M 3 : Manipulatives, Modeling, and Mayhem - Session I Activity #1

Transcription

1 M 3 : Manipulatives, Modeling, and Mayhem - Session I Activity #1 Purpose: The purpose of this activity is to develop a student s understanding of ways to organize data. In particular, by completing this hands-on activity, students will experience a visual interpretation of the median. Learning Objectives: Students who complete this activity will discover the concept of a median and outliers with minimal use of numbers, and gain a visual understanding of descriptive measures used in statistics. Required Materials: Five balls of varying sizes and weights. You may want to ensure that the median size differs from the median weight. Consider the five balls given below that are in no particular order. First consider the size of the balls. In order to organize these measurements, let s place them in order from smallest to largest. Smallest Ball Largest Ball 1

2 1. Which ball is in the middle? 2. What is this value called? Teacher s Note: Remind students that the median is the middle value of an ordered set of measurements (or the average of the middle two values if the dataset contains an even number of measurements). It is very important that the values in the dataset are placed in order with regards to a numerical measurement before finding a median. 3. If the beach ball were removed from this collection and replaced with the moon, what impact would this have on the median size of the balls? Explain. Now, let s organize these measurements according to weight. Put them in order from lightest to heaviest. Lightest Ball Heaviest Ball 4. Which ball represents the median weight? 5. Look back at your answer to Question 1. Does the same ball represent both the median weight and the median size? Teacher s Note: Students should go beyond an answer of because the middle balls are different. Students should understand that the median is determined after the balls have been sorted according to some numeric characteristic. Thus, two different measurement characteristics may produce two different medians. 6. Suppose that we ordered the balls alphabetically by name. Explain why a median has no meaning in this case. 2

3 Extensions: 1. Have five student volunteers sort themselves alphabetically by name. Explain why it is not appropriate to find a median for this group. 2. Line up a group of 5 girls from shortest to tallest. Do the same for a group of 5 boys. Have the person that represents the median height from each group step forward. Compare the group of girls to the group of boys using only the individuals that represent the medians. Teacher s Note: At this point, you may want to discuss why we use a median when summarizing data. When comparing these girls and boys, we could compare the shortest girl to the shortest boy, the second shortest girl to the second shortest boy, etc. However, this would be cumbersome with many data values. Obtaining the median for each group should capture much of the same essence of the data, and comparing the groups based on this single measurement is much easier than comparing all data points. 3. Consider replacing the tallest boy with Sultan Kosen of Mardin, Turkey. Kosen stands at 8 feet, 1 inch, and he currently holds the Guinness World Record for tallest living man. What impact would this have on the median height for boys? Teacher s Note: Students should understand that the median tells us some things about data, but not everything. Even if two groups have the same median value, there could still be large differences between the two groups. 3

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5 Bouncing Around Data Consider the five balls given below that are in no particular order. First consider the size of the balls. In order to organize these measurements, let s place them in order from smallest to largest. Smallest Ball Largest Ball 1. Which ball is in the middle? 2. What is this value called? 3. If the beach ball were removed from this collection and replaced with the moon, what impact would this have on the median size of the balls? Explain. 1

6 Now, let s organize these measurements according to weight. Put them in order from lightest to heaviest. Lightest Ball Heaviest Ball 4. Which ball represents the median weight? 5. Look back at your answer to Question 1. Does the same ball represent both the median weight and the median size? 6. Suppose that we ordered the balls alphabetically by name. Explain why a median has no meaning in this case. 2

7 M 3 : Manipulatives, Modeling, and Mayhem - Session I Activity #2 Purpose: The purpose of this activity is to enhance students' understanding of various descriptive measures. In particular, by completing this hands-on activity, students will experience a visual interpretation of a mean and the concept of distance-to-mean. Learning Objectives: Students who complete this activity will discover the concepts of a mean and distance-to-mean with minimal use of numbers, and gain a visual understanding of descriptive measures used in statistics. Required Materials: Poster board for puzzle pieces, colored paper, scissors, and tape for each group of 5 students. Give each person in your group one piece of the puzzle. Trace the shape of your puzzle piece onto a colored sheet of paper. Cut this shape out, and set your original puzzle piece aside. Next, put the colored puzzle pieces together to form a rectangle. Tape the pieces together (carefully tape all of the seams). This completed puzzle represents the total area for all of the puzzle pieces for your group. 1. What color is most prominent in this representation of the total area? 2. Why is this color most prominent? 3. Which color is least prominent? 4. Devise a method to obtain a representation of the average area of the puzzle pieces in your group. You are not allowed to use actual measurements to obtain this average. Explain your method. 1

