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Arrange students in groups of 2–4. Display the histograms for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three histograms that go together and can explain why. Next, tell students to share their response with their group, and then together to find as many sets of three as they can.
Which three go together? Why do they go together?
Histogram A, horizontal axis 45 to 155 by tens. Beginning at 75 up to but not including 85, height of bar at each interval is 7, 29, 30, 25, 3.
Histogram B, horizontal axis 45 to 155 by tens. Beginning at 55 up to but not including 65, height of bar at each interval is 7, 29, 30, 25, 3.
Histogram C, horizontal axis 45 to 155 by tens. Beginning at 55 up to but not inluding 65, height of bar at each interval is 1, 6, 15, 17, 26, 16, 6, 4, 0, 2.
Histogram D, horizontal axis 45 to 155 by tens. Beginning 75 up to but not including 85, height of bar at each interval is 4, 19, 25, 12, 7.
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Because there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology that they use, such as "center," "spread," or "distribution," and to clarify their reasoning as needed. Consider asking:
Students sort different histograms during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).
As students work, encourage them to refine their descriptions of the distributions using more precise language and mathematical terms (MP6).
Students may be familiar with the geometric meaning of symmetry. It is used in a similar way here to describe the shape of a distribution.
Math Community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms, or invite a student to read them out loud. Tell students that during this activity they are going to practice looking for their classmates putting the norms into action. At the end of the activity, students can share what norms they saw and how those norms supported the mathematical community during the activity.
Arrange students in groups of 2 and distribute pre-cut cards. Allow students to familiarize themselves with the representations on the cards:
Attend to the language that students use to describe their categories and distributions, giving them opportunities to describe their distributions more precisely. Highlight the use of terms like “symmetric,” “gaps,” or “clusters.” After a brief discussion, invite students to complete the remaining questions.
If the idea of symmetry is not brought up, ask students if they notice any symmetry in any of the distributions. If further discussion on the topic is helpful, 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.
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:
It is okay if there is some disagreement about how the histograms are categorized. Depending on the situation, it may be important to consider things like peaks more or less precisely. For example, if the data in Histogram F are very precise like from a chemistry experiment, it may be important to investigate the 3 possible peaks more closely. If the data are less precise like how people are rating a TV show, then it may be okay to say there is only 1 peak or that the data are fairly spread out and there are no peaks.
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.
Conclude the discussion by inviting 2–3 students to share a norm that they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:
Getting to School Handout
In this activity, students draw a bar graph and histogram, and then they describe the distributions shown on each display. Although the two visual displays may appear similar at first glance, there are important distinctions between the representations. Students notice differences in how we might characterize distributions in bar graphs and those in histograms, including how we describe typical values or categories. Along the way, students consolidate their understanding about categorical and numerical data.
Monitor for groups that draw a bar graph with the bars in a different order.
Students will need the data on their travel methods and times, collected at the beginning of the unit. Distribute or display the data collected for these questions from the survey given earlier in the unit. Alternatively, complete the tables in the blackline master ahead of time.
Arrange students in groups of 2. Give one copy of the blackline master to each group of students. Display the data from the prior survey or the completed frequency tables for all to see, or give a copy to each group of 2 students. Give students 5–6 minutes to complete the activity. Ask one partner to create a bar graph to represent the data on the class’s travel methods and the other to create a histogram to represent the data on travel times, and then answer the questions together.
Your teacher will provide you with some data that your class collected the other day.
Compare the histogram and the bar graph that you drew. How are they the same? How are they different?
The purpose of the discussion is for students to recognize the differences between histograms and bar graphs.
Invite previously selected groups to share an accurate histogram and bar graphs with bars in different orders. Then, solicit several observations about how the two graphical displays compare. Ask questions such as:
Then, invite students to share their descriptions of the distributions shown on each type of display. Ask questions such as:
Students should recognize that only the distribution of numerical data can be described in terms of shape, center, or spread. We cannot analyze these features for a distribution of categorical data on a bar graph because a bar graph does not use a number line. This means that the bars can be drawn anywhere, in any order, and with any kind of spacing, so shape, center, and spread would have no meaning.
We can describe the shape and features of the distribution shown on a histogram. Here are two distributions with very different shapes and features.
A histograms labeled “A” where the horizontal axis has the numbers 10 through 30, in increments of 2, indicated and on the vertical axis, the numbers 0 through 6 are indicated. The data represented by the bars on histogram are: 10 up to 12, 0; 12 up to 14, 1; 14 up to 16, 2, 16 up to 18, 4; 18 up to 20, 5; 20 up to 22, 6; 22 up to 24, 4, 24 up to 26, 3; 26 up to 28, 2; 28 up to 30, 1.
A histogram, labeled “B” where the horizontal axis has the numbers 10 through 30, in increments of 2, indicated and on the vertical axis, the numbers 0 through 6 are indicated. The data represented by the bars on histogram are: 10 up to 12, 5; 12 up to 14, 3; 14 up to 16, 2; 16 up to 18, 2; 18 up to 20, 1; 20 up to 22, 0; 22 up to 24, 1; 24 up to 26, 3; 26 up to 28, 2; 28 up to 30, 1.
Here is a bar graph showing the breeds of 30 dogs and a histogram for their weights.
A bar graph has three categories on the horizontal axis labeled "pugs," "beagles," and "German shepherds." The vertical axis has the numbers 0 through 12, in increments of 3, indicated. The bar labeled “pugs” has a height of 9. The bar labeled “beagles” has a height of 9” and the bar labeled “German shepherds” has a height of 12.
A histogram has a horizontal axis labeled “dog weights in kilograms.” The vertical axis has the numbers 0 through 10, in increments of 2, indicated. The data represented by the bars are as follows: 10 up to 15 kilograms, 5; 15 up to 20 kilograms, 7; 20 up to 25 kilograms, 10; 25 up to 30 kilograms, 3; 30 up to 35 kilograms, 5.
Bar graphs and histograms may look alike, but they have different uses.
Help us improve by sharing suggestions or reporting issues.
Your teacher will give you a set of cards. Each card contains a histogram.