# Reviewing Statistical Questions

## Overview

Students write statistical questions that can be used to find information about a typical sixth grade student. Then, the class works together to informally plan how to find the typical arm span of a student in their class.

# Key Concepts

- Statistical thinking, in large part, must deal with variability; statistical problem solving and decision making depend on understanding, explaining, and quantifying the variability in the data.
- “How tall is a sixth grader?” is a statistical question because all sixth graders are not the same height—there is variability.

# Goals and Learning Objectives

- Understand what a statistical question is.
- Realize there is variability in data and understand why.
- Describe informally the range, median, and mode of a set of data.

# Statistical Questions

# Lesson Guide

Have students read the questions. Have students talk to their partner and then share their responses with the class.

# Mathematics

During the discussion, ask the following questions:

- Mathematically, how can we determine the characteristics of a typical sixth grade student? (Answer: This can be determined by collecting data.)
- Is a typical sixth grader one person, or a mix of several people? (Answer: There is probably not one person with all of the “typical” characteristics.)
- How do we determine what is typical? (Answer: Accept any responses at this point.)

Discuss what a statistical question is, and that in order to determine what characteristics a typical sixth grader has, statistical questions must be answered.

- A statistical question can be answered by collecting data, and the data must have variability.
- “How tall am I?” is not a statistical question. Likewise, if every sixth grader were the same height, “How tall is a sixth grade student?” would not be a statistical question.
- For the purpose of our investigation, the statistical questions we will look at will be answered with numerical data, not categorical data. (“How tall is a typical sixth grade student?” works, but “What is the typical hair color of a sixth grade student?” doesn't.)

## Opening

# Statistical Questions

Think about these questions:

- When you think of a typical sixth grade student, who comes to mind?
- What characteristics does he or she have?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will create a list of questions about the characteristics of a typical sixth grade student.

SWD: Assist students with the meaning and use of terms such as *statistical question* and *variability*. Provide examples.

ELL: For this task, encourage students to explain their ideas to one another. Math language must be used. Encourage the use of English without discouraging students from using their language of origin(s). Make sure that students' questions can be answered using numerical data.

## Opening

Create a list of statistical questions about the characteristics of a typical sixth grade student.

# A Typical Sixth Grade Student

# Lesson Guide

Have students work individually. Give them 5 to 10 minutes to make their lists.

SWD: Make sure all students understand the task. Have students restate the task in their own words so you can assess their understanding of it.

ELL: For ELLs, the task of writing a statistical question can be challenging. Providing various examples and/or an explanation of a statistical question will benefit ELLs.

# Mathematics

Look for students who are writing categorical questions, and make sure they are thinking about numerical data. Questions could involve measurement (height, arm span, how long students can hold their breath), but could also involve frequency (number of pets per family, number of letters in first names).

Students should also list the units that will be used for the data. Height could be measured in inches or centimeters. Holding breath could be measured in seconds. Students should note the difference between questions that involve units and those that involve frequency.

# Interventions

**Student has difficulty getting started.**

- What is a characteristic you have that can be described with numbers or measurements?
- Does this characteristic vary for different students?
- What question can you ask to get information about this characteristic from your classmates?

**Student is listing questions with non-numerical answers**.

- Would an answer to your question be a number or measurement, or would it be a word?
- You will need to order the data you collect and do calculations with them. Will you be able to do this for the answers to your question?
- Is there a way to ask a question about your topic that does have a number for an answer? For example, instead of asking students what kind of music they like, you could ask them to rate how much they like country music on a scale from 1 to 10.

**Students do not specify units of measurement.**

- Is the answer to your question a measurement?
- What if some students answer in inches and others in centimeters? How will you decide what is typical?
- How can you be sure everyone gives their answers in the same unit?

# Answers

- Answers will vary.

## Work Time

# A Typical Sixth Grade Student

What characteristics does a typical sixth grade student have?

- Make a list of statistical questions that can be answered with numerical data to help you answer the question above.

## Hint:

What numbers could you use to describe the habits of sixth graders—such as the number of text messages sent per day, number of hours spent watching television per week, number of books read per month, and so on?

