So you want to mail a letter? Lets do the math.

Tom Ward posted work on one of Dan Meyer's makeover Monday problems and this got me spending hours researching the history of stamp prices as well as the USPS. I found a bunch of interesting stuff and wanted to share. First, I want to talk about my take on the original problem.

The original problem is a standard fixed growth mindset problem from a Pearson Text:

I like Tom's idea of setting this problem up around the concept of a wedding invitation sometime in the future. He posted this awesome invitation:

The lesson develops around the concept of a future wedding and the possible cost of having to mail out invitations.

My opening act 1 would certainly be similar. I am either going to stand by the door with a save the date notice and hand them to students as they enter or dress up as a postal worker and deliver the notices as students work on some type of introduction problem. The idea would be to generate some excitement around the lesson with a clear and fun theme. As a general goal, I try and set the atmosphere of my class to fit the theme of the lesson. So if I go full on wedding theme, there will be appropriate music and decor. If I focus on the theme of the postal service, I will set up the room like a large math mail room (whatever that is, I usually set that up the day before). 

For my lesson, I wanted to make my own save the date, but as I am already happily married (and very much in love) to my wife, I needed to pick two people that my students will respond to. So I chose two people that might get married someday (hopefully):

Christopher is my son:

Anna is my friend's daughter:

After a discussion of the stamp problem (and the fact that I am already trying to set my son up with my friend's daughter), the class will naturally start to ask about the cost of the stamps. 

I think am going to avoid Pearson's table because it doesn't capture the full spectrum of data around stamp prices. I want students to look at all stamp prices since the USPS started to use stamps. I might simply give them the list from USPS site, but Jo Boaler has me thinking I should give students a chance to see the information from their own perspective.  

This might be a bit too much, but I am thinking that students need a moment to see the stamps that correspond to the major shifts in price. They can always use functions, tables and graphs, but I want to give them a moment to make that decision. Here is a handout with all the stamps needed. I grabbed most of the images from this site: http://www.usstampgallery.com

I am thinking I will show stamps in a quick slide show and simply ask, "how do these stamps change?" 

Since there are quite a few stamps to look at, I will stop and ask students to share findings as we go and see if they can predict how and why the stamps change in price. 

The data on price over the full time span is so much fun, because it goes up and down and up again, increases at steady rates and then makes big leaps. It is the type of data that makes you ask, "what is going on here?" 

My act 2 would involve using the stamps to possibly generate a table, graph and/or function to help them find the cost of mailing those wedding invitations in 2038. I pulled some graphs from other sites around this data. 

I could show students this form the start and ask them to tell the story of each graph, but I want them to work with the data first. 

This one might be a good chart to discuss with the class. I would ask if they think the line should be constantly fluctuating or if it should be like Dan Meyer's step graph.

By including the full spectrum of the data (neither table above is complete), we could also discuss the line of best fit and the concept of an outlier. Students might notice how the price remains low for a very long time and then continues to jump. Hopefully they will debate whether or not we should include those points (which are possible outliers) in our calculations.  

Many students might also choose to find the cost of a future stamp by looking at time periods that most closely resemble our own.  Perhaps the stamp price growth after the great depression will help us now to predict the cost of a stamp after the great recession. 

The data and nature of the problem lends itself to debate and conjecture, which are (thankfully) now a part of the practice standards.

Here are some other thoughts and findings:

Wherever these lessons go, I know I have to talk to the class about the basic process of mailing a letter. I wonder how many of my students have ever mailed or been to a post office? Do they get anything in the mail other than junk mail? I might use this goofy video in the discussion (with some small edits to make it work for the class):

Instead of giving out stamp handouts or showing one stamp at a time, I might show the timeline here: http://www.priceofstamp.com. It might get students thinking about the general trend in a quick intuitive way. You can watch the timeline animation as a movie below:




Another fun expansion on the postal problem connects to linear equations. The cost of a stamp is based on weight, but the first ounce is often not the same cost as additional ounces. Students could review this wikipedia article and then graph the linear functions that represent costs for mail weights.

I really liked how the post office prices were first based on distance. This just screams for a lesson around a step function.


My favorite find was around the Simpson stamps.

Apparently the post office printed a billion stamps and only sold about 318 million stamps. The Bloomberg article mentions that "the inspector general criticized the process the services uses to decide how many stamps to produce, saying it's unscientific and too much of a judgement call."

The question then becomes how many stamps should the post office print? Data around this question can be found in this table from these USPS links. They also post the table below from this link:

Which the Wall Street Journal might have used:

I haven't done the math on these numbers, but I like this question. How many stamps should they print each year? This is something that is going to stir up some debate.

After all, I am wondering if there will even still be stamps by the time Chris and Anna (hopefully) get married.