Fall 2015 Math 46 Home Page

Fall 2015 Math 46 Home Page

Class 21, Thu., Dec. 3, 2015
We went over math practices from the Common Core and reviewed.

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.

Bring a scantron and your notebook (to be checked) and any homework you are need to turn in to the final exam.
The exam is open book, open notes, calculator allowed, and will have scantron and non-scantron questions.

Further study guide for final:

Eqyptian multiplication and Egyptian fractions.
Alternative multiplication, subtraction, and addition methods (nines complement, adding up subtraction, scratch addition, lattice multiplication, Russian multiplication, etc.)
Visualization methods for fraction multiplication, addition, and subtraction.
Proportional reasoning, as in the Barbie calculations.
The pigeonhole principle problems.
Repeating patterns in the plane (as on first day of class).
Divisibility tests for numbers divisible by 2,3,4,5,6,8,9,10 and 11.
Ancient mathematical systems.
Changing from another base to base ten and vice versa.
Venn diagrams, union, intersection, and complement.
Which fractions produce repeating decimals and which are terminating decimals, and converting from one form to another.
Fibonacci numbers, Pascal’s triangle.
Finding the GCD and LCM of two numbers.
Visualizations of number operations with manipulatives.
Simple modular arithmetic questions.
Closure for various sets of numbers.
Numbers of factors of a number.
Logic problems, as in the Fido problem.
Percentages.
Reasoning about fractions using alternative strategies: which fraction is larger?

Class 20, Tue., Dec. 1, 2015
We heard oral reports.

Class 19, Tue., Nov. 24
We did the Barbie activity and a bit about exponential functions.
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 decimals: all fractions are repeating decimals, and all repeating decimals are fractions. Also non-repeating, non-terminatiing decimals are irrational numbers - and vice versa. And fractions (in lowest terms) which have denominators that are products of powers of two and powers of five are terminating decimals, and vice versa.

On Tuesday, Nov 24, please bring Barbies (or Kens) for our activity on proportional reasoning.

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

Class 17, Tuesday, Nov. 17, 2015
We went over material on Egyptian fractions and Egyptian multiplication, and then started on Chapter 7 on percentages and decimals.

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

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 went over more on fractions, including playing Frack Jack (card game) and learning about Egyptian fractions.
Please print out and answer the few questions on this handout on Egyptian Fractions, to turn in on Thursday.
Take home exams due Tuesday.

Here is the set of slides of fraction problems.

Class 15, Tues., Nov. 10, 2015
We went over fraction material and worked on take home exams, due next Tuesday.

Class 14, Thursday, Nov. 5, 2015
We went over methods of using manipulatives to represent positive and negative numbers, and started chapter 6 on fractions.
We also worked on take home exams.

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.

Class 13, Tuesday, Nov. 3, 2015
We did the group problems at the board, and went over more from chapter 4.

Class 12, Thursday, Oct. 29, 2015
We saw the TED Dan Meyer video.
We worked on group problems from chapter 4 and went over the exam.
We will present group problems on Tuesday. Chapter 3 homework is due Tuesday.

	A	B	C	D
4.1, 24-26	Andrew	Kauru	Beatriz	Leticia
4.1, 31-32	Rachel	Kim	Angela	Stephanie
4.2, 17-18	Lisha	Joanne	Chen	Joseph
4.2, 21 and 4.3, 23	Adriana	Dawit	Leo	Weiwei
4.3, 16-17	Jeancarlos

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.

Second 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 5% 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.

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 11, Tuesday, Oct. 27, 2015
We worked on material from chapter 4, introduced modular arithmetic, and assigned group problems:

	A	B	C	D
4.1, 24-26	Andrew	Kauru	Beatriz	Leticia
4.1, 31-32	Rachel	Kim	Angela	Stephanie
4.2, 17-18	Lisha	Joanne	Chen	Joseph
4.2, 21 and 4.3, 23	Adriana	Dawit	Leo	Weiwei
4.3, 16-17	Jeancarlos

Class 10, Thursday, Oct. 22, 2015
Substitute Kejian Shi went over practice exams.

Class 9, Tuesday, Oct. 20, 2015
Substitute Kejian Shi went over practice exams.

Class 8, Thursday, Oct. 15, 2015
We went over section 4.1 and some of the study guides for the first exam.
We also spent some time making sure everyone has a journal site; if you have not sent it to me, do so now please!

Your first exam is next Thursday, Oct. 22. It is open book, open notes, calculator but not wifi capable devices allowed, one hour, BRING A HALF-PAGE SCANTRON.
Study guide for the first exam
Solutions to the study guide for the first exam problems
Here's another old exam 1.

Class 7, Thursday, Oct. 13, 2015
We went over chapter 3.3 through 3.5 on alternative calculation techniques, as well as approximation and mental arithmetic methods. We saw slides about the Brazilian street math study and new ideas about the psychology of learning mathematics.

Chapter 2 homework is due Thursday of this week.

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

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."

