Math 46, Mathematics for Elementary Education
Spring 2019 Home Page
Homework
Ch. 1.1: #10-13, 15
Ch. 1.2: #10, 11, 19
Ch. 1.3: # 1,3,6,8,12,13,17
Ch. 1.4: # 5,7-10,17,18,21
Ch. 1.5: # 5,8,13,14,20
Ch. 1.6: # 1,3,5,10,12,15
Chapter 1 homework is due on Thursday, Apr.
25
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: 2,7,11,16,25,30,31,34
Ch. 2.4: 1,2,4,10,25,38
Ch. 3.1: # 1,5,10,11
Ch. 3.2 # 1,6,9a,10a,11a,13a,14 a,b,c,19
Ch. 3.3 # 1a,7a,10,16,17,20a,b,24
Ch. 3.4 # 1a,7,13,17,21,23,24,25
Ch. 3.5 # 1,2a,c,f,3a,e,4a,c,8a,18,22,32
Ch. 4.1: # 6,8,11,15,16, 24, 25, 27, 31
Ch 4.2 # 1,2,6,9,10,12,14, 21
Ch 4.3 #1a, 2a, 3a, 4a, 5a, 6a, 9, 12,28
Read "The Chapter in Relation to Future Teachers," page 220.
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
Additional study guide materials for Math 46 final exam:
Class 22, Thu., Jun. 20, 2019
Final exam is Tuesday, June 25, 1:45-3:45 PM.
Final exam is cumulative and has many multiple choice questions (bring a half-page scantron).
It is open book, open notes, old exams OK, caculator but not communication capable device OK.
Bring Ch. 7 homework to the exam if you still need homework credit, I will check at start of exam and give back to you.
I will also check portfolio/notebooks durig the exam, worth 2% of your grade, so bring it to the exam and have it in order.
The evening of the exam I will do a final journal check, also worth 2% of your grade, so have that in order also.
Problems in sample tests below that we did not study or discuss are not included; no essay question.
Class 21, Tue., Jun. 18, 2019
We did the Barbie activity and went over some homework problems.
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 20, Thu., Jun. 13, 2019
We went over the take home exams and also learned about aspects of the decimal system: which rational numbers produce terminating and which repeating decimals?
(If the denominator of a fraction in lowest terms is the product of powers of two and five then it will terminate!) All other fractions produce repeating decimals.
Non-terminating and non-repeating decimals like pi or the square root of two actually comprise an infinity of decimals that is larger than the infinity of fractions!
On Tuesday, we will do an activity involving proportions and Barbie dolls. If you have a Barbie doll (or Ken doll or similar), bring it and we will use in class.
Chapter 6 homework will be due Tuesday, so I can return it to you by Thursday. Chapter 7 will be due at final exam time.
Class 19, Tue., Jun. 11, 2019
We changed the due date for chapters 4 and 5 homework to this Thursday, June 13.
We played Frac Jack again. Here is a Frac Jack handout with the rules - please print and place in your portfolio.
In the process we reviewed several new strategies for comparing fractions.
For example, suppose you want to decide which is larger, 7/9 or 6/7, without finding a common denominator or converting to decimals.
Note that 7/9 = 1 – 2/9 and 6/7 = 1 – 1/7 = 1 – 2/14. Since 2/9 > 2/14, it must be the case that 1 – 2/9 < 1 – 2/14, so 7/9 < 6/7.
We also began chapter 6. Homework for chapter 6 will be due next Tuesday.
Class 18, Thu., Jun. 6, 2019
We worked on more fraction problems, and learned the card game "Frac Jack."
Chapters 4 and 5 homework will be due next Tuesday, Jun. 11 - that's changed to Thu., June 13.
I will do a final check of journals this Sunday, June 9.
Class 17, Tue., Jun. 4, 2019
We went over problems on fractions, including a geometric model showing why the usual algorithm for fraction multiplication works.
We learned how to compare fractions with identical numerators.
We briefly went over the use of modular arithmetic to describe repeating patterns:
Here's a handout with problems on Patterns and Modular Arithmetic.
The Hidden Role of Modular Arithmetic reviews modular arithmetic and relates it to some other problems from chapter 1.
Here are some links about modular arithmetic, which connects to many things we have studied this 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.
Chapters 4 and 5 homework will be due next Tuesday, Jun. 11.
I will do a final check of journals this Sunday, June 9.
Class 16, Thu., May 30
We went over a take-away game example.
We also went over material from chapter 5.1 an 5.2 on adding and subtracting positive and negative numbers, using counters of two colors, and also modeling with walking forwards and backwards.
We decide that the Take Home Exam will be moved back one class, and is now due on Thursday, June 6 instead of June 4.
Homework for chapters 4 and 5 will be due on Tuesday, June 11.
I will begin the second journal check this weekend, so make sure they are in order and up to date - I will let you know if you still have work to do on your journal, with a deadline to be announced.
Class 15, Tue., May 28
We went over chapter 4 material on Least Common Multiple (LCM) and Greatest Common Divisor (GCD) and the Euclidean Algorithm.
We also went over base calculations - how to convert from base b to base ten, and how to convert from base ten to base b.
I showed you a "magic trick" or bar bet in which you ask someone to write down a four digit number, rearrange the digits in a new order, change a non-zero digit in the smaller of yor two numbers to zero, then subtract the smaller from the larger of your numbers. From the result of the subtraction it is easy to decide which digit was switched to zero (add up the digit sum).
We learned several divisibility rules.
Class 14, Thu., May 23
We worked some on take home exam, also went over some material from chapter 4.
Class 13, Tue., May 21
Class gave reports.
Class 12, Thu., May 16, 2019
We went over matrial from the rest of chapter 3, including several calculation methods: the scratch method of addition that simplifies carrying; the Gelosia or lattice method of multiplication which lines up partial products efficiently; the "Russian agrarian" method of multiplication, based on ancient Egyptian multiplication, which uses only multiplication and division by 2 as the only multiplication or division facts necessary.
I told you about the recent survey showing 56% of Americans do not want Arabic numerals taught in American schools!
We also went over several methods of doing mental arithmetic, and practiced creating rhythms by playing two rhythms at once - these are called polyrhythms, and this led to the concepts of Least Common Multiple (LCM) and Greatest Common Divisor (GCD, or greatest common factor GCF).
Please turn in chapter 3 homework next Thursday, May 23.
Your paper is due Tuesday, May 21, see below.
Class 11, Tue., May 14, 2019
We worked out the last five 5s problems: 13 = (55+5+5)/5, 17 = (55+5)/5 + 5, 18 = (5+5–5/5)/.5
We worked on the traffic jam problem: how many vehicles in a 2 mile long traffic jam?
We saw the video of Dan Meyer titled "Math Class Needs a Makeover."
We learned about base 2 arithmetic.
We learned about several methods of subtraction including Nines Complement subtraction.
We did the mental arithmetic problem: A friend of mine was born in 1937, how old is he? - and saw how creative we all are with mathemtics!
Here are some titles and subjects of students' papers in Math 44 and Math 46 from previous quarters.
Class 10, Thu., May 9, 2019
We went over chapter 2 homework and you handed it in.
We also went over chapter 3.1 and 3.2 on other place value systems, including base twelve and base six.
I asked you to figure out how base six numbering is used in college basketball - and why!
See this web page for a hint!
We also worked on the five 5s problem, and found ways to use five 5s to make calculations giving all numbers from 1 to 20 except 13, 17, and 18 - can you find these?
We will go over most of the rest of chapter 3 on Tuesday, so please read ahead and be working on the Chapter 3 homework.
Class 9, Tue., May 7, 2019
We went over the exam for the first hour.
If you have not yet brought your journal up to date do so by midnight tobight.
Please be prepared to show me your portfolio by Thursday (looseleaf notebook with sections for parts of the class.)
Please also turn in your chapter 2 homework on Thursday.
We started chapter 3. See the assigned textbook homework problems above.
Second Essay Assignment.
You have a short paper on a subject related to the course that catches your interest due Tuesday, May 21 at the start of class. 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 2 to 3 minutes in length at least) to the class will also be required. You may do a powerpoint if you wish.
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.
You may not do a report on the Fibonacci numbers or the golden ratio!
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 1-10 (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 8, Thu., May 2, 2019
We went over the study guide for exam 1 and had exam 1.
Homework for chapter 2 will be due next Thursday, May 9.
Over the coming weekend I will check journals, so please make sure they are up to date.
We will finish chapter 2 on Tuesday and start chapter 3.
Class 7, Tue., Apr. 30, 2019
First exam is this Thursday. Bring 1/2 page scantron. It's open book, open notes, calculator but not communication device allowed. See study guides below.
We went over homework from chapter 2, especially on sets and numbers of elements in sets.
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" or "Russell's Paradox."
New Math and how it grew out of the launch of Sputnik - see this article.
We played the Sorting Junk game and learned about inductive versus deductive thinking. (Please print this one page sorting Junk game explanation and include in your portfolio).
Inductive thinking moves from specific examples or cases to a general rule. Deductive thinking takes a general rule and applies it logically to specific cases.
We went over some material from chapter 2.3 briefly and learned, for example about how additon is commutative, but subtraction is "anti-commutative."
Chapter 2 homework will tentatively be due next Tuesday, depending on what we do the first hour of class on Thursday.
