Well, I had an interesting Pi Day of the Century over the weekend. Right when I was making my Pi Day post, at 9:26 AM, Channel 7 (KABC) here in Southern California had a story about Pi Day -- kudos to the TV station for timing their story so well. Right after this, I went on my weekly Saturday run -- normally I run two miles, but on that day I tried to make it as close to pi miles as possible. (I read that some cities actually held a so-called "Pi-K" run -- a misnomer, since it's actually pi miles, which is a little over 5K. A true 5K is about 3.11 miles.) For dinner, I ate some pizza "pi," and for dessert, I ate some homemade cherry "pi." I timed it so that I ate the cherry "pi" at 9:26 PM.
I also read another article on Pi Day, here on NPR:
http://www.npr.org/blogs/thesalt/2015/03/14/392589847/pi-day-isnt-just-magical-its-mathematics-and-theres-pie
A mini-debate occurred in the comments between Americans who said that the date was "3/14" and Europeans for whom that date was "14/3." It's often stated that 22/7, Pi Approximation Day, is preferable for those who put the date before the month -- sadly, 31/4 is impossible as April has only 30 days. Then someone happened to mention the thirteen-month Calendar Reform. Well, in addition to having Friday the 13th in all 13 months, the 13-month Calendar would also put the 14th of every month, including March, on a Saturday -- when Pi Day ought to be a school day. I think that the World Calendar does a better job with Pi Day -- March 14th is always on a Thursday, and it even assigns a 31st day to April (a Tuesday) to satisfy little-endians!
Now pi day is over, so it's back to the U of Chicago text for the next chapter. But I'm going to make some changes from what I originally planned back at the beginning of the semester. Originally, I was going to do Chapter 15, "Further Work With Circles," now. Placing the chapter on circles right after the lesson on pi would set up our unit very much like Dr. Franklin Mason's Chapter 9.
But this is a Common Core blog, and my main priority is to prepare the students for the upcoming PARCC and SBAC exams -- especially the PARCC, since students in PARCC states have to take this test every year from grades 3-11, unlike those in SBAC states who don't take it in either ninth or tenth grades. So a sophomore (or freshman) from some SBAC state will have completed the entire geometry course by the time he or she must take the SBAC, whereas a student from a PARCC state must begin the exam before completing the course. The question I must ask myself is, can a student do well on the PARCC exam based on the way I've paced my geometry course on the blog?
The PARCC exam for geometry (or any other course) consists of two parts. The first part is called the Performance-Based Assessment, or PBA, the other the End-of-Year Assessment, or EOY. The EOY, as it name implies, will probably be given around late April or May. But it's the PBA that I want to look at right now. The PBA is supposed to be given around the 75% mark of the year. As the beginning of this post implies, we are now three-quarters of the way through the year. This means that the PBA is being given right now.
So what exactly appears on the PARCC PBA exam? Let's look at some practice tests that are found on the PARCC website:
http://parcc.pearson.com/practice-tests/math/
When we look at the PBA and compare it to the U of Chicago chapters, we see that there is material on the test that we haven't covered yet on the blog -- in particular, surface area and volume, which are in Chapter 10 of the U of Chicago text. Indeed, the only topic that appears on the EOY in May that doesn't appear on the PBA are the circle topics of Chapter 15.
So this means that I didn't plan the second semester as well as I wanted to. Chapter 15 needs to be the final chapter -- the one given between the PBA and EOY dates. Chapters 9 and 10 will be moved up to now, so that students can learn about surface area and volume for the PBA.
Of course, part of the reason that I was confused about the PARCC exam is that I live in California, which is not a PARCC state. But there is a test that many sophomores in California do have to take this week -- the California High School Exit Exam, or CAHSEE.
The CAHSEE exam is administered several times a year. But many of these administrations are only for juniors and seniors who have failed it in the past. The sophomores who are first-time CAHSEE takers take it either the first week in February or this week in March. I am timing this blog so that this week is CAHSEE week.
The CAHSEE exam contains both ELA and math sections. The math section contains material from sixth grade up to Algebra I, so the Geometry content of this blog is actually beyond the material that will appear on the CAHSEE. Some people question the wisdom of California's continuing to have a CAHSEE exam, as many states are now using the PARCC or SBAC tests as exit exams.
Some readers might point out that if I had simply completed the U of Chicago text in its natural order, Chapter 15 would have been the final chapter anyway, and surface area and volume would have been covered well before the PBA. But then I would have been rushing Chapter 14, and the trig of Chapter 14 does appear on the PBA. (I even found a question on the practice PBA that requires inverse sine or cosine, while the U of Chicago only teaches inverse tangent.) Furthermore, I like the idea of having volume be the last thing before the test since the formulas are hard to remember -- it's only necessary that it be the last thing before the PBA. It also sets up Pi Day better -- simply going in order would put Chapter 14 around Pi Day, which I find undesirable unless we're doing trig in radians (and we don't in Geometry).
