Previously, students learned and reviewed how to construct box plots and dot plots. In this activity, students will practice constructing these. If they recall how to construct histograms from previous instruction, this is an opportunity to practice. If necessary, demonstrate how to construct a histogram. They will extend their thinking about data representations to understanding when and how each representation is useful when analyzing data.

Students reason abstractly and quantitatively (MP2) as they understand data in context as well as in data displays and make connections among the representations.

Monitor for students who choose different representations for the last 3 questions. While the questions about continuing analysis may lead students towards dot plots (because they can see each data point) the other questions depend on students’ approaches to data. Some students’ reasoning will be more mathematical (“The box plot is best because it shows the median clearly” or “The third histogram shows the shape without being cluttered with each piece of data”), while other students’ reasoning will be based on their opinions (“The first histogram is best because it is the easiest to make”). In the class discussion, acknowledge that opinion-based reasoning is acceptable if accompanied by mathematical support.

This is the first time Math Language Routine 7: Compare and Connect is suggested in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and they record their method for all to see. They then compare and identify correspondences across strategies by means of a teacher-led gallery walk with commentary or teacher think-aloud (such as "I notice _____. I wonder _____"). A typical discussion question is “What is the same and what is different?” prompting students to compare their own strategy to the others. The purpose of this routine is to allow students to make sense of mathematical strategies and, through constructive conversations, develop awareness of the language used as they compare, contrast, and connect other ways of thinking to their own.

Students sometimes forget the characteristics of each graph. Encourage them to ask people in their group for help, and provide previously constructed box plots, dot plots, and histograms as references.

Students often confuse histograms with bar graphs. Histograms are used for representing a distribution of numerical data along a continuum, while bar graphs are used for displaying categorical data, but it's not crucial that students can explain this difference. Make sure that students are creating proper histograms (with the intervals touching).

When constructing the intervals, students may not know which data points to include in a given range. In these materials, the lower bound of each interval is included and the upper bound is not. For example, if the intervals are 60–70 and 70–80, then 70 would be included in the second interval. Ensure that this rule is applied consistently.

The goal of this discussion is to show that different displays of the same data can be useful in different ways. Select several students or groups to share their preferred representations and their explanations. If not already mentioned by students, discuss the different insights that each display offers, or different challenges it poses. Here are sample questions to prompt class discussion:

• “What information can we get from using a dot plot to display the data?” (It displays everyone’s height in the class. This gives us more insight into the center of the data than a box plot does.)
• "What information can we get from using a box plot to display the data?” (It gives a summary of the data as well as where the median is located.)
• “Does the histogram give a meaningful snapshot of the distribution?” (A histogram gives meaningful insight into the spread of everyone’s height in the class because we can see how much the numbers range just by looking at the histogram.)
• “How do your graphs differ from one another?” (The histograms showed different shapes of the data sets. With each different interval size, the shape changes.)
• “Do all three types of graphs display the same information?” (No, the box plot shows the median, and the dot plot and histograms do not.)
• “How are all three histograms alike or different?” (They are alike because the same data points are included in each. They are different because they each use different intervals. Using a smaller interval size makes the distribution look more like the dot plot.)
• “To know the range and median at a glance, which would you use?” (box plot)

The goal of this activity is for students to understand that each display shows certain information, and they can choose a type of display to help them answer specific questions. Some of the displays cannot answer certain questions, and some questions can be answered using multiple displays, but may be more easily seen in one type than in another. Select students to share their solutions. Some questions for discussion:

• “What information can be easily found in both dot plots and box plots?” (The range of the data and clusters of data.)
• “Overall, what types of questions do box plots help answer? Histograms?” (Box plots help answer questions specifically about quartiles, interquartile range, range, median, lowest value, and highest value. Histograms help answer questions about ranges of data points and the shape of the distribution.)
• “If the histogram had interval sizes of 5, how would the graph’s display change? Would this graph be more useful for filling in the frequency table than the current histogram?” (If the interval size were 5, then the histogram would have fewer intervals than it currently does. No, it would not be more useful for filling in the frequency table, because the frequency table has intervals of 6, and the histogram would have intervals of 5. It would be difficult to find which data points to include from an interval of 5–10 in a frequency table that has 6–12.)

The goal of the lesson is for students to be able to decide which data representation will be most useful, given what information they need from a data set. Discuss how students know which data representation is most useful in answering the questions. Here are sample questions to promote a class discussion:

• “Are there any patterns in the questions you chose to use a dot plot (or histogram and box plot) to answer?” (I noticed that all the questions for which I used a dot plot to answer were questions about frequencies or accounting for each individual data point.)
• “How do you know which data representation is the most useful in finding your answers?” (I think about what the question asked, then connect that to the characteristics of each graph.)