# Arm Span

# Preparing for Ways of Thinking

Look for the following to be shared during Ways of Thinking:

- Interesting statistical questions about a typical sixth grade student that can be answered by collecting numerical data
- Questions about a typical sixth grade student that have non-numerical answers
- Questions that are unclear or that could not be used to identify a characteristic of a typical sixth grade student

# Interventions

**Student has difficulty getting started.**

- How could you collect data to answer this question?
- How would you do the measurements?
- What units should you use to measure?

**Student doesn't understand what "typical" means.**

- Is your arm span the average arm span?
- How many arm spans do you think you need to measure to get a typical arm span?
- What age group should you collect data from?
- How could you use the data you collected to determine a single typical arm span?

# Answers

- Answers will vary. Possible answer: You can measure the length of an arm span in centimeters.

# Challenge Problem

## Answers

- Answers will vary. Students will be able to find a range that reflects the typical sixth grader for some of the questions they wrote in Task 3.

## Work Time

# Arm Span

What is the typical arm span of a sixth grade student?

- Think about how this question could be answered.

# Challenge Problem

Choose several questions you wrote in Task 3.

- What is the set (or range) of possible answers for each question?
- What number in each range do you think reflects a typical sixth grader?

## Hint:

- The arm span of a sixth grade student varies, but the answer to this question is a single number.
- How could you collect data to answer this question?
- How would you do the measurements?
- How could you use the data you collected to determine a single, typical arm span?

# Make Connections

# Lesson Guide

Have students share the questions you selected during Work Time and ask other students to critique the questions by asking:

- Will the question help us find out something about a typical sixth grade student?
- Are the answers numerical?
- Will the answers vary?
- Is the wording of the question clear?

ELL: When calling on students, be sure to call on ELLs and to encourage them to actively participate. Understand that their pace might be slower or they might be shy or more reluctant to volunteer due to their weaker command of the language.

# Mathematical Practices

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

- Require students to explain, using precise language, what they did to meet the criteria of a statistical question.

# Mathematics

Discuss how questions with non-numerical answers are not appropriate for our purposes, because you need data that you can order and compute. Work as a class to rewrite poorly worded questions.

Ask the following:

- For each question, what measure would you use to decide what is typical?
- Can you use more than one measure?

Choose some of the questions students listed that were either common to many lists or just interesting statistically. Add them to the list of topics for Lesson 2.

## Performance Task

# Ways of Thinking: Make Connections

During the class discussion, take notes about interesting statistical questions that other students present.

## Hint:

As your classmates present, ask questions such as:

- Will the question help us find out something about a typical sixth grade student?
- Are the answers numerical?
- Will the answers vary?
- Is the wording of the question clear? Could respondents be confused or interpret it in different ways?

# Asking Statistical Questions

# A Possible Summary

To determine what is typical for a sixth grade student, a statistical question must be answered.

A statistical question involves variability in data and is answered by collecting numerical data.

ELL: When writing this summary, provide ELLs access to a dictionary and give them time to discuss their summary with a partner before writing, to help them organize their thoughts. Allow ELLs who share the same language of origin to discuss in their preferred language.

## Formative Assessment

# Summary of the Math: Asking Statistical Questions

- Summarize what you learned about asking statistical questions to determine the characteristics of a typical sixth grade student.

## Hint:

Check your summary.

- Did you say that we can ask statistical questions to investigate what is typical?
- Did you explain what a statistical question is, including the notion of variability of data?
- Did you give examples of questions that can be answered by collecting and analyzing numerical data?
- Did you discuss different ways of describing the typical arm span of the students in your class?

# Reflect on Your Work

# Lesson Guide

From *A Wrinkle in Time *by Madeleine L'Engle, read an excerpt in Chapter 6, “The Happy Medium.” Start partway through the chapter with the sentence “Below them the town was laid out in harsh angular patterns.” To set the scene, explain that the children have arrived on a planet. The excerpt describes what they see upon their arrival.

Have each student write a quick reflection before the end of the class. Review the reflections to learn what students think living in a world with no variability would be like.

## Work Time

# Reflection

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

**Living in a world with no variability, where everyone was the same, would be like …**