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Statistics and Probability for Middle-School Math Teachers: Addressing the Common Core Teacher Notes Pig Dice and Probability Logistics This lesson is intended for students in Grade 7 as an introductory lesson to the concepts of theoretical and empirical probabilities. Students will be introduced to the idea of computing probabilities of a chance process through observing frequencies of a desired event. They will also explore the idea of a subjective probability when a theoretical probability cannot be readily calculated. Materials: 1 – set of dot dice per four students (or number cubes) 1 – set of pig dice per four students Time: This lesson should be completed in two 45 – minute class periods. Objectives/Standards The objectives of this lesson are to: Develop a probability model for a fair die and use it to find probabilities of events related to rolling a fair die. Then, compare the model to the results of an experiment and explain the possible sources of the discrepancy CCSS.Math.Content.7.SP.7 References to Common Core are adapted from NGA Center/CCSSO © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. CCSS.Math.Practice.2 CCSS.Math.Practice.4 Approximate the probability of a chance event of rolling a pig die and obtaining a specific result by observing frequencies in the data. Also, explore the idea of probability as long-run relative frequencies. CCSS.Math.Content.7.SP.6 CCSS.Math.Content.7.SP.7 CCSS.Math.Practice.5 Summarize and describe the distributions of rolls of the standard die and pig die. CCSS.Math.Content.6.SP.4 CCSS.Math.Content.6.SP.5 CCSS.Math.Practice.6 Introduction For this inquiry, students will first explore the features of a standard, sixsided die and develop a uniform probability model to determine the probability of a specific value of the die appearing on a roll. They will then collect data to test their model and will examine the reasons why their observed frequencies may not match the theoretical model they initially proposed. Then, students will attempt to determine a theoretical model for a die in the shape of a pig. Due to the unusual shape of the pig dice, it is very difficult to determine the theoretical probability of any particular pig position. In fact, according to John C. Kern of Duquesne University, exact probabilities are unknown. Thus, students will reason that long-run 1 Statistics and Probability for Middle School Teachers: Addressing the Common Core Professional Learning Day; February 28, 2014 1 Statistics and Probability for Middle-School Math Teachers: Addressing the Common Core Teacher Notes relative frequencies must be used for this particular die to determine an approximation of the probability for a specific pig position to occur. Students should be grouped into teams of four students arranged in a square/rectangular pattern and then divided further into partner teams on the same side of the table (shoulder partners). For the first two activities in this lesson, students will work with their partner teams. Activity 1: Investigating a standard, six-sided die Is a standard, six-sided die “fair”? For this activity, students will: Examine the shape of the die to determine the “sample space” or list of possible outcomes if the die were rolled. Use the idea of the symmetric shape of the die to determine the probability of a specific value occurring on any given roll. Determine the theoretical probabilities for each possible outcome in the sample space. Develop a data collection strategy to test their theoretical model. Investigate their empirical data to determine if the data does/does not support their theoretical model and infer reasons why this occurred. Students may suggest changes to their models to reflect their changing ideas about long-run relative frequencies. Students will first examine the shape of a standard die and describe their observations. Sample student responses could be as follows: The die is a cube. There are six equal sides that are squares. There are values of 1, 2, 3, 4, 5, and 6 on the die (or dots if using dot dice). The die is a uniform weight – one side is not heavier than the other side. From these observations, students should then roll the die a few times to see the possible outcomes of each roll. From there, they can list the sample space of a single, six-sided die as 1, 2, 3, 4, 5 and 6. Assuming that every roll is independent of the previous roll, students can determine the theoretical probability of obtaining a specific value on a roll to be the uniform distribution of 1 desired outcome/ 6 possible outcomes. Upon completing the table in the student pages, they should conclude that the likelihood for any number being rolled is the same (uniform). Thus, they can create a table of theoretical probabilities as follows: 2 Statistics and Probability for Middle School Teachers: Addressing the Common Core Professional Learning Day; February 28, 2014 Statistics and Probability for Middle-School Math Teachers: Addressing the Common Core Teacher Notes Theoretical Probabilities of a roll of a Six-sided die Outcome 1 2 3 4 5 6 Probability 1 6 1 6 1 6 1 6 1 6 1 6 Now, student teams will collect data on 25 trials of rolling the die and recording the outcome of the roll. From these data, students will then calculate their experimental probabilities using the following formula: 𝑃(𝐸𝑣𝑒𝑛𝑡 𝐴) = # 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑤𝑎𝑠 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠 Following the data collection activity, students will be asked to compare their experimental data table to the theoretical table they generated and discuss the similarities/differences in the data. Ask students to share their observations. They will