Exam #1

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Hi everyone,

Our first exam will take place on Thursday, October 1st.

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 did 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

OpenLab #4: The MIU puzzle, continued

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This assignment is a continuation of OpenLab #3, on the MIU puzzle.  Your assignment this time is a little different.  Consider the following statement about the MIU puzzle:

Statement.  Any goal word (any word with first letter M, followed by a combination of U’s and I’s) can be obtained from the starting word MI using the rules of the MIU puzzle.

Assignment (Due Friday 9/25 – Because of the nature of this project, please do not post your answer before Wednesday 9/23). Do you think the statement above is true or false?  I would like you to think about it and see if you can convince yourself of the answer.  Try out different things!  If one idea doesn’t go anywhere, see if you can come up with a different approach.   This is an assignment about metacognition as well as puzzle-solving – pay attention to your thoughts and ideas, even (especially) the ones that don’t work out!

In the assignment below, you will be writing about your experience working on the MIU puzzle above.  Be sure to respond to EACH PART – detailed answers to part 2 and 3 below will gain the MOST credit.

  1. Write down what you think the answer is (is the statement TRUE or FALSE?).
  2. Describe in as much detail as you can the process of working on the assignment.  What did you think in the beginning? What are the different things you tried in order to solve the problem?  Describe each one, in the order that you tried them.
  3. Give a detailed list of the resources you used, and how you used each one (these could include anything – your brain, pen & paper, a computer (what applications did you use), the internet, other people, and so on).
  4. What activity or resource do you feel was most effective for you in working on the assignment (what helped you the most in understanding the assignment and figuring out an answer)?
  5. If you believe the statement is TRUE, post an example of a goal word that you encountered in your work that you think was most difficult to solve.  If you believe the statement is FALSE, post an example of a goal word that you believe is impossible to obtain by starting with the axiomatic word MI and following the rules.

Extra Credit.  Respond to one of your classmates’ submissions.

 

Week 4 Assignments

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Week 4 Assignments

HEADS UP: Next week there are no classes on Tuesday 9/22 or Wednesday 9/23.  We have class as usual on Thursday 9/24.  However, we also have class on Friday 9/25 as it is a Tuesday schedule.

Written work – none
WeBWorK – Assignment #3, due Thursday, September 24th, at midnight.
OpenLab – OpenLab #4, due Friday, September 25th (at start of class) – Because of the nature of this project, please do not post your answer before Wednesday 9/23.

OpenLab #3: The MIU puzzle

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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 “axiomatic” word MI and transform it using the following four rules, to obtain some “goal” word.  The rules state:

  1. You must always begin with the word MI, and attempt to reach the goal word.
  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

Here is a sample solution to Example 2:
Our goal is to reach the word MIIU.  According to rule 1, we begin with
MI to MII (rule 3)
MII to MIIU (rule 2)

Assignment (due Thursday, 10/17): 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.

Homework Hints: Intervals of real numbers – unions and intersections

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Hi everyone,

This is a followup to Thursday’s lecture, and should provide a little help with some of the homework problems (I’m looking at you, Problem 7).

Example.  Consider the intervals of real number A = [2, 5) and B = (4,\infty).  Find their intersection A\cap B and their union A\cup B.

One key idea is that these are intervals of the real numbers, so they include not just the whole numbers but all numbers between the endpoints.  The set A includes all numbers that are great than or equal to 2 and less than 5.  This means that A includes 2, 3 and 4, but also decimals such as 3.5 or 4.9998.  The set B includes all numbers greater than 4, such as 4.1 or six billion.

The intersection will be the places where these two overlap – it will include numbers greater than 4 but less than 5 (NOTE: it does not include the numbers 4 and 5 themselves, but it does include, for example, 4.3).  In interval notation, we write:

  • A\cap B = (4,5)

The union will include all numbers greater than or equal to 2, written:

  • A\cup B = [2,\infty)

WeBWorK Tip:  To enter the infinity symbol, just use the word “infinity” like this:
[2, infinity)

WeBWorK Tip: Sometimes in WeBWorK, your answer will consist of two different intervals – you want to include them both in the answer.  To do this, connect them with a union symbol (just use the capital U on your keyboard).  Here is a (made up) example:
[1,7] \cup (15,17]

Not sure if these will help, but they may give you a little more to go on – feel free to leave a comment here or send me an email if you have questions.

Best of luck!

Prof. Reitz

OpenLab #2: Mathography

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This assignment is due Thursday, September 10, at the end of the day

NOTE: Normally, OpenLab assignments are due on Thursday at the start of class. However, Thursday, 9/10, runs on a Monday schedule, so our class does not meet.  Instead, this assignment is due at the end of the day.

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 – 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’t know how.  The second is called metacognition, or “thinking about thinking.”  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’s why we’re getting started by writing a little about our past experiences with mathematics.