Date: Mar 24, 1995 8:12 PM Author: DavidBan1@aol.com 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

exchange.

David Bannard

Collegiate School

Richmond, VA

Davidban1@aol.com