1. Quantitative Reasoning
TMM011

2. What is College level? Examples
Each of these are examples of where I would like to go in a college-level QR course.
Not where the students will need to start.

3. Numeracy Example Personal Finance- Loan Payments
Melinda is purchasing a car and needs to finance $12000 to make the purchase. She needs to decide between two loans, both have an APR of 5%, but one has a term of 3 years while the other has a term of 6 years.
Compare the two loans by:
Estimating the monthly payment for each loan.
Estimating the interest she would pay for each loan.
Estimating the total amount she would pay for each loan.
Writing a few QR sentences to Melinda explaining the advantages and disadvantages of each loan.

4. Numeracy Example Answers
$ by the average balance method for 3 years.
b. $ for the 6 year loan.
c. $900 for the 3 year, $1800 for the 6 year.
d. $12,900 for the 3 year, $13,800 for the 6 year.
e. Hi Melinda, if you choose to purchase the car with the 6 year loan at a 5% APR, then your payment will be less each month, about $ as compared to $ for the 6 year loan. However, you will pay twice as much interest over the length of the loan on the 6 year loan, about $1800 as compared to $900. In addition, you may want to think about the value of the car after you have used it for a few years. Cars depreciate in value rather quickly, so your car may depreciate so fast that for the 6 year loan you may owe more than it is worth for a considerable amount of time. This means that you would not be able to sell it if you had to and receive enough money to pay off the loan.
Goal in part e is for the student to relate both the context and the math so that a reader unfamiliar with the situation can understand.

5. Numeracy Example – Building an Index
Gary and Anne are employees of a cell phone provider. Gary is a sales associate and Anne is his sales supervisor.  Their company offers a lot of opportunities for growth in the organization.  For example, Anne was promoted to supervisor from associate about three years ago.  Gary is starting some long-term planning and wants to decide if he wants to stay with this company and work for promotions, or if he wants to explore other options with another company or try a new career.  
Use the salary information provided below to create a salary index for each of the three positions.  Check with your instructor about which year to use as your base year.  Round each answer to the nearest tenth of a percent.
[Instructors:  Consider this opportunity to discuss the changes to an index by changing the base year.  You might also ask different groups to use different base years, then facilitate a discussion after the students complete the table and are able to compare results from different groups.  Or, you may ask each group to use 2009 as the base year.]

6. Numeracy Example – Building an Index
Salaries of Sales Personnel ( )
2009
2010
2011
2012
2013
2014
2015
2016
Sales Associate
$21,500
$22,600
$22,800
$22,000
$23,100
$23,600
$24,700
$25,800
Sales Supervisor
$46,500
$47,100
$48,700
$47,300
$50,500
$49,600
$50,700
$52,600
Regional Sales Manager
$75,000
$79,500
$77,400
$72,700
$77,100
$79,000
$76,100
$81,000
Salary Index Solutions (Base Year = 2010)
Here I have already filled in the table
2009
2010
2011
2012
2013
2014
2015
2016
Sales Associate
95.1
100.0
100.9
97.3
102.2
104.4
109.3
114.2
Sales Supervisor
98.7
103.4
100.4
107.2
105.3
107.6
111.7
Regional Sales Manager
94.3
97.4
91.4
97.0
99.4
95.7
101.9

7. Numeracy Example – Building an Index
 1. Use quantities in your index table to explain your observations and
ideas about what Gary should know about how the salaries change over time.
[Answers may vary.  Some observations are: the associate clearly has the greatest
relative increase from 2009 to 2016; the manager’s salary is the only one that ever
drops below its 2009 amount; every year from 2011 to 2015, the manager’s salary
was less than its 2010 amount; the manager had a 6% raise after 1 year,
and the associate reached a 6% raise after just 2 years.]  
2.  Make a graph of the indices that you created.  What are the advantages
and disadvantages of using this graph to interpret how the salaries change over time?
[If your students used different years for the base year, have students compare the
different graphs.  This graph shows that after about 5 or 6 years, the associate has the
greatest relative increases in pay.  The regional sales manager has the overall lowest
relative increases in pay.  One disadvantage is the absolute differences in pay between the
positions are lost.]
Numeracy Example – Building an Index

