2+Non-Euclidean+Geometries


 * (11 days)**


 * Days 1-5:** Taxicab Geometry

What happens when you change the assumptions of a well-known, widely-accepted system? do 3, 4, 9 baseball example (10 min)
 * Day 1: Intro**

What is Euclidean Geometry? Emphasize a set of assumptions that we will examine and contrast in the alternate geometries

video clip of development/history of ideas and brief discussion in class to get out main ideas and assumptions: How would you define geometry? (Interesting facts: Ptolemy discovered earth was round - not christopher columbus!, term Geometry ("earth measuring") coined before discovery of round earth http://www.math.ubc.ca/~cass/courses/m308-02b/projects/franco/index.htm) What are the fundamental objects you learned about at the beginning of 9th grade Geometry (if not mentioned yet)? Make a list T/F questions with basic Euclidean assumptions, e.g. "2 parallel lines will never intersect." take questions from taxicab word doc

-Students come up to poster and draw many different taxicab distances from A to B. See that they're all the same. Why? - Students work in groups to do practice sheets edited from our stuff (Class: pkt A #1-2 mushed with B#3 worked in, pkt B #3, day2 proc 2) HW: A #3, 4a-d, B #1 (add one more police car)
 * Define Taxicab geometry -** we're going to change the definition of distance. In taxicab geometry, you can only travel along the roads of the grid. How is this different from how you found distance freshman year?

Process HW Where can Matt eat lunch if he wants to stay 3 blocks from work? What should we call this set of points? Euclidean defn of circle, Taxicab defn of circle - let's make up a new word for it... B day 2 proc 8 draw any 4x4 square - are they all circles? Remember what we learned about pi? What's its equivalent in taxicab? How can we figure it out? (Use C/d, then check if it works with area formula - nope!) CW/HW: A pg 4 #1, 2, 4, "try to come up with an equation to describe one of your taxicab circles", pg 8 #1, pg 14 #1, 2
 * Day 2: Circles**

Process HW isosceles triangle investigation B #3, 4, 5 but modified so that it says 'draw a taxicab isosceles tri', then ask whether they are isosceles in euclidean too. can you come up with a generalization about whether a taxicab iso will also be iso in euclidean.' - word problem: show all places you could live so that your house are the same distance from the shopping mall that it is from the school. (make sure rectangle is even by even, and not squarish) & discuss - does anyone remember what this set of points is called in Euclidean geom? bring up how it's different from Euclidean perpendicular bisector (um, not perpendicular! make up a new name for it), what do you notice about this set of points and the points A and B? How would you describe them? (draw rectangle around A and B) - students work in groups on practice A pg 15 #3, 4, 5 (edit to make rectangles even by even) HW: A pg16 #6 broken into 3 separate problems & graphs - first find bis of AB, then BC, then AC, then challenge them to put them all onto one graph of a triangle - use colored pencils! What do you notice?
 * Day 3: Triangles / Perp Bisectors**

Process hw problem, then do part e, f - work in groups on A pg 16 #7 same questions as #6 spend rest of time doing a bit of work on each challenge problem in groups; - Fountain problem A pg 6 #7 - Ellipse problems A pg 9 #3, 4 - Points/Lines A pg 10-12 #1-4 - hard perp bis of triangle problem A17 #8 HW: pick one problem and work on it for 20 minutes
 * Day 4: finish perpendicular bisectors of triangle**

students meet in groups based on what problem they've chosen and spend class working together, write up work and findings for homework, even if they feel like it's not a finished solution.
 * Day 5: Work on hard problems of choice**

__Materials:__ register tape tape string transparency markers big rubber bands inflatable balls - Betty picked up 6 balls for class protractor photocopied onto transparency ring things to trace different sized circles onto a sphere (eg tape roll, disposable cups, sliced 2 liter bottle, etc.)
 * Days 6-8:** Spherical Geometry

What is a sphere? How is it related to circles? What is spherical geometry? How do you think it will be different from euclidean geometry?
 * Day 6**

