# Probability and Statistics

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1 Probability and Statistics Activity: TEKS: Your Average Joe (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: Materials: Grouping: Time: 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. Linking cubes Grid paper Transparencies 1-4 Handout 1 Calculator Large group and pairs of students Two 45-minute class periods Lesson: Procedures 1. (5 minutes) Discuss the meaning of the word average and how it is used in statistics. Perhaps have some newspaper or magazine articles that show different uses of average. Notes Possible question: What do you think of when you hear the word average? Your Average Joe Page 1

2 Procedures 2. (10 minutes) Ask students what they think of when they hear the expression 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 towers. Remark that the data is random and difficult to analyze. Suggest that arranging the data in some way may help. Have the students with a first name containing the smallest number of letters stand on one side of the room with their name tower and the students with the largest number of letters stand on the opposite side of the room with their tower. Ask the other students to come up with their name towers and line up between the least and greatest numbers in order according to the number of cubes in their towers. If there are students with the same number of letters in their first name, have them stand side by side. Identify the smallest and largest number of letters and discuss the spread of the data. 4. (5 minutes) Have students count off from both ends of the line simultaneously. When you reach the student in the center, the number of cubes in their tower is the median number of letters in the first names for this group. Remove one student from the line and talk about finding the median when two numbers are in the middle. Notes Possible question: What do you think of when you hear the phrase he s just your average Joe? Possible questions: How might we arrange the data to make it easier to analyze? Who has the least number of letters in their name? Who has the greatest number of letters in their name? What does this tell us about the data? Remind students that by looking at the greatest and least numbers in this or any set of data we can determine how much the data varies. The range of the set of data is the difference between the greatest and least numbers in the set and is one way to express the spread of the data. If the range is a small number, the data are close together. Possible question: How would we find the median of the data if there was not a number in the middle? 5. (5 minutes) Ask students with the same Possible questions: Your Average Joe Page 2

3 Procedures number of letters in their name to line up behind one another. Identify the longest line. The number of cubes that each person in the longest line has represents the mode of the number of letters used most often in first names for this group of people. 6. (10 minutes) To model finding the mean, have participants either give away or take cubes until all participants have towers of the same length. Hold extra cubes aside and discuss what to do with them. Discuss the mean of the group by looking at their even cube towers. Have students estimate the mean. This is particularly important if there are extra cubes. 7. (10 to 15 minutes) Have the students return to their seats. Place Transparency 2 on the overhead and define median, mode, and mean by referring to the physical activities. Calculate the mean using a calculator and compare to the earlier estimate. Notes Which number of letters occurs most frequently in this class? How do you think this would compare to other classes of 6 th graders? Possible questions: How might we use the cubes to find the mean number of letters in our names? What do we do with any extra cubes? Talk about the fact that we could have taken all the towers apart, 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. Possible questions: What did we do first to find the median? After we were arranged in order, how did we find the median? 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 such as the one on Transparency 3 for students to model. Ask students to work in pairs, use the linking cubes to model their thinking, and draw a diagram or sketch to explain their work. Provide grid paper for the drawings. Students may approach this problem in several different ways: (1) Some may build the five towers and then level them off by moving cubes one-by-one from the taller to the shorter towers. (2) Some may make one long tower and then break it into five Your Average Joe Page 3

4 Procedures Give several pairs of students the opportunity to share their work with the class. Notes equal towers. (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. Your Average Joe Page 4

5 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 Your Average Joe Page 5

6 Measures of Central Tendency Median Mode Mean Transparency 2 Your Average Joe Page 6

7 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 the Week Number of 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 Your Average Joe Page 7

8 Use your own paper to record your work. Finding the Mean, Median, and Mode 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. Does this data set have a mode? Explain. 2. Dave helped his dad with his landscaping business for 5 days. He earned \$9, \$11, \$14, \$15, and \$11. a. What was the mean amount he earned each day? Explain how you determined the answer. b. What is the median amount he earned? c. Does this data set have a mode? Explain. 3. Jacob scored an average (mean) of 18 points per game during the first 5 games of the season. Before the 6 th game he was injured and didn t get to play. What was his 6-game average? Explain how you determined your answer. 4. You want to convince your parents to raise your allowance. You ask several friends how much allowance they get each week. Their responses are recorded in the table. Name Weekly Allowance Andrew \$9.00 Collin \$7.00 Grayson \$8.00 Reid \$50.00 Sally \$8.00 a. What is the mean of their weekly allowances? Explain how you determined your answer. b. Does the mean describe the typical allowance for this group of friends? Explain. c. Does the median describe the typical allowance for this group? Explain 5. Yolanda has an average of 84 points on 3 assignments. She wants to bring up her average to 88 points. What must she score on her 4 th assignment? Use a diagram, sketch, and/or words to explain your answer. Handout 1 Your Average Joe Page 8