Class 6, Thursday, Oct. 8, 2015
We did the "How do we calculate" activity.
We also learned about two more parity tricks.
We went over material from chapter 3.1 and 3.2 on ancient number systems and other bases besides base ten.

I have sent a message to the etherpad folks asking why the site is down, hopefully will hear back. Otherwise we'll have to find an alternative. There are other free etherpad sites, but I'd rather not have to switch!

Class 5, Tuesday, Oct. 6, 2015
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 also learned about different methodes for understanding arithmetic operations: counting versus measuring versus tree-structures.

We saw two new parity puzzles, one a "bar bet" with three cups.

We went over chapter 2.2-2.4, and learned about various properties: commutative, associative, closure, etc.

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

Class 4, Thursday, Oct. 1, 2015
Homework for chapter one is due next Thursday, Oct. 8 - should be turned in on paper, NOT online.
On Tuesday, Oct. 6 turn in the Fibonacci assignment - should be turned in on paper, NOT online.

We did the "Frogs on a Log" problem.
We also played the Sorting Junk game - the link is a handout on it; please print and add to your notebook.

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, whose Incompleteness Theorem is based on the self-referential "Liars Paradox."

We talked about inductive versus deductive thinking, and I asked you to think about when we've used either during this class.

Class 3, Tuesday, Sep. 29, 2015

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 learned how to count Fibonacci numbers on pine cones.
On Tuesday, Oct. 6 turn in the Fibonacci assignment.

We also went over Pascal's triangle, why the sum of successive odd numbers produces a square number, and a visual proof of the Pythagorean Theorem.

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, Thu., Sep. 24, 2015
We played the game Ken Ken. Please go to the Ken Ken site and practice playing the game!
Here's a handout with a Ken Ken example, with solution explained, which may help you understand how the puzzle works.

You have your math autobiography due next Tuesday at the start of class (see green sheet above).

Also, write an entry in your journal about this week's classes. Talk about your expectations for the class, and how the class seems either to meet or contradict those expectations.

Class 1, Tue., Sep. 22, 2015
Homework:
(1) Get a looseleaf notebook and set up sections as described on the third page of the green sheet or below.
(2) Establish a journal 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 1.1-1.2. Chapter 1 homework will be due on Thursday, Jan. 15. The list of homework problems is below.
(4) I have registered you to the Turnitin.com class page with the email in my class record, so make sure you can access the class site at which you will be uploading your essays. Your first essay, your mathematical autobiography, is due on Tuesday, Jan. 13, by 4 PM. It is described above in the green sheet. Read the description of the assignment as you write it to make sure you include what it asks for!

Here is the pattern game we played in class. Print this out and include in your portfolio.

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.

Here are some of the vocabulary words we have used during class. 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 an odd number? (We'll learn about number "congruence" soon!)
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.

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 week of 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.

TEXTBOOK HOMEWORK:
Ch. 1.1: # 10,11,12,13,15,16
Ch. 1.2: # 4,10,11,19
Ch. 1.3: # 1,3,6,8,12,13,17
Ch. 1.4: # 1,3,5,7,8,9,10,17,18,21
Ch. 1.5: # 5,8,12,13,14,20
Ch. 1.6: # 1,3,5,10,12,15

Ch. 2.1: 7a,c,e,g, 10d, 12, 13,14a,e,15c,24,25
Ch. 2.2: 1,6a,9,13,24,29
Ch. 2.3: 2,7,11,16,30,31,34
Ch. 2.4: 1,2,4,10,25,38

Ch. 3.1: # 1,4,5,10,11
Ch. 3.2 # 1,6,9a,10a,11a,13a,14,19
Ch. 3.3 # 1a,7a,10,16,17,20,24
Ch. 3.4 # 1a,7,13,17,21,23,24,25
Ch. 3.5 # 1,2,3,4,8a,18,22,31

Ch. 4.1: # 6,8,11,15,16, group: 24-26, 27, group: 31-32
Ch 4.2 # 1,2,6,9,10,12,14, group: 17 & 18, group: 21 & section 4.3 # 23
Ch 4.3 #1a, 2a, 3a, 4a, 5a, 6a, 9, 12, 26, 32, group:16 &17, group:18-20, group:22, (group: 23, see section 4.2)

Ch. 5.1: # 1,4,7,8,17,18,22,23
Ch. 5.2: # 1,2,16,19,28,29,30,31
Ch. 5.3: # 5,6,22,23,24,27

Ch. 6.1: # 2,3,5,6,11,23,24,35,36,37
Ch. 6.2: # 1,2,9,13,21,22,23,25,26
Ch. 6.3: #1,2,10a,11a,18,26
Ch. 6.4: # 1,12,16,27,30,34,35

Ch. 7.1: # 1-7,9a,b,10a,b,18,20,23
Ch. 7.2: # 7,8,20a,b,31
Ch. 7.3: # 1,3a,b,7,9,12,29,40
Ch. 7.4: # 4a,b,8,19,23,27,37

Ch. 8.1: # 10,11,18.19,34,36,37

Below are links from previous classes:

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.

