Category Archives: OpenLab Assignments

Videos- Joseph Ruiz

OpenLab Assignments

1.a https://openlab.citytech.cuny.edu/2014-fall-mat-2071-reitz/?p=312#respond – “Doodling in Math class Infinity elephants”…  This video talked about a game you can play in almost any shape you decide to draw. You pretty much have to draw circles and circles until you fill up the entire shape inside.

b. https://www.youtube.com/watch?v=Gx5D09s5X6U&list=TLDmvTd50jxGk&index=10 – “How to snakes”….. In this video she talks about snakes and how they can be arranged in many different ways like in the game “SNAKE”. She represented the binary numbers by changing the colors of the snakeskin.

c. https://www.youtube.com/watch?v=jG7vhMMXagQ&index=6&list=TLDmvTd50jxGk “Pi is (still) Wrong” In this video she says how instead of PI we should use TAU. Its funny how she is baking a pie while describing PI

 

2. In this video I found it rather confusing. This lady talks so fast and in the video doodling in math class she draws a lot  of different animals and wierd shapes and talks about how doodling in math class is fun. I was a little confused because she jumps from talking about one thing to talking about another. One thing that I learned from the video was how she compared the “series” application into drawing elephants grabbing eachother’s tails infinetely. I found that very interesting. One question I still have is that if she can relate all of these videos to her own math life and how has it helped her?

3. What this video meant to me was inspiration. The way how she described doodling in math is a new form of way of looking at things from a different perspective. I know that sometimes I’ll begin doodling in my other classes like for example in “Number Theory” because I dont know whats going on in there. I will begin doodling about random things that dont relate to math in a sense, however I will keep track of each time I doodle and see how I can relate it to any math topic. This video wasnt really relevent to me in Prof. Rietz’s class.

 

OpenLab #10: Vi Hart

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Vi Hart describes herself as a “recreational mathemusician” – she has a unique approach to mathematics and its connections to the world.  In this assignment you’ll be exploring some of her videos (she has a YouTube channel here), and using them as a basis for creating a new blog post.

Assignment (Due November 20, 2014).  Create a new blog post responding to the instructions below. Creating a new blog post allows you more flexibility than simply leaving a comment. You have the ability to edit your work after you submit it, and to include photos, videos and other media. It also allows you to contribute to the public content of our course website.

You can get started by clicking the plus sign at the very top of our site (if you don’t see it, make sure you are logged in to the OpenLab). Detailed instructions on creating a new blog post can be found here (see “Writing a Post” in the middle of the page). You should create a new post including the following:

  • The title should include the word “Videos” and also your name.
  • Your post should include responses to all three parts of the assignment described below.
  • Under “Category,” select “OpenLab Assignment” (you will see this on the right side of the screen)
  • Under tags, enter “OpenLab 10”, “Vi Hart”, and any other tags you think describe the videos you watched (for example, you might choose “pi” if it’s a video about pi).  Don’t forget to click “Add” after entering your tags in the box.
  • When you’re done, click “Publish” (the blue button towards the top right on the screen).

Instructions:

Watch at least three different videos by Vi Hart from  https://www.youtube.com/user/Vihart/ .  You should:

  • choose videos at least 2 minutes in length
  • choose videos that are related to math in some way
  • choose three videos, at least two of which should not appear on her front page (older or less popular videos).  For a full list of her videos, click the “Videos” button near the top of her page – or click here.  Scroll to the bottom and click “Load more” to see older videos.

In your post, include a response to each of the following three Parts:

Part 1. Include a link to each video you watched (3 minimum), the title, and a one or two sentence description of what the video was about.

Now choose one video to focus on.  You MUST watch it 3 times. Use it as the basis for parts 2 and 3.

Part 2.  Write one paragraph discussing the contents of the video:

  • How did you feel watching it? Did you like it, or not? Were you confused? Inspired? Bored? Excited? Bewildered?  Why?
  • What is one thing you learned from the video?
  • What is one question you have after watching it?

Part 3.  Write a one-paragraph reflection discussing what the video could mean to your own math teaching.  Is it math? Is it teaching? Is it relevant to the work you will be doing in the classroom? Is there any connection to last week’s assigned reading (Lockhart’s Lament)?  Any other thoughts?

