Spring 2014 Math 46 Home Page

Green Sheet

Class 21, Tue., Mar. 18, 2014
We went over material on rational and irrational numbers and their relationship to repeating, terminating and non-repeating decimals:
Every rational number is representable as a terminating or repeating decimal. Irrational numbers are only representable as non-repeating decimals.
Repeating decimals are always rational numbers: a repeating decimal .abcabcabc, in which the repeating part is abc is also representable as the fraction abc/999, for example.
Terminating decimals must be fractions in lowest terms with denominators which have prime factor decomposition made up only of powers of 2 and 5.

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, Tue., Mar. 18, 2014
We did the Barbie activity, went over material from chapter 6 and 7.

Class 20, Thu., Mar. 13, 2014
We went over more material on fractions from chapter 6, including Egyptian fractions with numerator 1.
Next week we will go over chater 7 on decimal representation of numbers and ratios and proportion.
For the ratio and proportion section, please bring, if you have access to any, Barbie or Ken dolls, as we will do an activity relating to what these dolls would be like if they were human size.

Chapter 6 is due on Tuesday, if you want it back by Thursday to use to study for your exam, or Thursday, if it's OK not to get it back till the exam.
Chapter 7 will be due during the final (you'll get it back quickly.)

Class 19, Tue., Mar. 11, 2014
We worked on fraction questions the whole class, and will finish chapter 6 on Thursday and begin chapter 7.
Here is the set of slides of fraction problems.

Class 18, Thu., Mar. 6, 2014
We worked on problems from chapter 5, saw a solution to the first problem on the take-home, and started working on the fraction chapter.

Class 17, Tue., Mar. 4, 2014
We worked on problems presented at the board and take-home exams (due Thursday of this week!)

Class 16, Thu., Feb. 27, 2014
We did group problems at the board, and learned more about clock or modular arithmetic.
Chapter 4 homework due Tuesday of next week.

Class 15, Tue., Feb. 25, 2014
We did more fom chapter 4.

Class 14, Thu., Feb. 20, 2014
We practiced a bit more of the "Clap your name" activity, observing some of the mathematics behind the combination of two rhythms of different lengths. Here's a related handout on star polygons. Here are some materials on "modular arithmetic," which explains some of the properties we observed when looking at the rhythm patterns within circles:
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.

We worked on the group problems (see below), and will finish solving on Tuesday and try to present on Tuesday also. We briefly went over material from the last section of chapter 4 - chapter 4 homework will be due next Thursday, Feb. 27.

By the way, how is Halloween like Christmas?
Because Oct 31 = Dec 25 (do you know what this refers to? Hint: base systems...)

While we're at it, did you know there are 10 kinds of people: those who know binary, and those who don't.
And if you didn't like that, there are three kinds of people: those who can count, and those who cannot.

Math teacher jokes.

Class 13, Tue., Feb. 18, 2014
We heard oral reports and went over section 4.3. We also worked on groups a bit.
On Thursday we will work in groups again, then present the problems next Tuesday.
I do NOT have jury duty this week, we will hear the oral reports on Tuesday.
You have been emailed the take home exam. Please print it out and bring to class so you can work on it there.
Here's a handout with a Ken Ken example, with solution explained, which will help you with one of the take home problems.

Class 12, Thu., Feb. 13, 2014
We did the name-rhythm activity, selected group problems, and worked on material from chapter 41 and 4.2 on primes and divisibility tests.
Group problems:

Class 11, Tue., Feb. 11, 2014
We went over mental arithmetic and estimation technigues, reviewed Russian multiplication and lattice multiplication, and began chapter 4 by learning how to play the rhythm of vowels and consonants in our names - please practice your name rhythm!
Paper is due Thursday - however we might not do the oral report part of that assignment till next week.

Class 10, Thu., Feb. 4, 2014
We went over more calculation algorithms, saw some slides about the Brazilian street math study, and did an activity on how we calculate.
You have a paper due on Turnitin next Thursday, see the description below. The homework for chapter 3 will not be due until a week from Tuesday.
Please bring a calculator to our next class - either scientific or graphing calculator.

Class 9, Tue., Feb. 4, 2014
We went over the exam, also went over base systems other than ten. We learned several alternative addition and subtraction algorithms.

Please look at the paper assignment below!

Here are articles on Nines Complement subtraction and the Gelosia mulitplication method.
Here are some other methods of addition, etc. Here's another such web site.

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.

Class 8, Thu., Jan. 29
We had the first exam. We went over questions from chapters 1 and 2.
We also had a first introduction of base systems other than base ten: binary or base 2, and base six (used for college basketbal numbers. We also noted that base sixty was used by the Babylonians (and still in our timekeeping) and base a modified base twenty was used by the Mayans.

