# PBL Pathways

##### Content Area

Calculus, Nonlinear Functions, Modeling

##### Time Frame

4 to 5 weeks with Mini Lectures

December 4, 2009

##### Keywords

Rational functions, limits, modeling

#### Project Title

Students and Teachers 2

#### Author

David Graser, Yavapai College, Prescott, AZ (David_Graser@yc.edu)

#### Abstract

This project grew out of the highly publicized reports of teacher shortages in public schools. The data for this project are from the National Center for Education Statistics (NCES). Students look at national and state data and form student to teacher ratios for the time period from 2000 to 2006. Students initially attempt to model the student to teacher ratio by calculating the ratio in several years from 2000 to 2006. Polynomial models fail to give the asymptotic behavior. Once they understand this, students model the students and teachers individually with polynomials or exponential Functions and then use these functions to form a ratio function. Care must be taken to choose degrees that are consistent with the expected asymptotic behavior. All students complete a common component on the national data and then apply what they learn to state data that they are assigned.

#### Project Content

Project Letter: DOC | PDF
Project Data: Excel | PDF (Updated 8/16/09)

#### Scaffolding Resources

Technology Assignment : Scatter Plot DOC | PDF - In this tech assignment, students learn how to make a scatter plot on a graphing calculator and in Excel.

Technology Assignment: Regression Model  DOC | PDF - In this tech assignment, student use the scatter plot above and find linear and quadratic models of the student to teacher ratio data.

Technology Assignment: Rational Model DOC | PDF - To account for the asymptotic behavior that some datasets display, students model the number of students and teachers separately. Then the functions are combined to create a ratio function.

Technology Assignment: Limits at Infinity DOC | PDF - This tech assignment helps students to compute the asymptotic behavior of the ratio function they have produced.

#### Notes

• Most states have ratios that are decreasing. and above the NEA recommended ratio of 15 to 1. The expectations are that these states will continue to decrease, but level off. By choosing the degree of the polynomial appropriately, students can guarantee that their function will level off.
• Some models will lead to vertical asymptotes for years in the future. When this happens I suggest to the students that they try exponential models for both the students and the teachers. This could be done in the first place if your emphasis is the limits and not rational functions.