Math 44, Fall 2015, Home Page

Math 44
Fall 2015 Home Page
Green Sheet

Class 19, Tue., Nov. 24
We did more with fractals and learned about complex numbers and the Verhulst equation.
Ch. 6 homework is due next week, on Tuesday if you want it back before exam, otherwise on Thursday.

Paper is due on Tuesday on Turnitin.com.

Class 18, Thursday, Nov. 19, 2015
We learned more about fractals and practiced finding the fractal dimension.
We will be doing more with fractals next week as well.
Please turn in chapter 5 homework on Tuesday of next week, Nov. 24.

Class 17, Tuesday, Nov. 17, 2015
We went over new material on fractals, cellular automata, and Conway's Game of Life.

Please turn in chapter 5 homework next Tuesday, Nov. 24.

Here is a short handout on 1-dimensional cellular automata, like we did in class.

Here is the site with the 4 rows of 5 trick that I did in class.

Third Essay Assignment.
You have a short paper on a subject related to the course that catches your interest due Tue, Dec. 2 (two weeks later than originally assigned in green sheet), and worth 8.33% of your grade. You will turn the paper in via Turnitin.com. If you reported on the math/dance concert last time, you may do your math autobiography this time. You may NOT report on the Fibonacci numbers or the Golden Ratio!

Here's the description of the essay:
Report on an article or chapter from a popular book about mathematics or math education. The report will be one to two pages long, typewritten, (it must be at least 600 words), and will cover the mathematics from one to several chapters of a book from the following list; other books or sources may also be used. You must use published material, not just web sites, unless you get permission from the instructor, and you MUST cite your sources. A short oral report to the class will also be required.
You should include in what you write and talk about:
(1) why you chose this topic,
(2) what you learned, and
(3) what you think about the subject in question.
(4) What you might like to find out about the subject in the future.

Authors suggested: Martin Gardner, Ian Steward, Keith Devlin, Ivars Peterson.

Examples of books with mathematical content:
The Mathematical Tourist and Islands of Truth, by Ivars Peterson.
Any of the books of Martin Gardner on mathematics (over 15 titles).
Game, Set, Math and Does God Play Dice by Ian Stewart, or other titles on math by Stewart.
The Mathematical Experience by Davis and Hersh.
A Number For Your Thoughts and Numbers At Work and At Play by Stephen P. Richards.
Tilings and Patterns by Grunbaum and Shepard.
Mathematical Snapshots by Steinhaus.
Mathematics: The New Golden Age by Keith Devlin, or other titles by Devlin.
The Emperor's New Mind by Roger Penrose.
The Mathematics of Games by John Beasley.
Archimedes' Revenge by Paul Hoffman
What is Happening in the Mathematical Sciences, ed. by Barry Cipra, Vols 1-5 (on reserve in campus library)

Examples of books with cultural content:
Ethnomathematics by Marcia Ascher.
You can also consult this Multicultural Mathematics Bibliography. Many of the references are in our library, and the bibliography contains call numbers for those that are in the library.
A number of Martin Gardner's books are in the De Anza library.

Class 16, Thursday, Nov. 12, 2015
We made human knots, learned about math from non-European sources, and worked on take-home exams.

Class 15, Tues., Nov. 10, 2015
We worked on take-home exams and did more with symmetry and tessellations. We also learned about making polyhedra with loops of string.

Class 14, Thursday, Nov. 5, 2015
We worked on rhythm marches and built the 4 by 4 symmetry table using the 4 planar symmetries.
We also worked on take home exams.

Class 13, Tuesday, Nov. 3, 2015
We made mobius strips and learned how to do the rope trick.
Turn in chapter 4 homework on Tuesday, Nov. 10.

Here is the site with the 4 rows of 5 trick that I did in class.

Film about Martin Gardner and learned about Euler Circuits.

We also looked at the "Small World Phenomenon," and "six degrees of separation,"
Here are some more links:
Six degrees of Kevin Bacon.
Oracle of Bacon, where you can look up actor's links to other actors.
Kevin Bacon's charitable SixDegrees.org.
The Erdos-Bacon number site.
The Facebook Six Degrees site.
The Erdos Number project.

