AreYouFeelingLuckyProject.pdf

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UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science Partnership Grant Program NCLB Title II Part B Revised 8/15/12 1

Mathematics Senior Level Capstone Course Unit Overview

Title of Unit: Are you Feeling Lucky? Unit Designers:

Lisa Rozassa , Tiffany Comer, Kristin Vaughn, and Pam Walker Page County Schools

Laura Hansen Culpeper County Schools

Context: Students will investigate probability theory and apply this theory. They will create a math fair that consists of games based on probability. Number of Class

Hours: 10.5 hours

Unit

Design: ___Task Based

__x_Project Based

Other Subject Areas/Disciplines Addressed:

Art, English

Driving Question: How can we create, design, and play fair games based on theoretical probability?

Mathematics Content Addressed:

Theoretical probability, experimental probability, geometric probability, tree diagrams, combinations, fundamental counting

principal, fairness, simulations, ratios and percents, data collection, and display.

MPE Addressed:

Problem Solving Decision Making and Integration, Procedure and Calculation Assumption of Prior

Knowledge:

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College and Career Readiness/21st Century Skills to be taught (T) during this unit or expectation (E) for student use during this unit and assessed (A):

BIE Page 35-37

Collaboration E and A Research

Communication (Oral and/or Written) E and A Technology E

Critical Thinking/Decision Making T, E and A Other: (Describe)

Major Products and/or Performances:

Group: Each group will create two games that are based on probability for a math fair. Students will provide a written description of their game, including the rules of play and the theoretical probability involved.

Presentation Audience:

X Class

X School

Individual Expert

Community Other:

Launch: Event or experience used to engage the students interest and inquiry:

Video: http://blog.mrmeyer.com/?p=9553 This 1-minute video shows sprite and coke being poured in a glass and a dropper full taken from each and put in the other. Which glass has more of the original beverage.

Card Experiment: Students use 10 red and 10 black cards and run a simulation of mixing the two.

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(During the Unit) Mathematicians Journal Notes Preliminary

Plans/Outlines/Prototypes

Checklists

Rough Drafts X Concept maps

Field Tests Other:

Summative Assessment (End of Project)

Written Products, with a rubric X Peer Evaluation, with

a rubric

x

Oral Presentation with a rubric Self Evaluation, with

a rubric

x

Other Product(s) or

Performance(s), with a rubric

Other:

Resources Needed: On-site people, facilities:

Equipment/Technology: Graphing calculator or a computer with Excel. Computer with Internet access.

Materials: Coins, deck of cards, dice, and grid paper.

Community Resources:

Reflection Methods: Individual, Group, and/or Whole Class

Mathematicians Journal x Small/Focus Groups

Whole Class Discussions x Fishbowl Discussions

Survey Other:

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Template adapted from Buck Institute for Education: Project Based Learning for the 21st Century

U N I T C A L E N D A R

TITLE: Are You Feeling Lucky? Time Frame: 10.5 hours

M O N D A Y T U E S D A Y W E D N E S D A Y T H U R S D A Y F R I D A Y

P R O J E C T W E E K O N E

Notes: Calendar developed around 60 minute classes.

Introduce the unit with video and card trick simulation

Prepare students for working in groups to rotate through stations to play games to review important principles of probability

Students complete 4 rotations around 6 probability stations

Identified in Activity 1

Students complete last 2 rotations through 6 stations set up with games

Identified in Activity 1

Activity 2: Geometric Probability

Activity 3: The Birthday Paradox

Activity 4: The Monty Hall Problem

Prepare students for the project where they will be designing 2 games for a fair and composing a memo to the principal about the game, the mathematics in the game, and whether the game is fair.

P R O J E C T W E E K T W O Students working groups meet to

decide on the games they want to build for the fair.

Students brainstorm plan for constructing the games and what materials they will need

Students working groups identify the mathematics in the game and whether the game is fair. Students begin writing memo to principal.

Students develop plan for

constructing the games and who is responsible for each part

Student groups meet with teacher to review the memo.

Students work on games and complete the memo to principal.

Students complete their games and decide how they will facilitate their game at the class game day.

Class game day. Students will play and evaluate each others games.

(This may take more than 1 hour)

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Unit Title: Are You Feeling Lucky? Instructional Plan

Driving Question: How can we create, design, and play fair games based on theoretical probability?

Project: Students will create two fair games based on theoretical probability.

ENGAGE

Number of hours _1/2_

1. Show the 1-minute video, or the teacher can simulate the activity shown in the video inside the blog below.

http://blog.mrmeyer.com/?p=9553

After the video ask the question, Is there more Sprite in the Coke or more Coke in the Sprite? Have the student’s journal about this and make a choice. Then compare answers with the group.

