UNIT 4 ALGEBRA II TEMPLATE CREATED BY REGION 1 ESA UNIT 4

UNIT 4 ALGEBRA II TEMPLATE CREATED BY REGION 1 ESA UNIT 4

Algebra II Unit 4 Overview: Inferences and Conclusions from Data In this unit, students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data including sample surveys, experiments, and simulations and the role that randomness and careful design play in the conclusions that can be drawn. e: It is important to note that the units (or critical areas) are intended to convey coherent groupings of content. The clusters and standards within units are ordered as they are in the Common Core State s, and are not intended to convey an instructional order. Considerations regarding constraints, extensions, and connections are found in the instructional notes. The instructional notes are a critical attribute of the courses and should not be overlooked. For example, one will see that standards such as A.CED.1 and A.CED.2 are repeated in multiple courses, yet their emphases change from one course to the next. These changes are seen only in the instructional notes, making the notes an indispensable component of the pathways. (All instructional notes/suggestions will be found in italics throughout this document) Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol Template created by Region 1 ESA Page 2 of 13

Unit 4: Inferences and Conclusions from Data- S.ID.4 Cluster: Summarize, represent, and interpret data on a single count or measurement variable. Instructional es: While students may have heard of the normal distribution, it is unlikely that they will have prior experience using it to make specific estimates. Build on students understanding of data distributions to help them see how the normal distribution uses area to make estimates of frequencies (which can be expressed as probabilities). Emphasize that only some data are well described by a normal distribution. S.ID.4 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. I can use the mean and standard deviation of a set of data to fit the data to a normal curve. I can use the 68-95-99.7 Rule to estimate the percent of a normal population that falls within 1, 2, or 3 standard deviations of the mean. I can recognize that normal distributions are only appropriate for unimodal and symmetrical shapes. I can estimate the area under a normal curve using a calculator, table, or spreadsheet. #1 Make sense of problems and persevere in solving them. How can I communicate the properties of Assessments align to suggested learning targets. #2 Reason abstractly and quantitatively. a data set to illuminate its important #3 Construct viable arguments and critique the reasoning features? Statisticians summarize, represent, and #4 Model with mathematics Drill and practice interpret categorical and quantitative data in #5 Use appropriate tools strategically. Multiple choice multiple ways since one method can reveal or #6 Attend to precision. create a different impression than another. Short answer (written) Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Mean, standard deviation, data set, z-score, normal distribution, 68-95-99.7 Rule. Percent, population, unimodal, symmetric, distribution shape, area, normal curve Template created by Region 1 ESA Page 3 of 13

Unit 4: Inferences and Conclusions from Data- S.IC.1 Cluster: Understand and evaluate random processes underlying statistical experiments. Instructional es: For S.IC.2, include comparing theoretical and empirical results to evaluate the effectiveness of a treatment. S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. I can define population, population parameter, random sample, and inference. I can explain why randomization is used to draw a sample that represents a population well. I can recognize that statistics involves drawing conclusions about a population based on the results obtained from a random sample of the population. #1 Make sense of problems and persevere in solving them. How can a population be described when Assessments align to suggested learning targets. #2 Reason abstractly and quantitatively. it is so large, it would be nearly impossible #3 Construct viable arguments and critique the reasoning #4 Model with mathematics Drill and practice #5 Use appropriate tools strategically. Multiple choice #6 Attend to precision. population and make informed judgments. Short answer (written) Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Inference, population parameter, random sample, population, statistics Template created by Region 1 ESA Page 4 of 13

Unit 4: Inferences and Conclusions from Data- S.IC.2 Cluster: Understand and evaluate random processes underlying statistical experiments. Instructional es: For S.IC.2, include comparing theoretical and empirical results to evaluate the effectiveness of a treatment. S.IC.2 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? I can choose a probability model for a problem situation. I can conduct a simulation of the model and determine which results are typical of the model and which results are considered outliers (possible but unexpected). I can decide if the data collected is consistent with the selected model or if another model is required. I can pose a question that suggests a model and a means of collecting data and answer my question. #1 Make sense of problems and persevere in solving How can a population be described when Assessments align to suggested learning targets. them. it is so large, it would be nearly impossible #2 Reason abstractly and quantitatively. #3 Construct viable arguments and critique the reasoning Drill and practice #4 Model with mathematics Multiple choice #5 Use appropriate tools strategically. population and make informed judgments. Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Theoretical probability, experimental probability, simulation, model, event Template created by Region 1 ESA Page 5 of 13

