Probability+LPU

Probability G.1.1. Students will understand how probability works. G.1.2. Students will be able to explain why certain events are more possible than others. G.1.3. Students will understand the law of large numbers. G.1.4. Students will gain a relational understanding of probaility.
 * UNIT TITLE: ** Probability
 * LESSON TITLE: ** Probable Cause
 * APPROPRIATE GRADE LEVEL:** 7th
 * ENDURING UNDERSTANDING: ** When determining the probability of an event, what needs to be discovered about the event?
 * EXPECTATIONS & GOALS OF THE LESSON: **

O.1.1. Students will know how to compute probability. O.1.2. Students will be able to make accurate predictions about outcomes. O.1.3. Students will be able to identify how the amount of trials affects probability. O.1.4. Students will know the difference between experimental and theoretical probability.
 * MATHEMATICAL OBJECTIVES OF THE LESSON: **

Dice: Students will roll dice and record the observations to see the concept of an individual event. Roll Sheet: Students will record the outcomes of two different die on one sheet to see the conepts of an indivudal event. "Horse Race" Activity Sheets: Students will record outcomes of rolling one, then two die to see the outcomes of a possible sample space.Pennies: Students will flip coins and combine results to view how the "law of large numbers" works.
 * MULTIPLE REPRESENTATIONS (TOOLS): **


Students seem to understand probablitly when it is more closely related to their lives. Even young children can tell you if it is likley to snow in the middle of the summer. Probability starts to get increasingly harder as students are asked to predict and explore outcomes that are unknown to them in daily life. In order to help students with this concept, Van De Wall says that students can learn by making activities relate to their lives. For example, students can determine the likelihood of an event, flip a coin, or discover if a game is fair. By conducting experiments to understand probability, students are able to physically confirm their results as well as create some background knowledge for higher levels of probability learning (Van de Walle, 2007).
 * RESEARCH & BEST PRACTICES: **

According to Hollylynne S. Lee and Gemma Mojica, experiments and simulations are beneficial for teaching probability. In experiments, students can analyze random phenomena of interest by performing repeated trials and observing the outcomes of these trials. For example, a student can conduct an experiment by rolling two dice multiply times and observing the outcomes of the trials. A simulation is a special type of experiment in which a random generating device, physical or computer-based, models random phenomena and performs repeated trials. For example, two spinners could be designed and used to simulate the probability context of the original experiment of rolling two dice (Lee & Mojica, 2007).

Many phenomena that students encounter in school have predictable outcomes such as rolling a dice or flipping a coin. By repeating a random event many times, the distribution of outcomes forms a pattern. An important concept, which serves as a foundation for studying inferential statistics, is the idea that individual events are not predictable, but that a pattern of outcomes can be predicted. In order to reason statistically and aquire skills needed to be informed citizens and intelligent consumers, students must know about data analysis and related aspects of probability (NCTM, 2004).

7.S.2.1. (Comprehension) Given a sample space, find the probability of a specific outcome.
 * SD K-12 CONTENT STANDARDS: **

6-8 Data Analysis and Probability, Goal 4, Exp. 1: understand and use appropriate terminology to describe complementary and mutually exclusive events 6-8 Data Analysis and Probability, Goal 4, Exp. 2: use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations; 6-8 Data Analysis and Probability, Goal 4, Exp. 3: compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and array models.
 * NCTM EXPECTATIONS: **


 * RATIONALE AND JUSTIFICATION OF STANDARDS BEING MET:** As a result of our activities, students will meet these state and national standards. Students will directly experience probability through hands-on experiences with manipulatives.They will make and test out their own predictions about probability before finding experimental probability. In the dice activities, students will be given a sample space and challenged to find the probability of the outcome of an event. They will use organized lists to help them compute probability. By comparing complementary and mutually exclusive events with sheets of paper, the students will come to understand the distinct differences between the two. In addition, they will come to understand how probability plays a role in many aspects of their every day life through the continuum and class discussion.

