Author: Jonas Reitz (Page 4 of 5)

In-Class Group Project Activity 9/29/16 – Puzzle Making

September 29, 2016 / / 0 Comments

Group Activity (25 min).  Get into your groups (group assignments appear below), arrange your chairs in a circle, and take 25 minutes to:

1.  Share your responses to OpenLab #4.  Compare your answers to the given examples (do you all agree on the solutions?).  Now share your own puzzles with the group, and discuss.

2.  Create three new puzzles, as tricky as possible (try to stump Prof. Reitz!).

Group work due after 25 minutes:  Each group will hand in a sheet of paper with the names of the groups members, the date, and the three new puzzles created by the group.  You do not need to submit solutions, but you do need to be able to solve the puzzles on request.

Reflection:  To be completed individually after group work is complete, and submitted on paper with your name and the date.  Take 5 minutes to write on the following prompt:

Briefly reflect on the process of working in a group by responding to each of these points:
1.  Describe something you learned.
2.  Describe something you contributed to the group.
3.  How did today’s work change your understanding of your assigned game?

 

Group Assignments

Group 1: Hanan, Ismail, Jeron
Group 2: Jose, Violanda, Kevin, Carter
Group 3: Tyniqua, Sonam, Ahmad
Group 4: Gary,Leonardo, Armando

 

OpenLab #5: Lockhart’s Lament

September 29, 2016 / / 21 Comments

In 2002, a mathematician named Paul Lockhart wrote an essay called “A Mathematician’s Lament,” a passionate criticism of mathematics education in America.  It has become widely known among mathematicians and mathematics educators – not everyone agrees with everything he says (though many do), but everyone seems to have something to say about “Lockhart’s Lament,” as it is called.  For this week’s assignment, you will read a short excerpt (three pages) from his essay and respond to the prompts below.

Assignment (Due Friday, 10/14/16). Your assignment has three parts:

First, read the section titled “Mathematics and Culture” (pages 3-5) in Lockhart’s essay, (click here).  If you’re interested, I encourage you to read more, starting at the beginning – but this is not required.

Second, write a response to what you read and post it in the comments below.  Your response should be at least 300 words. Your response should represent your own thoughts and opinions on what you read, and can include responses to any or all of the following:

  • What is one thing that you agree with in the reading? Explain why.
  • What is one thing that you do not agree with? Explain.
  • Choose one quote that you think stands out in the reading.  Give the quote, and explain why you chose it.
  • Have you ever had an experience of mathematics as art?
  • On page 5, Lockhart describes mathematics in schools today as “heartbreaking”.  What do you think he means?  Do you agree? How do your own math experiences in school compare to his description?

Third, and most important, I want you to write down a conjecture about the Bridges and Walking Tours game, and bring it with you to class on Friday 10/14 (do NOT post it here).  Consider Lockhart’s example of a triangle drawn inside a rectangle.  He described the process of playing around with this picture, until he arrives at the basic idea for calculating the area of a triangle.  He contrasts this with a traditional math class, in which the formula is given to students without providing them any opportunity to explore the problem on their own.  The bridges and walking tours game is a little like the triangle-rectangle picture – it’s fun to play around with, but you may not be sure what the point is.  You’ve had a chance to play with it a bit, and try some different challenges.  Now what?  Your job is write down a conjecture (a guess!) or a question about your game. If you could have one question answered about your game, what would it be? If you wanted to be a master of your game, and be able to solve any challenge that was given to you, what would you need to know? Write down a conjecture or question about your game, and bring it with you to class on Friday 10/14 (do NOT post it here).

Here is an example: Let’s imagine that you have just been introduced to the game Tic-Tac-Toe.  After playing it for a while, you might come up with one of the following:
Conjecture: The person who goes first always wins.
Question: Is the corner the best move, or the center?
Conjecture: It’s impossible to win, no matter who goes first.

ps.  Paul Lockhart retired from being a first-rate research mathematician in order to teach math at a private elementary school here in Brooklyn, Saint Ann’s School, where he says “I have happily been subversively teaching mathematics (the real thing) since 2000.”

Week 6 Assignments

September 27, 2016 / / 0 Comments

HEADS UP: Next week, there are no classes on Monday 10/3 or Tuesday 10/4.  Also, Thursday 10/6 will run according to a Monday schedule.  Because of this, our class will not meet next week!

HEADS UP: The week after next there are no classes on Monday 10/10, Tuesday 10/11, or Wednesday 10/12.  However, Friday 10/14 will run according to a Tuesday schedule.   This means our class will meet on back-to-back days, Thursday 10/13 and Friday 10/14.

Written work – None.
WeBWorK – Assignment #5, due Thursday, October 13th, at midnight.
OpenLab – OpenLab #5, due Friday, October 14th, before class.

