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

**Abstract**

This is another in the series I have developed on modeling health insurance with piecewise linear functions. In this project, students model an insurance plan and prove that the model is continuous.

Content Area – Calculus

Time Frame – 5 to 6 weeks with Mini Lectures

Published – March 16, 2012

Keywords – piecewise functions, linear functions, continuity

**Project Content**

**Scaffolding Resources**

**Definitions** (DOC | PDF) This handout contains definitions of various terms you might find in a health insurance plan.

**Technology Assignment: Collect and Organize Data** (DOC | PDF) In this tech assignment, students learn how to format an Excel worksheet while collecting data about about a health insurance plan. They will put the data into an Excel worksheet with data from the plan they have chosen.- To help understand the terminology regarding health insurance, students fill out a table (DOC | PDF) of the relevant charges and costs in the plan they chose. The links above are to Word and PDF forms for the table. Although the students can save their data in the form using more recent versions of Adobe Reader, a small number have difficulty doing this. For those students, I recommend they fill out the Word table instead.

**Technology Assignment: Calculate the Total Annual Cost** (DOC | PDF) In this tech assignment, students modify the Excel worksheet from the technology assignment above to create a table for their health plan. Upon completion, they will have a table containing several values for total medical charges and the corresponding total annual medical costs. They will also have a graph of the function.

**Technology Assignment: Calculate the Total Annual Cost (Alternate Version)** (DOC | PDF) This tech assignment may be used to replace Find and Understand Costs and Calculate the Total Annual Cost above. These assignments used Excel extensively to create a table of values. In this assignment, students find three points on the piecewise linear model for health insurance. Then they make a graph of the function using Excel.

**Technology Assignment: Find Piecewise Linear Models** (DOC | PDF) This tech assignment demonstrates how to find the piecewise linear formulas for their health plan. It also shows how to use the equation editor in Word 2007/2010 to enter the model into a Word document.

**Notes**

- Students want to simply look at the graph of the model and say, “See, it is continuous!” I do not accept graphical evidence as the sole evidence for continuity. They must utilize the limit definition of continuity at the point where the deductible is met and where the out of pocket max is reached. At all other points, it is sufficient to not that the model is linear and linear functions are continuous.Typically, I need to mention this in class and on my class blog when we cover continuous functions.
- The Basic Plus Plan I use as an example is very easy to understand for most students. However, a majority of my students choose plans with copays that can be complicated. That is why I have the Deconstruction handouts below. The major hurdle for them to grasp is the charges the doctor send to the insurance company and the cost the insured party actually pays (the copays). Many students want to pay the charge AND the copays. They also have a difficult time excluding those charges when meeting the deductible.
- The website eHealthInsurance offers the ability to find and compare health insurance plans in all US states.
- Students have had trouble understanding what the entries in the handout for the Technology Assignment: Calculate the Total Annual Cost mean. I wrote a separate handout on
**Deconstructing the Premier Plan**(DOC | PDF) to help students see where each entry comes from. This is an updated version of this handout from the original handout. - I produced a video to help students fill out the table in Technology Assignment: Calculate the Total Annual Cost.

- If you would like to see a sample technical memo or research poster produced by a student for this project, you will need to email me.. Make sure you email me from your professional email account so I can verify your identity.
- The technology assignments have been updated to remove Mathtype instructions.
- Functions that model income taxes are also great for proving continuity. As in-class activities, I work with the Continuous Tax Worksheet or Handout: Continuity of a Piecewise Linear Tax Function. This forces them to realize that limits are required for proving continuity.