Math FAQ

This past May, I was faced with a problem regarding health insurance. My employer offers two different plans, the Basic Plus Plan and the Premier. In the past, I was insured under the Premier plan with no cost to me. However, this summer I added my wife and child to my health insurance. I was faced with a decision. Should I cover my family under the Premier plan or change to family coverage under the Basic Plus Plan?

The two plans are described in the summary plan (PDF). Examining the two plans you’ll see that there are several differences for family coverage.

• The out of pocket maximum for the Premier Plan is $6000 per year versus $12,000 per year for the Basic Plus Plan.
• The annual deductible for the Premier Plan is $600 per family versus $1200 per family for the Basic Plus Plan.
• In general, the Premier Plan covers 80% of medical charges and the Basic Plus Plan covers 60% of medical charges.

The plans differ in cost too. Family coverage under the Premier Plan is $514 per month compared to $46 per month under the Basic Plus Plan. There are other difference regarding in network and out of network, but these facts are what I used to make my coverage decision.

Under a number of assumptions (which I’ll detail in later posts), I came up with the following graph.

This graph shows the total annual cost for each plan as a function of the medical charges incurred by the insured. As you can see, each plan is described by a piecewise linear function. If my family incurs less than $26,880 in medical charges during the year, the Basic Plus Plan is cheaper. For more than $26,880 in medical charges, the Premier Plan is cheaper.

In this project, I want students to create a graph and accompanying mathematics for other insurance plans. Luckily, it is easy to get health insurance quotes online. Humana One will give you an instance quote when you click on the Get Quote link on their webpage. Although they ask for a lot of personal information, only the state, zip code and birthday information about the insured are required. For a family coverage of a 46 year old in Arizona, this PDF demonstrates that a number of plans are available with different levels of coverage, different out of pocket maximums, and different deductibles. Another great source for health insurance quotes from many different companies is eHealthInsurance. I want my students in College Algebra to enter in their pertinent information and compare two plans like I did in the graph above.

Over the next week, I’ll post the assignments I create to help point students in the proper direction as well as the project letter I come up with.

Now that the Spring semester is over, I have been working on a number of new projects. You might wonder why I would ever need to update projects…the answer is simple. Not only do I have students who repeat my courses for one reason or another, but I have also learned that over time a number of technical memos for my projects are publically available. By changing the projects (especially those for the beginning of the semester), I keep them fresh for me and defeat attempts by my students to not do their own work.

The Blending Whisky Project is designed to replace two projects I have used for several semesters. In the Gas Prices Project and the Arsenic Removal From Drinking Water Project, students set up a system of two equations in three variables. Typically, they use matrices to solve the system of equations and then analyze the resulting solutions similar to what is described in the Dependent System Handout (http://www.pblpathways.com/projects/gas/dep_sys_app.pdf). The Blending Whisky Projects utilizes the exact same type of solution strategy, but with mixing Scotch Whisky from several different casks at a distillery.

For this project, students read a little backgound material on Scotch Whisky (http://www.pblpathways.com/projects/whisky/whisky_background.pdf) before reading the letter which gives the problem statement (http://www.pblpathways.com/projects/whisky/whisky.pdf). This introduces them to the concept of alcohol by volume (concentration) and some of the unusual terms used to describe the casks in Scotland.

Like the other projects, students must come up with an equation for the total volume in the Scotch Whisky blend (the easy equation) and another for the total amount of alcohol in the Scotch Whisky blend (the challenging equation). The system has many solutions and using the information on cask sizes, they can come up with a range of acceptable solutions. I have also included a long list (http://www.pblpathways.com/projects/whisky/whisky_list.pdf) of commercially available cask strength Scotch Whiskies with prices you might use. Most of the prices are for a 70 centiliter bottle.

I plan to use this project in the Fall semester for my online and face-to-face Finite Math classes. This topic area is very rich for math applications. While researching this project, I came up with information for two other projects. The prices mentioned above might be used to create a linear programming project. The evaporation of alcohol and water from the aging casks can be used to come up with a function modeling the alcohol by volume as a function of time. The model can be a ratio of linear functions or a ratio of exponential functions and leads to some great graphs. Look for these new projects in the next few weeks. If you have any questions or corrections, feel free to comment or drop me an email at dave at symbol pblpathways.com.