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Algebra 1

COURSE: PROBABILITY AND STATISTICS (Advanced)

TOPIC: Descriptive Statistics

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: CT Frameworks: a) Real world problems may be analyzed using statistical techniques. b) Statistical Models can be used to describe and analyze sets of data. c) Technology can be used to interpret large sets of numbers. NCTM Standards: a) Know the characteristics of well-designed studies, including the role of randomization in surveys and experiments. b) Understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable. c) Understand histograms, parallel box plots, and scatter plots and use them to display data. d) Compute basic statistics and understand the distinction. e) For univariate measurement data, be able to display the distribution, describe its shape, and select and calculate summary statistics. f) Recognize how linear transformations of univariate data affect shape, center, and spread. g) Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
Enduring Understandings Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. Statistics is the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data. 2. The ability to read and understand statistical studies and to conduct research is essential in many professional fields. 3. Frequency distributions can be used to organize data. 4. Graphical representation of data from frequency distributions can be made using histograms, frequency polygons, and ogives. 5. Graphs of data can sometimes be misleading. 6. Measures of central tendency, such as the mean, median, mode, and midrange, can be used to describe and summarize data. 7. Measures of variation, such as the range, variance, and standard deviation can be used to describe and summarize data. 8. Measures of position, such as percentiles, and z-scores can be used to describe and summarize data.	What is statistics? What are the two branches of statistics? What are the different types of data? How are samples selected? How is data organized? How can data be represented graphically and which type of graph best represents the data? How can a graph be misleading? What are the measures of central tendency, how are they calculated, and what do they mean? What are the measures of variation, how are they calculated, and what do they mean? What are the measures of position and what do they mean? Which measure is most meaningful for the data being studied?
Knowledge and Skills What students are expected to know and be able to do
Students will know… Statistical terminology. The two branches of statistics The different types of data The basic sampling techniques. Techniques for organizing data. Graphing techniques for displaying data. Characteristics of misleading graphs. Measures of central tendency. Measures of variation. Measures of position.	Students will be able to… Demonstrate knowledge of statistical terms. Differentiate between the two branches of statistics. Identify and classify different types of data. Identify the four basic sampling techniques and discuss the advantages and disadvantages of each. Organize data using frequency distributions. Represent data graphically using histograms, frequency polygons, and ogives. Recognize how graphs can appear to be misleading Describe and summarize data using measures of central tendency. Calculate mean, median, mode, weighted average, and midrange and interpret their meaning. Describe data using measures of variation. Calculate range, variance, and standard deviation and interpret their meaning. Identify the position of data in a data set. Calculate z-scores and interpret their meaning. Construct a percentile graph and use the graph to locate the positions of data values from a data set.
STAGE 2: DETERMINE ACCEPTABLE EVIDENCE
Performance Task(s) Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)	Other Evidence Application that is functional in a classroom context only to evaluate student achievement of desired results
You are a statistician and you have been hired by the state consumer protection agency to conduct a study of the current price of regular unleaded gasoline in your area. You will randomly survey 50 gas stations in your assigned region to find their current price. You will submit a written report to the state consumer department at the completion of your study. You must include in your report a display of your data in a frequency distribution table and an appropriate graph that displays the data. You must calculate the mean, median, mode, and standard deviation of the prices. Your report should include a summary and an analysis of your findings. Your findings will be combined with the results from the other regions of the state to come up with state-wide results.	In addition to tests and quizzes, one or more of the following will be used: 1. Cooperative learning activities. 2. Calculator activities. 3. Informal and formal checks: homework checks, problem of the day and review worksheets.
STAGE 3: DEVELOP LEARNING PLAN
Learning Activities: Cooperative learning activities. Calculator activities. Practice exercises and worksheets. Review worksheets. Work stations. Project. Video Series “Against All Odds”.