Extra Credit.  You can earn extra credit by responding to one of your classmates’ posts.  As always, be kind, be respectful, be honest.

OpenLab #9: Proof Journal

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Your assignment for the next week is to try to prove the conjecture that your group created in class on Thursday, 11/6/14.  You must spend at least 90 minutes working on this (including 30 minutes in class on Tuesday, 11/11).  Trying to prove something can consist of many different activities, such as the following (you do NOT have to do all of these things – you can choose how to spend your time – they are provided for inspiration only).

  • coming up with ideas, and testing them out (for example, by creating puzzles and trying to solve them)
  • trying to understand what the conjecture says
  • trying to solve puzzles that other people created
  • trying to create puzzles (and solve them yourself)
  • communicating with other members of your group (talking, emailing, etc.)
  • trying to write down a proof
  • other stuff…

As you work, keep track of what you are doing, thinking, and feeling (this is metacognition – an idea that discussed way back in OpenLab #2).  What did you do during the time you spent?  Did you create any puzzles?  Did you solve puzzles?  Did you change your mind about whether the conjecture is true or false?   Did you have any new ideas about how to prove the conjecture?  Did you have any ideas that you gave up on?  How did you feel as you worked – were you frustrated/confused/happy/depressed? Why? Did your mood change along the way?

Assignment (Due Thursday, 11/13/14):  Submit a journal of your efforts in the comments below.  Your response should be at least 300 words.  Describe what you did during the 90 minutes you worked, and express in some way what you were thinking and feeling during the process.  Your response can include puzzles or other work you did along the way.

OpenLab #8: Lockhart’s Lament

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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 Thursday, 11/6/14). 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 or question about your game, and bring it with you to class on Thursday 11/6 (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.  Think about the game you worked on last week (the MIU game, the bridges and walking tours game, or the mutilated checkerboards game).  Each of these games 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 Thursday 11/6 (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:
Question: Is the corner the best move, or the center?
Conjecture: The person who goes first always wins.
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.”

OpenLab #5 Survey Results

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Thanks to everyone for completing the survey.  I want to share the results and make some observations.

QUESTIONS 1-9, RATE THE HELPFULNESS OF VARIOUS ACTIVITIES.

Data. I converted the ratings into a numerical scale from 4 = Extremely Helpful to 0 = Not At All Helpful.  I calculated the average “helpfulness rating” for each question – the results are presented below, with the questions listed in order according to their rating.

Question HELPFULNESS RATING 0-4
(Extremely Helpful = 4,
Not at all = 0,
Don’t Know = not counted)
Lecture 3.8
Professor answering questions in class 3.8
Email contact with Professor 3.7
WeBWorK assignments 3.6
Group Work in class 3.4
Office Hours 3.1
Working with peers (friends, classmates, other students) outside of class. 3.1
OpenLab Assignments 2.8
Tutoring at CityTech 2.7

Observations.  First, it’s interesting to see that the top two items are about me talking to you in the room.  I am curious to see if this shifts over the rest of the semester as a) the course becomes more challenging, and b) you have more opportunity to work with one another.  We’ll see!   The relatively low helpfulness rating for OpenLab assignments is not too surprising, as these are designed to supplement and contextualize the course content rather than contribute to it directly – however, I will put some thought into changes that might be made here (you will find, for example, that a number of the future OpenLab assignments will directly support completion of your class project).  I will also put some thought towards the timing of office hours, as I know the current schedule conflicts with other courses.

QUESTION 10, WHAT COULD BE DONE TO IMPROVE YOUR EXPERIENCE OF THE COURSE.

This question was short answer, but most of the responses fell into just a few different categories, summarized here.

Top Categories Number of responses
I’m happy with the class as it is 7
More advanced examples in class 2
More group work/more problem solving in class 2
Other 3

Observations. The biggest news here is that most of you are pretty satisfied with the class so far.  That’s great!  But don’t get complacent, and don’t hesitate to give feedback or ask for help as the semester continues, especially as we begin to explore proofs in earnest.  I’ll take the comments regarding examples and group work on board as I’m planning future classes.

I plan to revisit this survey later in the semester, and I look forward to seeing how the responses compare.  If you have any questions, feel free to post them in a comment here or send me an email.

