This Math Talk focuses on mental subtraction. It encourages students to think about patterns and to rely on structure to mentally solve problems. The strategies elicited here will be helpful when students compute the interquartile range.

To solve each expression, students need to look for and make use of structure (MP7).

This is the first Math Talk activity in the course. See the Launch for extended instructions for facilitating this activity successfully.

The goal of this discussion is to review students’ strategies for subtracting mentally.

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 ’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 ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to earlier problems do you see?”

In a previous course, students learned to find 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. Students reason abstractly and quantitatively (MP2) as they interpret the box plot that relates to data about their classmates.

The goal of this activity is to review how to find the five-number summary and create a box plot from a set of data. Draw the box plot, and display it 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, students’ drawings of the box plot may be considered rough drafts that are needed only to demonstrate understanding of the five-number summary and the range. Although some students may mention the interquartile range, a deep understanding is not needed 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. 

This is the first time Math Language Routine 2: Collect and Display is suggested in this course. In this routine, the teacher circulates and listens to student talk while jotting down words, phrases, drawings, or writing that students use. The language collected is displayed visually for the whole class to use throughout the lesson and unit. The purpose of this routine is to capture a variety of students’ words and phrases—including especially everyday or social language and non-English—in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language used in mathematics conversations.

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. Direct students’ attention to the reference created using Collect and Display. Ask students to share how they identified each part of the five-number summary in their classmates' box plots. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. 

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 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 no matter the order of the data. However, to calculate the quartiles, we 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 us 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 prompt 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.)