Category Archives: Finite Math

Regression Using Google Sheets

Like other spreadsheets, Google Sheets may be used to find a regression model for data. Following a few simple steps, we can graph a set of data in a scatter plot and find the corresponding model.

Suppose we have the demand data show in the table below.

sheets_reg_15

To find a linear model for the Average Price per Gallon as a function of the Weekly Demand, we need to make a scatter plot of this data and add the linear regression model to it. Continue reading Regression Using Google Sheets

Updates to the Medical Insurance Project

I have been absent from this page for a while. Over the winter break and throughout the beginning of the semester I have been updating the Medical Insurance Project and adapting it for my calculus class. In addition, I have a new project for college algebra on college costs. In this post I want to outline the changes I have made to the Medical Insurance Project.

Last semester, my students struggled with taking their plan details and creating a table of charges and costs. It seemed like they needed an intermediate step between entering the plan details in Excel and creating the table in Excel. In addition, creating models for two plans was quite a bit of work. To help mitigate these troubles, I have rewritten the project and named it Medical Insurance 2. This project includes a new technology assignment, Find and Understand Costs, where they fill out a table of charges and costs by hand. This forces them to confront the unfamiliar terms before they ever begin their cost calculations in Excel. Instead of finding two insurance plans to model, they only find one plan and compare it to the Basic Plus Plan. They should simplify the project a bit, but still preserve the same solution strategy.

In Medical Insurance 3, calculus students choose a health plan and model it with a piecewise linear function. Using three technology assignments, they complete the same strategy as college algebra students complete in Medical Insurance 2. Once they have their model, they use the definition of continuity to prove that it is continuous. This amounts to proving that the function is continuous at the charge where the deductible is met and the charge where the out of pocket maximum is reached. Great example of continuity in a real world context.

Both of these project utilize similar technology assignments. Since these assignments take some effort to create, reusing them makes better use of my time. My college algebra students are currently completing Medical Insurance 2. My business calculus students completed Medical Insurance 3 as their first project of the Spring semester. All of the students are frustrated with the non-mathematical content. This keeps me very busy in office hours and answering many emails each day. However, I get a lot of feedback about how important they think it is to understand all of this. After all, the debate over health care in the US and how to pay for it is far from over. Maybe these projects can do a small part to contribute to better understanding of the issues for my students.

Tech Assignment Updates

If you have been using my technology assignments from the Arsenic Removal from Drinking Water  or Doctors and Nurses Projects, you know that for projects early in the semester I give my students extensive directions on how to use the equation editor Mathtype. I use Mathtype because my college bought a site license years ago. It entitled us to use Mathtype 6.5c forever. It integrated with Word 2003 perfectly and even Word 2007 when it came out. However, it does not play nicely with Word 2010. It is possible to make it work, but it does not work as seemlessly as it did with older versions of Word.

Normally this would not be a big deal except many students are beginning to purchase Word 2010 as a part of Office 2010. I could certainly recommend that the college purchase a new site license for the latest version of Mathype, but in these trying economic times such purchases are not likely to go through. Instead, I decided to reevaluate the equation editor that is provided in Word 2007 and 2010 to see if it would be a reasonable substitute for Mathtype.

After several weeks of using the equation editor in Word 2007 and 2010, I decided that it was easy for students to use and produced professional looking equations. It certainly does not produce equations as well as Mathtype, but for College Algebra, Finite Math and Calculus students, it is good enough.

I have been adding new material in the Technology Assignments to cover the usage of the Word equation editor in addition to the existing Mathtype instructions. The affected technology assigments are

  • Technology Assignment: Row Operations on the Ti-83  DOC | PDF in the project Arsenic Removal from Drinking Water
  • Technology Assignment: Rational Model DOC | PDF in the Doctors and Nurses Project

Future projects with Mathtype components will also include a corresponding Word equation editor component also.

Modeling Health Insurance – Project Under Development

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.

Blending Whisky – A New Project for College Algebra and Finite Math

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.