8. Numeracy Example – Building an Index

9. Conditional Probability Example
Fifty men with a family history of prostate cancer participate in a study, and have their PSA level tested. Biopsies are then performed on all of them, to gauge the effectiveness of PSA screening, and determine when biopsies should have been performed. (Outside of the study, biopsies wouldn’t be performed on all of them). The following table shows the results of the study.
(a) Suppose the cutoff level is set at 2. What is the sensitivity for this group? What is the specificity?
[For the 10 men with prostate cancer, 7 are above the cutoff, so the sensitivity is 7/10 = 70%.  For the 40 men without prostate cancer, 25 are below the cutoff, so the specificity is 25/40=62.5%.]
PSA LEVEL
# WITHOUT PROSTATE CANCER
# WITH PROSTATE CANCER
3 and above 6 4
2-2.99 9 3
1-1.99 12 2
0-0.99 13 1

10. Conditional Probability Example (continued)
(b)  Suppose the cutoff level is raised to 3.  What is the sensitivity now for this group?  What is the new specificity?  
[(b)  For the 10 men with prostate cancer, now only 4 are above the cutoff, so the sensitivity deteriorated to 4/10=40%.  For the 40 men without prostate cancer, 34 will not be flagged, so the specificity improved to 34/40=85%.]
 (c)  What happened to sensitivity when the cutoff level was raised from 2 to 3?  Why?  Is this good or bad?
[(c)  It dropped from 70% to 40%, because 3 of the men with prostate cancer were between the two cutoffs, so they went from being told cancer was likely, to being told cancer was not likely.  This is unfortunate in their case, because their cancer is no longer detected at that PSA cutoff level.  In a normal situation, they wouldn’t be given a biopsy.]   

11. Conditional Probability Example (continued)
(d)  What happened to specificity when the cutoff level was raised from 2 to 3?  Why?  Is this good or bad?
[(d)  It raised from 62.5% to 85%, because 9 of the men without prostate cancer were between the two cutoffs, so they went from being told cancer is likely (and being unnecessarily worried) to being told cancer was unlikely.  This is good in their case, because they are well, and normally wouldn’t be given a biopsy.]
 (e)  Suppose the PSA cutoff level is dropped to 1.  Using just the PSA cutoff level, out of the 50 men, how many healthy men would have been told incorrectly that they may have cancer.  How many men with cancer would have been told incorrectly that they are probably well?  How many of the men would have received a correct diagnosis?
[ (e)  First, 27 healthy men will be above the cutoff and told they may have cancer.  Next, only 1 man with cancer will be below the cutoff and told that he is probably healthy.  Finally, the remaining 22 men will get the correct diagnosis, based on that PSA cutoff level.]   

12. An Example of Modeling PROBLEM SITUATION: WHICH OPTION?
Suppose you have an unexpected expense of $100 and do not have the money to pay for it. Usually banks do not make loans for this small of an amount of money. In this lesson, you will consider two options for short-term loans.
Investigate the two options below by considering how the unpaid balance accumulates over time.
Prepare a written presentation that you will make to another group discussing which option is best, assuming these are your only two options. Be sure to include equations, tables, and/or graphs to support your choice.

13. An Example of Modeling Option 1: Payday Loan
A payday loan is a small, short-term loan that is intended to cover a borrower's expenses until his or her next payday. Payday loan centers are often located in convenient locations such as shopping centers. Usually, they only require the borrower to have a checking account and a job. It is also assumed that the loan is short term, meaning it will be paid back in a few weeks. In many states, payday loans are somewhat regulated, with a maximum Annual Percentage Rate (APR) of 390%. Payday loan center justify this high interest rate because the loan is short term and high risk.
Although these loans do not fit the models we have studied exactly (because of loan fees and payment schedules), we can approximate what this loan costs the borrower.
There are also restrictions (usually around 45 days) for the amount of time you can hold the loan. However, many people just move the balance from center to center, letting the unpaid amount add up for quite a long time. That is, they use a new loan to pay off the old loan, and the balance continues to accumulate.

14. An Example of Modeling Option 2: Borrow from a Friend
You have a friend who offers to lend you the money. You agree to repay the original amount plus $10 for each week you owe her money.
[ At this point in the modeling curriculum we would expect students to create an exponential model for Option 1 and a linear model for Option 2. They would be expected decide which option is best , under what conditions, and to use tables or graphs to support their choice.

15. An Example of Modeling P = 100(1.075)t P = 100 + 10t
where t = time in weeks

16. Example of a Project