Subject:
Statistics and Probability
Material Type:
Lesson Plan
Level:
Middle School
6
Provider:
Pearson
Tags:
Language:
English
Media Formats:
Interactive, Text/HTML

# Reviewing How Data Affects A Histogram

## Overview

Students explore how adjusting the bin width or adding, deleting, or moving data values affects a histogram.

Students use the Histogram interactive to explore how the bin width can affect how the data are displayed and interpreted. Students also explore how adjusting the line plot affects the histogram.

# Key Concepts

As students learned in the last lesson, a histogram shows data in intervals. It shows how much data is in each bin, but it does not show individual data. In this lesson, students will see that the same histogram can be made with different sets of data. Students will also see that the bin width can greatly affect how the histogram looks.

# Goals and Learning Objectives

• Explore what the shape of the histogram tells us about the data set and how the bin width affects the shape of the histogram.
• Clarify similarities and differences between histograms and line plots.
• Compare a line plot and histogram for the same set of data.

# Lesson Guide

Have students spend a few minutes discussing the histogram of the quiz scores and trying to picture what would happen to the histogram if the bin size was changed.

ELL: In asking prompting questions, be sure to use adequate pace with ELLs, and be sure that they understand meaning of the questions.

# The Effect of Adjusting Bin Widths

Look at the histogram of scores from a quiz worth 60 points. With a partner, discuss the following questions:

• What do you think will happen if you change the bin width?
• Why would a smaller or larger bin width be helpful?

# Lesson Guide

Discuss the Math Mission. Students will investigate how changes to data values and bin widths affect the shape of a histogram.

## Opening

Investigate how changes to data values and bin widths affect the shape of a histogram.

# Lesson Guide

Tell students that today they will be using the Histogram interactive to explore how histograms change as data values or the bin width changes. Students can add or remove data in the table and adjust the bin width of the histogram.

Have students work in pairs. Make sure students save some of the histograms they make, in their Notebook, as they are investigating. They will need examples of the histograms they make in order to prepare a presentation.

ELL: Before assigning an investigation to students, you may want to model the investigation to ensure that all students understand the task. Have a student re-tell the instructions, and then have another student re-phrase the previous instructions.

# Mathematics

As students explore making histograms in the interactive, have them consider these questions:

• How does changing the bin width change the shape of the histogram?
• What kind of shape did your line plot of sixth grade students have? Would you want the histogram to have a similar shape?
• If there are outliers or gaps in the data, would you want to see that in the histogram?

# Mathematical Practices

Mathematical Practice 5: Use appropriate tools strategically.

• To work effectively, students must use the Histogram interactive strategically. They must think about whether they can create a given plot most efficiently by adjusting the bin width, editing the data points, or doing a combination of these things.

# Interventions

Students have trouble connecting the line plot with the histogram.

• Have students enter the data from the line plot of "How Long Can Sixth Grade Students Swim Underwater" into the Histogram interactive.
• What is the bin width? How many values are in each bin?
• Move the points at 13 and 14 in the line plot to 15.
• Now how many values are in each bin?
• Did the histogram change?
• Move some points in another bin so the histogram doesn't change.
• Now restore the line plot.
• Change the bin width to 7. How did the heights of the bins change? Why?
• Change the bin width to 3. How does the histogram change?

Students have difficulty hiding outliers in the histogram.

• Focus on the line plot first. How can you create a line plot with an outlier? Try it.
• Look at the histogram. Is there a gap between the outlier and the rest of the data?
• How can you adjust the bin width to get rid of this gap? Experiment and see.
• Can you explain why the gap has disappeared?

Students have difficulty creating a histogram with empty bins.

• What does it mean when a histogram has an empty bin?
• Start with any line plot. If the histogram doesn't have an empty bin, how can you move or remove points so it does?

• A histogram with a bin width of 1 would have a bar for each individual data value, with the height of each bar indicating the number of times that value occurs. This would be very similar to a line plot, except the height of the bar, rather than the number of dots, would indicate the frequency.
• A histogram with a bin width equal to the range would have one wide bar, with a height equal to the number of values in the data set.

