# Lesson 16

Heart Rates

## 16.1: Find Your Heart Rate (5 minutes)

### Warm-up

The purpose of this activity is to get familiar with the process of finding heart rate and collecting some initial data for the experiment in the next activity.

### Launch

Tell students they will find their pulse as part of an experiment to determine if counting while exercising affects heart rate. Demonstrate the process of finding your pulse for the class. Count the number of beats you feel in 10 seconds. Multiply the value by 6 to find your heart rate in beats per minute (bpm). Distribute stopwatches or instruct students to open a stopwatch app on an internet-enabled device by searching for "stopwatch" in an internet browser.

### Student Facing

Find your heart rate. One way is to place your index and middle fingers on your neck just below the jaw to feel your heartbeat. Count the number of heartbeats you feel in 10 seconds. Multiply the value by 6 to find your heart rate in beats per minute (bpm).

1. Write the value for your heart rate in bpm. This will be used as your resting heart rate.
2. Take some deep breaths, close your eyes, and repeat the process. Write the value for your heart rate in bpm. Has your heart rate changed?

### Activity Synthesis

The goal of this activity is for students to measure their resting heart rate and to begin thinking about how it changed after some deep breaths with their eyes closed.

Here are some questions for discussion:

• “How could we determine a typical resting heart rate for the class?” (We could find the mean.)
• “Raise your hand if your heart rate decreased after the deep breaths with your eyes closed.”
• “What was the treatment in this experiment?” (The treatment was taking deep breaths and having your eyes closed.)
• “What are some other reasons that your heart rate may have changed?” (I was nervous that I would take my pulse wrong.)

## 16.2: The Counting Experiment (10 minutes)

### Activity

The mathematical goal of this activity is for students to collect and summarize data from an experiment. Students will measure their heart rate after doing a light exercise and use the difference from their resting heart rate as data to analyze in a later activity.

### Launch

Arrange students in the 2 groups created at random in the activity Get Ready to Experiment. Tell students, “One group will count out loud while exercising and the other group will remain silent while exercising. Everyone will take their pulse silently after the exercising is finished.” For the exercise, consider asking the students to do 30 jumping-jacks or running in place for 1 minute. The intensity should be such that it raises student heart rates quickly without making them overly sweaty. Adjust the time or level of exercise to meet the needs of your class.

Action and Expression: Internalize Executive Functions. Provide students with a template so they can fill in the data for their heart rate and calculations. For example, include a line for each item students should record (resting heart rate, group, exercise heart rate for 10 seconds, multiply by 6 for beats per minute, exercise heart rate minus resting heart rate).
Supports accessibility for: Language; Organization

### Student Facing

Does counting while exercising affect your heart rate? Let’s perform an experiment to find out.

1. Your teacher will help divide you into two groups and lead the exercise. If you are selected to be in the group that will count during the exercise, count out loud together with your group while you do the exercise. If you are selected to be in the group that will remain silent during the exercise, remain quiet.
2. Immediately following the exercise, measure your heart rate again. Count the number of beats in ten seconds and multiply the result by 6 to get your heart rate in beats per minute. Record this result. Find the difference in heart rate by subtracting your heart rate after exercise from the resting heart rate you found during the warm-up.

### Activity Synthesis

Collect the differences in heart rate from each student and keep them separated by groups. Display the results for all to see. These results will be used later in the lesson.

The goal of this discussion is for students to think about how to analyze the data from the two groups. Here are some questions for discussion:

• “How could you graphically represent the data?” (You could make a histogram showing the distribution of the data from each group.)
• “What are some ways you could analyze the data?” (You could calculate and compare the mean and standard deviation of each data set, or compare the shapes of the two histograms.)
• “How would you find a randomization distribution for this data?” (You could record all of the values in a spreadsheet then choose half of them at random to be the counting group and the other half to be the silent group. Then you would find the means of those two groups and subtract them. You would repeat this many times.)

## 16.3: The Flapping Experiment (5 minutes)

### Activity

The mathematical goal of this activity is for students to collect and summarize data from an experiment. Students measure their heart rates under different conditions to collect data for analysis in a later activity.

