Date: Mar 24, 1995 8:12 PM
Subject: open-ended problems

I do not know if this is an appropriate posting for this list but as a high
school teacher, I am constantly searching for good problems to give to my
classes. What I try to give are open-ended problems that are appropriate for
group work. I would be glad to throw out two problems that I have used with
success this year and wonder if there are others who are interested in
sharing problems that they have found good. Bothe problems were used in my
precal class but would also be useful for calculus as well.
Lifeguard problem
A lifeguard (L) at the beach hears a swimmer (S) calling for help 150 yards
down the beach and 80 yards off shore. She knows she can run on the beach
at a rate of 320 yd./min. and swim at a rate of 60 yd./min. She decides to
enter the water at a point P which is x yards from the point T on the beach
perpendicular to the swimmer.
part 1: Solve the problem by determining how far down the beach she should
run before starting to swim. (If this is their first exposure to this kind
of problem, I will ask several leading questions about how to set the problem
up with a diagram, to estasblish what is being minimized and estimating
reasonable values for x and t, generating an equation etc. Equally
interesting is a discussion of the kind of accuracy that we should be
interested in. We have all had students give answers to questions like this
to the nearest hundredth. Questions I have asked to get at these issues go
something like this. Identify aspects of this problem that are either
unrealistic or are only approximations.

How much does it matter if the lifeguard does not start swimming at the
optimum point. Explain your answer with specific examples.

What effect does an error in estimating your swimming speed have on the
problem. How accurate do you think is the estimate of the swimmers distance
from the shore? The distance of the lifeguard from T?
Perhaps the best question to ask would go something like this. Suppose you
are in charge of training the lifeguards at a beach. You wish to instruct
your lifeguards on how they should decide where to run before they begin
swimming when they have to rescue a swimmer. Prepare a clear talk for your
lifeguards. Include a mathematical justification for any advice that you
choose to give your lifeguards

Problem 2:
I find that questions of this type are excellent for helping students to see
the relationship between a problem description and a graph. I find it
helpful to do classroom exercises that require students to estimate what
graphs will look like. These are also good problems to help students
understand the idea of parameters and to estimate the effect that changing
parameters will have on a graph. In addition, we can look at symmetry, the
effects that transformations of the original problem.

a) Sketch the graph of the function y = 4 - x^2. Include on the graph the
point P with coordinates (0, -3).
b) Write an equation for the function representing the distance of P from an
arbitrary point on the function in part (a) in terms of the x-coordinate of
the point.
c) Use your understanding of the problem and your common sense to sketch a
complete graph of the function in part (b). Be sure to label and scale your
axes. Indicate on your graph the coordinates of any relative maximum points
or relative minimum points. Check with your calculator
d) Is the graph in part (c) symmetric? If so, what symmetry does it
exhibit? Explain how you can tell from the equation in part (b) that the
function is symmetric.
e) In part (d) you explained how you could tell the graph was symmetric from
the equation. Explain how why you could predict this symmetry from the graph
and problem description in part (a).
f) What are the coordinates of the point on the graph of y = x2 that is
closest to the point (0, 7).
g) Compare this problem to the previous problem. Explain.
h) Give another problem that is equivalent to the original problem.
i) Describe what happens to the distance graph as the point P moves.
Include what happens to your graph as P moves along the y-axis. What
happens if P moves to the right?

I hope these problems spur some interest. I find that as a classroom
teacher, I am always trying to find better ways to encourage my students to
explore problems that really require them to develope solutions with
significant analysis. It is not always easy for me to come up with questions
that do this effectively. I hope there are others interested in this kind of
David Bannard
Collegiate School
Richmond, VA