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PATTERNS IN DATA

LESSON 1EXPLORING DATA

THIS PACKET WILL SERVE AS YOUR TEXT

THE THINK ABOUT THIS SITUATION IS COMPLETED TOGETHER AS A CLASS

THE INVESTIGATION IS COMPLETED IN ASSIGNED SMALL GROUPS

THE CHECKPOINT IS COMPLETED TOGETHER AS A CLASS

YOU MAY WRITE IN THIS PACKET (BACK OF EACH PAGE IS BLANK); YOUR ANSWERS MUST BE COMPLETE, NEAT, AND ORGANIZED! IF YOU DO NOT HAVE ENOUGH ROOM, FEEL FREE TO USE PAPER

Lesson 1

Exploring Data

The main theme of this text is that mathematics provides a powerful way of making sense of the world in which you live. Contemporary mathematics is rooted in the study of patterns involving data, shape, change, and chance-aspects of situations you experience every day. It involves investigating and expressing relationships among these patterns in ways that make sense to you. Once understood, the patterns and relationships can be applied in a variety of ways to other common and complex situations.

For example, patterns in data are used to make decisions that affect you every day. As a class, consider the kinds of data that are collected by the movie industry and by clothing manufacturers.

Think About This Situation

a)Examine the data on movie attendance.

How could theater owners use this information?
How could movie makers use these data?
How could you use this information?

b)How do you think a T-shirt manufacturer decides how big to make a size “large” T-shirt?

Decisions that are made based on data are only as good as the actual data. It is therefore very important that data collected be appropriate and accurate.

Investigation 1

Collecting and Analyzing Data

As you use this text, besides working as a whole class and individually, you often will work in pairs and in small groups. It is important that you have confidence in one another, share ideas, and help each other when asked.

Complete the data-gathering activities that follow by working together in your small groups. Each group will need a measuring tape or a measuring stick and a string.

Measure, in centimeters, the height and arm span of each person in your group.
Record your data in a table like the one shown here.

Student NameHeightArmspan______

______

What one height would be most typical or representative of your group?

Do you think your typical group height would be the same as the typical height of the entire class? Explain your reasoning.

How much variation (consistency) is there in the arm spans of your group?

Do you think the amount of variation in arm spans of your class would be less than or greater than the variation in your group? Explain your reasoning.

Combine your data for heights and arm spans with the data from other groups and record it in another table (on the Smartboard). Call it the Class Data Table.

Use the Class Data Table to find a typical height for the entire class. Do you still agree with your answer to activity 1.c?

After examining the Class Data Table, do you still agree with your response to activity 1.e?

On the basis of the data from your class, does that seem to be a relationship between height and arm span?

Measure the circumference (distance around) of the thumb and wrist of each person in your group.

Record your data in a table like the one shown here.

Student Name Thumb Circumference Wrist Circumference_

______

Combine your data of thumb and wrist circumferences with the data from other groups and record them in the Class Data Table. Again, record all measurements in the same unit.
Based on the class data, what relationship, if any, do you see between the wrist and thumb size of a person?

In the novel Gulliver’s Travels, the Lilliputians made clothes for Gulliver. They estimated that twice around the thumb is once around the wrist and twice around the wrist is once around the neck.

Does the first part of this statement seem true for your class data? Why or why not?

Describe how you could check the second part of the Lilliputians’ estimate.

Determine the shoe length and stride length for each person in your group.
Record the data in a table like the one shown here.

Student Name Shoe Length Stride Length___

______

Why do you think you were asked to determine shoe length rather than shoe size?

Combine your data for shoe lengths and stride lengths with the data from other groups and record it in the Class Data Table.

What is the smallest shoe length in the class? What is the largest shoe length?

Does there appear to be a relationship between shoe length and stride length?

Investigation 2

Describing Patterns in Data

To learn from one another in investigating situations, it is important that your group work together as a team. Following the guidelines below will help your group work well together. It will also help you build skills that have become very important for people who work in fields such as health care, business, and industry.

Group Guidelines

Each group member contributes to the group’s work.
Each member of the group is responsible for listening carefully when another group member is talking.
Each member of the group has the responsibility and the right to ask questions.
Each group member should help others in the group when asked.
Each member of the group should be considerate and encouraging
Work together until everyone in the group understands and can explain the group’s results.