also share their data with other groups to observe the variability in the recorded probabilities. From there, groups will develop a hypothesis on why they believe these differences are occurring and will design an experiment that they believe will give them data that more closely approximates the theoretical values that were calculated using the uniform probability distribution. Whole class discussion will focus on the sharing of student observations and conjectures. The following questions may be used to generate conversation: Did the experimental data you recorded exactly match the uniform distribution of 1/6 for each outcome on the first table? What reason(s) might there be for the differences you found, if indeed you found any? Was your experimental table exactly the same as other groups in the class? What could explain the variability from group to group? Do you think that your die is a fair die? Why or why not? What would it take for you to be convinced that your die was fair? How could you design an experiment to get closer to the actual theoretical frequencies of 1/6 for each value on the die? Following discussion, students may redesign their plans and then carry out their revised plans focusing on increasing the number of trials. They will then compare their new experimental data again to the theoretical data and their first set of data to determine what effect increasing the number of trials has on the data. This leads to a definition of probability as a long-run relative frequency. 3 Statistics and Probability for Middle School Teachers: Addressing the Common Core Professional Learning Day; February 28, 2014 3 Statistics and Probability for Middle-School Math Teachers: Addressing the Common Core Teacher Notes Activity 2: Investigating pig dice What is the probability of landing in a specific pig position with a single toss of the pig die? In the next part of this lesson, students will look at a die in the shape of a pig and will investigate the possible outcomes when this die is tossed. Students should rationalize that the geometry of the pig die makes it difficult, if not impossible, to theoretically calculate the likelihood of a particular pig position from occurring. They would need to generate subjective probabilities about their personal knowledge of what they believe is most likely to occur. For example, the pig may land on its side more often since this is the “largest” part of the pig die. Likewise, the pig landing on its back may occur more often than the pig resting on its ear. Following the completion of Page 6 of the student pages, pose the following questions to the class: What pig positions are possible with a single pig die? If there are six different pig positions, do you believe they are equally likely to occur as is the case with the standard die? Why or why not? What would you predict the probabilities of the different pig positions to be? What specifically about the pig die would lead you to your predictions? How could we attempt to determine the likelihood of a specific pig position? What things might you have to consider? Following the debrief on the initial section of this activity, students will formulate a plan that would allow them to calculate the experimental probabilities for each pig position. Allow partner teams time to formulate their plans, and then share their ideas. The class will decide, with the guidance of the teacher, a particular plan or combination of individual plans to follow. For the initial data collection, guide students to choose a plan in which the number of pig tosses is at least 25; data is recorded in a table listing the pig position that occurred at each toss or trial. Students may also take turns within their partner teams. This will generate additional data sets to use in the next part of the activity. Following the data collection phase, each partner team will calculate the relative frequencies, or experimental probabilities, for each pig position by using the following formula: 𝑃(𝑃𝑖𝑔 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛) = # 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑤𝑎𝑠 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠 4 Statistics and Probability for Middle School Teachers: Addressing the Common Core Professional Learning Day; February 28, 2014 Statistics and Probability for Middle-School Math Teachers: Addressing the Common Core Teacher Notes Students can adapt the formula from Activity 1 to calculate the experimental probabilities for the pig positions. Students should realize that they have a relatively small number of trials, so the confidence in their calculated probabilities should be quite low. One way to increase the number of trials without tossing the pig die more times, which may be impractical given time constraints, is to consolidate the class data. Student partner teams are now asked to gather class data for a particular pig position which they are assigned. There are six positions, so the pig positions may be analyzed by more than one group. This is desirable since students are not instructed on what type of graphical display to use. The most likely displays will be either a dot plot or a boxplot to represent the frequency of the assigned pig position. Frequency of Razorbacks Tossed (per 25 tosses) 3 6 9 12 Razorback (per 25 tosses) 15 Variable Mean Median Razorback (per 25 tosses) 7.800 7.500 Created using Minitab software Number of Razorbacks The above display shows an example of a dot plot depicting the frequency of razorbacks observed per 25 tosses of the pig die. Students could then revise their probability of a razorback to 7.8/25 or .312. Alternatively, they could add up the total number of razorbacks observed (234) and divide by the total number of tosses (750). **Please note that revised probabilities in the table should sum to 1!