The SodaStream Fizz Soda Maker

My family drinks a LOT of seltzer water. We've been intrigued by a new(ish) product that allows one to carbonate tap water at home. The touted benefits are that it's easy to use, is better for the environment than buying bottled seltzer water, and saves money.

I found the product, and its competition at a couple of local merchants.

Unfortunately, before I was able to take a photo of the replacement cartridge and its price, I was asked to leave by the store management, I guess because of my picture taking,  they thought I was either a corporate spy from Office Depot, or else I was casing the joint for an Oceans 11 style heist later that night...

I was not deterred in my quest, however... , I went online and found the price for the refill cartridge:

Your task is to analyze the manufacturers claim that using their product will be a money saver.  . Please create a clear and precise argument that either supports or refutes the manufacturer's claim.

Your analysis should start with fundamentals, that is the prices of the SodaStream and any additional components and supplies you need. Do not simply rely on someone else's analysis that merely states that the system will pay for itself in x-months.

Of course, if you are taking calculus I expect you will find a way to use a concept or two from calculus in your analysis!

That book costs HOW much??!!

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!

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.

A Deluge

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 offices 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?

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

[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

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?)