COURSE: PROBABILITY AND STATISTICS (Advanced)

TOPIC: Probability

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: CT Frameworks: a) Real world problems may be analyzed using statistical techniques. b) Statistical models can be used to describe and analyze sets of data. c) Principles of probability may be applied in a variety of situations. d) Functions are used in a variety of situations including to model data, to make predictions, and to find the rate of change. NCTM Standards: a) Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases. b) Understand the concepts of conditional probability and independent events. c) Understand how to compute the probability of a compound event.
Enduring Understandings Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. A tree diagram can be used to display and determine the number of possible outcomes of a sequence of events. 2. Multiplication rules of probability can be used to find the total number of possible outcomes of a sequence of events. 3. A permutation is an arrangement of n distinct objects in a specific order. 4. A combination is a selection of distinct objects without regard to order. 5. Classical probability and empirical probability can be used to find the probability of an event. 6. The probability of compound events can be found by using the addition or multiplication rules of probability. 7. Dependent events lead to conditional probability. 8. The law of large numbers states that when a probability experiment is repeated a large number of times, the relative frequency probability of an outcome will approach its theoretical probability.	What is a tree diagram? How many possible outcomes are there for a specific sequence of events? What are the multiplication rules of probability? What is a permutation? What is a combination? What is a sample space? How is the sample space used to determine the probability of an event? What is the difference between classical and empirical probability? How do you determine if two events are mutually exclusive or not mutually exclusive and how do you calculate their probability? What is conditional probability? How do you determine if two events are independent or dependent and how do you calculate their probability? What is the law of large numbers?
Knowledge and Skills What students are expected to know and be able to do
Students will know… Tree diagrams. Multiplication rules of probability. Permutations. Combinations. Classical probability. Empirical probability. The law of large numbers. The addition and multiplication rules of probability.	Students will be able to… 1. Determine the number of outcomes to a sequence of events using a tree diagram. 2. Determine the number of outcomes to a sequence of events using the multiplication rules. 3. Find the number of ways r objects can be selected from n objects using the permutation rule. 4. Find the number of ways r objects can be selected from n objects without regard to order using the combination rules. 5. Explain the differences between classical and empirical probability. 6. Determine sample spaces and find the probability of an event using classical or empirical probability. 7. Understand the concept of the “law of large numbers.” 8. Determine whether two events are mutually exclusive or not mutually exclusive. 9. Use the addition rules of probability to calculate the probability of mutually exclusive and not mutually exclusive events. 10. Determine whether two events are independent or dependent. 11. Use the multiplication rules of probability to calculate the probability of independent and dependent events.
STAGE 2: DETERMINE ACCEPTABLE EVIDENCE
Performance Task(s) Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)	Other Evidence Application that is functional in a classroom context only to evaluate student achievement of desired results
You are a mathematician who works for a company that develops lottery games. You have proposed a game in which the player has to select 5 numbers from 1 to 25 and then choose one of three colors (red, white, or blue). A player must get all five numbers correct and the correct color in order to win the jackpot. You are presenting this game to your boss and you must explain to him how you determined the number of possible choices there are and what is the probability for a player purchasing one ticket to win the jackpot. You also need to show your boss how the game would be effected if you changed the number of numbers you need to select to 6, or if you change the five numbers you need to pick to be from 1 to 30.	In addition to tests and quizzes, one or more of the following will be used: Cooperative learning activities. Calculator activities. Informal and formal checks: homework checks, problem of the day and review worksheets.
STAGE 3: DEVELOP LEARNING PLAN
Learning Activities: 1. Cooperative learning activities. 2. Calculator activities. 3. Practice exercises and worksheets. 4. Review worksheets. 5. Work stations. 6. Project. 7. Video Series “Against All Odds”.

COURSE: PROBABILITY AND STATISTICS (Advanced)

TOPIC: Simulations and Distributions

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: CT Frameworks: a) Real world problems may be analyzed using statistical techniques. b) Statistical models can be used to describe and analyze sets of data. c) Principles of probability may be applied in a variety of situations. d) Probability distributions can be used to make statistical inferences. e) A wide variety of functions can be used to model real world situations. f) Functions are used in a variety of situations including to model data, to make predictions, and to find the rate of change. NCTM Standards: a) Draw reasonable conclusions about a situation being modeled. b) Use simulations to explore the variability of sample statistics. c) Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases. d) Use simulations to construct empirical probability distributions. e) Compute and interpret the expected value of random variables in simple cases.
Enduring Understandings Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. Simulation is a procedure developed for answering questions about real-life problems by running experiments that closely resemble the real situation. 2. Simulations can be a useful alternative when studying actual situations that are too costly, too dangerous, or too time-consuming. 3. Mathematical simulation techniques use probability and random numbers to create conditions similar to those of real-life problems. 4. The expected value of a discrete random variable of a probability distribution is the theoretical average of the variable and can be used to determine the fairness of games of chance. 5. All lotteries and casino games favor the “house.” 6. A probability distribution consists of the values a random variable can assume and the corresponding probabilities of those values. 7. The mean, variance, standard deviation, and expected values are useful measures for a probability distribution. 8. A binomial distribution represents the outcomes of a binomial experiment and the corresponding probabilities of these events. 9. The multinomial distribution and the Poisson distribution are examples of other probability distributions that can be used under certain conditions. 10. The normal distribution is a continuous, symmetric, bell-shaped distribution of a variable. 11. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. 12. The z-value represents the number of standard deviations that a particular value is from the mean. 13. The area under the curve of the normal distribution corresponds to a probability. 14. Specific data values for given probabilities can be found using the standard normal distribution, 15. The Central Limit Theorem is used to solve problems involving sample means for large and small samples.	1. What is a simulation? 2. How does a simulation work? 3. How can you tell if a simulation is realistic? 4. What is the expected value of games of chance? 5. How do you determine whether or not a game is fair? 6. What is a probability distribution? 7. How do you calculate the mean, variance, standard deviation, and expected value of a probability distribution? 8. What is a binomial experiment? 9. When can you use the binomial distribution and how do you calculate the probability? 10. How do you calculate the mean, variance, and standard deviation of a binomial distribution? 11. What is the multinomial distribution? 12. What is the Poisson distribution? 13. When can you use the multinomial and Poisson distributions and how do you calculate the probabilities? 14. What is a normal distribution? 15. How do you calculate a z-value? 16. How do you use the z-value to determine probabilities for both individual scores and for sample means? 17. What is the Central Limit Theorem and how is it used to determine probabilities for sample means? 18. How do you find data values given specific probabilities?
Knowledge and Skills What students are expected to know and be able to do
Students will know… Simulations. Expected value. Probability distributions. The binomial distribution. The multinomial distribution. The Poisson distribution. Normal distributions. Standard normal distributions. Z-values. Central Limit Theorem.	Students will be able to… 1. Explain what a simulation is and how it works. 2. Calculate probabilities derived from simulations. 3. Assess whether or not a simulation is realistic. 4. Calculate the expected value for selected games of chance. 5. Determine whether or not a game is fair. 6. Construct and identify a probability distribution. 7. Calculate the mean, variance, standard deviation, and expected values of a probability distribution and interpret their meaning. 8. Determine if an experiment is a binomial experiment. 9. Calculate the mean, variance, and standard deviation of a binomial distribution. 10. Find the probabilities for outcomes of variables using the multinomial and Poisson distributions. 11. Define a normal distribution and list its properties. 12. Calculate the z-value for both individual and sample means. 13. Calculate the probability of an event using the normal distribution. 14. Find specific data values for given probabilities. 15. Use the Central Limit Theorem to determine probabilities for sample means for large and small samples.
STAGE 2: DETERMINE ACCEPTABLE EVIDENCE
Performance Task(s) Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)	Other Evidence Application that is functional in a classroom context only to evaluate student achievement of desired results
You are the chairman of the fundraising committee for a local charity. You have devised a scratch off ticket to be sold for $1 in which the player has three rows with 3 letters, A,B, and C, randomly arranged in each row ( a total of nine blocks). The player can only scratch off one block in each row and in order to win the prize, they must reveal three identical letters (AAA, BBB, or CCC). You need to determine what the minimum prize is that will theoretically assure that you will make a profit. Then, using this knowledge, determine how many tickets would need to be sold in order for you to expect to raise $2000.	In addition to tests and quizzes, one or more of the following will be used: 1. Cooperative learning activities. 2. Calculator activities. 3. Informal and formal checks: homework checks, problem of the day and review worksheets.
STAGE 3: DEVELOP LEARNING PLAN
Learning Activities: 1. Cooperative learning activities. 2. Calculator activities. 3. Simulations. 4. Practice exercises and worksheets. 5. Review worksheets. 6. Work stations. 7. Project. 8. Video Series “Against All Odds”.