Sets, Probability, Statistics I

Transcription

1 Sets, Probability, Statistics I Frank C. Wilson Real Lives. Real Learning. by Real Lives. Real Learning. Activity Collection ple m Fruit Snacks Fruit Snacks #2 Kinds of Candy Bars Menu Choices Phone Numbers Picking Plates - Canada Picking Plates - United States Playing Pig Playing Soccer Rolling the Dice Sa file Featuring real-world contexts:

2 Cover art: Blaine C. Wilson 2009 by Make It Real Learning Company With the purchase of this workbook, license is granted for one (1) teacher to copy the activities in this workbook for use in classes and professional development workshops. Copying pages in this workbook for any other use is prohibited without written consent from Make It Real Learning Company. For permissions, visit and complete the Contact Us form.

3 Table of Contents Introduction... 4 Activity Objectives... Fruit Snacks: Using Mean, Median, and Mode... 6 Solutions... 8 Fruit Snacks: Using Probability Solutions Kinds of Candy Bars: Working with Sets Solutions Menu Choices: Working with Combinations Solutions Phone Numbers : Using Principles of Counting Solutions Picking Plates - Canada: Using Principles of Counting Solutions Picking Plates - United States: Using Permutations Solutions Playing Pig: Using Expected Value Solutions Playing Soccer: Using Combinations and Probability Solutions Rolling the Dice: Using Probability Solutions About the Author Other Books in the Make It Real Learning Series... 46

4 Introduction When am I ever going to use this? It is a question that has plagued teachers and learners for decades. Now, with the help of the Make It Real Learning workbook series, you can answer the question. The Sets, Probability, Statistics Functions I workbook focuses on real-world situations that may be effectively analyzed using principles of probability and statistics. From determining a winning strategy for a dice game to predicting the contents of a bag of fruit snacks, learners get to use mathematics in meaningful ways. Rest assured that each activity integrates real world information not just realistic data. These are real products (e.g. fruit snacks, Almond Joy) and real world issues. The mathematical objectives of each activity are clearly specified on the Activity Objectives page following this introduction. Through the workbook series, we have consistently sought to address the content and process standards of the National Council of Teachers of Mathematics. There are multiple ways to use the activities in a teaching environment. Many teachers find that the activities are an excellent tool for stimulating mathematical discussions in a small group setting. Due to the challenging nature of each activity, group members are motivated to brainstorm problem solving strategies together. The interesting real world contexts motivate them to want to solve the problems. The activities may also be used for individual projects and class-wide discussions. As a ready-resource for teachers, the workbook also includes completely worked out solutions for each activity. To make it easier for teachers to assess student work, the solutions are included on a duplicate copy of each activity. We hope you enjoy the activities! We continue to increase the number of workbooks in the Make It Real Learning workbook series. Please visit for the most current list of activities. Thanks! Frank C. Wilson Author 4

5 Sets, Probability, Statistics I Activity Objectives Activity Title Fruit Snacks: Using Mean, Median, and Mode (p. 6) Fruit Snacks: Using Probability (p. 10) Kinds of Candy Bars: Working with Sets (p. 14) Menu Choices: Working with Combinations (p. 18) Phone Numbers: Using Principles of Counting (p. 22) Picking Plates - Canada: Using Principles of Counting (p. 26) Picking Plates - United States: Using Permutations (p. 30) Playing Pig: Using Expected Value (p. 34) Playing Soccer: Using Combinations and Probability (p. 38) Rolling the Dice: Using Probability (p. 42) Mathematical Objectives Calculate the mean, median, and mode of a data set Interpret the practical meaning of mean, median, and mode Determine the probability of an event Create a Venn diagram to represent a data set Find the intersection and union of a group of sets Find the complement of a set Use combinations to count different outcomes Use combinations and permutations to count possible outcomes Calculate percentages Use permutations to count possible outcomes Use principles of counting Use permutations to count possible outcomes Use principles of counting Calculate expected value Use numerical results to determine a game strategy Use combinations to count different outcomes Calculate theoretical probabilities Calculate the probability of an event Calculate conditional probabilities Use probabilities to develop game strategy

6 Fruit Snacks Using Mean, Median, and Mode M any people enjoy eating fruit snacks. Kirkland Signature Fruit Snacks are made from seven different fruit juices (Source: package labeling). A father opened five bags of the Fruit Snacks, counted the number of pieces in each bag, and recorded the results in the table. Bag #1 14 pieces 4 red, green, 4 orange, 1 yellow Bag #2 13 pieces 4 red, 3 green, 4 orange, 2 yellow Bag #3 12 pieces red, 3 green, 4 orange, 0 yellow Bag #4 14 pieces 2 red, 6 green, 4 orange, 2 yellow Bag # 13 pieces 2 red, 6 green, 4 orange, 1 yellow 1. What was the mean, median, and mode for the number of pieces in each bag? mean pieces The mean number of pieces in each bag is 13.2 pieces. To find the mode, we determine the value that is repeated most often. In this case, both 13 pieces and 14 pieces occur twice. So the data set has two modes: 13 pieces and 14 pieces. To find the median, we list the values in numerical order and pick the middle value. The values are 12, 13, 13, 14, and 14. The middle value is 13 so the median is Which is greater: the mean number of green pieces in each bag or the median number of green pieces in each bag? green pieces The mean number of green pieces in each bag is 4.6 pieces. To find the median, we list the number of green pieces in numerical order and pick the middle value. The values are 3, 3,, 6, and 6. The middle value is. The median value of is greater than the mean value of

7 3. For which fruit snack color are the mean, median, and mode all equal to each other? The mean, median and mode are all equal to 4 for the orange pieces. 4. Use the concepts of mean, median, and mode to explain why it is unlikely that there are 13 yellow pieces in a bag of Fruit Snacks mean pieces The mean number of pieces in each bag is 1.2 pieces. To find the mode, we determine the value that is repeated most often. In this case, both 1 piece and 2 pieces occur twice. So the data set has two modes: 1 piece and 2 pieces. To find the median, we list the values in numerical order and pick the middle value. The values are 0,1,1,2, and 2. The middle value is 1 so the median is 1. The fact that the mean, median, and mode are close to each other in value together with the fact that 13 is not at all close to the mean, median, and mode, makes it highly unlikely that there will be 13 yellow pieces in a bag.. Using the medians for the pieces of each color, predict the number of pieces of each color in an unopened bag of Fruit Snacks. Red: 2, 2, 4, 4, and red median: 4 Green: 3, 3,, 6, and 6 green median: Orange: 4, 4, 4, 4, and 4 orange median: 4 Yellow: 0, 1, 1, 2, and 2 yellow median: 1 We predict that there are 4 red pieces, green pieces, 4 orange pieces, and 1 yellow piece. 6. After predicting the number of Fruit Snacks in an unopened bag, the father opened up a sixth bag of Fruit Snacks. It contained 1 yellow piece, green pieces, 4 orange pieces, and 4 red pieces. How accurate was the prediction in Exercise? The sixth bag contained the exact number of red, green, orange, and yellow pieces predicted. (This really happened!) 7

8 Fruit Snacks Using Mean, Median, and Mode M any people enjoy eating fruit snacks. Kirkland Signature Fruit Snacks are made from seven different fruit juices (Source: package labeling). A father opened five bags of the Fruit Snacks, counted the number of pieces in each bag, and recorded the results in the table. Bag #1 14 pieces 4 red, green, 4 orange, 1 yellow Bag #2 13 pieces 4 red, 3 green, 4 orange, 2 yellow Bag #3 12 pieces red, 3 green, 4 orange, 0 yellow Bag #4 14 pieces 2 red, 6 green, 4 orange, 2 yellow Bag # 13 pieces 2 red, 6 green, 4 orange, 1 yellow 1. What was the mean, median, and mode for the number of pieces in each bag? mean pieces The mean number of pieces in each bag is 13.2 pieces. To find the mode, we determine the value that is repeated most often. In this case, both 13 pieces and 14 pieces occur twice. So the data set has two modes: 13 pieces and 14 pieces. To find the median, we list the values in numerical order and pick the middle value. The values are 12, 13, 13, 14, and 14. The middle value is 13 so the median is Which is greater: the mean number of green pieces in each bag or the median number of green pieces in each bag? green pieces The mean number of green pieces in each bag is 4.6 pieces. To find the median, we list the number of green pieces in numerical order and pick the middle value. The values are 3, 3,, 6, and 6. The middle value is. The median value of is greater than the mean value of

9 3. For which fruit snack color are the mean, median, and mode all equal to each other? The mean, median and mode are all equal to 4 for the orange pieces. 4. Use the concepts of mean, median, and mode to explain why it is unlikely that there are 13 yellow pieces in a bag of Fruit Snacks mean pieces The mean number of pieces in each bag is 1.2 pieces. To find the mode, we determine the value that is repeated most often. In this case, both 1 piece and 2 pieces occur twice. So the data set has two modes: 1 piece and 2 pieces. To find the median, we list the values in numerical order and pick the middle value. The values are 0,1,1,2, and 2. The middle value is 1 so the median is 1. The fact that the mean, median, and mode are close to each other in value together with the fact that 13 is not at all close to the mean, median, and mode, makes it highly unlikely that there will be 13 yellow pieces in a bag.. Using the medians for the pieces of each color, predict the number of pieces of each color in an unopened bag of Fruit Snacks. Red: 2, 2, 4, 4, and red median: 4 Green: 3, 3,, 6, and 6 green median: Orange: 4, 4, 4, 4, and 4 orange median: 4 Yellow: 0, 1, 1, 2, and 2 yellow median: 1 We predict that there are 4 red pieces, green pieces, 4 orange pieces, and 1 yellow piece. 6. After predicting the number of Fruit Snacks in an unopened bag, the father opened up a sixth bag of Fruit Snacks. It contained 1 yellow piece, green pieces, 4 orange pieces, and 4 red pieces. How accurate was the prediction in Exercise? The sixth bag contained the exact number of red, green, orange, and yellow pieces predicted. (This really happened!) 9