# PBL Pathways

##### Content Area

Calculus, Nonlinear Functions, Modeling

##### Time Frame

4 to 5 weeks with Mini Lectures

January 18, 2011

##### Keywords

Rational functions, limits, modeling

#### Project Title

Doctors and Nurses

#### Author

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

#### Abstract

This project is designed to use the same strategy as Students and Teachers 2. Students model the data for the number of nurses and teachers as a function of time with a polynomial function and then combine these models into a rational function. Once this model is formed and gives reasonable behavior, they locate when the ratio model gives a specific output and the location of any horizontal asymptotes.

#### Project Content

Project Letter: DOC | PDF (Updated 2/17/11)
Project Data: Excel | PDF

#### Scaffolding Resources

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

Technology Assignment: Regression Model  DOC | PDF - In this tech assignment, students use the scatter plot above and find linear model of the number of nurses and doctors as a function of time.

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 in the previous technology assignment. In this assignment, the functions are combined to create a rational function to model the nurse to doctor ratio. Students are given the option of using Mathtype or the equation editor in Word to create the formula of the rational function.

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

Video: How to Make a Scatter Plot in Excel (includes Regression too!)
Video: Graphing a Function in Excel
Video: Changing a Graph in Excel
Video: Graphing a Function on a TI-83 (or TI-84)
Video: How to Make a Scatter Plot on a TI-83 (or TI-84) includes Linear Regression too!
Video: Quadratic Regression on the TI-83 (or TI-84)
Video: Documenting Your Solution to an Equation Using Mathtype
Video: Documenting the Limit of a Rational Function in Mathtype
Video: Use Your TI-83 (or TI-84) to Generate a Table for a Limit

#### Notes

• For most states, the numbers of doctors and nurses is increasing overall which simplifies the analysis. For most states, linear models for the doctors and nurses give good results.
• 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 nurses and the doctors. This could be done in the first place if your emphasis is the limits and not rational functions.
• The data could be formed into a ratio in the first place and then used to introduce polynomial modeling in a college algebra class.