Lesson 4

The purpose of this Math Talk is to elicit strategies and understandings students have for finding distances on a number line. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to compute mean absolute deviation (MAD).

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?”

Students look at different examples of distribution shapes and their descriptions and determine whether or not they are accurate descriptions of the given distribution shapes. Whether the students agree or disagree with the descriptions, they will explain why. Explaining their answers allows students to engage in MP3 by focusing on and critiquing the explanations provided. This prepares students for a later lesson when they have to determine which data displays depict the same data set. In order to be successful with that task, students will have to remember the importance of distribution shapes. This activity also prepares students to create statistical questions or scenarios based solely on a distribution shape, because they will understand what the shape means about the data set and what types of data sets make sense for a given distribution shape.

Display for all to see one of the dot plots created as a class from a previous lesson. Ask students to come up with at least one piece of information they can see in the dot plot. Invite a few students to share. For example, they may say that each dot represents one person, or that the number of dots tells you how many people responded with a certain value. The purpose of this launch is just to quickly refamiliarize students with how dot plots represent a data set.

Allow students to work with a partner or as individuals. Each student should write their own explanation for each question.

Students may think that “symmetric” is equivalent to “bell-shaped”. Ensure that students understand that distributions can be symmetric, but still not bell-shaped as in the question that is labeled accurately as "symmetric," but is also bimodal. Compare this question to the question that is symmetric and bell-shaped. 

The goal of this activity is for students to be able to explain why a distribution shape is labeled as its respective shape. Review correct answers as a whole group, if time permits.

In this activity, students practice using mathematical language to describe the shape of distributions. Students engage in MP6 as they attend to the precision of their language, including using the term approximately as needed.

If desired, ask students to close their books or devices. Designate an area of the classroom for each distribution shape: symmetric, skewed, uniform, bimodal, and bell-shaped. Tell students that you will show them a diagram, and they will move to the area that corresponds to their answer.

Display each diagram one at a time, and give students a minute to move to the area of the room that they choose. After each diagram, ask a student who moved to the correct area to explain why they chose that area. If more than one answer is true for any problem, students can move to either area. Ensure that the explanations are addressed and students understand why it has two descriptions.

The goal of this activity is for students to recognize visual differences in distribution shapes. To help achieve this, highlight word meanings to help students remember some differences. Discuss some meanings (math-related or not) of each vocabulary word and how it relates to the distribution shape. Here are sample questions and sentence starters to promote a class discussion:

• “In geometry, what does symmetric mean? How does that relate to the data distributions that we labeled as such?” (Symmetric means parts of a figure match up after a transformation. Here, we specifically mean there is a line of reflection over which the left and right half match up. The data distributions that can be folded on a vertical line with the two sides matching up are called symmetric. It is important to note that distributions can be described as symmetric even if they do not have perfect symmetry. For example, the uniform histogram is also symmetric, even though a few bins are at slightly different heights than their counterparts on the other side.)
• “Uniform means the same, so data distributions are uniform when ________.” (the dots in dot plots are all stacked to about the same height; histogram bins are all about the same height)
• “What does it mean for a data distribution to be bell-shaped?” (It means the data distribution is shaped like a “bell”—higher in the middle, and lower at each end.)
• Skewed, in mathematics, means not symmetric, so skewed data distributions have data points that _________.” (are more frequent on one side of the distribution than the other side)
• “How do you know when a data distribution is skewed? What does this usually mean about the data set?”(You know it is skewed when the shape of the data is not symmetric, so there are more values on one side of the typical values than on the other.)
• “If a distribution shape is uniform, what can you say about all its data points?” (They are exactly the same or similar in frequency.)
• “Is the shape of a data distribution enough to fully explain a data set”? If not, what other characteristics can we use to describe data?” (No, since it does not say what the actual values are. Measures of center and variability, the five-number summary, and the frequency of values in certain intervals can also be useful in describing the data.)