Detective Curious got a lead on some missing compact CD players she was investigating. The informer hinted that the stolen CD players (and maybe even the culprit) could be found in an abandoned warehouse on the corner of Spruce Street and Elm. Detective Curious rushed to the scene. As she pulled up to the warehouse, she spotted someone coming out the front door of the warehouse. When the person saw Detective Curious pull up, he immediately bolted back into the warehouse, slamming the door shut behind him. Detective Curious raced to catch up to the now-apparent suspect. As she reached the door, she radioed the station for backup. While speaking to the dispatcher, Detective Curious learned that the building was once used as an amusement park maze. When Detective Curious entered the building, she immediately understood what the dispatcher meant. Standing in the lobby of the building, she noticed there were doors. She once again radioed the dispatcher and requested that her backup be alerted that she was going to take the door on the left and that her backup should take the door on the right. When Detective Curious entered the door on the left, she came upon new doors. Before choosing 1, she got a call on her radio that her backup had arrived. Her backup had entered the door on the right and came across new doors as well. Both Detective Curious and her backup decided to take the door on each of their left sides, and then they both found themselves 1 of 1
faced with new doors. As they consulted each other, they concluded that they had entered mazes that were mirror images of each other. Detective Curious took out her sketchpad and made a map of what she imagined to be the maze (see the attached sheet). If Detective Curious and her backup each continue entering 1 door at a time, eventually they should each come to the end of the maze (but there will be many paths unchecked). At the end of the maze, Detective Curious found that she had passed through a total of 10 doors. What is the mathematical chance that the suspect traveled the identical path that Detective Curious or her backup traveled? How likely is it that the suspect will get away? Teacher Note: See pages 9 10 of the PDF to print a complete worksheet with graphics. of 1
Suggested Grade Span 6 8 Grade(s) in Which Task Was Piloted 7 and 8 Task Detective Curious got a lead on some missing compact CD players she was investigating. The informer hinted that the stolen CD players (and maybe even the culprit) could be found in an abandoned warehouse on the corner of Spruce Street and Elm. Detective Curious rushed to the scene. As she pulled up to the warehouse, she spotted someone coming out the front door of the warehouse. When the person saw Detective Curious pull up, he immediately bolted back into the warehouse, slamming the door shut behind him. Detective Curious raced to catch up to the now-apparent suspect. As she reached the door, she radioed the station for backup. While speaking to the dispatcher, Detective Curious learned that the building was once used as an amusement park maze. When Detective Curious entered the building, she immediately understood what the dispatcher meant. Standing in the lobby of the building, she noticed there were doors. She once again radioed the dispatcher and requested that her backup be alerted that she was going to take the door on the left and that her backup should take the door on the right. When Detective Curious entered the door on the left, she came upon new doors. Before choosing 1, she got a call on her radio that her backup had arrived. Her backup had entered the door on the right and came across new doors as well. Both Detective Curious and her backup decided to take the door on each of their left sides, and then they both found themselves faced with new doors. As they consulted each other, they concluded that they had entered mazes that were mirror images of each other. Detective Curious took out her sketchpad and made a map of what she imagined to be the maze (see the attached sheet). If Detective Curious and her backup each continue entering 1 door at a time, eventually they should each come to the end of the maze (but there will be many paths unchecked). At the end of the maze, Detective Curious found that she had passed through a total of 10 doors. What is the mathematical chance that the suspect traveled the identical path that Detective Curious or her backup traveled? How likely is it that the suspect will get away? Teacher Note: See pages 9 10 of the PDF to print a complete worksheet with graphics. of 1
Alternative Versions of Task More Accessible Version: Detective Curious got a lead on some missing compact CD players she was investigating. The informer hinted that the stolen CD players (and maybe even the culprit) could be found in an abandoned warehouse on the corner of Spruce Street and Elm. Detective Curious rushed to the scene. As she pulled up to the warehouse, she spotted a someone coming out the front door of the warehouse. When the person saw Detective Curious pull up, he immediately bolted back into the warehouse, slamming the door shut behind him. Detective Curious raced to catch up to the now-apparent suspect. As she reached the door, she radioed the station for backup. While speaking to the dispatcher, Detective Curios learned that the building was once used as an amusement park maze. When Detective Curious entered the building, she immediately understood what the dispatcher meant. Standing in the lobby of the building, she noticed there were doors. She once again radioed the dispatcher and requested that her backup be alerted that she was going to take the door on the left and that her backup should take the door on the right. If Detective Curious and her backup each continue entering 1 door at a time, eventually they should each come to the end of the maze (but there will be many paths unchecked). At the end of the maze, Detective Curious found that she had passed through a total of 6 doors. What is the mathematical chance that the suspect traveled the identical path that Detective Curious or her backup traveled? How likely is it that the suspect will get away? (See the attached sheet.) Teacher Note: See pages 11 1 of the PDF to print a complete worksheet with graphics. More Challenging Version: Refer to the original version of the task, and... Is this task one whose circumstances could really exist? Why or why not? Support your conclusion mathematically. Teacher Note: See pages 1 14 of the PDF to print a complete worksheet with graphics. NCTM Content Standards and Evidence Data Analysis and Probability Standard for Grades 6 8: Instructional programs from prekindergarten through grade 1 should enable all students to... 4 of 1
