## The Devil and Daniel Webster

- Lesson

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.

*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.

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 amount left at the end of the previous day. So, at the beginning of the second day, I will pay you $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).

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.

Devil and Daniel Webster Activity Sheet

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.

The iterative process described in this problem can be executed on a calculator, and iterations can be achieved by pressing the enter key repeatedly. Similarly, the process can be investigated using a spreadsheet. The Devil and Daniel Webster Spreadsheet allows students to investigate the problem presented here, as well as to investigate the problem by changing the initial amount of Daniel's salary, the initial amount of the Devil's commission, and the factor by which the Devil's commission changes. (In the problem above, the initial amount of Daniel's salary is $1,000, the initial amount of the Devil's commission is $100, and the factor is 2 because all values double. When the spreadsheet is first opened, the simulation is set to these default values.)

Devil and Daniel Webster Spreadsheet

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.]

You may wish to continue the discussion with the questions in the **Questions for Students** section.

### Reference

Burke, Maurice, David Erickson, Johnny W. Lott, & Mindy Obert. (2005). Navigating Through Algebra in Grades 9 – 12. Reston, VA: NCTM.

- Daniel Webster Activity Sheet
- Spreadsheet software or graphing calculator, if needed
- Activity Sheet Solutions

**Assessment Options**

- 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.
- As students work in groups, listen to their discussions to determine the strategies they used to determine if they would be would be willing to work for the devil for a month.

**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.

**Questions for Students**

1. What is the closed form for the Devil's commission?

[c= 100 × 2^{n – 1}, wherenis the number of days.]

2. What is the closed form for Daniel's money at the end of the day?

[This formula can be difficult to determine, although it is easier to see if students attempt to investigate a pattern. The table below shows how the closed form can be found by using powers of 2 in each step:

Number of DaysMoney Remaining at End of Day1 2 ^{0}· 1000 – 2^{0}· 1002

2 · (2 ^{0}· 1000 – 2^{0}· 100) – 2^{1}· 1002 ^{1}· 1000 – 2^{1}· 100 – 2^{1}· 1002 ^{1}· 1000 – 2 · 2^{1}· 1003

2 · (2 ^{1}· 1000 – 2 · 2^{1}· 100) – 2^{2}· 1002 ^{2}· 1000 – 2 · 2^{2}· 100 – 2^{2}· 1002 ^{2}· 1000 – 3 · 2^{2}· 100: : n

2 ^{n – 1}· 1000 –n· 2^{n – 1}· 1002 ^{n – 1}· (1000 – 100n)Consequently, the closed form is

d= 2^{n – 1}× (1000 – 100n).]

**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?
- What did students like or dislike about 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?

### Learning Objectives

Students will:

- 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.

### NCTM Standards and Expectations

- Generalize patterns using explicitly defined and recursively defined functions.

- Understand relations and functions and select, convert flexibly among, and use various representations for them.

- Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.