# Lesson 16

Analyzing Data

## 16.1: Experimental Conditions (5 minutes)

### Warm-up

The mathematical purpose of this activity is for students to write a statistical question for dropping and catching a ruler under different conditions.

### Launch

Arrange students in groups of 2.

Demonstrate how to drop the ruler and how to measure the distance dropped. Show this video if necessary.

Explain that the ruler is being held by one person at the 12 inch mark and is caught by another person just below the 7 inch mark. The distance the ruler fell is about 6 inches. For groups struggling to think of conditions that might be interesting, here are some examples to help them get started:

• Standing and sitting
• Listening to music and quiet
• Listening to a favorite song and one that is less interesting
• Releasing quickly after they are ready and waiting at least 3 seconds after they are ready before dropping

### Student Facing

To test reaction time, the person running the test will hold a ruler at the 12 inch mark. The person whose reaction time is being tested will hold their thumb and forefinger open on either side of the flat side of the ruler at the 0 inch mark on the other side of the ruler. The person running the test will drop the ruler and the other person should close their fingers as soon as they notice the ruler moving to catch it. The distance that the ruler fell should be used as the data for this experiment.

With your partner, write a statistical question that can be answered by comparing data from two different conditions for the test.

### Activity Synthesis

The goal of this discussion is to make sure that everyone has a statistical question about reaction time that will require collecting data from two different conditions to test.

Check student questions and assist them in creating a question that meets the requirement. Here are some questions for discussion.

• “What are your two conditions?” (Standing on one foot and standing on both feet.)
• “How are you going to collect this data?” (I will hold and drop the ruler while my partner stands on one foot. Then I will repeat this while he is standing on two feet.)
• “How many trials do you think you should do under each condition?” (At least ten)

## 16.2: Dropping the Ruler (25 minutes)

### Activity

The mathematical purpose of this activity is for students to design an experiment to answer a statistical question, to collect data, to analyze data using statistics, and to communicate the answer to the statistical question using a display. It may be helpful to have multiple groups combine to allow students to experience the different conditions for many experiments. Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Notice groups that create displays that communicate their mathematical thinking clearly, contain an error that would be instructive to discuss, or organize the information in a way that is useful for all to see.

### Launch

Keep students in groups of 2. Provide each group with tools for creating a visual display. If students have access to statistical technology, suggest that it might be a helpful tool in this activity.

Explain to students that they will collect and analyze data using statistics to answer their statistical question from the warm-up.

Students will create a display showing the statistical questions, the data, a data display, and an answer to the statistical question with any supporting mathematical work.

Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem solving, chunk this task into more manageable parts. For example, show 1–2 features that the display must include instead of the entire list. Provide students with access to grid or graph paper to organize their supporting mathematical work for the visual display.
Supports accessibility for: Memory; Organization

### Student Facing

Earlier, you and your partner agreed on a statistical question that can be answered using data collected in 2 different ruler-dropping conditions. With your partner, run the experiment to collect at least 20 results under each condition.

Analyze your 2 data sets to compare the statistical question. Next, create a visual display that includes:

• the data you collected
• a data display
• the measure of center and variability you found that are appropriate for the data
• an answer to the statistical question with any supporting mathematical work

### Activity Synthesis

Select groups to share their visual displays. Encourage students to ask questions about the mathematical thinking or design approach that went into creating the display. For students who had an error in their display, ask “What error do you see in the display and how would you resolve it?” (Answers vary. Sample responses:

• I noticed that they used the mean instead of the median. I would resolve it by using the median since there was an outlier
• I noticed that the IQR was calculated incorrectly. I would have used technology to verify my statistics.)

Here are questions for discussion, if not already mentioned by students:

• “Once you collected your data, how did you answer your statistical question?” (I used two different dot plots to display my data and determined that the mean was the most appropriate measure of center because my dot plot was approximately symmetric. I then calculated the standard deviation using technology because it is appropriate to use with the mean. I then compared the mean and the standard deviation for the two sets of data and determined that standing on two feet gave my partner a slightly faster reaction time and showed less variability than standing on one foot.)
• “How did you choose which measure of center and which measure of variability to use?” (I used the distribution shape to determine the measure of center. My data was skewed so I used the median. I used the IQR as a measure of variability because it is based on the median. If I had used the mean, I would have used the standard deviation.)
• “Using the context of the two treatments, what did the measure of variability tell you?” (The measure of variability told me how spread apart the results were from the measure of center I chose. It let me know how consistent the reaction time was for each treatment.)
• “Imagine that you collected data for the same treatments from all the students in the class. How would this change how you displayed or analyzed your data?” (There would be a lot of data. I would probably have to use technology to find the statistics and to create the data display. I think that I might need to use a histogram to represent the distribution of the data.)
Writing, Speaking, Listening: MLR 1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their response to the answer to the statistical question that was picked earlier. Ask each student to meet with 2–3 other partners in a row for feedback. Display prompts for feedback that will help teams strengthen their ideas and clarify their language. For example,  “Can you explain how…?”, “You should expand on….”, etc. Students can borrow ideas and language from each partner to strengthen the final product. This will help speakers reflect on what they learned about data collection, data analysis, and answering a statistical question. Design Principle(s): Optimize output (for generalization)

## 16.3: Heights and Handedness (10 minutes)

### Activity

In this activity, students use a large data set to compare the size of students with different dominant hands. Students must select a variable that best represents the size of the students and compare the two groups. By selecting the relevant variable to analyze, students are engaging in aspects of mathematical modeling (MP4). Additionally, students must choose the appropriate tools (MP5) to analyze a large data set such as this.

### Launch

Arrange students in groups of 2. Distribute a copy of the blackline master to each group.

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts. Engage in a classroom discussion drawing attention to the relationship between height, foot length, and arm span of right- and left-handed students in the class. This will help students make connections between the large data set and specific students in the class.
Supports accessibility for: Conceptual processing; Memory

### Student Facing

Is there a connection between a student's dominant hand and their size? Use the table of information to compare the size of students with different dominant hands.

### Activity Synthesis

Select students to share their analysis of the data.

## Lesson Synthesis

### Lesson Synthesis

The purpose of this discussion is to help students reflect on what they learned about data collection, data analysis, and answering a statistical question. Here are some questions for discussion.

• “What did you find the most challenging about this lesson?” (It was really difficult to figure out what to do with the data that I collected because there were no directions. I had to look back to my statistical question and think about what tools I would need to use to answer it.)
• “What did you find interesting about this lesson?” (It was really interesting that I could actually use the statistics I learned about to answer a question. It seemed a lot like when we do experiments in science class.)
• “What mathematics do you need to know more about?” (I am still struggling with figuring out the shape of the data. Sometimes I think the data is roughly symmetric but it actually is skewed. I would like more practice about describing the shape of distributions.)