Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University Probability and Statistics Activity: What’s in a Name? TEKS: (8.12) Probability and statistics. The student uses statistical procedures to describe data. The student is expected to: (A) select the appropriate measure of central tendency or range to describe a set of data and justify the choice for a particular situation; (C) select and use an appropriate representation for presenting and displaying relationships among collected data, including line plots, line graphs, stem and leaf plots, circle graphs, bar graphs, box and whisker plots, histograms, and Venn diagrams, with and without the use of technology. (8.16) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. The student is expected to: (A) make conjectures from patterns or sets of examples and nonexamples Overview: Students will construct box-and-whisker plots to analyze and compare data sets. The teacher will pose the question: Do people with short first names have shorter last names than people who have long first names? To investigate this question, students first collect data about the length of the first names. They will line up in order from the shortest to the longest first name. The class will create a human box-and-whisker plot and locate the median, upper and lower quartiles, upper and lower extremes, and outliers. Adding machine tape is wrapped around the students to create the box portion of the plot. Yarn is used to create the whiskers. The median divides the students into two groups, the students with shorter first names and the students with longer first names. For each group, a boxand-whisker plot for the length of the last names will be created. These two box-and-whisker plots will be compared and analyzed. Based on these comparisons, students will reach their conclusions and be prepared to explain their reasoning. Materials: Index cards One roll of adding machine tape Scissors Colored yarn Graphing calculator (optional) Labels for Upper Extreme, Lower Extreme, Upper Quartile, Lower Quartile, Median, Outlier (2) How Old Where They? (1 copy for each student) Probability and Statistics What’s in a Name Grade 8 Page 1 Mathematics TEKS Refinement 2006 – 6-8 Grouping: Large group and small group Time: Two 45-minute class periods Tarleton State University Lesson: 1. Procedures Pose the following question to the class: Notes Do people with short first names have shorter last names than people with long first names? 2. Ask students to vote by a show of hands on their opinion: Yes, people with short first names have shorter last names than people with long first names. No, people with short first names don’t have short last names or the length of the first name makes no difference in the length of the last name. 3. Ask students how these questions might be investigated. Hopefully students will suggest collecting data about the first and last names of a group of people, organizing the data, analyzing the data, and drawing a conclusion. How might we use the names of the students in this class to investigate this question? 4. Have students record the number of letters in their first name on an index card. Tell the students that this data will be used to organize them into a human box and whisker plot. For this class, how could we determine which first names are short and which are long? Have students line up in order from those with the least number of letters in their first name to those with the most letters in their first name. Sample class data are provided, and construction of the box and whisker plot is modeled in the supplementary materials. Ask the students how we could find the median number of letters in the first names for this group. Probability and Statistics What’s in a Name Grade 8 Page 2 Mathematics TEKS Refinement 2006 – 6-8 Procedures Have students count off starting at both ends of the line until they reach the student who is in the middle. The number of letters in the student’s first name represents the median for this group. Give this student a label which says “median.” Those students who are above the median form the group with longer first names, and those below the median form the group with shorter first names. 5. Discuss each of the parts of the box-andwhisker plot with the students as it is being created. Give the students who represent the upper quartile and lower quartile a sign which labels them as such. Give the student with the shortest name a label that says lower extreme and the student with the longest number a label that says upper extreme. To create the box portion of the box-andwhisker plot, wrap the students from the lower quartile to the upper quartile in adding machine tape. Use the yarn to create the whiskers. One piece of yarn should be stretched from the lower quartile to the lower extreme and another from the upper quartile to the upper extreme. Check out any outliers (see Notes). Draw and label a box-and-whisker plot on the board so that students have a