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Go to Homework #1, 2, 3, 4, 7, 10, 14, 17, 20, 23, 26, 29, 32,
35. Due: Monday, May 5
a. Read pages 61 - 75 of Journey ..., a hyperbolic geometry party, and describe the evidence you got - in Lab 18 - for the "sum of interior angles of a hyperbolic triangle" described on page 63.
b. Complete Lab 18.
c. Write Q11 & Q12 , questions about two different project presentations.
d. Write R10, a reflection on something you found surprising about the nature of geometry, based on your experience in Math 120.
34. Due: Monday, Apr 28
Write R9, a reflection on your experience with Lab 18.
33. Due: Friday, Apr 25
a. Complete Lab 16, Inversion.
b. Read pages 47 - 51 of Journey ..., an introduction to a different model of geometry - a model not of a different geometry, but, rather, of Euclidean geometry itself. (Details contained in pages 52 - 58 will not be assigned.)
Read the Guide to Book I of the Elements, down to Logical structure ... and write R8, a reflection on the Guide, and Q10, one question you have about the reading.
31. Due: Friday, Apr 18
You should have completed at least 6 problem solution attempts.
30. Due: Wednesday, Apr 16
Read page 33 and the list of "things to remember about inversion" from the last paragraph at the bottom of page 40 through the top two lines of page 41 of Journey ...
a. Read the one-page web page, bulleted, Parallel Postulate Revisited.
b. Choose one of the 20 descriptions 0-19 in the above brief 'history,' and write Q9: a question about a detail of that one description you chose.
28. Due: Friday, Apr 11
Write R7: a reflection about something you learned and something you found surprising about the Euclidean symmetries or the rigid motions of the Euclidean plane.
27. Due: Wednesday, Apr 9
Write Q8: a question about a mathematical detail of the exposition on pages 31-32 of Journey ...
a. By experimenting, with paper or by using Sketchpad, gather evidence to answer the question: Is it always possible to execute EFM 5 in 4TH DRAFT OF ELEMENTARY FOLDING MOVES?
b. Read page 32 of Journey ...
25. Due: Friday, Apr 4
a. Submission of Portfolios (from students whose names are in alphabet range A-D)
b. Read pages 29-31 of Journey ...
24. Due: Wednesday, Apr 2
a. Submission of Portfolios (from students whose names are in alphabet range P-Z)
b. Read the third draft of Elementary Folding Moves, and show that the Proposed Elementary Folding Move to generalize the third fold, shown in the third diagram of the angle trisection process, cannot be used to do that third fold. (I.e., show that this proposed move does not generalize the third fold.) Revise the proposed move so that it generalizes that third fold.
a. Submission of Portfolios (from students whose names are in alphabet range F-M) The portfolio should be up-to-date in all homework assignments. Be sure to include all of your observations, questions, and reflections you made in and about classroom discussions and activities. Also, don't forget to respond to each of my post-its in which I asked a question or made a suggestion. For more details, go to Notes for Portfolio Check 2
b. Write R6, your reflections comparing and contrasting Elementary Folding Moves to Euclid's Postulates.
22. Due: Friday, Mar 28
a. Read pages 19-23 of Journey ...
b. You should have completed at least 5 problem solution attempts.
c. Complete Lab 10, constructions via straightedge and compass.
21. Due: Wednesday, Mar 26
a. Read pages 16-18 of Journey ...
b. Write Q7: a question about a mathematical detail of the exposition on pages 16-18 of Journey ...
One member of each EFM group should send to me [jm] the new EFM derived from one of the folds investigated in class Wednesday, Mar 12 - Dollar fold, Line onto line, Square fold, and Angle trisection fold - OR an explanation of your investigation showing that one of the previously proposed or accepted EFM's can be used to do the fold.
19. Due: Friday, Mar 14
a. Send to me [jm] your project title and a one-paragraph (or more) description of the project
b. Complete Lab 7, constructing a non-messupable square and a non-messupable diagram to depict the first two creases used to fold an equlateral triangle from a square.
18. Due: Wednesday, Mar 12
a. Complete Lab 9, Euclid's 5th Postulate. (#1 only)
b. Write R5 , your reflections on the thinking processes used in your problem-solving attempts.
a. Read pages 14-15 of Journey ...
b. Write Q6: a question about a mathematical detail of the exposition on pages 14-15 of Journey ...
c. You should have completed at least 3 problem solution attempts.
16. Due: Friday, Mar 7
Send to me [jm] Q5: one question - just one, but make it really a question, not something of the form, "I don't understand ..." - that you think would further your understanding of Elementary Folding Moves and/or the geometry of an EFM.
15. Due: Wednesday, Mar 5
Send to me [jm] your group's new EFM(s), started in class Monday, Mar 3, in the format prescribed in Lab 6. You may omit the "consequences" part.
a. You should have completed at least 2 problem solution attempts.
b. Identify a folding move - i.e., a single fold - you made as part of any one of the folding activities listed in Lab 6 that cannot be done by what I'll call Elementary Folding Move 1: Given two points A and B, one can fold point A onto point B. Describe your move in the format prescribed in Lab 6.
