Reviews for the Final Exam have been posted on the right side of the blog. We will review Graphing and Finance on Wednesday and Probability and Statistics on Friday. We will be reviewing Voting on Monday - you should bring both the Voting Review from the Final Exam Review folder and the worksheet title Voting - More Problems from the voting theory worksheet folder for class Monday.
The CAS is offering reviews this week (April 24-27) in Remington 117 from 4pm-6pm.
Monday - Graphing
Tuesday - Probability
Wednesday - Statistics
Thursday - Finance
I have my regularly scheduled office hours this week and students may also join me in Agenstein 123 MW 9-9:50am to review for the final.
Next week (May 1-5) my office hours will be as follows:
Monday, May 1: 10:30am - 12:30pm
Wednesday, May 3: 9:00am - 11:00am
or by appointment
Saturday, April 22, 2017
Wednesday, April 19, 2017
Week 13 - Voting
This week, we've discussed some fairness criteria on Monday, and started looking at how we might compare plurality versus plurality with elimination on Wednesday. You will receive full participation credit for this project by simply being in class on Wednesday and Friday and participating. I have posted the project and the Voting Methods summary on the right side of the blog.
Recall that the state of Maine voted in November to switch to plurality with elimination (called ranked choice voting) for many of their elections. Here are some interesting articles about this switch and about plurality vs plurality with elimination
Maine's Ranked Choice Voting: It's not Plurality (has a couple interesting videos)
Top 5 Ways Plurality Voting Fails
Maine Ranked Choice Voting Initiative, Question 5(2016) - Ballotpedia
Portland Press Herald
Podcast Meeting with Supporters on Both sides of Question 5 (scroll down to find this)
The Reasons for Maine to Adopt Ranked Choice Voting are Unconvincing
Yes on '5': No More Lesser of Two Evils in Maine
Maine became the first state in the country . . . to pass ranked choice voting
Can you find any interesting articles about Question 5 on Maine's ballot? How would you have voted on Question 5?
Recall that the state of Maine voted in November to switch to plurality with elimination (called ranked choice voting) for many of their elections. Here are some interesting articles about this switch and about plurality vs plurality with elimination
Maine's Ranked Choice Voting: It's not Plurality (has a couple interesting videos)
Top 5 Ways Plurality Voting Fails
Maine Ranked Choice Voting Initiative, Question 5(2016) - Ballotpedia
Portland Press Herald
Podcast Meeting with Supporters on Both sides of Question 5 (scroll down to find this)
The Reasons for Maine to Adopt Ranked Choice Voting are Unconvincing
Yes on '5': No More Lesser of Two Evils in Maine
Maine became the first state in the country . . . to pass ranked choice voting
Can you find any interesting articles about Question 5 on Maine's ballot? How would you have voted on Question 5?
Friday, April 14, 2017
Voting Theory Homework
There will be no Webwork assignments for the Voting Theory unit. Instead, there will be only paper homework assignments, which will be collected in class. The first assignment is posted on the link on the right side of the blog under Voting Worksheets, and will be collected in class on Wednesday, April 19. A selection of solutions will be available for you to check your answers in the CAS or my office before Wednesday.
Tuesday, April 11, 2017
Week 12 - Exam 3
Remember that our third exam is scheduled during class on Wednesday, April 12. Your Finance Worksheets 1-4 will also be due in class on April 12. There are two review sheets with answers linked on the right side of the blog. Make sure that you bring your calculator with you to class for the exam!
On Friday, we will begin our final unit with a discussion of voting theory. We will be discussing four ways of determining the winner of an election on Friday: Plurality, Plurality with Elimination, Pairwise Comparison, and Borda Count. Next week, we will discuss how we might compare these methods and determine one is "better" than another.
Challenge Problem: (Due Monday, April 17) In preparation for next week, research "fairness criteria for voting". In particular, describe the following criteria and give an example of an election and a voting method that fails to satisfy each criteria. (i.e., Give the results of a sample election under one of the voting methods and describe why this election violates the criterion. Do not use the examples in the notes.) Be sure to cite your sources.