concrete representation of the human plot they form. Tarleton State University Notes Order the data. Then the median of the data is the middle number if there is an odd number of data points or the average of the two middle numbers if there is an even number of data points. The upper quartile is the middle or median of the upper half of the data, and the lower quartile is the median of the lower half of the data. The highest value in the data set is called the upper extreme, and the lowest value in the data set is called the lower extreme. Values which are widely separated from the rest of the data are called outliers. An outlier is any value more than 1.5 inter quartile ranges above the upper quartile and/or more than 1.5 inter quartile ranges below the lower quartile. The inter quartile range is the difference between the upper quartile and the lower quartile. Questions to ask: How many students are represented by each part of the box plot? Look at the whisker from the lower extreme to the lower quartile – about what percent of the class is represented by it? Answer the same question about the “box” and the two sections of Probability and Statistics What’s in a Name Grade 8 Page 3 Mathematics TEKS Refinement 2006 – 6-8 Procedures Tarleton State University Notes the “box” formed when the median is drawn and the whisker drawn from the upper quartile to the upper extreme. It is important for students to note that each of these parts represents 25% of the group. Remind students that the big questions to answer are for this class: How many letters are in short first name? How many letters are in a long first name? Questions to ask: According to the data about our first names, how many letters are in a short first name? How many letters are in a long first name? See the step-by-step instructions on creating a box-and-whisker plot for a sample class roster. Steps to follow when constructing a box-and-whisker plot: 1. Write the data in numerical order. 2. Identify the five values you will use to construct the plot: • median • lower extreme • upper extreme • lower quartile • upper quartiles 3. Draw a number line which will accommodate the range of the data set. 4. Plot the five identified values from the data set below the number line. 5. Draw a box from the lower quartile to the upper quartile. 6. Draw a vertical line for the median. 7. Draw the whiskers from the box to the extremes of the data set. Probability and Statistics What’s in a Name Grade 8 Page 4 Mathematics TEKS Refinement 2006 – 6-8 6. 7. Procedures Collect the data about the number of letters in the last names. The data to be collected should be based on the two groupings so that students are able to compare the data (number of letters in the last name) of students with short first names with the data of students with long first names. Tarleton State University Notes Write “Short Names” and “Long Names” on the board and have students list the number of letters in their last name under the label that describes their first name. When the last name data has been recorded, Option: Students could use a have students work in small groups to graphing calculator to create the construct two box- and-whisker plots – one box-and-whisker plots. for the number of letters in the last names for students with short first names and a second one for the number of letters in the last names for students with long first names. Their task is to analyze the data to answer the original question: Do people with short first names have shorter last names than people who have long first names? Have students prepare a poster of their work which includes the box-and-whisker plot, their conclusion, and a justification for their conclusion. 8. Display the posters, and ask the groups to take turns presenting their work. 9. Student Reflection: How was the box-andwhisker plot helpful in comparing the length of last names? Were there questions about the length of last names that you couldn’t answer from the box plot? Homework: How Old Were They? Extensions: Make a scatterplot using the ordered pairs (number of letters in first name, number of letters in last name) for the students in this class. Can you draw any conclusions or make any predictions based on this data? Compare this graph to the box-and-whisker plots. What can you tell about the data from the scatterplot that you can’t tell from the box-and- Probability and Statistics What’s in a Name Grade 8 Page 5 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University whisker plot? What can you tell about the data from the box-andwhisker plot that you can’t tell from the scatterplot? Resources: Eric W. Weisstein. "Box-and-Whisker Plot." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Box-andWhiskerPlot.html http://www.purplemath.com/modules/boxwhisk.htm Probability and Statistics What’s in a Name Grade 8 Page 6 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University What’s in a Name? Sample Class Roster First Name Kyle Alissa Ryan Stephanie Kathy Logan Sanjuanita Andrew Arlene Angela Heather Layne Lee Bob Jeremy Collin Lisa Greyson Rafaeal Rosemary Juan Larry Joe Tricia Devin Jeff Macey Jonathan Brenda Susan Michael Probability and Statistics What’s in a Name Last Name Alexander Arendall Belford Box Ceeley Contreras Daroza Dominguez Gonzalez Green Hail Jackson Komel Littleton MacMillan McMurray Night Noske Ortega Orvera Padula Pizzini Renfro Rothenbury Smith Suhr Teran Thill Tindle Williams Wusterhausen Grade 8 Page 7 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University Steps to follow when constructing a box-and-whisker plot: (The example below uses first names from the sample class roster. The objective is to find out how many letters are in a short first name and a long first name for this class.) 1. Write the data in numerical order. The numbers of letters in the first names of the students on the sample roster are listed in order from least to greatest: 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10 2. Identify the five values you will use to construct the plot: median, lower and upper extremes, the lower and upper quartiles, and the outliers (if there are any). The median of the data is the middle number if there are an odd number of data points or the average of the two middle numbers if there is any even number of data points. This class has 31 students so the median is the data point that is in the 16th position in the ordered data set. The median is 6 letters in the first name. The upper quartile is the middle or median of the upper half of the data and the lower quartile is the median of the lower half of the data. For this set of data, the upper quartile is 7 and the lower quartile is 4. The highest value in the data set is called the upper extreme (for this set of data, 10) and the lowest value in the data set is called the lower extreme (for this set of data, 3). Values which are widely separated from the rest of the data are called outliers. An outlier is any value more than 1.5 inter quartile ranges above the upper quartile and/or more than 1.5 inter quartile ranges below the lower quartile. The inter quartile range is the difference between the upper quartile and the lower quartile. For this data, the inter quartile range is 7 – 4 or 3. 1.5 times 3 is 4.5. An outlier above the upper quartile for this data would be 4.5 added to 7 which is 11.5. Since 10 is the greatest number of letters, there are no outliers on the upper end of the data. There are no outliers on the lower end of the data as well 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10 Lower Extreme Lower quartile median Upper quartile Upper Extreme . Probability and Statistics What’s in a Name Grade 8 Page 8 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University 3. Draw a number line which will accommodate the range of the data set. 3 4 5 6 7 8 9 10 4. Plot the five identified values from the data set below the number line. See below LE is the lower extreme, 3; LQ is the lower quartile, 4; M is median, 6, UQ is upper quartile, 7; UE is upper extreme, 10. 3 • LE 4 • LQ 5 6 • M 7 • UQ 8 9 10 • UE 5. Draw a box from the lower quartile to the upper quartile. 3• 4• 5 6• •7 8 9 10 • 9 10 • 6. Draw a vertical line for the median. 3 • 4 • 5 6 • 7 • 8 7. Draw the whiskers from the box to the extremes of the data set. 3 4 5 Probability and Statistics What’s in a Name 6 7 8 9 10 Grade 8 Page 9 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University NOTES: After the box-and-whisker plot is constructed, there are many interesting questions to explore. Examples: How many people are represented by the shorter whisker? the longer whisker? How many people are represented by the box drawn between the lower and upper quartile? How many people are represented by the boxes inside the larger box? Why are the whiskers different lengths? What does that tell us about the first names in this class? Why is the larger box divided into two sections that are different sizes? What does that tell us about the first names in this class? Are there any outliers in this data? Remember that our objective was to divide the class into two groups according to the number of letters in the first names. One group would have short first names and the other long first names. Then we can compare the last names for these two groups to answer the original question: Do people with short first names have shorter last names than people who have long first names? For the sample class roster, short names had 3 to 5 letters, and long names had 6 to 10 letters. For each group, a box-and-whisker plot for the length of the last names will be created. These two box-and-whisker plots will be compared and analyzed. Number of letters of last names of students with short first names Number of letters of last names of students with long first names Probability and Statistics What’s in a Name Grade 8 Page 10 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University On the previous screen shot, the