NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 6•6 Lesson 14: Summarizing a Distribution Using a Box Plot Student Outcome . Students construct a box plot from a given set of data. Lesson Notes In this lesson, students transition from using dot plots to display data to using box plots. The lesson begins with exercises that lay the foundation for the development of a box plot. Students inspect dot plots of several sets of data and think about how to group or section the plots to get a sense of the spread of data values in each of the sections. When individual students determine how to make the sections, the results differ, and the process seems arbitrary and inconsistent. Thus, there is a need for a standard procedure for making a box plot. Using the median and the quartiles introduced in the previous lesson leads to a standard procedure. The lesson begins and ends with interactive activities. If time allows, students create a “human box plot” of the time it took them to get to school. Supplies are needed for this exercise: one sticky note for every student, one large piece of rope or yarn, and signs with the names of each of the five numbers included in the five-number summary. Classwork Students read the paragraph silently. A box plot is a graph that is used to summarize a data distribution. What does the box plot tell us about the data distribution? How does the box plot indicate the variability of the data distribution? These questions are explored in this lesson. Example 1 (5 minutes): Time to Get to School MP.3 The questions in this example are designed to help students begin to think about grouping data in order to get a sense of the spread of the data values in various parts of the data set. Let each student write an estimate of the time it took them to get to school on a sticky note. Teachers may want individual students to place their sticky notes on a dot plot that is displayed on the classroom board at the beginning of class, or they can make the dot plot as a class. Example 1: Time to Get to School Consider the statistical question, “What is the typical amount of time it takes for a person in your class to get to school?” The amount of time it takes to get to school in the morning varies for the students in your class. Take a minute to answer the following questions. Your class will use this information to create a dot plot. Write your name and an estimate of the number of minutes it took you to get to school today on a sticky note. Answers will vary. What were some of the things you had to think about when you made your estimate? Answers will vary. Some examples include: Does it count when you have to wait in the car for your sister? I usually walk, but today I got a ride. Does it matter that we had to go a different way because the road was closed? The bus was late. Lesson 14: Summarizing a Distribution Using a Box Plot 171 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G6-M6-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 6•6 . What does a dot on the dot plot represent? Each dot on a dot plot represents one value in a set of data. What is an estimate of the median time to get to school? Does the median do a good job of describing a typical time for students in your class? Answers will depend on the class data. What are the minimum and maximum times? Answers will depend on the class data. Exercises 1–4 (7 minutes) Students work individually on the exercises and compare their plots with their neighbors’ plots. Students should recognize that their divisions of the data are close but not always the same. If time permits, bring the class together for a discussion of the answers, and stress the idea that it is useful to see how the data values group together in different parts of the distribution. Are there a lot of values in the middle or at one end? The values in the lower half are closer together than the numbers in the upper half. What was the shortest time a student in Mr. S’s class took to get to school? The longest? A few students got to school in 5 minutes, which was the shortest. One student took 60 minutes to get to school, which was the longest. Looking at the plot, what is a typical time it takes for students in Mr. S’s class to get to school? Examining the dot plot, it looks like the typical time is 15 minutes. Exercises 1–4 Here is a dot plot of the estimates of the times it took students in Mr. S’s class to get to school one morning. 1. Put a line on the dot plot that you think separates the times into two groups—one group representing the longer times and the other group representing the shorter times. Answers may vary. Some might put the dividing line between ퟏퟓ and ퟐퟎ. 2. Put another line on the dot plot that separates out the times for students who live really close to the school. Add another line that separates out the times for students who take a very long time to get to school. Responses will be different. Some might put a line at ퟑퟎ and a line at ퟏퟎ. Lesson 14: Summarizing a Distribution Using a Box Plot 172 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G6-M6-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 6•6 3. Your dot plot should now be divided into four sections. Record the number of data values in each of the four sections. Answers will vary. Depending on the divisions, ퟕ or ퟖ in the lower one, ퟗ in the next, ퟓ in the next, and ퟓ in the upper section 4. Share your marked-up dot plot with some of your classmates. Compare how each of you divided the dot plot into four sections. Different responses; students should recognize that the divisions might be close but that some are different. Exercises 5–7 (8 minutes): Time to Get to School Students work individually on the questions. Then, discuss the answers as a class. Reinforce the idea of the five-number summary (the minimum, lower quartile (or Q1), median, upper quartile (or Q3), and the maximum) and how the five- number summary is used to create a box plot. Can you think of a way to divide the data set into four parts so that everyone would divide it up in the same way? Using the median and quartiles would divide a data set into four parts with about the same number of data values in each part. Exercises 5–7: Time to Get to School The times (in minutes) for the students in Mr. S’s class have been put in order from smallest to largest and are shown below. ퟓ ퟓ ퟓ ퟓ ퟕ ퟖ ퟖ ퟏퟎ ퟏퟎ ퟏퟐ ퟏퟐ ퟏퟐ ퟏퟐ ퟏퟓ ퟏퟓ ퟏퟓ ퟏퟓ ퟐퟐ ퟐퟐ ퟐퟓ ퟐퟓ ퟐퟓ ퟑퟎ ퟑퟎ ퟑퟓ ퟒퟓ ퟔퟎ 5. What is the value of the median time to get to school for students in Mr. S’s class? There are ퟐퟕ times in the data set, so the median is the ퟏ4th value in the ordered list. The median is ퟏퟓ. 6. What is the value of the lower quartile? The upper quartile? The lower quartile is the 7th value in the ordered list, and the upper quartile is the ퟐퟏst value in the ordered list. The lower quartile is ퟖ, and the upper quartile is ퟐퟓ. 7. The lines on the dot plot below indicate the location of the median, the lower quartile, and the upper quartile. These lines divide the data set into four parts. About what fraction of the data values are in each part? ퟑ There are about of the data values in each part. ퟒ Lesson 14: Summarizing a Distribution Using a Box Plot 173 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G6-M6-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 6•6 Example 2 (7 minutes): Making a Box Plot This example describes the procedure for finding a box plot. Consider having students read through the steps themselves, and then ask them to restate the directions. Example 2: Making a Box Plot A box plot is a graph made using the following five numbers: the smallest value in the data set, the lower quartile, the median, the upper quartile, and the largest value in the data set. To make a box plot: . Find the median of all of the data. Find Q1, the median of the bottom half of the data, and Q3, the median of the top half of the data.