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How Likely Is It?, Investigation 1, Problem 1.1 Completed Choosing Cereal Mathematical Goal National Standards State Standards NAEP 6NJ 4.4.B.1, • Develop an intuitive sense of probability through D4a, D4c, D4g 6NJ 4.4.C.3, a coin-tossing experiment. CAT6 6NJ 4.4.B.5 LV16.15 Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, CTBS Student Activity CD-ROM, www.PHSchool.com LV16.53 Materials: Student notebooks, Overhead projector, Coins, Paper ITBS cups (optional) LV12.PS Pacing: 45 minutes S10 Int2.DSP, TV LV16.15 1. LAUNCH (10 minutes) Targeted Resources Discuss the questions posed in Getting Ready. Introduce Problem 1.1 by Transparency 1.1A Getting telling Kalvin’s story about how he decided to toss a coin to decide his Ready breakfast. Have students predict, before they conduct the experiment, Transparency 1.1B Coin– how many days in June Kalvin will have Cocoa Blast for breakfast. Toss Results The following questions can help in a discussion of bias: Labsheet 1.1 Coin–Toss Table • What kinds of things could happen to affect the data you gather in this problem? • How should you toss a coin to be sure you have a fair trial? • What if you always start with tails facing up when tossing a coin? Do you think this introduces bias? Let students know that collecting data in a random way can help to better predict what to expect when a coin is tossed. In this case, random means that the coin’s behavior is not affected in a predictable way by how it is tossed. Distribute Labsheet 1.1 to each pair or group. • Your pair or group is going to toss a penny once for each day in the month of June and record the result, H or T, in the Result of Toss column. Have students work in pairs or small groups. 2. EXPLORE (20 minutes) Targeted Resources As students work, make sure they are not introducing bias and are recording their data correctly. You may want to illustrate how to fill the table in for some sample data for the first few days noting the cumulative nature of the table columns marked “so far”. Students can compute the fraction of heads after any number of trials by dividing the number of heads that occurred to that point by the number of trial days. When students are done with Question A, bring the entire class together to work on Question B of the problem. 3. SUMMARIZE (15 minutes) Targeted Resources Combine the class’s data. Ask one pair or group at a time to report how many heads they tossed in 30 trials, and record the results in a table. After you have collected all the data, combine it by making another table showing a running total of the number of trials and number of heads. Recompute the fraction and percent of heads each time. • As the number of trials increases, what is happening to the percent of heads? Have students do Question C as a class, in groups, or as homework 4. ASSIGNMENT GUIDE Targeted Resources Core 1,3–5, 19, 20 1ACE Exercise 2 Adapted Version Other Applications 2; Extensions 31 Answers to ACE and Adapted For suggestions about adapting Exercise 2 and other ACE Mathematical Reflections exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 19, 20: Bits and Pieces I How Likely Is It?, Investigation 1, Problem 1.2 Completed Tossing Paper Cups Mathematical Goals National Standards State Standards NAEP 6NJ 4.4.B.1, • Continue to develop an intuitive sense of probability through D4a, D4c, D4g 6NJ 4.4.C.1, a cup tossing experiment. CAT6 6NJ 4.4.B.5 • Understand that probabilities are useful for predicting what LV16.15 will happen over the long run. CTBS • Toss cups to find an experimental probability where the LV16.53 outcomes are not equally likely. ITBS LV12.PS Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, S10 Student Activity CD-ROM, www.PHSchool.com Int2.DSP, TV Materials: Student notebooks, Paper cups LV16.15 Pacing: 45 minutes 1. LAUNCH (10 minutes) Targeted Resources Tell the story of Kalvin’s idea for eating more Cocoa Blast. Ask students to predict if a paper cup is better than a coin for Kalvin to use. Talk with your students about the differences between using coins and using paper cups to generate random events. • Will paper cups behave like coins and land on an end or a side about the same number of times? Why or why not? • Do you think the paper cup is more likely to land on its side or on one of its ends? • What are some ways that you can toss a cup? Encourage students to handle the paper cup with care, so that it does not become misshapen. Changing the shape of the cup can affect the outcomes of the experiment. Help students develop a method for keeping track of their results. You want students, over time, to decide on record- keeping schemes for themselves. However, at this stage, they may need a few models. Have students work in pairs or small groups on the problem. 2. EXPLORE (20 minutes) Targeted Resources Encourage students to be careful about gathering their data and organizing their results. Ask groups: • Do you think your results and other groups’ results will be the same? Through your questions, help them to realize that there will be a great deal of variation among individual sets of 50 tosses, but less variation in the class’s combined results. When you notice that students have completed Questions A and B, bring the entire class together to talk about the rest of the problem 3. SUMMARIZE (15 minutes) Targeted Resources After pairs or groups have each tossed their paper cup 50 times, discuss their findings. • Did you all arrive at the same conclusion about which outcome (the paper cup landing on its side or the cup landing on one of its ends) is more likely? Combine the data from all the groups, and find the fraction (or percent) of times the paper cup landed on its side or an end as in Problem 1.1. This can help students see where the relative frequency of landing on an end begins to level off. After this discussion, return to the problem and ask students to think about Question D. 4. ASSIGNMENT GUIDE Targeted Resources Core 6–8 Answers to ACE and Mathematical Reflections Other Connections 21–23; Extensions 32; unassigned choices from previous problems Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 21, 23: Bits and Pieces III How Likely Is It?, Investigation 1, Problem 1.3 Completed One More Try