Course Name: Geometry/Math II Unit 9 Unit Title: Probability and Statistics
Enduring understanding (Big Idea):Students will be able to use probability practices to solve real-life statistics problems.
Essential Questions:
1. What is the difference between a permutation and a combination?
2. What is the difference between theoretical and experimental probability?
3. How do independence and dependence of events affect the computation of probabilities in two-stage experiments?
4. How is probability used in real-world settings?
BY THE END OF THIS UNIT:
Students will know…
· The difference between combination and permutation problems.
· The difference between independent and dependent probabilities. / Students will be able to:
· Determine if theoretical or experimental probability is the best course of action to solve a problem.
· Use dependent and independent computations to solve probabilities
Vocabulary:
Population, sample, convenience sample, self-selected sample, systematic sample, random sample, bias, observational study, controlled experiment, survey, Experimental probability, theoretical probability, geometric probability, simulation, sample space, equally likely outcomes, outcome, event, complement of an event, odds, conditional probability, relative frequency, probability distribution, uniform distribution, cumulative frequency, cumulative probability, two-way frequency table, Compound event, Independent and Dependent event, mutually exclusive events, overlapping events, Fundamental Counting Principal, permutation, combination, n factorial
Unit Resources:
Performance Task:
Pearon Alg 2 probability performance tasks.pdf
probability activity.pdf
Project: l21_probability_statements_beta_complete.pdf
Test Specification Weights for the Common Exams in Common Core Math II:
Suggested Order/Pacing:
Geometric Probability: Section 10.8
Theoretical and Experimental Probability: CC-21,
Algebra 1BK: Section 12.7, CB 12.7, ER 12.7
Algebra 2 BK: Section 11.2
Probability Distribution and Frequency Tables: CC-22
Algebra 2 BK: CB 11.3
Permutations and Combinations: CC-23,
Algebra 1 BK: Section 12.6, ER 12.6,
Algebra 2 BK: Section 11.1
Compound Probability and Probability of Multiple Events: CC-24, and Algebra 1 BK: Section 12.8, ER 12.8
Algebra 2 BK: Section 11.3
Contingency Tables: CC-25
Conditional Probability: CC-26,
Algebra 1 BK: CB 12.8 and Algebra 2 BK: Section 11.4 / Mathematical Practices in Focus:
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
6. Attend to precision
CCSS-M Included:
S.IC.2, S.IC.6, S.CP.1 – S.CP.9
Abbreviation Key:
CC – Common Core Additional Lessons found in the Pearson online materials.
CB- Concept Bytes found in between lessons in the Pearson textbook.
ER – Enrichment worksheets found in teacher resources per chapter.
Merge information from Geometry, Algebra 1, and Algebra 2 Books to complete this unit.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
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Course Name: Geometry/Math II Unit 9 Unit Title: Probability and Statistics
CORE CONTENT
Cluster Title: Understand and evaluate random processes underlying statistical experiments.Standard 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?
Concepts and Skills to Master
· Find the experimental probability of an event
· Find the theoretical probability of an event
· Use of a simulation to model an event
· Make decisions about probability based on simulated events
SUPPORTS FOR TEACHERS
Critical Background Knowledge:· Event, possibilities, successes
· Sample space, trials, outcomes
· Cards in a deck, faces of a die
Academic Vocabulary:
Experimental probability, simulation, sample space, equally likely outcomes, theoretical probability
Suggested Instructional Strategies:
· Utilized the TI-84 Probability Simulation App
· Remind students that they did simple probability in middle school
· Use a coin toss experiment to introduce the concepts but quickly move to simulations
· Use your calculator to do the simulations like coin toss and random number generator
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
1) Explain how well and why a sample represents the variable of interest from a population.
2) Demonstrate understanding of the different kinds of sampling methods.
3) Design simulations of random sampling: assign digits in appropriate proportions for events, carry out the simulation using random number generators and random number tables and explain the outcomes in context of the population and the known proportions. Use data-generating processes such as simulations to evaluate the validity of a statistical model.
Additional note from DPI for Level II:
Ex. Jack rolls a 6 sided die 15 times and gets the following results:
4, 6, 1, 3, 6, 6, 2, 5, 6, 5, 4, 1, 6, 3, 2
Based on these results, is Jack rolling a fair die? Justify your answer using a simulation. / Starting Resources:
Algebra 1 Textbook Correlation:
CB 12.7
Algebra 2 Textbook Correlation:
Section 11.2, ER 11.2, CB 11.3
Sample Assessment Tasks
Skill-based task:
On a multiple choice test, each item has 4 choices, but only one choice is correct. How can you simulate guessing the answers? Based on your simulation of at least 20 trials, what is the probability that you will pass the test by guessing at least 6 of 10 answers correctly? / Problem Task:
On a multiple-choice test, each item has 4 choices, but only one choice is correct. How can you simulate guessing the answer? What is the probability that you will pass the test by guessing at least 6 of 10 answers correctly?
CORE CONTENT
Standard: S-IC.6 Evaluate reports based on data.
Concepts and Skills to Master:
· Evaluate reports based on data
SUPPORTS FOR TEACHERS
Critical Background Knowledge:Academic Vocabulary:
Population, sample, convenience sample, self-selected sample, systematic sample, random sample, bias, observational study, controlled experiment, survey
Suggested Instructional Strategies:
What does this standard mean that a student will know and be able to do?
Read and explain in context data from outside reports. Evaluate reports based on data on multiple aspects (e.g. experimental design, controlling for lurking variables, representativeness of samples, choice of summary statistics, etc.) / Starting Resources:
Algebra 2 Textbook Correlation:
Section 11.7
Sample Assessment Tasks
Skill-based task:
A survey asks, “Aren’t handmade gifts always better than tacky purchased gifts?” Does this survey question have any bias? Explain. / Problem Task:
What sampling method could you use to find the percent of adults in your community who support building more nuclear power plants? What is an example of a survey question that is likely to yield unbiased information?
CORE CONTENT
Cluster Title: Understand independence and conditional probability and use them to interpret data.Standard: S.CP.1 Describe events and subsets of a sample space using characteristics of the outcomes, or as unions, intersections, and complements of other events ("or," "and," "not.")
Concepts and Skills to Master:
· The probability of an impossible event is 0, (0%), the probability of a certain event is 1 (100%), and all other probabilities are between 0 and 1.
· The probability that an event will occur + the probability it will not occur = 1.
· Define a sample space and events within the sample space. Identify subsets from sample space given defined events, including unions, intersections and complements of events.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:· Adding and multiplying fractions, Converting fractions to decimals, Calculating simple probabilities Find the area of polygons and circles.
Academic Vocabulary:
Experimental probability, theoretical probability, geometric probability, simulation, sample space, equally likely outcomes, outcome, event, complement of an event, odds
Suggested Instructional Strategies:
· Use Venn diagrams to remind students how to determine the difference between “and” and “or”.
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
1) Define the sample space for a given situation.
Additional note from DPI for Level II:
Ex. What is the sample space for rolling a die?
Ex. What is the sample space for randomly selecting one letter from the word MATHEMATICS?
2) Describe an event in terms of categories or characteristics of the outcomes in the sample space.
Additional note from DPI for Level II:
Ex. Describe different subsets of outcomes for rolling a die using a single category or characteristic.
3) Describe an event as the union, intersection, or complement of other events.
Additional note from DPI for Level II:
Ex. Describe the following subset of outcomes for choosing one card from a standard deck of cards as the intersection of two events: {queen of hearts, queen of diamonds}. / Starting Resources:
Geometry Textbook Correlation:
10.8, CC.21
Algebra 1 Textbook Correlation:
Section 12.7
Algebra 2 Textbook Correlation:
Section 11.2, ER 11.2
http://www.shodor.org Interactive Venn Diagram Shape Sorter
Sample Assessment Tasks
Skill-based task:
If the probability an event will occur is 74%, what is the probability it will not occur? / Problem Task:
What is the probability that a quarterback will complete his next pass if he has completed 30 of his last 40 passes?
