# The story of a flight

Alignments to Content Standards: F-IF.B.4

The graphs below show three different quantities recorded by the flight computer during a flight of a small airplane.

1. Make a guess what each graph might represent.

2. The quantities are altitude (in feet), vertical velocity (in feet/min) and propeller speed (in RPM, revolutions per minute). Which graph represents altitude vs. time? Which represents vertical velocity vs. time? Which represents propeller speed vs. time?

3. Match the graphs with these quantities and give several reasons why your answers make sense.

4. About 13 minutes into the flight there are some unusual features on the graphs. What might have happened during the flight at that point?

5. Tell a story for this flight.

## IM Commentary

This task uses data from an actual flight computer.  If the task is presented in the classroom, the teacher should show the graph and then ask the questions one at a time, rather than hand out the different parts all at once, since part b includes the answer for part a.

Since the data is from an actual flight computer and was not cleaned up, the graphs may appear untidy in places. This is part of the work of mathematical modeling, to interpret data and decide whether certain features make sense in the context of the situation or whether they should be treated with caution.

The three quantities shown in the graph are related and can be discussed together. In fact, for students in a precalculus class, it would be possible to talk about vertical velocity as the rate of change of the altitude. This is a foreshadowing of calculus ideas.

While the three quantities might be not familiar to the students at first, they are quite approachable. A discussion could include questions like: One of the graphs has negative output values. What happens when the values are negative? Why do the other graphs not have negative output values? Could they be negative for a different flight? Could you use information for one graph to predict/explain features of another graph?

As a resource with the task we give the Excel file with the recorded data. It has 813 data entries for each quantity. If appropriate, an extension of the task could be to have students find specific data entries in the file that correspond to certain features on the graph. For example: Find the exact time when the plane started its first descent. When did the loop take place? What was the highest altitude reached on the flight?

## Solution

a) The only information given is that the data shown comes from a plane’s computer, so they should be interesting and important to a pilot. It is reasonable to guess that the red graph shows altitude. The other two graphs are less clear.

The green graph starts with output values zero, then has positive output values while the plane is going up and negative output values while the plane is going down and the output values are zero when the plane is on the ground. This is consistent with vertical speed.

The blue graph might be the most difficult to guess. It only has positive output values, they are never zero and they are slightly higher when the plane is going up than when it is going down.  This is consistent with the speed of the propeller.

b) The red graph shows altitude (in feet), the green graph shows vertical velocity (in feet/min) and the blue graph shows the propeller speed (in RPM) as functions of time, in minutes. Since no scale on the vertical axis is given, it is not possible to know for sure what the units are, but thinking about what appropriate units would be could be part of the discussion. The reasons are already given in the answer to part a.

c) After 13 minutes there is a short blip in the altitude graph and the vertical velocity graph goes crazy with a rapid change from positive to negative values before it levels off again. The propeller speed on the other hand stays constant. There are many things that might have happened. At that point in the flight the pilot did a loop.  It only took a few seconds, the altitude did not change much compared to the altitude of the plane, but vertical speed changed quickly and drastically.

d) The plane taxis down the runway. It takes off after 6 minutes and gains elevation until the 12-minute mark. The plane then loses elevation (descends) and does a loop at 13 minutes. After 17 minutes it is approaching the airport and getting ready to land. The plane is on final approach at 18 minutes and lands shortly after 19 minutes. For some strange reason, the pilot decides to go back to the start of the runway to take off again, do a quick circle around the airport and land again (22-25 minutes). The data recording ends after 28 minutes.