Math In The News

That book costs HOW much??!!

2011-05-03T08:53:00.000-07:00

True Story:

In early April a graduate student was searching for a copy of this genetics text:

Upon searching for the book, and then locating it on Amazon, the researcher was stunned to find that the book was listed at well over a million dollars! The list price was $70.00 and the book was out print, but not rare. The tome should have commanded a price of somewhere in the neighborhood of $100.

Here's the screenshot from Amazon on the day this was noticed:

The student was curious and tracked the prices over the next several days:

Some good questions that arise from this:

How are these two booksellers determining thier new price each day?
What is a function that represents Bordeebook's price for this text d days after April 8th
How long had this pricing spiral been going on before it was discovered?
Why does Bordeebooks seem to be intentionally setting its price higher than Profnath each day?

Credit where credit is due...

(warning following these links will lead to answers to many of the above questions and may take away some of the fun you would have thinking for yourself!)

The original post about the discovery was on this blog written by Michael Eisen.

Let it Snow, Let it Snow, Let it Snow!

2011-01-21T17:06:00.002-08:00

Last week we had one of the biggest single day snowfalls I can remember. We went from basically bare ground to about 20 inches of snow on the ground in one day! The sheer volume and immensity of all that white stuff calls out for a mathematical analysis:

Imagine our soccer field. It is a rectangle, it is now covered with 20 inches of snow. What is the weight of the snow that lies directly over the field?

You're on your own to estimate the dimensions of the field...use any reasonable size for a regulation high school soccer field.

What I will tell you as you get started is how snow is classified with regards to its "fluffiness". When analyzing and predicting snowfall amounts, meteorologists refer to a snow's "snow:liquid ratio". This storm's ratio was about 18:1 . A wetter snow might be as low as 10:1 . A ratio of 18:1 means that 18 inches of snow on that soccer field is the equivalent of 1 inch of water on the field.

Using the ratio and some information you'll have to find with some online research you can find the surprising answer to how much that snow weighs.

2011-01-21T17:06:00.001-08:00

A Deluge

2010-10-02T10:00:00.001-07:00

This week saw massive rains in Vermont and elsewhere up and down the East Coast . The intensity of the rains and subsequent flooding are described in this article taken from the Brattleboro Reformer:

Flooding closes Vt. roads, state offi ces as storm slams East coast

Here in Putney, we were watching the Connecticut River pretty closely as our school keeps a number of rowing shells in a boathouse on the riverside and if the river floods over the banks, the results would be disastrous.

In looking into the situation, our crew coach used this graph, which she sent along to all the faculty to see if we might use it in our classes.

Since I was right at the end of a unit on functions, it provided an exceptional context to highlight and really use the concepts and vocabulary we had just encountered:

1. Find the average rate of change of the water level (stage) from 10AM Friday to 10AM Saturday.

2. During what 6 hour time period was the rate of change the largest, and what was that rate of change?

3. If the stage continued to decrease at the steady rate predicted by the green portion of the graph, when do you predict the level will return to the minimum level shown on September 29^th?

4. Notice that the ~~independent~~ dependent variable (the so-called “y-axis”) is scaled in two different ways: the left hand scale gives the rivers stage in feet as a function of time and the right hand scale gives the river’s flow in kcfs as a function of time. Find the average rate of change of flow during the 6 hour time period you picked in question #2. Please be sure to give your answer with a correct unit

5. Finally, make a table in which you use stage as the independent variable and flow as the dependent variable. Enter 10 pairs of values in the table, then graph this data with Geogebra.

6. In the INPUT BAR, enter the command FITPOLY[A,B,C,D,…,I,J,2] where A through J are the names Geogebra assigned to your points. This instructs Geogebra to make a 2^nd degree polynomial (a quadratic function) that fits the data you provided.

7. What happens to the rate of change of flow to stage as the stage gets greater?

8. Think about the physical mechanics of a river and give an explanation as to why this relationship should NOT be linear.

Now, I used a quadratic regression, just because I quickly saw that it fit pretty well. I have no idea right now if it's the best model, but it was a nice way to introduce the students to that command in Geogebra.

The document I gave the students to work from is available here.

How Important is Eating Locally?

2010-08-20T16:33:00.000-07:00

I was interested to read Stephen Budiansky's Op-Ed piece , Math Lessons for Locavores in the New York Times today. Budiansky posits that when the true carbon costs of shipping produce across or between continents is accurately analyzed, the impact is minimal, especially in relation to the other ways in which our food production systems use energy.

