Saturday, December 19, 2009

Zombie Dogs Attack!

An extra credit assignment for my calculus class:
A pack of zombie dogs (ZDs) is reproducing quickly and killing humans. Scientists and calculus students have determined that the the total number of human deaths due to ZDs t days after the start of 2010 is given by
zombie_dog_equation

1. At the end of January 1st (t = 1), how many deaths have occurred?

2. At the end of January, how many deaths will have occurred?

3. Determine the number of deaths/day after t days: h′(t).
Hint: you may want to go “outside-in” with the chain rule.

4. Graph both h(t) and h'(t) on the same axes:

5. On what day will 90% of the human population (6.8 billion) be annihilated by ZDs?

6. When does the death rate reach half a billion per day?

7. When does the death rate hit its peak?

Sunday, December 6, 2009

Nonlinear graph shapes

I collected a set of various nonlinear graphs and hid the labels and scales.  They're good for discussions (I'm teaching exponential growth now).  Here's the Word file.

Untitled-1b

Can you match them all? Height of a projectile, iTunes sales, World Population of gorillas, Transfer speed versus message size on several networks, Heights of a group of men, Population of fruit flies in a container, Decay of radioactive atoms, Heights of babies over time.

Sunday, November 29, 2009

Classroom Jeopardy!

I created a review game with this online tool today:



If you want to play it, here it is.

Tuesday, November 17, 2009

Position, Velocity, Acceleration

My students asked how you could have positive acceleration but negative velocity. The spacebus holds the answer: click on the bus, then use Page Up and Page Down keys.

Wednesday, November 11, 2009

The rational number line

I ran across this diagram on another math site recently:

RealLine

It's a good diagram, but a little misleading.  Between any two rational numbers is another rational number, so the rational number line would appear to be solid at any given magnification.  The fact that there is "room" for the irrationals continues to amaze me.  The fact that the irrationals are even more numerous than the rationals is inconceivable to me.

Tuesday, November 3, 2009

The cone problem

Here's a problem that I cooked up today to puzzle the math team.

cone

You must slice a cone parallel to the base to create two pieces such that one piece has twice the volume of the other.  If the cone has a height of h, where do you make the slice?  There are two answers.

Tuesday, October 20, 2009

100 Free math lectures

Here we have a newly-released hodgepodge of free lectures for your enjoyment.  Some reminded me how much less lecturers strive to engage their audiences than, say, school teachers.

100 Incredible Open Lectures for Math Geeks

Thursday, August 27, 2009

Millions in a trillion?

This video clip reminds me of an anecdote related in The Universe and the Teacup:
[A professor] likes to impress his students with the power of large numbers by drawing a line designating zero at one end of the blackboard and another marking a trillion on the far side.  Then he asks a volunteer to draw a line where a billion would fall.  Most people put it about a third of the way between zero and a trillion, he says.  Actually, it falls very near the chalk line that marks zero.



How Many Millions are in a Trillion? from Econ4U on Vimeo.

Tuesday, August 25, 2009

Subway Factorials

I was ordering a sandwich at Subway the other day when the checker said "Awesome shirt!  What are the exclamation marks for?"

I was wearing a shirt with one of Ramanujan's incredible formulas for 1/pi:

ramanujan-pi

I was able to explain to him what a factorial is (6! = 6 * 5 * 4 * 3 * 2 * 1 = 720) but ran out of time before getting to double-factorials:

double-factorial

So, 6!! = 6 * 4 * 2 = 48, while 7!! = 7 * 5 * 3 * 1 = 105.  While double factorials look more impressive, they are always less than or equal to normal factorials.

I learned that double factorials are one case of multifactorials, where the number of exclamation points indicates the difference between factors:

8!!! = 8 * 5 * 2 = 80

10!!!! = 10 * 6 * 2 = 120

Looking at the entries for "factorial" on Wikipedia and Wolfram, I was surprised at how little I know about them.  I did find a few cool facts to take away, though:

  • the factorial function growth faster than any polynomial or exponential function, but it can be approximated by an expression involving e and pi:
    factorial approximation

  • the gamma function is just like a factorial but can take real and complex arguments (not just integers).  The gamma function does not return 0! = 1, so it must be defined as a special case.
    GammaFunction_1000

  • 0.5! = sqrt(pi) / 2

What other stuff is cool about factorials?

Monday, August 24, 2009

Population graph

This could be illuminating for a class to generate: a map of the world where area is determined by population.  Can you find Australia?  Doing the US would be fun, with a huge California and tiny Alaska.

world_relative

Found at http://www.geohive.com/earth/gen_popsize.aspx

Wednesday, August 19, 2009

The Wooden Ratio

You've heard about the Golden Ratio, 1.618..., computed as the mean of 1 and sqrt(5), or as the limit of the ratios of successive terms in the Fibonacci Sequence.  This ratio rears its head in the natural world, music, art, and in the proportions within our own bodies.

I have recently become aware of another ratio with similar properties.  Computed as the mean of 1 and 3, or as the ratio of successive terms in the sequence {2^n}, I have dubbed this number the Wooden Ratio (after the two-by-four).  It begins 2.000... and we can represent it with the symbol þ (thorn).

While the Golden Ratio appears in architecture such as the Parthenon, just think about how many Wooden Ratios appear in all of the brick buildings made by humanity.

bricks-wooden-ratio


In each of these bricks, the length is þ times the width, and the width is þ times the thickness.