Class 6, Thu., Apr. 25, 2019
We did the frogs on a log problem (Ch. 1.6 #15) and several other homework problems.
We learned about set theory and how it entered the elementary school curriculum after the USSR launched Sputnik and the US was afraid it was "losing the space race."
Brief history of the New Math movement of the 1960s.
The game Set, see their daily puzzle.
Class 5, Tue., Apr. 23, 2019
We went over the pigeonhole principle problems, including the "geometric" problem, which is one of your homework problems.
We mentioned triangular numbers again: 1,3,6,10, 15,21,... The nth triangular number = n(n+1)/2, and the sum of two successive triangular numbers is a square number.
We did the "Where's Fido" logic problem. Here's a handout on the Fido puzzle. Please print and include in your portfolio (remember that!?)
We went over material on modular arithmetic.
Your first exam 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 4, Thu., Apr. 18, 2019
Chapter 1 homework is due on Thursday, Apr.
25
During the first hour we worked on the textbook problem about possible perimeters when we place nine unit squares together edge to edge. We found the perimeter must be (9)(4) – (2)(number of "inside edges"), since each edge that is "inside" the figure is formed by removing two edges of the total of 36 possible unit edges composing the nine squares' perimeters.
We then talked about possible extensions of the problem:
What if the polygons with joined edges are triangles or other polygons?
What kinds of shapes produce what length perimeters?
Is it the case that a shape composed with odd number of odd-sided unit-edge polygons will have perimeter that is an odd number, while a shape composed of an even number of odd-sided unit-edge polygons will have a perimeter that is an even number, as Emily proposed?
We also examined Pascal's triangle, which arose in finding the number of paths between two points in a rectangular grid, but also as the coefficients in the raising of a binomial like (a+b) to a positive integer power.
We also looked briefly at the pigeonhole principle, and its application to a number of "mind-reading" puzzles.
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.
We saw how the pigeonhole principle explained why properties 1 and 2 are true. For Tuesday, can you use the pigeonhole principle to explain the other properties? Remember, it's all in how you label to (six) pigeonholes! Property 5 is more difficult to explain; in this case the six pigeonholes may have different numbers of numbers assigned to them!
We went over the handshake problem and saw that the numbers of handshakes would fall in the series 1,3,6,10,15,21,... (these are called the triangular numbers).
Class 3, Tue., Apr. 16, 2019
We learned how to count Fibonacci numbers on pine cones.
On Tuesday, April 23, turn in the Fibonacci assignment.
Here are Vi Hart's Fibonacci number videos. The second video shows up at the end of the first, same with the third appearing after the second!
Here is a great site about Fibonacci numbers.
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.
We also spent a good deal of time going over the game of Ken Ken - please turn the Ken Ken handout in on Thursday of this week, Apr. 18.
Please work on the Chapter 1.1 and 1.2 homework, which we will discuss on Thursday.
Also, if you have not signed up a journal site do so. If you have but have not written an entry for last week, please do that now.
Class 2, Thu., Apr. 11, 2019
Your math autobiography is due Tue., Apr. 16. If you have trouble getting into Turnitin.com, let me know. I added everyone's email to the site so you should be able to upload your paper easily.
Today we played the 1 & 2 take-away game.
In the take-away game, in which each player removes 1 or 2 counters on each move, the last player to move winning.
We saw that the losing postions are all multiples of 3.
That is, if you can leave your opponent with a multiple of 3, then you can win - if you play the following strategy:
- If your opponent next removes 1, then you remove 2
- If your opponent next removes 2, then you remove 1
In both cases you complete a group of 3, again leaving your opponent with a multiple of 3 counters!
What happens if a player may remove 1, 2, or 3 counters on each move?
We also worked on the homework problem from the textbook on finding the possible perimeters when nine unit squares are placed together edge to edge.
We found that even number perimeters from 12 to 14 are possible. I asked you to figure our whether odd number perimeters are possible, and if not, why not.
Also, why does it seem that 12 and 20 are the largest and smallest perimeters possible?
We talked briefly about several sequences of numbers, including:
The odds: 1,3,5,7,...
The evens: 0,2,4,6,...
The squares: 1,4,9,16,...
Some of you noticed that the squares differ by successive odd numbers.
Here is a short explanation that we also went over in class.
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.
Class 1, Tue., Apr. 9, 2019
Today we played the pattern game and also learned to make our first names into a clapping slapping rhythm, as well as a marching rhythm. Please practice!
Chapter 1 homework which you should start working on as soon as you get the text book:
Chapter 1.1: #10-13, 15
Chapter 1.2: #10, 11 19
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 when we finish chapter 1.
(4) I will soon register 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, April 16. 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.
Here are some of the vocabulary words we have used or will use soon during class related to the pattern game. 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 or alternation: 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.