**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.

- Content Area – Calculus, Nonlinear Functions, Modeling
- Time Frame – 4 to 5 weeks with Mini Lectures
- Published -December 4, 2009
- Keywords – rational functions, limits, modeling

**Project Content**

**Project Letter** (DOC | PDF)

**Project Data** (Excel | PDF)

**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.

**How to Make a Scatter Plot in Excel (includes Regression too!)** (Video)

**Graphing a Function in Excel** (Video)

**Adding Axes Titles To Your Excel Graph** (Video)

**Adding a Chart Title to your Graph** (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** (Video)

**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.