Is there anything in Chapters 1 through 14 that doesn't appear on the PBA? Some readers may point out my folly of including tessellations, which don't appear on the PBA. But actually, when I tried to find a correspondence between practice PBA questions and U of Chicago chapters, the chapter that doesn't prepare students for PBA questions is Chapter 13. Section 13-5 (on tangents to circles) appears along with some other circle stuff on the EOY, but there wasn't a single question on the practice PBA that I'd call a Chapter 13 question. Chapter 13 -- just like Chapter 3 first semester -- is a chapter that can be broken up and its sections included in other chapters. I've already stated that Section 13-1, on logic, could fit with Chapter 2. The only problem is, I'm not quite sure when to include indirect proof, Section 13-4. I suppose I could include it with the Circumcenter Theorem (alluded to in Section 4-5), since the proof that the circumcenters are concurrent is indirect.
Once we break up Chapter 13, a possible second semester plan for next year may look like this:
First half of January -- Chapter 12 (Similarity)
Second half of January -- Chapter 11 (Coordinate Geometry)
First half of February -- Chapter 14 (Trigonometry)
Second half of February -- Chapter 8 (Perimeters and Areas)
First half of March -- Chapters 9-10 (Surface Areas and Volumes)
Second half of March -- review, prepare for CAHSEE and PBA
First half of April -- Chapter 15 (Circles)
Second half of April -- review, prepare for EOY
This keeps the three chapters related to similarity (12, 11, and 14) together. We can still set up Pi Day by having the Chapter 8 test remain up to 8-6 only (8-7, the Pythagorean Theorem, remaining included with the similarity block of 12, 11, and 14). Then March (i.e., Pi Month) begins with the lesson about pi and then moving on to surface area and volume, just as we're doing this year. A test on this section can occur right before Pi Day. I actually like this because then Pi Day could become a prize -- the period that performs the best on the test just before Pi Day will win the pizza party for Pi Day itself.
There's going to be a squeeze somewhere around chapters 8 through 10, though. It may be tough to include pi, surface area, and volume in the first two weeks of March. Another idea could be to move the surface areas and volumes that don't involve pi (prisms, pyramids) to the same unit as the first two-thirds of Chapter 8. Then the first half of March would mostly require pi itself plus applying the formulas involving pi to cylinders and cones. This might be preferable because some states actually gave the PBA this year as early as March 1st (or 2nd, as the 1st was a Sunday) -- part of this is to allow more time to reserve computers to take the test. So it's good for the students to have at least seen the word "volume" mentioned in the class by February 28th, before the earliest possible PBA date, even if it's just the volume of a box. But then we're squeezing all of 8-1 to 8-6 plus some surface area and volume into two weeks. (Well, I have a full year before I have to make that tough decision!)
Also, I want to point out that the CAHSEE is always given with ELA on a Tuesday and the math section on a Wednesday. In years when March 14th falls on a Tuesday or Wednesday, it may be the case that the CAHSEE occurs on Pi Day itself. Having a Pi Day party on Wednesday, right after the students have taken a hard CAHSEE math test, is a wonderful idea, but having it on Tuesday -- the day before that hard CAHSEE math test, isn't such a good idea. In either case, teachers would have to work around the awkward bell schedules both days.
The real problem, of course, is the insistence that the PBA be taken after only 75% of the year, when it covers more than 75% of the text. This has actually a problem with standardized testing since well before the Common Core era. I've discussed my mixed feelings about the Common Core Standards before, but this is a good time to consider my feelings about the Common Core assessments.
Anyone reading the news is familiar with the backlash against the PBA exam. Students in the PARCC states are walking out and boycotting the exam. So far, there hasn't been as much written about boycotting the SBAC exam. I suspect that the SBAC is marginally better than the PARCC, if only because Grades 9-10 don't have to take the SBAC, and the SBAC doesn't have PBA and EOY components, so there's less testing in SBAC states than in PARCC states. Because of this, most SBAC states haven't started testing yet. Some SBAC states begin in late March, but California won't start until April. (I bet part of this is to separate SBAC from CAHSEE here.) It's not hard to predict that SBAC boycotting will begin soon enough as well.
To me, I don't see why both a PBA and EOY exam for PARCC are necessary. If I were in charge, I'd at least drop the PBA and have an EOY exam only. Sometimes I wonder whether tests could be taken later in the year, closer to the last day of school, since they are computer graded -- but then again, the tests include questions that aren't multiple choice.
I have much more that I want to say about the Common Core exams. I will discuss this ideas in my posts throughout the week.
Anyway, I've decided to focus on surface area this week -- even though volume might appear on this week's (or month's) PARCC exam, it's nonetheless better to introduce the students to solid 3D figures first, then find their surface area. Dr. Franklin Mason's Chapter 12 is the guide for this week. We begin with Dr. M's Section 12.1, which simply introduces solids. The closest analog in the U of Chicago is Section 9-7, "Making Surfaces," with a hint of 9-2 and 9-3 thrown in. Surface area proper begins in Dr. M's Section 12.2, which we'll cover tomorrow.
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