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Grade/Course: Geometry (Second Semester) Instructional Unit 10: Understanding Probability Instructional Schedule: Fourth Nine Weeks (suggested for 15 days) Adapted from Timothy Kanold Scope-and-Sequence documents Note: The following statistics unit is listed as Algebra 2 curriculum, however it is recommended to be placed in the Geometry sequence of instruction to alleviate the pressure of the packed Algebra 2 course. There is a possible connection with implementing geometric probability contexts where applicable. Standards: Evidence Of Standard: Prerequisite Knowledge: Assessment Tools: (student should be able to…) (standards linked to content taught in ( formative assessments, quizzes, previous grades) mastery tasks/activities) Understand independence and conditional probability and use them to interpret data. (key content) (BA 5.3a) 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”). -Define event, sample space, union, intersection, and complements as it relates to probability. -Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes. -Describe events as unions, intersections, or complements of other events (“or”, “and”, “not”). (BA 5.3b) Understand that two events - I can show why two events A and B 𝐴 and 𝐵 are independent if the are independent if the probability of A probability of 𝐴 and 𝐵 occurring and B occurring together is the together is the product of their product of their probabilities with the probabilities, and use this equation P(A and B) = P(A) x P(B). characterization to determine if they are independent. (BA 5.3c) Understand the conditional -Explain the conditional probability of probability of 𝐴 given 𝐵 as A given B as P(A and B)/P(B), and 𝑃(𝐴 𝑎𝑛𝑑 𝐵)/𝑃(𝐵), and interpret illustrate this relationship using independence of 𝐴 and 𝐵 as saying examples. that the conditional probability of 𝐴 -Explain that independence of A is the given 𝐵 is the same as the probability conditional probability of A given B of 𝐴, and the conditional probability which is the same as the probability of of 𝐵 given 𝐴 is the same as the A, P(A), and the conditional probability of 𝐴. probability of B given A is the same as the probability of B, P(B). -Distinguish between events that are independent and those that are not. (BA 5.3d) Construct and interpret -Construct and interpret a two-way two-way frequency tables of data frequency tables of data when two when two categories are associated categories are associated with each with each object being classified. Use object being classified. the two-way table as a sample space -Use a two-way frequency table to to decide if events are independent decide if events are independent and and to approximate conditional to approximate conditional probabilities. For example, collect probabilities. 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 the tenth grade. Do the same for other subjects and compare the results. (BA 5.3e) Recognize and explain the -Collect data from a random sample concepts of conditional probability and determine if events are and independence in everyday independent. language and everyday situations. For -Explain the concepts of conditional example, compare the chance of probability and independence in having lung cancer if you are a smoker everyday language and everyday with the chance of being a smoker if situations. you have lung cancer. Use the rules of probability to compute probabilities of compound events in a uniform probability model. (key content) (BA 5.3f) 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. -Understand that the probability of A given B represents the outcomes remaining for A to occur once B has already occurred. -Understand that this probability is the fraction of outcomes of B that also belongs to A. (BA 5.3g) Apply the Addition Rule, -Understand and interpret the model of a situation that the true probability 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵), and interpret the answer of an event results from subtracting in terms of the model. the number of outcomes shared between A and B from each independent probability outcomes. Note: Any italicized text denotes portions of a given standard that do not apply to identified standard content in this unit. Resources/Exemplar Tasks: ( list possible task/activities students could engage in within this unit) Standards for Mathematical Practice: (highlight practice standards to be emphasized in the instructional unit) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of instruction. 8. Look for and express regularity in repeated reasoning. ( BA: Broken Arrow rigor standard; PASS: Priority Academic Student Skills standard; BA/PASS: Combination standard )