Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University

Probability and

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 (using concrete objects and pictorial models), , , 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.”

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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:

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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

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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.

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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

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Measures of

Median

Mode

Mean

Transparency 2

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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

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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. 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 6th 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 4th assignment? Use a diagram, sketch, and/or words to explain your answer.

Handout 1

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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

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Finding the Mean, Median and Mode Possible Solutions

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. 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.

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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 6th 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 6th 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 $16.00.

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.

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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 4th assignment? Use a diagram, sketch, and/or words to explain your answer. Possible Solution Method:

The first 3 towers drawn The fourth tower She needs to score 88 points plus 12 more with a solid black line represents a points to raise each of the first three represent the 84 points score of 88 on assignments to 88. That Yolanda has scored on the first three the fourth to score 100 on the fourth assignment to assignments. She needs to assignment. bring up her average to 88 points. She score 4 more points to raise better start studying! each of these assignments to 100

88 84

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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 = 42 42 – 5 – 5 – 8 – 6 – 4 – 10 = 4

The missing number is 4, so the person missing has 4 letters in his or her first name.

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