standard_description: Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
problem_activity: Activity
This activity is designed to expand both students’ exposure to various features of distributions and the language they could use to describe distributions. Students sort histograms based on features such as symmetry, gaps, clusters, and unusual values. In earlier grades, students used the term “symmetry” to describe geometric figures (4.G.3); here they use it to describe the shape of a distribution.
Launch
Display the image of the histogram here for all to see. Explain to students that a diagram of a distribution—a dot plot or a histogram—is described as symmetrical if you can draw a line on the diagram and the parts on one side of the line mirror the parts on the other side. Many distributions are not perfectly symmetrical, but are close to or approximately symmetrical.
The histogram here shows an approximately symmetrical distribution. When a line is drawn at the center (such as the line at 30) the two sides are roughly mirror images. If you were to fold the histogram at the line, the two sides would be close to matching.
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Tell students that symmetry is used to describe distributions, and that they will now look for symmetry and other features of distributions in a dozen histograms.
Arrange students in groups of 3–4. Give each group one set of pre-cut cards from the blackline master. Ask students to study the histograms and identify the features described on their task statement. Give groups 10–12 minutes to complete the activity. Explain that each group will need to discuss their work with another group after the first question and before completing the rest of the activity.
Representing, Conversing: MLR5 Co-Craft Questions.
Begin the launch by inviting students to write mathematical questions about the histogram displayed. Give students 1–2 minutes to share their questions with a partner. Listen for students that use terms center, spread, distribution, or frequency in their questions. Invite students to share their questions with the class, and to compare the language of their questions before continuing with the activity.
Design Principle(s): Optimize output (for generalization); Cultivate conversation
Student Facing
Your teacher will give your group a set of histogram cards. Sort them into two piles—one for histograms that are approximately symmetrical, and another for those that are not.
Discuss your sorting decisions with another group. Do both groups agree which cards should go in each pile? If not, discuss the reasons behind the differences and see if you can reach agreement. Record your final decisions.
Histograms that are approximately symmetrical:
Histograms that are not approximately symmetrical:
Histograms are also described by how many major peaks they have. Histogram A is an example of a distribution with a single peak that is not symmetrical.
Which other histograms have this feature?
Some histograms have a gap, a space between two bars where there are no data points. For example, if some students in a class have 7 or more siblings, but the rest of the students have 0, 1, or 2 siblings, the histogram for this data set would show gaps between the bars because no students have 3, 4, 5, or 6 siblings.
Which histograms do you think show one or more gaps?
Sometimes there are a few data points in a data set that are far from the center. Histogram A is an example of a distribution with this feature.
Which other histograms have this feature?
Student Response
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Activity Synthesis
Students will have had a chance to discuss the different features of a distribution in small groups. Use the whole-class discussion to prompt students to think about what the features might mean, and whether or how they affect the way we characterize a distribution. Remind students that we have been using the center of a distribution to talk about what is typical in a group. Discuss some of these questions:
“Look at the histograms that you think show symmetry. When a distribution is approximately symmetrical, where might its center be?”
“Now look at the histograms that you think are not approximately symmetrical. Where might its center be? How might we describe what is typical of a group that has one peak that is not symmetrical (such as that in Histogram B)?”
“Look at the histograms that show gaps. How might a gap (such as that in Histogram K) affect our description of what is typical in a group?”
“Look at the histograms that have values that are far away from other values. Do unusual values (such as those in Histogram G) affect our description of center and spread? If those unusual values weren't there, would our description of center and spread change?”
Expect students’ answers to be very informal. The goal of the discussion is to raise students' awareness that the shape and features of distributions may affect how we characterize the data. This experience provides a conceptual foundation that would help students make sense of measures of center (mean and median) and measures of spread (mean absolute deviation, interquartile range, and range) later.


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