2. Conduct the experiment described below with a partner.

1. Place 10 red cards and 10 black cards face down in separate piles.

2. Choose some cards at random from the red pile and mix them into the black pile.

3. Shuffle the mixed pile.

4. Return the same number of random cards (face down) from the mixed pile to the red pile.

5. Are there more red cards in the black pile or black cards in the red pile (count them)?

6. Repeat steps 1-5 several time choosing different numbers of cards.

7. As a class, discuss how this experiment helps answer the Coke/Sprite problem above.

Mathematician Journal

Prompts

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EXPLORE Teacher provides guidance for the explorations to prepare students with the knowledge and skills to engage in the task. Students will self-assess on the prior knowledge and skills assumed for the unit

Number of hours_3_

Activity 1 (HO#1a-f): Students will do a station activity where they explore and remediate probability and fairness. The teacher will circulate and help groups as needed.

For the stations to work smoothly our suggestion is to have circulating groups of three to four students. Students are to advance when they have completed the requirements for that station. For the stations that take longer than then other have two to three stations set up.

Stations HO#1 a-f: 1. Coin toss

2. Pick a Card (with and without replacement) 3. Paper, rock scissors

4. Rolling dice 5. How Many Ways? 6. Spinner

Answer Key for Stations 1-6 HO#1g

Activity 2: Geometric Probability:

At the amusement park, Tom’s favorite game is played by tossing a coin onto a large table that has been ruled into congruent squares. If a coin lands entirely within a square, he wins a prize. What is the probability that a randomly tossed coin will win given that the coin’s diameter is 2 cm. and the squares have sides of 5cm.

Have students simulate this and find the experimental probability.

Then have students calculate the theoretical probability. Encourage the students to draw a picture to help solve the problem and redirect as needed. (The theoretical geometric probability is the ratio of the area of the feasibility region over the area of the sample space or for this problem 9/25)

Activity 3: The Birthday Paradox. This activity and teacher instructions are described in the link below:

http://illuminations.nctm.org/LessonDetail.aspx?ID=L299

The problem involves finding the likelihood of two people in a random group having the same birthday. Random birthdates can be

Prompts

What is the difference between independent and dependent events? Decide which stations were

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generated on an excel spreadsheet using the ‘randbetween’ function or on a graphing calculator. The directions for the graphing

calculator can be found at the website above.

There is also a random birthday generator at the following website: http://www-stat.stanford.edu/~susan/surprise/Birthday.html

Activity 4: The Monty Hall Problem

Imagine that the set of Monty Hall’s game show Let’s Make a Deal has three closed doors. Behind one of these doors is a car; behind the other two are goats. The contestant does not know where the car is, but Monty Hall does. The contestant picks a door and Monty opens one of the remaining doors, one he knows doesn’t hide the car. If the contestant has already chosen the correct door, Monty is equally likely to open either of the two remaining doors. After Monty has shown a goat behind the door that he opens, the contestant is always given the option to switch doors. What is the probability of winning the car if he/she stays with his/her first choice? What if he/she decides to switch?

The following 4 minute clip from the television show Numbers

illustrate the mathematics behind the Monty Hall Problem.

http://www.youtube.com/watch?v=OBpEFqjkPO8&feature=related

The following websites might be helpful if a student needs more review of probability:

http://www.mathgoodies.com/lessons/toc_vol6.html

Fundamental counting principal:

http://www.basic-mathematics.com/fundamental-counting-principle.html

Combinations and Permutations

http://www.mathsisfun.com/combinatorics/combinations-permutations.html

Ratios and percents

http://www.aaamath.com/rat.htm

Geometric Probability

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EXPLAIN

Teacher introduces the main task of the unit and prepares students to in small group independent work...

Number of Hours_1/2_

Discuss going to a county fair and ask if students have ever played games. Discuss the games that they have played and how

successful they were in winning. Show one of the videos below. (Both videos describe how carnival games are often unfair. The first video is about 10 minutes and the second video is 2 minutes)

ABC 20/20 Carnival Game Expose (10 minutes) http://www.youtube.com/watch?v=PMSSYipFfg4

ABC News with Diane Sawyers-Carnival Scam (3 minutes) http://youtu.be/NIN6Zp6M5lk

After the video, have the students reflect on the games that they have played at fairs and carnivals in their journals. Discuss whether “fairness” was a problem in the games that they played.

The teacher will assign students to groups of three to four students.

The teacher will launch the project by sharing the following documents and providing the following instructions.

1. The final project is to create 2 games that will be used as part of a school-wide fair. In order to be accepted for the fair the students must build the game and have peers play the game. In addition, the principal who is a former high school math teacher wants to know if the game is fair.

2. Write a memo to the principal that includes:

 A written description of the games’ probability and the mathematics used to determine the probability.

 An explanation of whether the game is fair with supporting evidence.