top plot represents the number of letters in the last names of students with short first names and the bottom plot represents the number of letters in the last names of students with long first names. Possible questions to ask: How many letters are in the shortest last name of the students with short first names? How do you know? How many letters are in the longest last name of the students with short first names? How do you know? How many letters are in the shortest last name of the students with long first names? How do you know? How many letters are in the longest last name of the students with long first names? How do you know? What is the median length of a last name for each group? How do you know? Write three statements comparing the two sets of students that can be supported by analyzing these plots. Probability and Statistics What’s in a Name Grade 8 Page 11 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University How Old Were They? The following box-and-whisker plot shows the ages of the 17 Presidents of the United States elected in the twentieth-century at their inaugurations. 40 50 60 70 80 1. The youngest president was John F. Kennedy. How old was he at his inauguration? 2. The oldest president at inauguration was Ronald Reagan. How old was he at his inauguration? 3. What was the median age at inauguration? 4. What percentage of the presidents were over 60 (the upper quartile)? 5. What percentage of the presidents were under 51 (the lower quartile)? The following table gives the ages of the 21 Vice-presidents of the United States at the time they took office: Vice-president Age when he took office Charles Fairbanks 52 James Sherman 53 Thomas Marshall 58 Calvin Coolidge 48 Charles Dawes 59 Charles Curtis 68 John Garner 64 Henry Wallace 52 Harry Truman 60 Alben Barkley 71 Richard Nixon 40 Lyndon Johnson 52 Hubert Humphrey 53 Spiro Agnew 51 Gerald Ford 59 Nelson Rockafeller 65 Walter Mondale 49 George Bush 56 Dan Quayle 41 Albert Gore 44 Probability and Statistics What’s in a Name Grade 8 Page 12 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University 6. Make a box-and-whisker plot of the ages of the vice-presidents. Below the same number line, copy the box plot for the ages of the presidents. Compare the ages of the twentieth century presidents and vice-presidents. Write at least three things you learned from analyzing the data in the box plots. Include a statement about how old a person must be to become president or vice-president, and tell how this compares to the ages of the people who served in these offices in the twentieth century. Probability and Statistics What’s in a Name Grade 8 Page 13 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University How Old Were They? Solutions The following box-and-whisker plot shows the ages of the 17 Presidents of the United States elected in the twentieth-century at their inaugurations. 40 50 60 70 80 1. The youngest president was John F. Kennedy. How old was he at his inauguration? 43 years old 2. The oldest president at inauguration was Ronald Reagan. How old was he at his inauguration? 69 years old 3. What was the median age at inauguration? 55 years old 4. What percentage of the presidents were over 60 (the upper quartile)? 25% 5. What percentage of the presidents were under 51 (the lower quartile)? 25% The following table gives the ages of the 21 Vice-presidents of the United States at the time they took office: Vice-president Age when he took office Charles Fairbanks 52 James Sherman 53 Thomas Marshall 58 Calvin Coolidge 48 Charles Dawes 59 Charles Curtis 68 John Garner 64 Henry Wallace 52 Harry Truman 60 Alben Barkley 71 Richard Nixon 40 Lyndon Johnson 52 Hubert Humphrey 53 Spiro Agnew 51 Gerald Ford 59 Nelson Rockafeller 65 Walter Mondale 49 George Bush 56 Dan Quayle 41 Albert Gore 44 Probability and Statistics What’s in a Name Grade 8 Page 14 Mathematics TEKS Refinement 2006 – 6-8 Tarleton State University 6. Make a box-and-whisker plot of the ages of the vice-presidents. Below the same number line copy the box plot for the ages of the presidents. Compare the ages of the twentieth century presidents and vice-presidents. Write at least three things you learned from analyzing the data in the box plots. Include a statement about how old a person must be to serve as president or vice-president. 40 50 60 70 80 Vice-Presidents Presidents The youngest Vice-President was 40 years old while the youngest President was 43 years old. The oldest Vice-President was 71 years old while the oldest President was 69 years old. The median age for the Vice-Presidents was 53 years and for the Presidents the median was 55 years. The Presidents were slightly older than the Vice-Presidents since half of them were over 55 years old. According to Article II of the United States Constitution, you must be 35 years old to hold the office of President. The Twelfth Amendment makes this apply to the VicePresident as well. Probability and Statistics What’s in a Name Grade 8 Page 15