Spring 2013 Math 46 Home Page

Green Sheet

Class 16, Th., May 30
We will be going over material from chapter 6.

Take home exam due today!

Class 15, Tue., May 28
We went over the rest of chapter 5.
We also worked on the take-home exam.
We saw some slides about new understandings of how the mind "prevents us from learning simple multiplication facts," due to the strength of our associative memories.

Take home exam due date is postponed until Tuesday, June 4.

Class 14, Th., May 23
We went over material from chapter 5.
We also worked on the take-home exam.

Class 13, Tue. May 21
We went over the rest of chapter 4.
We learned more about codes, and the efficiency of the Euclidean Algorithm, also about Moore's law.
We also learned about prefixes for very large numbers or measurements, for example: kilo-, mega-, giga-, tera-, and peta-.
Also, for very small measurements: milli-, micro-, nano-, and pico-.
For example, 100 gigabytes is 100 billion characters. 30 nanometers is 30 billionths of a meter.

Here are group problem assignments:

Class 12, Th. May 16
We went over more material from chapter 4 on number theory.
We learned about modular arithmetic; here is the handout on magic squares and modular arithmetic which we discussed in class.
You should print it out, answer the questions, and bring to class Tuesday.
Here is another handout with problems on Patterns and Modular Arithmetic which you should also print out - we will go over soon.

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.

We learned about how the fact that multiplying two very large numbers together is easy (with a computer!),
but factoring the product of two very large prime numbers is difficult, is the base for the RSA code and other security schemes used to send credit card numbers over the internet.

We have gone over most of the material in chapter 4.

Class 11, Tue. May 14
We heard oral reports based on papers.
We also talked about prime numbers, the twin prime conjecture, and divisibility tests (see this link for some lesser known tests for 7, 19, etc.)

Here is a link to ana article about a result announced a few days ago about the twin prime conjecture.
It's a big break in the twin prime conjecture!

Class 10, Th. May 9
We went over section 3.6, on calculator use, and introduced chapter 4.1.
We spent a good deal of time on bases other than ten.
We also learned to clap the rhythm of our names, with clap = consonant, slap = vowel.
We also saw some slides about the Brazilian street math study.
Next paper due Tuesday at 4 PM on Turnitin!

Class 9, Tue. May 7
We learned about how we really do mathematical calculations: mentally, with paper and pencil, or using technology? And exactly or approximately?
We also learned that n! = n(n–1)(n–2)...(3)(2)(1) counts the number of ordered arrangements with n elements.

Class 8, Thu., May 2, 2013
We went over how to change from one base to another.
We also went over several unusual algorithms for adding, subtracting, multiplying, and dividing, including

"Adding up" or Austrian subtraction method.
Scratch method of addition
Nines Complement subtraction. (Homework: why does this method work?)
Gelosia or lattice mulitplication method and Russian peasant multiplication
Partial quotients method of division (and some other methods)

But here's a wonderful article recounting the invention of the first mechanical pocket calculator, the "Curta," conceived and designed by Curt Hertzstark while a prisoner in a German concentration camp during World War II. He might have been one of the first to come up with the "complement" subtraction method.

We also discussed a variety of Venn diagram problems.

Here's a site on Base systems, in chapter 3.
Site on number systems.
Number systems associated with languages.
Site with links to number system sites.
Site on number systems.
Number systems associated with languages.
Site with links to number system sites.

How things work on basketball players' numbers: "Each uniform must display one or two digits on the front and back of the jersey. The numbers on a jersey are used to identify a player when calling violations. In most cases, the digits can only be 0, 1, 2, 3, 4 or 5. While the NBA has allowed players to use numerals higher than 5, it is a rare allowance. This limitation on numerals allows referees to use their hands to signal player numbers to the game's official scorekeeper. Otherwise, a player wearing number 9 could be confused with a player wearing number 54."