Understand and apply basic concepts of probability. NCTM Evidence A: Understand and use appropriate terminology to describe complementary and mutually exclusive events. NCTM Evidence B: Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams and area models. Exemplars Task-Specific Evidence: This task requires students to determine the number of combinations possible. Time/Context/Qualifiers/Tip(s) From Piloting Teacher This task took my students one to two class periods to complete. Students whose benchmark samples are shown were studying combinations and permutations. They completed some activities from the Connected Math series, Clever Counting, which has activities designed around a Who Done It? mystery. The Exemplars task, "He Did It," was also designed to complement the activities in this text. To print a copy of the original version of the task with the graphic image, refer to pages 9 10. To print a copy of the more accessible version of the task with the graphic image, refer to pages 11 1. To print a copy of the more challenging version of the task with the graphic image, refer to pages 1 14. Links This task would complement a unit on reading and writing mysteries, as well as one on forensic science. Common Strategies Used to Solve This Task Students start with trying to create a tree diagram but soon realize that it will become too monstrous. They then tend to create a chart to record the number of doors and the number of new doors behind each. Students then employ the counting principle to determine the total number of paths. Finally, students determine the chance, as either a fraction or a percent, that the suspect will get away. 5 of 1
Possible Solutions Door Number Number of Choices Doors in that Row 1 1 1 6 4 1 5 6 6 7 7 16 8 4 9 1,96 10,59 To get the number of doors in any row, multiply the number of door choices in that row by the number of doors in the previous row. There is a /,59 chance of the suspect taking the same route as Detective Curious or her backup. This is equal to 0.08% chance, which is slim. More Accessible Version Solution: Door Number Number of Choices Doors in that Row 1 1 1 4 4 8 5 16 6 To get the number of doors in any row, multiply the number of door choices in that row by the number of doors in the previous row. There is a / chance of the suspect taking the same route as Detective Curious or her backup. This is equal to a 6% chance, which is slim. 6 of 1
More Challenging Version Solution: Door Number Number of Choices Doors in that Row 1 1 1 6 4 1 5 6 6 7 7 16 8 4 9 1,96 10,59 To get the number of doors in any row, multiply the number of door choices in that row by the number of doors in the previous row. There is a /,59 chance of the suspect taking the same route as Detective Curious or her backup. This is equal to 0.08% chance, which is slim. The circumstances in this task are unlikely. For a building to be designed in this fashion, it would have to be over 1.5 miles wide (assuming each door is 0 inches wide, and there are 10 inches between each door):,59 doors x 0 inches = 77,760 inches,59 spaces between the doors x 10 inches = 5,90 inches Total inches = 10,690 inches 10,690 inches 1 inches = 8,640.8 feet 8,640.8 feet 5,80 feet in a mile = 1.64 miles Task-Specific Assessment Notes General Notes Students need an understanding of combinations in order to solve this task. Novice The Novice will not have a strategy that leads to a correct solution. There will be little or no correct reasoning present, and no awareness of the audience will be communicated. The Novice will not attempt to make connections or use math representations. 7 of 1
Apprentice The Apprentice will be able to devise an approach that partly works, but will be unable to follow through to achieve a correct solution. Some formal math language may be used, and some awareness of the audience may be present. An attempt will be made to construct a math representation to record and communicate problem solving. Practitioner The Practitioner will be able to achieve a correct solution to all parts of the task and communicate a sense of audience and purpose. Accurate and appropriate math representations will be constructed and refined to portray solutions. The Practitioner will use formal math language throughout to communicate. Math connections will be recognized. For example, students may recognize that the counting principle is the underlying mathematics involved in the task. Expert The Expert will choose an efficient strategy, achieving a correct answer for all parts of the problem. The Expert will make deductive arguments and use evidence to support and justify decisions made and conclusions reached. Precise math language will be used to communicate ideas. A sense of audience and purpose will be communicated, and mathematical observations will be used to extend the solution. The Expert will construct symbolic representations to analyze relationships and extend thinking. 8 of 1
