- Subject:
- Statistics and Probability
- Material Type:
- Lesson Plan
- Level:
- Middle School
- Grade:
- 7
- Provider:
- Pearson
- Tags:

- License:
- Creative Commons Attribution Non-Commercial
- Language:
- English
- Media Formats:
- Interactive, Text/HTML

# Education Standards

# 20 Seconds Timer

# Comparing Sets of Data

## Overview

Students estimate the length of 20 seconds by starting an unseen timer and stopping it when they think 20 seconds has elapsed. They are shown the results and repeat the process two more times. The first and third times are recorded and compiled, producing two data sets to be compared. Students analyze the data to conclude whether or not their ability to estimate 20 seconds improves with practice.

# Key Concepts

- Measures of center and spread
- Line plots, box plots, and histograms
- Mean absolute deviation (MAD)

# Goals and Learning Objectives

- Apply knowledge of statistics to compare sets of data.
- Use measures of center and spread to analyze data.
- Decide which graph is appropriate for a given situation.

# How Long Is 10 Seconds?

# Lesson Guide

Give students a few minutes to explore the 10 Seconds Timer interactive. Conduct a classroom poll about how many attempts are correct for their five attempts at estimating 10 seconds; create a line plot for the number of correct estimations.

Lead a brief class discussion and show the line plot. Perhaps have a prize for the “winner."

# Mathematics

Discuss the Opening activity and show the line plot. Did students feel that they improved at estimating 10 seconds?

As students play, they should see that they are getting better at estimating how long 10 seconds is. The idea that measurement estimation will improve with practice is a key part of the lesson.

## Opening

# How Long Is 10 Seconds?

The 10 Seconds Timer interactive will give you 5 trials for estimating 10 seconds. On your screen are Start and Stop buttons for accurately predicting 10 seconds. When you have completed all 5 trials, all your time estimations will be shown. You will share these results with the class.

- Use the Start and Stop buttons to estimate 10 seconds.
- Repeat 4 more times until you have 5 estimates.

INTERACTIVE: 10 Seconds Timer

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will compare data sets using a variety of tools.

SWD: Using approximation and estimation can be challenging for some students with disabilities. Review and reinforce this skill with students prior to having them start the tasks in which they will be expected to approximate/estimate.

ELL: When working with ELLs, it is important to link new concepts to students' prior knowledge and experiences. This allows students to make more meaningful connections with the new mathematical concepts being learned.

## Opening

Compare data sets using a variety of tools.

# How Long Is 20 Seconds?

# Lesson Guide

Students will work individually.

Each student will receive a time estimation for their first, second, and third attempts for estimating 20 seconds. Conduct a classroom poll where you collect your students' time estimations for their first, second, and third attempts for estimating 20 seconds. Collect the data for all three attempts.

## Work Time

# How Long Is 20 Seconds?

Do you think your ability to estimate time will improve with practice? To answer this question, you first need to collect some data.

- Using the 20 Seconds Timer interactive, use the Start and Stop buttons to estimate when you think 20 seconds have passed.
- The 20 Seconds Timer interactive will record and display your time estimate.
- Repeat the process two more times.

INTERACTIVE: 20 Seconds Timer

# Analyze Data

# Lesson Guide

Students will work individually.

# Mathematics

Students need to decide how to analyze the data. They will need to think about the different types of graphs and what they are most useful for. Is there a wide range in the data? Would a line plot or a histogram be more appropriate? What does a box plot show?

The task is left open for students to decide what to do. Look for students who use different graphs and different methods to analyze the data, and have them share in Ways of Thinking. It is important that a variety of methods are shown not only to reinforce that different approaches work, but also to remind students of the statistical tools at their disposal (e.g., the line plot will show the mean, median, and mode, and the box plot can be used to show measures of spread). Students should also have a way to compare the graphs visually; although the range may be different, both graphs should use the same scale, covering the whole range between both data sets.

Look for ways that students deal with the quantities. Do they round them to whole numbers, or to tenths, or leave them in hundredths? Some students may leave the numbers as is and use a histogram with bins for each second. If students do round the numbers (most probably will), what is the reason? (Rounding won't affect the conclusions and whole numbers will be much easier to work with.)