### Launch

Rearrange students into 2 new randomly selected groups. Tell students they will do a similar experiment, but this time one group will do 30 jumping-jacks while the other group will just wave their arms 30 times and not jump or move their legs. Ask students if they think this treatment will affect the results of the experiment.

Action and Expression: Internalize Executive Functions. Provide students with a template so they can fill in the data for their heart rate and calculations. For example, include a line for each item students should record (resting heart rate, group, exercise heart rate for 10 seconds, multiply by 6 for beats per minute, exercise heart rate minus resting heart rate).
Supports accessibility for: Language; Organization

### Student Facing

Does the way you do an exercise affect your heart rate? Let’s perform an experiment to find out.

1. Your teacher will help divide you into two groups and lead the exercise. If you are selected to be in the group that will do jumping-jacks during the exercise, do the standard exercise using your arms and legs. If you are selected to be in the group that will only flap their arms, move your arms as in a jumping-jack, but do not jump or move your legs.
2. Immediately following the exercise, measure your heart rate again. Count the number of beats in ten seconds and multiply the result by 6 to get your heart rate in beats per minute. Record this result. Find the difference in heart rate by subtracting your heart rate after exercise from the resting heart rate you found during the warm-up.

### Activity Synthesis

Collect the differences in heart rate from each student and keep them separated by groups. Display the results for all to see. These results will be used in the next activity.

## 16.4: Analyzing the Heart Rates (15 minutes)

### Activity

The mathematical purpose of this activity is for students to perform a randomization experiment and analyze the results. Students use the difference in heart rates for the two experiments they did to determine whether the treatment is the likely cause of the difference between the two groups. In most cases, students should find that the difference in mean heart rates from The Counting Experiment should not be significant, but that the difference in mean heart rates for The Flapping Experiment should be significant.

### Launch

Arrange students in groups of 2. Assign half of the groups to work with the data from The Counting Experiment and the other half of the groups to work with the data from The Flapping Experiment.

Provide access to devices that can run GeoGebra or other statistical technology.

Allow students to work the first three questions and then pause to tell students “You are going to use a randomization distribution to analyze the data.” Then ask, “How did we do this in the previous lesson?” (We divided 10 students into two groups of 5 and then rearranged them into two new groups of 5 by selecting names at random from a bag.) Tell students to read question 4 and ask if they have questions. After question 4, collect class data from the simulations and display a histogram of the results for all to see.

### Student Facing

1. Using the data for the increase in heart rates, find the mean for each group.
2. Find the difference between the means for the two groups by subtracting the mean for the first group from the mean for the second group. What does a positive value mean? What does a negative value mean?
3. Does the difference in means tell us that there is an effect on heart rate? If so, explain the connection. If not, what else could account for the difference in means?
4. Since the groups were divided using a random process, the only reasons there should be a difference in means is if the treatment had an affect or if it was due to the random assignment to groups. To rule out random assignment as the explanation for the observed difference, assume for a bit that the different treatments had no effect on the increase in heart rate. In this case, we could shuffle around some of the results from the two groups and still have a similar outcome. Let’s examine what happens when we do this randomization experiment.
1. Cut a sheet of paper into pieces so there are as many pieces as students in the class. Write the data from the experiment on the pieces so that each piece has one value on it.
2. Shuffle the pieces and select half of them to represent the first group. Give the other pieces to your partner to represent the second group.
3. Find the mean for each group and record the difference in the means by subtracting the mean for the first group from the mean for the second group.
4. Repeat this process 4 more times so that you have recorded 5 differences in means from the randomization process.
5. Share your data with the class and examine the randomization distribution in the histogram of the difference in means. This distribution represents some of the differences that are possible due only to random assignment to experimental groups. Estimate the center of the distribution. What does this value mean in terms of heart rates for the two groups when reassigned at random?
6. Does the difference for your class from the actual experiment represent a typical value for this distribution, or is it unusual?

### Anticipated Misconceptions

Students may be confused about which values they are measuring and analyzing in the data. The original data is already a difference in heart rates between resting and exercise. Then, means of those differences are found. Then, differences between the means of the groups are analyzed. It may help students to keep track of the different levels of information that are used for this analysis if examples of the information are displayed in separate places so that the values students are currently working with can be pointed out.

