## Every Breath You Take

In the following open-ended exploration, students estimate, experiment, and display real-life data. Students use the number of breaths taken during a specified time period as the context for this exploration.

Begin the investigation by encouraging students to give an example of something that they do once a day such as eat breakfast, listen to school announcements, or play with a friend. Ask them to name something they do about ten times each day, which might include saying hello in the hall, changing the television channel, or writing their names on papers. Finally ask students to name something that they do at least one hundred times a day. A narrow range of responses often includes blinking and breathing. If not, take a deep breath and ask, "What about breathing?"

Ask students, "Do you take more than 100 breaths in a day? More than 100 or 1000 breaths in an hour?"

After a brief discussion, distribute the Every Breath You Take activity sheet and ask each student to estimate the number of breaths a person takes each hour. Have each student write down an estimate.

Every Breath You Take Activity Sheet

Students should work with a partner to answer questions 3 and 4. Have each pair share its response with another pair. As a class, discuss similarities and differences in data interpretation that surfaced when sharing observations.

To explore the data further, have each student use his or her estimate from question 1 and line up in order. Each students will determine his or her place in line without assistance.

The student with the lowest estimate should start by reading the estimate to the class. Determine the range by identifying the lowest and highest estimates, noting how much easier they are to find when the data points are so displayed. Ask students whether it is more obvious that some estimates appear to "bunch" or "cluster" around certain values or whether the estimates appear to be evenly distributed when students are lined up than when the numbers are listed on paper.

Each student should construct a graph of the class data using graph paper. Discuss an appropriate scale and label for the graph's vertical axis. Students will also need to determine whether to plot a scatter graph or a bar graph, depending on prior experience. Note that student estimates may vary widely, so you made need to use ranges of data for each of the bars in a bar graph. For example, you may have 0-60, 61-120, 121-180, and so on for the number of breaths estimated.

Explore the data in terms of, for example, an approximate mean, the median, and the outliers.

Discuss what makes a good estimate and more than one estimate may be good.

Encourage students to explore how they might use their estimates of the number of breaths taken in one hour to estimate the number taken in a day. Students who have been exposed to multiplication might suggest multiplying their estimates by 24, whereas other children might use a repeated-addition process on their calculators.

Discuss why this estimate might be considerably different from the actual number of breaths taken in a day.

To judge the reasonableness of student's estimates, ask, "How could you find out how many breaths a person actually takes in a day?" Allow students to brainstorm solving the problem. They might work with a partner and count the number of breaths in a given period of time—for instance, one minute—or determine the time it takes to breathe a given number of times, say, one hundred.

Have students implement their strategies and compare their answer with one another. On the activity sheet, students should also describe and defend the methods they chose to use.

Conclude the investigation by encouraging students to look for other things that are done many times every day in their world. By counting the number of times that we do these things, a sense of large numbers and a familiarity with them may be established.

- Stopwatches or clocks with secondhand
- Scissors
- Calculators
- Every Breath You Take Activity Sheet
- Graph paper
- Adding machine tape (optional)

**Extensions**

- From the information identified above, determine a class average of the number of breaths taken per hour.
- Ask a member of the class to use a meter stick and measure this number, in millimeters, and mark it on a roll of adding-machine tape. The tape should be folded over on itself twenty-three times, resulting in twenty-four sections each having the same length as the original length. Since each breath is represented by one millimeter, when unfolded, this model offers a physical representation of the number of breaths that a person takes in one day. Students could also investigate the number of breaths taken after performing some type of aerobic activity, such as running or doing jumping jacks, and then discuss how this activity might affect the number of breaths taken in an hour or a day.

### Making Your First Million

### Learning Objectives

- Make estimates about real-life data.
- Collect and analyze data, namely the number of breaths taken during certain time periods.
- Represent data through physical and graphical means.
- Develop number sense.

### NCTM Standards and Expectations

- Understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals.

- Develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results.

- Design investigations to address a question and consider how data-collection methods affect the nature of the data set.

- Collect data using observations, surveys, and experiments.

- Represent data using tables and graphs such as line plots, bar graphs, and line graphs.