Studies have shown that whether it’s grown in California or Maine, or whether it’s organic or conventional, about 5,000 calories of energy go into one pound of lettuce. Given how efficient trains and tractor-trailers are, shipping a head of lettuce across the country actually adds next to nothing to the total energy bill.

It takes about a tablespoon of diesel fuel to move one pound of freight 3,000 miles by rail; that works out to about 100 calories of energy. If it goes by truck, it’s about 300 calories, still a negligible amount in the overall picture. (For those checking the calculations at home, these are “large calories,” or kilocalories, the units used for food value.) Overall, transportation accounts for about 14 percent of the total energy consumed by the American food system.

Budianski has a blog, Liberal Curmudgeon, on which he posts this chart, taken from the University of Michigan's Center for Sustainable Systems.

Breakdown of Energy Consumption in U.S. Food System

What do you think? Does this chart demonstrate that the energy saved by growing and eating locally more often is so small as to be unimportant? Why or why not?

Integrating Current News Events into High School Math Teaching and Learning

2010-07-04T16:25:00.000-07:00

[Virtual Conference on Soft Skills]

We teachers of mathematics constantly have our legs kicked out from under us by the society we live and work in. We are charged with providing students the skills and understandings they need to be numerate citizens, while at the same time, those very same students are regularly bombarded with a not so subtle message that mathematics is difficult to understand and unrelated to their lives. How many times has an ADULT you know told you (or their own children) how much they hated math, how they only understood math before they "got to algebra", or otherwise derided mathematics.

In order to overcome these misconceptions about the relevance of mathematics to everyones' lives, I've tried for a long time to teach mathematics through a set of contextualized problems rather than as a set of isolated skills. Students are more apt to understand new (or not so new) concepts and to retain their understanding when they can make connections between the new mathematics and what they already know about the world they live in.

In the past, I have used hands on labs, "realistic" word problems, and graphing technology to help students make these connections. This year, I decided to try to make the connection even more direct by using real, current news stories as the basis for some of our problems.

The first story that really caught my attention in this process was the Deepwater Horizon oil disaster. Hearing about the situation unfolding on the radio each day I couldn't help but be reminded of the classic related rate problems about how the area of a circle changes as its radius changes by a known rate. It was late in the school year and my precalculus class was getting into a bit of a rut, so I made the decision to shake things up a bit with a math problem that really was based on a "real world problem".

After doing a bit of research on the details of the situation at the time, I decided to pose the problem in this blog post. I felt that to make the problem accessible, and to ensure it got at the concepts of related rates I needed to introduce some simplifications, which you see in the post.

My students LOVED this problem! Every student was deeply engaged, but I noticed one boy in particular, who had been kind of lackluster all year all of a sudden was completely hooked, and working as hard as I'd ever seen a student work at a math problem. He said that the realism of it made it worthwhile for him, that he never had bought into his teachers explanations of how math "could" be used, and that this problem really made the math matter for him.

In addition to the way the problem drew the students in, the problem was messy, which was really good for my students. The task required that the students really problem solve. The units were really unusual (barrels, micrometers,...) and the students really seemed to enjoy having to research what these units were.

Since I started this late in the school year, I didn't have time for many more problems. I managed to fit in two more. One, about the proposal for a "soda tax", I used the same way, using the story as a data source and writing some problems around the data. The other, about Vermont's legislation banning texting while driving I used simply as a hook for an experiment and data analysis activity I had already planned long before I saw the news item. It was good to see that this "Math in the News" process can be used in two ways: One is to provide for spontaneous problem solving opportunities in which students must decide what knowledge they must apply to a novel problem; and the second is to provide scaffolding around a learning activity that fits into a pre-determined place in my curriculum.

I've continued to look for news items and blog about how I would use them, in hopes of training myself to quickly see stories with potential for math problems. When I look for a story, the only criteria I have is that it has to be able to be analyzed with high school level mathematics. I don't want to force the mathematics onto a story, but rather to have the mathematics emerge clearly and naturally from the story.

Readers: Keep me in mind as you read the news this year, and send me a note if you see a story that could be integrated into high school mathematics problem solving. It's be fun to collaborate with some other teachers in the edublogosphere and create something together.