Unit 4: Inferences and Conclusions from Data- S.IC.3 Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Instructional es: In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For S.IC.4 and 5, focus on the variability of results from experiments that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness. S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. I can define sample survey, experiment, observational study, and randomization. I can describe the purpose of a sample survey, an experiment, and an observational study. I can describe the differences among sample surveys, experiments, and observational studies. I can explain the role of randomization in sample surveys, experiments, and observational studies. I can apply random sampling techniques to draw a sample from a population. #1 Make sense of problems and persevere in solving them. How can a population be described when Assessments align to suggested learning targets. #2 Reason abstractly and quantitatively. it is so large, it would be nearly impossible #3 Construct viable arguments and critique the reasoning #4 Model with mathematics Drill and practice #5 Use appropriate tools strategically. Multiple choice #6 Attend to precision. population and make informed judgments. Short answer (written) Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Sample survey, experiment, observational study, randomization Template created by Region 1 ESA Page 6 of 13

Unit 4: Inferences and Conclusions from Data- S.IC.4 Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Instructional es: In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For S.IC.4 and 5, focus on the variability of results from experiments that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness. S.IC.4 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. I can define population mean, sample mean, population proportion, and sample proportion. I can calculate the sample mean or proportion. I can defend the statement, The population mean or proportion is close to the sample mean or proportion when the sample is randomly selected and large enough to represent the population well. I can infer that the population mean or proportion is equal to the sample mean or proportion and conduct a simulation to determine which sample results are typical of this model and which results are considered possible outliers (possibly, but unexpected). #1 Make sense of problems and persevere in solving them. How can a population be described when Assessments align to suggested learning targets. #2 Reason abstractly and quantitatively. it is so large, it would be nearly impossible #3 Construct viable arguments and critique the reasoning #4 Model with mathematics Drill and practice #5 Use appropriate tools strategically. Multiple choice #6 Attend to precision. population and make informed judgments. Short answer (written) Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Population mean, sample mean, population proportion, sample proportion, sample survey, margin of error, simulation model, random sampling, confidence interval Template created by Region 1 ESA Page 7 of 13

Unit 4: Inferences and Conclusions from Data- S.IC.4 continued Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Instructional es: In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For S.IC.4 and 5, focus on the variability of results from experiments that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness. S.IC.4 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. I can choose an appropriate margin of error for the sample mean or proportion and create a confidence interval based on the results of the simulation conducted. I can determine how often the true population mean or proportion is within the margin of error of each sample mean or proportion. I can pose a question regarding the mean or proportion of a population, use statistical techniques to estimate the parameter, and design an appropriate product to summarize the process and report the estimate. #1 Make sense of problems and persevere in solving them. How can a population be described when Assessments align to suggested learning targets. #2 Reason abstractly and quantitatively. it is so large, it would be nearly impossible #3 Construct viable arguments and critique the reasoning #4 Model with mathematics Drill and practice #5 Use appropriate tools strategically. Multiple choice #6 Attend to precision. population and make informed judgments. Short answer (written) Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Population mean, sample mean, population proportion, sample proportion, sample survey, margin of error, simulation model, random sampling, confidence interval Template created by Region 1 ESA Page 8 of 13

Unit 4: Inferences and Conclusions from Data- S.IC.5 Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Instructional es: In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For S.IC.4 and 5, focus on the variability of results from experiments that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness. S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. I can calculate the sample mean and standard deviation of the two treatment groups and the difference of the means. I can conduct a simulation for each treatment group using the sample results as the parameters for the distribution. I can calculate the difference of means for each simulation and represent those differences in a histogram. I can use the results of the simulation to create a confidence interval for the difference of means. #1 Make sense of problems and persevere in solving them. How can a population be described when Assessments align to suggested learning targets. #2 Reason abstractly and quantitatively. it is so large, it would be nearly impossible #3 Construct viable arguments and critique the reasoning #4 Model with mathematics Drill and practice #5 Use appropriate tools strategically. Multiple choice #6 Attend to precision. population and make informed judgments. Short answer (written) Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Sample, mean, treatment, simulation, standard deviation, histogram, extreme, parameters, significant Template created by Region 1 ESA Page 9 of 13

