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

# Measures & Data Sets

## Overview

Students will apply what they have learned in previous lessons to analyze and draw conclusions about a set of data. They will also justify their thinking based on what they know about the measures (e.g., I know the mean is a good number to use to describe what is typical because the range is narrow and so the MAD is low.).

Students analyze one of the data sets about the characteristics of sixth grade students that was collected by the class in Lesson 2. Students construct line plots and calculate measures of center and spread in order to further their understanding of the characteristics of a typical sixth grade student.

# Key Concepts

No new mathematical ideas are introduced in this lesson. Instead, students apply the skills they have acquired in previous lessons to analyze a data set for one attribute of a sixth grade student. Students make a line plot of the data and find the mean, median, range, MAD, and outliers. They use these results to determine a typical value for their data.

# Goals and Learning Objectives

• Describe an attribute of a typical sixth grade student using line plots and measures of center (mean and median) and spread (range and MAD).
• Justify thinking about which measures are good descriptors of the data set.

# Lesson Guide

Each student can work on his or her own data set, or work with a partner to assist each other.

It may be helpful to choose certain topics for students based on ability, or to ensure that all topics are covered. Data sets with a narrow range will be easier than those with a wide range.

SWD: Decide how you will partner students together for this task. Partnering students by skill level will allow for more efficient provision of teacher support, while partnering students heterogeneously will promote cooperative teaching and learning opportunities for students with varying mathematical skills. Monitor partnerships to ensure all students' progress. It is often best practice to pair students with disabilities with typically developing peers.

Partner Work is designed to help students consolidate their learning by justifying their solutions to their peers. Students have an opportunity to collaborate with peers as they summarize, analyze, problem solve, explain, and apply new knowledge.

# Revisit Statistical Questions

Look at the questions and data your class recorded in the lesson "Data About Us" in order to answer the question “What are the characteristics of a typical sixth grade student?”

• Choose a question, that the class collected data on, to investigate further.
• Do you think you are a typical sixth grade student for this set of data?

# Lesson Guide

Discuss the Math Mission. Students will organize and analyze the data collected in order to describe a characteristic of a typical sixth grade student.

## Opening

Organize and analyze the data you collected in order to describe a characteristic of a typical sixth grade student.

# Lesson Guide

Tell students that their job today is to apply the concepts that they have acquired in previous lessons. They are to decide what is typical for their set of data by drawing a line plot and finding the mean, median, and range. If they have, time they can also calculate the MAD.

# Mathematics

Ideally the data sets will have a combination of wide and narrow ranges. This will give a variety of different line plots; ones where the mean is not a good measure and ones where both the mean and the median are the same. The line plots with a wide range will tend to be fairly flat, bringing up the question of whether there's a better or different tool to use (e.g., histograms). These line plots will also have a higher MAD and the measures of center are more likely to be different.

# Interventions

Students have trouble drawing the line plot.

• First figure out what values to show on the number line.
• What is the lowest value? What is the highest value?
• What in-between values should you label? Are all the values whole numbers?
• Now you can start recording the values on your line plot.
• What is the first value?
• How do you mark this on the plot?
• What if you have the same value several times? How do you show that on your plot?

Students have difficulty finding the mean, or median.

• How do you calculate the mean?
• You can list the data in order. How can this help you find the median?
• How can you use the line plot to help you find the median?

Students have trouble deciding on a typical value for their data.

• To decide if the mean is typical, look at the line plot. Are most of the data clustered near the mean or are they spread out?
• If the data are very spread out and there are not big peaks or gaps, then consider the median. Does it seem typical?
• If you don't think there is a typical value, then say that in your answer and explain why you think so.

• Line plots will vary.
• Measures of center, range and outliers will vary.

# Analyze My Data

Now that you have a set of data, you can create a line plot to help you analyze the data.

• Draw a line plot from your set of data.
• Find the following measures for your set of data:
• Mean
• Median
• Range
• Outlier(s)
• MAD (if you don’t have time to calculate it, make an estimate)
• What do you think is typical for your set of data? Why?
• What is your smallest and largest number?
• What is the range of your data?
• Are your tick marks evenly incremented?
• What range of numbers do you want to display on your line plot?

INTERACTIVE: Line Plot with Stats

## Hint:

• Consider the shape of your line plot. Do most of the data cluster around the mean? (If so, could the mean be typical?)
• Is the data spread out over a wide range? (If so, could the median be typical?)

# Lesson Guide

Student will prepare presentations on the characteristic they looked at.

