1. Topic-
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2. Content-
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The Devil and Daniel Webster is a short story by Stephen Vincent
Benet about a New Hampshire farmer who sells his soul to the Devil
and is defended by American statesman Daniel Webster. The problem
below uses this context, which provides an interdisciplinary connection,
although the problem given here is not taken directly from the story. |
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3. Goals: Aims/Outcomes-
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This problem is based on recursion. Graphing the results are challenging,
and understanding the type of change involved is also challenging.
The Principles and Standards (NCTM 2000) emphasize that recursive
formulas are used to solve many problems, and that students often
have a natural understanding of recursively defined functions.
Adapted from Navigating through Algebra in Grades 9"�12, this lesson
allows students to examine a recursive sequence in a game between
the Devil and Daniel Webster. |
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4. Objectives-
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Students will be able to:
Use recursive or iterative forms to represent relationships
Approximate and interpret rates of change from numerical data
Draw reasonable conclusions about a situation being modeled |
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5. Materials and Aids-
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Daniel Webster Activity Sheet
Spreadsheet software or graphing calculator, if needed
Activity Sheet Solutions |
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6. Procedures/Methods-
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A. Introduction-
Set up the problem by reading the following to students:
The devil made a proposition to Daniel Webster. The devil proposed
to pay Daniel for services in the following way:
On the first day, I will pay you $1000 early in the morning. At the
end of the first day, you must pay me a commission of $100; so, your
net salary that day is $900. At the start of the second day, I will
double your salary to $1800; but at the end of the second day, you
must double the amount that you pay me to $200. Will you work for
me for a month?
Ask students to describe the pattern for Day 3, Day 4, and so on.
Students should recognize that Daniel's salary at the beginning of
Day 3 will be $3200 (since the amount he had at the end of Day 2 was
$1600), and he will need to pay the Devil an amount of $400 at the
end of Day 3 (since he had paid $200 on the previous day). |
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B. Development-
Distribute the Devil and Daniel Webster activity sheet to all students.
Allow students to work individually for about two minutes, and then
have them work in pairs to compare their results and complete the
activity sheet. |
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C. Checking for understanding-
When students have completed the activity sheet, conduct a discussion
about the results. This discussion could focus on the following questions:
Why doesn't the salary scheme work? [Because the combined effect of
doubling the Devil's commission and subtracting it from Daniel's pay
causes the Devil's commission to increase more rapidly than Daniel's
pay.]
What types of curves result from the data in the table? [Both are
exponential; however, the graph for Daniel's net pay increases at
the beginning but then decreases rapidly, whereas the Devil's commission
increases from the beginning.] |
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7. Evaluation-
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Using the Devil and Daniel Webster Spreadsheet, ask students to
investigate other scenarios. You can randomize a different set of
values for the initial pay, commission, and factor for each student,
and then ask each student to write a journal entry about the results
when their set of randomized values is used.
Extensions
Have students consider this problem if the devil asks for a cut of
80% of the salary each day instead of the doubling scheme. Should
Daniel do the work for three weeks? For a month? At what rate is it
most advantageous for both Daniel and the devil? Should Daniel work
if there were a 1 percent cut for the Devil?
Advanced students might attempt to discover a closed formula for the
scenario. |
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8. Teacher Reflection-
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Teacher Reflection
Were students able to identify a closed form for the Devil's commission
or Daniel's net pay? What additional lessons or practice would help
those students who had difficulty?
Did students enjoy the lesson? Is there a scenario that would make
the lesson more enjoyable?
Were students able to accurately predict the types of graphs that
would result? What additional practice would be needed so that students
could reasonably predict the type of graph that results from a table
of values? |
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