Second Essay Assignment, due Tuesday, May 14 on Turnitin
You have a short paper on a subject related to the course that catches your interest due Tue., May 14, and worth 5% of your grade. You will turn the paper in via Turnitin.com.
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.

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.

If you have questions about topic or source contact me and ask first!

Class 7, Tue., April 30, 2013
We went over the first exam.
We also discussed ideas of closure and Cartesian product version of multiplication.

How is a form of base six used in college basketball?

Here is info on a recent controversy over the teaching and ideas of multiplication:
Keith Devlin's articles on multiplication as repeated addition.

Here's the video we watched in class:
TED Dan Meyer video.

Class 6, Tue., April 25, 2013
First exam was today, also new material on number systems from a variety of cultures.

Class 5, Tue., April 23, 2013
The game Set can be found here as a daily puzzle.
Here is a handout on what I call the Sorting Junk game. Please include in your portfolio.

Class 4, Thu., April 18, 2013
We learned how to find the Fibonacci numbers on pine cones.
Here is the Fibonacci assignment, which is due this Tuesday, April 23.
Here's a handout about the Fido puzzle from class: Where's Fido? Please include in your portfolio.

Study guide for the first exam
Solutions to the study guide for the first exam problems
Here's another old exam 1.

More links on Fibonacci numbers:
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.

We went over material from chapter 2.1 and 2.2.
Do homework through 2.2.
Chapter 1 homework will be due next Th. Apr. 25.

Class 3, Tue., April 16, 2013
We went over material from chapter 1, including certain kinds of number sequences: squares, triangular numbers, Fibonacci numbers, etc.
We also learned about Fibonacci (Leonardo of Pisa), Pascal's Triangle (or the Pascal-Khayyim Triangle), and the pigeonhole principle, and we worked on the "Frogs on a log" problem.
Do homework from sections 1.1-1.4.

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 are "relatively prime" (their only common divisor or factor is the number 1.)
We saw how labeling six pigeonholes with the pairs of numbers {1,12}, {2,11}, {3,10}, {4,9}, {5,8}, and {6,7} explained property 1. Can you do a similar labeling to explain properties 2, 3, and 4?
Here's a fifth property that is very tricky to explain (hint: the best explanation uses doubling and some pigeonholes with only one number assigned!)
(5) A pair of your numbers have the property that one divided the other equally.
We saw how the pigeonhole principle explained why properties 1,2, and 3 are true. 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! For example, one pigeonhole would have 3,6, and 12.

The "Frogs on a log" problem, which is a textbook homework problem, and in which we will find that with
1 frog per side it took a minimum of 3 moves to exchange places
2 frogs per side it took a minimum of 8 moves to exchange places
3 frogs per side it took a minimum of 15 moves to exchange places
Figure out the minimum number for 4 or 5 frogs on each side.
Figure out how to do these exchanges also, in the minimum number of moves; what is the numerical pattern?

We've used the so-called Gauss'es trick several times. Here is an article showing it should not be called Gauss'es trick!
Article by Brian Hayes on the history of Gauss's Trick, published in 2006 - you can see how the story came about, and how it was copied endlessly by other authors without evidence!

Class 2, Thu., April 11, 2013
We went over a problem from the chapter 1 homework that involves parity, and saw a card trick that uses parity also.
I asked you to try to figure out how the card trick works.

We played the 15 game, and saw how it relates to the 3 by 3 magic square that is part of the chapter 1 homework.
Please bring this handout to class Tuesday: magic squares and modular arithmetic, we will discuss in class.

We learned about the game Ken Ken - see their web site for daily puzzles.
Here is a Ken Ken example with solution explained.

Your math autobiography is due at the Turnitin site by 4 PM on Tuesday. I have added the names of those of you who are registered for the class and have not yet added themselves to the class list, so you just need to go to the site (see directions on the green sheet), and upload your paper.

Here are some of the vocabulary words we have used during classes. Try to use each one in a sentence, to make sure you understand them:
multiple: 12 is a "multiple" of 3 and of 4. 3 and 4 are "factors" of 12. Is 13 a multiple of 1? Is 0 a multiple of 13?
horizontal (row): parallel to the horizon. Often means we are thinking about right and left.
vertical (column): up and down
odd numbers : 1,3,5,7,... These numbers are congruent to 1, mod 2. Is -1 and odd number?
even numbers : 0,2,4,6,8, .... Is 0 even? These numbers are congruent to 0, mod 2. Is -6 and even number?
alternate: a pattern in which two "sub-patterns" are each displayed in every other section of the pattern.
parity: refers to the odd and even numbers, and how they relate to a given problem or situation. For example, we might say that 13 and 29 have the same parity.

Class 1, Tue., April 9, 2013
We played the pattern game (print out this handout and include in your portfolio!)
We also played the take-away game, in which each player removes 1 or 2 counters on each move, the last player to move winning.
Are you clear about the winning strategy in this game?

Homework:
(1) Get a looseleaf notebook and set up sections as described on the third page of the green sheet or below
(2) Write a journal entry for this class today and store it at an etherpad site. Email me the URL of your site. (See directions below or on the green sheet.)
(3) Read and work on homework from sections 1.1 and 1.2.
(4) Register or login to Turnitin.com to make sure you can access the class site at which you will be uploading your essays.

Portfolio. Put together your portfolio, a loose leaf notebook with these sections:
Homework
Handouts or articles provided to you at this site (for example the pattern game handout.)
Exams
Class notes
Articles
Your papers or essays

Write a journal entry for each class. It should be one long (6 or more sentences) several short paragraphs detailing your reflections on each day’s class. What struck you as interesting, useful, helpful, unhelpful, puzzling, etc.? How are you feeling about the class? What are your expectations of the class and your own participation? Imagine you are writing to your future self (as in a popular South Park episode?!) and mention those things most memorable! Keep your journal entries at a page you get at an etherpad site, for example, at the Mozilla etherpad site (at Mozilla click on "Create new public pad.")

Use this format for journal entries:
Stanley Student (keep your name at the top)

Th. Jan. 12 (most recent entry)
Blah, blah, blah (at least 1 long - 6 or more sentences - or 3 medium size paragraphs).

Tue. Jan. 10 (older entry)
Blah, blah, blah (at least 1 long or 3 medium size paragraphs).

If you have trouble using an etherpad site, try opening it with a different browser. I have no trouble using the (free) Google Chrome browser.

Join Turnitin.com by the next class. I have added everyone to the class list already.
See the green sheet for sign in information.

____________________________________________________________________________________________________________

Many links and handouts:

Study guide for the first exam
Solutions to the study guide for the first exam problems

Final Exam study guides:
Recent final exam study guide
Note that the study guide is from a previous year; you won't have an essay question.
Recent sample problem sheet.
Sample problems with some hints and solutions.

We will see several short videos about learning and teaching; you can find the links within Keith Devlin's recent online column.
Here is the set of slides of fraction problems.
Here is the set of slides of decimal/ratio problems

"PEMDAS" memory mnemonic

Prediction Card Trick handout
Painting the Pool
Britney Gallivan, who folded a "sheet" of paper 12 times.
Here is the set of slides of fraction problems.
Here is the set of slides of decimal/ratio problems.
Farey Sequences (skip the advanced part and the cute animation at the beginning!)
TED Dan Meyer video.

Nines Complement subtraction.
Gelosia mulitplication method
"Clap your name" activity.
Wikipedia entry on Turnitin.
Common Core Standards,
National Council of Teachers of Mathematics.

The triangle numbers,
Base systems
How things work on basketball players' numbers: "Each uniform must display one or two digits on the front and back of the jersey. The numbers on a jersey are used to identify a player when calling violations. In most cases, the digits can only be 0, 1, 2, 3, 4 or 5. While the NBA has allowed players to use numerals higher than 5, it is a rare allowance. This limitation on numerals allows referees to use their hands to signal player numbers to the game's official scorekeeper. Otherwise, a player wearing number 9 could be confused with a player wearing number 54."
Keith Devlin's articles on multiplication as repeated addition.
Brief history of the New Math.
The game Set, see their daily puzzle.
Sorting Junk game.

Here's a quote from Lewis Carroll's Through the Looking Glass. Alice is talking with the White Knight, who many commentators believe to be a stand-in for Carroll himself. We'll see it's relevance later in the course!
"The name of the song is called 'Haddock's Eyes'."
"Oh, that's the name of the song, is it?" Alice said, trying to feel interested.
"No, you don't understand," the Knight said, looking a little vexed. "That's what the name is called. The name really is 'The Aged Aged Man'."
"Then I ought to have said 'That's what the song is called?'" Alice corrected herself.
"No, you oughtn't: that's quite another thing! The song is called 'Ways and Means': but that's only what it's called, you know!"
"Well, what is the song, then?" said Alice, who was by this time completely bewildered.
"I was coming to that," the Knight said. "The song really is 'A-sitting on a Gate': and the tune's my own invention."

Voting methods and their history.
Where's Fido?

TED Dan Meyer video.

Here's a site on Base systems.
Site on number systems.
Number systems associated with languages.
Site with links to number system sites.
Site on number systems.
Number systems associated with languages.
Site with links to number system sites.

How things work on basketball players' numbers: "Each uniform must display one or two digits on the front and back of the jersey. The numbers on a jersey are used to identify a player when calling violations. In most cases, the digits can only be 0, 1, 2, 3, 4 or 5. While the NBA has allowed players to use numerals higher than 5, it is a rare allowance. This limitation on numerals allows referees to use their hands to signal player numbers to the game's official scorekeeper. Otherwise, a player wearing number 9 could be confused with a player wearing number 54."

Al Khwarizmi.

History of the Magic Square.

Triangle numbers, squares, Fibonacci numbers.

The game Ken Ken - this site has 6 new puzzles every day.

Fibonacci assignment
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.

Article by Brian Hayes on the history of Gauss's Trick, published in 2006.
Here are articles on Nines Complement subtraction and the Gelosia mulitplication method.
Here are some other methods of addition, etc.

The pattern game we played in class
Handout with problems on Patterns and Modular Arithmetic
The Hidden Role of Modular Arithmetic, that reviews what we did in the first two classes and relates it to some other problems from chapter 1.
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.

The 15-sum game, and how it is really "Magic Square tic-tac-toe."

Here's a handout about the Fido puzzle from class: Where's Fido?

The game Set can be found here as a daily puzzle.
Here is a handout on what I call the Sorting Junk game.

In the pigeonhole principle "magic trick," I will ask 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. 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! For example, one pigeonhole would have 3,6, and 12.

The "Frogs on a log" problem, which is a textbook homework problem, and in which we will find that with
1 frog per side it took a minimum of 3 moves to exchange places
2 frogs per side it took a minimum of 8 moves to exchange places
3 frogs per side it took a minimum of 15 moves to exchange places
You might have guessed that 4 frogs per side would require 24 moves, 5 frogs per side would require 35 moves.
Figure out how to do these exchanges also, in the minimum number of moves.

Records of previous classes:

Class 22
We went over take-home exam, talked about SLOs, went over some finals type problems. Also went over the pool handout.

Chapters 6 and 7 homeworks due during the final exam.

Final Exam study guides:
Recent final exam study guide
Note that the study guide is from a previous year; you won't have an essay question.
Recent sample problem sheet.
Sample problems with some hints and solutions.

Class 21
We worked on the "sports scam" and other topices from chapters 7 and 8.

Class 20
Pi Day! Also worked on handouts.

Class 19
Thursday is Pi Day. Please bring something appropriate!

Please bring these handouts to class Thursday:
Work on this one on magic squares and modular arithmetic, we will discuss in class.
We will work on this one, "Painting the Pool" in class Thursday.

We assigned chapter 4 group problems: