Document Sample

a056b847-f1cc-411a-a45e-428108f13468.doc This presentation uses an activity from this book: http://www.amazon.com/Project-Origami-Activities-Exploring- Mathematics/dp/1568812582 Room set up: Diagram of “folding business cards” is displayed on a projector screen. Workshop Intro: Script: General instructions: Our goal is to get the participants to collaborate. We want them to ask questions and learn something, by the end of the workshop. Each group leader should use pay attention to their group – are they participating, asking questions, getting stuck, and then decide when and what questions to ask. Team leaders should gauge their time so that all groups will have discussed Euler’s (pronounced oiler) Formula before leaving the workshop. When someone makes a conjecture, respond by saying “I want to believe what you’re saying. Do you buy that?” Ask other members of the team, “Do you buy that?” Step 1 Create building blocks: Now we are going to create the basic building blocks, which are displayed on the projector. First let’s create a Left Handed Unit. (Describe how you’re folding, while folding a business card.) We are now creating the left handed unit. Take the lower left corner and fold this to have it connect with the upper right hand corner. Make sure that your folds are crisp. We want crisp edges because this will be important later. I will now walk around to make sure that your shape matches mine. Next let’s create a Right Handed Unit. Notice that these simple folds produce equilateral triangles. Optional discussion topic 1: Are they really equilateral triangles and how can we tell? All sides equal, all 60 degree angles Optional discussion topic 2: What a polygon is? Possible answers: A closed plane figure with N-sides. A polygon can be either convex or concave. Page 1 of 9 a056b847-f1cc-411a-a45e-428108f13468.doc Step 2 Create a pyramid/tetrahedron: Optional discussion topic 3: Can you make a pyramid using 1 left hand and 1 right hand unit? Demonstrate the hugging technique - the small side hugs the larger size. It is free standing and if you throw it, it should not break apart. Now we need to think about vertices, edges, and faces. *On white board, begin following table Vertices Faces Edges Degree of Vertices The pyramid you just created is called a Tetrahedron. Optional discussion topic 4: Anyone know why it’s called a Tetrahedron Answer: It has four faces/sides and four vertices. Tetra comes from the Greek word for 4 If someone says at this point”Isn’t this the Euler’s formula?” We say that is where we are heading. Then you could say, would you be able to explain that to us? *On white board, update table. Recurring discussion topic How many vertices and edges? What is the degree of the vertices? Vertices Faces Edges Degree of Vertices Tetrahedron 4 4 6 3 Optional discussion topic 5: How many units/biz cards did you need to make a Tetrahedron? Answer: 2 Page 2 of 9 a056b847-f1cc-411a-a45e-428108f13468.doc Step 3 Create an octahedron: Optional discussion topic 6: How many left and right units do you need to create an octahedron? Answer: 4, (2 of each) Let’s try to build it. Optional discussion topic 7: How many faces does an octahedron have? (If stuck ask how many faces does the card have?) No card is ever covering a face *After build, on white board, update table. Recurring discussion topic How many vertices and edges? What is the degree of the vertices? Vertices Faces Edges Degree of Vertices Tetrahedron 4 4 6 3 Octahedron 6 8 12 4 Optional discussion topics 8: Are these convex shapes? Why? Answer: Optional discussion topic 9: Does anybody know what a convex polygon is? In the 2D planar domain Possible answers: Every line segment between two vertices remains inside or on the boundary of the polygon. Every internal angle is less than 180 degrees. *On white board draw examples of convex and concave polygons. Show the two sample shapes In 3D domain What is vertex? (Answer: Show on paper that a vertex is where edges come together) What is degree of vertex? (answer: the number of edges coming into a vertex) Page 3 of 9 a056b847-f1cc-411a-a45e-428108f13468.doc When talking about 3-d polygons made of 60 degree angle triangles what would be the maximum number of degree for a vertex that would not be flat? A flat surface in a 3-d polygon would have what degree? (Answer 360degrees). How many triangles can come together to in a vertex to make less than 360 degrees if all of the triangles are 60 degrees? (Answer is 5) Euler’s Formula: Discussion topic EF1: Is anyone familiar with Euler’s Formula? * On white board write the formula V - E + F = 2 Discussion topic EF2: Does Euler’s Formula work for the values in our table? Step 4 Create an icosahedron: Optional discussion topic 10: How many faces does an icosahedron have? Answer: 20 faces Let’s try to build it. *After build, on white board, update table. When you were building the icosahedron, did you notice how it would be impossible to have a vertex of 6 degree? Why is that? Remember when we talked about this earlier? (Answer: because 5 degrees is 300 degrees and 6 degrees would be 360 and we would no longer be convex) we would be flat.) Recurring discussion topic How many vertices and edges? What is the degree of the vertices? Vertices Faces Edges Degree of Vertices Tetrahedron 4 4 6 3 Octahedron 6 8 12 4 Icosahedron 12 20 30 5 Advanced Mathematics Continue as time permits Optional discussion topic EF3: Page 4 of 9 a056b847-f1cc-411a-a45e-428108f13468.doc What numbers/values will work? What numbers/values won’t work? Hints – How many faces does folded business card have? (2) How many edges does a business card have (3) What face combinations have we already done? Basically the shapes that we’ve created with these business cards are called deltahedra. A deltahedron is a polyhedron whose faces are all equilateral triangles. What other deltahedra can you build? Keep in mind that we can use Euler’s Formula to prove we’ve created a deltahedron.. Do they numbers/values have to be even? (No, we need an even number of face. Can have an odd number of edges and vertices) Do the vertices and edges have to be even? Let’s try to build something with 6 faces Optional discussion topic EF4: Could we build something with 22 faces? (We conjectured no, because degree of vertices would be greater than 5.) (Only deltahedra with 4,6,8,10,12,14,16,20 will work) Optional discussion topic EF5: Could we build something with 18 faces? (We conjectured no, because degree of vertices would be greater than 5.) Step 5 partially fill out the table for the participants: PASS OUT THE ONLY HANDOUT (or show on the projector) – the Johnson Solids. The definition of a Johnson solid is a bit recursive and requires a lot of definitions. It isn’t important but here it is in case someone asks. In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. Tell them: So, guess what. There are only 8 convex deltrahedron possible. Do you believe me? Let me give you some information and see what you think of it: Vertices Faces Edges Degree Cards V3 V4 V5 Total of V Page 5 of 9 a056b847-f1cc-411a-a45e-428108f13468.doc Vertices Tetrahedron 4 4 6 3 2 12 Octahedron 6 8 12 4 4 24 Icosahedron 12 20 30 5 10 60 Triangular 5 6 9 3 2 3 18 Dipyramid Pentagonal 7 10 15 5 5 2 30 Dipyramid Snub 8 12 18 6 0 4 4 36 Disphenoid Triangmented 9 14 21 7 0 3 6 42 Triangular Prism Dipyramid 10 16 24 8 2 8 48 Is this data correct? What convex deltrahedron shapes can we make with these business cards? What makes it convex? What numbers of faces are possible? Can we have odd faces? Each card adds how many faces? How many vertices? Is it true that each card adds three edges? Do you see any patterns in the data? Are any numbers missing? Under what conditions can these be built? Under what conditions are these impossible to be built? (Convex Deltrahedron) What if we told you that no vertex can have more than 6 degree? Is that true? Why? If we restrict the number of vertices to 5 at the max – can we use Euler’s formula to figure out possible shapes? Optional discussion topic EF6: What about 18 faces? 22 faces? Why aren’t these possible? What if you took an Icosahedron and pulled out one card? That would give you 18 faces. Why wouldn’t you be able to reconnect them? (Answer: because if you pulled out one card the surrounding vertices would be 4,4,5,5 and it is impossible to reconnect the shape without making a vertex connected with 6. And this isn’t possible) Think about how you would go from shape to shape? Can you pull a card out or add a card from one shape and get another shape? Page 6 of 9 a056b847-f1cc-411a-a45e-428108f13468.doc (Answers: pull one from snub disphenoid and get pentagonal dipyramid. Remove one from triaugmented triangular prism and get snub disphenoid. Add two to Gyroelongated square dipyramid and get an icosohedron) Think about the combinations of 4 and 5 vertex degree’s that are possible and those that are not. Groups can continue to work on creating different shapes. General instructions: Thank everyone for attending and participating Definitions: Convex Polygon – Every line segment between two vertices remains inside or on the boundary of the polygon. And every internal angle is less than 180 degrees. Deltahedron – a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedron, but only 8 of these are convex. Page 7 of 9 a056b847-f1cc-411a-a45e-428108f13468.doc Johnson Solids – our presentation deals with a subset of Johnson solids, the one with only triangle faces. Planar Graph – a graph with non-intersecting edges. A graph is planar if it can be drawn in a plane without the graph edges crossing. (If any of the edges intersect/cross then it is not a planar graph). Page 8 of 9 a056b847-f1cc-411a-a45e-428108f13468.doc Planar Graph examples Polygon – a closed plane figure with N-sides. Regular - all vertices have the same degree. Include or discard???? Other interesting questions we considered: Using the left and right basic building blocks what convex polygons can we build from these basic units? What can’t you build and why? Answer: You can build any polyhedron, with equilateral triangle faces and vertices of 5 or less. Do you really need two of each type of units make an octahedron? Answer: No, you can use only the left handed. All the folds will be going exactly the same way. If you use a left and a right, it may not be as difficult to recreate. Page 9 of 9

DOCUMENT INFO

Shared By:

Categories:

Tags:
SQL statement, dynamic cursor, Microsoft SQL, bug report, Open Source, Oracle database, June 27th, MS SQL, list of tables, HP-UX 10.20

Stats:

views: | 3 |

posted: | 4/26/2010 |

language: | English |

pages: | 9 |

OTHER DOCS BY runout

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.