Lesson 3

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?”

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 one is useful in analyzing data.

Either measure each student’s height, or ask students to report their height in inches. Poll the class and display the list of heights for all to see. (If you use a different question other than asking about height, ensure it is a question with a range of responses similar to student height in inches, something like 48–76.)

If student data is unavailable or if there are not enough students to create a useful display of data, use the data about the height of students in a small class:

• 60
• 60
• 60
• 63
• 63
• 65
• 65
• 65
• 65
• 67
• 67
• 70
• 70
• 70
• 70
• 70
• 72
• 72
• 72

Arrange students in groups of 3 or 4. Allow students to work together to create the different representations. If necessary, demonstrate how to create one of the histograms and allow students to practice by creating the others. If time is limited, encourage students to assign a graph to each group member and to take a look at each other’s graphs when they have completed them.

Students sometimes forget the characteristics of each graph. Encourage them to ask people in their group for assistance 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. However, ensure 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.

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. Ensure that students notice how each graph provides different information even when they are all displaying the same data set. 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 in to the center of the data than a box plot.)
• "What information can we get from using a box plot to display the data? (It provides a summary of the data as well as where the median is located.)
• “Does the histogram give 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.)
• “How did your graphs differ from one another?” (The histograms showed different shapes of the data sets. With each different interval size, the shape changes.)
• “Did all three types of graphs display the same information?” (No, the box plot showed the median, and the dot plot and histograms do not.)
• “How were all three histograms alike or different?” (They were alike because the same data points were included in each. They were different because they each used different intervals. Using a smaller bin 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)

Previously, students practiced constructing and interpreting data representations. In the first activity in this lesson, students were introduced to the idea that each representation can be used for different reasons. In this activity, students will further practice interpreting data by agreeing or disagreeing with statements about the information presented in data displays.

Ensure students understand that each question has two parts: agreeing or disagreeing with the claim, then explaining a reasoning, including identifying which data display was useful. 

Students should 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 can be seen in both dot plots and box plots.)
• “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 bin 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 bin sizes were 5, then the histogram would have fewer bins 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 determine 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 determine which data representation will be most useful, given what information they need from a data set. Discuss how students knew which data representation is most useful in answering the questions. Here are sample questions to promote a class discussion:

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