Empowering Women in Math

An interesting article about an educational non-profit in New York working to help women get higher paying jobs and better opportunities through math education:

http://www.nytimes.com/2009/04/19/jobs/19math.html

I actually hate using rote techniques when teaching math, but there is something to be said for being able to execute an arithmetical calculation correctly over and over again.

I thought the part in the article about differences in spatial reasoning were interesting, particularly the part about spatial reasoning being teachable. If you are looking for challenging spatial reasoning questions, a good place to look is Perceptial Ability section of the DAT, or Dental Admissions Test. From speaking to some dentists, being able to look into people's mouths upside down and identify what is going on takes a lot of spatial reasoning, and so they are tested on this in their entrance examinations. I've only found a few links to practice questions like here.

Funny Foxtrot Comic

Just a quick and interesting comic that involves math. Have fun!

All the Water In the World

I am trying to get the Google Earth plug-in working on a page somewhere, but for now here is the KMZ file of a demonstration that shows the volume of the water in the world compared to the volume of the Earth in a set of aspiring spheres. There is another demonstration I also found here, that illustrates the amount of CO-2 emissions geographically in a cool spherical bar chart. Reading graphs is an important skill learned in a mathematics education, and it is interesting that new forms of information management tools, like Google Earth, are also enabling new types of graphs and visualizations.

Upon seeing the graphic above, I was first struck by how small the largest sphere of water was to the total size of the Earth. If this relationship is actually to scale, it shows how spreading a substance evenly over the surface of an object that is much larger its size occupies far less volume than the object itself. A question to this end would be how to figure out the average depth of water it would take to cover the Earth if it were evenly spread over its whole surfaceusing the numerical values from the demonstration.

The relationships between volumes of geometric objects can be very useful to intuit when making rather ordinary calculations. It can help you decide which pail you should use to get sand to fill up a sandbox, or the size of a pot to use to make Jambalaya when all the sausage and rice are done cooking. In understanding this relationship, it is helpful to know how the formulas for volumes and areas vary with quantities like length, height and radius.

We discussed how to determine volumes from areas when a shape has a uniform cross section int the post "Rapelling Down Fissures" -- there we learned that for some objects, their volume changed linearly with the height of the solid. So if we were planning on building a new water tower in our village when our population had almost doubled, we could build a water tower with the same circular cross section but just twice as high. If we decided to change the radius of our tank, however, and keep the height the same, would we need a new tank whose radius was twice as big as our previous one? What do you think?

To understand how much our radius must grow to accomodate the new required volume of water, it is helpful to remember that while the volume of our cylindrical water tank will vary linearly with its height, it will vary quadratically with the radius of its circular base. So, increasing the radius by a certain amount would actually lead to a change in volume corresponding to the radius squared, so we wouldn't need to adjust our radius in a one-to-one proportion to our required volume.

QUIZ

Q1: Use the numerical values from the graphic to determine the radii of each of the spheres representing different quantities of water.

Q2: What is the volume of saltwater on the Earth? What would be the radius of a sphere used to contain this much water?

Q3: Draw a graph of the volume of each of the quantities portrayed in the aspiring spheres verses the radii obtained in Q2. Does your graph make sense? Why or why not?

Dynamic Currency Conversion

There is a very interesting article up on Washington Post that talks about dynamic currency conversion. Basically, this involves an additional transaction when using your credit card overseas to make purchases. It is supposed to be "convenient" for the consumer so that the cost can be seen in the consumer's home currency. But what is the cost of this convenience? That's what the article takes a look at. Here is an embedded currency converter:



Here are some ideas for a lesson plan around this article and currency conversion:

(1) What country would you like to visit most? Estimate the costs of a flight there, food, lodging, transportation, souvenirs, etc. This could all be done in dollars. This could be enhanced by looking up actual rates online, as this will give practice of finding and interpreting deals online for traveling.

(2) Compare the costs for each of the items with how much it will cost with each of the different modes of converting currency. These include, but aren't limited to, using direct cash conversion, using traveler's checks, credit card without DCC, and credit card with DCC. Remember that each of these come with their own conversion commissions. See which one offers the best price overall, and what the price difference is. What is the "cost for convenience?"

