Today and tomorrow's lessons serve to introduce the next two parts of the "Where Does My Stuff Come From?" project, even though in practice, the pacing will probably be different for every class.
It is likely that students will have some questions about the Texting vs. Social Media problem set that they began working on two days ago. Many will also need some time to finish it. I am prepared to slow down to give kids some extra time on that problem set. I'm also ready to speed up after seeing that students are able to complete both today's part of the "Where Does My Stuff Come From?" project and Part 3, which I introduce tomorrow. I make my copies in advance this week, so I'm ready to meet my students wherever they may be.
Today's Percentages Opener simply asks students to express each of a series of ratios as percentages. Each ratio is written in words, like "1 out of 5". This opener serves a pair of purposes:
To start, I have students calculate 1, 2, 3, 4, and 5 out of 5, because I'll take every chance I can get to extend the fluency of my students on percentages like these.
Then, the second column has four ratios of the form n/13, the first three of which will produce much messier decimal percentages than anything that's divided by 5. I do not indicate how I expect students to round, and when they ask I tell them to round however they'd like. As the "Where Does My Stuff Come From?" project continues, students will have to comfortable rounding the percentages that they calculate, so that work starts now.
Even though it's not required, many students will be naturally curious about having a complete list of "n/13" percentages as n goes from 1 to 13, so as soon as someone says they're thinking of doing this, I commend them on the great idea and encourage anyone who is interested to follow suit.
"What's going on here?"
When students have these percentages in their notes, I move through the next three slides of the opener, which will give us the opportunity to discuss rounding error. The first two slides are unproblematic, and raise the idea that if two ratios add up to a third ratio, then the sum of the percentages equivalent to each of those two ratios should be the same as the percentage equivalent to the third one. "When we're working with such nice ratios as 2/5 and 3/5," I say, "this should make sense." If I notice that students are unsure of my point, I'll throw another example up on the chalkboard that uses quarters, like "1 out of 4 plus 2 out of 4," which should help them make sense of this.
It's the third example (on slide #4) that presents the problem of rounding errors. If we take 1/13 + 4/13 + 8/13, it's clear that we'll get 13/13, which, like any "whole", is equivalent to 100%. The problem is that if we round each of those three ratios to the nearest percent, we get 8% + 31% + 62%, or a sum of 101%. Here, I turn it over to the students, simply by asking that great mathematical question, "what's going on here?"
I hope that someone will raise the idea of rounding error - or even better than that, that someone will say that they rounded to the nearest tenth of a percent rather than the nearest percent, and that, by doing so, they didn't up having the same problem. Try it for yourself to see what I mean.
Attention to rounding is going to be important as we begin the next part of the project, and I hope to have set the stage for that here.
In Part 2 of the "Where Does My Stuff Come From?" project, students summarize the data that they gathered in Part 1. Then, in groups of 3 or 4, they begin to combine that data, so that eventually we can look at results for the whole class. This screencast is an overview of Part 2 of the project.
This is pretty simple counting work. The real opportunity to learn something comes when kids start checking for errors. It's important to acknowledge that this can be annoying and that it takes patience. I say, “It’s pretty much a guarantee that every group is going to make a counting error or two. So the question is not, Did you make a mistake? but How did you respond when you found that there was an error in your work?”
Accounting for discrepancies is great because it requires collaboration and attention to precision. I am regularly surprised to see kids who demonstrate leadership skills that I was unaware of, as they grab the reigns of their group and figure out what's going on.
With a few minutes left in class, I ask for everyone's attention, and pose the simple question, "So, where does our stuff come from?" Invariably, everyone yells, "China!" and my response is, "Ok, so where else?" As students call out a few, I make a list of other countries on the board.
Next I ask, "Approximately what percentage of our stuff comes from the United States?" The answer tends to be near 33%, give or take. I ask everyone to note what they hear from the other groups in the class.
Finally, to set the stage for the next part of the project, I ask, "Do the same kinds of stuff come from both China and the United States?" It's pretty obvious to students that it's not the same. To conclude, I tell everyone what's coming tomorrow: "We'll break this data into the four categories of stuff, which will allow us to see further patterns in the data."