SFUSD Mathematics Core Curriculum Development Project€¦ · Let’s Make a Deal – Simulation (2 pages) 1 per pair 1 per pair Cards 5 Lesson Series 2 CPM Algebra 2 Connections

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SFUSD Mathematics Core Curriculum, Algebra 2, Unit X.1: Probability, 2014–2015

SFUSD Mathematics Core Curriculum Development Project

2014–2015

Creating meaningful transformation in mathematics education

Developing learners who are independent, assertive constructors of their own understanding

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Algebra 2

X.1 Probability

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task Rock, Paper, Scissors (2 pages) 1 per pair

6 Lesson Series 1 Oh Craps! The Game of Pig CPM Algebra 2 Connections 10.1.1 (4 pages) CPM Algebra 2 Connections 10.1.1 HW (2 pages) The Gambler’s Fallacy (2 pages) Expecting the Unexpected The Game of Little Pig

1 per pair 1 per student 1 per pair 1 per student 1 per pair 1 per pair 1 per pair

Dice Coins Index cards (optional) Bags with 3 cubes (red, blue, yellow)

2 Apprentice Task Let’s Make a Deal (2 pages) Let’s Make a Deal – Simulation (2 pages)

1 per pair 1 per pair

Cards

5 Lesson Series 2 CPM Algebra 2 Connections 10.1.2 (4 pages) The Carrier’s Payment Plan Quandary Choosing for Chores Pig Tails Little Pig Strategies

1 per pair 1 per pair 1 per pair 1 per pair 1 per pair

2 Expert Task A Fair Game Make It Fair Make It Fair: Extension Task A Fair Game (revisited)

1 per student 1 per pair 1 per pair 1 per student

7 Lesson Series 3 CPM Algebra 2 Connections 10.2.2–10.2.3 (4 pages) Should I Go On? Big Pig Meets Little Pig The Titanic 1 Probability Project (4 pages)

1 per pair 1 per pair 1 per pair 1 per pair 1 per pair

Dice Poster paper and markers

2 Milestone Task Constructed Response: Two-Spinner Sums Performance Assessment: The Titanic 3

Provided by AAO Provided by AAO

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Unit Overview

Big Idea

Students will extend their understanding of basic probability to calculate conditional and compound probabilities, permutations, combinations, and expected values in order to analyze decisions and strategies about real-world situations.

Unit Objectives

● Students will understand definitions of sample space, intersections and unions, and complements. ● Students will be able to use the general addition and multiplication rules to calculate probabilities of compound events. ● Students will understand and be able to determine independence of multiple events. ● Students will extend basic probability to calculate conditional probabilities. ● Students will be able to use two-way frequency tables to determine independence and approximate probabilities. ● Students will be able to calculate the number of permutations and combinations in a given situation. ● Students will be able to calculate the expected value of discrete variables (e.g., games, grades, the lottery). ● Students will be able to use probabilities and expected values to analyze decisions and strategies (about the fairness of games, whether to build a new

factory, or add another shift for personnel). ● Students will be able to use binomial distribution to calculate the probabilities of the outcomes of a series of binomial events.

Unit Description

The main goal of the unit is to use probability to make decisions. The unit starts by reviewing basic concepts of probability. The unit moves along to the definition of conditional probability as well as using it to determine whether events are independent or not. The unit also uses expected value to analyze whether games are fair or unfair. The unit culminates with a group project in which students use their knowledge of probability to solve a problem.

CCSS-M Content Standards

(All Statistics and Probability Standards are Modeling* Standards) Conditional Probability and the Rules of Probability Understand independence and conditional probability and use them to interpret data S.CP.1 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”). S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

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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 the probability of A, and the conditional probability of B given A is the same as the probability of B. S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way 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. S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Use the rules of probability to compute probabilities of compound events in a uniform probability model S.CP.6 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. S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

Calculate expected values and use them to solve problems S.MD.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. S.MD.3 (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. S.MD.4 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

Use probability to evaluate outcomes of decisions S.MD.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. S.MD.5a Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. S.MD.5b Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

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S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 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).

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Progression of Mathematical Ideas Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

Students have learned basic probability concepts in the seventh grade, but four years later in the eleventh grade, students will need to review those concepts and definitions. Students will need to review sample spaces, events, and calculating probabilities of single and multiple events.

Students will extend their understanding of basic probability to calculate conditional and compound probabilities, permutations, combinations, and expected values to analyze decisions and strategies about real-world situations. Students will be able to translate real-world situations into a probability context using diagrams and equations to model them.

Students will apply their knowledge of probability of a sample to make statistical inferences of the population in the next unit on Statistics. Students who choose to take AP Statistics will further build on the knowledge to explore distributions (normal and binomial) and be able to make statements with a certain level of confidence.

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Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are formative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit?

1 Day 6 Days 2 Days 5 Days 2 Days 7 Days 2 Days

Total Days: 25

Lesson Series 1

Lesson Series 2

Lesson Series 3

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Entry Task Rock, Paper, Scissors

Apprentice Task Let’s Make a Deal

Expert Task A Fair Game

Milestone Task Two-Spinner Sums and the Titanic 3

CCSS-M Standards

S.CP.1 S.MD.6 (+)

S.CP.5 S.MD.6 (+), S.MD.7 (+)

S.CP S.MD F.BF A.SSE

S.CP.1, S.CP.2, S.CP.3, S.CP.4, S.CP.5, S.CP.7 S.MD.2 (+), S.MD.5 (+), S.MD.5a, S.MD.6 (+), S.MD.7 (+)

Brief Description of Task

Students play a variation of Rock, Paper, and Scissors in groups of three. Player A earns a point if all three players match. Player B earns a point if two players match. Player C earns a point if no players match. A fourth group member may be the scorekeeper. Depending on the class numbers, the fourth member may be optional as the three players can keep score. Students discuss which player they think will have the most and least points before playing. Students will play the game and record the results to see if they match their predictions. Afterward, students list the possible outcomes (sample space) through the use of a tree diagram to calculate the theoretical probability.

Students simulate the Monty Hall problem and calculate the probability of staying with their original choice or switching. The Monty Hall problem is from a game show where there are three doors with a prize behind one of them. The contestant picks one door and then the host opens one door that he knows to not have the prize. The contestant can choose to stay with his original choice or to switch. Students will be introduced to the game, but then will play the game in groups to see what strategy the experimental probability leads them to choose.

Two friends, Dominic and Amy, are playing a game. There is a bag with black and white marbles in it and you pick out two marbles at the same time. If the two are the same color, then Amy wins. If you pick two that are different colors, then Dominic wins. Students will determine what combination of black and white marbles will make the game fair.

Project: Provide three different types of problems for students to analyze. Students can work in groups, pairs, or individually. It is up to you to decide. Presentation & Evaluation: You can elect to have students do class presentations or not.

Source CPM Algebra 2 Connections 10-1 IMP Year 3; pp. 341–343 Formative Assessment Lesson Modeling Conditional Probabilities 2

IMP Year 1, p. 151 Illustrative Mathematics http://www.illustrativemathematics.org/illustrations/951

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Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards

S.CP.1, S.CP.2 (+), S.CP.5 (+), S.CP.5a, S.CP.5b, S.CP.6 (+), S.CP.7

S.CP.1, S.CP.5 S.MD.5a, S.MD.5b, S.MD.6

S.CP.2, S.CP.3, S.CP.4, S.CP.5 S.MD.2 (+), S.MD.3 (+), S.MD.4 (+), S.MD.5 (+)

Brief Description of Lessons

Lesson Series 1 introduces the basic concepts of probability. Students learn to use tree diagrams, tables, and area models to visualize the sample space to calculate probability. Students are also introduced to the ideas of compound events.

In Lesson Series 2, students analyze different situations involving conditional probability. They learn to use expected value to make decisions.

In Lesson Series 3, students learn to use probability to make decisions.

Sources Oh Craps (SFUSD teacher created) The Game of Pig (IMP Year 1) CPM Algebra 2 Connections 10.1.1 The Gambler’s Fallacy (IMP Year 1) Expecting the Unexpected (IMP Year 1) The Game of Little Pig (IMP Year 1)

CPM Algebra 2 Connections 10.1.2 Carrier’s Payment Plan Quandary (IMP Year 1) Choosing for Chores (IMP Year 3) Pig Tails (IMP Year 1) Little Pig Strategies (IMP Year 1)

CPM Geometry Connections (10.2.2, 10.2.3) Should I Go On? (IMP Year 1) Big Pig Meets Little Pig (IMP Year 1) Titanic Problem 1 (Illustrative Mathematics)

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Entry Task Rock, Paper, Scissors

What will student do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will be re-introduced to the concept of probability and

chance. ● Students will draw tree diagrams to list out the possible outcomes of

an event. ● Students will be introduced to the idea of a fair/unfair game.

CCSS-M Standards Addressed: S.CP.1 Potential Misconceptions:

● Students may not list all 27 possible outcomes. Be sure to reinforce the idea of listing the outcomes systematically.

Students will play a variation of Rock, Paper, and Scissors. By drawing a tree diagram to list the possible outcomes, students will calculate the theoretical probability to determine if the game is fair or not. Launch: A suggested Do Now would be to re-introduce the idea of probability by asking students what the chances are of flipping heads in a coin, rolling 1 on a standard dice, or picking a boy (or girl) in the classroom. Have the students read the description of the modified version of Rock, Paper, Scissors. Check random students for understanding on the rules of the game. You may choose to play a practice game with the students. During: Monitor students’ progress. Rotate around the room and ask them for their opinions on which player has the advantage. Have each group write their results on the board so that a class total can be obtained. Make sure students list outcomes systematically so that they get all 27. Closure/Extension: Have a class discussion on which player has the advantage and why they have the advantage. Use this as an opportunity to re-introduce the idea of probability. Discuss the idea of a tree diagram as a means of systematically listing out all possible outcomes. Have a class discussion on what makes the game unfair. Takeaways: The idea of a fair game. Tree diagram used as a method of listing the possible outcomes.

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Rock, Paper, Scissors

How will students do this?

Focus Standards for Mathematical Practice: 3. Construct viable arguments and critique the reasoning of others.

Structures for Student Learning: Academic Language Support:

Vocabulary: tree diagram, theoretical probability, fair game Sentence frames:

Differentiation Strategies: You may choose to model a game with the students to ensure that they understand the rules. Participation Structures (group, partners, individual, other): This is a group activity to be done in groups of three or four. Each group will turn in a single sheet with a table of their results as well as the answers to the questions.

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Lesson Series #1

Lesson Series Overview: This series of lessons introduces the basic concepts of probability. Students learn to use tree diagrams, tables, and area models to visualize the sample space to calculate probability. Students are also introduced to the ideas of compound events. The Game of Pig is introduced (reappearing throughout the unit) to help students develop strategy based on probability concepts. The series also helps students understand that previous events do not necessarily affect the outcome of an event. In addition, students calculate the expected value of a game and use it to determine if a game is fair or not. CCSS-M Standards Addressed: S.CP.1, S.MD.2 (+), S.MD.5 (+), S.MD.5a, S.MD.5b, S.MD.6 (+), S.MD.7 (+) Time: 6 days

Lesson Overview – Day 1 Resources

Description of Lesson: Students learn the rules of the game of Craps and play multiple rounds to learn basic probability concepts including outcome, sample space, and the definition of probability. Students will also be shown how to use a table to organize the sample space as an alternative to a tree diagram. At the end, the class should have a discussion on what it means to have a probability of 0 and of 1 and why the probability must be between 0 and 1. Notes:

• This is a group activity that can be done in groups of twos or threes. Each group will require a die.

• You may choose to play a few games with the students to model the rules. • You should show the students how to use a table as an alternative to the tree

diagram as a means of organizing the sample space.

Oh Craps!


Description of Lesson: This lesson introduces the Game of Pig, which will be a recurring theme throughout the unit, culminating with the analysis of the strategy. In this lesson, students work on how to develop strategies and how to communicate it.

IMP Year 1, The Game of Pig

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Notes: • This is a group activity that can be done in groups of 2s or 3s. Each group will

require a die. • Play a game with the entire class. One possible way of doing this is to have the

entire class start by standing up. Roll the die. Students who choose to end the turn record their scores and sit down. Students who wish to continue remain standing. The game ends when you roll a 1 or when all students choose to sit.

• If students are having trouble coming up with strategies, guide them by asking, “When should you choose to end the turn, and when should you choose to continue?”


Description of Lesson: Students are introduced to the idea of using an area model as opposed to a tree diagram/table to calculate probability. In addition, they use expected value to determine whether a game is fair or unfair. Notes:

• Students may do these problems individually or as a group. • Students may or may not have learned about expected value already.

Depending on students’ prior knowledge, you may need to teach the concept of expected value (perhaps connecting it to how grades are calculated in a class that weighs different categories).

CPM Algebra 2 Connections 10.1.1, Problems 10-2 to 10-9


Description of Lesson: Through the Gambler’s Fallacy activity, students discover that the outcomes of prior events may not necessarily affect the outcomes of a future event, which is a common misconception. Notes: This is a group activity intended for students to work in pairs. Each pair will need a coin. You may choose to introduce the idea of independent events.

IMP Year 1, The Gambler’s Fallacy

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Description of Lesson: Students flip a coin 50 times and count the number of heads. The class’s data is tabulated. Through this, students understand that having a probability of 50% does not necessarily mean that the outcome appears 25 out of 50 times. Probability becomes more accurate as the number of trials increase (Law of Large Numbers). Notes:

• This is an individual activity but may be done in pairs. • You may choose to demonstrate the law of large numbers (probability

approaches 0.50 as the number of trials increases) through a technology simulation. For example, students may use Excel or a graphing calculator to randomly generate 0 (heads) or 1 (tails) and graph the fraction of flips being heads versus the number of trials.

IMP Year 1, Expecting the Unexpected


Description of Lesson: The Game of Little Pig is a variation of the Game of Pig. Instead of rolling a die, students draw three cubes from a bag. Despite the difference in rules, the underlying concepts are the same. However, students use expected value to analyze the situation to come up with best strategies. Notes: This is a group activity that should be done in twos or threes. The activity requires a bag and three blocks of different colors, but it can be done with a bag and any three distinguishable items with the same shape.

IMP Year 1, The Game of Little Pig

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Apprentice Task

Let’s Make a Deal!

What will students do?


Math Objectives: ● This POW describes a game show involving prizes behind

three doors and asks students to evaluate the probability of success with each of two possible strategies. In their write-ups and presentations, students focus on explaining the probabilities.

● Calculating the probabilities in this situation is particularly interesting because they are difficult to assess intuitively.

● While the answer to the dilemma posed in the POW is difficult to arrive at intuitively, this simulation will convince most students of which strategy is best. After the class briefly discusses what makes a good simulation, students work in their groups to design and carry out the experiment. Afterward, the class compiles a list of their simulation results.

CCSS-M Standards Addressed: S.CP.1, S.CP.5, S.CP.6 Potential Misconceptions:

● After one door is revealed, students may think that there is always a 50% chance of winning with either remaining door.

Students will simulate the Monty Hall problem and determine if the probability of winning by staying with your original choice is higher than the probability of winning by switching. Launch:

● Have students read the POW, and give them an opportunity to ask questions about how the game is played. Tell students that in Simulate a Deal they will do a simulation of the two strategies and discuss their results. (This simulation will probably reveal the answer to the “big question” of the POW, but students will still have to determine the specific probabilities and explain them.)

● Then have students switch to “Lets Make a Deal – Simulation” During:

● Discuss simulations before students begin the activity. You might first ask, “When else have you used simulations?” to elicit examples of simulations students have seen in other contexts.

● Ask, “What makes a good simulation?” Be sure that students note that randomness is essential in most simulations that involve probability. Therefore, a good simulation for this situation would be one in which the student playing the host chooses the winning door at random and the student playing the contestant makes the initial choice of door at random.

● If students need help setting up their simulations, you can propose this approach:

1. One student (the host) decides where the car is (perhaps writing this on a piece of paper).

2. The other student (the contestant) chooses a door (stating the choice aloud).

3. The host “opens” a specific door that does not have the car. (The host may or may not have a choice about which door to open.)

4. The contestant chooses the remaining closed door (in Question 1) or announces the decision to stay with the door originally chosen (in Question 2).

5. The host reveals which door actually has the car behind it.

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Closure/Extension:

● Have one or two teams describe exactly how they carried out their simulations, and then have all teams share their results. Compile a class list of the number of wins and losses for each strategy. This should convince most students that switching is a better strategy than staying. Point out to students that they will describe their work on this activity as part of their POW write-ups. Emphasize, however, that a key task of the POW is to explain why they (and the class) got the type of results they did.

● Then give students a class period to write up their POWs. ● Ask three selected students to give their presentations. From the simulations in

Let’s Make a Deal – Simulation, they should have experienced that by staying with their guess they win about ⅓ of the time, and by switching, they win about ⅔ the time. The focus of the discussion should therefore be on why the two strategies have these probabilities of success. Explanations of this vary widely, from elaborate tree diagrams to very concise arguments.

● There is a nice discussion of this problem in “Monty’s Dilemma: Should You Stick or Switch?” by Shaughnessy and Dick, in NCTM’s Mathematics Teacher (April 1991, pp. 252–256). The article suggests that the problem might be easier to understand in this more extreme form: Suppose there are 100 doors altogether, and that after you guess, the host opens all the doors except yours and one other. Should you stay or switch?

● In this situation, students should be able to see that it makes sense to switch.

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Let’s Make a Deal!

How will students do it?

Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 4. Model with mathematics.


Vocabulary: Simulation Sentence frames:

Differentiation Strategies: Participation Structures (group, partners, individual, other): Groups

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Lesson Series #2

Lesson Series Overview: Students analyze different situations involving conditional probability. They will learn to use expected value to make decisions. CCSS-M Standards Addressed: S.CP.1, S.CP.5, S.MD.5, S.MD.5b, S.MD.6 Time: 6 days


Description of Lesson: Students will calculate conditional probabilities. The problems in this lesson involve conditional probability and the use of the probability of the complement (the probability of every outcome other than the desired outcome). Students should be able to solve these problems using the tools that they already have. As they work on the problems and discuss the different strategies to use, introduce the term “conditional probability” and discuss the idea of finding the probability of the complement.

CPM Algebra 2 Connections 10.1.2, Problems 10-27 to 10-34


Description of Lesson: Students will employ their available tools to compute an expected value and compare it to a simulated result. In the typical payment plan, a customer pays the same amount each week or month for a given service. In this activity, a newspaper carrier is paid by a particular customer by choosing, each week, two bills from a bag containing five $1 bills and one $10 bill. The task is to determine the average weekly payment—that is, the expected value—and compare it to the results of a simulation.

IMP Year 1, The Carrier’s Payment Plan Quandary


Description of Lesson: In this activity, students examine a problem involving a conditional probability and expected value. Choosing for Chores gives students two problems, each involving conditional probability.

IMP Year 3, Choosing for Chores

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The discussion looks at each of the main approaches to solving this kind of problem; students will see that tree diagrams can be used not only for developing lists of possible outcomes, but also to compute probabilities. The activity serves as a basis for reviewing area diagrams, lists, and tree diagrams, as well as expected value.


Description of Lesson: This activity presents another simpler game to help prepare students to analyze Big Pig. This variation of Pig uses coin flips. In Pig Tails, if you get tails, your turn is over. Students will evaluate strategies that include flipping only once each turn, flipping twice each turn, and so on, and then look for a pattern to determine the expected value per turn for flipping n times.

IMP Year 1, Pig Tails


Description of Lesson: In this activity, students begin their analysis of the expected value per turn for the game of Little Pig. In order to eventually analyze the best strategy for Big Pig, students continue to focus on the much less complicated game of Little Pig. Following the problem-solving strategy of “working with a simpler problem” will help students to make good choices when analyzing a more complicated situation.

IMP Year 1, Little Pig Strategies

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Expert Task

A Fair Game



Math Objectives: This task is intended to help you assess how well students understand conditional probability, and, in particular, to help you identify and assist students who have the following difficulties:

● Representing events as a subset of a sample space using tables and tree diagrams.

● Understanding when conditional probabilities are equal for particular and general situations.

CCSS-M Standards Addressed: S.CP, S.MD, F.BF, A.SSE

Students will determine how to make a game fair. The situation involves a bag with black marbles in it. Students determine how many white marbles need to be added to make the game fair. Launch: Before the lesson, students work individually on an assessment task that is designed to reveal their current understandings and difficulties. You then review their work and create questions for students to answer to improve their solutions. During: During the lesson, students first work collaboratively on a related task. They have an opportunity to extend and generalize this work. The lesson ends with a whole-class discussion. Closure/Extension: In a subsequent lesson, students revise their individual solutions to the assessment task.

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A Fair Game

How will students do it?

Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics.


Vocabulary: Sentence frames:

Differentiation Strategies: Participation Structures (group, partners, individual, other): Individual, group, whole class

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Lesson Series #3

Lesson Series Overview: This series of lessons focuses on using expected value to determine whether a game is fair or unfair. In addition, students use conditional probability to determine whether events are independent or not. CCSS-M Standards Addressed: S.CP.2, S.CP.3, S.CP.4, S.CP.5, S.MD.2 (+), S.MD.3 (+), S.MD.4 (+), S.MD.5 (+) Time: 7 Days

Lesson Overview – Days 1-2 Resources

Description of Lesson: Students will review/practice how to find the expected value of a game of chance. Students will learn how to calculate expected value using the probability of each event occurring and will review solving equations by rewriting (also called “fraction busters”). Notes: Because this lesson is pulled from a geometry unit, some homework/practice problems will include geometry topics like arc length, sin, cos, and tan.

CPM Geometry Connections 10.2.2 CPM Geometry Connections 10.2.3


Description of Lesson: Students connect their investigation of strategies for playing Little Pig to the game of Big Pig. This activity shifts the focus from calculating expected value of a point or a draw strategy to deciding whether it is beneficial to draw again based on the points a player has already accumulated. Calculating a new expected value will answer the question, “Given what I have now, would I benefit in the long run from drawing again?”

IMP Year 1, Should I Go On?


Description of Lesson: Having developed an approach to analyze Little Pig, students now return to the unit problem.

IMP Year 1, Big Pig Meets Little Pig

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Many people initially think that the best strategy for Pig is to stop after a particular number of rolls. The analysis in the unit shows that, in fact, the best strategy is to stop after reaching a particular number of points.


Description of Lesson: This task guides students by asking a series of specific questions and lets them explore the concepts of probability as a fraction of outcomes, and using two-way tables of data. The emphasis is on developing their understanding of conditional probability. Students should understand the difference between P(A and B) and P(A|B), and notice that P(A|B) is not the same as P(B|A). Parts b and c require students to verbalize their understanding of probability. The last part of the task is open ended, and there are many possible questions we could ask, and answer, using the given table. For example, questions could be posed about second-class passengers. The task could lead to extended class discussions about the chances of events happening, and differences between unconditional and conditional probabilities. Special emphasis should be put on understanding what the sample space is for each question.

The Titanic Problem 1 http://www.illustrativemathematics.org/illustrations/949

Lesson Overview – Days 6-7 Resources

Description of Lesson: Students, as a group, will choose a probability problem to solve. Their final product will be a poster, a PowerPoint presentation, a report, or an alternative of their choice that requires teacher approval. • The Roulette Problem deals with expected value and using it to determine whether

games are fair or not. • The Birthday Problem deals with basic and conditional probability as well as

compound events. It has more of a conceptual requirement than a quantitative requirement.

• The Free Throw Contest deals with basic and conditional probability, as well as compound events. The solution involves an infinite geometric series. Connecting the probability to infinite geometric series will be the most difficult aspect of the problem.

You may assign problems to students or let students choose problems of their own. However, it is best to make sure that there are groups doing each problem. Students have two class periods to work on the project. During that time, you should circulate and offer students hints/guidance. Do not give away the problem.

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Milestone Task

Two-Spinner Sums and the Titanic



Math Objectives: ● Students demonstrate their probability knowledge by solving a

problem. CCSS-M Standards Addressed: S.CP.1, S.CP.2, S.CP.3, S.CP.4, S.CP.5, S.CP.7, S.MD.2 (+), S.MD.5 (+), S.MD.5a, S.MD.6, S.MD.7 Potential Misconceptions:

Launch: The constructed response questions (Two-Spinner Sums) and the performance assessment (The Titanic 3) for Part B of CLA 2. Students can do them both on the same day, but Part A of CLA 2 should be given on a different day. During: Students work on these tasks individually. Closure/Extension: You can review the tasks with the class after scoring them.

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Two-Spinner Sums and the Titanic

How will students do this?

Focus Standards for Mathematical Practice: 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 structure. 8. Look for and express regularity in repeated reasoning.


Vocabulary:

Differentiation Strategies: Participation Structures (group, partners, individual, other): Individual

Documents

SFUSD Mathematics Core Curriculum Development Project€¦ · Let’s Make a Deal – Simulation (2 pages) 1 per pair 1 per pair Cards 5 Lesson Series 2 CPM Algebra 2 Connections