MAT115R: Math for Liberal Arts

Assignments

Day Date Activity Assignment
Mon1/13 Welcome and Intro, including an intro to Mathemalchemy! Read From Fish to Infinity (and make sure you watch the video from Sesame Street -- which may not load from the article). (Let me know if you hit a paywall in trying to view anything!)

Also begin watching the Mathemalchemy documentary (you might just want to skim through it at the moment to get the big idea of it). I'll point out some of the highlights as we go along....

Tue1/14 Chipmunks: Primes and Prime Factorization; busting up rocks with trees

Read Rock Groups

Try computing the prime factorizations for some numbers (and draw their trees): or are they prime?

  • 64
  • 210
  • 229

Read The Loneliest Numbers.
Thu1/16 One-to-one correspondence and Rock Groups (Primes: the first great factorization) Homework:

  • Make sure that you can draw all of the Complete Graphs up to \(K_8\), using colors to illustrate the divisibility of the number.
  • Please read Early Concepts of Number and Counting for next time.

Mon1/20 Martin Luther King, Jr. Day  
Tue1/21 Primitive Counting Homework (referring to Early Concepts of Number and Counting for next time):
  • "...a messenger could be sent to another tribe with the message that they wish to trade 20 baskets of food for 15 pearl necklaces, say. This could have been done by simply indicating the point on his body that corresponded to the correct number of objects." Use the "Body counting" guide on page 3 to answer the following:

    • Which parts of the body would the messenger point to, for 20 and 15?
    • And if it were 33 baskets of food for 11 pearl necklaces?
    • Would you trade left foot little toe baskets of food for right knee pearl necklaces?
    • Do you notice any rhyme or reason to the ordering of the body parts?

  • Draw a cartoon of the Baker and the woman buying bread, comparing their tally sticks.
  • Draw a knotted string like that of "The Example 3643", but for the number 1294.
  • How do the South Africans mentioned count to numbers above 100?
Thu1/23 Binary numbers, The Great Fraudini, and the second great factorization

  1. How do we write the following numbers of sheep by partition (using the ternary trees?):
    • 9 sheep
    • 31 sheep
    • 54 sheep

  2. Can you go backwards? How many sheep is meant by the following:

    • 1,0,1,1
    • 1,1,0,1,1,0,0
    • 1,0,0,1,0,1,1,0,1
  3. Try the binary card trick on your friends and family. Are you successful in reading minds? (Don't give away your secrets!)

Mon1/27 The Binary Factorization

  • Write the first sixteen natural numbers as unique binary strings.
  • Given the binary string, tell what number it is:
    • 1011
    • 1101100
    • 100101101

  • Check out Vi Hart teaching us how to Binary hand-dance. See if you can dance along!
  • Word of the day: binary! Try out the Great Fraudini's trick on an unwitting victim; make sure that you can do it.

    If your victim were thinking of the following numbers, on which cards do the numbers appear, and why?

    • 63
    • 27
    • 42

    Can you draw the tree and write the number using the primitive counting trick?

Tue1/28 Pascal's Triangle Please check out some readings (found here -- view each image separately for easier reading):

  • Background History of "Pascal"'s triangle
  • Properties of "Pascal"'s triangle

You don't need to get into the weeds of the readings -- I'm not going to pick out nit-picky stuff for you to remember, and there is some mathematics that is a little beyond what we'll get into.

If you want to solve the following problems, we can use particular rows of Pascal's triangle:

  • If you toss a fair coin 7 times, what are the chances that it comes up
    0 heads 1 head 2 heads 3 heads 4 heads 5 heads 6 heads 7 heads

  • If there are three friends in a Facebook "group" (or graph), call them A, B, and C, draw all possible configurations of the Facebook by adding in arcs as "friendships" (and relate your results to a row in Pascal's triangle).
  • If you have 8 friends, but can only take 5 in your car, how many different carloads are possible?
Thu1/30 More Pascal Please read this short reading. Continue working on those previous problems.
Mon2/3 More Pascal

  1. To compute the number of Facebooks possible for five people, you would have to calculate the number of arcs possible (10), and then go to the 10 row of the table for Pascal's triangle. Either compute the 10th row (not a bad exercise), or look it up, and then answer these questions:

    • How many ways can you create a Facebook with 2 friendships between the people?
    • How many ways can you create a Facebook with 4 friendships between the people?
    • How many ways can you create a Facebook with 6 friendships between the people?
    • How many ways can you create a Facebook with 0 friendships between the people?

  2. Use the combinations formulas from class to find the following:

    \[ \begin{array}{c} C^3_{2} = \cr C^5_3 = \cr C^7_6 = \cr C^7_4 = \cr C^{10}_6 = \cr \end{array} \]

Tue2/4 The Egyptians: back to binary Please enjoy this reading which will help to prepare us for the next several "units": it includes some material which you've already seen elsewhere, but also includes information on the Egyptians.

