Thursday, January 12, 2017

Student Journals: Rotations (Day 79)

With so many blogging challenges going on now, today is one of the few posts that isn't part of one of the challenges. And as you know, in most of the non-challenge posts I write during challenge time, I end up linking to other participants. So expect me to mention challenges in every single post I write in January.

In particular, today I'll link to some of the other 2017 Blogging Initiative Posters. But first, let's start with the Pappas Question of the Day:

12 trillion = 12 * 10^?

One trillion is the 12th power of 10. So the answer is 12 -- and today's date is the twelfth. This isn't the actual question I gave today in any class (and you'll see why later on), but I could have given it in my eighth grade class earlier. Notice that this isn't true scientific notation (it should be 1.2 * 10^13), but the Illinois State text mentioned forms like 12 * 10^12 en route to teaching scientific notation.

I need to mention my eighth grade class today, especially since they're learning about transformations on this Common Core Geometry blog. In the end, I decided to delay the science lesson to tomorrow and teach transformations today. The main reason is that today's Bruin Corps member is a molecular biology major. But yesterday, I received a second Bruin Corps member whose major is atmospheric and oceanic sciences. This is more in line with the Green Team, and so I will wait until she's here tomorrow for the eighth grade science lesson.

This means that this week I had three full days to cover the three transformations. On the first two days, the translations and reflections went well, and most students appeared to understand. But I worried as today's lesson approached, because rotations are probably the most difficult of the three transformations for students to understand.

Now keep in mind that I'm using the Student Journals that are part of the Illinois State text. We know that rotations can be centered either at the origin or away from the origin. Rotations centered at the origin have easier formulas -- for example, the rotation of 180 degrees centered at the origin maps the point (x, y) to the point (-x, -y).

But none of the rotations mentioned in the Illinois State text are centered at the origin. Most of the questions direct a student to rotate a line segment around one of its endpoints. This at least makes it a little easier, since every rotation maps its center to itself.

And so here's what I dis today -- on the first page, the students are asked to rotate AB 90 degrees clockwise about point A, The coordinates are A(2, 3) and B(7, 3). I had the students change A to "the origin," and then I show them the 90-degree rotation about the origin. To do this, I had the students the paper 90 degrees counterclockwise -- that is, the opposite direction from the rotation. Then they drew the image A' by going 2 units on the new x-axis and 3 units along the new y-axis. They did the same to find B', and then they restored the paper to its original position. The new segment A'B' now appears to be the clockwise rotation image of AB.

Of course the students are confused by this at first, but in the end, I believe that they're starting to get the hang of this. I like teaching rotations this way because it sets them up nicely to learn the slopes of perpendicular lines later on. By the end of class, I think the most confusion came from changing all the questions in the Illinois State text, which were geared towards the rotation centered at A rather than the origin.

On the second page, I kept the original question intact. This time, the students are asked to rotate a segment 90 degrees around its midpoint rather than an endpoint. But they appeared to figure out quickly that the preimage and image together would form a cross.

At this point, you may be wondering why I didn't just create my own worksheets rather than modify the Illinois State text. (Recall that the worksheets I posted the first two years of the blog are not valid for this lesson as I didn't emphasize the coordinate plan enough.) The reason is that we teachers are required to use the Illinois State material as much as possible.

I must warn you that if you're a traditionalist, you may wish to stop reading this post, since you won't like anything that I'm about to say next. (Hopefully, traditionalists were already scared away by the word "rotations" in the post title, since they don't like transformation geometry in the first place.)

We are one of the first schools to use the Illinois State text. That's why the curriculum developers keep flying in over land and oceans to introduce the program to us. But we're not the very first school to pilot the program. A few other schools used it last year, and supposedly, those schools' scores on the Common Core tests skyrocketed from around 20% to near 80% proficiency. It doesn't matter whether I trust these claims or not -- what matters is that the administrators believe them.

And this was drilled home at yesterday's Common Planning meeting. We (and by "we," I mean the elementary teachers plus myself) have been told, basically, that if our test scores fail to rise by a significant amount this year, it's because we didn't fully implement the Illinois State program. And then next year we'd have to double down on our efforts to stick to the program. (I did warn the traditionalists that they won't like anything in this post!)

Each day, we should begin with an Illinois State "daily assessment," which I can find on the Illinois State website, and I use the projector to show the class. This is why I couldn't use the Pappas question today, since most of the time I'm using Illinois State questions whose answers aren't the date. (Today's question was to define "reflection," and so the answer isn't even a number, much less the date.)

