# Math 103 Spring, 2014 SUBJECT TO REVISION!

 Week\Day Monday Wednesday (Submit no later than by Friday Noon). 1 1-20  None 1-22 Reading 2 1-27 Reading 1-29 Reading I. Using the seven pieces of the tangram puzzle create the following figures and indicate with a sketch your solution. A. A rectangle and B. A right triangle. II. Suppose that the square made using the seven tangram pieces has a side of length 4. A. Find the area of each of the seven pieces from the figure without algebra! Explain your responses. B. Find the length of the sides of each of the seven pieces. Explain your responses. 3 2-3 Reading 2-5 Reading Ask your adviser or an instructor you know in your major the following questions. Submit a summary report of the interview responses.  A. How do you define mathematics?  (or...  What is mathematics?)  B. Can you give one example of how mathematics is used in your discipline? Ask for an explanation of the response. C. Has there been a paradigm shift in your discipline? Ask for an explanation of the response. 4 2-10 Reading 2-12 Reading Classifications by symmetry:    It is often useful to classify visual objects by their symmetries. For example, the letter "T" as it appears on this page has only a reflection symmetry determined by a vertical line, whereas the letter "I" has two reflection symmetries and one rotational symmetry of order 2 (a half turn). Consider the following list of letters with the specific font style: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z  1. Group the letters together in different classes determined by the number and types of symmetries they have as printed on this page. [It is up to you to determine the appropriate classes.] 2. Explain your classifications briefly by describing the common symmetry of the letters in each class. Tessellations with triangles: 3. Create a tessellation of the plane using a single isosceles triangle as the template. Is this pattern unique? If your answer is yes, explain what makes it unique. If your answer is no, create a distinct tessellation using the same triangle and explain how you distinguish it from your first tessellation. 5 2-17  Read Tessellation Day: Wear to class clothing that has a tiling pattern on it. 2-19 Read Submit the following assignment on line or on paper: The Sphere has brought the Torus to visit his new friend the Square in Flatland. Describe two possible successions of different planar shapes the Torus might  appear as while passing through Flatland. Compare how would these experiences would differ from what the square saw when the sphere passed through Flatland. You may draw the sequences as they would be seen in Flatland but you need to describe these drawings verbally in your response. Write a short essay that discusses one of the following two statements about the relation of perception to beliefs about truth and reality in the story of Flatland and the Cave. Use specific information from the stories to illustrate your discussion. A.  If what I believe is true about reality agrees with my perception, then my belief must be correct. B. If a person says something about reality that does not match what I believe is true about reality, then that statement must be incorrect. 6 2-24 Reading Project proposal due by 5 pm. 2-26 Portfolio Sample(s) due. 7 3-3 Reading Continued. 3-5 Investigating Shadows and Symmetry 1.The sphere is still trying to explain some of the features of the torus to a Flatlander. This time the sphere has decided to show the Flatlander different shadows that are cast by the projection of the torus onto Flatland. A. Draw three different shadows that the torus could cast. B. Do you think it is possible to make a torus that would cast a shadow on Flatland that covers a circle and its interior exactly? If so, describe some of the features of such a torus. If not, give some reasons for your belief. In other words, is it possible that a Flatlander might mistake a torus for a sphere based on the shadow it casts? 2. We are trying to describe the cube to a Flatlander, this time using the transformation of the framework made up of the edges and vertices of the cube onto the plane by central projections from a single light source. Show the image of the cube on the plane under the following projection situations: A.  One square of the cube is in Flatland and the center of projection is above the cube directly over the center of the cube. B.  Only one edge of the cube is in Flatland and the center of projection is above the cube directly over the center of the edge in Flatland. C.  One square of the cube is in Flatland and the center of projection is higher than the cube and not directly over any part of the cube. 3. Write a brief essay discussing a situation in the contemporary world where indirect experiences are used to make observations. How are the observations made? How are they connected to the actual situation? Do you think the inferences made from the observations are always accurate? [3 or 4 paragraphs should be adequate.] 