I love teaching logarithms!! I love modeling exponential decay and growth with students! It is like the Christmas holidays around here everyone growing exponentially. In fact I see this as essential learning. Of all the skills and concepts I teach, logarithms and all of the math behind them are the most important for all of our children to have a deep fundamental understanding.
The intentionally vague problem I start with is this one:
The average retail price of Bacon per pound in 1895 was around $0.13. The price of a pound of bacon in 2014 was around $5.78. What is the annual percentage rate of increase?
I sneak this problem at the end of a traditionally taught lesson reviewing the rules of exponents with the introduction to rational exponents. Here is a link to that worksheet. Usually students are grappling with their understanding of those rules from Algebra I and Algebra II that they probably forgot, when all of a sudden,Wham! problem 16 hits them. If your students are like mine a few attempt the problem and have something written down. Most leave it blank because there are scary words that bring on frozen fear. While another group does in fact very confidently come up with an answer.
Here is where the teaching magic happens. I have these students that are confident come up to the board and explain their thinking about the problem. This is almost always what I get from the few who try.
5.78-0.13 = $5.65
2014 – 1895 = 119 years
5.65/119 = .0475 price per year
At this point many do not know what APR means so they might quickly mumble 4.75% increase or something like that and suddenly sit down. This problem is vague for a reason. I am only giving them part of the story. The students that I teach have spent their entire math career working on proportional reasoning. Some will get this answer and not even realize that they found a slope, where rise is price and run is years. Some may not even realize that they are looking at the problem linearly. Students need to see their solution modeled for all of the years. As a class we go ahead and model their solution with Geogebra to show what their story of bacon would look like. Here is where we can ask the students. “Is this what the model of bacon prices over time should look like?” You’ll get a mix of answers and probably get some pretty good discussion going. But here is the actual data according to the Bureau of Labor Statistics. Combining this data with data from www.infoplease.com we can get a true story of the price of a pound of bacon from your local grocery store.
It is time to show them the true story of bacon. You cannot tell the story of anything with only two snapshots in time. Bacon’s story is filled with wars, depressions, booms, busts, supply, demand…. but most importantly inflation.
I did some of the research for you and put the raw data in Geogebra. Click here for the geogebra file. Now what? How can students manipulate this data? How can we model this data? What is the annual percent of increase for the price of bacon? Here is where I will leave the problem for now. I present this problem in a couple of days and hate to do more because this is where I like to leave a problem. What will students come up with? Will they care? Does it motivate a question from my audience? Will they dive into it and make it their own? These are the questions I am concerned with. Questions that all teachers should be concerned with. The products we as educators produce are only relevant if the students find them interesting and relevant in their own lives. It is time to test the problem and find out.
Where would I like to have students end up with this problem? I would like students finding out that their original linear proposal was a fine model as an initial guess and matches the simple interest formula. But after that initial view and looking at all the data, it looks like there could be a better model. Maybe it is y=ab^x for compounding annually. Maybe it is y = P(1+r/c)^(ct) for other compounding like quarterly, monthly, daily, hourly….. Maybe it is y = Pe^(rt) for growth compounded continually.
Ultimately I want students to see this model in our stock market, land prices, loans, heating, cooling, and other geometric series. The connection of this data to other naturally occurring phenomena is the beauty of mathematics. After all isn’t mathematics just a language used to describe our world? Shouldn’t all student speak that language?
Source data for average bacon prices 1895 – 2014