How to make polyhedra with our fingers, see these two short videos:
Handmade Math
Polyhedra at Your Fingertips

Class 12, Thursday, Oct. 29, 2015
We went over the graph theory alphabet, found all 11 "simple" graphs (no loops or multiple edges) with 4 vertices, learned about six degrees of separation, and handed in the symmetry handouts. We also learned more about polyhedra, including Euler's formula
v - e + f = 2.
Chapter 4 homework will be due on Tuesday of next week, Nov. 3.

Class 11, Tuesday, Oct. 27, 2015
We worked on symmetry and built all five Platonic solids.

Class 10, Thursday, Oct. 22, 2015

Class 9, Tuesday, Oct. 20, 2015
Substitute Kejian Shi had you do the oral report version of your mathematician biography.

Class 8, Thursday, Oct. 15, 2015
Here is the list of math bios. The list has links to articles, etc. If there's an X, then there is an interview in one of the two books on reserve in the campus library (Mathematical People and More Mathematical People). Description of the assignement is below. This is due on Turnitin.com on Tuesday, Oct 20. The report will be 600 words, and will also involve a short 2 to 3 minute oral report. Include in your paper some response by yourself to the person - is this someone you might like to have a conversation with, or is it someone who does not seem very interesting to you? Your main reference must be a printed source, not a web site, and you must cite any sources you use.

We learned about four planar symmetries:
translation: p --> p
reflection: p --> q
180 degree rotation: p --> d
glide reflection: p --> b

Link on pentominoes.
Here's a handout showing you how to find the symmetries in a tiling! Here are hundreds of links to some amazing tiling sites. Pentominoes each have five identical unit squares, linked complete edge to edge. For example, we might try to solve the same problem for the "hexomino:" there are 34 hexominoes, each having six squares. Here is a diagram indicating briefly how all 34 hexominoes tile the plane. These are not shown in complete detail; if you create a pentomino tiling it should extend further than these, and indicate how the pattern you found really could extend in all directions throughout the plane.

Here's an online article by Ivars Peterson on drivers license codes.
Here's Joseph Gallian's site on check digits in codes.
Here is the wikipedia article on UPC codes, with lots of detail.

Class 7, Thursday, Oct. 13, 2015
We went over the sample exam and had exam 1. Turn in chapter 2 homework on Thursday of this week.

Handout with problems on Patterns and Modular Arithmetic
The Hidden Role of Modular Arithmetic, that reviews material on modular arithmetic.
Here are some links about modular arithmetic, which we will learn more about throughout the quarter:
Here's a site on modular arithmetic.
Here's a site which will do modular arithmetic calculations for you.
Here's a site on modular arithmetic by Susan Addington.

Class 6, Thursday, Oct. 8, 2015
We went over material on modular arithmetic in chapter 2.4, including the ISBN error detecting code. Here is a short handout on modular arithmetic that we went over in class. We also began chapter 4 (we skip chapter 3) on geometry, including material on the Pythagorean Theorem and the Art Gallery Theorem. We also briefly looked at chapter 4.3 on the golden rectangle.

Here is the list of math bios. Some of the people you signed up for are not there because they did not get saved from Tuesday's class, so let me know if you missed class today and remember who you were assigned to report on. The list has links to articles, etc. If there's an X, then there is an interview in one of the two books on reserve in the library. Description of the assignement is below. This is due on Turnitin.com on Tuesday, Oct 20.

Your first exam is next Tuesday, Oct. 13. Here is a study guide for the first exam from a previous quarter.
Here is another practice exam from a previous quarter, more like what you will have, since it's a multiple choice/Scantron exam. Please bring the half page long scantron for your exam 1. This will be open book, open notes, calculator allowed, one hour.

Class 5, Tuesday, Oct. 6, 2015
We signed up for mathematician bios (to be turned in on Turnitin.com, due in two weeks) - will post the list of student assignements tomorrow. Here are Math biography assignment links to interviews and articles
The reference books, Mathematical People and More Mathematical People are on reserve in the campus library. The report will be 600 words, and will also involve a short 2 to 3 minute oral report. Include in your paper some response by yourself to the person - is this someone you might like to have a conversation with, or is it someone who does not seem very interesting to you? Your main reference must be a printed source, not a web site, and you must cite any sources you use.

We learned about the "billiard ball method" of solving water pouring problems.The water measuring problem appears in one of the Die Hard movies, and according to puzzle history expert David Singmaster and others, goes back to a book called "Annales Stadenses" compiled by the monk Abbot Albert in the year 1240 CE in Germany. The billiard ball solution method is apparently due to M.C.K. Tweedie in 1939. If given 3 containers holding x, y, and z gallons of water, there is no known simple method for determining which amounts can be measured if z happens to be smaller than x + y - the billiard method is not that helpful in this case. For example, if you have three containers holding 6, 10, and 15 gallons, and an unlimited supply of water, which amounts 1 through 14 is it possible to measure exactly?
Here's a site where you can get free software for implementing the billiard ball approach to the water pouring problem. You have to download the "Mathematica Player" (also free) first.
Here's an article explaining the method.

We learned a little about modular arithmetic. Here is a short handout on modular arithmetic; please print and birng to class Thursday.
We went over the star polygon constructions, and saw some art work/icons which use star polygons.

Class 4, Thursday, Oct. 1, 2015
On Tuesday, Oct. 6 turn in the Fibonacci assignment.
Bring the "clocks" used for making star polygons to class Tuesday and we will complete them if you haven't already - see if you can draw star polygons {8/7} and {8/4} as part of homework. Also, on the reverse side, {12/n} polygons for various n between 1 and 11.

We went over a few homework problems. I'll find a nice explanation of the 12 coin problem for you...
We also saw a film about Martin Gardner and I mentioned the next writing assignment due Oct. 20, which we'll assign specific mathematicians to each student in the class during the next class.

Here are two videos on the finger polyhedra:
Scott Kim talking about and demonstrating the four-finger tetrahdron.
The four-hand tetrahedron, and others.

Here is someone doing that "tie a knot without letting go of the ends" trick!

We also made rhythms out of our names, and saw how to put two rhythms together to make a star polygon.
Here's the "Clap your name" activity.

Class 3, Sep. 29, 2015
We learned how to count Fibonacci numbers on pine cones.
On Tuesday, Oct. 6 turn in the Fibonacci assignment.

We learned about the pigeonhole principle. In the pigeonhole principle "magic trick," I asked you to choose seven numbers from the list 1,2,3,...,12. The properties each of your lists had were:
(1) A pair of your numbers had a sum of 13.
(2) A pair of your numbers had a difference of 6.
(3) A pair of your numbers had a difference of 3.
(4) A pair of your numbers had the property that their only common factor was 1.
(5) A pair of your numbers had the property that one divided the other equally.
We saw how the pigeonhole principle explained why properties 1, 2, 3, and 4 are true. For Thursday, can you use the pigeonhole principle to explain property 5? Remember, it's all in how you label to (six) pigeonholes! Property 5 is more difficult to explain; in this case the six pigeonholes have different numbers of numbers assigned to them!

We also learned about continued fractions and how they relate to the Fibonacci numbers.
Here's a long intro to continued fractions.

Homework for chapter 1 is due Thursday of this week.

Here are Vi Hart's Fibonacci number videos.
Here is a great site about Fibonacci numbers.
Here is a Fibonacci site with lots of pictures and interactive applets.
Here is an interactive site that helps explain phyllotaxis, which is the pattern of spirals in many plants.
Here is a site about Pingala's possible use of the Fibonacci Numbers in ancient India.
Look up Rachel Hall who credits Indian mathematician/musician with the Fibonacci numbers - see her article "Math for Poets and Drummers" listed at her site.

Class 2, Sep. 24, 2015
We did the Comedians and Officers puzzle, and also the Fido puzzle.

Work on chapter 1 problems 1-15, due at the fourth class, Thursday of next week.
Your math autobiography is due next Tuesday (see the green sheet above.)
Please get your textbook right away!

