Lesson 1

The purpose of this Math Talk is to elicit strategies and understandings students have for mental subtraction. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to compute interquartile range.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

In a previous course, students learned to identify the five-number summary of data sets (minimum, maximum, quartile 1, median, and quartile 3) and construct box plots from this data. They also calculated the range and interquartile range of distributions. In this activity, students recall the meaning of the values in the five-number summary and construct a box plot. This will be useful when students analyze data in a later lesson. They explore this new representation of data by creating a human box plot to represent class data.

Before the lesson, use thin painter’s tape to make a number line on the ground. If the floor is tiled with equal-sized tiles, consider using the tiles for the intervals of the number line. Otherwise, mark off equal intervals on the tape. The number line should cover at least the distance between the least data value (the smallest shoe size) to the greatest (the largest shoe size).

Distribute a slip of paper to each student and ask students to write down their shoe size in men's or youth size (students wearing a women's size should subtract 2 from their shoe size). (You could also ask for a different metric, if there is one more appropriate for your students. Examples might be the number of letters in their first and last name or the number of minutes it takes them to travel to school.)

Poll the class to create a list of these numbers for all to see. It is recommended that you not put these numbers in order.

Ask students to stand up, holding their slip of paper in front of them, and place themselves on the point on the number line that corresponds to their number. (Consider asking students to do so without speaking at all.) Students who have the same numbers should stand one in front of the other. Ask about each component of the five-number summary in regard to the class’ data set. Ask the students, “What are the maximum and minimum values? What value splits the group in half?” Remind students that the value that splits the group in half is called the median of the data set. Ask students, “If we split the class into quarters (or fourths), how many students would be in each group? What value splits the lower half of the data in half again?” Students should understand that splitting the lower half of the data in half gives the first quartile (Q1), and splitting the upper half of the data in half gives the third quartile (Q3).

Record and display the five-number summary for all to see.

Hold up the index card that has been labeled with “minimum.” Ask students who should claim the card, then hand the card to the appropriate student. Do the same for the other labels of the five-number summary. If any of the quartiles fall between two students’ numbers, write that number of the index card and have both students hold that card together.

Now that the five numbers are identified and each is associated with one or more students, use wide painter’s tape to construct a box plot.

• Form a rectangle on the ground by affixing the tape around the group of students between Q1 and Q3. If a quartile is between two people, put the tape down between them. If a quartile has the value of a student’s number, put the tape down at that value and have the student stand on it.
• Put a tape segment at Q2, from the top side of the rectangle to the bottom side, to subdivide the rectangle into two smaller rectangles. If Q2 is a student’s number, have the student stand on the tape.
• For the left whisker, affix the end of tape to the Q1 end of the rectangle; extend it to where the student holding the “minimum” card is standing. Do the same for the right whisker, from Q3 to the maximum.
• Tape the five-number summary cards and students’ cards that correspond to them in the right locations. This image shows an example of a completed human box plot.

  

  

Explain to students that they have made a human box plot. Consider taking a photo of the box plot for reference and discussion later.

Have a plan for students who struggle to get started finding the five values in the task. Some approaches might include suggesting that they:

• Look up the terms in the glossary.
• Collaborate with other students.
• Write all the values in the data set in order.

Draw the box plot and display for all to see. Label the number line and each value in the five-number summary. Discuss:

• “Where can the median be seen in the box plot?” (It is the line inside the box.) "Where are the first and third quartiles?" (The left and right sides of the box.)
• “Where can the interquartile range be seen in the box plot?” (It is the length of the box.)
• “The two segments of tape on the two ends are sometimes called ‘whiskers.’ What do they represent?” (The lower one-fourth of the data and the upper one-fourth of data.)
• “How many people are part of the box, between Q1 and Q3? Approximately what fraction of the data set is that number?” (About half. Note that the number of people that are part of the box may not be exactly one half of the total number of people, depending on whether the number of data points is odd or even, and depending on how the values are distributed.)
• “Why might it be helpful to summarize a data set with a box plot?” (It could help us see how close together or spread out the values are, and how each fourth is distributed.)

Explain to students that we will draw and analyze box plots in upcoming activities and further explore why they might be useful.

To construct their box plots, students calculate the five-number summary: minimum, maximum, median, first quartile, and third quartile. At this stage, student drawings of the box plot may be considered rough drafts that are only required to demonstrate understanding of the five-number summary and the range. Although some students may mention the interquartile range, a deep understanding is not required at this stage.

As the groups display their box plots for the class to see, students observe the different box plots to answer the questions. 

Arrange students in groups of 2–4. Provide each group with tools for creating a visual display. Assign each group a data set:

• Data set A: The height, in inches, of each plant in a garden: 7, 3, 4, 1, 7, 10, 3
• Data set B: The age of each dancer at a dance camp: 3, 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 8, 11, 13
• Data set C: The number of points a player scored in each of their basketball games: 19, 16, 15, 21, 13, 16, 17, 19
• Data set D: A student’s score on several quizzes as a percentage: 81, 81, 82, 83, 83, 83, 84, 86, 86
• Data set E: The temperature in each sauna at the spa in degrees Fahrenheit: 132, 126, 129, 130, 132, 137, 134, 128
• Data set F: The high temperature, in degrees Fahrenheit, each day during a family vacation: 50, 70, 70, 73, 75, 75, 80

Encourage each group to make a rough draft of their box plot, and then create a visual display of their box plot. As each group finishes its box plot, post them around the room. Once all the displays are ready, invite students to observe each posted box plot and answer the questions in the task statement.

Students may not remember how to find the five-number summary. You can:

• Draw attention to the human box plot and ask student to think about how the five-number summary was found.
• Refer students to the glossary.

The goal of this activity is for students to remember how to construct and interpret a box plot by creating one and then interpreting several. Ask students questions to prompt discussion about their experiences constructing box plots:

• “There is a Q1 and Q3. Why do you think there is no Q2?” (The median is a quartile itself. There are three quartiles that divide the data set into four groups, and the median of the data set is technically Q2.)
• “What was the most difficult part to plot? Why?” (The quartiles were most difficult, because the highest and lowest values of the data set are straightforward to find. However, to calculate the quartiles, you need to list the data set in order and know which values to look for.)
• “Does it make sense that the range for the temperature on the family vacation is greater than the range for the temperatures in the saunas? Explain your reasoning.” (Yes. The high temperature can vary from day to day, but saunas are generally kept at around the same temperature.)

The goal of the lesson is for students to connect components of a box plot to interpretations of the data set. For example, the box plot shows the median and quartiles, which helps you to calculate the interquartile range and determine how spread out the data is. Discuss how to construct box plots and when they are useful. Here are sample questions to promote a class discussion:

• “How do box plots help to analyze data?” (They show variability of a data set. By glancing at a box plot, the range is straightforward to observe, and the interquartile range is the distance between the upper and lower quartiles. Also, the median is a center of the data set.)
• “When are box plots useful?” (Box plots are useful for visualizing some characteristics of large data sets and comparing two data sets.)
• “What are other ways to represent data?” (We can represent data with tables, histograms, and dot plots.)
• “What are some strategies you found useful when creating a box plot?” (It was easier for me to find the median first, and then each quartile.)