Here is the video we saw in class: TED Dan Meyer video.

Here's the "Clap your name" activity.
Here are the
Common Core Standards,
National Council of Teachers of Mathematics

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

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

In the last class 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.

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

Please keep your journals up to date. For today's entry, please make up an alternative to "Please Excuse My Dear Aunt Sally" for a mnemonic device to remember the order of operations usually called PEMDAS.

On Thursday, Oct. 9 turn in the Fibonacci assignment.
The game Set, which we played in class, can be found here as a daily puzzle.
We also played the Sorting Junk game - the link is a handout on it; please print and add to your notebook.

Today we went over the pigeonhole principle and also several other aspects of sequences of numbers, especially the Fibonacci numbers. We also looked at the Pascal-Khayyim triangle.

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. 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.

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 Thursday, on Thursday, Oct. 9. 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.

We played the 15-sum game, and how it is really "Magic Square tic-tac-toe."
We played the game Ken Ken. Please go to the Ken Ken site and practice playing the game!
Here's a handout with a Ken Ken example, with solution explained, which may help you understand how the puzzle works.

You have your math autobiography due next Tuesday at the start of class (see green sheet above).

A student asked about turning in homework on Turnitin.com. Only the two 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 turn in all sections of your chapter 1 homework at one time, not section by section each class session.

https://etherpad.mozilla.org/

We played the pattern game. Here's a handout about the pattern game, please print and include in your portfolio.
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.

Some links from recent classes:

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.

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!)

We worked on this handout with problems on Patterns and Modular Arithmetic.

Ken Ken Example

Second Essay Assignment.
You have a short paper on a subject related to the course that catches your interest due Thu., May 22, and worth 5% 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.

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.
We saw the TED Dan Meyer video.

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.

We did the Fido puzzle - please print out this Where's Fido handout and include in your portfolio. Include the extra problems in the handout as part of homework!

Study guide for the first exam
Solutions to the study guide for the first exam problems
Here's another old exam 1.
Here is the Fibonacci assignment, which is due next Tuesday, April 29. Read it CAREFULLY and do everything it asks!!

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

We also played what I call the Sorting Junk game and also the online game Set.
The game Set can be found here as a daily puzzle.
Here is a handout on the Sorting Junk game. Print out such handouts to include in your portfolio for the class.

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 played the game Ken Ken. Please go to the Ken Ken site and practice playing the game!

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 cubes: 1,8, 27, 64, 125, 216,...
The primes: 2,3,5,7,11,13,...
The Fibonacci numbers: 1,1,2,3,5,8,13,21,24,...
We also learned about the Online Encyclopedia of Integer Sequences (OEIS), started by mathematician Neil Sloane, at which you can find just about any number sequence you can imagine!

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.

Here is the set of slides of fraction problems.
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.

By the way, how is Halloween like Christmas?
Because Oct 31 = Dec 25 (do you know what this refers to? Hint: base systems...)
Here's a handout with a Ken Ken example, with solution explained, which will help you with one of the take home problems.

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.

Base ten:	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Base six:	0	1	2	3	4	5	10	11	12	13	14	15	20	21	22	23	24	25	30

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!!

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.

We also learned about a connection between the patterns in this game and the 3 by 3 magic square. 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.

You should print out such handouts as the one above and include in your portfolio.

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.

Al Khwarizmi.

History of the Magic Square.

Triangle numbers, squares, Fibonacci numbers.

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

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.

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.

We played the game Ken Ken. Please go to the Ken Ken site and practice playing the game!

We went over some homework problems from chapter 1, focusing attention on some number sequences that we'll be seeing a lot of this quarter:
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,... (see Triangle numbers for more on these.)
The cubes: 1,8, 27, 64, 125, 216,... (whooops, forgot to mention this one!)
The primes: 2,3,5,7,11,13,...
The Fibonacci numbers: 1,1,2,3,5,8,13,21,24,...
By the way, there is something called the Online Encyclopedia of Integer Sequences (OEIS), started by mathematician Neil Sloane, at which you can find just about any number sequence you can imagine!
We also looked at the Pascal (also called the Pascal-Khayyim) triangle.
We saw visualizations of some of these sequences, and noted that visual mathematics has become more popular recently as technology has supported more visual displays, and this has even led to changes in language taking us back in the direction of ancient hieroglyphics! (Think emojis or emoticons.)

Today we also went over 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 a difference of 1.
(5) A pair of your numbers had the property that one divided the other equally.
(6) Two subsets of your numbers had the same sum.
We saw how the pigeonhole principle explained why properties 1-5 are true. Number 5 was tricky, and involved labeling the six pigeonholes with different numbers of numbers!
Property 6 is more difficult to explain; in this case the we have to show that there are more possible subsets than there are possible sums.

We worked on the "Frogs on a log" problem, which is a textbook homework problem in section 1.6, 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.