Best regards,
Prof. Reitz

OpenLab #7: Let the games begin

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Hi everyone,

Based on your responses to OpenLab #6, I have assigned each of you a game to work on (everyone got their first or second choice).  Check the “Who is doing what” list below to see which game you will be working on.  Then follow the appropriate link from the list below to view your assignment (detailed instructions are provided for each game).

Best,
Prof. Reitz

Links to assignments (Due TUESDAY, OCTOBER 28th – end of day):

Who is doing what:

Chiu,SinFong Bridges and Walking Tours
Conyers,Sarah Bridges and Walking Tours
Felamon,Marina MIU Game
Garcia,Yanira Bridges and Walking Tours
Hamza,Syed Mutilated Checkerboards
Jones,Stacy Mutilated Checkerboards
Kamath, Neil Bridges and Walking Tours
Laing,Latina Mutilated Checkerboards
Lee,Victor MIU Game
M Rafeek, Rushdha Mutilated Checkerboards
Mongo,Joseph Mutilated Checkerboards
Perez,Leonardo MIU Game
Rivera,Julia Mutilated Checkerboards
Ruiz,Joseph Mutilated Checkerboards
Sao,Sidney Mutilated Checkerboards
Shati,Farjana MIU Game
Sun,Jian Mutilated Checkerboards

OpenLab #6: Choose a game!

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What is it that mathematicians really do?  They prove theorems, it’s true – but that’s just the end result.  Along the way comes the fun part – playing around with a problem, trying to see if you can understand the rules of the game, and find some kind of pattern that “makes the whole thing make sense”.  Over the next few weeks, you will be working individually and in groups on a puzzle or game.  Of all the things you do in this class, this work will be most similar to the work that real mathematicians do.

OpenLab #6: Due this Thursday 10/16

Read through the descriptions of the three puzzles below.  Then respond to this comment, ranking them in order of preference.  That’s all you have to do.

For example, you might respond like this:

First choice: Mutilated Checkerboards
Second choice: Bridges and Walking Tours
Third choice: MIU Game

NOTE: This project is about the process of playing around with a problem when you do not know what the answer is (or even what the problem is, really).  BUT all of these puzzles are well-documented on the internet.  I request that you do not Google them, or look at any other resources related to them, until this project is over.

RULES:

  • Read the puzzle descriptions below.  Do NOT Google these topics for more information.
  • If you are already familiar with one of these puzzles, pick another one as your first choice.
  • Puzzles will be assigned next week, with more information and specific instructions.
  • You do NOT need to solve these puzzles for now.

The Puzzles

Puzzle 1: The MIU Game

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. In the MIU game, we always start with the “axiomatic” word MI, and transform it using the following four rules, to obtain some “goal” word.  The rules state that you may:

  1. Add a U to the end of any string ending in I. For example: MI to MIU, or MUUII to MUUIIU.
  2. Double any string after the M (that is, change Mx, to Mxx, where ‘x’ represents any string of symbols). For example: MIU to MIUIU
  3. Replace any III with a U. For example: MUIIIU to MUUU
  4. Remove any UU. For example: MUUU to MU

EXAMPLE 1
Goal Word: MIIU
Solution:

Start with MI (axiomatic word)
MI to MII (rule 1)
MII to MIIU (rule 2)

EXAMPLE 2
Goal Word: MIIIIIIU

 

Puzzle 2: Mutilated Checkerboards

We are going to play a game with checkerboards and dominos.  So that we all have the same picture in our heads, a checkerboard is an 8×8 grid of squares (64 squares total), colored alternatingly black and white:

A domino is a 1×2 block, which is exactly the size of two squares on the checkerboard (the dots on the domino do not mean anything in this game):

Your goal is to cover the checkerboard with dominos, according to the following rules:

  1. You can use as many dominos as you wish.
  2. Dominos can be arranged horizontally or vertically, as long as each domino covers exactly two squares on the checkerboard.
  3. Dominos cannot overlap each other, and cannot extend off the edge of the board.


EXAMPLE 1:  Can you cover the checkerboard with dominos?

EXAMPLE 2: What if we remove one of the corner squares from the checkerboard (a “mutilated checkerboard”) – now can you cover it with dominos?

 

Puzzle 3: Bridges and Walking Tours

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.

 

 

OpenLab #5: Feedback on the Course

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We are currently in the fifth (or sixth, depending on how you count) week of the semester – about 1/3 of the way through the course.  For this OpenLab assignment, I am asking you to provide some feedback on your experience of the course so far.  We will revisit this survey later in the course.

Assignment (Due Thursday, October 9th, at the start of class).  Your assignment consists of two parts:

  1. Complete the survey by clicking here.  The survey consists of 10 questions, and should take 5-10 minutes to complete.  It is anonymous (I will see the answers, but I will not know who submitted them).
  2. Confirm that you have completed the survey by leaving a comment in reply to this post (for example, you could just write “I completed the survey just now”).

Extra Credit.  Tell me what music you are listening to these days.  What’s in heavy rotation on your iPod?  Give us a brief description.  I’m listening to Parov Stelar (“compulsively listenable electroswing”) and Shpongle (“a unique concoction of lounge driven downtempo encased in an endless sphere of production”).    A few ground rules:  No offensive lyrics, be considerate of others, feel free to include a link but only to a legal site such as YouTube or SoundCloud.

OpenLab #4: The importance of “Why?”

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Kids love to ask “Why?”

If you’re like most people who attended public school in the U.S., most of the math you’ve done in your life has been primarily computational, or algorithmic – you learn a kind of problem, you learn a method to solve it, you practice it a few times, and you demonstrate that you’ve learned it by solving a problem on an exam.  Maybe the solution involves memorizing a formula, or a set of steps (an algorithm).  Very seldom are you asked to consider the reasons behind the method – what I call the “why” questions.  And if you ever asked these kind of questions, there is a good chance that you were not often given a very satisfying answer.

Questions you may not have spent much time on in math class:

  • Why does the formula look like this, instead of something else?
  • Why do we do these steps, in this order? How do we know this method actually works?  Why does it work?
  • Why is this fact (“theorem”) true?  I don’t believe it – can you convince me?
  • What’s the big deal about this fact (or formula)?  Why is it important?

To begin this assignment, I’d like you to spend some time thinking about the different kinds of math you’ve learned in your life.  Look at math textbooks or class notes from past and current classes.  Think about the  math skills you learned at different ages – from early experiences with counting, up through things like algebra or geometry, and on to Calculus or Linear Algebra or other recent courses.   If you still have some old math work from your high school, middle school, or elementary school years that might help too (maybe your parents kept some of your childhood work?).  If you’re in the Math Ed program, you can think about the math that you are working on in your MEDU courses.  As you consider these topics, think about the “why” questions – when you were learning this topic, did you spend time in class talking about “why?”   Did you do activities or other work that helped you understand the reasons for the formulas and methods that you learned?

Assignment (due Thursday 10/2/14). The assignment consists of two parts.  Be sure to answer each part completely (they are equally important).

1.  Describe a time that you DID learn about the “why” behind a mathematical idea (why something works, or where it came from, etc.).  Tell us about it – how old were you, what was the class (or did it happen outside of class?), what was the topic, and what did you learn?

One example that came up in a class discussion was the activity introducing the number \pi by having students measure the diameter and circumference of various circular objects, and showing that the ratio was always a little more than 3.

2.  State three questions about math that you do not know the answer to (but would like to!).  For each question, tell us why you find it interesting.  The questions…

  • …should involve “why” or “how”
  • …should be about math topics that you are already familiar with – things you’ve studied (you need not be an expert)
  • …should come from different math topics, at different levels (some examples:  arithmetic, fractions, algebra, geometry, functions, calculus, probability, … )

Some examples of “why” questions. 

  1. Why does a \times b equal b \times a?
  2. Why do we need a common denominator when we add fractions?
  3. How do we know that \sqrt{2} is irrational (not equal to a fraction)?
  4. Where does the quadratic formula come from?  How do we know it always works?
  5. The fundamental theorem of calculus gives us a way to evaluate an integral – for example, to find the area under a curve.  How do we know this actually works, and gives the correct answer?

 

OpenLab #3: “Sentences”

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Due Thursday, 9/18/14.  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 by Prof. Park in class on Thursday )?  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