I just finished giving this presentation at the AMATYC Conference in Austin, TX. In this 17 minute video, I show how to use tables to recognize the objective function for optimization problems in calculus. I do a number of traditional problems as well as examples on economic order quantity and economic lot size from Business Calculus.
Last week I gave a presentation on the Medical Insurance Project at the Fall 2011 ArizMATYC Conference at Estrella Mountain College. This video summarizes the posts I have been making on this project and is a good introduction to the project.
The Medical Insurance Project now has a web page with all of the resources for it in one place. You can click on the link above or access it the the Projects page on the menu at the top of this page. In the notes for this project there is a link to an example of a technical memo for this project. You’ll need to contact me to be able to access this PDF since the file is password protected.
My next conference presentation will be at the AMATYC Conference in Austin, Texas. I’ll be presenting on “A Different Strategy for Solving Optimization Problems in Calculus” on Thursday, November 10 from 11:20AM to 12:10PM. Hope to see you there! If you are not able to make it, I’ll post a Flash video of the session immediately afterward.
In the last two posts, I described the first steps in modeling the total annual cost as a function of the total medical charges. In the post Medical Insurance Project, I described the basic idea behind the project and how students can collect and organize information on health insurance plans. In the post Medical Insurance Project – Calculate the Total Annual Cost, I described how students create table of values including several total medical charges and the corresponding cost.
Once the students have an appropriate table, they need to graph the data.
Excel Table for Basic Plus Plan
In the next technology assignment, student learn how to make a line graph of the total medical charges versus the total annual charges.
Technology Assignment: Graph the Total Annual Cost DOC | PDF
Upon completing this assignment, students will have a graph for each health insurance plan like the ones below.
Total cost as a function of medical charges.
To compare these graphs, it is convenient to have them on the same graph. In the next technology assignment, students put these two graphs together and find the piecewise linear functions corresponding to each health plan.
Technology Assignment: Find Piecewise Linear Models DOC | PDF
With the graphs together, it is much easier to compare the total costs.
Both graphs for the health plans on the same axes.
Once students have put these graphs together, hopefully they will make the next logical step which is to find the point of intersection using the piecewise functions.
If the students get to this point successfully, they are over the major hurdles of this project. Next they need to document their solution strategy. I’ll describe the document they turn in for this project in my next blog post.
In an earlier post, I discussed how to have students collect information to model two insurance plans. The second step in modeling the total cost in the Medical Insurance Project is to get the students to create a table of annual medical charges and the corresponding total annual cost using the information they have gathered on their two plans. Using Excel with formulas and fills, they complete a table for each plan.
For 17 and 18 year old students, this is a daunting task. Not only are they struggling with using Excel, they are also struggling with the terminology. In the technology assignment handout, I take great pains to spell out exactly how to use Excel to create the table.
Technology Assignment: Calculate the Total Annual Cost DOC | PDF
As you can see, each step is carefully outlined. You would think that it would be a very simple task for students to create the table for each plan.
Oh how wrong I was about that! Students focus on the steps of using Excel and completely shut down thinking about where the numbers come from. Here are a few of the glaring errors I saw the first time I used this assignment.
After giving this assignment for the first time, I had to clean up the carnage so that they could continue working on the project successfully. I created another handout for them on the specific entries in the Premier Plan.
Handout: Deconstructing the Premier Plan DOC | PDF
In this handout I tried to link the formulas in the cells with the assumptions in the plan. Using this, they could modify the formulas to fit their own plans. It is not uncommon that an assignment does not work as anticipated. Sometimes it is just too much too handle in one assignment or your assumptions about your students is way off base. For this assignment, my students had no idea how insurance worked so assuming they could track the numbers through the handout easily was doomed. However, next semester this might work just fine for a group of older, more experienced, students.
Even after I had posted the Deconstructing the Premier Plan Handout, there were a few stragglers still fighting with their Excel spreadsheets. I decided to make a short video on putting together the table of values for total annual cost as a function fo the total medical charges.
Video Constructing the Table for an Insurance Plan
Next semester I plan to put together a short worksheet which the students can complete in class and a corresponding video for online students. This worksheet will give them three different charges (one from each of the three regions on the model) and ask them to caclulate the corresponding charge by hand. This way they will have some familiarity with the concepts and numbers before they start putting the formulas into Excel. I’ll post that worksheet once it is complete.
In a previous post (Modeling Health Insurance – Project Under Development), I described a choice I had to make regarding health insurance during the summer. In short, I needed to choose between a high premium/low deductible plan and a low premium / high deductible for my family. The analysis I completed on the two plans led me to a new project for College Algebra students.
To present the components of the project, let me explain my thought process in developing the project. The goal of the project is to have students develop models for two medical insurance plans of their choosing and to compare them. The models should give the total annual cost of each plan to the insured as the total medical charges increase. With these models they construct a graph to compare the plans like the one below.
Based on the graph, students can make a choice of the two insurance plans based on their own lives. As a template, I lead them through a potential solution using the information from my two health plans, the Basic Plus Plan and the Premier Plan. They will need to follow the same strategy with two plans of their own choosing.
This means the first task they need to accomplish is selecting two health plans. The technology assignment
Technology Assignment : Collect and Organize Data DOC | PDF
is customized to their unique data (state, age, dependents). This accomplishes one of my requirements of projects. Each student does a unique project based on the theme of comparing health plans.
Once they have this information in hand, they use Excel to document their plan. The primary goal of using Excel in this assignment is to get them familiar with the interface and to demonstrate how to format cells. Later assignments utilize Excel’s capabilities more fully.
When I grade the assignment, I make sure that they have at least given me the most basic information about their plans such as copays, coinsurance, deductibles and out of pocket maximums.
I used this assignments this semester and students had little trouble completing it. There were many comments in class about why some plans are cheaper than others. I was surprised at how many students were completely ignorant about how insurance works. I guess I should not be too surprised since my classes are mainly 17 to 21 year olds with a handful of older students. Opening their eyes to the important issues and terms regarding healthcare was very satisfying.
In my next blog post I’ll talk about the second technology assignment I give students. In this assignment they make a table of increasing medical charges and the corresponding costs to the insured.
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
Future projects with Mathtype components will also include a corresponding Word equation editor component also.
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 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 (http://www.pblpathways.com/projects/whisky/whisky.html) is designed to replace two projects I have used for several semesters. In the Gas Prices Project (http://www.pblpathways.com/projects/gas/gas_prices.html) and the Arsenic Removal From Drinking Water Project (http://www.pblpathways.com/projects/arsenic/arsenic.html), 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@pblpathways.com.
For the last several years PBL Pathways has been a fairly static website…a few new projects each semester, a couple posts about conferences I attended, ect. For the future, I am going to use this blog style entry page in an attempt to be more dynamic.
A number of users have requested more details on how I run my classes, how I come up with new projects, and other similar information. Next semester (starting in August) I’ll document what is going on in my classrooms (I’ll be finished with my sabbatical year) and what I am doing to update projects.
You’ll still find all of the pages you are used to on this site…they can be accessed through the navigation links across the top. Some of the pages maybe spruced up a bit, but they are basically the same.
I’ll caregorize my blog posts so you can find the information on the blog easier…I’ll have a Projects category to hightlight new and existing projects, and categories for each of the classes I teach, College Algebra, Finite Math, and Survey of Calculus. I may add more categories as the need arises.