Unit 4: Inferences and Conclusions from Data- S.IC.5 continued Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Instructional es: In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For S.IC.4 and 5, focus on the variability of results from experiments that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness. S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. I can use the confidence interval to determine if the parameters are significantly different based on the original difference of means. I can pose a question regarding the means or proportions of two populations, use statistical techniques to estimate the difference, and design an appropriate product to summarize the process and report the estimate. #1 Make sense of problems and persevere in solving them. How can a population be described when Assessments align to suggested learning targets. #2 Reason abstractly and quantitatively. it is so large, it would be nearly impossible #3 Construct viable arguments and critique the reasoning #4 Model with mathematics Drill and practice #5 Use appropriate tools strategically. Multiple choice #6 Attend to precision. population and make informed judgments. Short answer (written) Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Sample, mean, treatment, simulation, standard deviation, histogram, extreme, parameters, significant Template created by Region 1 ESA Page 10 of 13

Unit 4: Inferences and Conclusions from Data- S.IC.6 Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Instructional es: In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For S.IC.4 and 5, focus on the variability of results from experiments that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness. S.IC.6 Evaluate reports based on data. I can identify the variables as quantitative or categorical. I can describe how the data was collected. I can indicate any biases or flaws. I can identify inferences the author of the report made from the sample data. I can write or present a summary of a databased report addressing the sampling techniques used, inferences made, and any flaws or biases. #1 Make sense of problems and persevere in solving them. How can a population be described when Assessments align to suggested learning targets. #2 Reason abstractly and quantitatively. it is so large, it would be nearly impossible #3 Construct viable arguments and critique the reasoning #4 Model with mathematics Drill and practice #5 Use appropriate tools strategically. Multiple choice #6 Attend to precision. population and make informed judgments. Short answer (written) Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Report, variables, quantitative, categorical, bias, inferences Template created by Region 1 ESA Page 11 of 13

Unit 4: Inferences and Conclusions from Data- S.MD.6 Cluster: Use probability to evaluate outcomes of decisions. Instructional es: Extend to more complex probability models. Include situations such as those involving quality control, or diagnostic tests that yield both false positive and false negative results. S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). I can use probability to create a method for making a fair decision. I can use probability to analyze the results of a process and decide if it resulted in a fair decision. #1 Make sense of problems and persevere in solving them. How is probability used to make informed Assessments align to suggested learning targets. #2 Reason abstractly and quantitatively. decisions about uncertain events? #3 Construct viable arguments and critique the reasoning The rules of probability can lead to more valid and reliable predictions about the likelihood of #4 Model with mathematics Drill and practice an event occurring. #5 Use appropriate tools strategically. Multiple choice #6 Attend to precision. Short answer (written) Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Sample space, probability, event, simulation, fair Template created by Region 1 ESA Page 12 of 13

Unit 4: Inferences and Conclusions from Data- S.MD.7 Cluster: Use probability to evaluate outcomes of decisions. Instructional es: Extend to more complex probability models. Include situations such as those involving quality control, or diagnostic tests that yield both false positive and false negative results. S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). I can analyze data to determine whether or not the best decision was made. I can analyze the available strategies, recommend a strategy, and defend my choice. #1 Make sense of problems and persevere in solving them. How is probability used to make informed Assessments align to suggested learning targets. #2 Reason abstractly and quantitatively. decisions about uncertain events? #3 Construct viable arguments and critique the reasoning The rules of probability can lead to more valid and reliable predictions about the likelihood of #4 Model with mathematics Drill and practice an event occurring. #5 Use appropriate tools strategically. Multiple choice #6 Attend to precision. Short answer (written) Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Sample space, probability, event Template created by Region 1 ESA Page 13 of 13