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8th Grade Mathematics: Unit 8: Examining Chances Ascension Parish Comprehensive Curriculum Concept Correlation Unit 8: Examining Chances Time Frame: 3 Weeks Big Picture: (Taken from Unit Description and Student Understanding) The interaction of events affects probability. Probability is the mathematics of chance. Sampling affects the relationship between experimental and theoretical probability. Activities Documented GLEs Guiding Questions Essential Activities are denoted GLE’s Date and with an asterisk GLEs GLEs Method of Concept 1: Probability Bloom’s Level Activity 106: Let Me Count Assessment the Ways 42 Select random samples 33. Can students recognize and that are representative of discuss ways that GQ 34 DOCUMENTATION the population, including randomness contributes to Activity 107: How Many sampling with and surveys, experiments, and Ways? 42, 43 without replacement, games of chance? GQ 34 and explain the effect of 41 Activity 108: What does the sampling on bias (D-2- 34. Can students calculate and Cookie Thief Look Like? 43 M) (D-4-M) interpret single- and GQ 34 (Evaluation) multiple-event probabilities *Activity 109: Independent in a wide variety of Events 45 situations, including GQ 33, 34 Use experimental data independent, mutually presented in tables and exclusive, and dependent, graphs to make outcome non-mutually exclusive Activity 110: Dependent predictions of settings? 44 Events 45 independent events (D- GQ 33, 34 5-M)(Evaluation) 8th Grade Mathematics: Unit 8: Examining Chances 8th Grade Mathematics: Unit 8: Examining Chances Activity 111: Is It Fair? Calculate, illustrate, and 45 GQ 33, 34 apply single- and Concept 2: Sampling and *Activity 112: Selecting a multiple-event Experimental Data Sample 41 probabilities, including GQ 33, 35 mutually exclusive, 45 33. Can students recognize and Activity 113: Experimental independent events and discuss ways that Probabilities non-mutually exclusive, randomness contributes to GQ 33 44 dependent events (D-5- surveys, experiments, and M) (Synthesis) games of chance? *Activity 114: Who Did It? 41, 44 35. Can students suggest ways GQ 33, 35 of minimizing bias in sampling or surveys through the use of random samples? Activity 115: Replacement to 41, 44 Sample Set GQ 33, 35 8th Grade Mathematics: Unit 8: Examining Chances 8th Grade Mathematics: Unit 8: Examining Chances Unit 8 Concept 1: Probability GLEs *Bolded GLEs are assessed in this unit. 42 Use lists, tree diagrams, and tables to apply the concept of permutations to represent an ordering with and without replacement (D-4-M) (Synthesis) 43 Use lists and tables to apply the concept of combinations to represent the number of possible ways a set of objects can be selected from a group (D-4-M) (Analysis) 45 Calculate, illustrate, and apply single- and multiple-event probabilities, including mutually exclusive, independent events and non-mutually exclusive, dependent events (D-5-M) (Synthesis) Purpose/Guiding Questions: Vocabulary: 33. Can students recognize and discuss ways Dependent Events that randomness contributes to surveys, Experimental Probability experiments, and games of chance? Independent Events 34. Can students calculate and interpret Multiple Event Probability single- and multiple-event probabilities Single Event Probability in a wide variety of situations, including Theoretical Probability independent, mutually exclusive, and dependent, non-mutually exclusive settings? Key Concepts Calculate single-event and multiple- event probability, including occurrence of mutually exclusive and independent events, and of non- mutually exclusive and dependent events. Assessment Ideas: Resources: See end of Unit 8 Spinners Number Cubes Activity Specific Assessments: Cards Activity 107, 109, 110, 111 Paper Clips Styrofoam Plates Independent Events Handout Dependent Events Handout Is It Fair Handout Teacher-Made Supplemental Resources Writing Strategies See the Teacher-Made Supplemental Resources for Daily Problems to use with journal writing. 8th Grade Mathematics: Unit 8-Examining Chances 126 8th Grade Mathematics: Unit 8: Examining Chances Instructional Activities Note: Essential activities are key to the development of student understandings of each concept. Substituted activities must cover the same GLEs at the same Bloom’s level. Activity 106: Let Me Count the Ways (LCC Unit 8 Activity 2) (GLE: 42) Materials List: pencil, paper, math learning log, How Many Ways? BLM Have students work in groups of four and determine how many ways they could possibly line up in a single-file line. Ask them to record each of the ways that this could occur. Discuss the student results and have the groups make observations about the relationship of the results and the number of ways they could line up. Answer: There are four students, so there are four possible students eligible for position 1; 3 possible for position 2; 2 possible for position 3; and only 1possible for position 4 giving a total of 6 different ways for them to line up with student 1 first. Therefore, there are a total of 24 different ways for the students to line up when students 2 – 4 are included in first place. The following are ways to show students an organized manner to determine the answer. The following lists are all possible ways the four students can line up: ABCD ACDB BACD BADC CABD CBDA DABC DBCA ABDC ADBC BDCA