Somewhere in a New Jersey elementary school:
The students in Mrs. Chaplain's fifth grade class eagerly
return from recess, excited by the prospect of working on another of
her famous Chaplain's Challenges. Mrs. Chaplain regularly uses a
Challenge with her math class, and today she has promised the children
that the problem would be a great one. She believes that all of her
students will be up to the Challenge and expects that it will engage
them in an exploration and discussion of the relationship between area
and perimeter.
Suppose you had 64 meters of fencing with which you were going
to build a pen for your large pet dog. What are some of the
different pens you could build if you used all of the fencing? Which
pen would have the most play space? Which would give the most
running space? What would be the best pen?
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As the students file into the classroom, they stand, scattered
around the room, reading the Challenge from the board even before
finding their seats. Then they begin to ask questions about it.
What is fencing? Wouldn't the pen with the most
play space be the same as the one with the most running space? What
shapes are allowed? Mrs. Chaplain answers some of these
questions directly (she has brought a sample of fencing to class so
she could show the students what it looks like), but, for the most
part, she tells the students that they can discuss their questions in
their regular working groups.
The groups begin their exploration by discussing how to organize
their efforts. One of the first questions to arise is what kinds of
tools would help
solve the problem. The students have used a variety of materials
to deal with Chaplain's Challenges, and they have often found
that the groups that fashioned the best models of the problem
situations were the ones that found it easiest to find solutions.
Today, one group decides to use the 10by10 geoboards, figuring that
they can quickly make a lot of different "pens" out of
rubber bands if they let the space between nails equal four meters.
Another group decides to get some graph paper on which to draw their
pens, because that gives them a lot of flexibility. Still another
group, reluctant to be limited to rectangular shapes and work spaces,
thinks that the geometry construction software loaded on the computers
in the back of the room would let them draw a variety of shapes and
even help them measure various characteristics of the shapes. One
last group, striving for realism, decides to use a loop of string
sixtyfour inches long. As the session progresses, the groups of
students make many sample pens with whatever materials they have
chosen to use. Some groups switch materials as they perceive other
materials to be less restrictive than the ones they are using.
Keeping the perimeter a constant 64 meters, they measure the areas of
the pens using some of the strategies they developed the week before.
Mrs. Chaplain circulates around the room, paying careful attention to
the contributions of individual students, making notes to herself
about two
particular children, one who seems to be having difficulty with the
concept of area, and another who is doing a nice job of leading her
group to a solution.
Gradually, the work becomes more symbolic and verbal and less
concrete. The students begin to make tables to record the dimensions
and descriptions of their pens and to look for some
kind of pattern, because they have learned from experience that
this frequently leads to insights.
One group follows the teacher's suggestion and enters their
table of values for rectangular pens into a computer, generating a
brokenline graph of the length of the pen versus its area.
Toward the end of the class, the students become comfortable with
their discoveries. Mrs. Chaplain reflects again on how glad she is
that the faculty decided to organize the school schedule in such a way
as to allow for these extended class sessions. When she sees how
involved and active the students are, how they try to persuade each
other to follow one path or another, how their verbalizations either
cement their own understandings or provide opportunities for others to
point out flaws in their thinking, she realizes that only with this
kind of time and this kind of effort can she do an adequate job of
teaching mathematics.
The summary discussion at the end of the session allows the
students an opportunity to see what their classmates have done and to
evaluate their own group's results. Mrs. Chaplain learns that
everyone in the class understands that if you hold the perimeter
constant, you can create figures with a whole range of areas.
Moreover, she feels that a majority of the class also has come to the
generalization that the more compact a figure is, the greater its
area, and the more stretched out it is, the smaller its area.
But the students still have very different answers to the question,
What would be the best pen? That fits her
plans perfectly. For homework, Mrs. Chaplain asks each student to
design the pen that he or she thinks is best, draw a diagram of it,
label its dimensions and its area, and write a paragraph about why
that particular pen would be best for the dog. Mrs. Chaplain plans to
move on from this activity to others where the students concentrate on
more efficient strategies for finding the areas of some of the
nonrectangular shapes they explored in this Challenge.

Somewhere in a New Jersey high school:
Ms. Diego's algebra class and Mr. Browning's physical
science class are jointly investigating radioactive decay. The two
teachers, with the support of the school administrators, have worked
out a schedule that enables their classes to meet together this month
to explore some of the mathematical aspects of the physical sciences.
Both teachers regularly incorporate some content from the other's
discipline in class activities, but this month was specially planned
to be a kind of celebration of the relationship between the two areas.
By the end of the month, they expect that the students will really
appreciate the role that mathematics plays in the sciences, and the
problems that are presented by the sciences that call for innovative
mathematical solutions.
The classes are average. Nearly every student in the high school
takes these two classes at some point during their stay and, over the
past few years, because of exciting realworld problems like the one
on which they are working this week, the classes have become two of
the most popular in the school.
Monday's class begins with a presentation by Mr. Browning
about the process of carbon dating. He describes the problem that
archaeologists faced in the 1940s with respect to determining the age
of a fossil. They knew that all living things contained a predictable
amount of radioactive carbon that began to diminish as soon as the
organism died.
If they could measure the amount that remained in some discovered
fossil and if they knew the rate
at which the carbon "decayed," they could figure out the
age of the object. An American chemist
named Willard Libby developed a technique that
allowed them to do so. Ms. Diego explains that the classes will
spend the next few days exploring the concept of radioactive decay
and, toward the end of the week, they will be able to solve some of
the same kinds of problems solved by those archaeologists.
On Tuesday, working at stations created by the teachers, the
students begin to explore both the mathematical and scientific aspects
of radioactive decay. Working in groups, the students use sets of 50
dice to simulate collections of radioactive nuclei. Each roll of the
collection of dice represents the passage of one day. Any time a die
lands with a "1" showing, it "spontaneously
decays" and is taken out of the collection. The students plot
the number of radioactive nuclei left versus the number of days
passed in an effort to determine the halflife of the element 
the amount of time it takes for half of the element to decay. Because
the experiment is relatively well controlled, each group working on
the task produces a graph that effectively illustrates the decay, but,
because the process is also a truly random process, each group's
results are slightly different from those of other groups.
On Wednesday, in a very different kind of activity, students use
graphing calculators in a guided activity to discover properties of
exponential functions, and the effects on the graphs of various
changes to the parameters in the functions. Working from a worksheet
prepared by the teachers, they start with the general form of an
exponential function, y = ab^{x}. Using the
values a = 1 and b = 2, they input the equation into the
calculators and study the resulting graph. Then, they systematically
change the values of a and b to discover what each
change does to the graph. They are directed by the worksheet to pay
particular attention to the effect of changing b to a value
between zero and one, because graphs of that type will be especially
important for their work with radioactive decay. The culminating
problem on the worksheet is a challenge to try to find the values of
a and b that produce a graph that looks like the ones
that resulted from the experiment with the dice. The students enjoy
the problem and use their calculators to quickly check and refine
their solutions, zeroing in on the critical numbers. There is a lot
of discussion about why those numbers might be the correct ones.
On Thursday, the students discuss a reading that was assigned for
homework the night before, focusing on carbon dating and addressing
some of the mathematical processes used to determine the age of
fossils. This discussion is led by the two teachers, who have brought
in some fossilized samples to better acquaint the students with the
kind of materials they read about. Ms. Diego then leads a session to
develop the computational procedures for solving the carbon dating
problems using exponential functions. The students will be given some
homework problems of this type and will spend tomorrow's class
discussing those problems and wrapping up the unit.
The teachers are very pleased with what the classes have
accomplished. The active involvement with a handson experiment
simulating decay, the symbolic manipulations and graph explorations
made possible by the graphing calculator, and the study of a
particular scientific application of the mathematics have been very
productive. By working together as a team, the teachers have been
able to relate the different aspects of the phenomenon to each other.
The students have learned a great deal of both mathematics and science
and have seen how strongly they are linked.