Mathematical Goals . National Standards State Standards NAEP 6NJ 4.4.B.2, • Develop strategies for finding experimental probabilities for a D4a, D4c, D4g 6NJ 4.4.B.5 situation that involves tossing two coins. CAT6 • Begin to explore the notion of fair and unfair. LV16.15 CTBS Vocabulary: probability, experimental probability LV16.53 Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, ITBS Student Activity CD-ROM, www.PHSchool.com LV12.PS Materials: Student notebooks, Coins S10 Pacing: 45 minutes Int2.DSP, TV LV16.15 1. LAUNCH (10 minutes) Targeted Resources After discussing the vocabulary and notation, you will want to talk to students about how Kalvin uses two coins to determine his breakfast. If the coins match, he gets to eat Cocoa Blast and if they do not match, he eats Health Nut Flakes. • If Kalvin uses this method for the month of June, how many days in June do you think Kalvin will eat Cocoa Blast? Save these conjectures on the board or a transparency to revisit in the summary. Demonstrate how to record the data for tossing two coins. Have students experiment in pairs or groups for Question A. 2. EXPLORE (20 minutes) Targeted Resources After the students have collected their data in Question A, bring the class back together to compile the data in order to discuss Question B. Because the two coins are the same, students may not see heads-tails as being different from tails-heads. It is probably better to wait until the summary to address this concern, when students will be seeking an explanation for why match and no-match are equally likely. Give the class time to complete Questions C and D on their own and save the discussion for the summary. 3. SUMMARIZE (15 minutes) Targeted Resources Question C encourages students to think about theoretical probability. This term will be defined in Investigation 2. For now, try to encourage students to seek explanations for the probabilities in this problem. • So what did you find out about the probability of a match and the probability of a no-match? • Why is the probability of a match the same as a no-match? If students are struggling with the realization that there are two ways to get a no-match, consider the suggestions in the Mathematics Background on pages 4 and 5 on strategies for finding outcomes. Question D revisits the Law of Large Numbers with a new and unfamiliar experiment. Discuss this as a class. • What does the numerator in each probability tell us? • What does the denominator in each probability tell us? • Which probability is greater? 4. ASSIGNMENT GUIDE Targeted Resources Core 9, 10, 24, 25 Answers to ACE and Other Connections 26; unassigned choices from previous problems Mathematical Reflections Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 24: Prime Time; 25: Bits and Pieces I How Likely Is It?, Investigation 1, Problem 1.4 Completed Analyzing Events Mathematical Goals . National Standards State Standards • Understand the concepts of equally likely and not equally NAEP 6NJ 4.4.B.1, likely. D4a, D4c, D4g 6NJ 4.4.C.3, CAT6 6NJ 4.4.B.5 Vocabulary: equally likely LV16.15 Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, CTBS Student Activity CD-ROM, www.PHSchool.com LV16.53 Materials: Student notebooks, Overhead projector ITBS Pacing: 45 minutes LV12.PS S10 Int2.DSP, TV LV16.15 1. LAUNCH (10 minutes) Targeted Resources Tell the story of Kalvin’s coin and his mother’s suspicion that it is not a fair Transparency 1.4 Table coin. Ask students: • Why do you think Kalvin’s mother is suspicious of the coin? • What do you think it means for a coin to be “fair”? Review his mother’s example (names chosen from a hat) to illustrate the difference between equally likely and not equally likely. Students might need help understanding how the table in the problem is structured. • The “Action” column explains the situation; the “Possible results” column describes the things, or outcomes, that can happen. You may want to read the first entry in the table together. Have students work on the problem individually and then gather in pairs or small groups to discuss their answers. 2. EXPLORE (20 minutes) Targeted Resources Make sure groups understand that they should try to reach consensus about each situation. Listen to how students defend their answers, and be on the lookout for inventive ways of arguing for a particular answer. In parts (5)–(7) of Question A, first determine the set of possible results for each action, and then determine if the results are equally likely. Note the results each pair or group generates. Encourage them to convince themselves that they have found all the possible results for each action. 3. SUMMARIZE (15 minutes) Targeted Resources Discuss the groups’ answers as a class. There may be some disagreement. Encourage students to explain their reasoning as they share their answers. Record some of the actions that students make up to illustrate equally likely events. 4. ASSIGNMENT GUIDE Targeted Resources Core 11–17 Answers to ACE and Other Applications 18,Connections 27–30; unassigned choices from Mathematical Reflections previous problems Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 27–30: Data About Us How Likely Is It?, Investigation 1 Completed Completing the Investigation Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, Student Activity CD-ROM, www.PHSchool.com Pacing: 65 minutes 1. MATHEMATICAL REFLECTIONS (25 minutes) Targeted Resources Students reflect on and summarize their learning at the end of the Answers to ACE and investigation. Mathematical Reflections 2. ASSESSMENT (40 minutes) Targeted Resources Assessment and Practice Additional Practice: Investigation 1 Use these resources to assess students’ understanding of the mathematics taught in this investigation, to create additional worksheets, Multiple Choice Items or to provide additional practice problems. Question Bank Additional Practice and Skills These questions can also be found on the ExamView CD-ROM. Workbook Answers Assessment Answers ExamView CD–ROM Questions ExamView CD–ROM Adapted Questions 3. SPANISH ASSESSMENT Targeted Resources Assessment and Practice Spanish Additional Practice: Investigation 1 Use these resources, in Spanish, to assess students’ understanding of the mathematics taught in this investigation, to create additional