Math 44, Winter 2013, Home Page

Math 44
Fall 2014 Home Page
Green Sheet

Class 20, Tue., Dec. 4, 2014
We worked on fractal dimension and did some review.

Here's the 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.
And here are some solutions to the problems in the lists above.

Class 19, Tue., Dec. 2, 2014
We heard project reports. Good work!

Class 18, Tue., Nov. 25, 2014
We worked on projects, which are due on Tuesday. We also learned a little more about fractals, from chapter 6.

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.

Class 17, Thu., Nov. 20, 2014
We worked on cellular automata and fractals. We discussed the final projects, which are due the last day of class; however, a prior report on what you plan to do for your project is due next Tuesday, Nov. 25.

Please turn in the cellular automata and fractal handouts on Tuesday.

Class 16, Tue., Nov. 18, 2014
We heard reports on the 3rd paper, and also learned 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.

Class 15, Tue., Nov. 11, 2014
We learned more about graph theory, including Hamiltonian cycles.
We also learned how to do the parity "full house" card trick.
We also learned about surfaces like the Mobius strip.

Most of you turned in the take-home exams; those who did not may still turn them in on Thursday at the Physical Sciences Division Office in S-31 (I will be absent that day, and it does not appear that there will be a sub for that class) - however you must get it to the secretary on Thursday, she will put it in my mailbox.

For your third written report, due on Tuesday, Nov. 18, 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 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.

Please check out one of the books and do a report on one chapter due on Tuesday, Nov. 18. You can email me which chapter you want to do a report on, if you have any question. 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.

Class 14, Thu., Nov. 6, 2014
We worked on take-home exams and also learned the graph theory alphabet. We also built several polyhedra.

Class 13, Tue., Nov. 4, 2014
We learned about polyhedra and more symmetry.

Class 12, Thu., Oct. 30, 2014
We learned about symmetries in the plane, and worked on several handouts. We also started learning about polyhedra.
Please complete the handout in which you find all 24 ways to construct 4-fold rotational symmetries in a 5 by 5 grid, and hand in Tuesday.

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.

Class 11, Tue., Oct. 28, 2014
We spent most of class hearing reports. We will work on symmetry and polyhedra (chapter 4) next class.
Also bring your take home exams, we will spend some time on them.

Class 10, Thu., Oct. 21, 2014
We went over the exam and also the next math biography assignment due next Tuesday.

Take home exam will be emailed to you shortly.

Your mathematician biography has been assigned.
The reference books, Mathematical People and More Mathematical People willl be placed 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.

Class 9, Tue., Oct. 21, 2014
We had exam 1. We also went over a variety of problems from previous chapters.

Your mathematical biography is due next Tuesday. Here is a web page with the assignments and links to interviews and articles about some of the mathematicians who've been assigned. For those still needing resources, I'll help you find some shortly.

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

Class 8, Th., Oct. 16, 2014
We went over section 4.1 on the Pythagorean theorem, 4.2 on the Art Gallery theorem, and 4.3 on the Golden Ratio.
We also learned about the pentominoes, polyominoes, and tessellations and practiced using pentominoes to make tessellations.

Your first exam is this Tuesday, Oct. 21, see the sample questions below.
Chapter 2 homework is due this Tuesday also.

Tuesday, Oct. 21 is the 100th birthday of Martin Gardner, and here is an article from the New York Times about Martin Gardner, in honor of his 95th birthday.

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'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.

Class 7, Tue., Oct. 14, 2014
Your mathematician biography has been assigned. Here are the mathematician bios assigned.
We postponed the first exam until next Tuesday, Oct. 21.
Homework for chapter 2 will be due then also.
The Mathematical Biography paper will be due a week from Tuesday, Oct. 28.
Today we did more with number theory and saw the relationship between polyrhythm and star polygons.
We also learned a little about the RSA code.

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

Class 6, Thu., Oct. 9, 2014
We went over material from chapter 2.3 and 2.4.
First exam is next Thursday, when chapter 2 homework will be due.
We learned about modular arithmetic, also Fermat's last theorem and the twin prime conjectiure.

We learned to clap the sequence of vowels and consonants in our name - please practice!!

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.

Class 5, Tue., Oct. 7, 2014
We went over material from chapter 2.3 and 2.4 and introduced the idea of "modular arithmetic."
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.
Please print them out, we will go over in class on Thursday.

Your mathematician biography has been assigned. Here are the mathematician bios assigned (those not in class Tuesday will have one assigned on Thursday.)
The reference books, Mathematical People and More Mathematical People willl be placed 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.

Here is a study guide for the first exam from a previous quarter. We have not gone over all the material yet.
Here is another practice exam from a previous quarter, more like what you will have, since it includes a multiple choice/Scantron exam.

Class 4, Thu., Oct. 2, 2014
We went over chapters 2.2 and part of 2.3.
Work on homework from chapters 2.1 and 2.2.
On Tuesday, Oct. 7 turn in the Fibonacci assignment. (Note: this is the correct due date - had it incorrect for a few hours!)

Class 3, Tue., Sep. 30, 2014
Chapter 1 homework is due on Thursday.

Today we went over the pigeonhole principle and also several other aspects of sequences of numbers, especially the Fibonacci numbers.

Here is a link to the MoSAIC event this Friday and Saturday in Berkeley. You can use it as the subject of the second paper.

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.

A student asked about turning in homework on Turnitin.com. Only the written papers will be turned in on Turnitin.com, the homework will be turned in on paper. The chapter 1 homework is not due until next Thursday, Oct. 2. (You will be turning in all sections of each chapter homework at one time, not section by section each class session.)

The first paper, your math autobiography, worth 5% or your grade, is due this coming Tuesday by 1:30 PM on Turnitin. Each one of you is registered on Turnitin already under the email address that De Anza has for you. Check to see that your name is already there.

Class 2, Thu., Sep. 25
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!

We acted out the $7,$8,$9,$10 problem (#3 in the homework) and worked out the "moat and castle" problem, the problem on the two weighings, and the river crossing problem (#8 in homework). Do problems 1-15 at the end of chapter 1 (on pages 33-37). These will be due next Thursday.
You have your math autobiography due on Tuesday at the start of class (see green sheet above).

Class 1, Tue., Sep. 23
We played the pattern game. Here's a handout about the pattern game, please print and include in your notebook.
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 developed 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 next Thursday. Your math autobiography is due next Tuesday (see the green sheet above, which has a few corrections.) Please get your textbook right away!

Some links from previous quarters:

The problems on the final from the material in chapter six cover the following:
(1) 1- and 2-dimensional cellular automata.
(2) Calculate the dimension of a fractal - please bring your calculator!
(3) What does chaos mean (see section 6.5)

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.

We taked about projects. Your project description is due on Thursday, and the project is due the last day of class, next Thursday.

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.

Here is your project description, for the project which is due the last week of class.

For your third written report, you are to report on a mathematical subject connected to this course, that has caught your interest. 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.

We saw part of the film about Martin Gardner and learned about Euler Circuits.

We learned about numbers of graphs.
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.

We went over chapter 4.3, and did a movement activity on symmetry. Here are the handouts we worked on at the end of class, which you should turn in next Tuesday:
Geoboard paths
Rotational Symmetry
Reflection symmetry

We had exam 1, and went over 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.

We also heard a bit of Sophie Germain's story. Here's a nicely written bio about Germain.

Here is a study guide for the first exam from a previous quarter. We have not gone over numbers 4(b), 4(c), and 10, that material is not on the test.
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.

Your mathematician biography is due a week from Thursday, Feb. 7. Here are the mathematician bios assigned.
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.

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 are the mathematician bios which were assigned in class. If you were absent, I assigned you someone also.

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

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 your f inal project form, for the project which is due the last week of class.

For your third written report, you are to report on a mathematical subject connected to this course, that has caught your interest. 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.

Please check out one of the books and sign up to do a report on one chapter. You can email me which chapter you want to do a report on, if you have any question. 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 Th., Nov. 6.

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.

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.

Please 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.