Interested in writing a story
or nominating a friend? Yes
Translate this story into Spanish? Yes
|
Making math relate to the real world
A math teacher challenges his students with interesting story problems By Jim Bowman |
 ince the National
Council of Teachers of Mathematics (NCTM) Standards
for mathematics were released in 1988,
most schools have tried to increase the amount of mathematics taken by
most students. More advanced topics have also been incorporated into most
courses. I believe that the most effective way to get students to solve
problems involving advanced material is to make technology, in the form of
computers and graphing calculators, an integral part of every high school
mathematics course. These tools have allowed my students in the
regular mathematics classes to solve problems that in the past were
considered calculus problems. |
 like to
use special problems, sometimes done as a class project, to
help create interest in a concept we are studying or to overcome the
student's perception that, since he/she is not taking calculus, the math must
not be as important or as useful. I recently gave the following problem to
my second year algebra class as a special project. The problem is a
modification on one printed by Texas Instruments in their newsletter
Eightysomething!"The local cable television company needs to run cable to three new housing developments. They can install only one junction box in the existing main line because of the very high cost of a junction box. The first development is 4 miles north of the main line, the second development is 5 miles east and 1 mile north of the first development, while the third is 6 miles east and 1 mile south of the first development. Each developer insists on a private line directly to each housing area. There are no obstructions anywhere so that we can run a line directly from the junction box to each housing area. Where along the existing main cable line should we install the junction box so that the least possible length of cable is needed?" |
 exas
Instruments' original problem involved pipelines. My students
related better to cable television so I modified the problem. Students were
given one week to develop their best solution to the problem.Jerry immediately asked "What is the formula?" No one in class could find one, although Jason did suggest the standard distance-rate-time relationship but almost everyone agreed that it did not apply in this situation. I then asked, "What other distance formula have you worked with?" Alyson responded "distance in a plane," remembering the formula used in Geometry. Another student suggested that we might use the distance on a line relationship. Alyson then remembered that in Geometry the points she had to find the distance between were either graphed or given as ordered pairs. I replied "That's right, Alyson, so see if you can find a way of graphing the situation described in our problem." At this point I suggested that each group try drawing a graph of the problem. Most students were concerned about where to put the origin since that point is always given in their mathematics books. My answer was "Just try some different places on your graph and see what happens." By the next day after some experimenting, most groups had an expression for the total length of the cable needed. We spent a few minutes at the end of class the second day to allow each group to explain to the rest of class how they arrived at their expression and to talk about what went on in their groups as they worked on the problem. |
 Jim uses a scientific calculator to provide real world examples to his class. |
 t this point
everyone had an expression or a "formula" for the length
of the required cable, but no one knew how to minimize that length.
Most groups had decided to place the origin of their graph at the first
housing development, while two others had chosen a point on the main cable
line that was directly south of that housing development. I was somewhat
surprised that no other locations were tried. Most groups began trying a
guess-and-check method of solving the problem. No more class time was
spent on the problem at this point.During the next class Tim asked, "Can we get help from students in the calculus class or some other advanced mathematics class?" I responded with "Sure, as long as you can explain both how and why you solved the problem using the method that you used." I allowed about ten minutes at the end of class for discussing progress toward a solution. At this point no one really had made much progress beyond writing the function for the total length of cable that was required. Since the function was the sum of three expressions with square root signs most of the class was completely intimidated by the thought of having to manipulate the expression. |
|
t the next
class session, Tim said, "I asked a calculus student how to
solve the problem and he told me that I needed to find the derivative of the
function, but he could not do it. He said that you should show me how to find
the derivative. Will you"? I then told the class "That is correct. The traditional way of finding minimum or maximum values for a function does involve working with the derivative. Since the derivative is a calculus concept and we are not studying calculus, I will not show you how to solve the problem that way. However, you can take advantage of either a computer or a graphing calculator to solve the problem using only the mathematics that you already know." I gave the class about ten minutes at the end of class to discuss their ideas for a solution within their group. |
 y the
deadline for solving the problem, there were three different
methods used that proved to be at least somewhat successful. The first, and
least accurate, was just to plug values into the formula that the group had
developed and calculate the total length of cable needed. Three groups used
this method and they all had the same answer, which was that the junction
box should be on the main line four miles east of the first housing
development. Two of those three groups had decided that 4 was the correct
number or answer, but could not explain the location in terms of 4 miles
from what point. All three of these groups had tried only whole numbers.
For some reason, it had not occurred to any of them to use a fraction of a
mile. Since the numbers were small whole numbers this method was not too
difficult nor time consuming. However, after considering fractions of a
mile, they all agreed that we would actually waste some wire. |
 ason's
group had used a spreadsheet on his computer at home to
create a table of distances along the main cable in addition to the total length
of cable required to reach the three housing developments. They came up
with 4.1 miles east of the first housing area and on the main cable. When
asked to tell the class why they had chosen a table to solve the problem,
Jennifer said, "I remembered that we had used a table to approximate the
zeros of a polynomial and thought that we could calculate a lot of distances
quickly doing that."They had first calculated the total length of cable needed for each whole mile from the first development to 6 miles east of that development. Jennifer said, "We knew that the junction had to be between the housing developments. The first table showed that the smallest amount of cable was a little over 14 miles and that put the junction between 4 and 5 miles east of the first development. We then redid our table using distances between 4 and 5 adding 0.1 miles each time. Since you did not tell us where to round our answer we stopped at 4.1 miles east of the first development." They still had some extra wire but agreed that it would be easy to recalculate their table to hundredths, thousandths, or whatever was required. |
|
This problem created enough interest in using technology to solve
problems that many in the class asked for another problem. |
he third
group used a graphing calculator to solve the problem. They
created a function by adding the three expressions that had been obtained
using the distance formula, but rather than simplifying that complex
expression, they let the calculator graph it and then zoomed in on the vertex
of the parabola to find both the distance east of the first housing
development and the total length of cable needed. Their answer was also 4.1
miles. They also agreed that they could easily obtain the answer to however
many decimal places needed. They would simply use the zoom function on
the calculator. After discussing the three successful methods of solving the problem, most of the class agreed that the graphing method was the quickest method. Some, however, still preferred the table on the spreadsheet. No one thought the guess-and-check method was preferable. |
|
his problem
created enough interest in using technology to solve
problems that many in the class asked for another problem. I first reminded
them that we could not throw out our district-adopted course syllabus; and
so, would have to continue with our regular course work as well. We agreed
that the next problem would be extra credit, that we would not discuss
it in class until after the deadline for completion and that the regular
assignments must be completed before they could receive credit for this
problem. |
 Using calculators and computers, Jim's students have found that the "guess-and-check" method of problem solving is not always the best. |
ince
I teach a course in the School of Business at Washburn
University called Quantative Analysis, I decided to give them a problem
from that class. "Suppose that you are running a pizza parlor and normally sell a large pizza for $13.00. At this price you have been selling an average of 200 pizzas each night. Based on previous experience you have determined that for each $.50 increase in price you will sell 5 fewer pizzas. At what level should we set the price of a pizza to bring in the greatest revenue? Give the price to the nearest penny. In other words you are not limited to just multiples of 50 cents in lowering the price." Bill immediately said, " Let's charge $100." Although we had agreed not to discuss this one in class before the students had a solution, I felt that Bill's quick response involved a basic and common misunderstanding of the situation created by our problem, and that a short response was necessary, so I asked Bill, "How many $100 dollar pizzas have you purchased recently?" I asked the class to think about the relationship between how many pizzas they buy when the price is low as compared to how many they buy when the price goes up. With that, the students were left on their own or with their group to set the price. Again, a one week deadline was set for credit on the problem. |
|
he
students really did quite well on this problem. Since they had
been solving algebra problems using equations, most of them did not seem to
have much trouble finding a function for the revenue at the new sale price.
The ideal price was $16.50. About half of the class had that solution. Four
students had other answers and the rest did not have any solution. Three
methods were used in obtaining the student's solution. |
 ne method
was to seek help from students in more advanced classes,
especially the calculus class, or help from parents or other adults. Most of
the adults and non-calculus helpers used a guess-and-check method. They
just raised the price and calculated the revenue. After narrowing the
answer to the nearest fifty cents, they raised the price by ten cents and
then finally by a penny at a time. Since they are required in my class to
explain their method and justify the answer, most of these student filled at
least two pages before they had a solution.The calculus students were much more helpful on this problem because the revenue function turned out to be a polynomial and all of them could find the derivative and then solve the resulting equation. They solved the problem with only a short paragraph of work on the page. |
|
ome students
again solved the problem by creating a table using a
spreadsheet. They started at the regular price and increased the price by
one cent, then calculated the revenue. It took them about a page and a half
of output to find the best price, but since the computer did all of the work
after the spreadsheet was created, most students agreed that the method
was workable. Another group of students used a graphing calculator to graph the revenue function. It turned out to be a parabola and some of them used a formula that they remembered from other courses to find the vertex. Others, either because they did not remember the vertex formula or chose not to use it, used the zoom function on the calculator to find the coordinates of the vertex. These students used about a half page to explain their method and give the solution to the problem. |
|
hese
problems increased the interest in mathematics for most of the
students in this class. Most of the students spent quit a bit of their own
time working on the problems and seemed to enjoy the challenge. They were
especially excited about being able to solve a mathematics problem that
even the so-called "math geniuses" in calculus could not solve. The problems
really fit into the material we were studying at the time because they
involved the vertex of a parabola and the class was studying the unit on
conic sections in the text. After having worked these two problems and some similar ones, the question that almost every math teacher hears frequently, "Where will I ever use this?" takes on a different meaning. Instead of assuming that the concept was useless, many students developed a genuine interest in learning how and where mathematics is used by people "in the real world."  |
|