BDAC CADB CDAB DACB DCAB ACBD ADCB BCAD BCDA CBAD CDBA DBAC DCBA The diagram below illustrates the same part of the problem as the list above, but in a tree diagram. This is only for Student A, and there will be the same number of arrangements for students B, C, and D. 3rd S tudent C 4th S tudent D 2nd S tudent B 4th S tudent C 3rd S tudent D 3rd S tudent C 4th S tudent D 1st S tudent A 2nd S tudent B 3rd S tudent D 4th S tudent C 3rd S tudent C 4th S tudent D 2nd S tudent B 3rd S tudent D 4th S tudent C Stress that there are four student positions possible when the first person lines up, then there are only 3 people left for the second spot, then two people left for the 3rd spot and at this point only 1 person for the last spot. A permutation is an arrangement or listing in which order is important. A combination is an arrangement or listing in which order is not important. As in this example, the 1st, 2nd, 3rd, and 4th place in line is different as determined by which student is in each position, making this a 8th Grade Mathematics: Unit 8-Examining Chances 127 8th Grade Mathematics: Unit 8: Examining Chances permutation. If we were forming a group of four students for a project, it would not matter which order the students were picked, making it a combination. The number of permutations possible when all members of the initial set are used without replacement can also be found mathematically by multiplying the number of members available for each place in the order. Example: 4 x 3 x 2 x 1 = 24 4 people for 1st place, 3 people for 2nd place, 2 people for 3rd place and 1 person for 4th place. This is represented by factorial notation. A factorial (n!) is the product of a whole number and every positive whole number less than itself Write: 4! = 4 x 3 x 2x 1 Say: Four factorial equals four times three times two times one. Challenge students to use what they know about permutations and determine the number of ways that Pepperoni Pizza, Hamburger Pizza, Canadian Bacon Pizza, Vegetable Pizza and Extra Cheese Pizza (order) can be listed on a menu for the local restaurant. Allow students to use lists, tables, or tree diagrams to aid them in determining the number of permutations. Have them share the diagram they used with others. Answer: There are 5 possible choices for the first pizza listed, 4 possible choices for the second pizza listed, 3 possible choices for the third pizza listed, 2 possible choices fore the fourth pizza listed and then only one will be left for the last position. 5! (5x4x3x2x1 = 120 ways) Explain that the problems done thus far are permutations without replacement and all the members of the initial set are used. Tell students that it is also possible to find permutations without replacement using only some members of the initial set. For example, if there are four students, it is possible to find all the different ways only 2 of the students can line up.. Put students in groups. Have half of the groups create a list and the others a tree diagram to find the different ways 2 out of the 4 students can line up. There are 4 students so 4 students are possible for position 1, and 3 students possible for position 2. This gives 3 possible ways to line up with student 1 first. Therefore there are a total of 12 different ways for 2 out of 4 students to line up. List: AB BC CD DA AC BD CA DB AD BA CB DC Tree Diagram: The diagram below is only for Student A, and there will be the same number of arrangements for students B, C and D. Stress that there are four students who can take the first position when lining up and then there are only 3 people left for the second spot. Reminder, only 2 of the 4 students available are being lined up. 2ndstudent B 1st student A 2nd student C 2nd student D 8th Grade Mathematics: Unit 8-Examining Chances 128 8th Grade Mathematics: Unit 8: Examining Chances Mathematically: The number of permutations possible when some members of the initial set are used without replacement can be found mathematically by multiplying the number of members available for each place in the order. Example: 4 x 3 = 12 4 people for 1st place and 3 people for 2nd place make possible permutations or arrangements. Distribute How Many Ways? BLM and have the students work individually or in pairs to determine the number of possible outcomes for the different situations given. Have students discuss answers with larger groups or have a class discussion. Have students use their math learning logs (view literacy strategy descriptions) to explain in their own words the difference in determining the possible number of combinations for placing 3 pictures out of a set of 5 pictures in a certain position on a wall and the possible number of ways three people can finish running a race when six people are running. They should use a tree diagram, chart or list, or mathematical way to justify their answers in their math learning logs. FYI – You can‟t use the factorial notation because you are not using all members of the set for your line up, only two of them at one time. As you monitor students working on this problem, question them about the similarities and differences of these situations to the previous situation. This website can be used as an introduction to probability and has an interactive spinner, die, and a collection of colored marbles. http://www.mathgoodies.com/lessons/vol6/intro_probability.html Activity 107: How Many Ways? (LCC Unit 8 Activity 3) (GLEs: 42, 43) Materials List: paper, pencil, chart paper, marker, Which is it? BLM, calculators Begin class using SQPL (view literacy strategy descriptions) by having partners brainstorm (view literacy strategy descriptions) two to three questions they would like answered about the following statement. There would be more possible combinations of officers for a class (President, Vice President, Secretary, and Treasurer) than there would be combinations of four- person committees from a class of ten students. Write the SQPL statement on the board or overhead for students to see. Have pairs then share their questions with the class. The class will make a list of questions that it hopes to be answered during the lesson. Post this list of questions as the lesson progresses. An internal summary can be made by pointing out to the students that they can now answer certain questions that they had. As the lesson begins, pose a situation where a five-person committee must be formed from seven individuals to plan for an upcoming event. Challenge students to determine the number of different committees that could be formed from these seven students. This will be different from 8th Grade Mathematics: Unit 8-Examining Chances 129 8th Grade Mathematics: Unit 8: Examining Chances those problems done previously, because in these, order is not important. The combinations are shown in the list at the right. Next, pose a scenario where five individuals must fill the five roles of officers: one person is the president, one is the vice president, one is the secretary, one is the treasurer, and one is the historian. Ask if any one of the five students could serve in any of the positions, then ask how many different ways this group of five officers could be selected. Lead discussion about the similarities and differences in these situations and whether or not order is important. Answer: First scenario – 21 different 5 person committees. Order does not change the make up of the committees. Answer: Second scenario – 120 different ways. 5×4×3×2×1 (Order is important because if the person is selected for President, it is different than if that person is chosen for Secretary.) Make sure the discussion of these scenarios involves some brainstorming by students of situations in which order is important (permutations) and not important (combinations). Distribute Which Is It? BLM and have students practice determining whether the situation involves a combination or a permutation and provides practice for the students in solving these problems. Discuss student responses on the BLM as a class to clarify any misconceptions. 1,2,3,4,5 1,2,5,6,7 2,3,5,6,7 1,2,3,4,6 1,3,4,5,6 2,4,5,6,7 1,2,3,4,7 1,3,4,5,7 3,4,5,6,7 1,2,3,5,6 1,3,4,6,7 1,2,3,5,7 1,3,5,6,7 1,2,3,6,7 1,4,5,6,7 1,2,4,5,6 2,3,4,5,6 1,2,4,5,7 2,3,4,5,7 1,2,4,6,7 2,3,4,6,7 Assessment Secure menus from a restaurant that advertises several ways its product can be purchased (e.g., Burger King, Baskin-Robbins Ice Cream), and the student will determine the validity of the claim. Activity 108: What does the Cookie Thief Look Like? (LCC Unit 8 Activity 4) (GLE: 43) Materials List: Who Stole the Cookies? BLM, paper, pencil, newsprint or chart paper, markers Provide students with Who Stole the Cookies? BLM and read the situation aloud to the class. Jackie worked at a restaurant in the evening. She had a locker in the back where she put all of her personal belongings. One night she bought a big box of cookies to take to her grandmother the next day. She put this box of cookies in her locker so that she could take it home after work. When she went back to the locker at 10:00 P.M. after work, the cookies were gone! One of her friends saw a stranger at the lockers about 9:30 P.M. 8th Grade Mathematics: Unit 8-Examining Chances 130 8th Grade Mathematics: Unit 8: Examining Chances Jackie and her friend talked to the store manager, and they were given a list of possible characteristics to help in identification. The characteristics were given to the friends in a chart like the one on Who Stole the Cookies? BLM. Challenge pairs of students to come up with all the different descriptions possible for the cookie thief. Have the pairs of students determine the different combinations of descriptions that could have described the thief. Then have them display their findings using some type of chart or graph. Once the student pairs have completed their description, randomly select one group to be professor-know-it-all (view literacy strategy descriptions) and have it explain to the class the different descriptions and its method of organizing their descriptions. Allow class members to ask questions of the group that is professor-know-it-all. *Activity 109: Independent Events (LCC Unit 8 Activity 6) (GLE: 45) Materials List: number cubes, pencil, paper Have groups of four students create a game of chance like Yahtzee® using number cubes. Have students determine the rules for their game, the materials (die, spinner, cards, etc) and then justify how the theoretical probability of winning makes their game a fair game. After playing the game several times, explore the experimental probabilities of obtaining each of the required outcomes. For example, explore the possibility of rolling all number cubes and getting the same number on each. The roll of each die is independent. Have students exchange games with another group and follow the rules determined by the game‟s creator. Compare experimental results with the theoretical results. Lead classroom discussion about the independent events involved in each of the games created. (See Teacher-Made Supplemental Resources) Assessment The student will create a game of chance in which player 1 has twice the chance of winning as player 2. Assessment The student will play several different games of chance and then analyze the probabilities of winning. Activity 110: Dependent Events (LCC Unit 8 Activity 7) (GLE: 45) Materials List: styrofoam plates, paper clips, Dependent Events BLM, pencil, paper Create a multiple-event experiment where the events are dependent, and have the students determine the probability of a result. Have each group of four students make two spinners with sturdy plates that have the thumbprints or dimples around the edge such as the Hefty® brand of plate. Secure a paper clip as the spinner by using a second paper clip through the bottom of the plate. These plates are already divided into 36 thumbprints so the students can easily divide the plate into thirds or fourths. Divide one of the plates into thirds and let this plate represent the number of coins. Allow students to determine the numbers in each section but encourage them to use numbers less than 10. It will make it easier for class discussion if the groups use the same 8th Grade Mathematics: Unit 8-Examining Chances 131 8th Grade Mathematics: Unit 8: Examining Chances three numbers (possibly-2, 5, 10). Divide the second plate into fourths and write the name of a coin in each of the four sections (possibly-penny, nickel, dime, quarter). Distribute Dependents Events BLM and explain to the students that they must figure the theoretical probability of spinning less than, more than or exactly fifty cents. The groups will then collect experimental data and record their data on a chart. An example of a possible chart is shown below: Spin # # of Coin Total >, < or Spin # # of Coin Total >, < or coins value Value = to coins value Value = to of spin $.50 of spin $.50 1 10 2 11 3 12 ... ... Have students compare their experimental results with their theoretical results. Then have groups of students compare results with other groups. Discuss how the results might be different if the spinners were not fair spinners. Sample size should also be part of the discussion. Relate the situation to a possible game at the fair or some other carnival. Discuss the probability of winning prizes at the fair. (See Teacher-Made Supplemental Resources) Assessment The student will prepare directions and make a game that involves dependent events. The student will describe the game using the theoretical probability of outcomes to describe how the game is won. Activity 111: Is It Fair? (LCC Unit 8 Activity 8) (GLE: 45) Materials List: two number cubes of different color, paper, pencil Provide pairs of students with two number cubes of different colors. Ask students to roll the number cubes and find the product of the two cubes. Player 1 will be the tallest person, and will roll both number cubes first. If the product of the numbers rolled is odd, player 1 will receive two points. If the product is even, player two will get 1 point. Have play continue until one of the players reaches 20 points. Repeat the game exchanging positions of player 1 and player 2. After the students have played the game at least two times, have the students create a table showing the theoretical probability of each product‟s occurring. Ask students then to determine the probability of an odd or even product and whether the game rules were fair. Challenge the groups to determine rules that would create a fair game using number cube products. To do this, have the students form groups of four to make a modified story chain (view literacy strategy descriptions). Student 1 will be the person closest to the teacher, and the students will be numbered clockwise from Student 1. Student 1 will write the first in a set of rules for making a fair game with the number cubes, pass the paper to Student 2 who will in turn write the second rule or step, Student 3 and then Student 4. This will continue until the group has completed their rules for the game. Each student in the group should then get a chance to read and challenge any of the rules or steps written so that their game is fair. Have groups follow the steps or rules that have been written to play the game and determine if each player has an equal 8th Grade Mathematics: Unit 8-Examining Chances 132 8th Grade Mathematics: Unit 8: Examining Chances chance of winning. An exit ticket is a student summary of the lesson as they respond to a prompt or questions from the teacher. Have the students use an exit ticket to provide written individual explanations of why their rules created a fair game. Lead discussion with the class about whether the events involved in the game were independent or dependent events. (See Teacher-Made Supplemental Resources) Assessment The student will prepare a presentation to explain how theoretical probability is used to make predictions like the weather forecast. Assessment The student will make four different sketches of polygons with a shaded area inside or outside of the polygon that would illustrate a 25%, 50%, 75% and 60% probability of an object falling randomly on each figure and landing on the shaded area. Example: the figure at the right would represent a 50% probability of a randomly dropped object that would fall on the figure landing on the shaded area. Assessment The student will complete a probability project assessed by a teacher-created rubric. 8th Grade Mathematics: Unit 8-Examining Chances 133 8th Grade Mathematics: Unit 8: Examining Chances Unit 8 Concept 2: Sampling and Experimental Data GLEs *Bolded GLEs are assessed in this unit. 41 Select random samples that are representative of the population, including sampling with and without replacement, and explain the effect of sampling on bias (D-2-M) (D-4-M)(Evaluation) 44 Use experimental data presented in tables and graphs to make outcome predictions of independent events (D-5-M)(Evaluation) Purpose/Guiding Questions: Vocabulary: 33. Can students recognize and discuss ways Population that randomness contributes to surveys, Random experiments, and games of chance? Sample 35. Can students suggest ways of Survey minimizing bias in sampling or surveys through the use of random samples? Key Concepts Analyze data considering random sampling, sample size, bias, and data extremes. Understand the concept of a sample and sampling with/without replacement. Use experimental data presented in tales or graphs to make outcome predictions based on the probability of independent events; and explain predictions based on an understanding of the logic of probability. Assessment Ideas: Resources: See end of Unit 8. Glencoe Book 3 (Eighth Grade) Textbook, Concept Summary Pages Activity Specific Assessments: 406-407 Activity 112, 113, 115 Spinners Paper Bags Multi-Colored Blocks Selecting a Sample Handout Experimental Probabilities Handout Who Did It Handout Teacher-Made Supplemental Resources 8th Grade Mathematics: Unit 8-Examining Chances 134 8th Grade Mathematics: Unit 8: Examining Chances Writing Strategies See the Teacher-Made Supplemental Resources for Daily Problems to use with journal writing. Instructional Activities Note: Essential activities are key to the development of student understandings of each concept. Substituted activities must cover the same GLEs at the same Bloom’s level. *Activity 112: Selecting a Sample (LCC Unit 8 Activity 1) (GLE: 41) Materials List: Random or Biased Sampling Opinionnaire BLM, Random or Biased Sampling BLM, pencil, paper, brown paper bags (1 per group), color tiles (10 in each bag: 5 of one color, 3 of another color, 1 of a third color, and 1 red) Begin class by having students work in pairs to complete the Random or Biased Sampling Opinionnaire BLM. Opinionnaires (view literacy strategy descriptions) are tools used to elicit attitudes about a topic. A modified Opinionnaire is being used to generate some thinking about biased and unbiased sampling. Once the pairs of students have completed the survey, have the pairs of students get into groups of four and discuss their answers and reasons. Have the groups of four students write a summary statement giving their idea(s) about random sampling. Have students share their ideas with the class prior to the discussion of surveys. Lead a discussion about the need for surveys. Such questions as: When would a survey be done? What would be gained from the survey? What can be found from a survey? Does it matter who is surveyed? Have students design a survey about an issue that interests them and survey a sample from the eighth-grade student body. Lead a discussion about the pros and cons of just surveying their own class. Ask students if it would be better to survey all students; however, sometimes it is impossible to survey all members of the population. In such cases, a sample must be taken. Help students understand that their sampling population should be random and discuss how to ensure this randomness. Have students determine a way to randomly select a sample from the population of the entire eighth-grade student body. Lead a discussion about why random selection will help keep out any bias and provide a sample that is representative of the entire eighth-grade student body. To help students see how a random sample is selected, provide them with a bag containing 10 cubes or color tiles of 4 colors (5 of one color, 3 of another color, 1 of a third color, and 1 red) and have them shake the bag and then remove a cube or tile and note its color (do not allow the students to look in the bag prior to their data collection). To simulate a large population, replace the cube drawn and then shake the bag and draw another cube, note its color, and then replace it. Have each student determine the number of times that the process should be repeated to allow them to make a good guess as to what the colors of the tile in the bag are and how many of each color are in the bag, if there are 10 total cubes or tiles in the bag. Have students make predictions as to what the next randomly selected sample color will be from their collection. Discuss how certain they are about their prediction and then have them collect the sample. Ask how closely each student‟s sample of 10 matched the population - this is a good time to discuss the importance of sample size. Combine all the results in the class and then determine how closely the aggregated data match the actual color proportions in the population. A website with an interactive „Let‟s Make A Deal‟ probability page is available at http://matti.usu.edu/nlvm/nav/frames_asid_117_g_4_t_2.html 8th Grade Mathematics: Unit 8-Examining Chances 135 8th Grade Mathematics: Unit 8: Examining Chances Tell students that they will design a survey that is based on a random sampling population. Lead a discussion about the need for surveys. When would a survey be done? What would be gained from the survey? What can be found from a survey? Does it matter who is surveyed? Ask student how they could determine which color T-Shirt to order for the 8th grade party without asking each and every student in the 8th grade. Discuss how the sample population affects the results. Distribute the Random or Biased Sampling BLM. Have the students complete the questions independently prior to assigning them their survey to assure understanding of biased and random sampling. State that it would be better to survey all students; however, sometimes it is impossible to survey all members of the population. In such cases, a sample must be taken. Help students understand that their sampling population should be random and discuss how to ensure this randomness. Have students determine a way to randomly select a sample from the population of the entire 8th grade student body. Have students design a survey about an issue that interests them and survey a sample from the 8th grade student body. Lead a discussion about the pros and cons of just surveying their own class. Lead a discussion about why random selection will help keep out any bias and provide a sample that is representative of the entire 8th grade student body. Student groups should conduct their surveys after the teacher has verified their survey question and bring results to class. Have students complete their survey and prepare a presentation for the class by writing a paragraph explaining their survey question, the sample population and the results of their survey. Student groups should present these paragraphs to the class. The presentations should give the reason for the survey and what results were gathered. (See Teacher-Made Supplemental Resources) Have the students review their Random or Biased Sampling Opinionnaire BLM and make any changes to their responses that can now be answered with a better understanding of random and biased sampling. Assessment The teacher will provide the student with a survey topic and the students will describe in his/her journal what population will be surveyed, the sample size, and the sample questions. The student will also explain how the survey will be used. The student will conduct the survey and prepare their results with an explanation as to how the survey results will be used. Activity 113: Experimental Probabilities (LCC Unit 8 Activity 5) (GLE: 44) Materials List: styrofoam plates, paper clips, pencil, paper Have students make a prediction based on the results of spinning a spinner that has been divided into equal sections of three colors. These spinners can be made with the foam plates that have the thumbprints around the circumference. Hefty plates have 36 thumbprints and are easy to divide into equivalent sections. Have students determine the theoretical probability on any given spin. The theoretical probability would be that on any given spin, the chances of getting any one of the three colors is one-third; however, have students perform the experiment of spinning the spinner twenty to thirty times and then use their experimental results to make the prediction of the next spin. 8th Grade Mathematics: Unit 8-Examining Chances 136 8th Grade Mathematics: Unit 8: Examining Chances Use a real-life example: The local mall is having a grand opening celebration. They are using a spinner like the one used in the experiment to determine the prizewinners every fifteen minutes. They display the results throughout the day. When you get to the mall, the spinner result display looks like the one below. The mall official will randomly select an audience member to call a color and if that color wins, the member will win a prize. red blue yellow Have the students work in pairs to determine what their choice would be if they were selected as the next lucky person to spin. Students should explain answers with a sketch or diagram. Assessment The student will develop an experiment and then determine the experimental probability associated with the event taking place. *Activity 114: Who Did It? (LCC Unit 8 Activity 9) (GLEs: 41, 44) Materials List: Who Did It? BLM, brown lunch bags (4 for each group), 10 color tiles of four different colors (in each of 4 bags for each group), pencil, paper Begin the class with a discussion about sampling. Tell the students that today they will collect results without replacement. Discuss what this means. Distribute the Who Did It? BLM to each student, and give four brown lunch bags filled with the following (unknown to them) tile or cube combinations: Bags should be labeled A – D and should each contain a total of 10 tiles or cubes of four different colors. Let the students know that there are 10 tiles in each bag and four different colors. Bag A and one of the other bags identical (have the same number of each color of cubes. Tell the students that the sample in Bag A was found at the scene of a crime. CSI investigators explored the contents of Bag A. When the investigators bagged the contents of Bag A, they duplicated the items from bag A and made a second bag. A new CSI trainee was also on the scene, and he took these bags back to the lab without labeling the second one. These bags were placed on a box with bags from another crime scene. When the CSI went to get the bags, there were four bags and only Bag A was labeled. The CSI knew there were two identical bags, and these bags are very important to their case. They want to try to determine which are the identical bags and not touch the items any more than necessary. Discuss as a class that when sample are examined without replacement, the sample size is constantly changing. Suppose this is what happens when the first tile is selected from the bags: 8th Grade Mathematics: Unit 8-Examining Chances 137 8th Grade Mathematics: Unit 8: Examining Chances Bag A – red; Bag B – red; Bag C – green; Bag D – red. Now there are only nine tiles in each bag to select from. Challenge the students to devise a plan to sample contents of the bags without replacement so that they can make the best prediction based on experimental probability without looking at the contents of the bags. Suppose a red tile is selected from Bag A on the first selection, a red tile from Bag B on the first selection, a green tile from Bag 3 on the first selection and a red tile from Bag 4 on the first selection, based on the information collected so far, can a good prediction be made as to the matching bags? Now there are only nine tiles in each bag to select from. Have students record their results and make a prediction after the 6th selection from each bag, justifying why this is their selection. Lead a discussion about whether the predictions give enough information to make the prediction. Ask if all four bags have to be completely empty to make a valid prediction? Have them explain their thinking and their results. Teacher Note: Student results will be different and they will have to use some logical reasoning as they compare the results they gather. Activity 115: Replacement to Sample Set (LCC Unit 8 Activity 10) (GLEs: 41, 44) Materials List: Who Did It? BLM (Activity 114), brown lunch bags (4 for each group), 10 color tiles of four different colors (in each of 4 bags for each group), pencil, paper Begin class with a discussion about sampling without replacement that was done in Activity 114. Tell the students that today they will collect results of the same problem with replacement. Have students determine whether or not they think this method will be a better way to make a prediction as to contents and why they made that choice. Have students take out their Who Did It? BLM (Activity 114), and distribute the brown bags to each group of four. Have student groups make a plan for determining the contents with replacement. They should write this plan on the back of their Who Did It? BLM sheet. Once they have devised a plan, they should collect their data and answer questions 4 and 5 on the Who Did It? BLM. Discuss as a class the results and the difference in sampling with and without replacement. Assessment The student will prepare a poster proving that his/her prediction is based on experimental probability after the 6th selection. The student will also use the actual contents of the bag to compare the theoretical probability of his/her prediction after the 6th selection. The student will include an explanation of how sampling without replacement affected their prediction. 8th Grade Mathematics: Unit 8-Examining Chances 138 8th Grade Mathematics: Unit 8: Examining Chances Unit 8 Assessment Options General Assessment Guidelines The student will create a game of chance in which player 1 has twice the chance of winning as player 2. The student will prepare a presentation to explain how theoretical probability is used to make predictions like the weather forecast. The student will make four different sketches of polygons with a shaded area inside or outside of the polygon that would illustrate a 25%, 50%, 75% and 60% probability of an object falling randomly on each figure and landing on the shaded area. Example: the figure at the right would represent a 50% probability of a randomly dropped object that would fall on the figure landing on the shaded area. The student will play several different games of chance and then analyze the probabilities of winning. The student will develop an experiment and then determine the experimental probability associated with the event taking place. Whenever possible, the teacher will create extensions to an activity by increasing the difficulty or by asking “what if” questions. The student will create portfolios containing samples of their experiments and activities. The student will complete a probability project assessed by a teacher-created rubric. Activity-Specific Assessments Concept 1 Activity 107, 109, 110, 111 Concept 2 Activity 1112, 113, 115 8th Grade Mathematics: Unit 8-Examining Chances 139 8th Grade Mathematics: Unit 8: Examining Chances Name/School_________________________________ Unit No.:______________ Grade ________________________________ Unit Name:________________ Feedback Form This form should be filled out as the unit is being taught and turned in to your teacher coach upon completion. Concern and/or Activity Changes needed* Justification for changes Number * If you suggest an activity substitution, please attach a copy of the activity narrative formatted like the activities in the APCC (i.e. GLEs, guiding questions, etc.). 8th Grade Mathematics: Unit 8-Examining Chances 140