13. Due: Friday, Feb 29
Write R4 , your reflections on the process of constructing a list of Elementary Folding Moves (Lab 6) BACK TO TOP
12. Due: Wednesday, Feb 27
a. Write your thoughts on resolving the angle trisection impossibility paradox. See the handout, A Brief History of Impossibility and Geometric Problems of Antiquity
b. Complete Lab 4, the Star Unit
11. Due: Monday, Feb 25
Complete Lab 6a, the Categorization Lab, which includes responses to Question 1 (one collection per group) and to Question 4 (also one collection per group), as well as a revision of the categories in the CategoriesEdited document in Resources (may be done individually or by your group). Only the group question responses need to be e-mailed to me.
a. Complete Lab 5
b. Write a question Q4 about the method of trisecting a line segment by folding.
c. Submission of Portfolios (from students whose names are in alphabet range P-Z) Be sure to include all of your observations, questions, and reflections you made in and about classroom discussions and activities.
9. Due: Wednesday, Feb 20
Submission of Portfolios (from students whose names are in alphabet range F-M) Be sure to include all of your observations, questions, and reflections you made in and about classroom discussions and activities. BACK TO TOP
8. Due: Monday, Feb 18
a. Write your observations and reflections about the construction of origami cubes.
b. Submission of Portfolios (from students whose names are in alphabet range A-D) Be sure to include all of your observations, questions, and reflections you made in and about classroom discussions and activities.
Complete Lab 3 and write your reflections R3 about the Sketchpad techniques you learned from doing the lab.
6. Due: Wednesday, Feb 13
a. Assignment Q3: Write at least one question you have about the mathematics involved in Lab 2 and send just one of them to me [jm].
b. Read pages 7 - 10 of Journey.
5. Due: Monday, Feb 11
Complete Lab 2 and write your reflections R2 about the geometry you learned from doing the lab.
The dollar bill folding done in class is purported to use the 1:2 ratio for the shortest:longest sides of the type of triangle described in Homework Assignment 3 below. (This ratio is one of the properties that could be stated in your response to Homework Assignment 3.) Below is a depiction of the dollar after the second folding step. BACK TO TOP
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You can see where the bill was first folded in half, a dashed crease, and you can see where the left upper corner of the bill was, at point A, before the second fold and where it is, at point D, after the second fold. Point E isn't actually located in the folding process; it's there to help with the reasoning about what happens. E is located so that angle BED is a right angle. |
Which two segments must be equal (congruent) based on the fact that the corner that was at point A is folded so that it ends up at point D? Identify two segments that are in a 1:2 ratio.
The next step, the third fold, is to fold "over" the line segment CD and its extension, which produces something that looks like this:
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You can now see where the left lower corner of the bill was, at point B, before the third fold, and where it is, at point G, after the third fold. The dashed lines show both creases and where edges and corners were before folding took place. |
The final two steps are to fold "over" the line segment FG and then "tuck" the upper right corner in.
But there remain some things to be explained! First, why does the corner at point B end up at point G on the upper edge of the bill when it is folded over CD?
Secondly, something rather amazing, to me, happens when the corner at point C is folded over FG. What is amazing? To try to answer this question, you might try the same folding procedure on a bill of a different currency or just any rectangle that is not similar to a U.S. bill.
Q2: Write one question that you have remaining about the ideas involved in the dollar bill folding.
The above diagrams were made using the Geometer's Sketchpad software, which is what you will learn to use in Math 120. For a dynamic Sketchpad illustration of the dollar folding, click here. (Drag the red corner point to make the fold.)
a. From one member of each group: Send to me [jm] your group's list of categories that were based on the notes, observations, and questions from Wednesday, the 30th's, Geometry Walk- Lab 1, and Friday, the 1st's group discussion. For each category, list one object that you have included in that category.
Please cc the other members of your group so that I get their e-mail addresses, too.
b. List all the properties you can think of for the two congruent triangles resulting from folding an equilateral triangle "in half." Justify your thinking. (Note that this item, due to be completed by Wednesday, Feb 6, is not to be turned in by that date. It should go into your portfolio, to be checked there.) BACK TO TOP
R1: Send me [jm] two or three paragraphs reflecting on your experience in mathematics classes and what you think are important characteristics of the discipline of mathematics. I'd like to hear about your hopes for the class and personal interests you have and anything else you think it might be useful for me to know.
Q1: Browse the seminar web pages. (Be sure to take a look at SYLLABUS.) Then send me [jm] at least one question you have about the pages. (You should also include this question in your PORTFOLIO.)
I've been revising the pages, which were used in a previous version of Math 120; I've undoubtedly not made all the necessary changes, so, in addition to your Question, I'd appreciate hearing from you about anything that needs to be revised. Thanks!
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