Satisfying Topics and Objectives and A Note on the Commonalities of Mat 1010s

More recently we applied the reasoned strategies that mathematics illustrates as a powerful tools that give us surprising strength for analyzing and conquering life's issues in the statistics and finance segments.

All Mat 1010s must satisfy the three designators, and must cover the statistics segment. Most also cover the financial mathematics segment. Hence, for most of the semester, most 1010s look alike. We differed slightly by taking an interdisciplinary approach, combining financial mathematics with political science and history in the Ben Franklin lab, and statistics with a real-world view via class activities, and the stock market.

Using File/New Window, open up a new web browser and enter WebCT. Compose a WebCT posting message to Dr. Sarah by choosing the forum containing you and Dr. Sarah and using the subject "peer review". You may copy the following and paste it into WebCT by highlighting the text, and going to Edit/Copy here, and then Edit/Paste in WebCT.

Final Exam

The final exam will cover What is Mathematics? and Finance.

While much of your grade was determined by lab work, writing projects, and participation, your performance on the final exam is important since it demonstrates your mastery of the recent course material and the What is Mathematics? theme. I do take improvement over the course of the semester into consideration.

One 8.5*11 sheet with writing on both sides allowed and one calculator mandatory. More details on this in class tomorrow, where you will also receive the take home portion of the test.

Geometry of the Earth and Universe

As a reminder, skim through the Shape of our Universe readings again.

What is a Mathematician?

Using the control key, do control and then a mouse click on this link Homer's blackboard equations and then open the link in a new window and read through it.


1. Write down the 2nd equation from Homer's blackboard (see the text below the pictures) involving the power of 12.
   
   
   
   
2. What is the statement of Fermat's Last Theorem? If you don't remember, look it up in your notes or on the web.
   
   
   
   
3. Use Fermat's Last Theorem (which Andrew Wiles proved was true) to explain why the 2nd equation on Homer's blackboard cannot be true.
   
   
   
   
4. Yet, we do not need the full strength of Fermat's Last Theorem. Explain how we can show the equation is false if we know only that Fermat's Last Theorem doesn't hold when the power is 3 (proven by Euler) or 4 (proven by Fermat).
   
   
   
   
5. Can an odd/even argument be used to justify that the 2nd equation is false? If so, then give the details, and if not, explain why.
   
   
   
   
6. Give a different mathematical reasons why the equation cannot hold. Hint: Use the fact that 9 divides a number if and only if 9 divides the sum of the digits.
   
   
   
   
7. I decide to calculate the 12th root of the left hand side on my TI-83 Plus calculator. When I do this, it tells me that the answer is 4472, which would imply that the equation was valid. This really did happen on my calculator. Resolve the apparent conflict.

Statistics

Stock Update

1. Take out your stock packet and do the stock update and show it to me so that I can check it off on your sheet. Recall that the stock market is unpredictable in the short term, but, in the past, over the LONG TERM (20 years), stocks that don't go out of business have historically performed better than savings accounts. The stock sheet has additional updates listed so that you can check it again over the long term (after the semester is over) if you like.
   
   

   Various Perspectives Using Statistics

   We have seen that the media often misrepresents statistics, and that one must dig deeply to fully understand the (often contradictory) issues involved.
   
   

   Bin Laden

   For example, you may have heard about the 25 million dollar reward offered for information leading to the capture of Bin Laden. After searching for +reward   +"Bin Laden" from www.google.com, I found that this reward is actually worded as "up to $25 million", which means that the reward could actually be any amount of money less than or equal to $25 million. By searching further, I happened across an NPR report on this that clarified that the state department will decide on the actual amount of the reward after Bin Laden is captured.
   
   

   Flu

   A second example relates to statistics of deaths from a disease. I searched with
   +2000   +"number of deaths"   +influenza   +"united states"
   to find out that there were
   91,871 deaths from influenza and pneumonia (which sometimes results from influenza) in the United States in 2000. I also found that there are
   94,800,000 flu/pneumonia cases each year.
   I found out that since the late 1940s, the flu vaccine has been the best way to prevent influenza or lessen its severity and that the vaccine prevents over 90% of the influenza/pneumonia deaths, but in 2004, there was a shortage of the flu vaccine. I was surprised by these statistics since I didn't realize that so many people died from complications of the flu. I then looked up the population of the United States via a search with
   +"census 2000"   +"total population"   +"United States",
   and obtained
   281,421,906 people in the United States.
   
   One statistic is that the
   # deaths from the flu / total population of the united states = 91,871 / 281,421,906 = 3.26...E-4 (E-4 is scientific notation meaning that I go 4 decimal places over to the left - ie .000326...) = .0326%. The is the percentage of the United States population that dies from influenza and pneumonia each year.
   Try this on your calculator and make sure you understand before moving on.
   
   There are also other perspectives to examine:
   
   
2. What is the percentage of cases of flu out of the total population of the United States? SHOW WORK.
   
   
3. What is the risk of dying from the flu (as a percentage) once you already have it? Calculate this by taking the number of deaths, dividing by the number of cases and then turning this into a percentage. SHOW WORK.