The Illinois State text also provides some "Interactive Homework." Most of the rotation questions are of segments being rotated about an endpoint. I'm now wondering whether it was better for me to keep the first page in the journal intact (since its rotations are also centered at an endpoint) and change the second page to the origin (since rotations centered at a midpoint don't appear in the homework). Of course, I couldn't create my own worksheet since I must use the text.

But there are some poor questions included in the homework as well. One type of question gives students two segments and asks whether a translation, reflection, or rotation maps one segment to the other segment.

Now here's the problem -- the answer usually isn't unique! In particular, if AB and A'B' are segments and there exists a reflection mapping AB to A'B', then there must also be either a translation or a rotation mapping AB to A'B'! Proof: Let m be the mirror of the reflection mapping AB to A'B'. Now, we know by the Segment Symmetry Theorem (found in the U of Chicago text, mentioned during the first two years of the blog) that the reflection image of AB over its own line, AB, is itself. And so we have two mirrors, line AB and m, and reflecting AB first over line AB and then m produces A'B'. This is a composite of reflections mapping AB to A'B'. If line AB is parallel to m, this composite is a translation mapping AB to A'B', otherwise it's a rotation mapping AB to A'B'. QED

Most of the mirrors in these problems are either the coordinate axes themselves or at least parallel to an axis, and most of the rotations are centered at an endpoint of the segment being rotated. So if the segment and its image have a common endpoint, the intended answer is "rotation," even though reflection over the bisector of the angle formed by the two segments also works. In particular, if the segment and its image are parallel, then the intended answer is "translation," even though a reflection may sometimes work as well. (Most of the time it doesn't, since the composite of a translation and a reflection is usually not a reflection, but a glide reflection instead.)

One blatant error gives the students the line x + y = 1 and asks them which one of three given lines is the reflection image of the original line. As it turns out, all three lines are the images of x + y = 1 -- one is the image over the x-axis, one the image over the y-axis, and one the image over y = -x. In fact, we can show that in 2D, there exists a reflection mapping any line to any other line! If the two lines are parallel, the mirror is parallel to both and halfway between them. If the lines intersect, then there are two mirrors possible, each one a bisector of an angle formed by the lines. (This fails in 3D, because the lines could be skew.) Even one of my students figured out that there was no single correct answer to this homework question!

One of the curriculum developers provided us with a list of the "major content" (MC) standards for each grade level -- that is, the standards most likely to be test on the SBAC. And I have a huge problem -- I haven't covered enough of the standards yet, especially not in sixth grade.

The problem is that my pacing plan was to cover one STEM project every two weeks -- and each STEM project is linked to various standards. But too many projects that I've covered link to standards that aren't MC, and some that do link to MC standards are near the end of the STEM text. At the meeting, the administrators told that we must submit to them a new pacing plan to demonstrate how we plan to cover all the MC standards before the SBAC.

As it turns out, in sixth and seventh grades, all of the MC standards are either Ratios and Proportional Thinking, Number Sense, or Expressions and Equations -- none are from the Geometry or Statistics and Probability strands. In fact, in seventh grade this is simple -- every single RP, NS, or EE standard is MC, while again, no Geo or SP standard is MC.

On the other hand, some of the sixth grade NS standards are not MC. To my surprise, decimal division -- after I made such a big deal about it in previous posts -- is not MC, and neither is whole number long division or any decimal arithmetic. The only NS standards that are MC are fraction division and introduction to negative numbers.

In eighth grade, the NS standards are not MC. All of the EE standards are MC, as are most of the Function standards, Eighth grade is the only year with Geometry standards that are MC -- the only one that isn't MC is volume -- and we can see why, since this is the introduction to transformation geometry that is critical to high school Geometry classes.

The big culprit for my failure to cover enough MC standards in time are the first four Learning Modules -- the infamous Unit 0: "Tools for Leaning" projects that are only linked to Mathematical Practices rather than any content standard (much less any MC standards). In fact, there are 15 modules in the eighth grade text (and fewer in the other grades) that are linked to MC. Hey, didn't I say earlier that there's just enough time for 15 projects? Of course, none of these are the "Tools for Learning" projects -- and yet I wasted nearly an entire trimester on them!

So far, I've only covered three modules (numbered 5-7) linked to MC standards. This leaves me with 12 MC modules to cover -- forcing me to speed up to one module per week! Here is the pacing guide that I plan to send to my administrator:

Week of 1/17: Module 8 (MC standards covered: G3, G4, G5)
Week of 1/23: Module 9 (G6, G7, G8)
Week of 1/30: Module 10 (EE7a)
Week of 2/6: Module 11 (SP2 -- not MC, but appears on 2nd Trimester Benchmarks)
Week of 2/21: Module 12 (EE8a, EE8b)
Week of 2/27: Module 13 (EE8c)
Week of 3/6: Module 20 (F3 -- appears on 2nd Trimester Benchmarks)
Week of 3/13: Module 14 (EE5, EE6)
Week of 3/20: Module 15 (EE4)
Week of 3/27: Module 17 (F1)
Week of 4/3: Module 18 (F2)

This leaves us with a little extra time to cover a few more modules before the last Benchmark Test leading up to the SBAC. Notice that Module 16 is skipped -- I unwittingly included EE1 and EE2 during "Tools for Learning," in order to prepare for EE3 in Module 5.

Also, we see that EE7a is included, but not EE7b. The STEM text mentions EE7b very late -- it's in the last two modules, 24 and 25, But those STEM projects are on matrices, which I feel are not appropriate for eighth grade. (I'm glad they're at the end of the text so I don't have to teach them.) So I must sneak in EE7b in somehow. After EE7a is logical, but keep in mind that EE7b (which is on solving multi-step equations with like terms, parentheses, etc.) is much more difficult than EE7a (on solving one- and two-step equations.) As a student teacher, I've seen freshmen struggle with the multi-steppers -- how much more trouble, then, will my eighth graders have? In this case, it's okay for me to bleed EE7b into the following week -- SP2 is only for Benchmarks, not MC, and notice that I put some leeway in around the five-day President's Day weekend.

A typical module week will look like this:

Monday: Coding
Tuesday: STEM Project (however much can be finished in one day)
Wednesday: Student Journals
Thursday: Student Journals
Friday: Quiz or Test, followed by Science

I also have to figure out how the quizzes and tests are going to work out. I'm almost considering going back to my original three-week rotation of Dren Quiz, Regular Quiz, Test (though it won't be staggered among the three grades).

The sixth grade pacing plan looks even worse, since I covered so many standards -- long division, and the current standard (GCF and LCM) -- that are not MC. I'll begin with Learning Modules 8, 9, and 10, then skip to 24 and 25. Recall that Modules 24 and 25 are about water, which is related to the Green Team. I actually had a meeting today with the Green Team leader. She tells me that the Green Team projects should begin in earnest in February, so I schedule Modules 24-25 for those weeks.

In seventh grade, we must skip around immediately -- tomorrow I'll begin Module 14. The only MC module remaining before 14 is 9 -- and since it includes circumference and area of a circle, and we're skipping around anyway, I might as well save it for March, close to Pi Day. Notice that circle measures are not MC, but Module 9 also includes EE6, which is MC.

By the way, you may be wondering about my Algebra I pacing plan with my new eighth grader. Well, she still hasn't been given an IXL account, so I can't start yet. Actually, today would have been a bad day for Algebra I anyway -- for reasons that I choose not to disclose on the blog, some of my seventh graders were in my classroom for the first half of IXL time along with the eighth graders. With over 30 kids in the room, there weren't enough laptops for everyone -- and the class was so noisy that the new girl wouldn't have been able to concentrate on algebra. This at least gives us an extra week for her to be given the IXL account.

Okay, now that I've gotten all of this out of the way, let's see those Blogging Initiative links. As it turns out, there are three fellow middle school teachers participating in the Initiative:

Cheryl Leung is a seventh grade teacher. (I couldn't find her state easily.) The Week 1 topic was "My Favorite" -- and her favorite is programming robots! My school doesn't having robots, but programming would fall under the purview of our Monday coding teacher.

Melynee Naegele teaches all three middle school grades (again, state unknown). Her favorite is a "bellringer" activity at the start of each class. I also have "bellringers," and I call them Warm-Ups -- these are either a Pappas date question or an Illinois State question.

Anna Pacura is a New Yorker who teaches all three middle school grades. (Wow, three of us four middle school teachers cover all three grades!) Her favorite consists of online resources. This again highlights the problem I have with Illinois State -- if I try going to online resources (including those on MTBoS), I'm made to feel guilty for not choosing an Illinois State resource instead!

I will make my own 2017 Blogging Initiative post for Week 2 tomorrow.

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