Your answers should be typed or at least written very neatly. Clearly indicate the item number of each problem with your answer. If you do any hand calculation, show your work and/or explain your...

Your answers should be typed or at least written very neatly. Clearly indicate the item number of each problem with your answer. If you do any hand calculation, show your work and/or explain your process.
The test is marked out of 100, with points for each item indicated in curly brackets {}
Partial credit will be given for good attempts and apparent effort—so, even in the short answers, showing your work may lead to partial credit if appropriate.

Section A {65}: Answer all of the following four questions


1.
{14} You saw the following questions on the first day of class. Use your experience in the class to answer them anew. You may refer to your “culture” from the oral presentation and/or any other aspects of your culture—and (in part c) the cultures of your students, past or present. Specific examples will enhance your answers.
a. {4} How could you
use
your own cultural heritage in your teaching?
b. {5} How does your own cultural heritage
influence
your teaching? Should it?
c. {5} Can you use your students' cultural heritages as resources for your teaching? How?

2.
{24} Bishop stated six activities done by all societies. All can lead to mathematical thinking.

1. {12} Give an example for each of the six activities drawn from your reading of Ascher, describing the activity and the mathematics that comes from it.


2. {12} Give an example for each of the six activities from what you saw in the presentations of your classmates for "Math in MY OWN Culture,"again pointing out the mathematics in the activity. Do NOT use examples from your own presentation.

3.
{12} Briefly describe the sense of the origin and meaning of the concept of “circle” from the point of view of (a) a Platonist; (b) a Formalist; and (c) a Socio-Cultural-ist, in a sentence or two each.

4.
{15} Mario Livio reports that Nobel Laureate Eugene Wigner marveled at the “unreasonable effectiveness of mathematics”—wondering why mathematics works so well! How would his amazement be answered by (a) Platonism, (b) formalism, and (c) social-culturalism philosophies of mathematics?

Section B (answer
any five
out of the following seven questions)--35% {each is 7 points}


5.
{7} Ramanujan told Hardy that 1729 was a very interesting number:
the smallest number that can be expressed as the sum of two perfect cubes in two different ways. Find
the smallest number that can be expressed as the sum of two perfect squares in two different ways. Note that 0-squared cannot be used and that no perfect square can be used twice.

6.
{7} From the basic definition of perfect numbers, demonstrate that 33,550,336 is a perfect number without using Mersenne’s formula. Hint: Try it with 496 first, to see the pattern; it isn’t a messy as it might seem. Also, you can assume 8191 (which does come from Mersenne’s formula) is prime—trust me, it is!

7.
{7}
(notice this has parts a, b, and c)

(a) {2} Explain how to physically construct a Möbius ring.
(b) {2) Explain how to physically construct a Klein bottle.

[in both (a) and (b), include a clear statement of the materials and actions needed to do these constructions.]

(c) {3} Explain the feature of your three-dimensional Klein bottle that is actually not a feature of a “real” four-dimensional Klein bottle—a place where we have to “cheat” to make it in our limited three dimensional world.

8.
{7} President Barack Obama was born on 4 August 1961. His wife, Michelle Obama, was born on January 17, 1964. If they had been born in Ghana, what would their
Akan day names
be?

9.
{7} Set up a 5-by-5 number chart like the 602 chart in the magic square handout (not a regular magic square), that forces a sum of 2014.

10.
{7} Calculate an estimate of the percentage of the U.S. population that eat at McDonald’s on an average day. Explain your thinking and calculation and any estimates you make. (Hint: You can assume there are about 15,000 McDonald’s stores in the US, which is approximately correct)
Your process is more important than your final answer.


11.
{7} Chanukah and Christmas are usually fairly close together in Kislev/December. Since the important Muslim holiday of Eid-al-Fitr moves through the calendar, it usually does not coincide with the Chanukah/Christmas season. This year, 2013, Eid-al-Fitr will come on August 10. In approximately what year (using the civil year, not the Jewish or Muslim year) will Eid-al-Fitr again be in December?

Bonus questions: next page



BONUS

A. Prime-numbered years are not very common. The most recent were 2003 and 2011.
Find the
next four
prime-numbered years
{1 point for each, up to 4}
B. {3} What is “special” about the whole number 1385 ? What I’m thinking of is actually an irrational number that rounds to 1385. (This was my car’s license plate number in Nigeria)
Hint: Think of exponents and/or logarithm values using “famous” numbers.
(“8^th
Euler number” will not be accepted; also telling me its factors are 5 and 277 is not enough)