SWD: Some students with disabilities can get frustrated when working with decimals. If so, provide them access to a calculator or encourage them to round their times to the nearest second so they can focus on learning and practicing the more complex, conceptual elements of this task.

ELL: As in other lessons, when academic language is reviewed (or introduced, for that matter), show it in writing and leave it in a place that is visible so that all students can refer to it. Make sure students copy academic language into their notebook.

# Interventions

**Student has trouble getting started.**

- What are some of the tools you can use to analyze data?
- Did you check the Hints?
- Have you tried using any of the graphing tools?
- What graph will show the data best?
- Have you looked at the range?
- What are the measures of center?

# Possible Answers

- Most students will probably find tables and graphs useful.
- Answers will vary based on actual data sets.
- A simple line graph or bar graph which plots time (or deviation from 20 seconds) on the y-axis and trial number on the
*x*-axis would be useful for analysis—any graph that clearly separates the trials and shows their time/accuracy. - Answers will vary based on the tools and graph students choose to use.

## Work Time

# Analyze Data

Analyze the data to make a conclusion about whether your ability to estimate 20 seconds improves with practice. Think about and answer the following questions.

- What tools can you use to analyze the data?
- What evidence can you use to justify your conclusion?
- What type of graph would be the most useful for analyzing the data?
- How can you compare the data sets?
- What graph will show the data best?
- Have you looked at the range?

# Prepare a Presentation

# Interventions

Student has a solution but does not provide an explanation.

- Why did you approach the problem the way you did?
- Explain your strategy for solving the problem.
- How do you know that this is the typical score?

# Challenge Problem

## Answers

- Answers will vary, and are dependent on the actual data sets.

## Work Time

# Prepare a Presentation

- Prepare a presentation where you analyze your data sets and their results and conclusions.

# Challenge Problem

- Find the mean absolute deviation for the data. What does this value tell you about the data sets?

# Make Connections

# Mathematics

Have students share their conclusions and evidence to justify their conclusions. Remind students that everyone used the same sets of data.

Some questions for discussion:

- Did you round the numbers to whole numbers? Why or why not?
- What was the typical time for estimating 20 seconds for the first attempt? For the third attempt?
- What was the range of data each time? Was it wider or narrower after practice?
- Where is the data clustered in your graphs? Where is the middle 50% of the data?
- Why did you use the graph you did?
- Why did you use the measures you did?
- Do seventh grade students improve their estimation of 20 seconds with practice?
- Can we generalize this conclusion to all seventh grade students? Why or why not?
- Even though everyone used the same data sets, why didn't we arrive at the same conclusions?

SWD: At this point in the course, students should be comfortable in presenting. Extend student responses by using prompts and targeted questions that are appropriate to the students' ability.

## Performance Task

# Ways of Thinking: Make Connections

- Take notes about the methods your classmates used to analyze the data sets and their results and conclusions.

As your classmates present, ask questions such as:

- Did you round the numbers to whole numbers? Why or why not?
- What is the range of data for each trial? Does the range get wider or narrower from the first trial to the third trial?
- Where is the data clustered in your graphs? Where does the middle 50% of the data lie?
- Why did you use that graph to represent the data?
- Why did you use those tools to analyze the data?
- Based on your results, can you conclude that your estimation skills improve with practice?
- Can you generalize this conclusion to students in your grade? Why or why not?
- Suppose that you analyzed the data from the second trial. How do you think this analysis would compare to your analyses of the first and third data sets?

# Comparing Data Sets

# A Possible Summary

You can calculate measures of center to compare data sets, or graph the data to see how the shapes compare. We already have tools to use to compare data sets, like line plots, box plots, and histograms.

## Formative Assessment

# Summary of the Math: Comparing Data Sets

Write a summary about comparing data sets.

Check your summary:

- Do you explain how to compare data sets?
- Do you describe the methods you can use to analyze data and make comparisons?

# Reflect on Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out what reasoning students have for comparing the first and third trials to conclude whether practice improves the ability to estimate time.

ELL: This 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.

## Work Time

# Reflection

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

**To draw a conclusion about whether practice improves the ability to estimate time, I think comparing the third trial to the first trial is the best comparison to make because…**