For example, here's how you write the base ten numbers 1 through 18 in base six. For example, 21 base six represents 2 sixes and 1 one, so it is equivalent to "our" base ten number thirteen.

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."

Here is the Fibonacci assignment, which is due next Tuesday, Feb. 4. Read it CAREFULLY and do everything it asks!!

We have now gone through chapter 3.2, and will probably finish chapter 3 next class.

Second Essay Assignment.
You have a short paper on a subject related to the course that catches your interest due Thu., Feb. 13, 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.

Class 7, Tue., Jan. 27
We learned how to find two successive Fibonacci numbers on pine cones.

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 played the 15-sum game, and learned how it is really "Magic Square tic-tac-toe."
Here's a handout on magic squares and modular arithmetic. We haven't covered all of the material in this handout yet, but will soon.

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

Here's a handout with a Ken Ken example, with solution explained.

You should print out these handouts and include in your portfolio.

Here's a site on Base systems, which we'll get to very soon.
Site on number systems.
Number systems associated with languages.
Site with links to number system sites.
Site on number systems.

Class 6, Thu., Jan. 23
We covered the last sections of chapter 2 and most of the first section of chapter 3.
Homework for chapter 2 is due next Thursday after the exam.
First exam is Thursday of next week, please bring a scantron for that exam.
Here is an older
Study guide for the first exam
Solutions to the study guide for the first exam problems

Class 5, Tue., Jan. 21
We went over some chapter 1 problems, saw two more applications of the pigeonhole principle to a geometric problem and to school buses! We also worked on the problem of finding the number of vehicles in a 2 mile long traffic jam.
We learned about the game Set -it can be found here as a daily puzzle.
I talked about the New Math and how it grew out of the launch of Sputnik - see this article.
I also talked about how the symbolism of set theory and logic was developed during the late 1800s and later to try to make all mathematics into an algebraic activity, and how that failed totally due to the work of Kurt Godel.
We also learned about parity and saw a magic trick in which it explains how the trick works.
Homework for chapter 1 is due Thursday. Please turn in your flag design with it - you may simply place your PEMDAS example at your web site.

Class 4, Th., Jan. 16
Sub showed part II of the Story of Maths video, and did some games and puzzles, also showed the Pigeonhole Principle.

Class 3, Tue., Jan. 14
The sub showed part I of the Story of Maths video and worked on ch. 1 problems.

Note: some of you to whom I gave an add code have had trouble registering. However one person was able to add with the code I gave her. I am not sure what the problem is. Perhaps you are trying to register for Educ 46 instead of Math 46 (the two courses are identical). You might need to go to the registrar on campus and get their help in registering. remember that Jan. 18 is the deadline to register, no exceptions.

Class 2, Th., Jan. 9, 2014
We played the game Ken Ken. Please go to the Ken Ken site and practive playing the game!
We also did the five 5s activity, which also uses basic calculations in a creative manner. It also allows us to practice correct order of operations. Please write your own "PEMDAS," that is a mnemonic for PEMDAS like "Please Excuse My Dear Aunt Sally" for next class.
We learned about several number sequences which are important, including
The odds: 1,3,5,7,...
The evens: 0,2,4,6,...
The squares: 1,4,9,16,...
The triangular numbers: 1,3,6,10,15,...
The primes: 2,3,5,7,11,13,...
The Fibonacci numbers: 1,1,2,3,5,8,13,21,24,...

Please work on chapter 1 homework for Tuesday.
Your math autobiography is due Tuesday, uploaded to Turnitin.com

I also asked you to create a design for a US flag with 51 stars (not one big star and 50 in the current design!)

Class 1, Tue., Jan. 7, 2014
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 for the following class. I will add everyone to the class list very soon.
Watch this site for sign in information.

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.

Many Links and Handouts:

Handout on magic squares and modular arithmetic.
Handout: "Painting the Pool".
Ken Ken example with solution explained.
Handout on star polygons.

Here's the wikipedia site on Nines Complement subtraction. Here's a link for the Gelosia mulitplication method.
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 this "complement" subtraction method.

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.

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.
You have a short paper on a subject related to the course that catches your interest due Tue., Feb. 12, 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.

Here's another old exam 1.

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

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!
Here are articles on Nines Complement subtraction and the Gelosia mulitplication method.
Here are some other methods of addition, etc.

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

Here's a site on Base systems, in chapter 3; we went over some of chapter 3.1.
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.

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

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.

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

The game Ken Ken. See their web site for daily puzzles.

If you have not gotten an etherpad site for your class journal, do so now.
If you have not entered a journal entry for Tuesday also, please do so now, as I am already checking journals. Your first journal check will be complete by Tuesday, April 24, and will count 2 points of the 10 points for the "Portfolio" portion of your grade.

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.

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.