CORE CONTENT
Cluster Title: Understand independence and conditional probability and use them to interpret data.Standard S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the products of their probabilities, and use that characterization to determine if they are independent.
Concepts and Skills to Master
· Explain properties of Independence and Conditional Probabilities in context and simple English.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:· Understand basic properties of probability. (7.SP.5)
· Approximate probabilities of chance events through experiment. (7.SP.6)
· Use Venn diagrams (II.4.S.CP.1) and two-way frequency tables. (I.S.ID.5)
· (A ∩ B) is the equivalent of the probability of event A and event B occurring together. (II.4.S.CP.1)
Academic Vocabulary:
Conditional probability,
Suggested Instructional Strategies:
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
1) Understand that two events A and B are independent if and only if P(A and B)= () ∙ ().
2) Determine whether two events are independent using the Multiplication Rule (stated above).
3) Explain properties of Independence and Conditional Probabilities in context and simple English.
Additional note from DPI for Level II:
Ex. For the situation of drawing a card from a standard deck of cards, consider the two events of “draw a diamond” and “draw an ace.” Determine if these two events are independent.
Ex. Create and prove two events are independent from drawing a card from a standard deck. / Starting Resources:
Geometry Textbook Correlation:
CC-26
Algebra 1 Textbook Correlation:
CB 12.8
Algebra 2 Textbook Correlation:
Section 11.4
Sample Assessment Tasks
Skill-based task:
C and D are independent events, , and . What is P(C and D)? / Problem Task:
Suppose you randomly select a shape from this circle. What is the probability
that the shape is black and has five points?
CORE CONTENT
Cluster Title: Understand independence and conditional probability and use them to interpret data.Standard S.CP.3 Understand the conditional probability of A given B as P(A and B)/ P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as P(A) and the conditional probability of B given A is the same as P(B).
Concepts and Skills to Master:
· The probability that an event B will occur given that another event, A, has already occurred.
· Conditional probability occurs when two events are dependent.
· Define and calculate conditional probabilities. Use the Multiplication Principal to decide if two events are independent and to calculate conditional probabilities.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:Adding and multiplying fractions, Converting fractions to decimals, Calculating simple probabilities
Academic Vocabulary:
Conditional Probability
Suggested Instructional Strategies:
Use Venn diagrams to explore and compute conditional probabilities.
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
1) Understand that the conditional probability of event A given event B has already happened is given by the formula: PAB=P(A and B)P(B)
2) Understand that two events A and B are independent if and only if PAB=PAand PBA=PB. In other words, the fact that one of the events happened does not change the probability of the other event happening.
3) Prove that two events A and B are independent by showing that PAB=PAand PBA=PB.
Additional note from DPI for Level II:
Ex. For the situation of drawing a card from a standard deck of cards, consider the two events of “draw a spade” and “draw a king.” Prove that these two events are independent.
Ex. Create and prove two events are dependent from drawing a card from a standard deck. / Starting Resources:
Geometry Textbook Correlation:
CC-26
Algebra 1 Textbook Correlation:
CB 12.8
Algebra 2 Textbook Correlation:
Section 11.4
Cut the Knot – Conditional Probability and Independent Events:
http://www.cut-the-knot.org/Curriculum/Probability/ConditionalProbability.shtml
Texas A&M – Conditional Probability Applet:
http://www.stat.tamu.edu/~west/applets/Venn1.html
Sample Assessment Tasks
Skill-based task:
Given the following Venn
diagram, determine whether
events A and B are
independent.
/ Problem Task:
A box contains 10 blue cubes, 5 red cubes, 5 blue marbles and 10 red marbles. You randomly pick a blue shape from the box. What is the probability you picked a cube? ANS: 67%
CORE CONTENT
Cluster Title: Understand independence and conditional probability and use them to interpret data.Standard S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the table as a sample space to decide if events are independent and to approximate conditional probabilities.
For example, collect 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 tenth grade. Do the same for other subjects and compare the results.
Concepts and Skills to Master:
· Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified.
· Use a two-way table as a sample space to decide if events are independent
· Use a two-way table to approximate conditional probabilities.
SUPPORTS FOR TEACHERS