HO#4 Are You Feeling Lucky?

Project Criteria and Summative Assessment (End of Project) Go over the guidelines for the project and to provide students with the criteria for an exemplary project.

Prompts

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HO#2 and HO#3: Share with students and let them know that these will be used for peer evaluation

ELABORATE The student groups are working independently with teacher consultations. Number of Hours_5__

Students develop their games of chance, calculate the probabilities, and test for fairness. Students then write their memo to the

principal.

Students review the rough draft of the memo to the principal with the teacher, before completion of the project. Students should refer to the rubrics listed below to be sure they know what is expected of them.

Prompts

What is

necessary for a game to be appealing and interesting for fair participants? EVALUATE Working groups submit products or make presentations Number of Hours_1.5_

Students play the games developed by other groups and rate the games using the Peer Game Evaluation Rubric. HO#2

Students evaluate themselves and their group members using the Peer/Self Teamwork Evaluation Form. HO#3

Teacher grades the project using the Summative Assessment. HO#4

As an extension activity the games could be used as a math fair during lunch shifts or at a local elementary school. Students would play the games and then rate them.

Prompts

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Map the Unit

What do students need to know and be able to do to complete the task/project/problem successfully? How and when will they assess their own necessary knowledge and skills? How will they remediate their own gaps or weaknesses in knowledge and skills? Look at each major task for the unit and analyze the tasks necessary to produce a high-quality product.

Task: Students will create two fair games based on theoretical probability.

KNOWLEDGE AND SKILLS NEEDED

Assumed already learned

Students will self-assess

Will be taught during the unit

1. Theoretical Probability x x

2. Experimental Probability x x

3. Geometric Probability x

4. Fundamental Counting Principal x x

5. Combinations x x

6. Ratios and Percents x x

7. Data collection x

8.

What project tools will student’s use?

 Know/need to know lists

 Daily goal sheet

X Mathematician’s Journals

 Briefs/Memos

 Task lists

 Planning Calendar

□ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________

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HO #1a

Student Response Sheet Name__________________

Station 1 - Coin Toss

Toss a coin 20 times. Use tally marks to record your results after each toss.

Toss # Heads Tails

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1. Find the ratio of the number of times it landed on heads to the number of tosses. Example: # of heads

Total # tosses ___________

2. Find the ratio of the number of times it landed on tails to the number of tosses.___________

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HO#1b

Station 2- Pick a Card

Before you pick a card, answer the following questions given that there are 13 hearts and 4 aces in a 52 card deck.

1. What is the likelihood that the card you picked is a heart?_______ Why?

2. What is the likelihood that the card you picked is an ace? _______ Why?

3. What is the likelihood that the card you picked is an ace of hearts? _______ Why?

4. a. Go ahead, pick your card. Write down what type of card it is. __________________

b. Now replace your card in the deck. What is the likelihood that on your second pick, you will get the same card? ______

5. Suppose you draw two cards from this deck, what is the probability that both cards are aces if you don’t replace the first card before drawing the second? _______

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HO#1c

Station 3 – Rock, Paper Scissors

Play this game with a group of four people

Play the rock, paper, scissors game with three players and one recorder. All players make a fist and on the count of four, each player shows either:

 Rock (by showing a fist)

 Paper (by showing an open hand)

 Scissors (by showing two fingers)

Decide who is player A, B and C and play 20 times with the following rules:

 Player A gets the point if all players show the same sign.

 Player B gets the point if only two players show the same sign

 Player C gets the point, if all players show different signs.

Tally the winning points: (only one player gets a point on each of the 20 times..

Player Tally Total

A B C

A fair game is a game where all outcomes have a reasonable chance of occuring.

1. Is this game fair? _____________

2. Which player would you rather be? Explain your reasoning.

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HO#1d

Student Response Sheet Name__________________

Station 4 – Rolling Dice Game

Play this game with a partner

1. Take turns rolling the dice for at least 10 trials and find the sum for on each roll. .

a. Player A scores a point if the sum is even. Player B scores a point if the sum is odd.

Number of times Player A wins with an even sum_____ Number of times Player B wins with an odd sum _____

b. Is the game fair? If not, how could you make the game fair? Explain your reasoning.

2. Play the game again, this time calculating the product. Take turns rolling the dice for at least 10 trials.

a. Player A scores a point if the product is even. Player B scores a point if the product is odd.

Number of times Player A wins with an even product_____ Number of times Player B wins with an odd product _____

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HO#1e

Station 5 - How Many Ways?

Consider a chance experiment where Harry and Sally are trying to decide what to have for dinner at the Oh So Good Diner. The limited menu includes three entrées: hamburger, pizza, and chef’s salad and two desserts: apple pie and carrot cake. Although you don’t know what their selections will be, can you describe the complete set of possible outcomes?

