Heart Rate Monitoring

Task

Serena is starting a new workout routine and wants to keep track of her heart rate to make sure that she is exercising at the optimum level. First she did a warm-up, then she did her training exercises, then she did a cool-down.

• Before beginning her workout, Serena's resting heart rate was 60 beats per minute.
• She started her workout with a warm-up. While warming up, her heart rate increased at a constant rate of 8 beats per minute each minute.
• She warmed up for 10 minutes.
• After her warm up, her heart rate held steady throughout her training exercises, which lasted for 30 minutes.
• After her training exercises, she walked for 20 minutes as a cool-down and her heart rate decreased at a constant rate, finally returning to her rest rate of 60 beats per minute by the end of her cool-down.

1. Construct a graph of Serena's heart rate, $h$ in beats per minute (bpm), as a function of time, $t$ in minutes, where $t=0$ is when she started her warmup. Make sure to include the times before, during, and after her workout.
2. For about how many minutes total was her heart rate at 100 beats per minute or above?
3. Compare how quickly her heart rate changed during her warm-up versus how quickly her heart rate changed during her cool-down.

IM Commentary

In this task, students are asked to draw a graph that represents heart rate as a function of time from a verbal description of that function. Then they use the graph to draw conclusions about the context, for instance they have to understand that a heart rate greater than 100 beats per minute occurs when the graph is above the line $y=100$. Students who work together on this task will benefit from discussing and comparing their graphs.

A good introduction into this activity would even be a self-experiment, where students take their own pulse and then do jumping jacks and take their pulse again. Students might notice that it is not very realistic that the heart rate increases (or decreases) at a steady rate, but this simplification makes it possible to ask more interesting questions with 8th grade tools. Also, this is the type of simplification that we often make in modeling contexts.

The units for the rates of change in the problem are a little tricky. The units for the function are "beats per minute" which is often abbreviated "bpm." Since the function itself is a rate, the rate of change of the function has units "beats per minute per minute" or "bpm per minute." This is analogous to the relationship between velocity and acceleration: velocity is a rate, and acceleration is the rate of change of the velocity. Suppose we choose to measure velocity in "meters per second" (which we often write "m/s"); then acceleration is typically measured in "meters per second per second" which often gets abbreviated to "m/s²." If we followed this convention for the current task, then we might write "beats per minute" as "b/m" in which case the units for the slopes of the line segments in the graph could be abbreviated "b/m²."

Solution

1. In constructing our graph, we will have $h$ (her heart rate) on the vertical axis, in beats per minute, and $t$ (time) on the horizontal axis, in minutes since she started her workout.

   Our first observation is that her heart rate was at a resting rate of 60 bpm when she started her workout, so for at least a few moments before $t=0$, we know that $h=60$, and we can indicate this by putting a point at $(0,60)$ on our graph and making a horizontal line to the left of this point.

   Next, during her 10 minute warm-up, her heart rate increased at a constant rate of 8 bpm per minute. This translates into a total increase of 80 bpm for this 10 minute interval. Her heart rate increased from 60 to (60+80)=140 bpm between $t=0$ and $t=10$, and we can put another point at (10,140) and connect (0,60) to this point with a line segment.

   She trained hard for 30 minutes with a constant heart rate, meaning her heart rate was a steady 140 bpm between 10 and 40 minutes. So, $h=140$ for $10\leq t \leq 40$ and we indicate this with a horizontal line segment.

   For the 20 minutes of her cool-down, her heart rate decreased at a constant rate until it reached 60 bpm. Since her heart rate was at 140 bpm up until $t=40$ minutes and dropped to 60 bpm after another 20 minutes, we can place a point at (60,60) and connect it to (40,140) with a line segment.

   Finally, her heart-rate is back at resting rate after the 60 minute workout, so $h=60$ for at least a few moments after her workout, so we can add a horizontal line to the left of the point (60,60). Here is the graph:

   
2. Looking at our graph, we see that she first went above 100 bpm at about 5 minutes in, and then dropped back under 100 bpm at about 50 minutes in. So her heart rate was above 100 beats per minute for about 50-5=45 minutes total. If we wanted to, we could find the exact equations for each line and find the exact time when her heart rate went above 100 bpm, but the purpose of this task is to make reasonable judgements from looking at the graph, so this isn't necessary (and it would turn out that we get the same exact values anyway).

3. Her heart rate changed during her warmup by 8 bpm for each minute, as given to us in the observations. During her cool down, her heart rate dropped a total of 80 bpm in 20 minutes, meaning that for every one minute, her heart rate decreased by $80 \div 20=4$ bpm per minute. The slope of this line segment is -4 to reflect this.

   Although one rate is positive and one is negative, we can still compare the absolute values of these rates, because this will tell us how quickly her heart rate was slowing down and speeding up. Since 8 is greater than 4, her heart rate changed more quickly when she was warming up than it did while cooling down, although in one case it was increasing and in the other it was decreasing.