1. Majority Criterion
2. Condorcet's Criterion
3. Monotonicity Criterion
4. Independence of Irrelevant Alternatives Criterion
On Friday, we will begin our final unit with a discussion of voting theory. We will be discussing four ways of determining the winner of an election on Friday: Plurality, Plurality with Elimination, Pairwise Comparison, and Borda Count. Next week, we will discuss how we might compare these methods and determine one is "better" than another.
Challenge Problem: (Due Monday, April 17) In preparation for next week, research "fairness criteria for voting". In particular, describe the following criteria and give an example of an election and a voting method that fails to satisfy each criteria. (i.e., Give the results of a sample election under one of the voting methods and describe why this election violates the criterion. Do not use the examples in the notes.) Be sure to cite your sources.
1. Majority Criterion
2. Condorcet's Criterion
3. Monotonicity Criterion
4. Independence of Irrelevant Alternatives Criterion
Tuesday, April 4, 2017
Week 11 - Amortized Loans
This week, our focus will be on amortized loans. Most loans that you take out will be amortized loans - mortgages, student loans, car loans, furniture loans, etc. Last week we learned how to determine the payment on an amortized loan. This week, we'll be focusing on answering several questions.
1. If you know how much you can afford to pay each month and you know the rate and term of the loan, how much can you afford to borrow?
2. Suppose you are several years into a given loan and you are offered to refinance to another loan at a different rate. What does it mean to refinance? Will this particular offer save you money?
3. Suppose you pay extra each month on your amortized loan. How much will you save in interest by paying ahead? How soon will you pay off the loan?
These are all very interesting questions to ask when you are looking at a loan. Question 1 simply requires you to solve for P in the formula below. Questions 2 and 3 will require a little more thought before we can solve them.
\[ P = R\left[\dfrac{1-\left(1+\frac{r}{m}\right)^{-mt}}{\frac{r}{m}}\right] \]
We will be learning how to use the formula to answer the questions above by hand. However, there are many calculators online to help answer these questions and more about taking out a loan. Below are links to some calculators that you may find useful when you want to consider taking out a loan in the future.
Bankrate.com
Amortization Schedule Calculator
Mortgage Payoff Calculator
Cost of Living Calculator
Is it Better to Rent or Buy?
We will only be considering fixed rate loans in all of our examples. But lenders may also offer you an adjustable rate mortgage (ARM). As the name suggests, with an adjustable rate mortgage, your rate can change, depending on market rates. These loans can be risky - if rates go up, so does your monthly payment.
Challenge Problem: (Due Wednesday, April 12) Determine what kind of job you want to have after you graduate. Research salaries for this job in your dream town. Using a home affordability calculator, determine how expensive a house you could afford based on this salary. Then, go to zillow.com, and choose a home to purchase (in your budget) in your dream town. Looking at current interest rates, choose a loan. Given this loan, calculate the following by hand. (You may use the online calculator to check your answers, but you must also work out the answers by hand.)
1. Your monthly payments.
2. The amount of interest you would pay by just making the minimal monthly payments.
3. The amount of interest you would pay by paying an extra $100 each month and how soon you would be able to pay off the loan.
4. The amount of interest you would pay by paying an extra $250 each month and how soon you would be able to pay off the loan.
Include the resources you used to determine your future salary, home affordability, loan terms, and the zillow page for the home you chose.
1. If you know how much you can afford to pay each month and you know the rate and term of the loan, how much can you afford to borrow?
2. Suppose you are several years into a given loan and you are offered to refinance to another loan at a different rate. What does it mean to refinance? Will this particular offer save you money?
3. Suppose you pay extra each month on your amortized loan. How much will you save in interest by paying ahead? How soon will you pay off the loan?
These are all very interesting questions to ask when you are looking at a loan. Question 1 simply requires you to solve for P in the formula below. Questions 2 and 3 will require a little more thought before we can solve them.
\[ P = R\left[\dfrac{1-\left(1+\frac{r}{m}\right)^{-mt}}{\frac{r}{m}}\right] \]
We will be learning how to use the formula to answer the questions above by hand. However, there are many calculators online to help answer these questions and more about taking out a loan. Below are links to some calculators that you may find useful when you want to consider taking out a loan in the future.
Bankrate.com
Amortization Schedule Calculator
Mortgage Payoff Calculator
Cost of Living Calculator
Is it Better to Rent or Buy?