worksheets, Spanish Multiple Choice or to provide additional practice problems. Items These questions can also be found on the ExamView CD-ROM. Spanish Question Bank Spanish Assessment Answers Spanish ExamView CD–ROM Questions How Likely Is It?, Investigation 2, Problem 2.1 Completed Predicting to Win Mathematical Goals National Standards State Standards NAEP 6NJ 4.4.B.2, • Find the theoretical probability by analyzing the possible D4d, D4e, D4f 6NJ 4.4.B.4 equally likely outcomes involved in a game of guessing the CAT6 color of a block. LV16.15 • Compare the experimental and theoretical probabilities. CTBS LV16.53 Vocabulary: outcomes, theoretical probability ITBS Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, LV12.PS Student Activity CD-ROM, www.PHSchool.com S10 Materials: Student notebooks, Opaque container with 9 red, 6 Int2.DSP, TV yellow, and 3 blue block (you may substitute other LV16.15 objects) Pacing: 45 minutes 1. LAUNCH (10 minutes) Targeted Resources The answers for Problem 2.1 are based on a bucket containing 9 red, 6 yellow and 3 blue blocks. Tell the story about the Gee Whiz Everyone Wins! game show. Then ask the following questions: • What do you think random means? • Suppose you are a member of the audience. Would you rather be called to the stage first or last? Why? • We are going to play this game as a class. I will be the host, and you will be the contestants. You will take turns choosing from the container. We need to keep careful track of the outcome of each choice. After each turn, we will return the container to its original state by replacing the block and then mixing the blocks. Why do we need to do this? Do Question A as a whole-class activity. Ask students one at a time to predict the color, then let them choose a block. After finding the color of the block, have them replace the block. Have one student keep track of the colors chosen with a chart with tally marks at the board. • Are the data that are accumulating on the board useful for making a prediction for the block color? Why? When all students have had a turn ask the class: • What is the experimental probability of choosing each color? Have students complete Questions B, C, and D as a whole class or in pairs. 2. EXPLORE (20 minutes) Targeted Resources If parts of the problem are done in groups, be sure that students are analyzing the data correctly. Since there are 9 red blocks, 6 yellow blocks and 3 blue blocks, there is a total of 18 blocks. So the probability of 9 1 choosing a red is or . 18 2 3. SUMMARIZE (15 minutes) Targeted Resources Explain that the probabilities found in Question B are called theoretical probabilities. Talk about why this is a sensible name. Specifically, they are probabilities derived from theory (what we think ought to happen) without having performed any experiment. Discuss parts (1) and (2) of Question C: • Are the probabilities of choosing a blue block, a red block, or a yellow block equally likely? Discuss Question D. Ask students once again: • Is there an advantage to going first or going last? Students should recognize that there is an advantage to going last because you have more information about what is in the bucket. A guess based on good experimental probabilities can be more predictive of what will happen over time. 4. ASSIGNMENT GUIDE Targeted Resources Core 1–2, 14–16 Answers to ACE and Mathematical Reflections Other Connections 13; Extensions 34 Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 13, 16: Bits and Pieces I; 14, 15: Bits and Pieces II How Likely Is It?, Investigation 2, Problem 2.2 Completed Exploring Probabilities Mathematical Goal National Standards State Standards • Observe some properties of probability. NAEP 6NJ 4.4.B.1, D4d, D4e, D4f 6NJ 4.4.B.2, Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, CAT6 6NJ 4.4.B.3 Student Activity CD-ROM, www.PHSchool.com LV16.15 Materials: Student notebooks, Opaque containers, Colored CTBS blocks LV16.53 ITBS Pacing: 45 minutes LV12.PS S10 Int2.DSP, TV LV16.15 1. LAUNCH (10 minutes) Targeted Resources Briefly review the ideas in this Investigation, including theoretical probability, notation, and the context from Problem 2.1. You might want to have a brief discussion about the use of the words “or” and “and”. An “or’ statement is true if one or both of the phases (or probabilities) occur. So, 6 in Problem 2.1, P(red) or P(yellow) occurring is P(red) + P(yellow) = + 12 2 8 = . An “and” statement is true if both events (probabilities) occur. 12 12 Then let students work on the problem in pairs. 2. EXPLORE (20 minutes) Targeted Resources As students work, pay attention to which ideas seem to be difficult for them. If students struggle with the probability of something not happening, you may want to ask them about what could happen, rather than pointing them to a formula. After students have found that choosing red or choosing yellow is the same as not choosing a blue, you might ask: • What happens if we add P(blue) and P(not blue)? 3. SUMMARIZE (15 minutes) Targeted Resources Since there are lots of small questions in this problem, you should focus your summary around one or two of the big ideas your students have found interesting or difficult. You might ask questions like the following: • What does it mean to find the probability of something not happening? • How can you compute the probability of something not happening? • How can you find P(red or yellow)? • Can a probability ever be a value greater than 1? • Can a probability ever be 0? Give an example from this problem. • Can a probability ever be 1? Give an example from this problem. Questions C and D provide you with an opportunity to assess students’ understanding of equivalent fractions and probability. After discussing these and any other ideas from your class, you may want to finish by having students individually work on ACE Exercise 3. 4. ASSIGNMENT GUIDE Targeted Resources Core 3–6 Answers to ACE and Other Connections 17–24; unassigned choices from previous problems Mathematical Reflections Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. How Likely Is It?, Investigation 2, Problem 2.3 Completed Winning the Bonus Prize Mathematical Goals National Standards State Standards NAEP 6NJ 4.4.B.4, • Use organized lists and tree diagrams to find theoretical D4d, D4e, D4f 6NJ 4.4.B.2 probabilities. CAT6 • Understand that experimental probabilities are better LV16.15 estimates of theoretical probabilities when based on larger CTBS numbers of trials. LV16.53 ITBS Vocabulary: counting tree LV12.PS Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, S10 Student Activity CD-ROM, www.PHSchool.com Int2.DSP, TV Materials: Student notebooks, Overhead projector, Two opaque LV16.15 containers filled with 1 red, 1yellow, and 1 blue block, Chart paper or blank transparencies (optional) Pacing: 45 minutes 1. LAUNCH (10 minutes) Targeted Resources Use the opening paragraphs of Problem 2.3 to introduce tree diagrams. Transparency 2.3 Tree Demonstrate how to make a tree diagram and ask students questions Diagrams modeling those they will need to ask themselves: If I toss two coins, • What are the possible outcomes for the first coin? • Are these outcomes equally likely? • If you get heads with the first coin, what are the possible outcomes for the second coin? Are these outcomes equally likely? • If you get tails with the first coin, what are the possible outcomes for the second coin? • If you toss two coins, what is the probability that the coins will match? What is the probability that they won’t match? Discuss the bonus game with students. Make sure that students realize that contestants must make a prediction before choosing a block from each bag. So they must say, for example, “blue from Bag 1 and red from bag 2.” Ask students to make a prediction: • What are the contestant’s chances of winning this game? This activity works well either as a whole-class experiment or in groups, each group having two containers of blocks. 2. EXPLORE (20 minutes) Targeted Resources Allow students to experiment and decide when they have enough data to make a good estimate of a contestant’s chances of winning. If students do the activity in groups, pool class’s data before students answer Question B. Some may need help making a tree diagram. Have them ask themselves: • What are the possible outcomes for the first choice? For the second choice? • How do you know when you have all possible outcomes? Students may need support in determining which of the outcomes represent a win for the contestant. Remind students that contestants are given one guess to correctly predict the color of block chosen from both bags. Give chart paper or transparencies to each group to record their strategies for finding the theoretical probabilities 3. SUMMARIZE (15 minutes) Targeted Resources •What are the experimental probabilities for predicting each pair of colors? • How did you determine the theoretical probabilities? Compare the theoretical probabilities to the experimental probabilities. Have a class discussion about the kind of outcomes in Problem 2.1 and the kind of outcomes in this problem. 4. ASSIGNMENT GUIDE Targeted Resources Core 7–9 2ACE Exercise 7 Adapted Other Connections 25–31; Extensions 35, 36; unassigned choices from Version previous problems Answers to ACE and Adapted For suggestions about adapting Exercise 7 and other ACE Mathematical Reflections exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 25–28: Bits and Pieces I; 29: Data About Us; 30, 31: Bits and Pieces II How Likely Is It?, Investigation 2, Problem 2.4 Completed Pondering Possible and Probable Mathematical Goals National Standards State Standards NAEP 6NJ 4.4.C.1, • Deepen understanding of equally likely and not equally likely. D4d, D4e, D4f 6NJ 4.4.B.2, • Understand that a game of chance is fair only if each player CAT6 6NJ 4.4.B.4 has the same chance of winning, not just a possible chance LV16.15 of winning. CTBS LV16.53 Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, ITBS Student Activity CD-ROM, www.PHSchool.com LV12.PS Materials: Student notebooks, Coins S10 Pacing: 45 minutes Int2.DSP, TV LV16.15 1. LAUNCH (10 minutes) Targeted Resources A good launching strategy is to play the coin-tossing game with the class so students understand how to score and take turns. Initiate a discussion about what it means for a game to be fair. • What do you think it means to say that a game is a fair game? Make sure students understand that you are referring to games of chance (such as bingo) rather than games of skill (such as tennis). The coin-tossing game is a two-person game, so students need to think in terms of two players. In fair games of chance, each player has an equal chance, or probability, of winning. You may want to collect experimental data as a whole class. If you do this, record the probabilities on the overhead or board. Let students know that they are to examine the coin-tossing game for fairness. Let them work on the problem in pairs. 2. EXPLORE (20 minutes) Targeted Resources As you listen to students discuss the game, remind them that they must be prepared to explain why they think the game is fair or unfair. 3. SUMMARIZE (15 minutes) Targeted Resources If you collected experimental data, then you can use this to help students see the variability in the results of all the games that were played. From this data, you and the class can determine relative frequencies of getting all heads, all tails, or a matching pair. To determine the theoretical probabilities, ask the class to share their strategies. Be sure that both an organized list and tree diagram are discussed. The outcomes are: TTT, TTH, THT, HTT, THH, HTH, HHT and HHH. Each of the eight possibilities has the same chance of occurring. However, in this game we are 1 interested in three of a kind or a pair. P(three heads or three tails) = 2( ) 8 2 1 1 6 3 = = and P(two heads or two tails) = 6( ) = = . This means that 8 4 8 8 4 one player has a much better chance of scoring and therefore of winning. • How can you make the game fair? You can use the outcomes to ask other questions about this experiment. For example, • What is the probability of getting exactly 0 heads? Exactly 1 head? Exactly 2 heads? Exactly 3 heads? Two heads and one tail? 4. ASSIGNMENT GUIDE Targeted Resources Core 10–12 Answers to ACE and Mathematical Reflections Other Connections 32, 33; Extensions 37; unassigned choices from previous problems Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. How Likely Is It?, Investigation 2 Completed Completing the