Draw 2 pts on sphere, not too close together, but not opposite. Draw the shortest distance between them. Describe your result. Now keep it going. What happens? Describe the result. Do you think this will always happen? Repeat 2 more times with different points. Have poster/graphic organizer to fill in comparing Euclidean v. spherical so far... point = same in both; line segment = arc; line = great circle *note: line is infinite, but great circle is finite How should we measure these lines since they're circular? length vs angle (if kids don't come up with angle, draw cross section/great circle and ask again if there's any other way to describe the distance from A to B with reference to the circle) -- what are the advantages of each? Move forward with angle measure to describe distance on a sphere. So, we need a new measuring device. Make a ruler. Measure your segments. Next, how many intersection points are there between two geodesics? compare to intersection of 2 lines in euclidean. draw examples for both. How many parallel geodesics can you draw given a geodesic and a point not on that geodesic. Compare to euclidean. draw examples for both. What does this mean? Add info to poster at this point. HW: watch first basic video from discovery here - write down 5 things from video you remember learning in class. write down any questions you have so far about spherical geometry. Longer term assignment: as we complete our unit on non-euclidean geometries, we ask you to participate in an online/classroom discussion on the following questions. Please post 3 original comments, and reply to 3 classmate's comments before the end of the unit. First comment due tomorrow. - Q1: Why do you think has Euclidean geometry prevailed so long given that humans have known for quite some time that the earth and the nature of space is not flat? - Q2: Select one modern day application of non-euclidean geometry that you think has been the most important to humans. Explain why. - Q3: Describe one thing from this unit that you thought was the most fascinating and explain why.

What is a circle in spherical geometry? Choose a center point and a radius, and draw a circle on your sphere. Now, choose an object and trace a circle onto your sphere. Find the center and radius. Actually, there are two centers and radii for any circle on a sphere. Be sure to find both. Contrast this with Euclidean. Polygons: what is the minimum number of sides needed to draw an enclosed figure in euclidean vs. spherical? called a lune. Draw one on your sphere. How long are the sides? What is the sum of the angles? What do you think the range is of the sum of the angles in a lune? Draw 3 noncollinear points on your sphere. Connect them to make a triangle. Use your protractor to find the sum of the angles of your triangle. Post answers on board. Try to generalize the sum of the angles on a spherical triangle. can you have more than one right angle? show images from this page: http://euler.slu.edu/escher/index.php/Spherical_Geometry - the cross-section Euclidean triangle vs. spherical triangle, and at bottom the tessellations of sphere with polygons HW: D id you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are south of Florida - why is flying north to Alaska a short-cut? Use your globe to draw the shortest path. choose an article to read: NYT article here about shortest flight paths or NYT science article here about bird migration along great circles - be prepared to discuss in class tomorrow!
 * Day 7: Circles, Triangles**

Kids read, watch, explore and then answer questions about the basics of Hyperbolic geometry. Resources: Hyperbolic Intro page: @http://www.math.hmc.edu/funfacts/ffiles/30001.2.shtml Draw lines on poincare disk and move them around applet: http://www.math.umn.edu/~garrett/a02/H2.html Thesis with background information
 * Day 8: Hyperbolic Geometry**

Good visual comparison of euclidean, hyperbolic, elliptical geometry http://mathforum.org/mathimages/index.php/Summary_of_Geometries Good basic description - click on "More Mathematical Explanation" as well http://mathforum.org/mathimages/index.php/Hyperbolic_Geometry#Basic_Description Hyperbolic Geometry Youtube Song http://www.youtube.com/watch?v=B16YjC9OS0k Questions: Describe 3 key ways that hyperbolic geometry is different from the other geometries we've studied. Also: What is a Poincare disk? Given a line and a point, how many parallel lines can you draw? What does a triangle look like? Conjecture the sum of its angles.

Now, do one from each category below: Youtube video lego hyperbolic surface: @http://www.youtube.com/watch?v=8YikT9DtrLQ&feature=related Youtube video origami pringle chip: @http://www.youtube.com/watch?v=HDqCG2bUFIQ&feature=fvwp TedTalk hyperbolic knitting models video : or this one on coral reef here PBS video Fabric of the Cosmos: Episode 1: What is Space? from 16:23 - 26:25 (Ch. 3 The Problem with Gravity)
 * Watch a video**:

Download and play the maze game: Link to hyperbolic games download: http://www.geometrygames.org/HyperbolicGames/ Watch the video of another Game played on hyperbolic poincare disc: http://www.youtube.com/watch?v=nwIiT_uJOj8&feature=fvwp Download NonEuclid program and play around with drawing things on a Poincare disk: http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html
 * For Fun & Games:**

article - may be more relevant in the art unit http://blog.theautry.org/2011/04/27/crochet-as-an-act-of-mathematics-to-save-the-environment/
 * Read an article:**







HW: finish above, and be prepared to share/discuss in class tomorrow, and also second discussion comment due online.