Payday Lenders

2010-06-22T17:05:00.000-07:00

I went on kind of a long drive last week (3200 mile round trip to Barkus, MN to pick up my Scamp!) On the way there I heard a story on the radio about payday lenders and check cashing shops. I've seen these places around and even remember using places like these myself when I was in my early twenties...I cant remember why I used a check cashing shop rather than a bank. I certainly wouldn't have used it had I taken a minute to compare the fees with what a real bank would have charged!

Here is a video from PBS Newshour that explains some of the ins and outs of payday lending:

I think it'd be a great exercise in financial literacy (and it could easily be expanded to include issues of social justice) to have my students go to a payday lender's website (for instance My Cash Now) and use the information provided by the lender to answer questions like these:

If a borrower took out a 14 day loan for $500 and then paid it back early after only 5 days, what would the APR be?
It's quite likely that a person who needs to borrow money 14 days before they are going to be paid will spend the money he borrowed and then need to use the new paycheck to pay back the loan, and so need to immediately take out another loan. If this happened for an entire year before the borrower was finally able to pay off the loan and not need another loan, how much will the borrower have paid in interest on that $500?
What if in the above situation, the person found they needed to take out a new loan to pay back the previous loan and its fees? So at the end of the first 14 days, they took out a loan of more than $500 because they needed the $500 plus the interest they had to pay. If they coninued in this fashion for an entire year, how much would the borrower pay in interest on that original $500?
Devise a fee structure that a payday lender might use if they decided that they really wanted to make exactly 1000% APR from their customers (victims?)

Fill 'er Up --are the oceans getting filled with oil?

2010-06-07T09:44:00.001-07:00

The Deepwater Horizon oil disaster has stayed in the forefront of the news lately, so it has me wondering what more I might do with it in terms of high school mathematics. The latest news is about the "containment cap" that has been used to gather up some portion of the escaping oil.

I'm going to try to seek out some more global sources for the news I use, so here is a nice peice from the Guardian, a U.K. news organization summarizing the current state of the attempt to contain some of the spill: Guardian Article

Here's some video showing how much oil is still escaping around the so called "containment cap":

Some questions I think I would pose to my students (they're gone for the summer now, so this is just to build a database of ideas for the next few months) are:

Research the current information available about the spill and make an estimate of what volume of oil is still escaping each day and then calculate what percent of the total volume of the world's oceans and seas that is.
After oil has leaked into the ocean, we can think of the oceans as a mixture that is pat seawater and part crude oil. Make a function that will give the percent of the oceans' volume that is of crude oil as a function of time.
Find a meaningful way to graphically display this information.
How might you reframe the question to make the information found in answering the question more powerful if your goal is to inform the public about the immense scope of this disaster?
How might you reframe the question if you were working for BP and wanted to give the impression that the sheer size of the oceans give's them an enormous capacity to "absorb" human scale problems?

Trillion Dollar Daze

2010-05-31T19:26:00.000-07:00

Last weekend the United States' military passed the one trillion dollar mark in spending on the wars in Iraq and Afganistan. There seemed to be little mention of this in the news; a cursory search on Google revealed few hits from the mainstream media outlets. I did find this piece in the Huffington Post about it, though.

I became aware of the passing of this sad milestone when a student of mine sent along this great graphic created on Google Sketchup showing a visual representation of what a trillion dollars looks like: http://www.pagetutor.com/trillion/index.html. You'll have to click through to see a trillion, but this is a billion dollars, in stacks of hundred dollar bills stacked on pallets.

The story and the graphic got me wondering what we could do with this in a math class:

Ask students to come up with their own ways to visually represent how large a trillion dollars is in some kind of way that makes sense to a viewer.
How much does a trillion dollars worth of hundred dollar bills weigh? How much would a trillion dollars worth of pennies weigh?
Research the cost of other wars and the rate of inflation and compare the cost of other wars to the cost of the Iraq/Afganistan war.
What else ???

Vermont Bans Texting While Driving

2010-05-25T19:57:00.000-07:00

One of the items the VT state legislature concerned itself with during the 2010 legislative session is the use of electronic devices by drivers. The legislature passed bills making it illegal for anyone to text while driving a motor vehicle. The lawmakers stopped short of prohibiting handheld cell phones entirely, and instead, only made it illegal for drivers under the age of 18 to talk on a handheld phone while driving. You can read the Burlington Free Press editorial on the legislation here.

The basis of these laws is that a distracted driver has slower reactions than a non-distracted driver.