Investigation Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, Student Activity CD-ROM, www.PHSchool.com Pacing: 65 minutes 1. MATHEMATICAL REFLECTIONS (25 minutes) Targeted Resources Students reflect on and summarize their learning at the end of the Answers to ACE and investigation. Mathematical Reflections 2. ASSESSMENT (40 minutes) Targeted Resources Assessment and Practice Additional Practice: Investigation 2 Use these resources to assess students’ understanding of the mathematics taught in this investigation, to create additional worksheets, Skill Practice: Probability or to provide additional practice problems. Check–Up Multiple Choice Items These questions can also be found on the ExamView CD-ROM. Question Bank Additional Practice and Skills Workbook Answers Assessment Answers ExamView CD–ROM Questions ExamView CD–ROM Adapted Questions 3. SPANISH ASSESSMENT Targeted Resources Assessment and Practice Spanish Additional Practice: Investigation 2 Use these resources, in Spanish, to assess students’ understanding of the mathematics taught in this investigation, to create additional worksheets, Spanish Skill Practice: or to provide additional practice problems. Probability These questions can also be found on the ExamView CD-ROM. Spanish Check–Up Spanish Multiple Choice Items Spanish Question Bank Spanish Assessment Answers Spanish ExamView CD–ROM Questions How Likely Is It?, Investigation 3, Problem 3.1 Completed Designing a Spinner Mathematical Goal National Standards State Standards • Develop strategies for finding experimental and theoretical NAEP 6NJ 4.4.B.4, probabilities in situations involving spinners. D4b, D4c, D4d, D4j 6NJ 4.4.B.1 CAT6 Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, LV16.15 Student Activity CD-ROM, www.PHSchool.com CTBS Materials: Student notebooks, Overhead projector, Bobby pins or LV16.53 ITBS paper clips, Angle rulers LV12.PS Pacing: 45 minutes S10 Int2.DSP, TV LV16.15 1. LAUNCH (10 minutes) Targeted Resources Discuss the three spinners in the Getting Ready. Discuss Kalvin’s idea Transparency 3.1A Getting about determining his bedtime. Then pose the questions: Ready • Which spinner will give Kalvin the best chance of going to bed at Transparency 3.1B Kalvin's 11:00? Explain your reasoning. Spinner Labsheet 3.1 Kalvin's Spinner Discuss the need for students to make sure that the way they conduct the experiment won’t affect, or bias, the outcome. Ask the class: • What kinds of things can happen to affect the data you gather in this problem? • How should you spin the pointer to be sure you have a fair trial? Explain to students that Kalvin decided to make his own spinner to use in determining his bedtime. If possible, demonstrate a couple of spins on Transparency 3.1B. Record your data, demonstrating how to keep a tally for each spin. After spinning two or three times, ask whether you have a large enough sample. Distribute Labsheet 3.1 to each pair or group of students. 2. EXPLORE (20 minutes) Targeted Resources • Is there a tendency for the pointer to land in the same area each time? Students should find that Kalvin’s chances of going to bed at 11:00 are about (37.5%), the same as for 9:00. Ask them to compare their experimental and theoretical probabilities. Focus on how students are finding the theoretical probabilities. Many students will want to compare the number of sections rather than the relative sizes of the sections. If students are struggling with thinking about the sizes of the angles of the sections, ask them the following question: • Look at one of these 9:00 sections. Is the pointer as likely to land there as in the 11:00 section? • What is the angle size associated with each section? Encourage students to be specific about why the 11:00 section is more likely than the 9:00 section. 3. SUMMARIZE (15 minutes) Targeted Resources Discuss the data that groups collected and their answers to the questions. Transparency 3.1C Another Combine the data for all the groups, recalculating the fractions or percents Spinner after each groups’ data is added. Students should conclude that Kalvin’s chances of going to bed at 11:00 are about 38%. To check understanding of determining theoretical probabilities, put up Transparency 3.1C. • If Kalvin uses this spinner, what is the probability that he will go to bed at 10:00? At 11:00? Why? 4. ASSIGNMENT GUIDE Targeted Resources Core 1, 3, 4, 11–17 Labsheet 3ACE Exercise 1 3ACE Exercise 2 Adapted Other Applications 2, 5, 6; Connections 18–21; Extensions 31, 32 Version Labsheet 3ACE Exercise 1 is provided if Exercise 1 is assigned. Answers to ACE and Adapted For suggestions about adapting Exercise 2 and other ACE Mathematical Reflections exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 11–16, 19–21: Bits and Pieces I How Likely Is It?, Investigation 3, Problem 3.2 Completed Making Decisions Mathematical Goals National Standards State Standards NAEP 6NJ 4.4.B.5, • Analyze probability situations. D4b, D4c, D4d, D4j 6NJ 4.4.D.1 • Use probability to make decisions. CAT6 • Decide whether the probability situations are fair or unfair. LV16.15 CTBS Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, LV16.53 Student Activity CD-ROM, www.PHSchool.com ITBS Materials: Student notebooks, Overhead projector, Paper clips or LV12.PS bobby pins, number cubes, coins, colored blocks, and S10 bags (optional) Int2.DSP, TV Pacing: 45 minutes LV16.15 1. LAUNCH (10 minutes) Targeted Resources Discuss the Getting Ready question with students. They will quickly see Transparency 3.2 Getting that tossing a coin once to decide which of three people should go to the Ready office makes no sense. Students may have suggestions for how to use a number cube (e.g., 1 and 2 represent Billie, 3 and 4 represent Evo, and 5 and 6 represent Carla), colored blocks in a bag or a spinner. In each case, it is important that there are three equally likely outcomes. Help students to see this and to express precisely what those outcomes are. Suggested Questions When students make suggestions, ask them: • What are the outcomes? • Are these outcomes equally likely? Make sure students understand that they are to evaluate each student’s suggestion in each part of the problem. Allow them to work in pairs or small groups. You may want to have blank spinners (Labsheet 3.2), number cubes, coins, colored blocks, and bags available for students who want to test their ideas. 