# Preparing for Ways of Thinking

Look for line plots with a variety of shapes to share during Ways of Thinking.

• Choose some for which the choice of a typical value is clear (probably a data set with a narrow range), and some for which the typical value is ambiguous (particularly ones with a wide range).
• Choose at least one plot with an outlier, if possible.

ELL: Encourage students to use the academic vocabulary that they have learned. As they participate in the discussion, monitor for knowledge of the topic. Stay alert to follow up on statements that seem unclear. When ELLs contribute, focus on content, and don't allow grammar difficulties to distract you from understanding the meaning. Help ELLs who make grammar mistakes by rephrasing, but do it only when your rephrasing will not become an interruption or interfere with their thinking. As with other discussions, make sure that all academic language is posted in the room on the word wall or anchor charts.

# Prepare a Presentation

Prepare a presentation on your definition of what's typical of a sixth grader for the characteristic you looked at. * Use your data to justify your conclusion.

# Challenge Problem

• If another student joined the class tomorrow, where do you think he/she would fit into your set of data? Why?

# Lesson Guide

There won't be time to share all of the line plots. Make sure that a wide variety of line plots are shared. Discuss plots for which the choice of a typical value is clear and plots for which it is not. Students should explain how they decided which value is typical or why they concluded that there is no typical value.

ELL: As with other oral instructions, ensure that the pace of your speech is appropriate for ELLs. Pause frequently to allow students to pose questions. Alternatively, ask questions, as your explanation unfolds, to monitor understanding.

# Mathematics

For the plots with outliers, discuss with the class why the outliers may have occurred. Could they be the result of measurement errors? Could they be real, but unusual, values? For example, someone in the class may be able to hold his or her breath for an extremely long time. Discuss how the outliers affect the measures of center. You can have students ignore the outliers and recompute the mean. Is the new mean a better representation of what is typical?

Consider these questions:

• Why do you think the typical value for your set of data is __ ?
• Is the mean a good descriptor of what is typical for your set of data?
• If it was difficult to draw a conclusion about the data, why was that?
• If your conclusion was unclear, what could help to make it clearer? (Answer: more data)

Be sure to comment on the “flat” line plots that don't really show the shape of the data. Ask students to speculate that maybe the line plot isn't always the best representation and there might be other ways to display the data.

# Mathematical Practices

Mathematical Practice 6: Attend to precision.

• As presenters explain how they decided what value is typical for their data, encourage them to use clear, precise language and correct mathematical terms.

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

• After students present their reasoning for how they chose a typical value, ask other students to critique their reasoning and to offer ideas or suggestions if they disagree.

Mathematical Practice 7: Look for and make use of structure.

• Ask students how they used the structure and patterns in their line plots to decide what value is typical.

Mathematical Practice 5: Use appropriate tools strategically

• Have students discuss how the Line Plot with Stats interactive made their work easier or helped them to determine a typical value for their data set.

# Ways of Thinking: Make Connections

Take notes as your classmates present their line plots, measures, and conclusions about the characteristics of a typical sixth grade student.

## Hint:

• Is the mean a good descriptor of what is typical for your set of data?
• Why do you think the typical value for your set of data is what you have concluded it is?
• Did you find it to be easy or difficult to draw a conclusion about your data? Why?
• If your conclusion is unclear based on the data, what could you do to help make it clearer?

# Lesson Guide

Students will write a summary of what they learned about analyzing data.

# A Possible Summary

Making a line plot of a data set and finding the measures of center and spread can help you decide on a typical value. If the data are spread out over a very large range, it may not be possible to name one typical value. However, if the data have a fairly small range, then one of the measures of center might be typical. If the data are clustered close to the mean, then the mean may be typical. If there are some outliers or if the values are spread from the mean, then the median might be a typical value. If one or two data values occur much more than the others, then the mode might be typical.

When analyzing a set of data, the graph and all of the measures are necessary to draw a valid conclusion. If the measures are the same or close, and the data are clustered around that point, the conclusion is probably valid. If the data are spread out and the MAD is large, it may be necessary to have more data to draw a valid conclusion.

# Summary of the Math: Tools for Analyzing Data

Write a summary of how you can use mathematical tools to analyze data and describe what the typical characteristics of a sixth grade student are.

## Hint:

• Do you include the typical value for your data and how you found it?
• Did you provide suggestions about how to decide what a typical value in a data set is?

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out what students think a typical sixth grade student is.

# Reflection

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

I think the typical characteristics of a sixth grade student are…