Class 1, Tue., Sep. 22, 2015
We played the pattern game.
We played the take-away game, in which each player removes 1 or 2 counters on each move, the last player to move winning. We will develop a winning strategy involving leaving your opponent with a multiple of 3 on each turn, if possible.
Work on chapter 1 problems 1-15, due at the fourth class, Thursday of next week.
Your math autobiography is due next Tuesday (see the green sheet above.)
Please get your textbook right away!

I've uploaded everyone's name and email address to turnitin.com for your first paper.
ID for the class on Turnitin:
Class name: Math46, Fall 2015
Enrollment password: M46ForFall15
Class ID: 10773286

All of these puzzles in chapter 1 are famous ones in what is known as "recreational mathematics." The water measuring problem, according to puzzle history expert David Singmaster and others, goes back to a book called "Annales Stadenses" compiled by the monk Abbot Albert in the year 1240 CE in Germany. The billiard ball solution method is apparently due to M.C.K. Tweedie in 1939. If given 3 containers holding x, y, and z gallons of water, there is no known simple method for determining which amounts can be measured if z happens to be smaller than x + y. For example, if you have three containers holding 6, 10, and 15 gallons, and an unlimited supply of water, which amounts 1 through 14 is it possible to measure exactly?

Here's a reference on the weighing problems. (Problem 15 in the chapter 1 homework is quite difficult.) Here's a link to the New York Times puzzle column (called Numberplay). Unfortunately, they only allow you to read it a few times before charging for access!

Here's a solution to the 12 coin problem from chapter 1 written up by John Conway - in case you want to see a solution!

Papers for this class:
(1) Math Autobiography (see second page of green sheet). Due third class session.
(2) Math biography assignment: links to interviews and articles
The reference books, Mathematical People and More Mathematical People are on reserve in the campus library. The report will be 600 words, and will also involve a short 2 to 3 minute oral report. Include in your paper some response by yourself to the person - is this someone you might like to have a conversation with, or is it someone who does not seem very interesting to you? Your main reference must be a printed source, not a web site, and you must cite any sources you use.
(3) For your third written report, due on Tuesday, May 12, you are to report on a mathematical subject connected to this course, that has caught your interest. You may do a report on the dance concert The Daughters of Hypatia: Circles of Mathematical Women, Th., Apr. 30, 2015 at the De Anza College Visual and Performaing Arts Center. You may not report on the Fibonacci numbers or the Golden Ratio! The report must be at least 600 words, and will cover the mathematics from one to several chapters of a book from the following list; other books or sources may also be used. You must use published material, not just web sites, unless you get prior permission from the instructor, and you MUST cite your sources. A short oral report to the class will also be required.

You should include in what you write and talk about:

(1) why you chose this topic,
(2) what you learned, and
(3) what you think about the subject in question.
(4) What you might like to find out about the subject in the future.

I suggest that you read a chapter or two from one of Martin Gardner's books on mathematics, he covers a wide variety of topics. Here is a list of a few of the 70 books by Martin Gardner. Choose a chapter (or two) that is of interest to you and report on it:
The Colossal Book of Mathematics
Mathematical Circus
Mathematical Magic Show
The Magic Numbers of Dr. Matrix
Mathematical Carnival
Knotted Doughnuts and Other Mathematical Entertainments
Wheels, Life, and other Mathematical Amusements
Time Travel and Other Mathematical Bewilderments

You might also want to look at books by Ian Stewart, Keith Devlin, or Ivars Peterson.

This will be a 600 word paper, as before, submitted to Turnitin.com. The main criteria for how you select which chapter to report on is that it should be of interest to you. You will also give a brief 2-3 minute oral report on what you learned to the class, if we have time, on Tuesday, Nov. 17.

Study guides for first and final exam:
Here is a study guide for the first exam from a previous quarter.
Here is another practice exam from a previous quarter, more like what you will have, since it's a multiple choice/Scantron exam. Please bring the half page long scantron for your exam 1.

Here's a final exam sample problem list from a previous quarter. Here's another list of sample problems.
And here are some solutions to the problems in the lists above.

Some links from previous quarters:

Here is a short handout on 1-dimensional cellular automata, like we did in class.

Here is the site with the 4 rows of 5 trick that I did in class.

Handouts on symmetry
Geoboard paths
Rotational Symmetry
Reflection symmetry

The "billiard method" of solving water pouring problems.
Here's a site where you can get free software for implementing the billiard ball approach to the water pouring problem. You have to download the "Mathematica Player" (also free) first.

Sophie Germain's story. Here's a nicely written bio about Germain.

Here's an online article by Ivars Peterson on drivers license codes.
Here's Joseph Gallian's site on check digits in codes.

Here is a short handout on modular arithmetic; please print and birng to class Tuesday.

Here is the wikipedia article on UPC codes, with lots of detail. The reason it did not "work" in class the other day: we were looking at the bar code for the ISBN, not the UPC!

We also made rhythms out of our names, and saw how to put two rhythms together to make a star polygon. Here's a solution to the 12 coin problem written up by John Conway.

Here is the Fibonacci assignment,

In the pigeonhole principle "magic trick," I asked you to choose seven numbers from the list 1,2,3,...,12. The properties each of your lists had were:
(1) A pair of your numbers had a sum of 13.
(2) A pair of your numbers had a difference of 6.
(3) A pair of your numbers had a difference of 3.
(4) A pair of your numbers had the property that their only common factor was 1.
(5) A pair of your numbers had the property that one divided the other equally.
We saw how the pigeonhole principle explained why properties 1,2, and 3 are true. For Thursday, can you use the pigeonhole principle to explain property 4? Remember, it's all in how you label to (six) pigeonholes!
Property 5 is more difficult to explain; in this case the six pigeonholes have different numbers of numbers assigned to them!

Here is the Fibonacci assignment, which is due next Thursday, Jan. 24.

Here are Vi Hart's Fibonacci number videos.
A great site about Fibonacci numbers.
Fibonacci site with lots of pictures and interactive applets.
Interactive site that helps explain phyllotaxis, which is the pattern of spirals in many plants.
Pingala's possible use of the Fibonacci Numbers
Rachel Hall who credits Indian mathematician/musician with the Fibonacci numbers - see her article "Math for Poets and Drummers" listed at her site.

Here is an article published in the New York Times about Martin Gardner, in honor of his 95th birthday.

Symmetry leads us to the field of "tilings" or "tessellations." There is an enormous amount of recent mathematics and art on this subject. Take a look at David Eppstein's enormous collection of links on tilings. He calls his collection of such links the Geometry Junkyard; it's very entertaining, in a mathematical and arts sort of way!

Link on pentominos.
Here's a handout showing you how to find the symmetries in a tiling! Here are hundreds of links to some amazing tiling sites. Pentominoes each have five identical unit squares, linked complete edge to edge. For example, we might try to solve the same problem for the "hexomino:" there are 34 hexominoes, each having six squares. Here is a diagram indicating briefly how all 34 hexominoes tile the plane. These are not shown in complete detail; your penomino tiling should extend further than these, and indicate how the pattern you found really could extend in all directions throughout the plane.

Rotational symmetry handout.
4-Fold rotational symmetry handout.

Here's a list of "distributed" mathematical projects now being conducted throughout the internet.
Here's the "World Community Grid" site listing many online projects in which you (and your computer) can participate!
Here's another site for "Volunteer Computer Grids."

Patterns and Modular Arithmetic.
Modular Arithmetic Intro handout.

Here's a site where you can get free software for implementing the billiard ball approach to the water pouring problem. You have to download the "Mathematica Player" (also free) first.

We also played the "take-away" game, and saw an analysis of it that introduced modular arithmetic. And we learned a little about the history of modular arithmetic.

Visit Scott Kim's homepage and look at the many examples of Inversions, some from his book by the same name, some animated. Recently other artists have produced these letterform images that exhibit symmetry, for example here is a site for ambigram tattoos by Mark Palmer. The artist John Langdon has also been creating Ambigrams for many years, and produced those used by Dan Brown in his book Angels and Demons. See Langdon's site for examples of his work, or look for his book Wordplay. Here's a site by the mathematician and artist Burkard Polster with lots of playful examples.

Symmetry leads us to the field of "tilings" or "tessellations." There is an enormous amount of recent mathematics and art on this subject. Take a look at David Eppstein's enormous collection of links on tilings. He calls his collection of such links the Geometry Junkyard; it's very entertaining, in a mathematical and arts sort of way!

We also played the "take-away" game, and saw an analysis of it that introduced modular arithmetic. And we learned a little about the history of modular arithmetic.

We went over a number of problems from chapter 1, including the water pouring problem turned into a" billiards" problem. Here's a site where you can get free software for implementing the billiard ball approach to the water pouring problem. You have to download the "Mathematica Player" (also free) first.

All of these puzzles are famous ones in what is known as "recreational mathematics." The water measuring problem, according to puzzle history expert David Singmaster and others, goes back to a book called "Annales Stadenses" compiled by the monk Abbot Albert in the year 1240 CE in Germany. The billiard ball solution method is apparently due to M.C.K. Tweedie in 1939. If given 3 containers holding x, y, and z gallons of water, there is no known simple method for determining which amounts can be measured if z happens to be smaller than x + y. For example, if you have three containers holding 6, 10, and 15 gallons, and an unlimited supply of water, which amounts 1 through 14 is it possible to measure exactly?

Here's a reference on the weighing problems. Here's a link to the New York Times puzzle column (called Numberplay). Unfortunately, they only allow you to read it a few times before charging for access!

Here is the Fibonacci assignment, which will be due one week from Tuesday on Tuesday, Oct. 7. Read it CAREFULLY and do everything it asks!!

Here are some more links on Fibonacci numbers:
Vi Hart's Fibonacci number videos.
A great site about Fibonacci numbers.
Fibonacci site with lots of pictures and interactive applets.
Interactive site that helps explain phyllotaxis, which is the pattern of spirals in many plants.
Pingala's possible use of the Fibonacci Numbers
Rachel Hall who credits Indian mathematician/musician with the Fibonacci numbers - see her article "Math for Poets and Drummers" listed at her site.

Here is a short handout on 1-dimensional cellular automata, like we did in class.
We had project presentations, went over take-home exam.
Here's a sample problem list from a previous quarter. Yours will be similar, but there are some things here we did not go over, or did not go over very much. Here's another list of sample problems. Again, some similarities, some differences.

Here is the site with the 4 rows of 5 trick that I did in class.
Chapter 5 homework is due next Thursday, the last chapter homework is due on the day of the final exam.

How to tie a knot without letting go of the ends of a piece of string, and how to make a tetrahedron, five pointed star, octahedron, cube, pentagram, and six-pointed star with a loop of rope. Here are some links to videos showing how to make some of these figures.

Please print Patterns and Modular Arithmetic, work on the problems, and bring to next class.

Here is a site where you can communicate about study groups for the take-home, etc.

We will spend more time on polyhedra and symmetry next class. Here are two videos on the finger polyhedra:
Scott Kim talking about and demonstrating the four-finger tetrahdron.
The four-hand tetrahedron, and others.

Here is a set of puzzles created for the 100th birthday of Martin Gardner by the Grabarchuk family.

Link on pentominoes.
Here's a handout showing you how to find the symmetries in a tiling! Here are hundreds of links to some amazing tiling sites. Pentominoes each have five identical unit squares, linked complete edge to edge. For example, we might try to solve the same problem for the "hexomino:" there are 34 hexominoes, each having six squares. Here is a diagram indicating briefly how all 34 hexominoes tile the plane. These are not shown in complete detail; your pentomino tiling should extend further than these, and indicate how the pattern you found really could extend in all directions throughout the plane.

Here's an online article by Ivars Peterson on drivers license codes.
Here's Joseph Gallian's site on check digits in codes.

Here's an article about Yitang Zhang's work on the twin prime conjecture.

Here's a list of "distributed" mathematical projects now being conducted throughout the internet, which we discussed last class. Here's the "World Community Grid" site listing many online projects in which you (and your computer) can participate! Here's another site for "Volunteer Computer Grids."

Here's an online article by Ivars Peterson on drivers license codes.
Here's Joseph Gallian's site on check digits in codes.
Here is the wikipedia article on UPC codes, with lots of detail.

We also looked at prime numbers. Here is a 2 minute slightly nutty but mathematically accurate music video (professionally produced by Nova) of SJSU's Dan Goldston's discoveries about the Twin Prime Conjecture.

See modular arithmetic. And we learned a little about the history of modular arithmetic. Here are two further handouts:
Patterns and Modular Arithmetic.
Modular Arithmetic Intro handout.

Fibonacci assignment.

Here's a final exam sample problem list from a previous quarter. Here's another list of sample problems.
And here are some solutions to the problems in the lists above.

How to make polyhedra with our fingers, see these two short videos:
Handmade Math
Polyhedra at Your Fingertips

Making tessellations with tetrominoes and finding their symmetries (see this handout).

There is an enormous amount of recent mathematics and art on the subject of "tilings" or "tessellations." Take a look at David Eppstein's enormous collection of links on tilings. He calls his collection of such links the Geometry Junkyard; it's very entertaining, in a mathematical and arts sort of way! Here's his page on "Polyominoes and other Animals." Link on pentominoes.
Here are hundreds of links to some amazing tiling sites. Pentominoes each have five identical unit squares, linked complete edge to edge. There are 34 hexominoes, each having six squares. Here is a diagram indicating briefly how all 34 hexominoes tile the plane.

Symmetry hanbdouts:
Geoboard paths
Rotational Symmetry
Reflection symmetry

For your third written report, you are to report on a mathematical subject connected to this course, that has caught your interest. You may NOT report on the Fibonacci numbers or the Golden Ratio! The report must be at least 600 words, and may cover the mathematics from one to several chapters of a book from the following list; other books or sources may also be used. You must use published material, not just web sites, unless you get prior permission from the instructor, and you MUST cite your sources at the end of your paper. A short oral report to the class will also be required.

You should include in what you write and talk about:

(1) why you chose this topic,
(2) what you learned, and
(3) what you think about the subject in question.
(4) What you might like to find out about the subject in the future.

Here is a study guide for the first exam from a previous quarter.
Here is another practice exam from a previous quarter, more like what you will have, since it's a multiple choice/Scantron exam. Please bring the half page long scantron for your exam 1.

Four symmetries:
translation: p-->p
reflection: p-->q
180 degree rotation: p-->d
glide reflection: p-->b

Here's an online article by Ivars Peterson on drivers license codes.
Here's Joseph Gallian's site on check digits in codes.
Here is the wikipedia article on UPC codes, with lots of detail.

Here is a 2 minute slightly nutty but mathematically accurate music video (professionally produced by Nova) of SJSU's Dan Goldston's discoveries about the Twin Prime Conjecture.
Patterns and Modular Arithmetic.
Modular Arithmetic Intro handout.
We'll complete these soon, partly in class, so continue working on them outside of class.

Here's the Fibonacci assignment

Here's an article about Yitang Zhang's work on the twin prime conjecture.

The Fido puzzle, Vi Hart's Fibonacci number video.

Here's the Fibonacci assignment, which will be due a week from Thursday, April 23.
Here are some more links on Fibonacci numbers:
Vi Hart's Fibonacci number videos.
A great site about Fibonacci numbers.
Fibonacci site with lots of pictures and interactive applets.
Interactive site that helps explain phyllotaxis, which is the pattern of spirals in many plants.
Pingala's possible use of the Fibonacci Numbers
Rachel Hall who credits Indian mathematician/musician with the Fibonacci numbers - see her article "Math for Poets and Drummers" listed at her site.

We went over some of the homework problems, used the counters to work on the "cops and comedians" river crossing problem as well as the "frogs on a log" problem, and saw a mathematical magic trick that uses parity (odds and evens). We also briefly learned about the pigeonhole principle and saw another "magic trick" based on it.

When we played the "take-away" game we actually used ideas from modular arithmetic.