** The enduring understanding that students should take away from this part of the lesson is that not every outcome can be assigned a theoretical probability, and often the best we can do is an empirical probability that depends upon the number of trials. Pig dice images from: http://www.softcom.net/users/mikey719/pass_the_pigs/PassThePigs.html Activity 3: Putting it all together For this final part of the lesson, students will again be working with their partners in order to summarize the concepts that were explored in the first two activities. Initially they will work with their partner to check the box which indicates if they believe the statement is true for the standard die, 5 Statistics and Probability for Middle School Teachers: Addressing the Common Core Professional Learning Day; February 28, 2014 5 Statistics and Probability for Middle-School Math Teachers: Addressing the Common Core Teacher Notes the pig die, both, or neither. Then, student teams may compare their answers with the other team at their table and discuss the items where they do not have consensus. Following this discussion, students will write a final summary of the main ideas they learned in this lesson. These ideas will be debriefed in a whole-class discussion. The following will serve as a guideline for facilitating discussion on this part of the activity: The sum of the probabilities for all possible outcomes in a sample space is 1. Both theoretical probability and empirical probability are ratios. A uniform probability distribution has all outcomes equally likely. A chance process that is uniform is considered “fair”. Since theoretical probability is based on a calculation of what is expected to occur given all possible outcomes, it does not always accurately predict what does occur in a given trial or series of trials. Experimental probability is an approximation of a theoretical probability and tends to approach the theoretical probability after a large number of trials. In cases where the theoretical probability cannot be calculated, we need to rely on experimental probability. Conclusion This lesson is intended to allow students the time and opportunity to explore the concepts of theoretical and empirical probability. Furthermore, students will investigate the idea of fairness as equally likely and will look at the effect of long-run relative frequencies as an approximation of the probability of a chance event. Using a guided inquiry model, students are asked to explain their reasoning, justify their thoughts, and predict outcomes based on both prior knowledge and what they are learning throughout the inquiry. The Common Core Mathematics Practices, as outlined in the Common Core State Standards Initiative, are integrated throughout the lesson. Students made sense of the tasks they were assigned and analyzed relationships between their calculated probabilities for the standard die and experimental (MP.1) and revised their thinking as new information was presented. They constructed a uniform probability model to describe the outcomes of rolling a standard die (MP.4) and used a similar quantitative relationship to define the experimental probabilities for both the pig die and the standard die (MP.3). Students looked at patterns in the data to discover that the sum of the probabilities for a sample space was 1 (MP.7) and used the models obtained through experimentation with manipulatives (MP.5) to communicate their findings precisely through written and graphical formats (MP.6). 6 Statistics and Probability for Middle School Teachers: Addressing the Common Core Professional Learning Day; February 28, 2014 Pig Dice and Probability Name: _________________________________ Partner Name: ____________________________ For this inquiry, you and a partner will investigate a standard die or number cube and a die in the shape of a pig and determine the likelihood of various rolls occurring for each of the different-shaped dice. Activity 1: The Standard Die Investigate a standard die or number cube. In the space below, write down all observations you and your partner make about the number cube. Be ready to share your observations with your classmates! Roll the die or number cube several times. What observations can you make about the results of your rolls? Based on your observations, do you believe this is a “fair” die or number cube? Explain your reasoning. Pig Dice and Probability Student Pages 1 1 If we can consider the die to be “fair”, then each value on the die has an equal likelihood, or chance, of occurring. So, to figure out what the probability would be, we first need to list all of the possible outcomes of rolling a single six-sided die. We call this list the “sample space.” In the space below, list the possible outcomes of rolling a single die or a number cube: Now, to calculate the probability for a specific value on the die, say a roll of the number 5, the following formula would be used: 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑟𝑜𝑙𝑙𝑖𝑛𝑔 𝑎 (5) = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑑𝑖𝑒 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑎 5 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 Use this idea to complete the following table: Probability for a specific roll of the die Outcome Probability What can you conclude from looking at the table above? Now, you and your partner are going to test whether or not the theoretical probabilities you generated in your table are actually a good model for what happens if you roll a single, six-sided die. For this first test, you are going to roll the die 25 times and record in a table on the next page the results of your