How well did your group work together in completing the projects in the last investigation? In your group, discuss responses to the following questions.

How could your group encourage everyone to contribute ideas?

How could your group ensure that everyone listens while someone is talking?

How could your group ensure that each person understands and agrees with the group’s decisions?

Making sense of information and data is a common feature of many careers. It is also important for a thorough understanding of radio, television, and newspaper reports, which often include information gathered from opinion polls or surveys. As an example, consider the following report of a survey of the pop music industry.

The Los Angeles Times once asked 25 important people in the pop music industry to determine the pop world’s “hottest properties.” Among the 25 people were the presidents of Motown Records and Capitol Records. The artists and groups were ranked by giving 10 points for each first-place vote, 9 points for a second lace vote, and so on. The following is the list of the 20 artists and groups who received the most points.

Industry Leader’s Hot Properties

Rank / Artist / Points / Rank / Artist / Points
1 / U2 / 165 / 11 / En Vogue / 30
2 / R.E.M. / 95 / Janet Jackson / 30
3 / Pearl Jam / 83 / 13 / Madonna / 29
4 / Metallica / 81 / 14 / Michael Jackson / 27
5 / Garth Brooks / 75 / 15 / Bruce Springsteen / 22
6 / Whitney Houston / 69 / 16 / Michael Bolton / 20
7 / Guns N’ Roses / 54 / Mariah Carey / 20
8 / Boyz II Men / 44 / Red Hot Chili Peppers / 20
9 / Arrested Development / 40 / Nirvana / 20
10 / Nine Inch Nails / 38 / 20 / George Michael / 18
Source: Hilburn, Robert. 1993. "Who are pop's hot properties?" Los Angeles Times Calendar,Feb. 7.

In your group, discuss each of the following questions. Record the answer that your group decides is the best response.
U2 could have received its 165 points by getting 15 first-place votes, a second-place vote, and a fifth-place vote. Find two other ways that U2 could have received its 165 points.

What is the largest possible number of points an artist or group could have received?

How popular are these artists today? Does your group believe that these record executives were good at predicting “pop’s hottest properties”? Give some evidence to back up your position.

Does it appear that the rankings of the 25 executives were a lot alike, or do you think there was substantial disagreement in their rankings?

Students in one class at City High were asked to describe the distribution of points for the top 20 artists and groups.

Joshua’s response: “U2 was highest with 165 points and George Michael was last with 18 points. My favorite singer, Mariah Carey, came in tried for 16th place. I don’t think that is an accurate assessment of how ‘hot’ she is. She definitely will be a bigger star than some of those rated above her.”

Sarah’s response: “U2 was much higher than the other artists and groups. The next highest group, R.E.M. received an average ranking of less than 4 from the 15 raters. Most of the artists and groups received low scores, under 40 points.”

Whose response does your group think was the most complete and helpful? Why?

What would you do to improve the response that you chose as the best?

When asked to describe the distribution of points for the top 20 artists and groups, Paul and Maria first made a stem-and-leaf plot like the one shown here.

Industry Leader’s Hot Properties
1 / 8
2 / 0 0 0 0 2 7 9
3 / 0 0 8
4 / 0
5 / 4
6 / 9
7 / 5
8 / 1 3
9 / 5
10
11
12
13
14
15
16 / 5
1 / 8 represents 18

In your group, discuss the organization of this stem-and-leaf plot. Is the plot an accurate display of the data in the chart? If not, explain how to correct it.

Paul’s description: “Based on the stem-and-leaf plot, most artists and groups were ranked low. Only one of the artists and groups had a high ranking.”

Maria’s description: “You can see in the stem-and-leaf plot that there were a whole bunch in the 20s. The rest were spread out.”

What are the strengths and weaknesses of Paul’s response?

What are the strengths and weaknesses of Maria’s response?

As a group, write what you think would be a better response, either Paul’s or Maria’s. Compare your response with those of other groups.

Fifteen students from the Student Council at City High ranked their top ten artists and groups from the Los Angeles Times list in a similar manner. Their results are shown in the chart below. Make a stem-and-leaf plot for these data.