We will only be considering fixed rate loans in all of our examples. But lenders may also offer you an adjustable rate mortgage (ARM). As the name suggests, with an adjustable rate mortgage, your rate can change, depending on market rates. These loans can be risky - if rates go up, so does your monthly payment.
Challenge Problem: (Due Wednesday, April 12) Determine what kind of job you want to have after you graduate. Research salaries for this job in your dream town. Using a home affordability calculator, determine how expensive a house you could afford based on this salary. Then, go to zillow.com, and choose a home to purchase (in your budget) in your dream town. Looking at current interest rates, choose a loan. Given this loan, calculate the following by hand. (You may use the online calculator to check your answers, but you must also work out the answers by hand.)
1. Your monthly payments.
2. The amount of interest you would pay by just making the minimal monthly payments.
3. The amount of interest you would pay by paying an extra $100 each month and how soon you would be able to pay off the loan.
4. The amount of interest you would pay by paying an extra $250 each month and how soon you would be able to pay off the loan.
Include the resources you used to determine your future salary, home affordability, loan terms, and the zillow page for the home you chose.
Thursday, March 30, 2017
Make-up Assignment for Friday March 31
We will not have class on Friday, March 31. Instead, you should complete the worksheet below. You may work in groups of 3-4 people. I will collect the completed worksheets in class on Monday, April 3. The videos below may also be useful.
Worksheet for March 31
Worksheet for March 31
Tuesday, March 28, 2017
Week 10 - Annuities and Amortized Loans
Last week, we looked at what happens when you take a fixed amount of principal, P, and invest it in an interest-bearing account for a set amount of time, t. We looked at simple interest and compound interest. You will need to be very comfortable with those formulas when it comes time for the exam. So far, the formulas you've seen have been
Simple Interest \[ A=P(1+rt)\]
Compound Interest, compounded m times per year \[A=P(1+\frac{r}{m})^{mt}\]
Compound Interest, compounded continuously \[A=Pe^{rt}\]
This week, we'll be looking at what happens when, instead of depositing once into your account, you make regular deposits of equal amounts for a set amount of time. This is called an annuity. The formula for the amount, A, in the annuity after t years, if you make m regular payments of R dollars per year is given by
\[A=R\left[\dfrac{(1+\frac{r}{m})^{mt}-1}{\frac{r}{m}}\right] \]
When we know A and want to solve for R, we sometimes call the account a sinking fund. On Wednesday, we'll start looking at amortized loans. Your school loans, car loans, and mortgages are all usually amortized loans. The formula we'll be using for this will be
\[PV = R\left[\dfrac{1-(1+\frac{r}{m})^{-mt}}{\frac{r}{m}}\right]\]
where PV (sometimes left as just P) is the present value of the loan (i.e., the amount you are borrowing).
We are taking a very simplistic look at these accounts. Other factors such as inflation, tax rates, and cost of living also will have a big impact on your investments and loans. The financial crisis in 2008 was precipitated by many factors. Your parents' or grandparents' retirement accounts may have been quite affected by the events surrounding 2008.
Challenge Problem: (Due Monday, April 3) Research the events leading up to the financial crisis in 2007-2008 and/or the subprime mortgage crisis that occurred during this same period. Some questions you might consider are: What caused so many banks to close in 2008? What is a subprime mortgage? What was the investment bank Lehman Brothers and how did its collapse occur? What impact did the collapse of Lehman Brothers have on global markets? What is Fannie Mae and Freddie Mac? What role did they play in the subprime mortgage crisis?
Write at least one page (typed) on your findings, being sure to appropriately cite your sources.
Simple Interest \[ A=P(1+rt)\]
Compound Interest, compounded m times per year \[A=P(1+\frac{r}{m})^{mt}\]
Compound Interest, compounded continuously \[A=Pe^{rt}\]
This week, we'll be looking at what happens when, instead of depositing once into your account, you make regular deposits of equal amounts for a set amount of time. This is called an annuity. The formula for the amount, A, in the annuity after t years, if you make m regular payments of R dollars per year is given by
\[A=R\left[\dfrac{(1+\frac{r}{m})^{mt}-1}{\frac{r}{m}}\right] \]
When we know A and want to solve for R, we sometimes call the account a sinking fund. On Wednesday, we'll start looking at amortized loans. Your school loans, car loans, and mortgages are all usually amortized loans. The formula we'll be using for this will be
\[PV = R\left[\dfrac{1-(1+\frac{r}{m})^{-mt}}{\frac{r}{m}}\right]\]
where PV (sometimes left as just P) is the present value of the loan (i.e., the amount you are borrowing).
