Teaching & Learning Plans Plan 3: Fair Trials with Two Dice Leaving Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve. Prior Knowledge points to relevant knowledge students may already have and also to knowledge which may be necessary in order to support them in accessing this new topic. Learning Outcomes outline what a student will be able to do, know and understand having completed the topic. Relationship to Syllabus refers to the relevant section of either the Junior and/or Leaving Certificate Syllabus. Resources Required lists the resources which will be needed in the teaching and learning of a particular topic. Introducing the topic (in some plans only) outlines an approach to introducing the topic. Lesson Interaction is set out under four sub-headings: i. Student Learning Tasks – Teacher Input: This section focuses on teacher input and gives details of the key student tasks and teacher questions which move the lesson forward. ii. Student Activities – Possible and Expected Responses: Gives details of possible student reactions and responses and possible misconceptions students may have. iii. Teacher’s Support and Actions: Gives details of teacher actions designed to support and scaffold student learning. iv. Checking Understanding: Suggests questions a teacher might ask to evaluate whether the goals/learning outcomes are being/have been achieved. This evaluation will inform and direct the teaching and learning activities of the next class(es). Student Activities linked to the lesson(s) are provided at the end of each plan. Teaching & Learning Plan 3: Fair Trials with Two Dice Aims • To learn that not all results are equally probable • To consider a given set of rules and the outcomes that produce a ‘win’ for one of two players and to be able to determine whether the game is ‘fair’, i.e. whether each of the two players has the same chance of winning • To consider the theoretical probability of each result and devise a set of rules to make the game ‘fair’ Prior Knowledge Students should have prior knowledge (from T and L Plan 1 and/or from primary school) of some terms associated with chance and uncertainty. They should be familiar with probability expressed as fraction or decimal in the range 0 to 1, or as a percentage in the range 0% to 100%. Learning Outcomes As a result of studying this topic, students will be able to • investigate further what the concept of fairness means in a game with 2 dice • list all the possible outcomes for throwing 2 dice using a two way table • relate the number of outcomes to the fundamental principle of counting • come up with rules for a game which make it fair/unfair • construct a probability table • understand the relationship between an event and its complement • determine the probability of an event using the results of an experiment and use this to predict the result of a repetition of the experiment, for equally likely outcomes © Project Maths Development Team 2009 www.projectmaths.ie 1 Teaching & Learning Plan 3: Fair Trials with Two Dice Relationship to Leaving Certificate Syllabus Sub-topics Foundation Level Ordinary Level 1.1 Counting List outcomes of an experiment. Count the arrangements of distinct objects. Higher Level Apply the fundamental principle of counting. 1.2 Concepts of Recognise that probability probability is a measure on a scale of 01- of how likely an event is to occur. Discrete probability (as relative frequency) Probability of desired outcomes in problems involving experiments, such as, dice throwing. 1.3 Outcomes of random processes Apply the principle that, in the case of equally likely outcomes, the probability is given by the number of outcomes of interest divided by the total number of outcomes. Recognise the role of expected value in decision making with a focus on fair games. Extend the students’ understanding of the basic rules of probability. Resources Required Counters, 2 dice and a Game Sheet for each pair of students © Project Maths Development Team 2009 www.projectmaths.ie 2 Student Activities: Possible and Expected Responses © Project Maths Development Team 2009 www.projectmaths.ie »» Students may think that »» Think about this for a few since 11 numbers have minutes and write down been listed, and since A which player you think will can win on 6 of these and win the game most often if B only on 5, that A is more it is played many times, and likely to win than B. why you think this. »» B wins if sum i.e. is 5, 6, 7, 8, 9. Play the game on Game Sheet 1. »» A wins if sum (i.e. outcome) is 2, 3, 4, 10, 11 or 12. »» Students A and B alternately roll the die, each time adding the scores on each die to get the outcome. They place a counter on each outcome. »» In each group of two, one person is nominated as A and one as B. Players A and B take turns to roll the die, and the winner is determined