# Explore Histogram Bin Widths

Explore making histograms with the Histogram interactive by playing with a set of data and bin sizes. You can make the bins wider or narrower, and you can add or remove data from the table.

• Create a set of data so that the histogram has an empty bin (and/or adjust the bin width to create an empty bin).
• What happens if you make the bin width 1?
• What happens if you make the bin width equal to the range of your line plot?
• Make a set of data with an outlier giving you more than one empty bin on the histogram. Now adjust the bin width to "hide" the outlier.

INTERACTIVE: Histogram

## Hint:

• Can you move the data points but keep the number of points in each interval the same?
• Can you adjust the bin width?

# Lesson Guide

Students will prepare presentations on how they satisfied the conditions in the questions and what happened to their histogram when they changed bin width.

# Preparing for Ways of Thinking

Select histograms for each of the questions to share and discuss in Ways of Thinking.

• How does changing the bin width change the shape of the histogram?
• What kind of shape does the line plot show? Would you want the histogram to have a similar shape?
• If there are outliers or gaps in the data, would you want to see that in the histogram?

SWD: If you know that some students may need additional time and/or prompting to participate in this discussion, provide them several of the questions ahead of time (printed out or digitally).

# Mathematical Practices

Mathematical Practice 2: Reason abstractly and quantitatively.

• The Challenge Problem requires students to think abstractly about a data set with a wide range and thousands of points to determine an appropriate bin width.

# Challenge Problem

• The bins should be fairly wide, but it is open to interpretation. The decision on bin width is based partly on convenience (10, 20, 25) and partly on what is best for showing the shape of the data. For this situation, a bin width of 50 would be good if the data are fairly evenly distributed, but 100 could also work given the amount of data.

# Prepare a Presentation

• Discuss how you made each histogram so that it satisfied the conditions in the questions.
• Explain what happened to your histogram of sixth grade student data when you changed the bin width.

# Challenge Problem

• If the range of a data set is 500, and the data set includes thousands of data points, what do you think a good bin width would be for a histogram representing the data?
• Explain why you think the bin width you chose would be a good fit for the data.

# Lesson Guide

Have students share the different histograms.

# Mathematics

Consider these questions:

• Is the bin width that shows the shape of the data best related to the range of the data?
• How can a histogram be deceptive in showing the shape of the data?
• How does an outlier affect the histogram?
• How could you construct a different line plot that would give the same histogram?

# Mathematical Practices

Mathematical Practice 6: Attend to precision.

• Require students to explain the strategy they used to meet the criteria. Students should use precise language and incorporate the following terms in their explanations: histogram, bin, median, and mode interval.

Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

• Have students who did the Challenge Problem defend their choice of bin width.

# Ways of Thinking: Make Connections

Take notes as your classmates present their histograms.

## Hint:

• Can you explain how you made your histogram for that question?
• Which histograms did you have more difficulty making? Which were easier to make?
• How did changing the bin width in the histogram of your sixth grade student data change the histogram?

# Lesson Guide

The students write summaries about bin width and histograms.

# A Possible Summary

The shape of the data in a line plot can affect the way the data look as a histogram, especially if there are gaps in the data. The choice of the bin width can highlight, or hide, the shape of the data. Narrower bin widths can show more detail and highlight gaps, while wider bins can give a better idea of the general shape of the data. Thinking about the spread and range of the data can determine the best bin width.

# Summary of the Math: The Importance of Bin Width

Write a summary of what you learned about bin width and histograms.

## Hint:

• Does your summary explain how the choice of bin width can highlight or hide the shape of the data?

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out what students can learn by looking at a histogram.

ELL: The “Reflect on Your Work” section provides opportunities for ELLs to develop literacy in English and proficiency in mathematics. Make sure students use both academic and specialized mathematical language when reflecting on their learning at the end of each session. Give students time to discuss the summary before they write.

# Reflection

Write a reflection about the ideas discussed in class today. Use this sentence starter if you find it to be helpful.

When I look at a histogram, I can tell these things about the data …