Monthly Archives: October 2015

In-Class Group Project Activity 10/20/15 – Make and Test Conjectures

Assignments, Project

NOTE: As a component of OpenLab #6, each person should bring to class a conjecture or question about the Bridges and Walking Tours game.

Group Activity (30 min).  Get into your groups, arrange your chairs in a circle, and take 30 minutes to:

1. Each person should share their conjecture with the group.  For each conjecture, the group should decide if they think it is true or false (or don’t know).  The group should record their conclusion for each conjecture.

2. Choose one conjecture (or create a new one) to focus on as a group.  Your goal for the next few weeks will be to try to prove or disprove this conjecture.  Come up with several ideas about how you might prove it.

Group work due after 30 minutes:  Each group will hand in a sheet of paper with the names of the group members, the date, and the following:
– Each member’s conjecture, along with a brief description of what the group thinks – is it true or false?
– Be sure to clearly indicate which of the conjectures the group has chosen to work on – or, if you have created a new conjecture to work on as a group, include that as well.
– Two different ideas about how you might try to prove the chosen conjecture.

Reflection:  To be completed individually after group work is complete, and submitted on paper with your names 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?

Week 9 Assignments

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Written work, Due Tuesday, November 3rd, in class:
Chapter 6 p.116: 3,4,5,8,9
      **NOTE: this assignment is not due until November 3rd, HOWEVER I strongly recommend that you spend some time working on the first problem before Tuesday October 27 – this is a tricky topic and tends to give people some trouble, so early preparation will help you!
WeBWorK – none
OpenLab – OpenLab #7 due Thursday, October 29th at the start of class

Class work: There will be group & individual work completed & submitted in class on Tuesday 10/20/15, which will count towards your “Project” grade.

Homework Chp 5 Update – problem #20

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Hi everyone,

Problem #20 in Chapter 5 uses an idea that we have *not* yet covered in class.  It is no longer a required problem – you don’t have to do it – but I will give you extra credit if you turn in a solution.  This is excellent practice in reading and applying a definition (just as we have been doing for the definitions of odd, even, divides, and so on).  The problem relies on a new definition, that of congruence mod n – it appears in the book as Definition 5.1 on page 105, but I will also give it here:

Definition.  Given integers a and b, and n \in \mathbb{N}, we say “a is congruent to b mod n”, or a \equiv b \pmod n, if
n | (b-a).

For example, if are told that x \equiv 7 \pmod 3, then we can conclude:
3|(7-x)  (by the definition of congruence), and
7-x = 3k for some integer k (by the definition of divides)

Hope this helps!  Please write back and let me know if you have any questions.

Best of luck,
Prof. Reitz

Week 8 Assignments

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Written work, Due Tuesday, October 20th, in class:
Chapter 4* p.100: 1, 6, 15, 16
Chapter 5 p.110: 1, 4, 20**
Odd problems are worth 4 points, even problems worth 8 points.
*(NOTE: If you wish to turn these in on Thurs 10/15 to receive feedback before the exam, I will return them on Tuesday 10/20)
** (Chapter 5 Problem 20 is optional – solutions will receive extra credit)   
WeBWorK – none
OpenLab – none

Exam #2 will take place on Thursday, 10/22 (first half of class).

OpenLab #6: 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, 10/15/14). 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 the Bridges and Walking Tours game, and bring it with you to class on Tuesday 10/20 (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 onTuesday 10/20 (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.”

In-Class Group Project Activity 10/8/14 – Puzzle Making

Assignments, Project

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 #5.  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: Samuel, Kenny, Rahat, Justin
Group 2: Deborah,Josiel, Sanaya, Irania
Group 3: Xiong, Abdelmajid,Sarah, Mei
Group 4: Daniela, Ryan, Yuyu, Fuzail

 

Week 7 Assignments

Assignments

Written work: none
WeBWorK – extension granted: WeBWorK #5 is now due Tuesday, 10/13, at midnight
OpenLab – OpenLab #6 due Thursday, 10/15* before class
(* the final part of the assignment, to create a Conjecture and bring it to class, is due in class on Tuesday, 10/20)