  1. Demonstrate Egyptian multiplication by multiplying the following (write out the table, and check your work):
    1. 13*34
    2. 23*79
    3. 81*123
    4. 255*256

Thu2/6
  1. Demonstrate Egyptian division by dividing:
    1. 9/4
    2. 13/8
    3. 30/5
    4. 105/7
    Try these using the same sort of "doubling/halving" table that we use for multiplication.

Mon2/10 The Babylonians and the Mayans (other bases) You might re-read on number systems, for an overview of some of the things that we've discussed lately, and a reminder of those Babylonians.

Please read the following from Georges Ifrah's "The Universal History of Numbers":

For practice, try writing the following numbers in Babylonian and Mayan:

  • 21
  • 171
  • 360
  • 400
  • 3600
  • 47331

Tue2/11 Please make sure to read Georges Ifrah's introduction to the Babylonian's Positional System.
Thu2/13 Make sure that you've read Georges Ifrah's introduction to the Mayan's Positional System. You might also be interested in this lovely history of the Mayan calendar we study.

You should finish the translation of the Mayan calendar, on your own, so that you know how to write and interpret Mayan numbers.

Sean noticed that the first four numbers at the top were just multiples of 177. But that eventually fails. In which column?

Mon2/17 Nim: the third great factorization
Tue2/18 Fibonacci numbers

  1. Please read this short description of the Fibonacci numbers.

  2. As usual, Vi Hart makes math fun! Doodling in Math: Spirals, Fibonacci, and Being a Plant [Part 1 of 3]; here is Doodling in Math Class: Spirals, Fibonacci, and Being a Plant [2 of 3] and Part 3.

  3. Write the following numbers as a sum of non-consecutive Fibonacci numbers:

    1. 155
    2. 24
    3. 99
    4. 128

  4. If you were playing me in Fibonacci Nim, two questions: do you go first or second, and, if first, what's your first move?

    1. 47
    2. 24
    3. 34
    4. 19

Thu2/20  
Mon2/24 Golden Rectangles: from numbers to geometry Use the Fibonacci Spiral Fractal maker, with your own image, to make a photo Fibonacci spiral.

You might try some of the "art" features.

Email me this, and we'll hold an art show of spirals, with "get out of quiz free" prizes.... (Due Wednesday, 3/5)

Tue2/25 Review  
Thu2/27 Exam 1  
Mon3/3 Symmetry

  1. Read this symmetry handout (pages 1-6, 8).

    Carry out the following exercises on that page:

    1. A1, A4 (p. 1)
    2. B1, B3, B4 (p. 2)
    3. C1 (p. 3)

  2. Also read this article (from The New York Times, March 22, 2022), as we move into geometry: Is Geometry a Language That Only Humans Know? (here's the source -- the New York Times). Neuroscientists are exploring whether shapes like squares rectangles and and our ability to them recognize are part of what makes our species special.

    (There are two fun games described in this article, to test your instinctual geometric abilities. Look for them on the left side-bar.)

Tue3/4  
Thu3/6 Cut and create Platonic solids out of paper, using this template.

You will be allowed to use these as a cheat sheet for the next exam.

Mon3/10 Spring Break  
Tue3/11 Spring Break  
Thu3/13 Spring Break  
Mon3/17 Platonic Solids and Symmetry  
Tue3/18 Platonic Solids and Graphs (Euler's formula) Please read The Enemy Of My Enemy.
  • Find examples of company logos which involve Platonic solids (a Google search, e.g. cube logos will get you started)

  • Draw 2-dimensional projections of each of the Platonic solids. That is, a realistic view of a platonic solid on 2-dimensional paper. Try your hardest to do this well!

  • Count vertices, faces, and edges for a soccer ball, and verify that it satisfies Euler's formula.
Thu3/20 Please read this short summary of some of our graph theory material.

  • Construct floor plans of houses that have Euler paths through them (remember that doors are "bridges" and rooms are "land masses", in terms of the Konigsberg bridges problem).

Mon3/24 Fractals

Tue3/25  
Thu3/27 Please read this short summary of fractals.

Can you identify some fractals in nature, or in human society? Can you create some fractals of your own, using sticks, rectangles, triangles, etc.? Try using the on-line gasket maker to make some fractals, and try out the L-system fractals. What can you create?

You've already created a fractal of your own, if you tried my Fibonacci Spiral Fractal maker!

Mon3/31 Links and Knots: all tied up Enjoy these videos from Vi Hart:

Tue4/1 Review  
Thu4/3 Exam 2  
Mon4/7 Distinguishing Knots

Tue4/8    
Thu4/10    
Mon4/14 Distinguishing Knots  
Tue4/15 Angels and Miracles  
Thu4/17  
Mon4/21 To Infinity, and beyond!  
Tue4/22  
Thu4/24  
Mon4/28 Review  
Tue4/29 Logo Day!  
Thu5/1 Project Day!  
Mon5/5 Rest up...  
Tue5/6 Rest up...  
Thu5/8 Comprehensive Final Exam 4:40pm