Tuesday, September 6, 2016

Traditional Lesson: Rational and Irrational Numbers (Days 14-15)

The first project in my classes -- the mousetrap cars -- are complete. After each project, the Illinois State text provides us with a traditional textbook from which we should give lessons.

But over the long Admission-Labor Day weekend, I've been thinking long and hard about how I should be assessing the students. As I mentioned in my last post, my original plan was to rotate among a Dren Quiz, general quiz, and test, which the eighth graders starting with a test. Yet I believe that the students were not prepared for this test, since the days leading up to it were spent on the mousetrap project rather than the traditional text.

As it turned out, of my 14 eighth graders, exactly half passed the test and half failed. I'd argue that later in the year -- when the math starts to get tough -- as many as half passing is great. But at the start of the year, during the Willis (or Wong) Unit, I don't want half the class feeling frustrated. I want to set the tone for success, and that means most of the class should pass the first test.

Indeed, let's think back to the Wongs and their book, The First Days of School. According to the Wongs, there are several poor reasons for giving assessments, such as "the passage of time" and "points on a curve." Now my rotation plan sounds dangerously like giving tests just because of "the passage of time" -- tests were to be given every three weeks. My rationale for this schedule was that I didn't want too many tests to grade at one time -- but that's yet another pro-teacher reason for giving the tests. Instead, according to the Wongs, I need to show how my testing benefits the students and their learning.

I also said that I gave tests every three weeks as a student teacher. But my master teacher did warn me that I shouldn't just blindly give tests every three weeks or after every chapter -- in fact, it just so happened that those students were ready for tests every three weeks. On the other hand, my students were clearly not ready for a test right after spending all week on the project.

Notice that "because I don't want too many tests to grade at once" could be a pro-student reason for the timing of a test -- for example, because the teacher wants to give the students prompt feedback on how well they are doing. But, as I wrote before, that doesn't apply to my situation -- I work at a small charter school, so it's not as if I have nearly 200 students or anything like that.

Regarding the Wongs and their comments about "points on a curve," I would argue that transparency in grading is a pro-student reason for making sure that there are the correct number of tests worth the proper number of points. I still think back to my student teaching and the computer grading system, where the homework could be worth 10 points and a test worth 100, yet the test is worth much more than 10 homework assignments because one homework point is not equal to one test point (as points are instead weighted so that homework is 10% and tests 40% of the grade).

I've decided that in my class, homework will remain 10%, but classwork, quizzes, and tests will each comprise 30% of the grade. Classwork has been elevated to equal tests since there are so many projects that require considerable effort. The idea is that the whole trimester is worth 1000 points, and so I can give three tests each worth 100 points, which total 300 points, or 30% of the grade.

So here's my plan -- unfortunately I already began this rotation plan, so I should follow it through one complete rotation. This means that eighth graders will rotate to a Dren Quiz this week and a general quiz next week. Meanwhile, a test will be given to the seventh graders this week and the sixth graders next week. I will make sure through this week's traditional lessons that my younger students are prepared for the tests.

After my sixth graders take their test, I will abandon my rotation and switch to what I hope will be a more student-friendly assessment plan. Notice that the Illinois State text itself provides assessments that line up with each Learning Module. But simply giving a test at the end of every Module will result in too many tests -- for example, I wish to finish Module 5 this trimester, but I only want to give three tests this trimester, not five.

Instead, I could come up with a new sort of rotation -- one which shows which types of assessments occur during each module:

(Module 1: Dren Quiz, General Quiz)
Module 2: Dren Quiz, Test
Module 3: General Quiz, Test
Module 4: Dren Quiz, General Quiz
Module 5: Dren Quiz, Test

and so on. This plan has several advantages. It reemphasizes what I wrote last week -- if I'd followed this plan from the start, all the students would have started with a Dren Quiz, which is more logical than a test after spending so much time on the project (and I probably should have waited until today to give said Dren Quiz, so that all students had the opportunity to work of the project). In fact, here Dren Quizzes always occur in the middle of a unit, while tests always occur at the end -- after traditional lessons. If a General Quiz occurs in the middle of a module, I will make sure that all information needed for the quiz can be derived from working on the project.

Because I gave an extra test at the start of the year, this means that there will be four tests this trimester rather than three. But I don't mind having a fourth test, since whenever there are four tests, I can drop the score of the lowest test. This could help that half of my eighth grade class that failed this first test.

According to the pattern, Module 16 will contain a Dren Quiz and a general quiz, but definitely not a major test. This is good, because Module 16 will be taught during the SBAC. And so this appears to be a strong assessment plan that will help the students learn throughout the year.

All three grades test at the same time on this plan. The eighth grader who had been the most upset on Friday about the test asked me today whether the other grades have tested yet or not. Technically, I'm within my rights to push the eighth graders harder and test them more because they're older. But actually, after this first rotation, all three grades will test at the same time.

This week, the students will work out of a traditional text. Notice that while the first four modules of the main Illinois State text (the STEM project book) are identical for all three grades, the traditional texts are different for sixth, seventh, and eighth grade. The traditional texts are arranged by naively preserving the order of Common Core Standards themselves. Therefore the sixth and seventh grade texts begin with Ratios and Proportional Thinking followed by the Number System, while the eighth grade text, which doesn't have a Ratios strand, goes directly to the Number System.

Illinois State provides the teachers with a "pacing plan" which tells us which traditional lesson corresponds to each STEM project. But not only is this pacing plan cumbersome to read, but the first four modules ("Tools for Learning") -- despite appearing in the STEM text for all three grades -- are only assigned traditional lessons in the sixth grade text!

I have no problem with jumping around in the traditional text -- after all, consider the order in which I posted lessons to the blog last year! Still, starting at the beginning of the text is logical for both sixth and seventh grades, since Module 1 is The Need for Speed, and speed definitely is a ratio -- namely the ratio of distance to time.

The eighth grade traditional text doesn't have a lesson on ratio. But the lesson at the start of the book is about rational numbers -- specifically the existence of numbers that aren't rational. So in a way it's at least somewhat related to ratios. As usual, this blog focuses on my eighth grade class, so let's look at this lesson in more detail. The first Common Core Math 8 standard is:

CCSS.MATH.CONTENT.8.NS.A.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Back in my Square Root Day posts, I wrote about the irrationality of sqrt(2) on the blog. Before I took Algebra I, I had always heard that sqrt(2) was irrational, but I never realized that this was something that we could prove until I saw a form of the proof in my Algebra I text.

I might give my students a proof of the irrationality of sqrt(2). In fact, I'd love to tell the story such as the math YouTube star Vi Hart does:

In her video, Hart discusses how Pythagoras and his disciples eschewed beans. I definitely like Hart's pun about how one of the followers "spilled the beans" about the irrationality proof!

Today I start to tell the story about Pythagoras, but my eighth graders are a little confused as to why I'm telling them this story. I ask them what sqrt(2) is, and one student tries 1, until I tell them that 1 times 1 is 1. And of course they shouldn't even bother with 2, since 2 times 2 is 4. Someone suggests 1 1/2, or 3/2, which is a good guess, but its square is 9/4, which is a little more than 2.

But then the story falls apart when I try to suggest 7/5, or 1.4, as the next guess, since the students are confused where these numbers come from. And besides, the story works better when sticking to fractions, rather than decimals -- otherwise, students may wonder why sqrt(2) is "impossible" when 1.414213562... is apparently sqrt(2). (Though in the end, one student does figure out what my story is leading up to.)

The Common Core Standard requires students to know that rational numbers have repeating decimals as well as how to convert these repeating decimals into fractions. The Illinois State text teaches the following method:

x = 0.333...
10x = 3.333...
9x = 3
x = 3/9
x = 1/3

Perhaps my students would have appreciated the story of Pythagoras more if I had waited to tell the story after having them convert between fractions and decimals. Then it would have been more obvious where 7/5 and 1.4 come from.

As I wrote earlier, the lower two grades are working on ratios. Because I don't have enough textbooks for all of my sixth and seventh graders yet, I have these students take foldable notes instead. (The eighth graders don't need foldables since all Illinois State texts are consumable.) The students are required to copy what I write on the board into the foldable -- but then some students claim that they can't see what I'm writing!

Recall that the idea behind foldables is that students are more likely to take notes on the foldable than on notebook paper. But here we see the students are still coming up with excuses. Another idea is for the students to take guided notes where they fill in the blanks -- the words to fill in are right in front of them, so that's no longer an excuse (but of course, some students will come up with any reason why they can't take notes). But I want to avoid guided notes since they take extra time to prepare as well as require overuse of the Xerox machine.

In all classes, this is the song that I sing:

RATIOS:
Ratios are everywhere.
Ratios surround you,
Probably here and there.
For every "for every"
There's a ratio.
Divide at the colon,
And away we go.
That's all there is
To a ratio!

Bridge:
Ratios are everywhere,
But not everything's rational.
Square root of two and pi,
Are proved to be irrational.

By the way, since this is a two-day post, I'm thinking ahead to Wednesday's lesson, when I want to return to science. In the past few days my go-to for science, Sarah Carter, posted some lessons on accuracy vs. precision and scientific notation:

http://mathequalslove.blogspot.com/2016/09/accuracy-precision-error-and-percent.html
http://mathequalslove.blogspot.com/2016/09/scientific-notation-inb-pages.html
http://mathequalslove.blogspot.com/2016/09/scientific-notation-ordering-cards.html

I'm torn between giving these lessons from Carter and continuing with the online website that I've been using for science. On the website I was hoping for some simple speed and velocity questions, but most of the mechanics questions for eighth grade deal with force -- a concept that will confuse and frustrate students if I teach it this week. Indeed, force will make sense if I teach it along with Module 3, where the STEM project involves measuring force. So it's probably better for me to give Carter's lessons this week.

Notice that Carter -- who originally championed the idea of using foldables -- has now switched to using what she calls an "interactive notebook." In many way, an interactive notebook combines some of the ideas of foldables and guided notes -- students cut and paste pages into their notebooks. As of now I have no plans to use interactive notebooks, but I may consider it in the future.

My next post will be on Thursday.