Exam #1 grades are posted

September 25, 2016 / / 0 Comments

Hi everyone,

Your grades for Exam #1 are posted on the Grades page (email me if you don’t remember the password decided on in class).  Current grades for the Homework and OpenLab portions of the class also appear.  Exams will be returned on Tuesday.

Best regards,
Prof. Reitz

OpenLab #4: Bridges and Walking Tours

September 22, 2016 / / 39 Comments

The assignment below is due BEFORE CLASS on Thursday, September 29th (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’s Mountain, which is built on four land masses – both shores of a river and two islands in midstream – 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 “a walking tour that crosses each bridge once” is now matter of tracing out all the edges without lifting our pencil (and without repeating any edge).

Assignment, Due Thursday 9/27 (beginning of class)

Warm up (This Warm Up is just for practice – you do NOT need to submit your answers – see below for the three-part Assignment to be submitted).  The following examples build on EXAMPLE 1 above.

WARM-UP 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!

WARM-UP 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!

WARM-UP 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.  Leave a comment responding to EXAMPLE 4 (above), telling us for each one of the 8 graphs whether a walking tour is possible or not.  You only have to state whether it is possible or impossible for each one.

PART 2.  Challenge your friends:  Now it’s 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’re 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

September 20, 2016 / / 0 Comments

Week 5 Assignments

Exam #1 will take place on Thursday, 9/22

Written work – none
WeBWorK – Assignment #4, due Tuesday, September 27th, at midnight. You are encouraged to start working on Assignment #5, which will be due one week later.
OpenLab – OpenLab #4, due Thursday, September 29th, at the start of class

 

Exam #1

September 13, 2016 / / 0 Comments

Hi everyone,

Our first exam will take place on Thursday, September 22nd.

The Review Sheet and Answer Key have been posted to the “Classroom Resources/Exam Reviews” page of the OpenLab.

IMPORTANT: Problems #7c and #7d on the Review Sheet have images of Venn diagrams which do not print well – take a look at the online version to see which regions are shaded in.  Same goes for the Answer Key #7a and #7b.

If you get stuck or have questions about a problem, feel free to leave a comment in reply to this post – I’ll check it regularly and respond here.

Best of luck,
Prof. Reitz

Week 4 Assignments

September 13, 2016 / / 0 Comments

Week 4 Assignments

Written work – none
WeBWorK – Assignment #3 and Assignment #4, due Tuesday, September 20th, at midnight.
OpenLab – none

STUDY – for your first exam, taking place next Thursday, 9/22, during the first hour of class.

OpenLab #3: The MIU puzzle

September 8, 2016 / / 51 Comments

We are going to play a game with strings of symbols.  This game was invented by a man named Douglas Hofstadter and found in his book Gödel, Escher, Bach. Here are the rules:

Suppose there are the symbols ‘M’, ‘I’, and ‘U’, which can be combined to produce strings of symbols called “words”, like MUI or MIUUU. The MIU game asks one to start with the word MI and transform it using the following rules, to obtain some goal word (which is given to you).  The rules state:

  1. You must always begin with the word MI.
  2. You may add a U to the end of any string ending in I. For example: MI to MIU, or MUUII to MUUIIU.
  3. You may double any string after the M (that is, change Mx, to Mxx, where ‘x’ represents any string of symbols). For example: MIU to MIUIU
  4. You may replace any III with a U. For example: MUIIIU to MUUU
  5. You may remove any UU. For example: MUUU to MU

WARM UP.  In each example, start with the axiomatic word MI and show, step-by-step, how to obtain the goal word (in each step, state which of the rules you used). These are just for practice (you do NOT need to submit your answers).

Example 1: Goal word MIU
Example 2: Goal word MIIU
Example 3: Goal word MIIUIIU
Example 4: Goal word MUUII
Example 5: Goal word MUUIIUIIU

Example: Goal word MUI
Solution:
Step 1: MI  (we always start with this word)
Step 2: MI to MII (rule 3)
Step 3: MII to MIIII (rule 3)
Step 4: MIIII to MUI (rule 4)
DONE!

Assignment (due Thursday, 9/15): Your assignment has three parts.

PART 1.  First, create an MIU puzzle — that is, make up a goal word, and post it in the comments. Your goal word should be between 8 and 16 letters long.  Try to make it tricky to reach, requiring at least four steps to reach (but the more the better!).  See if you can find a clever use of the rules!

PART 2.  The second part of your assignment is to solve someone else’s puzzle.   Type your solution step-by-step, indicating which rule you used at each step.  Leave your comment as a response to their puzzle.  Only one solution per puzzle!

PART 3.  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, MIU puzzle:  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.

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