October 2010

Coffee and Cream

I was recently reminded of an excellent math problem involving mixtures.

Imagine yourself sitting in front of a cup of coffee and a cup of cream.

Suppose you take a spoonful of cream, pour it into the coffee, and stir it up. Now once that’s thoroughly mixed, you take a spoonful of the mixture and pour it back into the cream. Then you mix that up. After all of this, is there more coffee in the cream, more cream in the coffee, or equal amounts in both?

I encourage you to think about the problem before perusing the several solutions below!

Challenge Teaching

Paddling Upstream

The Canoe in the River problem is an algebra classic. You know how it goes: “Paddling upstream, it takes Betty Boater 6 hours to travel up the river to Point Apex. It takes only 3 hours for the return trip downstream to Point Bellows. If the distance between Point A. and Point B. is 15 miles, what would Betty Boater’s speed be in still water?”

Below is wonderful retelling of the Canoe in the River problem created by Dan Meyer. Using a video camera, an ipod, a quiet morning in a mall, and some great editing, this problem is brought new life in this modern and engaging context.

Check it out at http://blog.mrmeyer.com/?p=7649. Meyer seems to be focused on modernizing mathematics curricula, and the more stuff he does like this, the better.

And for Betty Boater’s speed, click here.

By patrick honner, 14 years9 years ago

Geometry Technology

The Perfect Parallelepiped

In general, it’s unusual for a rectangle to have sides and diagonals whose lengths are all integers (i.e., whole numbers). Consider the following three rectangles, all of width 3:

Looking at the different lengths, we see one place where the diagonal length is an integer, but in the other cases, the diagonal length happens to be a non-terminating, non-repeating decimal (i.e., irrational). Indeed, the diagonal length will be an integer exactly when the length and width are part of a Pythagorean triple, but compared to the alternative, this is uncommon. (While there are infinitely many occurrences of this, we can still meaningfully consider it uncommon).

Now, imagine the situation in three dimensions. A rectangular prism (think of a cardboard box) has 12 sides, 12 face diagonals, and four space diagonals. It would be extremely unusual for all of those 28 lengths to be integers. Even if we didn’t limit ourselves to rectangular prisms, but we allowed for the box to be slanted in all directions (that is, a parallelepiped), it would still be a numerical miracle for all those lengths to be integers.

Well, meet the perfect parallelepiped!

This was discovered by a couple of mathematicians at Lafayette College in Pennsylvania, using brute-force computer trials. It looks like they found some others, too. So thank you, Clifford Reiter and Jorge Sawyer, for giving me an extra credit problem for my next exam!

Related Posts

CDs, Prisms, and Parallelepipeds

By patrick honner, 14 years8 years ago

Application Statistics

Benford’s Law

This is an article about the discovery of new sets of data that seem to obey Benford’s Law–a curious mathematical characteristic of the numbers we collect from the world that is really more conjecture than law.

http://www.newscientist.com/article/mg20827824.700-curious-mathematical-law-is-rife-in-nature.html

It seems that in scores of data sets collected from natural phenomena, the numbers we see tend to start with the digit 1 far more often than, say, with the digit 6. Indeed, statistical analysis shows that when you look at population numbers, death rates, street addresses, lengths of rivers, stock prices, and more recently, depths of earthquakes and brightness of gamma rays, the observed numbers start with the digit 1 about 30% of the time. The occurences of other digits as the leading digit fall as you go up the scale.

Apart from being a natural curiosity, Benford’s Law has proven to have some very useful applications. Scientists can use Benford’s Law to help predict phenomena and look for trends in data, as the rule gives number-crunchers an idea of what they might be looking at from the start.

Additionally, Benford’s Law has been successfully used to identify all kinds of numerical fraud–tax fraud, voter fraud–because when people are faking numbers, they tend to evenly distribute leading digits. Benford’s Law tells the data-police that if approximately 1/9 of the numbers they are looking start with 1, then something fishy is going on.

Keep that in mind next April.

By patrick honner, 14 years9 years ago

NYT Resources Teaching

Math Lesson: Predicting the Vote

I am very excited to have my first Lesson Plan published by the New York Times Learning Network. I wrote a mathematics lesson built around profiling the upcoming presidenital election, using data and analysis from Nate Silver’s 538 Blog at the Times.

The lesson is titled “Predicting the Vote: Analyzing Election Data”, and can be found here:

http://learning.blogs.nytimes.com/2010/10/18/predicting-the-vote-analyzing-election-data/

Trying to write a lesson plan for general use was much more challenging than I imagined, but it was an interesting and educational experience for me. Hopefully it will produce some interesting educational experiences for others.

By patrick honner, 14 years9 years ago

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