   Anthrax

   Let's see why people were so afraid of the anthrax mailing when the chances of getting the disease are so small: I used effective web searching techniques to find the number of anthrax cases in the United States since September 11th. There were 11 cases of inhalation anthrax infection and 5 deaths. There were 12 cases of cutaneous anthrax infection and 0 deaths.
   
   
4. Use this information and the population of the United States (found above) to determine the percentage of cases of anthrax out of the total population of the US. This statistic could be used to calm the public. Note that this is much, much smaller than the corresponding flu percentage. SHOW WORK.
   
   
   
5. Determine the risk of dying once you have the disease as a percentage. Notice that this is much, much larger than the percentage of risk of dying once you have the flu, which explains why people are so afraid. SHOW WORK.
   
   
   
6. We could perform a similar analysis of any disease. Name some other diseases that have been mentioned in the recent news that people are afraid of contracting.
   
   

What is Mathematics?

We have examined this theme and have also looked at many examples of different kinds of math and some applications of mathematics to real-life.


1. Read through Adding up Fuzzy Math: Teaching the Basics. This is a new reading.
   
   
   
   
2. Review the Creative Inquiry Lessons for Life list.
   
   
   
   
3. Review the Quantitative Literacy reading.
   
   
   
   
4. Review the What is Mathematics? reading.

Finance

Keep America Rolling

Since September 11th, you may have seen television commercials from the car companies offering 0% interest on new cars. GM was first to start this with its "Keep America Rolling" program, which offers interest-free financing on all new cars and trucks for certain loan periods. Zero-percent financing can save customers thousands of dollars over the life of the loan. Many of these loans were even extended into 2003. I used our effective web searching techniques in order to find the details on the length of an interest free car loan. By searching with
"Keep America Rolling Program" 0%
I found out that GM cancelled the program at one point, but brought it back again, and that the loan was often for 3 years. Let's assume that the loan would have compounded monthly at 8%. In the "Keep America Rolling" program, these 3 year loans charge 0% over the 3 years.

Loan Payment Formula:

Monthly Payment =



Loan (rate/12)

__________________

1 - (1 + rate/12)(- years*12)


Total Interest Over the Life of a Loan: Interest = Monthly Payment * 12 *years - Loan


1. Use the loan payment and interest formulas to determind how much a customer will save - ie calculate the monthly payment assuming an 8% interest loan of $20,000 compounded monthly for 3 years and then calculate the interest of the life of the loan - this interest is the amount the customer would save in the "Keep America Rolling" program. SHOW WORK
   
   
   
   

   Ben Franklin

   Superscripts in Word Open up Microsoft Word and type in
   1000(1+rate)100=131000
   Highlight 100 and under Format, release on Font. Then check the box next to superscript and hit OK.
2. Show Dr. Sarah your professionally formatted equation so that she can check off that you were able to do this. You will need similar professional formatting in your report.
   
   Lab answers Take out your Ben Franklin lab and look at your answers for. Compare your answers to the following:
   With a lump sum formula there are no additional principal payments made, while a periodic payment formula would be used when principal is put in each year or month. While it seems as if money is coming into this account periodically (each year in this case), in actuality, Ben Franklin only put $1000 pounds in as a lump sum principal. The rest of the new money comes in as INTEREST on this lump sum. When we derived the lump sum formula in class, we added the annual earned interest to the amount that was in the account at the beginning of the year to solve for the savings at the end of the end of the year. We repeated this process with each successive year and found a general lump sum formula. By looking at the derivation of the formula, it is clear that earned interest is already accounted for within the lump sum formula, so this is the appropriate formula to use in this case.
   
   The rate of approximately 5% is not the actual earned rate of the fund because it was difficult to find as many borrowers to completely utilize the funds. Hence, even though the money that was loaned out was loaned at 5%, the money that was not loaned out was earning 0%. Without enough borrowers, you can't count on that interest that was expected to be paid by them. This would bring down the average earned rate of the entire fund, which is determined by the average earnings of each bit of money in the fund. In addition, all of the borrowers did not pay back the money they owed. This affects the rate by making the ending amount in the account less than what was expected. Not paying back the money borrowed drops the total sum of the account in two ways. First, the amount is subtracted from the total, and second, the amount that should have been in there is not gaining any interest. This means that there is a loss of all this money, resulting in a negative earned rate. When this is averaged into the 0% or 5% rates (see above), this also brings down the average earned rate of the fund. To summarize, because of these problems, the fund earned less money than it would have if the money had been completely utilized, and so the earned interest rate was lower. For example, if half the money was lent out at 5% and was successfully repaid at this rate, but the other half of the money was not earning any interest, then the average earned rate of the entire fund would be 2.5%. If 75% of the money was lent out at 5% and was successfully repaid at this rate, but the remaining 25% of the fund was not earning any interest, then the earned rate of the entire fund would be .75(5) +.25(0) = 3.75%. In other words, the average earned rate of the fund is the weighted average of how much each part of the fund earned.
   
   Next go to the main web page and look at homework for the rest of the week. As usual, I'm happy to answer any questions you have as I make my way around the room, in office hours, or on WebCT. We'll talk more about the final in class tomorrow.