Llama 2
Posted December
8, 1999 as a Math
Forum EPoW
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Introduction: In the Llama
puzzle series, students explore
the relationship between area and
perimeter by experimenting with
an applet.
In this part of the series, students
experiment with the applet while observing a graph that models the situation.
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Where's the Math: This
problem challenges students to estimate,
hypothesize, experiment and draw
conclusions using geometry, ratio/proportion
and measurement sense. Students
encounter concepts of maxima as
they experiment with various heights
and widths of the lama's pen. Open-ended
questions encourage students to
revisit their assumptions, targeted
questions encourage directed exploration,
and process oriented questions encourage
students to rethink their answers.
This part of the series builds
on the previous by offering students
the opportunity to experiment with
concepts of "maximum"
graphically, and students are challenged
to compare the affordances of different
representations.
Standards: Measurement,
algebra,
geometry,
data
analysis & probability
Role of Components: The
Llama problem series uses Geometer's
Sketchpad allow students to
interact with a simulated barn and
pen. Height and widths are inputted
via number
entry fields (labeled with ESCOT
labels) and triggered by a button
panel. Instructions are displayed
via the HTML
viewer, and solutions are entered
in a
text entry box. Logoscript
handles communication among components.
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Try the applet!
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Sample submitted solutions:
From:
Amber, age 13
Katie, age 13
School: Issaquah Middle School, Issaquah,
Washington
1. What does the graph
say about the area of the rectangle?
The graph tells you what the area of
the rectangle is going to be.
2. How does your graph
help you answer the following question:
Is there just one solution for the largest
area? (from question #3 in Llama, part 1)
The graph shows you whether or not there
is more than one solution. no there is not
more than one solution.
3. What are the tradeoffs
for using the area calculator (from Llama,
part 1) and the graphing tool (from this
problem)?
you can use your calculater to plug in the
numbers on your graph to see what area is
going to be.
BONUS 4. What type of
geometric curve do you see in the graph?
a parabola
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Reflections:
In part II of the llama series, we wanted
students to see the idea of a maximum graphically.
Three groups of students submitted correct
answers, which described how the shape of
the graph is related to the area of the
rectangle. In posing the question, we should
have been more specific... When students
encounter open ended questions such as "What
does the graph say about the area of the
rectangle?" should make an effort to think
the question through and try to give a detailed
answer... Students should try to imagine
that they are writing to someone with no
knowledge of the problem and supply the
level of detail necessary for their audience
to understand their answer.
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