# Fall 2011

Whew! I finally got all my Fall 2011 syllabi posted here on the blog. Hover over the class name at the top of the blog and a menu will drop down where you can see the long version and the short version of the syllabus for each of my regular classes. I’ll add course homepages and course calendars over time.

See the Golden Ratio at work in the Apple logo.

h/t World-Shaker

# Found Math

I was playing with a friend’s hourglass and the few grains at the top made these lovely triangles that reminded me of the Sierpinski Triangle.

# MathFest Day 1, or rethinking grading part 1

As always, MathFest is an exciting and instructive experience. I signed up for the mini course on alternatives to traditional grading. I’ve been really interested in this since I started reading Think Thank Thunk regularly and have wanted to implement such ideas in my college classes. In fact, last semester in my liberal arts math and calculus 2 classes, I included a quiz component where I tested the various concepts and skills individually and assessed on a 0-5 scale, a grading scale much like what the mini course facilitators are using at the Air Force Academy. I would give failing marks for significant errors from prerequisite courses, like the “
Freshman’s dream” or forgetting to use the product rule. The nicest aspect for me is changing the conversation from “how many points is this worth” to a more concrete “how much of the concept did the student understand and how well did they communicate it?” Responses from my students in this experiment were overwhelmingly positive.

After the mini course I attended the contributed paper session on history of mathematics and classroom uses, my new passion. Some highlights were learning about Cicero’s visit to Archimedes’ grave (recounted in the Tusculan Disputations of all places – thanks again to Mrs. Dr. Hall’s high school Latin classes for learning to appreciate the classical world), and an admonition from Thomas Drucker to have a theme for a history of math class. I think my theme for the fall will be the “.9 repeating is 1″ debate/controversy. This simple equation bothers or intrigues almost all of my students, so using that to frame my walk through history in my history-focused liberal arts math class this fall might be quite useful. The concept of infinity through history, as Drucker used, would also be cool. I was also particularly intrigued by using historical papers on the concept of trees in graph theory, as brought out by Jerry Lodder in his talk.

On a tech note, I’m enjoying using an iPad to keep track of my notes and thoughts. The keyboard works well enough, the Bamboo Paper app is good for taking quick sketches and handwritten notes, and the camera is nice for getting quick snaps of a important slides (especially those bibliographic slides that are so hard to preserve without asking for copies the PowerPoint). Any other ideas out there for taking good notes? Finally, blogging about things has helped me think about what I want to come away with from the conference. More from Lexington later!

# MathFest

I’m in Lexington, KY at MathFest 2011 (http://www.maa.org/mathfest/). I went to several interesting talks this morning – talks on math history and teaching. I’ll try to post some things I find particularly interesting.

If you don’t know already, I’m fascinated by language and grammar, so one tidbit I heard in a talk ( “Excursions to and from Semantic Oblivion” by David Easdown) about mathematics and syntax vs. semantics issues is the sentence “Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo.” (see http://en.wikipedia.org/wiki/Buffalo_buffalo_Buffalo_buffalo_buffalo_buffalo_Buffalo_buffalo)
This sentence makes no sense on first reading, but for those who understand, the sentence makes perfect sense. One must decode, must understand the meaning of the words and their usage in the sentence in order to be admitted to the secret society of those who get it. Such is the way of math (as well as other subjects), but I find it challenging to remember the difficulties of our students and truly sympathize, especially when I’ve taught a class multiple times and the content becomes more and more obvious to me. Sometimes it’s easy to maintain that sympathetic connection and sometimes not, but I’m convinced it’s worth the effort.

# Homework for 7/22

1. Consider the relation “the integer a is related to the integer b if a-b is positive.” Is this an equivalence relation? Why or why not?

2. Create addition and multiplication tables for $\mathbb{Z}_9$.

3. Consider the equivalence relation given by “a rational number a is equivalent to a rational number b if they have the same denominator when in reduced form (treat negative signs as part of the numerator so that all denominators are positive).” Is the operation of addition defined by $[a]+[b]=[a+b]$ well-defined for this set of equivalence classes? Why or why not?

4. Prove that a fraction of complex numbers written in standard form (with nonzero denominator) can be written in standard form.

1. Give the algebraic formula for a function whose graph has been obtained from the graph of $f(x)=\sqrt{x}$ by the following sequence of transformations: (1) shifted to the left 3 units, (2) reflected across the y-axis, (3) shifted up 3 units, and then (4) reflected across the x-axis.
2. Suppose the point (2,4) is on the graph of $y=f(x)$. Give the coordinates of the corresponding point on the graph of the transformation $y=-2f(6x+3)+4$.
3. Where is the vertical asymptote of $y=\ln(3x-1)$. Where are the vertical asymptotes of $y=\ln(x^2-4)$? Explain how you know for sure.
4. Create addition and multiplication tables for the single digit sums and products in the base 9 number system. Then calculate $23_9\times 56_9$ in the base 9 system.