Note: Students do not meet the expectation of using tree diagrams and array models to compute probabilites.
 * text could not be crossed out without being deleted

Why are some events more probable than others? Why is my experimental probability very different from my prediction? Do results come closer to theoretical probability as more trials are combined? Will the outcome of the first dice have an affect on the roll of the second dice? When does one event affect another? When does one event not affect another? When will I use probability to help me?
 * SESSION-RELATED QUESTIONS: **

Formative Observation: Observe whether students can identify events of probability and predict their probability on the continuum. Formative Observation: Ask quesions and listen whether students gain an understanding of the law of large numbers through the penny flipping activity. Formative Assessment: View the actions of students in dice activities. Do they just guess a number or do they think critically about the outcomes?
 * IMBEDDED/FORMATIVE ASSESSMENT OPPORTUNITIES: **

White Board Markers Dice Continuum Chart (See Appendix C below)
 * INSTRUCTOR MATERIALS: **

Horse Race "Activity Sheet" for one dice (See Appendix D below) Horse Race "Activity Sheet" for two die (See Appendix D below) "Roll Sheet" (See Appendix D below) Pennies Dice
 * PARTICIPANT MATERIALS: **


 * ACCOMODATIONS:**
 * To assist students with special needs, dice with numbers written instead of dots will be provided.
 * To assist visual learners, continuums, interactive activities, and other content will be displayed on the Smart Board.
 * To assist ELL students, they will be paired with a partner on activities.

The lesson will begin with the idea of a continuum from impossible to certain. Students will organize the likelihood of events on a continuum and determine where to put their own real-life examples on the continuum. After putting the real life experiences in order students will use a spinner to observe the probability of spinning a color in race to the top. The lesson will then take a shift towards explaining the "law of large numbers" in an indirect way. Students will predict and test the probability of flipping a coin. They will combine, compare, and analyze results in small groups and then as a class. Finally, students will further explore probability through rolling dice activitites. Using one and two dice, the students will be challenged to explore possible outcomes for different events.
 * TEACHING NOTES / LESSON SCRIPT: **

Summative Assessment Problem - Appendix E: Include the chart you created during the horse race activity. Explain the results you got from the activity and why. What do you now understand about probability from this activity? Explain three ways you can use probability in your life and provide an example probability problem for each. How do you think the results of rolling a dice 10 times would compare with the results of rolling a dice 100 times? What questions do you still have about probability? REFERENCES: ** -Joseph, L. (2000). //Let's make a deal: The study of probability//. Multimedia Schools, 7 (2): 31. -Honeycut, B, & Pierce, B. (2007). Illustrating probability in genetics with hands-on learning: making the math real. //American Biology Teacher//, 69(9), 544-551. -Lee, H, & Mojica, G. (2007). Teachers' use of experiments and simulations in middle school probability lessons. //Conference papers: Psychology of mathematics and education in// // north america // , // -NCTM Principles and Standards of School Mathematics // (2004). Retrieved November 29, 2009, from [] -Van de Walle, J.A. (2007). //Elementary and middle school mathematics: Teaching developmentally.// Boston, MA: Pearson Education, Inc.
 * ASSESSMENT / HOMEWORK: **

List of Appendices: Appendix A: Lesson Script (long version) Appendix B: Research Articles Informing the Lesson Appendix C: Instructor Materials Appendix D: Participant Materials Appendix E: Assignment/Homework
 * APPENDICES:**


 * Appendix A: Lesson Script (long version) **

Continuum

Show the students a continuum from impossible to certain. Have students organize the probability of different events. Ask the students to come up with some real-life events and determine where they would go on the continuum. Show the students a spinner that is half purple and half yellow. Ask the students to predict what they think a continuum of spinning a purple might look like. Have the students organize spinners to create the continuum for spinning a purple.

Spinners

Students will be able to create a spinner that is one fourth one color and three fourths of another color. Students will then spin the spinner in teams to race colors to the top of a chart. The students can predict which color they think will make it to the top first, and then observe the results. As the student's understanding progresses, students can change the spinners to add more colors and see how the race changes as the amount of color or number of colors on the spinner changes.

Penny Flipping Activity

Inform the students that they are going to flip a penny ten times and record the number of heads and tails they get. Ask the students what they predict might happen and what the probability of flipping a coin is. Inform the students that this is what is called the theoretical probability, or what should occur. Tell students that they are going to find the experimental probability, or what actually occurs from an experiment. Allow the students to individually flip their coin ten times. Have students compare their results with a partner. Ask students to combine their results in a small group and disuss their observations. Finally, have the entire class combine their results. Discuss the results as a class. Ask the students to give their ideas on why these results may have occurred.

Dice Rolling Activity (Horse Racing)

Start off the second activity by giving each group of students a pair of dice and a worksheet for the "Horse Race" activity. Explain to them that the “Horse Race” activity sheet is a called a sample space. This shows students every possible event outcome that is possible.

