learn all about the normal model and the normal distribution. They will use the normal distribution to describe a variety of different variables, and also use it to solve a wide variety of different problems.


Normal distributions Applications of the normal distribution The Central Limit Theorem

Focus Standards Addressed in this Unit: CC.2.4.HS.B.7 – Apply rules of probability to compute probabilities of compound events in a uniform probability model


Misconceptions: Accidently trying to apply the normal model to a skewed or non-normal distribution Mistakes with calculating z-scores and/or finding their corresponding percentages

Concepts/Content: Symmetric distribution Skewed distribution Normal distribution z-score Standard normal distribution Sampling error Sampling distribution Central limit theorem Standard error of the mean

Competencies/Skills: Identify distributions as

symmetric or skewed Identify the properties of a

normal distribution Find the area under the standard

normal distribution, given various z values

Find probabilities for a normally distributed variable by transforming it into a standard variable

Find specific data values for given percentages using the standard normal distribution

Use the central limit theorem to

Description of Activities: Direct teacher-led instruction Large group Q & A discussion Independent student work Small group collaboration Online remedial/enrichment videos Project: “Investigating Distributions” – The

students will each select a topic to research and gather data for (e.g. sports statistics, stock prices, income, populations, etc.). From this data, the students will calculate mean and standard deviation, as well as create a histogram. The histograms will then be analyzed for symmetry/skewness and modality. The students will make comparisons between their histograms and the normal distribution. The data used in this project will also be revisited in

solve problems involving sample means for large samples

the near future when class topics involve confidence intervals and hypothesis testing.






Unit Title: Confidence Intervals and Sample Size

Unit Overview/Essential Understanding: In Unit 7, the students will learn how to make a true estimate of a parameter, what is meant by the margin of error, and whether or not the sample size is large enough to properly represent the population. The very beginnings of inferential statistics will be explored with the introduction of confidence intervals.


Confidence intervals for the mean when σ is known and unknown Confidence intervals and sample size for proportions Confidence intervals for variances and standard deviations



Misconceptions: Confusion between the differences in confidence intervals for means and proportions Failure to appropriately justify all assumptions and conditions for inference

Concepts/Content: Assumptions Estimation Point estimate Confidence interval Confidence level Consistent estimator Unbiased estimator Relatively efficient estimator Interval estimate Margin of error Robust

Competencies/Skills: Find the confidence interval for

the mean when σ is known Determine the minimum sample

size for finding a confidence interval for the mean

Find the confidence interval for the mean when σ is unknown

Find the confidence interval for a proportion

Determine the minimum sample size for finding a confidence interval for a proportion

Find a confidence interval for a variance and standard deviation


Assessments:

Informal questioning In-class formative assessments Review of homework Periodic quizzes Unit test





Unit Title: Hypothesis Testing

Unit Overview/Essential Understanding: Unit 8 introduces students to a process called hypothesis testing. This will allow students to answer certain types of statistical questions and arrive at statistically backed conclusions. Three different methods of hypothesis testing will be explored and used throughout the unit.


The steps in hypothesis testing

Tests for a mean Tests for a proportion Tests for a variance or standard deviation Additional topics regarding hypothesis testing



Misconceptions: Confusion between the differences in the different types of tests for means and proportions Failure to appropriately justify all assumptions and conditions for inference

Concepts/Content: Null hypothesis Alternative hypothesis Hypothesis testing Test value Type I and II errors Alpha Beta One-tailed test Two-tailed test Critical value Critical region z test t test Level of significance P-value

Competencies/Skills: Understand the definitions used

in hypothesis testing State the null and alternative

hypotheses Find critical values for the z test State the five steps used in

hypothesis testing Test means when σ is known,

using the z test Test means when σ is unknown,

using the t test Test proportions, using the z test Test hypotheses, using confidence

intervals


Assessments:

Informal questioning In-class formative assessments Review of homework Periodic quizzes Unit test





Unit Title: Correlation and Regression

Unit Overview/Essential Understanding: The main themes of Unit 9 are correlation and regression. Students will collect and analyze data to determine if there is a linear relationship between two variables. If so, the students will then determine how strong that relationship is, as well as how to make predictions based on this relationship.


Scatter plots and correlation Regression Coefficient of determination and standard error of the estimate

Focus Standards Addressed in this Unit: CC.2.4.HS.B.2 – Summarize, represent, and interpret data on two categorical and quantitative variables


Misconceptions: Attempting to fit a linear model to a curved relationship Extrapolating, or trying to make predictions beyond the bounds of the collected data

Concepts/Content: Independent variable Dependent variable Scatter plot Positive relationship Negative relationship Correlation Correlation coefficient Regression line Influential point Extrapolation Lurking variable Residual plot Standard error Coefficient of determination

Competencies/Skills: Draw a scatter plot for a set of

ordered pairs Compute the correlation

coefficient Test the hypothesis: Ho: p = 0. Compute the equation of the

regression line Compute the coefficient of

determination Compute the standard error of

the estimate Find a prediction interval


Assessments: Informal questioning In-class formative assessments

Review of homework Periodic quizzes Unit test





Unit Title: Sampling and Simulation

Unit Overview/Essential Understanding: In the 10th and final unit of this course, students will study various sampling techniques for gathering information from populations. There will also be some work with using simulation and random numbers to model real-life situations.


Common sampling techniques

Surveys and questionnaire design Simulation techniques

Focus Standards Addressed in this Unit: CC.2.4.HS.B.5 – Make inferences and justify conclusions based on sample surveys, experiments, and observational studies


Misconceptions: Using biased or inappropriate methods for gathering a sample from a population Applying random numbers inaccurately when attempting to model real-life events

Concepts/Content: Random sample Biased sample Unbiased sample Systematic sample Stratified sample Cluster sample Convenience sample Multistage sampling Simulation technique

Competencies/Skills: Demonstrate a knowledge of the

four basic sampling methods Recognize faulty questions on a

survey and other factors that can bias responses

Solve problems, using simulation techniques

Description of Activities: Direct teacher-led instruction Large group Q & A discussion Independent student work Small group collaboration Online remedial/enrichment videos Project: “Sampling and Surveys” – The students

will design and conduct surveys to gather information from their peers relating to a specific chosen topic (sports, music, politics, school, etc.). They will describe the appropriate sampling techniques and randomization strategies that they used during this process. Special attention will be paid to the design of the survey questions in order to avoid unnecessary bias whenever possible. The students will share the results of their surveys with the class and discuss how information like this could be used in a meaningful way by businesses and/or retailers.

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