Written work, Due Tuesday, October 22, in class:
Chapter 4 p.100: 4, 5, 10, 11
Odd problems are worth 4 points, even problems worth 8 points.
WeBWorK – none
OpenLab – Recall: OpenLab #5 due at the beginning of class on Thursday, 10/24. NOTE: Please complete the writing portion on the OpenLab, but bring your Conjecture to class on 10/24 (do not post your Conjecture to the OpenLab)
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, 10/24/17). 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:
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 Thursday 10/24 (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. Write down a conjecture or question about the bridges and walking tours game, and bring it with you to class on Thursday 10/24 (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 conjectures about Tic-Tac-Toe:
Conjecture: The person who goes first always wins.
Conjecture: The center is the best move.
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.”
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 the bridges and walking tours game?
Group 1: Song Yu, Randy, Aurkaw
Group 2: Youshmanie, Dylan
Written work, Due Thursday, October 17th, in class:
Chapter 4 p.100: 1, 6, 7, 15, 16
Odd problems are worth 4 points, even problems worth 8 points.
WeBWorK – none
OpenLab – OpenLab #5 Due Thursday 10/24. NOTE: Please complete the writing portion on the OpenLab, but bring your Conjecture to class on 10/24 (do not post your Conjecture to the OpenLab)
Hi everyone,
Just a heads up – the fifth WeBWorK assignment is now available, due this Thursday 10/10 at midnight.
Best,
Prof. Reitz
NOTE: Next week Tuesday 10/8 there are no classes at City Tech.
Written work – None.
WeBWorK – Assignment #5, due Thursday, October 10th, at midnight.
OpenLab – OpenLab #4, due Thursday, October 10th, before class.
If you are logged in to the OpenLab, you can find them under “Dashboard / OpenLab Gradebook” (start by clicking the course name in the black bar at the very top of the page). Exams will be returned in class on Thursday.
NOTE: Your grades will only appear on the OpenLab if 1) you have an OpenLab account, and 2) you have previously Joined this course. Once you Join the course, your name will appear in the Gradebook.
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’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.
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).
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 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 – G4), 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!
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
You will have 45 minutes to complete the exam, ending at ___
You may use: pen/pencil. NO: Scratch paper, phones, laptops, tablets, ec.
Please spread out as much as possible in the room.
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