 * Day 9-10:** **Design 10-15 min lesson for 9th graders as final product**

Last comment due online.
 * Day 11: Do mini-lesson to freshman class on this day - any free block - coordinate**


 * Resources for Grace & Betty**
 * Resources for Grace & Betty**

blog entry re: William Thurston who proposed a bunch of famous topology questions in the 80's that have since been answered/proven before this death in 2012. [] 50min youtube video about noneuclidean geometry overview: - not one that we would show students, but gives good background info. http://www.youtube.com/watch?v=zHh9q_nKrbc&feature=related

lots of resources on this blog post [|here]
 * Taxicab Reference:**
 * http://www.taxicabgeometry.net/triangles/index.html**

Article from American Mathematical Society: - discusses why we consider different systems and gives good overview of important taxicam geometry ideas: http://www.ams.org/samplings/feature-column/fcarc-taxi

another introduction, assessment and and construction of taxicab ellipse activity http://web.pdx.edu/%7Earturog/TaxicabLesson.html

another worksheet with scenario and practice problems and true/false difference between taxicab and euclidean geometry http://www.docstoc.com/docs/83468591/Taxicab-Geometry-Questions# -- use the T/F questions comparing taxicab v. Euclidean http://www.docstoc.com/docs/22105749/Taxicab-Geometry

Good introduction to vocabulary of spherical geometry and overview of big principles
 * Spherical Geometry**


 * note the spherical geometry applets at the bottom of the article. . .they seem kinda cool

Cool 2 min video clips explaining various ideas of spherical geometry, but has commercials in between >< also has some stuff about ancient mathematicians and stuff. . . http://videos.howstuffworks.com/discovery/28013-assignment-discovery-spherical-geometry-video.htm http://dsc.discovery.com/tv-shows/other-shows/videos/assignment-discovery-spherical-geometry.htm

http://dsc.discovery.com/tv-shows/other-shows/videos/assignment-discovery-spherical-trigonometry.htm

Overview worksheet: http://mathteachermambo.blogspot.com/2010/11/spherical-geometry.html

Grace has a worksheet download on her computer


 * Non-Euclidean Geometry Overview:**
 * http://www.youtube.com/watch?v=tn0MYbCkNZY**


 * Very brief summary of non-euclidean gemoetries:**
 * http://www.cs.unm.edu/~joel/NonEuclid/noneuclidean.html**

1.4 Hyperbolic Geometry: **
 * Spherical geometry is used by pilots and ship captains as they navigate around the globe. Working in spherical geometry has some non-intuitive results. For example, did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are **south **of Florida - why is flying** north **to Alaska a short-cut? The answer is that Florida, Alaska, and the Philippines are collinear locations in spherical geometry (they lie on a great circle). Another odd property of spherical geometry is that** the sum of the angles of a triangle is always greater then 180°**. Small triangles, like those drawn on a football field, have very, very close to 180°. Big triangles, however, (like the triangle with veracities: New York, L.A. and Tampa) have significantly more than 180°.**

hyperbolic geometry is the geometry of which the NonEuclid software is a model. Hyperbolic geometry is a "curved" space, and plays an important role in Einstein's General theory of Relativity. hyperbolic geometry is also has many applications within the field of Topology. Hyperbolic geometry shares many proofs and theorems with Euclidean geometry, and provides a novel and beautiful prospective from which to view those theorems. Hyperbolic geometry also has many differences from Euclidean geometry. The following sections discuss and explore hyperbolic geometry in some detail.

Concise websites for non-euclidean geometry explanations: (must click different frames to get to the other information - maybe we can just use the information and make our own wiki-page because the information here is good, but it is a difficult site to navigate) What is it: http://noneuclidean.tripod.com/whatisit.html

Good spherical applet (seems like basic equivalent of geometer's sketchpad for spherical) http://merganser.math.gvsu.edu/easel/applet.html

Very good summary of non-Euclidean, focusing on hyperbolic with very cool links to articles and related info http://www.math.cornell.edu/~mec/mircea.html