Today's case study has the potential to get pretty big (and even a little unruly), but at its root this is a chance for students to demonstrate and reinforce some fundamental skills.
We're going to look at data from the Economist's Big Mac Index. Published since 1986, the Big Mac Index aims to be "a lighthearted guide to whether currencies are at their 'correct' level." To give students all the economic background knowledge they need to understand what that means is beyond the scope of this course, but some students will be curious enough to want to know more, and they might even identify their own interests in that field. Additionally, there are plenty of ways to extend this case study, both in terms of how we follow up in this math course, and by making connections to other courses if your school values an interdisciplinary approach.
With all that in mind, here are some links to background reading. The official Economist post includes some interactive charts (today, students will create their own plot of GDP per person vs. Big Mac Price) and data for download. Wikipedia articles about the Big Mac Index and Purchasing Power Parity provide an overview of how to interpret the data, if you and and your students want to extend this project.
I keep all of that in the back of my mind, but I don't force any extensions. If students want to know more, I'll go there with them, but with most of my students, my primary goals are to use this task to assess what they know so far and to draw some connections to previous activities. In the lesson narrative that follows, I stick to these objectives, before offering a few general ideas for how to extend the project.
Students will spend most of today's class working hard in groups. Sometimes for classes like these, I like to employ a brief, conversational start. On the second slide of the lesson notes, I post this discussion prompt:
What is one thing that all countries have in common?
As students arrive I ask them to think about it. As they come up with ideas, debates might pop up, or the conversation might tend toward sweeping generalizations. Often, students will recognize that they have an easier time saying what is different about certain countries or regions. I try to keep the conversation lighthearted, because that's how I expect students to respond to the answer I propose on the following slide.
For better or for worse, one thing you can count on almost anywhere you travel is that you're going to see some golden arches. When you do, you might find some local embellishments on the menu, but you can count on seeing the Big Mac included on the list, with pretty much the same recipe, wherever you go.
"So if you can get a Big Mac anywhere you go," I say, leading students into today's task, "do you think it will cost the same amount of money everywhere you go?" My students already know the answer to that question - anyone who has been to a McDonald's in midtown Manhattan knows that prices are higher there, and kids like to point out that the "dollar menu" is really the "$1.29 and up menu" at many locations. In our own city, prices for the same item are different depending on your neighborhood, so it's not surprising to learn that prices are different from country to country.
"Today you're going to look at data for 50 different countries," I say. "You'll look for a relationship between the average price of a Big Mac and the average wealth of the people in each country." It doesn't take much for kids to imagine that in wealthier countries, people pay more for their food, so with that, we're off and running.
About the Task:
Students will work in groups of 3. They will research currency conversion rates, make some calculations, and construct a scatter plot of data that includes a regression line. To run the lesson, I need computers, graph paper, graphing calculators and two handouts.
I provide students with these handouts: the Big Mac Index Assignment Description and the Big Mac Index Table. The assignment description includes learning targets and an overview of how students will demonstrate mastery of each SLT. I write on the assignment that "specific instructions will be given in class," and I give the instructions for most of this task verbally. I find that it's more helpful to describe each step as students work through the assignment. When I do, I try to say as little as possible to get them going - because I want to see how much kids can figure out on their own - but I'm always able to provide impromptu examples when needed.
The Big Mac Index Table lists a series of countries, providing the per capita GDP, the name of currency, and the local price of a Big Mac for each. Students will fill in the other three columns: exchange rate (per $1), Big Mac price in $, and % of average daily pay.
On the 7th slide of the lesson notes, I name the group roles and provide a first step for each person. The researcher should get a computer and start looking up exchange rates on xe.com. These exchange rates are to be written in the 4th column of the Big Mac Index Table, which is the initial job of the recorder. The graph maker gets a sheet of graph paper and looks at the per capita GDP data to decide how to set up an x-axis that will accommodate it.
Students might need a little help reading the currency exchange rates. I project the xe.com homepage at the front of the room, and make sure that everyone understands that they're looking for the value of $1 in every other currency. In order to show students the end result, I'll also model how to use these conversion rates to calculate the price of a Big Mac in USD. Working all the way through a first example helps students grasp the meaning of the conversion rates as they work through the chart. Here, I also remind students to pay attention to how they round each number to the nearest cent.
As students get to work, I circulate to share the next instruction with the graph maker. The Big Mac Price in USD will be the response variable in this data, so we'll want the y-axis of our scatter plot to accommodate that. By the time the graph maker has the axes set up, he should be able to start plotting points from the table.
% of Average Daily Pay
The final column of the table requires students to calculate the "% of Average Daily Pay" that it would cost a person in each country to buy a Big Mac. Now, there are all sorts of assumptions we make in this calculation - and to note and to question them is a great learning opportunity - but I don't worry too much about forcing any of that. The real point is to give students a chance to make these calculations, and the results are illustrative enough to be useful. On other hand, when I have students who understand that dividing "per capita GDP" by the number of days in a year might not provide a fair measure of a regular citizen's average pay, or kids who know that western-style fast-food is a luxury good in some countries, I'm more than willing to join them in exploring those ideas.
The Linear Regression
The meat of this assignment is in producing a scatter plot and a linear regression model of this data. When the Big Mac Index Table is complete, all three group members work together to complete that task. From here, I leave students on their own, because I want to assess how well they understand that process, which has been our topic of study for the last few days.
I do make sure to recommend that students think about the GDP per capita in 1000's of USD. Without doing so, it's much more difficult to interpret the slope of the regression model. If we do count by 1000's, a least-squares regression will yield:
y = 0.031x + 3.03
When students have that model, I ask them to interpret it. As everyone works on the task, I might ask groups to explain it verbally, or if we're crunched for time I'll have them write it down. What I'm looking for is something along the lines of, "Around the world, we can expect a Big Mac to cost about $3.03, plus about 3 cents for $1000 in per capita GDP."
Here is an Excel spreadsheet of the data, with exchange rates current as of the first week in December, 2014. Here's what the scatter plot looks like: Scatter Plot and Least Squares Line, and as food for thought, here's a subset of the data that shows that the correlation is much weaker when we old consider countries with lower GDP per capita.
If you're pressed for time, I recommend doing parts of this activity while using more powerful technology, like Excel, Fathom, or r. I continue to use paper and TI-83's for this activity because of the nature of my class, which is designed for juniors and seniors who have struggled in math, but who can grasp bigger concepts. I use this as a chance to remediate some basic skills while considering big ideas.
There are so many more places to go with this activity, and on any giving year, I might touch on some of them. It's always useful to tie this case study back to the Where Does My Stuff Come From? project, and our work on Gapminder. I want students to continue to ask questions about the world, based on the data.
Beyond just using this activity as a one-day performance, it can make a great launchpad into other investigations. One question I always raise is about the outliers - what kinds of countries are far above the regression line? How about far below? Can we draw any conclusions about these places? I might have students do some further research along those lines, or to explore other indices that address purchasing power parity.
Students can take a cue from Gapminder and color the dots by geography, to determine if there are any geographical trends at play here. Going deeper into economics, I might try to help students understand what the Economist means by undervalued and overvalued currency. Then there's all the softer stuff, where students gain awareness of their world: I'll hear kids wondering - the GDP is how low in certain countries? We never know where wonder might lead.
At the end of class, I'll lead a short debrief where kids share their thoughts and provide an update on their progress. If it's clear that everyone needs more time, I say that we'll be able to finish up tomorrow. Otherwise, I say that everyone should keep this activity in mind as we continue to analyze regression models.