It would be interesting to do an experiment to gather some evidence to support this hypothesis. Here is one procedure you might use to explore this using just a ruler:

While one person holds a ruler at the 30 cm position your thumb and index finger close to, but not touching, the first line mark at the bottom of the ruler (zero line). You should be lining up the bottom of your fingers with the 0 cm. mark.
As your partner releases the ruler, try to catch it as quickly as you can, using only your thumb and index finger.
Record the number of centimeters the ruler fell . Repeat the procedure a total of ten times recording your results each time.
Repeat the experiment 10 more times. This time, however, your classmate should distract you by asking you questions or possibly by telling you jokes while she releases the ruler. Record these results separately.
Change places with your partner and repeat the entire investigation.
Find enough other people to do the experiment that you get around 100 or more results for both distracted and undistracted distances.

This experiment doesn’t directly measure reaction times, but rather measures the distance a ruler falls during the reaction times.

Use the formula for the distance a freely falling object falls in t seconds to change these distances into times. If you don’t already know this formula, do some research!

Find what you think are the best way to represent the data graphically and use appropriate statistical measures to justify your support for, or opposition to, the hypothesis.

Many of my students wondered about the law giving more restrictions on cell phone use to younger drivers than to older drivers. They argued that they believed younger people have faster reaction times than older folks, so it didn’t make sense to them to make a law based reaction times that was more restrictive to those with better reaction times.

Could you design an experiment that would test the hypothesis that reaction times are negatively correlated with age?

Soda is How Cheap?!?!

2010-05-23T18:49:00.000-07:00

I recently watched the documentary film Food Inc.

It was a good look at how badly mishapen our food productions systems have become. The central message of the film is how we have given over the control of our nourishment to the hands of a handful of huge coorporations whose sole goal is certainly not public health or sustainability,but only maximizing short term profits.

One of the central messages of the film was that the low price of unhealthy foods is largely caused by U.S. goverment subsidies given to corn growers, which in turn have made high fructose corn syrup (among countless other corn based items) artificially cheap.

About a week after watching the film I came across a great article in the New York Times about a proposal to tax soda as a means to raise revenue to fund public health efforts in combating childhood obesity.

The online version of the story leads with this graph:

Change in price of items since 1978, relative to overall inflation, as measured by the Consumer Price Index. The price of carbonated drinks, for example, has fallen 34 percent relative to all other prices.

The graph confirms my feelings that since I graduated from high school, fresh fruits and vegetables have become quite expensive, while soda has become nearly free.

It leads to some great questions:

This month, Walmart is selling a case of 24 cans of name brand soda for about $5. Find the average rate of inflation (CPI) since 1978 and determine what the price of soda would have been in 1978 had the cost of soda matched the overall rate of inflation.
Again, use the average rate of inflation (CPI) since 1978, but this time use the information from the graph to estimate the cost of a can of soda in 1978 based on todays cost. Do some research and find the actual cost of soda in 1978 and speculate about why your estimate might be off by a bit from the actual price you find.
Assume there exists some food item with a price that has changed linearly relative to inflation since 1978. What would its graph look like if it was just steep enough for the price of the item had actually have decreased throughout this time period. Again, use the actual average rate of inflation given by the CPI as you answer this question.

BP Oil Spill

2010-05-01T21:03:00.000-07:00

An offshore oil rig off the coast of Louisiana burned and sank in late April 2010, causing a massive leak of crude oil into the gulf of Mexico. The New York Times reported on it here, and you can watch a video clip of a news report on the spill here:

Mathematics Problems:

We can make some interesting mathematics problems by taking some information from the actual situation and making some simplifying assumptions to make the problems solvable on a reasonable amount of time

The well is leaking 5,000 barrels of oil per day (Thought to be true initially, but later raised a great deal)
The well is located about 5 miles from the coast (True)
The oil will form a circle as it reaches the surface of the water. (Not True)
The oil slick will have a consistent thickness of 100 µm (micrometres) (Not True)

Please find the radius of the spill after 1 day,2 days,3 days,4 days,5 days,10 days, and 20 days. (Do each calculation by first finding the volume of the oil released, then finding the radius of the cylinder 100µm thick that contains that much oil.
Find an explicit equation that gives the radius of the oil slick as a function of time.
Please graph the radius of the spill as a function of time.
Please estimate how long it will take the spill to reach the coast.
Write a clear explanation of why the volume of the oil and the radius of the slick are not directly proportional.