We are taking a very simplistic look at these accounts. Other factors such as inflation, tax rates, and cost of living also will have a big impact on your investments and loans. The financial crisis in 2008 was precipitated by many factors. Your parents' or grandparents' retirement accounts may have been quite affected by the events surrounding 2008.
Challenge Problem: (Due Monday, April 3) Research the events leading up to the financial crisis in 2007-2008 and/or the subprime mortgage crisis that occurred during this same period. Some questions you might consider are: What caused so many banks to close in 2008? What is a subprime mortgage? What was the investment bank Lehman Brothers and how did its collapse occur? What impact did the collapse of Lehman Brothers have on global markets? What is Fannie Mae and Freddie Mac? What role did they play in the subprime mortgage crisis?
Write at least one page (typed) on your findings, being sure to appropriately cite your sources.
Wednesday, March 22, 2017
Week 8 - Reminders for Friday
I appreciate your patience this week with my missing class Monday and Wednesday. If you did not turn in your worksheet in class today, I will accept it at the start of class on Friday. If you would like to make corrections to a worksheet that you turned in, I will accept that on Friday as well.
Be sure that you watch the video from Monday’s class if you haven’t already.
Your midterm grades include all the homework submitted up to this point, 2 exams, participation, and 2 comments on the blog. If you have not commented on the blog yet, please make sure that you start doing so.
Below are some links to articles that describe the power of compound interest and why it's better than simple interest. Click on the links and comment with your thoughts.
I will be having additional office hours this week on Thursday from 12:15 pm - 2:00pm. If you have questions over the material covered on Monday or Wednesday or questions about your midterms, please come see me.
Friday's office hour at 9:00am will be moved to Agenstein 126.
I will be handing your exams back on Friday or you may pick them up in my office on Thursday.
Sunday, March 19, 2017
Week 8 - Simple and Compound Interest - Class Makeup Video
This week, we will begin a new unit on Financial Math. Please make sure you print out the notes and bring them with you to class. We will start this week with a discussion of interest - simple interest and compound interest. There will be a lot of formulas in this unit, and you will need to practice with them to get comfortable using them. I will provide you with the formulas (unlabeled) for the exam, so you need to focus on identifying what formulas apply to what problems. In addition, it will be very important to remember your calculator for class every day in this unit - you don't want to have to do these calculations by hand!Those of you in MAT099 should have been reviewing logarithms and exponentials in that class recently. We will be using logarithms to help solve problems in this unit; if you need additional review of logarithms, please see me. The webpage below is also a good source of review for problems involving logs and exponents.
Exponential and Logarithmic Equations
We will be starting with simple and compound interest on Monday. Unfortunately, due to a nasty stomach bug that's been taking down my family one-by-one this weekend, I need to cancel class on Monday. To make up for this, I have recorded a video lecture over simple and compound interest. Please watch the video below and then complete the worksheet linked below. I will be collecting the worksheet in class on Wednesday, March 22.
Challenge Problem (Due Monday, March 27): A PayDay loan company charges $25 in (simple) interest for every $100 borrowed for a 14 day loan. The total borrowed, plus interest must be paid back at the end of 14 days. Suppose you want to borrow $400 to pay for your Spring Break trip. How much will you be required to pay back at the end of 14 days? What is the interest rate on this loan?
Exponential and Logarithmic Equations
We will be starting with simple and compound interest on Monday. Unfortunately, due to a nasty stomach bug that's been taking down my family one-by-one this weekend, I need to cancel class on Monday. To make up for this, I have recorded a video lecture over simple and compound interest. Please watch the video below and then complete the worksheet linked below. I will be collecting the worksheet in class on Wednesday, March 22.
Challenge Problem (Due Monday, March 27): A PayDay loan company charges $25 in (simple) interest for every $100 borrowed for a 14 day loan. The total borrowed, plus interest must be paid back at the end of 14 days. Suppose you want to borrow $400 to pay for your Spring Break trip. How much will you be required to pay back at the end of 14 days? What is the interest rate on this loan?
Friday, March 3, 2017
Lecture Video
I made a video last semester over the most recent material covered in MAT110(E). I have linked the video below if you would like to use this to review this material.
Remember that our Probability and Statistics exam will be in class on Friday, March 10. An exam review is posted on the right side of the blog.
Remember that our Probability and Statistics exam will be in class on Friday, March 10. An exam review is posted on the right side of the blog.
Monday, February 27, 2017
Week 7 - Normal Distributions
This week we will be continuing with the statistics notes with a discussion of normal distributions. You will need your calculators and the z-score table from the end of the notes in class with you every day. The Math is Fun website has a nice interactive Standard Normal Distribution and introduction to the Standard Normal Distribution. I recommend that you play around with this to get a better understanding of what we are discussing in class.
There is a review worksheet over the standard normal distribution linked on the right side of the blog. We will start on these problems in class on Monday; you should complete them for the start of Wednesday's class.
Challenge Problem: (Due Monday, March 6) Scores on a standardized test were normally distributed with mean \[\mu = 70\] and standard deviation \[\sigma=11.\] What must you score to be in the top 5% of scores?
There is a review worksheet over the standard normal distribution linked on the right side of the blog. We will start on these problems in class on Monday; you should complete them for the start of Wednesday's class.
Challenge Problem: (Due Monday, March 6) Scores on a standardized test were normally distributed with mean \[\mu = 70\] and standard deviation \[\sigma=11.\] What must you score to be in the top 5% of scores?
Monday, February 20, 2017
Week 6 - Statistics
We will be reviewing counting and probability in class today (Monday). I have extended the second WebWork assignment for Probability and Statistics to Tuesday, February 21. Make sure you are keeping up with the WebWork homework.
Please make sure you print the Counting and Probability Review linked on the right side of the blog and bring it to class with you today. Wednesday, we will move in to statistics. You'll definitely want to make sure you have your calculator in class with you this week and next. Statistics is the study of data. We will start by looking at some ways to organize numerical data and compute some basic numerical summaries.
Challenge Problem: (Due Monday, February 27) The website www.fivethirtyeight.com computes all sorts of statistics. They also have a weekly "riddle" that uses math, logic, or probability in an interesting way. Choose one "Riddler" question and read the solution. (Or try to solve it yourself.) Were you able to understand the solution? What didn't you understand? Were you surprised by the answer? Print out the riddle and its solution from the website and write a paragraph discussing the riddle and the questions above. You can go directly to the archive of "Riddler" articles at https://fivethirtyeight.com/tag/the-riddler/.
Please make sure you print the Counting and Probability Review linked on the right side of the blog and bring it to class with you today. Wednesday, we will move in to statistics. You'll definitely want to make sure you have your calculator in class with you this week and next. Statistics is the study of data. We will start by looking at some ways to organize numerical data and compute some basic numerical summaries.
Challenge Problem: (Due Monday, February 27) The website www.fivethirtyeight.com computes all sorts of statistics. They also have a weekly "riddle" that uses math, logic, or probability in an interesting way. Choose one "Riddler" question and read the solution. (Or try to solve it yourself.) Were you able to understand the solution? What didn't you understand? Were you surprised by the answer? Print out the riddle and its solution from the website and write a paragraph discussing the riddle and the questions above. You can go directly to the archive of "Riddler" articles at https://fivethirtyeight.com/tag/the-riddler/.
Tuesday, February 14, 2017
Week 5 - Counting and Probability
This week we will finish up methods of counting and start with probability. You'll need to apply your methods of counting to many of the probability problems, so make sure that you've got the counting methods down this week. Make sure that you are bringing your calculator to class every day.
On Monday, we counted the number of 5 card hands that can be dealt from a standard 52 card deck. In one of my sections, I had a student ask about counting cards. The podcast This American Life has a fun episode about counting cards for Blackjack. You can listen to it here.
Challenge Problem (Due Monday, February 20): Option 1: In the game of Euchre, 5 cards are dealt from a deck consisting of the 9, 10, J, Q, K, A of each of the four suits. Find the probability of getting a hand containing three of a kind.
Option 2: Listen to the podcast linked above and write a 2-3 paragraph summary of the podcast.
(You may only choose one option to complete.)
On Monday, we counted the number of 5 card hands that can be dealt from a standard 52 card deck. In one of my sections, I had a student ask about counting cards. The podcast This American Life has a fun episode about counting cards for Blackjack. You can listen to it here.
Challenge Problem (Due Monday, February 20): Option 1: In the game of Euchre, 5 cards are dealt from a deck consisting of the 9, 10, J, Q, K, A of each of the four suits. Find the probability of getting a hand containing three of a kind.
Option 2: Listen to the podcast linked above and write a 2-3 paragraph summary of the podcast.
(You may only choose one option to complete.)
Monday, February 6, 2017
Week 4 - Exam 1 and Counting
We have our first exam of the semester on Monday. This concludes our unit on Graph Theory. Wednesday, we will start discussing Probability and Statistics. Make sure that you print the notes for this unit. There are a lot of pages in these notes, so you might want to print multiple slides per page. (The link to the notes is available on the right side of the blog or on the right side of the WebWork.) You'll also want to make sure you bring your calculators to class for this unit. We'll kick off the Probability and Stats unit with some discussions of methods of counting this week. Consider the question below.
Look at the menu of your favorite pizza shop. How many different types of crusts, sauces, and toppings are there? How many different three topping pizzas could you make if a pizza consists of a crust, sauce, and three toppings? What about four topping pizzas?
These are the kinds of questions we will be able to answer after Friday.
Challenge Problem (Due WEDNESDAY, February 15): Print the menu from your favorite pizza shop. How many different types of crusts, sauces, and toppings are there? How many different three topping pizzas could you make if a pizza consists of one crust, one sauce, and three toppings? What about four topping pizzas? Include the pizza menu with your answer.
Look at the menu of your favorite pizza shop. How many different types of crusts, sauces, and toppings are there? How many different three topping pizzas could you make if a pizza consists of a crust, sauce, and three toppings? What about four topping pizzas?
These are the kinds of questions we will be able to answer after Friday.
Challenge Problem (Due WEDNESDAY, February 15): Print the menu from your favorite pizza shop. How many different types of crusts, sauces, and toppings are there? How many different three topping pizzas could you make if a pizza consists of one crust, one sauce, and three toppings? What about four topping pizzas? Include the pizza menu with your answer.
Friday, February 3, 2017
Wednesday, February 1, 2017
No Class on 02/03/17
I will not be on campus on Friday February 3, so class and office hours are canceled for this Friday. To make up for the missed class, you should complete the worksheet linked below. You may work in groups of up to 4 members and turn in one solution set per group. I will collect your work on Monday February 6. I will still be available via email on Friday and throughout the weekend.
Out of Class Assignment for 02/03/17
Don't forget that Exam 1 will be in class on Monday February 6. I will also collect your graded Graph Theory Worksheets 1-4 on Monday. You can grade your worksheets in the CAS.
Out of Class Assignment for 02/03/17
Don't forget that Exam 1 will be in class on Monday February 6. I will also collect your graded Graph Theory Worksheets 1-4 on Monday. You can grade your worksheets in the CAS.
Monday, January 30, 2017
Week 3 - Spanning Trees and Kruskal's Algorithm
This week we will be finishing up the graph theory material with a discussion of trees and spanning trees. A tree is just a connected graph which contains no circuits. We will be concerned with finding subgraphs of graphs which contain all the original vertices and are trees. These are called spanning trees. Kruskal's algorithm gives us a way to find a spanning tree of minimal weight in a weighted graph. This algorithm should feel familiar to you; it is very similar to the Side Sorted Algorithm, except now our goal will be to end up with a spanning tree rather than a Hamiltonian circuit.
Exam 1 will be in class on Monday, February 6. Make sure to bring your graded worksheets to class with you on Monday to turn in. I have posted an exam review on the right hand side of the blog and below. Please print this out and bring it to class with you on Wednesday February 1. (Don't forget you must be logged in to your MWSU email account to access this.)
Graph Theory Exam Review
Challenge Problem: A connected graph G has 18 vertices. How many edges does a spanning tree of G have? How many vertices does a spanning tree of G have? What can one say about the number of edges G has?
Exam 1 will be in class on Monday, February 6. Make sure to bring your graded worksheets to class with you on Monday to turn in. I have posted an exam review on the right hand side of the blog and below. Please print this out and bring it to class with you on Wednesday February 1. (Don't forget you must be logged in to your MWSU email account to access this.)
Graph Theory Exam Review
Challenge Problem: A connected graph G has 18 vertices. How many edges does a spanning tree of G have? How many vertices does a spanning tree of G have? What can one say about the number of edges G has?
Wednesday, January 25, 2017
Beginning of the year survey
Please fill out the survey below honestly. You must be logged into your Missouri Western account to access the survey, but your answers will be collected anonymously.
Tuesday, January 24, 2017
Graph Isomorphisms and Making Mistakes
I've received several questions about problems 12 and 13 in the first WebWork graph theory assignment. These questions give you a graph and ask you to determine which of the three graphs below is the same graph, represented differently. It then asks you to give the one-to-one correspondence that identifies the two graphs. This correspondence is called a graph isomorphism. These problems can be tricky, and require some thought and creativity to find the right correspondence. However, it's a valuable exercise to struggle with problems sometimes, and it makes finding the right solution all the more sweet!
Imagine if you had to do this same thing, but for a graph with thousands of vertices! It might be a daunting task to imagine. Of course, you wouldn't want to have to try to find this isomorphism by hand, but even asking a computer to do this would be difficult. In fact, recent research has been trying to determine if there is a way to determine if two graphs are isomorphic "quickly". A computer scientist, Laszlo Babai, from the University of Chicago, announced in November 2015 that he had found an algorithm to determine if two graphs were isomorphic in quasipolynomial time. But, earlier this month, another mathematician, Harold Helfgott, posted that he had found an error in the proof. It can be frustrating for you to find that you've answered a problem incorrectly in MAT110(E); imagine if you had announced to all of your colleagues that you had solved a problem, only to find that your solution was incorrect. This happens to mathematicians all the time. (I, too, have submitted a paper to a mathematics journal claiming to have proven something, only to find that there was a subtle error in my reasoning. I've still been unable to correct my error!) Even when we get the answer wrong, there is value in the struggle that it takes to work with a problem. Maybe those skills that you learned to work with this problem will help you with a problem in the future. Maybe you'll remember how to correct it just a little bit better because you had to struggle to find the right answer. Don't focus too much on worrying about making mistakes - everyone will make them - instead, make sure you learn from your mistakes and are ready to apply what you learned in the struggle to the next problem.
Don't worry about Babai - he's corrected the problem in his algorithm.
You can read more about graph isomorphisms and Laszlo Babai's algorithm at the American Mathematical Society's blog on math and the blog Computational Complexity.
Monday, January 23, 2017
Week 2 - Fleury's Algorithm and Hamiltonian Circuits
This
week we will be finishing up with Euler circuits and moving into Hamiltonian Paths and Circuits and weighted graphs. Fleury's algorithm will give us a systematic way to find an Euler circuit in a graph that contains one. Not every graph has an Euler circuit, but we can Eulerize a graph by duplicating existing edges in such a way that the resulting graph contains an Euler circuit. Why do you think it might be useful to Eulerize a graph?
A
weighted graph is just a graph with numbers (weights) on the edges. Our
goal will be to use weighted graphs and Hamiltonian circuits to solve the
Traveling Salesman Problem. We will see three
algorithms for solving this: The Nearest Neighbor Algorithm, The Side-Sorted
(or Best Edge) Algorithm, and the Repetitive Nearest Neighbor Algorithm.
We will also discuss how to solve this using Brute Force. You will
need to memorize each of these algorithms.
Don't forget
to keep up with the homework on WebWork and the associated worksheets. Once you complete a worksheet, you should take it to the CAS to grade and correct your work. I will collect all the graded/corrected worksheets in class on the day of the unit exam.
Challenge
Problem: (Due Monday, January 30) The
picture below is the floor plan for the Plaza Level of the Nelson-Atkins Museum in Kansas City. Doorways are represented by a break in the wall. Is it possible to start in one room on this floor, travel through every doorway, closing the door behind you, and return to where you started? Use graph theory to answer this question. Your solution should include a graph, explain how the graph is related to the floor plan, and explain how you used graph theory to determine if such a path exists. If the path exists, give it; otherwise explain why it is not possible.
Thursday, January 19, 2017
How to Email Your Professor
Every semester, I receive an email that looks something like the following:
hey mrs. mccune. did i miss anything important in class today? also, i don't know how to do number 2.
There are several issues with this email. First, you should address your instructors as Professor or Dr, unless they have asked to be addressed differently. (I am Dr. McCune.) Second, if you must miss class, please acknowledge that you are responsible for the material covered in your absence. I am happy to give a brief recap of what you missed, but please remember to first consult the syllabus and the notes before composing your email.
If you need help with a specific homework problem, please be clear about which problem you need help with. For example, "I need help with number two on the webwork Graph Theory set 1". You should tell me what you've tried so far and specifically what part of the problem is giving you trouble. It is often useful to include an attachment with a picture of your work so that I can best point you in the right direction to solve the problem.
Finally, be sure to sign your email with your name, course number, and section. It is especially difficult to respond to an email if I don't know who the sender is.
Wikihow has a nice summary of how to email a professor, which you should apply when emailing any of your professors on campus.
How to Email a Professor
hey mrs. mccune. did i miss anything important in class today? also, i don't know how to do number 2.
There are several issues with this email. First, you should address your instructors as Professor or Dr, unless they have asked to be addressed differently. (I am Dr. McCune.) Second, if you must miss class, please acknowledge that you are responsible for the material covered in your absence. I am happy to give a brief recap of what you missed, but please remember to first consult the syllabus and the notes before composing your email.
If you need help with a specific homework problem, please be clear about which problem you need help with. For example, "I need help with number two on the webwork Graph Theory set 1". You should tell me what you've tried so far and specifically what part of the problem is giving you trouble. It is often useful to include an attachment with a picture of your work so that I can best point you in the right direction to solve the problem.
Finally, be sure to sign your email with your name, course number, and section. It is especially difficult to respond to an email if I don't know who the sender is.
Wikihow has a nice summary of how to email a professor, which you should apply when emailing any of your professors on campus.
How to Email a Professor
Thursday, January 12, 2017
Week 1 - Welcome to MAT110(E)
Welcome to MAT110/MAT110E! You should be enrolled in MAT110 if you have a 22 or higher on the ACT or passed the math placement exam (MPE) with a 70 or higher. Otherwise, you should be enrolled in MAT110E and MAT099. Keep in mind that if you fail MAT099, you will fail MAT110E as well, regardless of your exam and homework scores in MAT110E. Hence, it is imperative that you attend and actively participate in your MAT099 section.
Our semester will be broken into three major units: Graph Theory, Financial Math, and Probability and Statistics. We will have a fourth shorter unit over Voting Theory at the end of the semester. I hope that you will enjoy learning a little bit about each of these topics and how they are used in your everyday life and the world around you. I'm looking forward to a great semester!
We will be kicking off the semester this week with an introduction to graph theory. A graph is a collection of vertices (think dots) and edges (think lines) between the vertices. We can use graphs to study many things in the world around us. For example, a graph can represent streets and intersections from a map (see The Traveling Salesperson Problem), computer networks, social networks, or even be used to study DNA (see A Graph Theoretical Approach to DNA Fragment Assembly). By the end of this week, you should know what a graph is and be able to describe several properties of a graph.
A little bit about me: I am in my fifth year as an Assistant Professor of mathematics here at Missouri Western State University. Before coming to MWSU, I spent a year as a visiting assistant professor at Ashland University. I received my PhD from the University of Nebraska-Lincoln in 2011. (Go Big Red!) My husband is also a mathematician at William Jewell College. We have a 2.5 year old daughter and a 9-month old son who are both a bundle of energy.
Please ask for help as soon as you are having trouble with this class. You can visit me in my office (Agenstein 135K). Peer tutoring is also available (for free) through the Center for Academic Support.
Challenge Problem #1: (Due at the start of class on Monday, January 23) Sketch several examples of graphs. Determine the degree of the vertices in each graph. When you add the degrees of all the vertices, you will always get an even number. Why is that?
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