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Part 1: An Opening
Discussion
- Discuss
the following scenario in small groups.
Imagine that one of your students comes to class
very excited. She tells you that she has figured out a theory that you
never told the class. She explains that she has discovered that as the
perimeter of a closed figure increases, the area also increases. She shows
you this picture to prove what she is doing:
- How would you respond to this student?
- Is this student’s “theory” valid? After
initial discussions in small groups, write a short journal entry about
your reaction to this student and a brief mathematical discussion
either validating or disconfirming this “theory.”
Part 2:
Exploring perimeter with a fixed area
with manipulatives
- With 12 square tiles, arrange the tiles (flat on
a surface) so as to obtain the maximum perimeter. The tiles should
touch completely side-to-side, not by corners.
Examples include:
-
What is the maximum perimeter? What general observations did you make about how to make your
figure have a larger perimeter? Is there more than one figure that can
have the same maximum perimeter? Why or why not?
-
Restrict the figure to a solid rectangle. Find all possible rectangles with an area of 12 square tiles.
Record the respective dimensions and perimeter for the
rectangles. Which rectangle has the maximum perimeter? Why?
-
Given any fixed area for a rectangle, make a
general statement about which dimensions maximize the perimeter.
Part
3:
Exploring area with a fixed perimeter
using manipulatives
Farmer Ted wants to build a rectangular pigpen
for his pigs. He has 24
meters of fencing left from his last project. What size rectangle should
he build so that the pigs have the maximum amount of play area?
-
Use square tiles to model different-sized solid
rectangular pigpens. Build
all possible rectangular pigpens that can be enclosed with 24 meters
of fencing. Record the
corresponding dimensions and areas for those pigpens. Which pigpen
gives the pig the most play area?
-
As the length of the rectangular pigpen changes,
how does that affect the width and area of the pigpen? What patterns
do you notice? Explore this relationship numerically and express this
algebraically. Use this relationship to explain why the maximum area
occurs with a square pigpen.
-
Given any amount of fencing, make a general
statement about the dimensions of the rectangular pigpen that give the
maximum area.
Part
4:
Exploring the problem with a spreadsheet
template
-
Open the Excel file called maxarea.xls.
(You will need to ENABLE MACROS when prompted.) This spreadsheet
allows the user to change the perimeter (P) and then vary the length
of side of X using the scroll bars.
As these values change, the yellow box will display the current
values of P, X, Y, and the Area (A) of the rectangle. The rectangle
and the data point (X, A) on the graph of the Length of side X vs.
Area will also change accordingly.
-
Fix the perimeter (P) to 24 as in the pigpen
problem. Then incrementally increase the length of side X from 1 to
12. Describe the changes that occur in the numerical, graphical, and
geometrical representations.
-
Click on the Show
All Points tab to display the next worksheet. What is the shape of
the graph of all the data points (X, A)? Why does this make sense?
Which Length of X gives the maximum value for A? Why?
-
How does this experience differ from the
manipulative experience of physically building each rectangle? Discuss
the benefits and drawbacks of the physical and spreadsheet approach to
the pigpen problem.
Part
5:
Exploring the relationship with algebra and calculus
-
If Farmer Ted had 158 meters of fencing, predict
what the dimensions would be of the rectangle with the maximum area.
Explain how you made this prediction. Create a rule as a function of P to obtain the length of side X
which gives the maximum area. Confirm this rule with several other examples.
-
Create a general equation for determining the
Area of a rectangle given both a fixed P and length of side X. How
does this equation relate to the shape of the graph in the
spreadsheet? With this Area equation, use the first derivative to
verify the value of X which gives the maximum value of A.
How does this compare to the rule you created above (a function
of P to obtain the length of side of X which gives the maximum area)?
-
Discuss the non-calculus and calculus based
approaches to making this generalization, including students’
accessibility, advantages, and disadvantages.
-
Why is the maximum value for Area achieved when
the first derivative is equal to zero? For a graphical exploration of
this, use the following java applet:
http://www.ies.co.jp/math/java/calc/doukan/doukan.html
Part
6: Revisiting the Opening Scenario
-
Revisit the scenario and subsequent group
discussions from Part 1. Now that you have explored the relationship
between perimeter and area at several different levels and with
various tools, how would you use the student’s “theory” to
provide relevant classroom experiences if you were teaching:
1.
A sixth grade class
3.
A Calculus course
Either discuss this in small groups or write an
individual journal entry that addresses how you would approach this
topic at each of the three levels, including justifications as to why
you believe your approach is pedagogically and mathematically
appropriate.
Extensions:
-
Consider the generalization you made in Part 2
when a rectangle has a fixed area and you want to maximize perimeter.
Create an interactive spreadsheet template that shows the numerical
and graphical representations of this problem. Use scroll bars to fix the value of A and change the values of
X, and create a graph of X vs. P.
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View the following java applet:
http://www.ma.utexas.edu/users/kawasaki/mathPages.dir/calculus.dir/max-min.dir/ problem1.html
Discuss the benefits and drawbacks of using the applet or the
spreadsheet template to explore the relationship between perimeter,
area, and the length of side X.
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