tosses. Pig Dice and Probability Student Pages 2 2 Results of Rolling a Single Six-Sided Number Cube Trial Value Trial Value Trial Value Trial Value Trial Value # on die # on die # on die # on die # on die 1 6 11 16 21 2 7 12 17 22 3 8 13 18 23 4 9 14 19 24 5 10 15 20 25 Let us consider rolling the die a random experiment. Then, let A be an outcome of the random experiment (a particular roll of the die) . Then A is called an event (similar to an outcome). The experimental probability of the event A is given by the following formula: 𝑃 (𝐴) = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠 where P(A) represents the probability of event A occuring. Experimental probability is also known as the relative frequency definition of probability. Using the above definition, complete the following table: Experimental Probability for a specific roll of the die Event Probability Now, compare the probabilities in the table above for each value on the die to the ones in the previous table. On the next page, discuss the similarities and differences in the probabilities for the two tables. Pig Dice and Probability Student Pages 3 3 Similarities and Differences Between the Theoretical Probability Distribution (first table) and the Experimental Probability Distribution (second table): Share your experimental data with other groups. Write down some observations you make as you compare your experimental probabilities to those of your classmates. Based on your experimental data, is the six-sided die fair? With your partner, design another experiment that might give you data that better approximates the theoretical probabilities of rolling a six-sided die. Be ready to share your observations and design with your classmates! Pig Dice and Probability Student Pages 4 4 Carry out your experiment in the space below. Record any necessary data and make any necessary calculations. What conclusion(s) can you form based upon your revised experiment? Pig Dice and Probability Student Pages 5 5 Activity 2: The Non-standard Pig Die Given a die in the shape of a pig, toss the pig die several times to determine the ways that the pig die can land on the table. Record your observations below: Compare your list above with the other team at your table. Add any additional “pig positions” to the list. Do you believe that all of the different pig positions are equally likely? In other words, is the pig die a “fair” die? Justify your response. Based on the shape of the pig die, is it possible to calculate the probability of each of the different pig positions shown. If yes, describe how you might go about calculating the probabilities. If no, describe why it would be impossible. Be prepared to share your responses with the class. Pig Dice and Probability Student Pages 6 6 For the following experiment, we are going to assume the following pig positions: Pink sider Dot sider Razorback Trotter Snouter Leaning Jowler With your partner, create a plan for determining the probability of rolling each specific pig position. We will be deciding as a class which plan to use to conduct the experiment. Carry out the plan decided upon by the class. Record all necessary information. Pig Dice and Probability Student Pages 7 7 Based on your data, complete the following table: Experimental Probability for a specific roll of the pig die Event Pink sider Dot sider Razorback Trotter Snouter Leaning Jowler Probability How confident are you in the results of your experiment? What could be done to increase the confidence in the probabilities for the pig positions? Your partner team will be assigned a particular pig position. Using the class data, you will create a graphical display for the observed frequency of the assigned pig position for each team on a sheet of paper provided to you by your teacher. Your display should include the following information. Title (Pig Position Investigated) Graphical Display Numeric Summary of key features of graphical display Based on the class data, revise your probabilities for each of the pig positions. Revised Experimental Probability for a specific roll of the pig die Event Pink sider Dot sider Razorback Trotter Snouter Leaning Jowler Probability What conclusions can you make based on the above table? Pig Dice and Probability Student Pages 8 8 Activity 3: Putting it all together In the following activity, you will compare and summarize the two experiments that were conducted using both the six-sided die and the pig die. For the first part of this activity, place a check mark in the box if the statement applies to the experiment with the standard die or the pig die. Both boxes may be checked if the statement applies to both types of die, or neither box may be checked if the statement does not apply to both types of die. Statement It is possible to calculate a theoretical probability for determining the likelihood of a specific roll of the die. It is possible to calculate an empirical (experimental) probability for determining the likelihood of a specific roll of the die. The sum of the probabilities for all of the possible outcomes for the die was always 1. All of the groups in my class had the same results when tossing the die a fixed number of trials. There was variability in the experimental probabilities from group to group when tossing the die a fixed number of trials. All of the possible outcomes are equally likely. The probability can only be approximated from the observed frequencies. Increasing the number of tosses of the die (trials) results in the experimental probability more closely approximating the theoretical probability. The die is a “fair” die. Based on the above table, what key ideas can you conclude about the probability of a chance event? Pig Dice and Probability Student Pages 9 9