9 Think About It! For a group of 7 students, the number of letters in their last names is 5, 5, 8, 6, 4, and 10. As you can see, one piece of data is missing, but we know that the median is 6. Can we determine the missing number? Why or why not? Suppose we know the mean is 6, can we determine the missing number? Why or why not? Transparency 4 Your Average Joe Page 9

10 Use your own paper to record your work. Finding the Mean, Median and Mode Possible Solutions 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. Possible solution: Move one cube from the first tower to the second, move two cubes from the last tower to the third tower. The towers are level at four cubes in each, so the mean number of runs is four. b. What is the median number of runs and how does it compare to the mean? First I put the runs in order. (2, 3, 5, 6) Because there are 4 numbers, there is no middle number. The median is the average of the two middle scores, 3 and 5. The median is 4 which is the same as the mean. c. Does this data set have a mode? Explain. No, there is no mode because each number of runs occurs only once. 2. Dave helped his dad with his landscaping business for 5 days. He earned \$9, \$11, \$14, \$15, and \$11. a. What was the mean amount he earned each day? Explain how you determined the answer. Possible Solution Method: Since he earned the most on the fourth day, I started there and shared some of that money with day 1 and day 5. Then I moved some of the money from day 3 to day 1 and day 2. Once the cubes were level, I could see that the mean amount he earned each day was \$12. Your Average Joe Page 10

11 b. What is the median amount he earned and does it compare to the mean? First I put the amounts of money in order. (9, 11, 11, 14, 15) Because there are 5 numbers, the median is the one in the middle or 11. The median is \$11 which is less than the mean. c. Does this data set have a mode? Explain. Yes. The mode is \$11 because it occurs most often in the list of the amounts he earned. 3. Jacob scored an average (mean) of 18 points per game during the first 5 games of the season. Before the 6 th game, he was injured and didn t get to play. What was his 6-game average? Explain how you determined your answer. If he scored an average of 18 points for 5 games, that is a total of 90 points. If he didn t score during the 6 th game, the 90 points now must be divided by 6. So, his 6- game average is 15 points. 4. You want to convince your parents to raise your allowance. You ask several friends how much allowance they get each week. Their responses are recorded in the table. Name Weekly Allowance Andrew \$9.00 Collin \$7.00 Grayson \$8.00 Reid \$50.00 Sally \$8.00 a. What is the mean of their weekly allowances? Explain how you determined your answer. I added the allowances together and divided by 5. The mean is \$ b. Does the mean describe the typical allowance for this group of friends? Explain. No, because the mean is \$16 which is not close to any of the weekly allowances. c. Does the median describe the typical allowance for this group? Explain Yes, because the median is \$8 which is close to 4 of the 5 allowances. Your Average Joe Page 11

12 5. Yolanda has an average of 84 points on 3 assignments. She wants to bring up her average to 88 points. What must she score on her 4 th assignment? Use a diagram, sketch, and/or words to explain your answer. Possible Solution Method: The first 3 towers drawn with a solid black line represent the 84 points scored on the first three assignments. She needs to score 4 more points to raise each of these assignments to The fourth tower represents a score of 88 on the fourth assignment. 100 She needs to score 88 points plus 12 more points to raise each of the first three assignments to 88. That means Yolanda has to score 100 on the fourth assignment to bring up her average to 88 points. She better start studying! Your Average Joe Page 12

13 Mathematics TEKS Refinement 2006 K-5 Think About It! Possible Solutions For a group of 7 students, the number of letters in their last names is 5, 5, 8, 6, 4, and 10. As you can see, once piece of data is missing, but we know that the median is 6. Can we determine the missing number? Why or why not? No. All we know is that the missing number must be greater than or equal to 6 in order for 6 to be in the middle. Suppose we know the mean is 6, can we determine the missing number? Why or why not? Yes. Since we know the mean of the numbers is 6 we can find the sum of the numbers. Then we can subtract the numbers we know it find the missing number. 7 X 6 = = 4 The missing number is 4, so the person missing has 4 letters in his or her first name. Your Average Joe Page 13

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