Horse Race Activity Sheet for One Dice
 * Horse || First Lap || Second Lap || Third Lap || Fourth Lap || Fifth Lap || Sixth Lap ||
 * 1 ||  ||   ||   ||   ||   ||   ||
 * 2 ||  ||   ||   ||   ||   ||   ||
 * 3 ||  ||   ||   ||   ||   ||   ||
 * 4 ||  ||   ||   ||   ||   ||   ||
 * 5 ||  ||   ||   ||   ||   ||   ||
 * 6 ||  ||   ||   ||   ||   ||   ||

Have each student pick a number on the sheet for his or her "horse" and then give the dice a roll. When the dice has stopped rolling, have the students mark down which number was facing up on their "Horse Race" activity sheet. Keep rolling the dice until one horse has finished the sheet. Have the students notice that what they are keeping a record of is called a sample. The individual X’s that they placed on the sheet is called an event (something that has occurred) and is a subset of the sample space mentioned previously.

When the students have finished, have them discuss in their group any findings they have. Then have the students share their information with the rest of the class. Discuss which “horses” won each group and have them explain why different horses won. After the students have shared their guesses and their reasoning, if they have not answered correctly, explain to them that each horse the same chance in winning and therefore multiple horses won.

Then give them the second “Horse Race” activity that will be used with 2 dice and the “roll sheet” that will allow them to write down what each die result was. Horse Race Activity for Two Die


 * Horse || First Lap || Second Lap || Third Lap || Fourth Lap || Fifth Lap || Sixth Lap ||
 * 2 ||  ||   ||   ||   ||   ||   ||
 * 3 ||  ||   ||   ||   ||   ||   ||
 * 4 ||  ||   ||   ||   ||   ||   ||
 * 5 ||  ||   ||   ||   ||   ||   ||
 * 6 ||  ||   ||   ||   ||   ||   ||
 * 7 ||  ||   ||   ||   ||   ||   ||
 * 8 ||  ||   ||   ||   ||   ||   ||
 * 9 ||  ||   ||   ||   ||   ||   ||
 * 10 ||  ||   ||   ||   ||   ||   ||
 * 11 ||  ||   ||   ||   ||   ||   ||
 * 12 ||  ||   ||   ||   ||   ||   ||

Roll Sheet

Have the students pick a horse again and then have them roll two dices. Have two different students roll each die and then have them record on the “roll sheet” the result of each die. Once they have written down the result of each individual die, have them add the two numbers together and then place an X in the correct number and lap on the “Horse Race” activity sheet.
 * |||||||||||||| ** Die #1 ** ||
 * ** Die #2 ** || || ** 1 ** || ** 2 ** || ** 3 ** || ** 4 ** || ** 5 ** || ** 6 ** ||
 * ^  || ** 1 ** || || || || || || ||
 * ^  || ** 2 ** || || || || || || ||
 * ^  || ** 3 ** || || || || || || ||
 * ^  || ** 4 ** || || || || || || ||
 * ^  || ** 5 ** || || || || || ||  ||
 * ^  || ** 6 ** || || || || || || ||

Once they have finished the race, have them continue rolling die and recording the individual results. Ask each group how many of each total number they had in accordance with the “Horse Race” activity sheet. Add it on the white board to illustrate how many of each number the whole class achieved. Allow them to work together in a small group to come up with some findings about the horse race activity and the information on the whiteboard. Then, have the students share their information with the rest of the class.

If students have not noticed, ask them why some numbers appeared more often than others? The explanation is that there are more outcomes to produce a total number of 7 compared to the total numbers of 2. The dice have three different combinations to obtain a 7 (1,6; 2,5; 3,4) and the dice only have one combination for a total of 2 (1,1). Because the events are independent, there are actually SIX possible combinations to obtain a 7 (1,6; 2,5; 3,4; 4,3; 5,2; 6,1) as well as a total of 36 possible boxes in accordance with the roll sheet. Therefore, to find the probability of rolling a total of 7 would be 6/36 or 1/6 and not 3/36 or 1/12.

Also, if the students have not noticed, explain to them the events of two dice rolls are independent events. This means that no matter what the first dice rolled was, the fist roll would not affect the roll of the second dice.

