# Hoisting the Flag 1

## Task

Every morning at summer camp, one of the campers has to hoist the camp flag to the top of the flagpole.

For each graph below, describe the action of the flag raiser that might have led to this graph that shows the height of the flag as a function of time. Is any situation more realistic than another? Why or why not?

## IM Commentary

In this task, students are given a scenario and different graphs that could describe the relationship of the quantities in the situation, the height of a flag as a function of time. The scenario is open ended enough, so that most of the graphs could fit the situation. Therefore, the task is much richer than simply a matching problem. Students have to realize how different features of the graph connect to the physical situation.

A follow-up question to this task could be to ask students to draw their own graph and swap with a partner, who has to describe the corresponding motion of the flag. An instructor could also modify the task to include graphs that are increasing and decreasing. Then the flag would go up the flagpole and down again. Students could also be asked to draw the height of the flag over several days. This way the graph would exhibit additional mathematical features: increasing and decreasing intervals, maxima, minima, and periodicity.

## Solution

- In this scenario, the flag is raised at a constant rate. This means that every second, the height increases by the same amount. This looks like a very steady, fluid motion.
- Here the flag is raised quickly at first and then more slowly towards the end, almost as if the camper ran out of steam.
- In this scenario, one might imagine the camper to reach up and pull down, then reach up with the other arm and pull down, etc. In between pull-downs, the flag stays at the same height for a few seconds. Then it goes up another foot or so and stays, goes up and stays, etc.
- Here the flag goes up slowly at first and then picks up speed.
- Here the flag goes up slowly, then picks up speed and slows down again at the end.
- The last picture is not the graph of a function. In this scenario nothing happens for some time, the flag is nowhere. Then the flag is at every height at the same time, then it is nowhere again.

Different students could make arguments that different graphs are more or less realistic.

(a) would be realistic if the flag had a crank as a hoisting system or if the person hoisting the flag is very experienced and one hand takes over immediately after the other hand has pulled down all the way.

(c) might be quite realistic if you pulled down with both hands at the same time of if you briefly rest between pulls.

(f) is the least realistic since it does not reasonably capture the situation at all. The picture is not the graph of a function. As time flows, the flag has to be at some height, which this picture does not show. Then all of a sudden, the flag is at every height. That is not possible either. Then the flag is at no height again.

## Hoisting the Flag 1

Every morning at summer camp, one of the campers has to hoist the camp flag to the top of the flagpole.

For each graph below, describe the action of the flag raiser that might have led to this graph that shows the height of the flag as a function of time. Is any situation more realistic than another? Why or why not?