MAT115: Math for Liberal Arts

Assignments

Please check out this video An Infinity Paradox - How Many Balls Are In The Vase?
Day Date Activity Assignment
Tue1/9 Welcome and Intro; meet the textbook: Mathemalchemy! What is a number? Read From Fish to Infinity (and make sure you watch the video from Sesame Street!)

Also please begin reading the Mathemalchemy comic book (in particular the part about the chipmonks), which will give you an important mathematical overview of the installation as well as detail on the prime factorizations and the sieve of Eratosthenes. These readings (and video) will be the focus of Thursday's quiz.

Thu1/11 Chipmonks: 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.
Tue1/16 Primitive Counting Please read Early Concepts of Number and Counting.

Check out Vi Hart teaching us how to Binary hand-dance. See if you can dance along! Then you might continue on, and see how she uses binary trees to prepare her Thanksgiving feast (vegetarians and vegans: serious meat warning....).

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?
Thu1/18 Binary Factorization, and Quiz! Homework:
  • Imagine that you're playing Fraudini's trick with a friend: for each of the following numbers, indicate which cards contain that number by writing each number as a sum of powers of 2 (you might use a tree to create the factorization):

    1. 53
    2. 49
    3. 31

  • For the following use the method of "primitive counting" described in class:

    1. Turn the following into the appropriate string of 1s and 0s by drawing the appropriate ternary tree:

      1. 32
      2. 63
      3. 97

    2. Turn the following strings of 1s and 0s into the appropriate number of sheep. Drawing the ternary tree is desired!

      1. 1,0,1,0,1,0
      2. 1,0,1,0,1,0,1
      3. 1,0,1,1,0,0,0,1

Tue1/23 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?

Thu1/25 Pascal's triangle. For more on the development of Chinese positional number system and bamboo counting rods, take a look at this reading.

This one-page reading gives a third version of the triangle, from the Arab world, and a little background on the influence of Indian culture on our number systems.

Tue1/30 Babylonians and Mayans (Other Bases): Nothing solves the place value problem Today you received two "artifacts" in class, one Babylonian and one Mayan. Your job: to "translate" those artifacts into our numbers systems. The Mayan has been started for you; the Babylonian one shouldn't be too much trouble, since we pretty well did it in class.

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

Thu2/1 Quiz!
Tue2/6 Fibonacci Numbers Please read this short description of the Fibonacci numbers.

You must check out this great video by Vi Hart, featuring Fibonaccis in Nature. There are three of them actually, and you'll probably enjoy all three!

Thu2/8 Quiz!
Tue2/13 Review Make sure that you can beat me at Fraudini Nim. (I don't want to take your money....)
Thu2/15 Exam 1
Tue2/20 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 (but only match to sample is working). Look for them on the left side-bar.)

Thu2/22 More symmetry Please read this short (and fun) reading on symmetry.
Tue2/27 Platonic solids Please watch the following two videos:

Please read these short essays by Georges Hattab:

  • Geometry (a local copy)

    This one introduces you to another generalization of the Platonic solids -- to the Archimedian solids, and then to the more general Johnson solids, of which there are 92.

  • Symmetry (a local copy)

For Thursday: explore the 17 wallpapers patterns, and try to find ways to distinguish them.

This site can help out a lot! Distinguishing the 17 wallpaper groups. In particular, take a look at the glide reflection symmetry.

Thu2/29 Quiz!
Tue3/5 Spring Break
Thu3/7 Spring Break
Tue3/12 Fractals Please create your very own Spiral Fractal, and submit it to me by Thursday (it will be half of your quiz). You'll need a small square photo (that's best, although the program allows you to crop to a square portion of an image); but please use your creativity to create beautiful things!

Email the resulting image to me (taking a screenshot of it may be easiest).

Then watch Vi Hart draw fractals! She's always fun to watch....

Thu3/14 Quiz! Please read the article Pi Day: How One Irrational Number Made Us Modern: The famous mathematical ratio, estimated to more than 22 trillion digits (and counting), is the perfect symbol for our species' long effort to tame infinity.
Tue3/19 Knots and Links Please watch Vi Hart

Thu3/21 Quiz! Prepare a collection of bands, links, and knots.

  • Unknot ("Wedding Band"; untwisted paper)
  • (Pair) of mirrored right-handed and left-handed Mobius Bands
  • Hopf Link (cut a twice-twisted band in two; or cut a Mobius band into "thirds", to create a short mobius band linked with a twice-twisted band in a Hopf Link)
  • Trefoil knot (cut a thrice-twisted band "in two")
  • "Solomon's knot" (cut a four-twisted band in two, to make a link)
  • Cinquefoil knot (cut a five-twisted band "in two")

Take a picture of your collection, attractively displayed, and email it to me. This will be a small fraction (5/100) of your exam grade next week.

We'll have another gallery of those, and more prizes.... Mixed media collections are encouraged (you can make them out of fruit roll-ups if you wish...).

Please watch this video from Vi Hart, about snakes and graphs.

  • This was assigned because it relates to knots and links, but it also relates to graphs. Vi is doing double duty for us here! (I hope that you also watched her video about the Mobius Music Box).
  • Notice her graphs in the first part of the video, including the graph of an octahedron!
  • By making a closed-curve squiggle graph, you're creating a knot. (A knot is a single piece of string, in which the two cut ends are ultimately tied back together to create a single continuous piece of string -- which may be "knotted"!)
  • "Borromean snakes": "No two snakes are actually linked to each other" (1:22).
  • Just before the Borromean snakes there is a trefoil snake (1:14).
  • "What kind of knots are you drawing and can you classify them?"

Tue3/26 Review
Thu3/28 Exam 2
Tue4/2 Distinguishing Knots Please read this short piece about the Reidemeister moves and Tri-colorability.

  • Decide whether the Borromean rings are tricolorable or not.
  • Decide whether the two five knots are tricolorable or not.
  • Identify the knots or links in this "story", which I call A Knotty Tale. You may need to apply the Reidemeister moves to convince yourself that a picture of a knot is really the unknot, say. You can also try tricoloring pictures, to eliminate some candidates, or bolster the claims of others.

Thu4/4 Quiz!
Tue4/9 Infinity Please read The Hilbert Hotel.
Thu4/11 Quiz!
Tue4/16 Angels and Miracles Please watch our friend Vi Hart describe Doodling in Math: Sick Number Games. In particular, I'm interested in her observation at 2 and a half minutes that Pascal's triangle lives in a sea of invisible zeros....

Here's another version of that, with a transcript, and links to more of her videos.

Then watch the Numberphiles episode which covers the material from today's lecture.

Thu4/18 Quiz!
Tue4/23 Logos
Thu4/25 and Projects
Tue4/30 Rest up
Thu5/2 Final Exam