Original Version Worksheet Detective Curious got a lead on some missing compact CD players she was investigating. The informer hinted that the stolen CD players (and maybe even the culprit) could be found in an abandoned warehouse on the corner of Spruce Street and Elm. Detective Curious rushed to the scene. As she pulled up to the warehouse, she spotted someone coming out the front door of the warehouse. When the person saw Detective Curious pull up, he immediately bolted back into the warehouse, slamming the door shut behind him. Detective Curious raced to catch up to the now-apparent suspect. As she reached the door, she radioed the station for backup. While speaking to the dispatcher, Detective Curious learned that the building was once used as an amusement park maze. When Detective Curious entered the building, she immediately understood what the dispatcher meant. Standing in the lobby of the building, she noticed there were doors. She once again radioed the dispatcher and requested that her backup be alerted that she was going to take the door on the left and that her backup should take the door on the right. When Detective Curious entered the door on the left, she came upon new doors. Before choosing 1, she got a call on her radio that her backup had arrived. Her backup had entered the door on the right and came across new doors as well. Both Detective Curious and her backup decided to take the door on each of their left sides, and then they both found themselves faced with new doors. As they consulted each other, they concluded that they had entered mazes that were mirror images of each other. Detective Curious took out her sketchpad and made a map of what she imagined to be the maze (see the attached sheet). If Detective Curious and her backup each continue entering 1 door at a time, eventually they should each come to the end of the maze (but there will be many paths unchecked). At the end of the maze, Detective Curious found that she had passed through a total of 10 doors. What is the mathematical chance that the suspect traveled the identical path that Detective Curious or her backup traveled? How likely is it that the suspect will get away? 9 of 1
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More Accessible Version Worksheet Detective Curious got a lead on some missing compact CD players she was investigating. The informer hinted that the stolen CD players (and maybe even the culprit) could be found in an abandoned warehouse on the corner of Spruce Street and Elm. Detective Curious rushed to the scene. As she pulled up to the warehouse, she spotted a someone coming out the front door of the warehouse. When the person saw Detective Curious pull up, he immediately bolted back into the warehouse, slamming the door shut behind him. Detective Curious raced to catch up to the now-apparent suspect. As she reached the door, she radioed the station for backup. While speaking to the dispatcher, Detective Curios learned that the building was once used as an amusement park maze. When Detective Curious entered the building, she immediately understood what the dispatcher meant. Standing in the lobby of the building, she noticed there were doors. She once again radioed the dispatcher and requested that her backup be alerted that she was going to take the door on the left and that her backup should take the door on the right. If Detective Curious and her backup each continue entering 1 door at a time, eventually they should each come to the end of the maze (but there will be many paths unchecked). At the end of the maze, Detective Curious found that she had passed through a total of 6 doors. What is the mathematical chance that the suspect traveled the identical path that Detective Curious or her backup traveled? How likely is it that the suspect will get away? 11 of 1
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More Challenging Version Worksheet Detective Curious got a lead on some missing compact CD players she was investigating. The informer hinted that the stolen CD players (and maybe even the culprit) could be found in an abandoned warehouse on the corner of Spruce Street and Elm. Detective Curious rushed to the scene. As she pulled up to the warehouse, she spotted someone coming out the front door of the warehouse. When the person saw Detective Curious pull up, he immediately bolted back into the warehouse, slamming the door shut behind him. Detective Curious raced to catch up to the now-apparent suspect. As she reached the door, she radioed the station for backup. While speaking to the dispatcher, Detective Curious learned that the building was once used as an amusement park maze. When Detective Curious entered the building, she immediately understood what the dispatcher meant. Standing in the lobby of the building, she noticed there were doors. She once again radioed the dispatcher and requested that her backup be alerted that she was going to take the door on the left and that her backup should take the door on the right. When Detective Curious entered the door on the left, she came upon new doors. Before choosing 1, she got a call on her radio that her backup had arrived. Her backup had entered the door on the right and came across new doors as well. Both Detective Curious and her backup decided to take the door on each of their left sides, and then they both found themselves faced with new doors. As they consulted each other, they concluded that they had entered mazes that were mirror images of each other. Detective Curious took out her sketchpad and made a map of what she imagined to be the maze (see the attached sheet). If Detective Curious and her backup each continue entering 1 door at a time, eventually they should each come to the end of the maze (but there will be many paths unchecked). At the end of the maze, Detective Curious found that she had passed through a total of 10 doors. What is the mathematical chance that the suspect traveled the identical path that Detective Curious or her backup traveled? How likely is it that the suspect will get away? Is this task one whose circumstances could really exist? Why or why not? Support your conclusion mathematically. 1 of 1
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