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Probability and Statistics Activity: Your Average Joe TEKS Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University Probability and Statistics Activity: Your Average Joe TEKS: (6.10) Probability and statistics. The student uses statistical representations to analyze data. The student is expected to: (B) identify mean (using concrete objects and pictorial models), median, mode, and range of a set of data; (6.11) Underlying processes and mathematical tools. The student applies Grade 6 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school The student is expected to: (D) select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems. Overview: In this activity, students will answer a question about the average number of letters in a first name. Using linking cubes, students will physically model the mean, median, and mode of the number of letters in their first names. Definitions will be formulated for these terms by referring to the physical activities used to determine their value. Example: to find the median, we first arranged ourselves in order from the shortest first name to the longest first name. Finally, students will be given a problem to solve in pairs, and then solution strategies will be shared with the class. Materials: Linking cubes Grid paper Transparencies 1-4 Handout 1 Calculator Grouping: Large group and pairs of students Time: Two 45-minute class periods Lesson: Procedures Notes 1. (5 minutes) Discuss the meaning of the Possible question: word “average” and how it is used in statistics. Perhaps have some newspaper or What do you think of when you magazine articles that show different uses of hear the word average? “average.” Probability and Statistics Grade 6 Your Average Joe Page 1 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University Procedures Notes 2. (10 minutes) Ask students what they think Possible question: of when they hear the expression “he’s just What do you think of when you your average Joe.” hear the phrase “he’s just your average Joe?” Place Transparency 1 on the overhead. Give students time to use linking cubes to make a tower with the number of cubes that represents the number of letters in their first name. (If there is an even number of students in the group, the teacher should participate.) 3. (10 minutes) Ask students to hold up their Possible questions: towers. Remark that the data is random and difficult to analyze. Suggest that arranging How might we arrange the data to the data in some way may help. make it easier to analyze? Have the students with a first name Who has the least number of containing the smallest number of letters letters in their name? Who has the stand on one side of the room with their greatest number of letters in their name tower and the students with the largest name? What does this tell us number of letters stand on the opposite side about the data? of the room with their tower. Ask the other students to come up with their name towers Remind students that by looking at and line up between the least and greatest the greatest and least numbers in numbers in order according to the number of this or any set of data we can cubes in their towers. If there are students determine how much the data with the same number of letters in their first varies. The range of the set of name, have them stand side by side. data is the difference between the Identify the smallest and largest number of greatest and least numbers in the letters and discuss the spread of the data. set and is one way to express the spread of the data. If the range is a small number, the data are close together. 4. (5 minutes) Have students count off from Possible question: both ends of the line simultaneously. When you reach the student in the center, the How would we find the median of number of cubes in their tower is the median the data if there was not a number number of letters in the first names for this in the middle? group. Remove one student from the line and talk about finding the median when two numbers are in the middle. 5. (5 minutes) Ask students with the same Possible questions: Probability and Statistics Grade 6 Your Average Joe Page 2 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University Procedures Notes number of letters in their name to line up Which number of letters occurs behind one another. Identify the longest line. most frequently in this class? The number of cubes that each person in the How do you think this would longest line has represents the mode of the compare to other classes of 6th number of letters used most often in first graders? names for this group of people. 6. (10 minutes) To model finding the mean, Possible questions: have participants either give away or take How might we use the cubes to cubes until all participants have towers of the find the mean number of letters in same length. Hold extra cubes aside and our names? What do we do with discuss what to do with them. Discuss the any extra cubes? mean of the group by looking at their even cube towers. Have students estimate the Talk about the fact that we could mean. This is particularly important if there have taken all the towers apart, are extra cubes. put the cubes in a pile, and let each person take cubes from the pile until everyone had the same number of cubes. This is more like the “add and divide” method of finding the mean with which most students are familiar. 7. (10 to 15 minutes) Have the students return Possible questions: to their seats. Place Transparency 2 on the What did we do first to find the overhead and define median, mode, and median? After we were arranged mean by referring to the physical activities. in order, how did we find the median? Calculate the mean using a calculator and compare to the earlier estimate. How did we determine the mode? How did we determine the mean? How does the mean we found using the cubes compare to the mean we computed with the calculator? 8. (20 to 30 minutes) Pose another problem Students may approach this such as the one on Transparency 3 for problem in several different ways: students to model. (1) Some may build the five towers and then level them off by Ask students to work in pairs, use the linking moving cubes one-by-one from cubes to model their thinking, and draw a the taller to the shorter towers. diagram or sketch to explain their work. (2) Some may make one long Provide grid paper for the drawings. tower and then break it into five Probability and Statistics Grade 6 Your Average Joe Page 3 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University Procedures Notes Give several pairs of students the equal towers. opportunity to share their work with the class. (3) Others may break the towers into a large pile and rebuild them into five equal towers. Experience with leveling off towers of cubes, describing their methods, and listening to others’ methods helps students to develop a strong visual model for the meaning of average or mean. Questions to ask during sharing: How is your solution different from this one? How is your solution the same as this one? 9. Student Reflection: How will this activity help you remember how to determine the median, mode and mean for a set of data? Homework: Handout 1 can be used as homework. Extensions: See Transparency 4. Probability and Statistics Grade 6 Your Average Joe Page 4 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University Your Average Joe You’ve heard the expression “your average Joe.” Is Joe really average? The name Joe only contains three letters. Does the average name contain three letters? How many letters do you think an average name might have? Let’s look at the names of the people in this class and see what we can determine about the average name. Count the number of letters in your first name. Use the linking cubes to make a tower with this number of cubes. Transparency 1 Probability and Statistics Grade 6 Your Average Joe Page 5 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University Measures of Central Tendency Median Mode Mean Transparency 2 Probability and Statistics Grade 6 Your Average Joe Page 6 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University Missing from Math Class The table shows the number of students absent from a mathematics class each day last week. What was the mean number of students absent per day last week? Day of Number of the Week Students Absent Monday 2 Tuesday 6 Wednesday 4 Thursday 2 Friday 1 Find the median and mode for the number of days absent. How do these measures of the data compare to the mean? Explain why this happens. Transparency 3 Probability and Statistics Grade 6 Your Average Joe Page 7 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University Finding the Mean, Median, and Mode Use your own paper to record your work. 1. In the last four softball games, Michelle scored 5, 3, 2, and 6 runs. a. Use concrete objects to model the mean and draw of diagram of your thinking. b. What is the median number of runs and how does to compare to the mean? c.
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