(3) Find out what the foreign exchange fees are for yours or their own credit card and compare with the average that's given in the article (~3%). How much money does this amount to?

(4) An investment type of question that requires research: Imagine that you had the equivalent of $10,000USD in a major (or maybe not so major) foreign currency. Based on the current economy and historical fluctuations (click here for viewing historical data), do you think it would be better after 2 years to have exchanged it back to USD (with the exchange fee) or not exchanged it back at all? How does this compare with how 2 years ago would compare to the present?

Calculating Carbon Footprints.

Finding ways to teach mathematics through the context of environmental awareness and climate change is an interesting and important challenge. Mathematics can be helpful to understand things on scales much larger than many people are used to, particularly geological scales of size and time.

Quantifying climate change is an interesting mathematical challenge, since weather and climate are such complicated systems. There are a number of interesting quantities we can consider, however, to assign quantitative measure to detrimental changes in the Earth's ecosystem. The rates of species extinction and deforestation, as well as the amount of water and food that will continue to be available are interesting metrics to help us quantify the problems we are facing.

Perhaps the most talked about number regarding climate change is a person or organization's Carbon Footprint, or the amount of CO-2 they emit based on their lifestyle. That is what we will consider using the Berkeley Institute of the Environment's Carbon Calculator.

The video below is (I believe) part of the 25 million dollar Virgin Earth challenge, which calls for teams to suggest strategies to specifically remove CO-2 from the atmosphere.

The Berkeley Institute of the Environment has developed a very interesting tool called the CoolClimate Carbon Footprint Calculator which enables a user to enter information related to their lifestyle, and determine how much carbon dioxide they are emitting compared to the national and international averages. There are a number of interesting ways to teach high school level mathematical concepts through this interactive environment.





Some ideas:

(1) Have students use the Carbon Calculator to enter values related to their household. This will likely require them to talk to their parents about their household's monthly utility bills, miles driven and the square footage of their house. This seems like a good way to help promote awareness of issues like home and auto ownership for the students, and facilitate discussion between them and their parents.

(2) Once a student uses the Calculator to determine their household's CO2 emissions, this number will appear in the upper right corner of the carbon calculator. With this number, ask them to compute the size of a building it would take to store all of the carbon they will emit this year. This is an opportunity to discuss the notion of density and volume, how to determine the weight of a molecule (i.e. CO2) using its chemical formula, and converting from units of linear measure (inches, meters, etc.) to units of cubic measure (cubic inches, cubic meters, etc.). To give a standard of measure, you might ask a student to measure the volume of their bedroom, the volume of their school and the volume of a sports stadium given its dimensions. Once they do this, they can compare this to the volume of CO2 they emit.

(3) The calculator also gives bar graphs to indicate how much CO2 is emitted by an individual compared to national averages and international averages. These quantities can be used to discuss percentages and percent increases. A question might be as simple as having them calculate what percentage of these other averages is their emission footprint? What is the percent increase of their footprint compared to the other averages??

(4) Given the average carbon footprint of a US household and that of an international household, ask a student to calculate how much total carbon is emitted by humans in the world, and what percentage of that is emitted by the US?? This would involve them doing some independent research on population sizes, and to realize that they need to know those values to get to the answer.

(5) The interactivity of the Carbon Calculator enables us to explore the underlying functions being used by the BIE to generate these estimates. The equations are given in a document here, for the purposes of helping students check their answer. What I might ask them is to calculate the functions for metric tons of C02 vs. amount of money spent on furniture, food, square footage of their house, etc. This would require them to change those values and see what the resultant change is by the calculator. Since exact values aren't given, and no decimal significant digits are kept for the tons of CO2 emitted, this would require them to make big changes in a value (say 100$ on furniture, 1000$ on furniture, $5000 on furniture) and use these big jumps to estimate a slope of character to the graph they obtain.

(6) An interesting feature in calculating carbon footprint based on your utility supplier (at least for California) could lead to an interesting research project by students to understand what kinds of energy sources are used for the power grid, which utility companies are more progressive in pursuing "green" and non-polluting alternatives, and how where you live and the choices made by your public utilities and local government affect your carbon footprint regardless of your personal efforts of conservation.