4. Imagine you are a Flatlander talking to a Linelander. Write an explanation of symmetry to a Linelander from the point of view of a Flatlander. In your explanation discuss the kinds of symmetry that are possible in Lineland. Discuss which Flatland symmetries (if any) would you associate with Lineland symmetries? Explain the association briefly. Here are some terms you might use in your discussion: Reflection  Rotation  Translation  Orientation 8 3-10 Reading TBA 3-12 1. The fourth dimension can be used to visualize and keep track of many things involving four distinct qualities that can be measured in some fashion. A. For example, a 13 card bridge hand can be thought of as a point in four dimensions where the coordinates represent the number of cards of each suit present in the hand. In this context the point with coordinates ( 2, 4, 6, 1) might represent a hand with 2 clubs, 4 diamonds, 6 hearts and 1 spade. Using this convention discuss briefly the following representations of bridge hands: (0, 0, 0, 13),  (0, 0, 6, 7),  (3, 3, 3, 4). Suppose a bridge hand is represented by the point with coordinates (x, y, z, w). Explain why x + y + z + w = 13. B. Describe another context where four dimensions can be used in representing some features of the context. 2. Hypercubes in Higher Dimensions. The 16 vertices of the 4-dimensional hypercube can be described by the collection of ordered quadruples (a,b,c,d) where the numbers  a, b, c, and d are either 0 or 1. Write a description of the vertices of the 5-dimensional hypercube. How many vertices does the 5-dimensional hypercube have? How many vertices does the 6-dimensional hypercube have?  How many vertices does the 10-dimensional hypercube have? What can you say about the vertices for the hypercube of dimension N. 3. Find  and "copy" 3 different world maps. Describe how each map deals with lines of longitude, latitude, and the poles. 9 3-17 No Class - Spring Break 3-19 no class 10 3-24 Read 3-26 Read - work on Portfolios 11 3-31 No class CC Day 4-2  1. Find a Sudoku puzzle and show how to add 4 numbers to the puzzle solution. "Prove" that the 4 numbers you have added must be a part of the solution to the puzzle. [Note: Since you can find a solution to most puzzles, the assignment here is to provide the explanation/ proof. Without the proof you will not be given credit for this work.] 2. a. Following the proof that the square root of 3 is not a rational number, give a proof that the square root of 5 is not a rational number. b.Discuss why the outline of the proof to show that the square root 3 and 5 is not a rational number will work for other numbers as long as they are not perfect squares like 4, 9, 16, 25, 36, ... 3. Make a torus from a single sheet of material using the pattern . 12 4-7 Reading 4-9 Work on Portfolio Entries, 13 4-14 Reading. 4-16 1. Classifications by topological equivalence:    It is often useful to classify visual objects using topological equivalence. For example, the letter "T " as it appears on this page has a single point which separates the letter into three disconnected pieces, whereas the letter "I " has only two disconnected pieces for any single point. Group the following letters as printed on this page together in different classes determined by topological equivalence. Indicate what distinguishes the resulting classes from each other. [It is up to you to determine the appropriate classes.] A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 2. G-Coding the rational numbers.  a.Write an explanation for the following statement: "Every positive natural number is a rational  number." Consider the following G-code for positive rational numbers of the form a/b where a and b are positive counting numbers: The number a/b is G-coded by the natural number 2a 3b. b. i.Find the rational numbers G-coded by the number 2, 4, 6, 9, 12, 16. ii. List the next ten numbers that G-code rational numbers. iii. Explain why 1000000 is not on the list of numbers G-coding rational numbers. c. Write an explanation for the following statement: "There are fewer positive rational numbers than there are natural numbers." d. Discuss the following statement: "The statements in part a and c contradictory." 14 4-21 4-23 Portfolios are due! 15 4-28 Reading 4-30 Continue to work on Projects- Due Monday May 5th by noon. 16 5-5 Projects due by Noon. Reading 5-7 Final Assignment.[May be submitted on paper or on-line until 5-13] I. Surfaces: A.Describe 3 distinct physical objects which have topologically different closed surfaces. For each surface discuss the number of holes and the Euler characteristic for that surface. B. Draw two different networks on a map on the mobius band: one that has five regions that requires five colors. and one with six regions that requires six colors. Explain your  why each of your sketches is correct. Compute the Euler characteristic for these networks. II.Over the term we have covered many topics in class and through the readings. Choose two topics we have studied for examples in writing a paper (1-3 pages) discussing one of the following statements:      A. The study of  mathematics is powerful in showing what can be done and also what cannot be done.      B. The amazing thing about mathematics is how it is able to turn even the simplest things into abstractions and that very abstract things can often be coded using just the counting counting numbers. BELOW THIS LINE ALL ASSIGNMENTS ARE  TENTATIVE! 15 Unused old Assignments!
General Reading Assignments (Revised 1-26-2014)
Week Assnm't Source  Chapter, pages and Investigations for Reading/Thought Comments, Web Sites to Visit, and other things
1and 2
Flatland

Truth...

Introduction, Preface, and Part I.

Preface, Navigating.
Ch.1 Doubt: 1.1
(14-22), 1.3 (35-36; 37-39)

Flatland is available on the web.
Over 30 proofs of the Pythagorean theorem!
Many Java Applets that visualize proofs of the Pythagorean Theorem

Tangram Introduction

Web references related to scissors congruence- dissections.
2 and 3
Flatland

Truth...

Finish reading Flatland.

Ch.1: 1.2 (23-25; 29; 31-32)
1.5 (53,54, 56)

A wealth of materials can be found by going to this Tesselation Tutorial.
This might be a good time to visit Rug patterns and Mathematics exhibit plus...

Temple Grandin: The world needs all kinds of minds | Video on TED ...
Ames Room Temple Grandin Movie Link on YouTube
4 and 5
Truth...
Ch.3: 3.1, 3.2, 3.3 (1-4), 3.5 (19-21)
Ch. 4: 4.1 (18-23), (25,26)
Ch.5: 5.4.4 (66-70)
Review: A wealth of materials can be found by going to this Tessellation Tutorial.
This might be a good time to visit Rug patterns and Mathematics exhibit plus...
You might want to visit the Geometry Center's Introduction to Tilings as well as the  Kali: Symmetry group page now .

You might want to look at Penrose tilings by downloading Winlab by Richard Parris.

6 and 7
Truth... Ch. 4: 4.2 (40-53); 4.3
Ch 5: 5.8 (95-103)
How We Classify Border Patterns
Wallpaper groups.
Rug patterns and Mathematics exhibit plus...
You can look at polyhedra by downloading Wingeom by Richard Parris
The Platonic solids  is an interesting site with Java viewers for interactive manipulation created by Peter Alfeld of Univ. of Utah.

8 and 9
Truth...
Ch. 5: 5.1 (1-5); 5.2 (16-26)
More to come. :)

10 ,11,and 12! Truth... Ch 4: 4.2 (40-53 again!) , 4.3
5.1 (1-15)
5.5 (80-84)
5.8 (TBA)
6.2
Maps and coordinates/ Cartesian coordinates
Models for geometry.

The Fourth dimension  A Visualization of 4d hypercube (YouTube).
Coding and numbers [wikipedia on Codes]
Prime Numbers
The Fundamental Theorem of Arithmetic
13-15
Functions of Proof
Truth...
THE ROLE AND FUNCTION OF PROOF : Introduction (on MOODLE)
5.8.  (95-100)
5.9
7.2 (assignment revised 4-30)
7.6 (pp 109-119) We will do many of these activities in class.
(Optional: 7.6 p120)

The Four-Color Theorem at mathworld.wolfram.com
"proof " summary of the four color theorem
Proof!
knights and knaves
Surfaces in topology

The Moebius strip,  The Klein bottle, orientability, and dimension.
Constructing surfaces in general

BELOW THIS LINE ALL ASSIGNMENTS ARE  TENTATIVE!

Perspective drawing
Projective Geometry
Configurations

13 and  14

There is material that reviews proportion, lines, and parabolas at the Purple Math website.
 Ratio & Proportion Graphing Overview Slope of a straight line Slope and Graphing Slope and y-intercept Function Notation Functions graphing quadratics Graphing: Quadratic Equations   How to Derive the Vertex Formula

## Resource List for Portfolio Entries

The following list contains suggestions  for finding resources as well as the names of resources that may be used for one or more portfolio entries. Before reading an article in one of these resources thoroughly it is advisable to scan it quickly to see that it contains something of interest to yourself. Your portfolio entry can report on the content of your reading, illustrate it by examples, and/or follow up on it with your own response and creativity.

The content of the portfolio entry should relate specifically and directly to some visual mathematics. Personal observations , philosophical musings, and aesthetic judgments are not adequate connections to something visual by themselves to qualify as mathematical content.

These articles may also be useful in developing a deper level of understanding on a topic which will support your term project. I will add to this list as the term progresses.

• Several chapters from the course text will not be covered in class but can be used for portfolio entries. An entry based on our text should report on a selection of the included exercises along with the content of the chapter.
• Use my collection of Visual Mathematics web sites for surfing visual mathematics and geometry.
• Use articles from old Scientific American magazines (located outside my office at Library 48 and available from HSU on-line)
• (Older issues) Martin Gardiner's articles are usually short and clear enough to provide material for one or even two even entries.
• (More recent issues) Ian Stewart 's articles are similar and about as playful as the Gardner pieces.
• Some issues  have had articles on special topics that are relevant to our interests. These are usually longer and require a little more effort to digest - though well worth the effort.
• "Topology" by Tucker and Bailey, 1950, pp 8-24.
• A number of liberal arts / mathematics textbooks contain chapters that would be suitable for reporting.
• Mathematics: the Man-made Universe by Sherman Stein.
• Excursions into Mathematics by Beck, Bleicher, and Crowe.
• What is Mathematics? by Courant and Robbins.
• The World of Mathematics by Newman.
• The library has a collection of films and videos that are relevant to our interests.
• For All Practical Purposes (COMAP)
• Some of the history of mathematics videos from the Open University Series (BBC)
• There are several non-text mathematics books and collections of essays.
• Mathematics: The Science of Patterns  by  K. Devlin
• Beyond the Third Dimension by T. Banchoff.
• Martin Gardiner has many books full of puzzles and recreations many of which are relevant.
• The Problems of Mathematics by Ian Stewart.
• The Mathematical Experience by Philip Davis and Reuben Hersh