The Devil and Daniel Webster
http://illuminations.nctm.org/LessonDetail.aspx?ID=L288
A lesson plan for grades 9–12 Mathematics
This interdisciplinary lesson allows students to examine a recursive sequence in a game between the Devil and Daniel Webster. Illuminations provides an activity sheet, solutions for the activity, a link to a helpful spreadsheet, and detailed instructions for completing the lesson.
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.
- Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
- Understand relations and functions and select, convert flexibly among, and use various representations for them.
North Carolina Curriculum Alignment
Mathematics (2004)
Grade 9–12 — Advanced Functions and Modeling
- Goal 2: Algebra - The learner will use functions to solve problems.
- Objective 2.05: Use recursively-defined functions to model and solve problems.
- Find the sum of a finite sequence.
- Find the sum of an infinite sequence.
- Determine whether a given series converges or diverges.
- Translate between recursive and explicit representations.
- Objective 2.05: Use recursively-defined functions to model and solve problems.
Grade 9–12 — Algebra 1
- Goal 1: Number and Operations - The learner will perform operations with numbers and expressions to solve problems.
- Objective 1.02: Use formulas and algebraic expressions, including iterative and recursive forms, to model and solve problems.
Grade 9–12 — Discrete Mathematics
- Goal 3: Geometry and Measurement - The learner will describe and use recursively-defined relationships to solve problems.
- Objective 3.01: Use recursion to model and solve problems.
- Find the sum of a finite sequence.
- Find the sum of an infinite sequence.
- Determine whether a given series converges or diverges.
- Write explicit definitions using iterative processes, including finite differences and arithmetic and geometric formulas.
- Verify an explicit definition with inductive proof.
- Objective 3.01: Use recursion to model and solve problems.
Grade 9–12 — Integrated Mathematics 1
- Goal 1: Number and Operations - The learner will perform operations with numbers and expressions to solve problems.
- Objective 1.02: Use algebraic expressions, including iterative and recursive forms, to model and solve problems.
Grade 9–12 — Integrated Mathematics 2
- Goal 1: Number and Operations - The learner will perform operations with numbers, algebraic expressions, and matrices to solve problems.
- Objective 1.02: Use algebraic expressions, including iterative and recursive forms, to model and solve problems.
Grade 9–12 — Integrated Mathematics 3
- Goal 1: Number and Operations - The learner will perform operations with numbers and algebraic expressions to solve problems.
- Objective 1.02: Use algebraic expressions, including iterative and recursive forms, to model and solve problems.
Grade 9–12 — Integrated Mathematics 4
- Goal 4: Algebra - The learner will use relations and functions to solve problems.
- Objective 4.02: Use recursively-defined functions to model and solve problems.
- Find the sum of a finite sequence.
- Find the sum of an infinite sequence.
- Determine whether a given series converges or diverges.
- Translate between recursive and explicit representations.
- Objective 4.02: Use recursively-defined functions to model and solve problems.
Grade 9–12 — Pre-Calculus
- Goal 2: Algebra - The learner will use relations and functions to solve problems.
- Objective 2.07: Use recursively-defined functions to model and solve problems.
- Find the sum of a finite sequence.
- Find the sum of an infinite sequence.
- Determine whether a given series converges or diverges.
- Translate between recursive and explicit representations.
- Objective 2.07: Use recursively-defined functions to model and solve problems.
