Title
Linear??? Quadratic??? Cubic??? (John Long)
Goals
(1) Students will make an input/output table for each problem
(2) Using the table of information the students will find a recursion formula for each problem
(3) Using finite differences and system of equations the students will find a closed form equation for each problem
(4) Students will be able to explain their method(s) in finding solutions to each problem.
Abstract
This activity set in the context of (a) finding the number of seats at an air show, (b) finding the number of lines to connect a certain number of dots, and (c) how many basketballs are there in a stack shaped like a pyramid of n layers, focuses on using finite differences to go from a recursive formula to a closed formula.
Problem Statement
Many natural occurrence can be describe be a polynomial function. Today we will look at three different type of problems which can be describe not only with a recursive formula, but a closed formula as well. In finding the closed formula we will use our tools of finite differences and systems of equations to solve.
Instructor Suggestions
(1) Discuss the problem statement with your students
(2) In small groups, allow the students to work on the activity sheet.
(3) Allow time for student to present their solutions and discuss their methods.
Materials
Linear??? Quadratic??? Cubic??? Activity Sheet, Calculators
Time
Introduction (5 min.), small group work (25 min.), discussion (15 min.)
Mathematical Concepts
Discrete Mathematics Concepts
Recursion, Closed form equations, method of finite differences.
Related Mathematics Concepts
Polynomials, Systems of Equations
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections, Algebra, Discrete Mathematics
Colorado Model Content Standards Addressed
Algebraic Techniques (2), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be introduced in an Algebra I or Algebra II course when working with systems of equations. In doing so, students must be introduced to recursion. In a Pre-Algebra course the students are generally able to find the recursive formula. Using a graphing calculator, they could find the closed form formula.
Further Investigation
Using a TI-82 or 81 calculator to find the line of best fit could be used to find the closed form formula.
Variations/Comments
References/Resources
Crisler, N., Fisher,P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W.G. Greeman and Company.
Herr, Ted, Johnson, Ken (1994). Crossing the River With Dogs. Berkeley, CA.: Key Curriculum Press.