Student Council’s Hot Properties

Rank / Artist / Points / Rank / Artist / Points
1 / Pearl Jam / 105 / 11 / Whitney Houston / 41
2 / R.E.M. / 102 / 12 / En Vogue / 33
3 / Boyz II Men / 98 / 13 / Mariah Carey / 11
4 / Arrested Development / 87 / 14 / Garth Brooks / 7
5 / Metallica / 59 / 15 / Madonna / 5
Red Hot Chili Peppers / 59 / 16 / Michael Jackson / 4
7 / U2 / 56 / 17 / Bruce Springsteen / 2
8 / Nine Inch Nails / 55 / 18 / Nirvana / 1
9 / Guns N’ Roses / 52 / 19 / Michael Bolton / 0
10 / Janet Jackson / 48 / George Michael / 0
Student Council’s Hot Properties

Students were asked to make a stem-and-leaf plot for the Student Council rankings and to compare the plot to the one in 4.

Desmond’s response: “The new stem-and-leaf plot has more ratings in the middle of the plot.”

Regina’s response: “They both have a gap. The new plot has a cluster of groups ranked high.”

What are the strengths and weaknesses of Desmond’s response?

What are the strengths and weaknesses of Regina’s response?

As a group, write what you think would be a better response than either Regina’s or Desmond’s. Compare your response to those of other groups.

For homework, students at City High were asked to explain why stem-and-leaf plots are helpful in understanding data such as the rankings and points from the Los Angeles Times.

Cecilia wrote this explanation: “Stem-and-leaf plots help you understand the information because you can see quickly how the data are spread out or clumped together. You can see any gaps where there are no data. By using the numbers instead of tally marks or Xs, you can still read the number of points that the judges assigned. Since these data sets are small, the stem-and-leaf displays were concise, easy to make, and easy to read.

Ms. Thomas, the mathematics teacher, decided that Cecilia’s explanation was excellent, so she gave her 5 points, the maximum points possible.

Here is Jesse’s explanation: “The stem-and-leaf plot has the stems on the left and the leaves on the right. For the numbers that has two digits, the tens digit it the stem, and the ones digit is the leaf. For numbers that have three digits, the hundreds and tens digits together are the stem and the ones digit is the lead. The numbers are in order from smallest at the top to largest at the bottom.

As a group, use Cecilia’s response as a guide to evaluate Jesse’s explanation on a scale of 1 to 5, with 5 being the top score. Explain why you assigned the score you did.

By now you may be wondering: “What’s all the fuss about writing good explanations? This is a math course!” Being able to describe your conclusions and explain your reasoning is important for at least two reasons. (1) it helps you better understand your own thinking and the mathematics you are studying, and (2) it is a skill, like teamwork, that businesses and industries look for in new employees.

The final activity in this lesson will help you and your group becomes more familiar with these new aspects of doing mathematics.

Reassign your group roles. Study the chart below giving average monthly earnings in 1990 for adults in the United States.

Does Education Pay?

Average Monthly Earnings (Adults 18 and Over)

Level of Education / Average Monthly
Earnings (in dollars)
No high school diploma / 856
High school diploma only / 1,357
Vocational degree / 1,568
Associate’s degree / 1,879
Bachelor’s degree / 2,489
Master’s degree / 3,211
Doctorate degree / 4,545
Professional degree
(ex. Medicine) / 5,554

Discuss each question and then come to a single response that each member of your group agrees with and understands.

On the average, what would a U.S. worker make in a year if the person was not a high school graduate?

On average, what is the difference in yearly earnings between a recent high school graduate and a person without a high school diploma?

Now it’s your turn. Write a question that interests your group and can be answered using these data. Write the answer next to your question, too.

From which level of education to the next is the largest jump in average monthly earnings?

Did you use the table or bar graph in answering part a? Explain why you chose the display you did.

Write a summary of the conclusions you can draw using the information in the table and the graph. There is a minimum of three significant conclusions and explanations.

In this lesson, you have reviewed how to make stem-and-leaf plots and how to measure lengths and record your results efficiently. You also have begun to learn how to analyze and write descriptions of distributions of data. In the next lesson, you will learn more about plotting and describing distributions.