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HO#1f

Station 6 – Spinner

Go to the website http://www.shodor.org/interactivate/activities/AdjustableSpinner/

EXPLORATION QUESTIONS : Note your responses to each question.

1. Pick one color and try to make it more likely that you will spin that color. Test the spinner 20 times. Did you really spin that color more often?

2. Create a game that is fair for 5 different colors. What fraction is each color?

3. Create a game that is fair for 4 different colors. What fraction is each color? Do you notice a pattern?

4. What percentage is each color on the spinner in question 3? Why is that?

5. Describe how can you ensure that a spinner is fair no matter how many colors or sections it contains?

6. Explain why the theoretical probability is often different from the experimental probability?

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Handout 1g

Probability Stations (1-6) Answer Key

Station 1 Coin Toss 1. Answers will vary 2. Answers will vary

3. The outcomes probably will not occur equally, but with many repetitions the ratio will come close to ½.

Station 2 Pick a Card 1. 13/52

2. 1/13 3. 1/52

4. The probability that both cards are Aces if you don’t replace the 1st card will be; (4/52)(3/51) or 12/2652

Station 3 Rock, Paper, and Scissors Player A- the probability of a win is 3/27 Player B- the probability of a win is 18/27 Player C- the probability of a win is 6/27

The game is not fair. You would want to be player B.

Station 4 Rolling Dice

1. The probability of an even or odd sum is ½. The game is fair.

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Station 5 How Many Ways

There are a total of (3·2·1) =6 possible combinations

Station 6 Spinners 1. Varies 2. 1/5

3. ¼

4. 25%, the entire spinner represents 1 whole

5. Each possible choice has to represent an equivalent part or percent of the whole which is the spinner

6. Theoretical probability is calculated based on the chance of an event happening under ideal conditions.

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HO #2 Game: Peer Evaluation Rubric

Game Name: ______________________________

Directions: Play the game and rate using the rubric below. Circle the statement and number that describes the game that you played best.

4 3 2 1

Visual Appeal The game is highly organized,

neat, and appealing.

4

The game is neat, organized, and

appealing.

3

The game is somewhat organized, neat,

and appealing.

2

The game is not organized, neat and appealing.

1

Entertainment The game is highly entertaining to

play.

4

The game is entertaining to

play.

3

The game is somewhat entertaining to

play.

2

The game is not entertaining to

play.

1

Overall Game Appeal

The game is highly appealing.

4

The game is appealing.

3

The game is somewhat appealing.

2

The game is not appealing.

1

Fairness of the game

The game is very fair

4

The game is somewhat fair

3

The game is not very fair

2

The game is highly unfair

1

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HO #3

Teamworker: Peer/Self Evaluation Form

The following is a list of statements to be answered by you and about each of your group members. Think carefully about assigning rating values for each of the statements.

4- Strongly Agree 3- Agree 2- Disagree 1- Strongly Disagree

Self: Teammate: Teammate: Teammate: Was dependable in

attending class

Willing to accept assigned tasks

Contributed positively to group discussion

Completed work on time or made alternative arrangements

Helped others with their work when needed

Did work accurately and completely

Worked well with other group members

Overall was a valuable member of the team

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HO#4

Are You Feeling Lucky?

Project Criteria and Summative Assessment (End of Project)

Teacher Name: _______________ _______________

Student Name: ________________________________________

CATEGORY 4 3 2 1

Thoroughness of game directions

Directions are clear, complete and easy to understand.

Directions are mostly clear, complete and easy to understand.

Directions are somewhat clear, complete and easy to understand.

Directions are unclear, hard to understand, incomplete or missing. Mathematical

correctness

The description of the probability was mathematically correct and thorough.

The description of the probability was mostly

mathematically correct and thorough.

The description of the probability was somewhat

mathematically correct and thorough.

Little of the description of the probability was mathematically correct or it was incomplete. Mathematical

Explanation

Explanation is detailed and clear to justify their conclusion mathematically.

Explanation is clear to justify their conclusion

mathematically.

Explanation is a little difficult to understand, but includes critical components to justify conclusion. Explanation is difficult to understand and is missing several components OR was not included.

Game Appeal The presentation of

the game highly motivates people to want to play. It is neat, organized, and well made.

The presentation of the game motivates people to want to play. It is mostly neat, organized and well made.

The presentation of the game

somewhat

motivates people to want to play. It is somewhat neat, organized and well made.

The presentation of the game does not motivate people to want to play. It is not neat, organized and well made.

Game: Peer Evaluation

Average peer score is 3.1- 4.

Average peer score is 2.1 - 3.

Average peer score is 1.1-2.

Average peer score is 0-1.

Team member: Peer Evaluation

Average score is 3.1- 4.

Average score is 2.1 - 3.

Average score is 1.1-2.

Average score is 0-1.