Title
Red Rocks Amphitheater (Dan Snook)
Goals
1. Students will determine terms of a sequence defined recursively and in closed form.
2. Students will use recurrence relations to solve the problem.
Abstract
Students will find the closed form for finding the nth term of a list of numbers that they have generated. They will need to solve a quadratic equation the number of terms, given the sum of the values. Students will then use the closed form to find where a specific value lies in the list.
Problem Statement
This examples deals with seating at Red Rocks amphitheater. As rows rise from the stage, the number of seats they contain increase. Generally, in this type of example, one would have the students find the capacity of the venue, given the number of rows. This problem is more difficult in that it gives the total seating and asks students to find the number of rows. Students are also asked to find where specific tickets will be seated.
Instructor Suggestions
1. Make sure students understand what is happening in the problem.
2. It may be helpful to review certain facts about arithmetic sequences first. For instance, how you might find the nth term, how to find the sum, etc.
3. Allow students time to work on the problem in small groups.
4. Have each group present their results to the class.
5. Compare and discuss the results.
Materials
"Red Rocks" worksheet, dry erase board
Time
Problem introduction (5 minutes), Small group work (25 minutes), Small group presentation and class discussion (25 minutes)
Mathematics Concepts
Discrete Mathematics Concepts
Recursion
Related Mathematics Concepts
Arithmetic sequences and series, quadratic equations
NCTM Standards Addressed
Problem Solving, Reasoning, Communication, Connections, Algebra, Discrete Math
Colorado Model Content Standards Addressed
Number Sense (1), Algebraic Techniques (2), Data Collection and Analysis (3), Geometric Techniques (4), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity would fit into an Algebra I or Algebra II course after work on arithmetic sequences had been developed. Students should also know how to solve quadratic equations.
Further Investigation
In writing an equation to that equates the total number of seats, students may use the fact that the average of the first and last terms, multiplied by the number of terms will equal the total number of seats. Ask students what planar shape the rows of seats will form. Compare their equation for the total number of seats to the area formula for a trapezoid.
Variations/Comments
Avoid allowing students to attempt this problem via number crunching. It certainly is possible to add 50+54+58+62+... until one reaches 9,234, but this is not the point of the activity. Also assure the students that this is a "suppose" type of problem, and that Red Rocks does not number its seats in this manner!
References/Resources
Colorado Model Mathematics Standards Task Force. (1995) Colorado model content standards for mathematics.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author:
Crisler, Nancy, Fisher, Patience, Froelich, Gary. (1994) Discrete Mathematics Through Applications. New York: W.H. Freeman and Company