Today's opener is on the first slide of today's Prezi, and it simply asks students to express each of a series of ratios as percentages. Each ratio is written in words, like "1 out of 5". On the surface, I want two things to happen: I want students to be comfortable with the idea of a ratio written in words, and I want them to see how some denominators make for "cleaner" or "nicer" decimal representations than others. To that end, I have students calculate 1, 2, 3, 4, and 5 out of 5, because I've often found myself disappointed with how many of my students are unfamiliar with the ratios represented by 20%, 40%, 60%, and 80%, and I'll take every chance I can get to extend their basic numeracy skills.
Also to that end, 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. I don't tell them that we're going to be talking about rounding errors a lot today, starting in the mini-lesson that's coming right up.
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.
Mini-Lesson: When Percentages Don't Round Right
The second purpose of today's opener is prepare students for the work they're about to do on Part 3 of the "Where Does My Stuff Come From?" Project. You can see the questions I use to initiate these notes on slides 2 through 4 of today's Prezi. 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 the question over to the students, simply by asking, "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.
I tell students that we're going to continue on the "Where Does My Stuff Come From?" Project that we've been working on for the last couple of days, and that we're going to revisit the data from the previous class.
As students find their notes and the first two parts of the project, I post the Excel spreadsheet that we completed in the previous lesson. I ask for a volunteer to distribute Stuff Part 3 Relative Frequency, (this is printed on legal-sized paper, double-sided with the map that appears in the next section of this lesson).
Looking at the New Student Learning Target
SLT 2.2 says, "I can calculate relative frequencies in a two-way table, and I can interpret these values in the context of the data."
This learning target is atop the new handout, and it's on slide #6 of today's Prezi. As always, I ask for a volunteer to read the SLT aloud, then I give all students a chance to shout out the key vocabulary words from this target.
I expect that at least one student will point to the words "relative frequencies," and I'll just have to reiterate that the key word here is "relative". We'll have a brief conversation about what the word relative means, who our relatives are, and that the word indicates relationships. Then I'll explain that one tool we have for understanding relationships between numbers is percentages. One example that always makes sense to my students is when I say, "Suppose my little brother has $20 in his pocket, and my uncle Moneybags has $1000 in his pocket. If I offer them both $10 for cleaning my kitchen, who do you think is going to want it more?" Although it's a fault-laden argument, it makes sense to students (especially those with little brothers) that the kid without much money is going to want ten bucks more than the rich uncle. And herein lies the use of relative values expressed as percentages. Ten dollars is 50% of $20, but just 1% of of $1000. In relative terms, my $10 bill is 50 times more valuable to my brother than to my uncle. There are other ways to approach this conversation, and they can be tailored to the interests of your students.
Part 3 is a Relative Frequency Table
On the front side of Part 3 (Stuff Part 3 Relative Frequency), the task is to fill in a two-way relative frequency table of the "personal stuff" data that the class has collected (see the previous lesson for details). I tell students that this is really a simple percentage exercise, and they should work with their group-mates to divide and conquer the work.
We have a total number of objects and we have the data, which I display on the same Excel spreadsheet that we produced during the previous class. The only editorial change I've made to that spread sheet is that I've sorted it from greatest total number of items to least.
Just like yesterday, students should copy the list of countries into the first column on their table. Then they should work through the chart, dividing the number in each joint frequency cell of the two-way table by the total number of items, and recording the relative frequency as a decimal. This is another change: we're not converting to percentages, but leaving the relative frequencies as decimal numbers. I check for understanding by saying, "So how would you write 1% as a decimal? How about 2.5%?" Then we all acknowledge that it's actually easier not to take the extra step of converting decimals to percentages. Finally, I say that we're going to round everything to three decimal places, or the nearest tenth of a percent.
Group Work Time
Students get to work, dividing and conquering the work. This is busy work with a purpose. Yes, a computer could do this pretty quickly, but students are getting the feel for the numbers and they're gaining comfort with how relative frequencies are written. As they work, I circulate to see how everything is going. I ask each group to tell me their "quality control plan," and if I feel like their plan is lacking, I'll give them some tips, but in general, this a pretty straightforward exercise.
Error Checking with Marginal Frequencies
When the joint frequencies are complete, it's time to fill in the marginal frequencies on both the right side and bottom of the table. I ask the class to explain two ways that they could determine marginal frequencies. I'm looking for a) taking the sum of each row and column and b) simply dividing the quantity by the total number of items to get a relative frequency. With these methods settled, I tell everyone that I'd like them to use both methods to fill in the marginal values, and that they should all play close attention any inconsistencies.
What's interesting here is that columns are more likely to have errors than rows, because there are more values to add in these - especially more smaller values - and these will contribute to more rounding errors.
After combing through and analyzing the data in a two-way frequency table, it's time to construct another representation and see where it leads. On the Stuff Part 3 Map that comprises the back of the Part 3 handout, students will find and label the top importers to the United States, color-coded by category of stuff (apparel, electronics, produce, other).
Geography is another variable, and this map is a representation that may allow us to spot some new trends. One that we can certainly expect, for example, is that while the Far East may be a major player in overall trade, apparel and electronics, produce will not be concentrated there.
I instruct students to get five different-colored pencils, and to complete the key at the bottom of the map by indicating the color for each category of stuff. There are five colors are for each of the four categories plus the total.
For each category, then, they should find and label the top five importers of each kind of stuff to the homes of all students in our classroom. What we'll have is a fairly basic infographic that will show us where our stuff come from.
As we'll see tomorrow, this graphic is limited by the fact that this data is limited to a small population of teenagers in one American city. It might turn out to be little different when we look at U.S. imports as a whole. But for now, I don't ruin that surprise.
On an additional note, I have used this project in collaboration with the Global History teacher, as we've talked about the forces behind and implications of world trade. Consider sharing this idea with a colleague if some of your students are also enrolled in a global history or economics course.
To finish class, students take out their Record Sheets for the "Where Does My Stuff Come From?" Project, write the date, and respond to the following 3, 2, 1 prompt: