Month: September 2019

OpenLab #4: Bridges and Walking Tours

The assignment below is due BEFORE CLASS on Thursday, October 10th (it is essential that you complete it before class, as we will be doing a class activity building on the assignment).

We are going to play a game creating walking tours of cities with bridges. ¬†We begin in the city of King‚Äôs Mountain, which is built on four land masses ‚Äď both shores of a river and two islands in midstream ‚Äď connected by a total of seven bridges (shown in green).

EXAMPLE 1:  Can you create a walking tour of the city that crosses every bridge exactly once?  You can begin anywhere you like, and end anywhere you like, as long as you cross each bridge just once.

Background –¬†Graph Theory

We can simplify the picture of King’s Mountain to make it easier to deal with:

The key elements of the map are the four land masses (let’s label them A, B, C, and D) and the seven bridges (p,q,r,s,t,u and v) (thanks to mathisfun.com for the images):

For the purposes of our problem, we can simply think about each land mass as a point (A, B, C, and D), and the bridges as lines connecting the points (p,q,r,s,t,u and v) – like this:

We call this kind of picture a graph – the points are called vertices and the the lines are called edges. ¬†Our goal of finding ‚Äúa walking tour that crosses each bridge once‚ÄĚ is now matter of tracing out all the edges without lifting our pencil (and without repeating any edge).

Assignment, Due Thursday 10/10 (beginning of class)

Warm up (This Warm Up is just for practice Рyou do NOT need to submit your answers Рsee below for the three-part Assignment to be submitted).  The following examples build on the example above.

EXAMPLE¬†2: If you are given the freedom to build one new bridge in King’s Mountain (“make one new edge in the graph”), can you do it in such a way the walking tour becomes possible? ¬†Do it!

EXAMPLE 3: If you are given the freedom to destroy one bridge (“erase one edge”), can you do it in such a way that the walking tour becomes possible? Do it!

EXAMPLE 4: Construct walking tours for each of the following graphs (or decide if it is impossible).


Assignment.  Your assignment has 4 parts.

PART 1.  For each of the four graphs below (G1 РG4), decide whether it is possible to create a walking tour crossing each bridge exactly once.  Post your solutions here (TO POST A SOLUTION, JUST LIST THE POINTS OF YOUR WALKING TOUR IN ORDER).  If it is not possible to create a create a solution, say so!Euler Paths G1-G4

PART 2. ¬†Challenge your friends: ¬†Now it‚Äôs up to you to build your own graph, and challenge your classmates to construct a walking tour (or to determine if it is impossible). ¬†It can consist of as many points as you wish, and as many bridges (edges) connecting them. ¬†You MUST label your points¬†“A, B, C…” etc. ¬†When you‚Äôre finished, decide for yourself if a walking tour crossing each bridge exactly once is possible. ¬†¬†Remember, the most challenging puzzles are the ones where the answer is difficult to determine. Post two puzzles in the comments. ¬†See the note ¬†“POSTING YOUR PUZZLE ONLINE” below for instructions on how to draw and share graphs online.

PART 3. ¬†Solve a friend’s puzzle. ¬†Leave a response to a friend’s posted puzzle, giving a solution. ¬†TO POST A SOLUTION, JUST LIST THE POINTS OF YOUR WALKING TOUR IN ORDER.

Example:
Here is a puzzle: http://sketchtoy.com/67467551
Here is a solution: (start at A) –¬†A, B, D, A, E, B, C, E

PART 4.  The third part of your assignment is to write a short paragraph (at least 3 sentences) responding to the following prompt.  Be sure to respond to each part:

Writing Prompt: ¬†Did you enjoy this assignment? Why or why not? ¬†Describe a connection between this assignment and our work in the class. ¬†(If you don’t believe there is a connection, try to imagine why we are doing this). ¬†Leave your response in the comments.

POSTING YOUR PUZZLE ONLINE. ¬†I recommend the site sketchtoy.com¬†– it allows you to draw something, then click “SAVE” and get a link to your drawing. ¬†You can post the link in a comment, and we’ll be able to click on it and view your drawing. ¬†¬†Don’t worry if it’s not pretty! ¬†For example, here is a graph that I drew (can you find a walking tour that crosses all edges?):¬†http://sketchtoy.com/67467556

 

Week 5 Assignments

Week 5 Assignments

NOTE 1: Exam #1 will take place on Thursday,  9/26 (first half of class).
NOTE 2: Next week Tuesday 10/1 there are no classes at City Tech.
NOTE 3: The following week Tuesday 10/8 there are no classes at City Tech.

Written work¬†‚Ästnone
WeBWorK РAssignment #4, due Thursday, October 3rd, at midnight. You are encouraged to start working on Assignment #5, which will be due one week later.
OpenLab РOpenLab #4: Bridges and Walking Tours due AT START OF CLASS on Thursday, 10/10.

 

Exam #1 Review Sheet is posted

The first exam will take place on Thursday, September 26. ¬†The review sheet is posted under “Classroom Resources/Exam Reviews” – it is the same as last year’s Exam #1 Review. ¬†Please let me know if you have any questions, or to report an error.

Regards,
Prof. Reitz

Week 4 Assignments

Week 4 Assignments

UPDATE: WeBWorK Assignment #4 will be due next week, Thursday 10/3.

Written work¬†‚Ästnone
WeBWorK РAssignment #3 and Assignment #4, due Tuesday, September 24th, at midnight.
OpenLab Рnone

STUDY Рfor your first exam, taking place on Thursday, September 26th, during the first hour of class.  A review sheet will be posted at least 1 week prior to the exam.

OpenLab #3: “Sentences”

Due Thursday, 9/19/19. ¬†For this week’s writing assignment, take a look at the picture below called “Sentences.” ¬†Read every sentence in the picture. ¬†As you read, pay attention to your own stream of consciousness – what are the thoughts that pop into your head? ¬†For full credit, respond to all 4 of the following items.

  1. Record¬†two observations¬†about the sentences in the picture – what do you notice / what’s something you find interesting / what popped into your head / what stood¬†out.
  2. Choose one of the sentences in the picture (do NOT choose the same sentence as anyone else). ¬†Type the sentence out “in quotes”, and then answer the following questions:
    a. Is it a statement (as discussed in class)?  Explain why or why not.
    b. Is it true or false? Explain in everyday English why or why not.
  3. Make up a sentence that you believe would fit into this picture.  Tell us the sentence, and then tell us whether it is a statement, and whether it is true or false.
  4. What connection (if any) does this assignment have to do with the work we are doing in class?

“Sentences” by Flickr user Eldeem

Week 3 Assignments

Week 3 Assignments

Written work ‚Äď Sec 1.8 p.29: 3, 5, 6, 8, due Tuesday, 9/17 (in class)
WeBWorK РStart on WeBWorK 3, (it will not be due until Tuesday, 9/24 at midnight).
OpenLab РOpenLab #3, due Thursday, 9/19 (at start of class).

 

OpenLab #2: Mathography

This assignment is due Thursday, September 12, at the start of class.

Assignment.  Choose ONE of the following two topics.  Write a reply to this post, responding to the topic.  Begin by telling us which topic you chose. (1-2 paragraphs).

Topics.

  1. Sometimes people can recognize a time when their opinion of math dramatically changed either for the better or the worse. If such a time happened to you, tell us about it.
  2. Choose an experience you had in which you suddenly understood a math concept (it could be any kind of math, from elementary school up through college).  Describe what happened.  Do you think you could explain it to others in a way that they could have the same flash of understanding?

Extra Credit. ¬†For extra credit, write a response to one of your classmates’ comments. ¬†Do you feel the same, or different? ¬†Did you learn anything? ¬†Did you get any ideas about teaching, or about learning?

Why are we doing this, anyway?¬† We are following two ideas that have come up already in class — things that may¬†not¬†seem related to learning math, but research shows that engaging in these activities can¬†dramatically¬†increase the amount that you learn, and change the way you learn it. ¬†The first is¬†writing¬†‚Äď something not typically associated with mathematics. ¬†When you express your ideas in words, it forces you to think them through very carefully, detail by detail. ¬†A great way to check and see if you really understand something is to try to explain it to someone else, either out loud or in writing. ¬†Example: if you know how to add fractions, try teaching it someone who doesn‚Äôt know how. ¬†The second is called¬†metacognition, or ‚Äúthinking about thinking.‚ÄĚ ¬†This happens when you think about what was going on in your head while you were working on a problem or trying to learn a new idea. ¬†What train of thought did you follow? ¬†Where did you get stuck, and what did you do next? ¬†What were you feeling at the time? and so on. ¬†Combining writing and metacognition can be a tremendously powerful tool in identifying the ways we learn best and the ways we make mistakes, and learning to improve. ¬†However, like any skill, it takes practice. ¬†That‚Äôs why we‚Äôre getting started by writing a little about our past experiences with mathematics.