Today's opener continues our work of kicking off class with percentage problems. This time the task is to calculate percentages based on data given in a two-way frequency table, which is part of what students are asked to do in Part 5 of the "Where Does My Stuff Come From?" project.
I've taken this simple example of a two-way table from Stat Trek (see http://stattrek.com/statistics/two-way-table.aspx). I hope that the questions posed here will help students think about the nature of percentages. To see that both 8 males and 8 females identified TV as their favorite leisure activity, but then to see that 8/20 yields a greater percentage than 8/30 should help students to further grasp the utility of percentages as a comparative tool.
The first and last questions are conversation starters. My goal is to get students thinking about what real data looks and feels like, and posing these questions yields different results with each class.
For the previous lesson, students were in the computer lab learning how to navigate around the Trade Map web site. It is likely that many students will still have some data gathering to finish up, so before class I print some of the charts that they'll need to finish up and post them around the room.
I remind the class of the steps we followed yesterday, just to make sure that everyone understands that the data on the walls is the same data that they would have seen if they'd completed the work on Trade Map. Even if kids are done with the chart on Part 4 of the project, it's useful to set up this gallery walk anyway, because it will ensure that the data is available in the classroom whenever we need to refer back to it.
I tell all students with incomplete work on Part 4 to get right to it. As soon as each student is ready, it's time to move them on to Part 5.
The fifth and final part of the "Where Does My Stuff Come From?" project is all about analyzing the data that students have gathered over the first four parts. This part of the project consists of four pages. I print these as two double-sided pages, and to begin, I only provide the first handout (with pages 1 and 2). I tell students that they have to complete the first two pages in order to get the next two.
My delivery of this part of the lesson changes, depending on whether or not my students have access to computers today. Ideally, the work for both today and tomorrow would happen in a place where students have access to computers. If it's impossible to do both, then today's lesson should happen in a classroom, and students should have computer access tomorrow.
For this lesson, my students are in the classroom, and they should have about 20 minutes today to get started on Part 5. In this narrative, I'll outline the first two pages of Part 5, as students see them in my classroom. I tell students that today we're going to do as much as possible to prepare for our time in the computer lab tomorrow.
On the first page, problem #1 is an analysis of Parts 1 through 3 of the project. For this, students will need the combined data of all their classmates on Part 3. This was my responsibility to collect and present in a two-way table. Here is an example of that whole-class data. To share this with students, I post a few copies of this table around the room, so it becomes part of the gallery walk. This way, as students complete Part 4, they can check in with me to get Part 5, and then I can set them to continue their gallery walk by getting started on the first problem. Some students will wonder if they should include stuff from the United States in their calculations, so I also have a few copies of a table omitting the US-made stuff at the ready, so everyone can make their own decision. I also post both documents on the class web site, so students can reference this data on their smartphones or on computers.
On problem #1b, students calculate percentages of their stuff listed in the total column; on problem #1c they do the same thing, but for each joint frequency column.
On problem #2, students use the percentage of stuff in their class that comes from China, and they find that percentage of the approximately $2.3 trillion dollars that the United States spent importing goods in 2012. They then compare that dollar amount to the actual dollar amount of US imports from China, as was found in Part 4. Usually, the value extrapolated from the class data is a lot higher than the actual amount. There is not a formal interpretation question on this part of the assignment, but as I circulate, I ask students if they can explain the disparity between these two numbers.
Next, students begin to analyze the two-way table that they completed on Part 4. When they filled in the columns for each product category, students only saw the top 24 suppliers of each product to the United States. If a country is not in that top 24, students left that cell of the table blank. Now, in a quick look at that table, we can see which of the top exporters of stuff to the United States are players in all, some, or just one of these categories. Again, I'm not explicitly asking students to interpret this yet, but while they're working, I ask kids to explain what they see. "What does it mean if a country is a top exporter of all four categories of stuff? What if most of their export value comes from just one product category? And what if just category is missing?" I want questions like these to become implicit when students do work like this, and I want to cultivate curiosity in each of them. Today we're doing some training along those lines.
On the fourth problem, students calculate the percentages of US imports that fall under each of the top four product categories. This involves dividing numbers in the hundreds of billions by 2.3 trillion. This gets interesting because most of the calculators that students bring to class can't quite handle numbers this big. This problem is cause for some strategic use of tools, and I treat it in two ways. Significant digits are the first tool at our disposal. I ask students how they'll want to round their answer to these questions, then I show them that, just as we can cancel zeros in a fraction, we can just cut off a whole bunch of extra digits in each, if all we're looking for is an answer to two or three decimal places.
Many of my students just don't buy this - it just feels too simple, how can we just get rid of number like that? - so here I employ the next tool, calling up WolframAlpha on the screen. I say that this is a very powerful web site, and that it's like a cross between a calculator and a search engine. I type the exact numbers that we found on Trade Map into the search bar, and everyone sees that WolframAlpha can handle number of this magnitude. It also provides some extra information and visualizations. I share this tool with students now so that they know it's available to them tomorrow in the computer lab. I'll also make sure during tomorrow's lesson to let them play with WolframAlpha for a little while. In any case, this powerful tool proves that it's just as easy to lop off a bunch of digits as it is to worry about the entire number.
Extension Note: On the other hand, it's fun to compare differences in rounding percentages. For example, I ask students to calculate 18.57%, 18.6% and 19% of $2,333,805,200,000. I'll leave you to play with those calculations. Measured in dollars, are the differences significant?