2. EXPLORE (20 minutes) Targeted Resources As students work, listen for whether they are being precise about the Labsheet 3.2 Blank Spinner outcomes in each suggestion. For example, in Questions B, Tony’s suggestion is that an outcome of 3 or 6 corresponds to his team while an outcome of 1, 2, 4, or 5 corresponds to Meda’s team. These are two outcomes and the latter is more likely. Pay attention to whether students are able to fill in the missing gap in Sal’s suggestion in Question C. If the number from the first bag is the tens digit and the number from the second bag is the units digit, then the suggestion is a fair way to choose the winner. This is a different simulation from what has been done in the unit so far. Students may need help thinking about this. If a student has the number 0, then zero must be chosen from both bags. So 00 represents the number 0 and 05 (choosing 0 from the first bag and 5 from the second bag) represents the number 5. 3. SUMMARIZE (15 minutes) Targeted Resources Base your summary on what students did and did not understand in the problem. Make sure that students can specify the outcomes in each suggestion and determine whether each is a fair way to make a decision. If your students struggled, you will want to spend time carefully discussing the various outcomes and ways to determine whether they are fair. Suggested Questions You might ask a series of questions like the following: • What are the possible outcomes of when you roll a number cube? (1, 2, 3, 4, 5, or 6) • What are the possible outcomes of Ava’s suggestion? (There are 5: kickball, soccer, baseball, dodge ball, or roll again.) • Are all five of these outcomes equally likely? (No, roll again is most 2 likely. The probability is .) 6 • Are all four of the games equally likely? (Yes, on any roll, the 1 probability of each game is .) 6 • Is this a fair way to make the decision? (Yes, because the probability for each game is equally likely.) Have students talk about Huey’s suggestion in Question C. Here, the outcomes are not equally likely. The easiest way to see this is to note that the smallest possible sum is 10. Therefore, students 1 through 9 would not even have a chance. • How does Student 60 win? (If all ten number cubes came up with a 6.) • How does Student 50 win? (Possible answers: All ten number cubes could come up 5 or eight of them could come up 5, one comes up 6, and one comes up 4.) • How does Student 10 win? (If all the number cubes come up with a 1.) • Does every student have an equal chance of winning? (No; Student 50 has lots of ways to win while Student 60 has only one and Student 1 has no chance.) • What is the probability of Student 1 winning? [P(Student 1 wins) = 0] Have students discuss Sal’s method. You may want to demonstrate how to choose and determine the numbers. • Is it possible to choose each number from 0 to 59? [Yes; to choose 0 you must choose zero from both bags. So 00 represents the number 0 and 05 (choosing 0 from the first bag and 5 from the second bag) represents the number 5. You may want to make an organized list to show that choosing each number is equally likely.] Check for Understanding Ask students to pick one question from Questions A–C. Determine one more way to make the decision fairly. 4. ASSIGNMENT GUIDE Targeted Resources Core 8, 9 Answers to ACE and Mathematical Reflections Other Applications 7; Connections 22–26; unassigned choices from previous problems Adapted For suggestions about adapting Exercise 2 and other ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 22 –25; Bits and Pieces I How Likely Is It?, Investigation 3, Problem 3.3 Completed Scratching Spots Mathematical Goals National Standards State Standards • Develop strategies for finding both experimental (using a NAEP 6NJ 4.4.B.4, simulation of a scratch-off prize card) and theoretical D4b, D4c, D4d, D4j 6NJ 4.4.B.2 CAT6 probabilities (analyzing the scratch-off prize card). LV16.15 Vocabulary: simulation CTBS Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, LV16.53 ITBS Student Activity CD-ROM, www.PHSchool.com LV12.PS Materials: Student notebooks, Paper clips or bobby pins, number S10 cubes, coins, colored blocks, and bags (optional) Int2.DSP, TV Pacing: 45 minutes LV16.15 1. LAUNCH (10 minutes) Targeted Resources Describe the situation to the students and discuss possible ways of finding the experimental probability of winning a prize with one scratch-off prize card. This will involve doing a simulation. Collect various simulation suggestions. Analyze their suggestions with them. NOTE: You may want the class to try different simulations, but it might be easier if the class uses the same simulation. One way to simulate the game is to put two red blocks, one white block, one yellow block, and one blue block in an opaque container, then choose two blocks at a time. When you choose the two red blocks, you simulate scratching off the two matching spots. Another possibility is to have five students act as the “spots” in the front of the room. They hold tags that are hidden from the rest of the class. Two students hold tags with matching symbols, and three hold tags with three other non-matching symbols. A seated student chooses two of the five students to reveal their tags. After each guess, students mix up their tags. Let the class explore the problem in groups of two to four students. (The way you decide to arrange it depends on whether you do the same simulation as a class or if students all do different simulations.) 2. EXPLORE (20 minutes) Targeted Resources If you let the groups decide on their own simulation, then each group must Labsheet 3.2 Blank Spinner decide for themselves how to collect the data and when they have collected enough data to find a reasonable experimental probability. Be sure that each method is mathematically equivalent to the scratch-off prize card. Allow the groups to plan and carry out their simulations, collect their data, and find the experimental probability. When they have completed their experiment, students should move on to Question B in which they analyze the outcomes and find theoretical probabilities and then do the rest of the problem. 3. SUMMARIZE (15 minutes) Targeted Resources Have the groups share their solutions and strategies with the class. If groups chose their own simulation, discuss the different methods. • When you were finding the theoretical probability, what strategy did you use to find all the outcomes? A list shows that there are ten ways to scratch off the spots. One of these 1 ten combinations is the matching pair, so the probability of winning is . 10 Notice that the order in which the spots in a pair are chosen does not matter. Students may want to list each pair twice with the letters reversed (to represent two cards). There would be 20 pairs (2 of which are winners) 2 1 which also gives a probability of or . 20 10 4. ASSIGNMENT GUIDE Targeted Resources Core 10, 27 Answers to ACE and Mathematical Reflections Other Connections 28–30; Extensions 33–35; unassigned choices from previous problems Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 29: Covering and Surrounding How Likely Is It?, Investigation 3 Completed Completing the Investigation Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, Student Activity CD-ROM, www.PHSchool.com Pacing: 65 minutes 1. MATHEMATICAL REFLECTIONS (25 minutes) Targeted Resources Students reflect on and summarize their learning at the end of the Answers to ACE and investigation. Mathematical Reflections 2. ASSESSMENT (40 minutes) Targeted Resources Assessment and Practice Additional Practice: Investigation 3 Use these resources to assess students’ understanding of the mathematics taught in this investigation, to create additional worksheets, Partner Quiz or to provide additional practice problems. Multiple Choice Items Question Bank These questions can also be found on the ExamView CD-ROM. Additional Practice and Skills Workbook Answers Assessment Answers ExamView CD–ROM Questions ExamView CD–ROM Adapted Questions 3. SPANISH ASSESSMENT Targeted Resources Assessment and Practice Spanish Additional Practice: Investigation 3 Use these resources, in Spanish, to assess students’ understanding of the mathematics taught in this investigation, to create additional worksheets, Spanish Partner Quiz or to provide additional practice problems. Spanish Multiple Choice These questions can also be found on the ExamView CD-ROM. Items Spanish Question Bank Spanish Assessment Answers Spanish ExamView CD–ROM Questions How Likely Is It?, Investigation 4, Problem 4.1 Completed Genetic Traits Mathematical Goals National Standards State Standards • Use class and survey data to find the experimental NAEP 6NJ 4.4.B.2, probabilities for certain genetic traits. D4b, D4d, D4e, D4f, 6NJ 4.4.B.5 D4g, D4j Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, CAT6 Student Activity CD-ROM, www.PHSchool.com LV16.15 Materials: Student notebooks, Overhead projector CTBS Pacing: 45 minutes LV16.53 ITBS LV12.PS S10 Int2.DSP, TV LV16.15 1. LAUNCH (10 minutes) Targeted Resources Discuss each of the characteristics described in the Student Edition with Transparency 4.1 Generic your class. Collect your class data for these four traits and record the data Traits in a table like the one in the Student Edition. Students will use this data to complete Question B. Let students work in groups of two or three. 2. EXPLORE (20 minutes) Targeted Resources Parts (3) and (4) of Questions B are proportional scaling questions that some students at this level may find difficult. Since they have not formally been taught the strategies needed to solve these problems, note student reasoning for these questions. Help students make connections to their work with equivalent fractions in Bits and Pieces I, their work with percents in Bits and Pieces III, and to any other novel solution strategies that students in your class may offer. 3. SUMMARIZE (15 minutes) Targeted Resources As a class, discuss Question B, part (1). Ask students about one of the traits that had a relatively high probability of occurring within your 3 classroom (for example, say attached earlobe had a probability of .) 4 • Do you feel confident that a student selected at random from this school will have an attached earlobe? You want students to realize that their classroom data serves as an estimate of the distribution of the trait in the larger school population. You may ask: • What could we do to increase our confidence that a student selected at random from this school will have an attached earlobe? Students may see that a larger sample would be more predictive and may want to collect additional data. Discuss this last point in the context of Question C, part (2) about the national data. • Is our class representative of the nation as a whole? 4. ASSIGNMENT GUIDE Targeted Resources Core 1, 2 4ACE Exercise 2 Adapted Version Other Connections 13–17 Answers to ACE and Adapted For suggestions about adapting Exercise 2 and other ACE Mathematical Reflections exercises, see the CMP Special Needs Handbook. How Likely Is It?, Investigation 4, Problem 4.2 Completed Tracing Traits Mathematical Goals National Standards State Standards NAEP 6NJ 4.4.B.4, • Appreciate the power of probability for making predictions. D4b, D4d, D4e, D4f, 6NJ 4.4.B.2 • Develop strategies (for example, using a chart or table) for D4g, D4j finding theoretical probabilities involving genetics. CAT6 LV16.15 Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, CTBS Student Activity CD-ROM, www.PHSchool.com LV16.53 Materials: Student notebooks, Overhead projector ITBS Pacing: 45 minutes LV12.PS S10 Int2.DSP, TV LV16.15 1. LAUNCH (10 minutes) Targeted Resources Discuss what a geneticist is and does. The Student Edition presents a Transparency 4.2 Tracing summary of the genetics information students need to know. Have Traits students read the genetics information in the Student Edition, or study the information yourself and present it to them. They may already be familiar with the term gene. An allele is a certain form of a gene and is the more correct term to use when discussing the different forms of a gene, such as dominant and recessive. You may want to recommend further reading for students who are especially interested in this topic. As you talk about the example of Bonnie and Evan, emphasize that the four possibilities are equally likely, and therefore we can conclude that the 2 1 probability of the baby having attached earlobes is or , and the 4 2 1 probability of having nonattached earlobes is also . 