8 Once you have devised a method to obtain the average area, cut the colored paper so that each member has a colored piece of paper that represents the average area. Teacher s note: Help students realize that several different colors were combined to obtain the total area. Each group member contributes to the total area and hence the average, which is why more than one color may show up in a piece representing the average. 5. Are all of the pieces representing your group s average the same size? Should they be? Teacher s note: Students should understand that even though each person in their group is holding a piece that represents the average, the data set has only one mean. 6. What color appears most prominently in the representations of the average? Why does this happen? 7. Which group member has the smallest original puzzle piece? How does this person s puzzle piece compare to your group s average? 8. Which group member has the puzzle piece that is most typical in area? How does this person s puzzle piece compare to your group s average? Using a mean to give an idea of a typical value in a dataset is only one aspect of summarizing measurements. An equally important concept is variation in measurements. Place your original white puzzle piece under/over your piece of colored paper that represents the average area. Cut whichever piece is necessary so that the two pieces are of the same size. Keep the piece of paper that is left-over (i.e., the residual piece). Teacher s note: This residual piece is what is used in the computation of both the mean absolute deviation (MAD) and standard deviation, which are commonly used measures of variation. 2

9 9. What does your residual piece of paper represent? 10. Why do some of your group members have white residual pieces and others have colored residual pieces? 11. Will the person with the smallest residual piece necessarily have the smallest original puzzle piece? Explain. 12. Who has the smallest residual? Is this what you expected based on your answer to Question 11? 13. Suppose a group of four people decided to construct a puzzle the size of this room. Would all of the group members end up with white residual pieces? Explain why or why not. Extensions: 1. Have students explain how they could determine the average area of all puzzle pieces in the entire class. 2. Ask students whether the average from the entire class would be drastically larger than the average from an individual group. Have them explain why this would or would not be the case. 3. Suppose a group of four people decided to construct a puzzle the size of a post-it note. Ask whether all group members would end up with colored residual pieces, and ask them to explain why or why not. 4. Consider these two scenarios: (1) an original puzzle the size of a post-it note, and (2) an original puzzle the size of this room. Ask students which scenario would result in larger residual pieces. Use this discussion to help students understand that measures of variability should always be considered in the context of the scale of the data. When comparing the variability of two groups with different units or very different means, one should consider making the comparison based on the coefficient of variation. 5. Consider the residual pieces. Repeat the process of putting these pieces together and finding their mean. This result is a representation of the Mean Absolute Deviation (MAD) which is a common measure of variation. 3

10 6. Purposefully create the puzzles so that some groups have more variability in their puzzle pieces than do other groups. As long as there are an equal number of students in each group, students can compare their group s total residuals to another group s total residuals to decide which group had more variability in their puzzle pieces. If the group sizes differ, then these comparisons should be made based on the Mean Absolute Deviation (MAD). 7. The total area of the white residual pieces will match the total area of the colored residual pieces. This fact can be used as a visual proof that the sum of the residuals will be zero; hence, it is necessary to take the absolute value (as in the computation of the MAD) or square the residuals (as in the computation of the variance) before obtaining a sum. 4

11 Putting the Pieces Together Give each person in your group one piece of the puzzle. Trace the shape of your puzzle piece onto a colored sheet of paper. Cut this shape out, and set your original puzzle piece aside. Next, put the colored puzzle pieces together to form a rectangle. Tape the pieces together (carefully tape all of the seams). This completed puzzle represents the total area for all of the puzzle pieces for your group. 1. What color is most prominent in this representation of the total area? 2. Why is this color most prominent? 3. Which color is least prominent? 4. Devise a method to obtain a representation of the average area of the puzzle pieces in your group. You are not allowed to use actual measurements to obtain this average. Explain your method. Once you have devised a method to obtain the average area, cut the colored paper so that each member has a colored piece of paper that represents the average area. 5. Are all of the pieces representing your group s average the same size? Should they be? 6. What color appears most prominently in the representations of the average? Why does this happen? 1

12 7. Which group member has the smallest original puzzle piece? How does this person s puzzle piece compare to your group s average? 8. Which group member has the puzzle piece that is most typical in area? How does this person s puzzle piece compare to your group s average? Using a mean to give an idea of a typical value in a dataset is only one aspect of summarizing measurements. An equally important concept is variation in measurements. Place your original white puzzle piece under/over your piece of colored paper that represents the average area. Cut whichever piece is necessary so that the two pieces are of the same size. Keep the piece of paper that is left-over (i.e., the residual piece). 9. What does your residual piece of paper represent? 10. Why do some of your group members have white residual pieces and others have colored residual pieces? 11. Will the person with the smallest residual piece necessarily have the smallest original puzzle piece? Explain. 12. Who has the smallest residual? Is this what you expected based on your answer to Question 11? 13. Suppose a group of four people decided to construct a puzzle the size of this room. Would all of the group members end up with white residual pieces? Explain why or why not. 2