(7) Ask students to use the calculator to determine what lifestyle changes they and their family would have to undertake to get to the international average. Are these changes possible? What are some alternative means to reduce our individual impact or community's impact? How many trees would they need to plant to offset their carbon footprint? Do the indirect costs of such efforts override the apparent benefits?

Getting a Piece of the Pie

Providing students with the tools to interpret quantitative data and information they encounter on the Internet or through news sources is an important component to promoting mathematical literacy. I have recently been teaching my students how to read the information found in Google Finance while implementing important concepts in fractions and ratios, as well as scientific notation.

A typical lesson might look like this:

(1) Ask the students about companies they know about, especially related to things they are interested in. Some of my students have picked sporting goods companies (Nike, Reebok, etc.), entertainment companies like Disney, or clothing companies like the Gap. Also ask them about what companies they think might be doing well or not so well in the current financial crisis, and why. Use this information to bring up a few Google Finance summaries of some of these companies.

(2) Given their imaginary bankroll of $10,000, I ask students to decide how much they want to invest in their first company. Using that amount, students can use the share price to determine how many shares they can buy. I then ask them to determine what percentage of the company they now own, given how many shares they have by directing them to the Outstanding Shares entry of the Finance summary. Because that number is very large, I have them compute this percentage without a calculator using scientific notation.

(3) Many students don't realize that the price of a share isn't that important, since different companies are broken up into a different number of shares. This point is well expressed by using Google's stock price vs. a company like Microsoft. Google's price is currently in the high 300's, while Microsoft is just below 20, making it seem like they can buy MORE of Microsoft with the same amount of money. By having them compute the percentage of the company they own for the same amount of money, this fallacy can be debunked, and the idea of market capitalization (i.e. the total value of a company) can be introduced.

(4) In continuing to analyze a particular stock, the idea of positive and negative slope can be introduced in the context of linear regression -- i.e. approximating a trend in a stock price using a line. The slope of that line corresponds to the gain or loss of value of the stock, and can thus help them to understand why we spend so much time discussing the slope. If a stock changes directions many times, this is also an opportunity to introduce how when using polynomials to approximate graphs, the degree of the polynomial determines how many slope changes our approximation can accomodate.

Listening to Math

Finding ways to engage students sensory and creative faculties in the cause of teaching math is, in my opinion, essential to helping increase their interaction and long term retention of otherwise abstract concepts. Although much of the material on the Internet focuses on visualizing mathematics, developing techniques to relate audio to the topics they are learning seems equally important.

The applet I have been using most often to achieve this is the one found here:

Demonstrating Audio Beat Frequencies

This tool allows you to control the amplitude and frequency of two sinusoids being added together, and to listen to the resulting output. It is immediately obvious that this can be used to teach the relationship of frequency and amplitude to the resulting sound generated, which is important for topics in the Math Level IIC SAT, as well as pre-calculus and physics. It is interesting to incorporate a phase shift in the sinusoids, and see how that affects both the resulting sound as well as the resulting waveform.

One of my favorite uses of this tool, however, is to demonstrate the difference between linear functions and non-linear functions through sound. Typically the argument of a sinusoid is LINEAR in time, such as sin(10t) or cos(3t). Consequently, there is a character to the sound that is even and repeating. The frequency stays the same over the length of the tone, making the sound uniform. If, however, we change the argument in the sin wave to one that is NON-LINEAR in time, such as sin(10t^2), the sound changes noticeably. No longer is the frequency the same over time, and the sound takes on a very psychedelic nature. Try it yourself and see what it sounds like!

A sample lesson for this applet might be something like:


(1) Using only one sine wave, ask students to guess how changing the number within the sine's arguments (i.e. the frequency) and the coefficient in front of the sine (i.e. the amplitude) should affect the resultant sound. Pick a few different amplitudes and frequencies, and have them guess which one is which from just the sound. What is the effect of a negative amplitude or frequency?

(2) Try applying a phase shift to the wave. Does this affect the resulting sound? Why or why not?

(3) Instead of using a linear function of time in the argument, try using sin (10t^2) or sin (10t^3 - 10t). How does this affect the tone? Use this as a way to affirm the notion that something that is linear has the same general character over its entire domain, but something that is non-linear does not.

(4) Using two sine waves, what relationships of the frequencies will produce a beat? Help students to see that when a beat forms, there are TWO frequencies that are occuring: one is the reference frequency of the underlying tone, and the other is the frequency of the "beat", i.e. the pulsing of the tone's amplitude. One should be at the sum of the two frequencies, and the other at the difference of the two frequencies. Have the students use the trigonometric identities for angle addition and angle subtraction to generate particular beat patterns or tones.

Creating A Fantastic Contraption

Everyone likes games, especially our students. Fun and educational games are hard to come by, but the internet has everything doesn't it? While discussing a recent "indie" game, I was told of a game called "Fantastic Contraption." This game involves the the sandbox concept, which means you can solve the puzzles in any manner that you wish. There's just a simple goal of getting the pink ball to the pink area. I've included a screenshot, since embedding the flash for the game has the sound on and can be quite annoying.



The levels require some ingenuity and engineering experimentation to solve. After solving a level, you can also look at how other people completed it, so you can get more ideas! See how many you can do!

The Perfect Sunset

Recently, I showed a student of mine a Flash applet from the Astronomy Department at the University of Nebraska-Lincoln. It very generally demonstrates how the sunset changes its North-South location on the horizon during the course of the year (due largely to the tilt in the Earth's axis).

In talking about the sunset's movement, my student expressed to me a lifelong desire: to see the sun setting over the Golden Gate Bridge. It takes a little figuring out, since not only does the sunset move during the year but where it happens depends on where on Earth you are (eg If you are far North of the Equator during the summer, then sunset will appear to happen much farther North than if you were standing at the Equator)!

After thinking about different locations in the East Bay to view the Golden Gate Bridge from, I've determined that Cesar Chavez Park is the optimal place (or possibly the nearby Berkeley Pier).


Latitude is a measure of how far North or South a place on the Earth is from the Equator, so knowing the latitudes of both Cesar Chavez Park and the Golden Gate Bridge will help us to find the date of our particular sunset. Latitude is given in degrees, like the way you measured angles in Geometry.

The Flash applet below gives you control over the Time of Day, Date, and Latitude of the observer. Given that

Latitude of Chavez Park: 37.87^oN
Latitude of Golden Gate Bridge: 37.82^oN

can you use the Flash applet below to find the date of the perfect sunset?

[There are some hints below the applet that may help you.]



Key (for 3-D Observation Simulator)

• Yellow Line: Sun's Declination (Daily Path)
• White Line: Sun's Ecliptic (Annual Path)
• Blue Line (E-W): Celestial Equator
• Blue Line (N-S): Prime Hour Circle

Hint 1

If you're feeling stuck, play around with the Date and Latitude. See what happens in different places at different times of year. What patterns can you find?

Hint 2

Pay close attention to the Sun's Declination. Go ahead and read about it on Wikipedia. There's a continuous measure of declination in the box in the lower left of the applet.

Hint 3

The latitudes of Cesar Chavez Park and the Golden Gate Bridge are just 0.05^o apart. How significant is that in our calculations, considering that our accuracy is limited by that of the applet?

[Cross-posted at Roland's Miscellany.]

Falling Into a Black Hole

So if you've ever wondered what it might look like to fall into a Black Hole, wonder no longer! From an article at New Scientist, researchers at the University of Colorado - Boulder

built a computer code based on the equations of Einstein's general theory of relativity, which describes gravity as a distortion of space and time.

They follow the fate of an imaginary observer on an orbit that swoops down into a giant black hole weighing 5 million times the mass of the sun, about the same size as the hole in the centre of our galaxy.

[Note: There are more details on what exactly is happening in the video in the article linked above.]



[Cross-posted at Roland's Miscellany.]

Planetary Orbits, Ellipses

From the Astronomy Department at the University of Nebraska-Lincoln: