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Algebra II Module 4 Inferences and Conclusions From Data Topic A Probability 13 Days Topic B Modeling Data Distributions 12 Days Topic C Drawing Conclusions Using Data From a Sample 10 Days 1 Overview The concepts of probability and statistics covered in Grade 11 build on students’ previous work in Grades 7 and 9. Topics A and B address standards S-CP.A.1–5 and S-CP.B.6–7, which deal primarily with probability. In Topic A, fundamental ideas from Grade 7 are revisited and extended to allow students to build a more formal understanding of probability. More complex events are considered (unions, intersections, complements) (S-CP.A.1). Students calculate probabilities based on two-way data tables and interpret them in context (S-CP.A.4). They also see how to create “hypothetical 1000” two-way tables as a way of calculating probabilities. Students are introduced to conditional probability (S-CP.A.3, S-CP.A.5), and the important concept of independence is developed (SCP.A.2, S-CP.A.5). The final lessons in this topic introduce probability rules (S-CP.B.6, S-CP.B.7). Topic B is a short topic consisting of four lessons. This topic introduces the idea of using a smooth curve to model a data distribution, describes properties of the normal distribution, and asks students to distinguish between data distributions for which it would be reasonable to use a normal distribution as a model and those for which a normal distribution would not be a reasonable model. In the final two lessons of this topic, students use tables and technology to find areas under a normal curve and interpret these areas in the context of modeling a data distribution (S-ID.A.4). Topics C develops students’ understanding of statistical inference and introduce different types of statistical studies (observational studies, surveys, and experiments) (S-IC.B.3). In Topic C, students explore using data from a random sample to estimate a population mean or a population proportion. Building on what they learned about sampling variability in Grade 7, students use simulation to create an understanding of margin of error. Students calculate the margin of error and interpret it in context (SIC.B.4). Students also evaluate reports from the media using sample data to estimate a population mean or proportion (S-IC.B.6). 2 Lesson Big Idea Emphasize Suggested Problems Exit Ticket # of Days TOPIC A Probability 2 Students calculate probabilities given a two-way table of data. Students interpret probabilities in context. Data organized in a two-way frequency table can be used to calculate probabilities. Exercises 1-14 Problem Set 1a-f Yes 3 Students calculate conditional probabilities given a two-way data table or using a hypothetical 1000 two-way table. How to calculate conditional probability using a two way table (not formula yet). Exercises 1-6, 10-14 Problem Set 1, 2, 4, 6 Yes Two events are independent when knowing that one event has occurred does not change the likelihood that the second event has occurred. Exercises 1-11 Problem Set 2-3 Yes 2 Combinations of events s using “and,” “or,” and “not” are represented by different regions of a Venn Diagram. Opening Exercise Examples 1-3 Exercises 1-3 Problem Set 1-2 Yes 2 2 2 Students interpret probabilities, including conditional probabilities, in context. 4 Students use two-way tables to determine if two events are independent. 5 Students represent events by shading appropriate regions in a Venn diagram. Students use a Venn Diagram to calculate probabilities. 3 Lesson 6 Big Idea Emphasize Students use the complement rule to calculate the probability of the complement of an event and the multiplication rule for independent events to calculate the probability of the intersection of two independent events. Suggested Problems Examples 1-3 Exercises 1-2 Problem Set 1, 3-6 Exit Ticket Yes # of Days 3 Students recognize that two events 𝐴 and 𝐵 are independent if and only if 𝑃(𝐴 and 𝐵) = 𝑃(𝐴)𝑃(𝐵) and interpret independence of two events 𝐴 and 𝐵 as meaning that the conditional probability of 𝐴 given 𝐵 is equal to 𝑃(𝐴). Students use the formula for conditional probability to calculate conditional probabilities and interpret probabilities in context. 7 Students use the addition rule to calculate the probability of a union of two events. Opening Exercise Exercises 1-3 Examples 1-2 Problem Set 1-3, 5 4 Yes 2 Lesson Big Idea Emphasize Suggested Problems Exit Ticket # of Days TOPIC B – Modeling Data Distributions *S.D. Topic B assumes that students are familiar with standard deviation, which students may or may not have learned in Algebra 1. Teachers may need to teach the concept of standard deviation prior to lesson 8. How to calculate mean and standard deviation using a graphing calculator. eMath Instruction Algebra 1 Unit 10 Lesson 4 8 Students describe data distributions in terms of shape, center, and variability. The mean of a distribution is interpreted as a typical value and is the average of the data values that make up the distribution. Exercises 1-3, 7-9 Problem Set 1-2 eMath Instruction Unit 13 Lesson 2 Yes 2 Students use the mean and standard deviation to describe center and variability for a data distribution that is approximately symmetric. The standard deviation is a value that describes a typical distance from the mean. Exercises 1-9 Problem Set 1-3 eMath Instruction Unit 13 Lesson 3 Yes 2 9 Students draw a smooth curve that could be used as a model for a given data distribution. Students recognize when it is reasonable and when it is not reasonable to use a normal curve as a model for a given data distribution. A normal curve is symmetric and bell shaped. The mean of a normal distribution is located in the center of the distribution. Areas under a normal curve can be used to estimate the proportion of the data values that fall within a given interval. When a distribution is skewed, it is not appropriate to model the data distribution with a normal curve. 5 2 10 Students calculate 𝑧 scores. z score is defined as Exercises 1-4 Examples 1-2 Problem Set 1-2, 5 eMath Instruction Unit 13 Lesson 4 Yes 3 Probabilities associated with normal distributions can be found using 𝒛 scores or directly without using zscores. Examples 1-2 Exercises 1-2 Problem Set 2-3 Yes 3 To avoid bias in observational studies and surveys, it is important to select subjects randomly. Exercises 1-3 Problem Set 1,2,4 eMath Instruction Unit 13 Lesson 1 Yes 2 Students use technology and tables to estimate the area under a normal curve. Probabilities associated with normal distributions are determined using scores and can be found using a graphing calculator or tables of standard normal curve areas. 11 Students use tables and technology to estimate the area under a normal curve. Topic C -Drawing Conclusions Using Data from a Sample 12 Students distinguish between observational studies, surveys, and experiments Students explain why random selection is an important consideration in observational studies and surveys and why random assignment is an important consideration in experiments. 6 Lesson 13 Big Idea Students differentiate between a population and a sample. 14 15 18 19 Suggested Problems Examples 1-3 A population is the entire set of subjects in which there is an interest Exercises 1-3 Problem Set #1 Emphasize Students recognize statistical questions that are answered by estimating a population mean or a population proportion. A sample is a part of the population from which information (data) is Students understand the term “sampling variability” in the context of estimating a population proportion. The vast majority of the statistics that you've done so far has been descriptive. With descriptive statistics, we summarize how a data set "looks" with measures of central tendency, like the mean, and measures of dispersion, like the standard deviation. But, the more powerful branch of statistics is known as inferential where we try to infer properties about a population from samples that we take. We do this by using probability and sampling variability to estimate how likely the sample is given a certain population. Students understand that the standard deviation of the sampling distribution of the sample proportion offers insight into the accuracy of the sample proportion as an estimate of the population proportion. Exit Ticket Yes # of Days 2 gathered, often for the purpose of generalizing from the sample to the population. 7 eMath Instruction Unit 13 Lesson 5 eMath Instruction Unit 13 Lesson 6 eMath Instruction Unit 13 Lesson 7 No 6