Box O’ Numbers Teachers need to give students numerous opportunities to engage in probabilistic thinking about simple situations from which students can develop notions of chance. They should use appropriate terminology in their discussions of chance and use probability to make predictions and test conjectures. Students will be presented with the following problem: Suppose you have a box containing 100 slips of paper numbered from 1 through 100. If you select one slip of paper at random, what is the probability that the number is a multiple of 5? A multiple of 8? Is not a multiple of 5? Is a multiple of both 5 and 8? Students will figure the probability of each of these to compare the results. Students should be able to use basic notions of chance and some basic knowledge of number theory to determine the likelihood of selecting a number that is a multiple of 5 and the likelihood of not selecting a multiple of 5. In order to facilitate discussion, help students learn commonly accepted terminology. Students should know that "selecting a multiple of 5" and "selecting a number that is not a multiple of 5" are //complementary // events and that because 40 is in the set of possible outcomes for both "selecting a multiple of 5" and "selecting a multiple of 8," they are not //mutually exclusive // events.

Monty Hall Problem Students will be presented with 3 doors; only one of the three has a prize behind it. After a student chooses a door, the host will show them the one of the doors that does not have the prize behind it. After doing so, ask them if they think that the probability of winning would change if they switched the door that they originally chose. It is quite common for students to think that switching will make no difference on the probability of picking the right door, when in all actuality it does. By eliminating a losing door, the contestant is left with a probability of 2/3 and since there is only one door to choose from, the probability is raised to 2/3 chance of winning by switching.

Joseph, L. (2000). //Let's make a deal: The study of probability//. Multimedia Schools, 7 (2): 31. Retrieved December 1, 2009, from []
 * Appendix B: Research Articles Informing the Lesson**

Honeycut, B, & Pierce, B. (2007). Illustrating probability in genetics with hands-on learning: making the math real. //American Biology Teacher//, 69(9), 544-551.Retrieved December 2, 2009, from [|**http://web.ebscohost.com.ezproxy.usd.edu/ehost/pdf?vid=6&hid=7&sid=2578e535-38c0-4d91-aba8-b8cf46272aa7%40sessionmgr14**]

Lee, H, & Mojica, G. (2007). //Teachers' use of experiments and simulations in middle school probability lessons//. Conference papers: Psychology of mathematics and education in north america: 1-184. Retrieved November 28, 2009, from []


 * Appendix C: Instructor Materials**

Smart Board Presentation: [|Math LPU.notebook]


 * Appendix D: Participant Materials**

Horse Race Activity Sheet for One Dice

Horse Race Activity for Two Die
 * Horse || First Lap || Second Lap || Third Lap || Fourth Lap || Fifth Lap || Sixth Lap ||
 * 1 ||  ||   ||   ||   ||   ||   ||
 * 2 ||  ||   ||   ||   ||   ||   ||
 * 3 ||  ||   ||   ||   ||   ||   ||
 * 4 ||  ||   ||   ||   ||   ||   ||
 * 5 ||  ||   ||   ||   ||   ||   ||
 * 6 ||  ||   ||   ||   ||   ||   ||

Roll Sheet
 * Horse || First Lap || Second Lap || Third Lap || Fourth Lap || Fifth Lap || Sixth Lap ||
 * 2 ||  ||   ||   ||   ||   ||   ||
 * 3 ||  ||   ||   ||   ||   ||   ||
 * 4 ||  ||   ||   ||   ||   ||   ||
 * 5 ||  ||   ||   ||   ||   ||   ||
 * 6 ||  ||   ||   ||   ||   ||   ||
 * 7 ||  ||   ||   ||   ||   ||   ||
 * 8 ||  ||   ||   ||   ||   ||   ||
 * 9 ||  ||   ||   ||   ||   ||   ||
 * 10 ||  ||   ||   ||   ||   ||   ||
 * 11 ||  ||   ||   ||   ||   ||   ||
 * 12 ||  ||   ||   ||   ||   ||   ||


 * |||||||||||||| ** Die #1 ** ||
 * ** Die #2 ** || || ** 1 ** || ** 2 ** || ** 3 ** || ** 4 ** || ** 5 ** || ** 6 ** ||
 * ^  || ** 1 ** || || || || || || ||
 * ^  || ** 2 ** || || || || || || ||
 * ^  || ** 3 ** || || || || || || ||
 * ^  || ** 4 ** || || || || || || ||
 * ^  || ** 5 ** || || || || || || ||
 * ^  || ** 6 ** || || || || || || ||

Answer the following questions on the bottom or back of your horse race chart. 1. Explain the results of your horse race activity and why you got those results. 2. What did you learn about probability by doing this activity and why? 3. Explain three ways you can use probability in your life and provide an example probability problem for each. 4. How do you think the results of rolling a dice 10 times would compare with the results of rolling a dice 100 times? 5. What questions do you still have about probability?
 * Appendix E: Assignment/Homework**