2 Students may wonder how someone could figure out what his or her parents’ alleles were. One possibility: suppose that Dasan has attached earlobes and both of his parents do not. Dasan’s earlobe alleles must be ee, which means that both of his parents must have at least one e allele. Since both parents have nonattached earlobes, each must also have an E, giving them both Ee. Have students work in pairs or groups of three. 2. EXPLORE (20 minutes) Targeted Resources Have students analyze each family situation in the problem. Encourage groups to make a chart as shown in the example about Bonnie and Evan to help them list the possibilities. 3. SUMMARIZE (15 minutes) Targeted Resources Review the groups’ charts. Make sure students understand how to analyze the possibilities. Any combination containing the dominant allele E 1 means the child will have nonattached earlobes. There is only a 4 probability that the child will inherit the ee combination and have attached earlobes. In the general population, having attached earlobes is more rare than having nonattached earlobes. After Question C, you may want to ask: • What if Eileen’s parents had nonattached earlobes? Can you find the probability that Eileen has nonattached earlobes? Students may be interested in knowing that these charts or tables are called Punnett Squares in biology. It is an analysis tool for simple genetic inheritance problems and is a common topic in middle–school biology. 4. ASSIGNMENT GUIDE Targeted Resources Core 3–7, 18 Answers to ACE and Mathematical Reflections Other Connections 19; Extensions 27; unassigned choices from previous problems Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 19: Shapes and Designs How Likely Is It?, Investigation 4, Problem 4.3 Completed Roller Derby Mathematical Goals National Standards State Standards NAEP 6NJ 4.4.D.1, • Appreciate the power of probability in determining strategies D4b, D4d, D4e, D4f, 6NJ 4.4.B.5 for winning a game. D4g, D4j • Develop strategies (i.e. using a chart or table) for finding CAT6 both experimental and theoretical probabilities of winning a LV16.15 game. CTBS LV16.53 Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, ITBS Student Activity CD-ROM, www.PHSchool.com LV12.PS Materials: Student notebooks, Overhead projector, Two number S10 cubes, Game markers Int2.DSP, TV Pacing: 45 minutes LV16.15 1. LAUNCH (10 minutes) Targeted Resources Discuss the directions for the game. Illustrate the game using Transparency 4.3 Roller Transparency 4.3. Explain that if the column corresponding to a roll is Derby empty, no marker is removed. • What is your best guess at a strategy for placing your markers so that you will be able to remove them before your opponents remove their markers? Try to remain neutral when students make their suggestions. As they play they will become more aware of which strategies work. Remember the goal of the game is to remove all markers. Refer students to the Roller Derby rules. Divide the class into teams of two, and then pair teams to play against one another. Tell students to discuss strategies with their teammates, not their opponents. 2. EXPLORE (20 minutes) Targeted Resources While students play the game, observe the strategies they use and the Labsheet 4.3 knowledge they display about sums of the numbers on the cubes. Take note of the strategies they use. 3. SUMMARIZE (15 minutes) Targeted Resources After students have played the game a few times and have had time to work through Question A, ask them again about winning strategies. • What was the first strategy your team used for placing markers in the 12 columns? • Did you change your strategy the second time you played? Why or why not? If you were going to play the game again, what strategy would you use now? Why? Question C asks students to list all the possible outcomes when they roll two number cubes. The list should allow students to determine all the possible sums and to see all the ways each sum can occur. • What strategies did you use for making your list? One way to see all 36 sums is to use a table with possible outcomes of one number cube down the side and possible outcomes of the other number cube along the top. Once the class has made the table, you may want to ask: • When you roll two number cubes, how many different number pairs are possible? Are these pairs equally likely? • How many different sums are possible? Are these sums equally likely? • When you roll two number cubes, what is the probability that the sum will be 12? That the sum will be 1? That the sum will be 0? • How is the table in this problem like the chart in Problem 4.2? 4. ASSIGNMENT GUIDE Targeted Resources Core 8–12 Answers to ACE and Mathematical Reflections Other Connections 2–26; unassigned choices from previous problems Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 20–23: Prime Time How Likely Is It?, Investigation 4 Completed Completing the Investigation National Standards NAEP Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, Student Activity CD-ROM, D4b, D4d, D4e, D4f, www.PHSchool.com D4g, D4j Pacing: 40 minutes CAT6 LV16.15 CTBS LV16.53 ITBS LV12.PS S10 Int2.DSP, TV LV16.15 1. ASSESSMENT (40 minutes) Targeted Resources Assessment and Practice Additional Practice: Investigation 4 Use these resources to assess students’ understanding of the mathematics taught in this investigation, to create additional worksheets, Skill Practice: Experimental or to provide additional practice problems. and Theoretical Probability Multiple Choice Items These questions can also be found on the ExamView CD-ROM. Question Bank Additional Practice and Skills Workbook Answers Assessment Answers ExamView CD–ROM Questions ExamView CD–ROM Adapted Questions 2. SPANISH ASSESSMENT Targeted Resources Assessment and Practice Spanish Additional Practice: Investigation 4 Use these resources, in Spanish, to assess students’ understanding of the mathematics taught in this investigation, to create additional worksheets, Spanish Skill Practice: or to provide additional practice problems. Experimental and Theoretical Probability These questions can also be found on the ExamView CD-ROM. Spanish Multiple Choice Items Spanish Question Bank Spanish Assessment Answers Spanish ExamView CD–ROM Questions