by the sum of the numbers on the faces as follows: Student Learning Tasks: Teacher Input Checking Understanding KEY: » next step »» Ask for a show of hands on who thinks A will win and who thinks B will win. »» Distribute counters and Game Sheet 1 and dice to each pair, checking that students have a written prediction and a reason for it. • student answer/response »» Distribute Student Activity »» Can students record their 1 so that students can write initial results in their their prediction (Student copies? Activity 1A) and keep a count of the number of times each player wins (Student Activity 1B). Teacher’s Support and Actions Lesson Interaction Teaching & Learning Plan 3: Fair Trials with Two Dice 3 »» The students should find that A wins only about 1/3 of the time. »» Students check this against their prediction, analysing any difference and fill in Student Activity 1C. »» The game is played until one person has reached the bottom of the grid on Game Sheet 1 and students count the number of times A won and the number of times B won and fill out the Master Sheet on the board also. »» When you are finished fill out Student Activity 1B and fill in your results on Master Sheet 1 (results from the whole class) on the board. www.projectmaths.ie • student answer/response »» Did they react to this as being unfair? »» How many students in the class were able to make accurate predictions with valid reasons? »» Were students able to conduct the game successfully and did they notice B winning much more often? Checking Understanding KEY: » next step • No. Fair – if A and B have »» Give students a minute to an equal chance of winning think and then ask one otherwise unfair. student. »» Does this game appear to be fair? © Project Maths Development Team 2009 • “1 “can never be an outcome as the smallest sum possible is 2. »» As you circulate check predictions against outcomes of the experiment, and ask students about their justifications. »» Place Master Sheet 1 (results from the whole class) on the board. »» Check that each pair understands what they are doing and occasionally, as you circulate, ask “hands up how many A’s are winning?” and then “hands up how many B’s are winning?” so students are aware of an overall trend. Teacher’s Support and Actions »» Which number on each die cannot be a possible outcome? »» The relative frequency is now closer to 1/3. Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input Teaching & Learning Plan 3: Fair Trials with Two Dice 4 © Project Maths Development Team 2009 »» Students fill up the table as per example in Student Activity 2A. www.projectmaths.ie »» Individually, fill in the 2-way table for the sample space for the sum achieved on throwing 2 dice (Student Activity 2A). »» Try this for a few minutes. • student answer/response »» Are all students able to fill in the table successfully and with understanding? »» How many of the students were able to come up with a suitable table? »» Do students understand the concept of fairness? Checking Understanding KEY: » next step »» Circulate to check that students understand how to fill it in. »» Distribute Student Activity 2. »» Circulate looking at student suggestions and giving helpful hints. On finding a suitable table ask students to show and explain it to the class. Students may need help designing the table. »» Could you design a table to »» Some students will probably come up with the show you all the possible 2 way table and show it to outcomes. the class. »» The set of all the possible outcomes is called the ‘Sample Space’. »» Give students a minute to think and then ask one student. »» Give students a minute to think and then ask one student. • No, because the first time 4 is from die 1 and the second time it’s from die 2. »» In how many ways could an • 4+5 outcome of 9 be achieved? • 5+4 • 3+6 • 6+3 »» Is the outcome 4+5 the same as the outcome 5+4? • All outcomes are not equally likely. • Students might say that 2 »» Give students a minute to can only be got 1 way (1+1) think and then ask one whereas 9 for example can student. be got from (3+6), (4+5). »» Why is it not fair? Teacher’s Support and Actions Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input Teaching & Learning Plan 3: Fair Trials with Two Dice 5 • If there are 6 different outcomes from die 1, and 6 different outcomes from die 2, then the total number of possible outcomes is 6x6 = 36. • 12 of the 36 outcomes give a win for A whereas 24 of the outcomes give a win for B. »» Students articulate their misconceptions. »» Can you relate this back to the fundamental principle of counting in a previous lesson? »» Going back to the rules of the game, for how many outcomes will player A win and for how many outcomes will player B win? Student Activity 2C. »» Go back to your prediction again, Student Activity 1C, and if it was not consistent with the outcome write down why it was different. www.projectmaths.ie • 36. »» How many possible outcomes are there? Student Activity 2B. © Project Maths Development Team 2009 Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input • student answer/response »» Are students able to understand in cases where their predictions were in error why this was so? »» Have students been able to recall and use the fundamental principle of counting? Checking Understanding KEY: » next step »» Circulate and note misconceptions, and ask a selection of students who thought A would win to explain why they thought that and why they now think differently. »» Ask the question and wait for someone to volunteer an answer. If it’s the same person answering all the time, ask another student, leading them to the answer if they don’t know it, by referring back to an example they did when learning the fundamental principle of counting. »» Circulate and look at the answers being filled in, asking questions where necessary. Teacher’s Support and Actions Teaching & Learning Plan 3: Fair Trials with Two Dice 6 © Project Maths Development Team 2009 www.projectmaths.ie »» In the theoretical approach, when throwing a die, there are six possible outcomes which are all equally likely to appear due to the symmetry of the die, and given that it is a fair die (not loaded) so we have no reason to assume that any number will appear more often than another. A2 is one of the 6 possible outcomes, so the likelihood of it turning up is 1 out of 6. »» The approach in Teaching & Learning Plan 2, and above in Student Activity 1B, is known as the ‘experimental’ or ‘empirical’ approach to calculating probabilities. In this case the probability of an event is the value that the relative frequency tends to in an infinite number of trials. • For example. Player A wins if the outcome is 2, 3, 4, 5, 8 or 10 and B wins if the outcome is 6, 7, 9, 11 or 12. »» Different groups present their new rules for class approval. »» Write down new rules for the game, Student Activity 2D, which will make it fair and an explanation of why it is now fair. »» Is more than one set of rules possible? Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input • student answer/response »» Are students able to make up more than one set of rules satisfying the criterion for a fair game? Checking Understanding KEY: » next step »» Circulate and when groups are finished ask a couple of groups to explain their new rules. Teacher’s Support and Actions Teaching & Learning Plan 3: Fair Trials with Two Dice 7 • As the number of trials increases the relative frequency becomes almost equal to the theoretical probability. • How many times it occurs divided by the total number of possible outcomes. • All outcomes are equally likely. »» What is the relationship between the experimental approach to calculating probability and the theoretical approach? »» How do we calculate the theoretical probability of each outcome in this sample space? »» What assumption are we making? © Project Maths Development Team 2009 www.projectmaths.ie »» What is the sum of the • 1 probabilities for the sample space? »» Assuming that both of the • Using the 2 way table (Student Activity 2A) dice are fair, and all 36 showing the sample space outcomes are equally likely, constructed for the game, what is the probability of the sum being 5? there are 4 outcomes for 5, so the probability of getting a 5 is 4/36. »» Construct a probability table (Student Activity 3) for the sum of 2 dice using Student Activity 2A. »» Are all 36 outcomes equally • Yes. likely here? Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input • student answer/response »» Did all students get 1? »» Did students fill out the table correctly? »» Can students recall how to calculate probability for equally likely outcomes? Checking Understanding KEY: » next step »» Ask students if they remember this happening in the last lesson with outcomes from 1 die? »» Walk around observing students as they fill out the probability table. »» Asks students to recall and then after a short pause asks an individual. Teacher’s Support and Actions Teaching & Learning Plan 3: Fair Trials with Two Dice 8 © Project Maths Development Team 2009 »» Fill in answer on Student Activity 3. • student answer/response »» Do students see that the experimental results approach the theoretical values for probability as the number of games increases? »» Do students understand the relationship between an event and its complement? »» How many students got the correct answer without having to add up all the other probabilities? Checking Understanding KEY: » next step »» Circulate checking that Student Activity 3 is being completed, asking questions of students who are having difficulty. • They are very close and the relative frequency becomes closer to the theoretical probability as the number of games increases. »» Compare this with experimental result for relative frequency. »» Fill in answer on Student Activity 3. www.projectmaths.ie »» If a student cannot get the answer without adding up all the other probabilities, tell them to do that first and then see if they can then find the other way by inspecting their answer. • 1-1/3=2/3 »» Use the probability that A wins to calculate the probability that B wins, without adding. »» Fill in answer on Student Activity 3. »» For students who have difficulty: ask them to count how many of the outcomes give A a win, and how many outcomes are there in total. • 12/36=1/3 »» Going back to the original rules for the game, what is the probability that A wins? Teacher’s Support and Actions »» If a student cannot get the answer without adding up all the other probabilities, tell them to do that first and then see if they can then find the other way by inspecting their answer. Student Activities: Possible and Expected Responses »» If the probability of getting • 1-2/36=34/36 a 3 is 2/36, what is the probability of not getting a 3 without adding up all the other probabilities? Write down the answer. Student Learning Tasks: Teacher Input Teaching & Learning Plan 3: Fair Trials with Two Dice 9 Student Activities: Possible and Expected Responses © Project Maths Development Team 2009 »» Write down any questions you may have. »» Write down anything you found difficult. »» Write down 3 things you learned about probability today. Reflection »» Emphasise that you are dealing with equally likely outcomes. Teacher’s Support and Actions www.projectmaths.ie • student answer/response »» Are they using the terminology with understanding and communicating with each other using these terms? »» Have all students learned and understood these items? »» Can students calculate an expected value for a large number of trials? Checking Understanding KEY: » next step • How to: »» Circulate and take 1. list all possible outcomes note particularly of any (sample space) for questions students have throwing 2 dice and help them to answer 2. use the list of all possible them. outcomes to judge fairness 3. construct a probability table 4. calculate the complement of an event 5. calculate expected value for a large number of trials. »» If the probability of a sum • 100x1/6 i.e. approx 17 of 7 occurring is 6/36=1/6, (1/6 of the time you expect how many 7’s would you a 7.) expect to get if the dice are tossed 100 times? Student Learning Tasks: Teacher Input Teaching & Learning Plan 3: Fair Trials with Two Dice 10 Teaching & Learning Plan 3: Fair Trials with Two Dice Game Sheet 1 1 A wins A wins A wins B wins B wins B wins B wins B wins A wins A wins A wins 2 3 4 5 6 7 8 9 10 11 12 © Project Maths Development Team 2009 www.projectmaths.ie 11 Teaching & Learning Plan 3: Fair Trials with Two Dice Student Activity 1 Student Activity 1A Prediction Player _____ will win most often because: Student Activity 1B Play the game and record the results below: Use Tally Marks to Help Total (Frequency) You Keep a Score Relative Frequency (total no. of wins/total no. of games) Player A Wins Player B Wins Totals Student Activity 1C Did your predicted results agree with your actual results? © Project Maths Development Team 2009 www.projectmaths.ie 12 Teaching & Learning Plan 3: Fair Trials with Two Dice Master Sheet 1 (Results from the Whole Class) Play the game and record the results below: Use Tally Marks to Help Total (Frequency) You Keep a Score Player A Wins Relative Frequency (total no. of wins/total no. of games) 2+3+... Player B Wins Totals © Project Maths Development Team 2009 www.projectmaths.ie 13 Teaching & Learning Plan 3: Fair Trials with Two Dice Student Activity 2 Student Activity 2A Complete the table to show all of the possible outcomes. Two way table showing the sample space. Number thrown 1 1 2 3 4 5 6 1+1=2 2 3 4 5 5+2=7 6 Student Activity 2B How many possible outcomes are there?__________________ Student Activity 2C Original Rules: Player A wins when the sum is 2, 3, 4, 10, 11 or 12. Player B wins when the sum is 5, 6, 7, 8 or 9. For how many outcomes will player A win?__________________ For how many outcomes will player B win?__________________ Student Activity 2D New Rules: Player A wins when the sum is __________________________ Player B wins when the sum is __________________________ Why I chose these new rules: © Project Maths Development Team 2009 www.projectmaths.ie 14 Teaching & Learning Plan 3: Fair Trials with Two Dice Student Activity 3 Sum Frequency 2 1 Probability no. of outcomes in the event = no. of outcomes in the sample event 1 36 3 4 5 6 7 8 9 10 11 12 © Project Maths Development Team 2009 www.projectmaths.ie 15

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