### Activity Synthesis

The goal of this discussion is for students to share their analysis of the results of the randomization experiment. For groups that are too small to create a good histogram, display the example here for all to see, otherwise display the histogram from the class data.

Here are some questions for discussion:

• “Where does the difference between the mean heart rates of the 2 treatment groups fall in the randomization distribution?” (For The Counting Experiment, it falls right in the middle which means it is very likely that the difference occurred by chance. For The Flapping Experiment, it is far from the typical values which means it is very likely that the difference is not due to chance.)
• “What size difference between the mean heart rates of the 2 treatments would cause you to conclude that the difference did not occur by chance? Explain your reasoning.” (For The Counting Experiment, I think a difference greater than 8 or less than -8 would have allowed me to say it did not occur by chance. This is because most of the values resulting in the randomization distribution were greater than -8 or less than 8. For The Flapping Experiment, I think a difference greater than 10 or less than -10 would have allowed me to say it did not occur by chance.)
• “Do you think counting while exercising affects heart rate? Explain your reasoning.” (I do not think it affects heart rate because the small difference in means we found is likely to happen due to chance alone.)
• “Do you think doing only part of the exercise affects heart rate? Explain your reasoning.” (I do think it affects heart rate because the relatively large difference in means we found is not likely to happen due to chance alone.)
• “Were these experiments failures? Explain your reasoning.” (No, they are not failures. We found evidence that counting after exercise did not significantly affect heart rate. Just because we did not see a change does not mean that the experiment failed.)
Conversing: MLR8 Discussion Supports. Use this routine to help students analyze the results of the randomization experiment. Before the whole-class discussion, allow time for pairs to discuss their ideas for the question, “What size difference between the mean heart rates of the 2 treatments would cause you to conclude that the difference did not occur by chance? Explain your reasoning.” Invite Partner A to begin with this sentence frame: “The difference is _____, because . . .” or “I notice _____ which means . . . .” Invite the listener, Partner B, to respond by saying, “I agree/disagree because . . . .” This will help students justify their reasoning about whether or not the difference occurred by chance.
Design Principle(s): Support sense-making; Cultivate conversation

## Lesson Synthesis

### Lesson Synthesis

Here are some questions for discussion:

• “How do randomization distributions help us draw conclusions from our results?” (Randomization distributions allow you to see how likely it is that you would arrive at the difference of means due to chance. Knowing that it is likely or unlikely allows you to be more or less confident in your results.)
• “Could you use the mean and the standard deviation to compare two groups? Explain your reasoning.” (You could use the standard deviation to compare two groups but you would have to take into account the shape of the data. In addition, just because the means plus or minus one standard deviation overlap or do not overlap that still does not tell us whether the difference occurred by chance.)
• “What questions do you have about randomization distributions?” (Is there a way to get randomization distributions with technology? Who came up with the idea of using simulations in this way?)
• “What have you learned about statistics in this unit?” (I have really learned that data can vary due to chance or factors that you may not have thought of. An experiment helps us to determine if the factor being studied is the cause for what is being observed in the data.)

## Student Lesson Summary

### Student Facing

A randomization distribution is used to determine if the difference between the means of different treatment groups could be due to chance.

A company farms a type of fish called tilapia. They conduct an experiment to determine if a freshwater environment, treatment A, or a slightly salty environment, treatment B, causes the fish to grow at a faster rate. The table displays the weight gain, in grams, of a random sample of 11 tilapia from treatment A and a random sample of 11 tilapia from treatment B.

treatment A weight gain (grams) treatment B weight gain (grams)
120 120
125 130
115 135
135 125
110 135
125 125
130 130
120 130
125 125
120 135
125 130

The mean weight gain of the sample from treatment A is approximately 122.7 grams and the mean weight gain of the sample from treatment B is approximately 129.1 grams, a difference of approximately -6.4 grams.

The results for 100 trials of simulating redistributing the data are summarized in the histogram.

Only 2 out of 100 trials for the simulation show a weight gain difference between the groups of at least 6 grams. Since the difference between the mean weight gains from the treatment groups is -6.4 grams, we can say we have good evidence that the difference did not occur by chance. Therefore, there is evidence that the saltiness of the water in which the tilapia are grown does have an effect on the weight gain of the fish grown in that environment.