Friday, March 20, 2015

Activity: Pythagorean Theorem Practice (Day 136)

Today is the last day of school before spring break. Thankfully, our Long March through a tough part of the school year is finally over.

Now many of you are saying -- what? Easter isn't until April 5th, and most schools take either the week before or the week after the holiday off. And even a school taking the week preceding Easter off still has a full week left before spring break. So how can today be the last day before the vacation?

As it turns out, the school calendar that I'm following happens to be one where spring break isn't tied to the Easter date at all. Many schools around the country wish to avoid the high variability associated with the Easter date and the havoc it wreaks on the school calendar. For example, consider a school that takes the week after Easter off. A few years ago, Easter was so early that it was still March when the students returned from spring break. Students were greeted with end-of-quarter exams (at schools with Labor Day starts) and state testing right when they came back. But then in 2011, Easter was so late that it was already May when the students returned from spring break. Students were greeted with AP exams right when they came back -- indeed, the AP Chemistry exam was the very first morning after the students had taken a week off from thinking about academics.

If one wants to avoid the volatility of the Easter date and select a fixed week in the school calendar during which to schedule spring break, a natural choice would be at the end of the third quarter. In fact, this makes very much sense at schools on the Early Start Calendar, since just as the winter break divides the year into semesters, the spring break can divide the second semester into quarters.

But one may recall that the fourth quarter of my calendar began earlier this week. If the school's spring break was intended to divide the quarters, spring break should have been scheduled this week, not next week. My guess is that the CAHSEE exam prevented this week from being spring break, since it would be awkward to make sophomores come to school to test during a vacation week. So next week is the closest that spring break can get to the actual midpoint of the semester without interfering with the state test.

So what does this say about my so-called "Long March"? I spent the day after President's Day lamenting the tough stretch of the school year, as there would be no holidays until spring break. Well, our Long March is five weeks long. This is still longer than the maximum of three weeks that occurred between holidays from Veteran's Day to President's Day, but it is shorter than the six- or seven-week Long Marches that occur at schools that observe spring break at Easter.

What this does mean is that the longest stretch of my school year without a holiday is actually from Easter to Memorial Day. Many schools that don't tie spring break to Easter still have a long weekend for the holiday, including mine. This year, there is a seven-week stretch from the long Easter weekend to the long Memorial Day weekend. But "Long April/May" doesn't have the same ring to it as "Long March" does.

With spring break -- and the spring equinox -- upon us, now let's consider what effect this has on our math lessons. This was, in fact, part of the reason that I originally wanted to do Chapter 15 of the U of Chicago text this week. I knew that there would be only one week left before spring break, and the last important section of the chapter was Section 15-3 -- nothing after this lesson usually appears on standardized tests, including the practice PARCC exam that we've been discussing all week. I could have covered the lessons up to Section 15-3 by Tuesday, and then have a short quiz covering the last part of Chapter 8 (on pi) and the first part of Chapter 15 (on circles and tangent lines) by Thursday -- this would have been very similar to Chapter 9 of Dr. Franklin Mason's text, as Dr. M also combines pi and tangent lines in a circle chapter. Then we could have introduced surface area and volume after spring break without being interrupted by the vacation period.

But I changed my plans after taking a peek at that practice PARCC exam -- especially after I had compared the PBA and EOY versions of the test. Combining pi and tangent lines into a single circle chapter, as Dr. M does, is no longer feasible because pi, along with surface area and volume formulas involving pi, appear on the PBA, while tangent lines don't appear until the EOY. So Chapter 15 needs to be the final unit and be taught between the PBA and EOY, while pi, surface area, and volume should be taught before the PBA.

Notice that I want to time my lessons to both CAHSEE, for the sake of Californian readers, and the PARCC, for the sake of readers from PARCC states. But the dates aren't quite compatible, as for CAHSEE, all I'd have to do is introduce surface area and volume of boxes (and their unions) by the second week of March, before the CAHSEE. But PARCC requires that I teach surface area and volume before the PBA, which could be given as early as March 1st.

Instead, my plan for next year is to compromise -- I teach the surface area and volume of boxes at the end of February, and finish pi, surface areas, and volumes by the second week in March. Then the last week before spring break could be a full activity week. The Dan Meyer lesson that I posted earlier could become an opening activity for surface area and volume in late February, while this week I could have posted another lesson -- Meyer also has a volume activity involving meatballs.

But that's next year. This year, I'm left with a need to give a lesson for today. Just as I wrote before Thanksgiving break, I want to avoid teaching a brand new lesson right before a week-long break. So instead, just as I did before Thanksgiving, today's lesson is a preview activity of what's to come after the vacation. Coming up will be the surface areas and volumes of pyramids, cones, and spheres.

Today I ended up subbing in a special education course -- but it was in fact a Geometry class. The class is now in Chapter 8 of the AGS text that I mentioned during the second week in January -- and this is on the Pythagorean Theorem. Considering that Chapter 12, the last chapter of the text, is not covered on standardized tests, Chapter 8 is more or less on pace to reach Chapter 11 by the end of the year, although I'm not sure about reaching it before the SBAC for juniors.

This is significant because Chapter 11 is on volume and surface area -- that is, it corresponds to Chapter 10 of the U of Chicago. The volume lesson will be given too late to help sophomores on the CAHSEE, so let's see whether it will be taught in time to help juniors on the SBAC.

Indeed, as we look at the surrounding chapters of the AGS text, we notice that Chapter 10, on circles, includes both pi and tangents -- both the tangent of a circle and the trig tangent! Indeed, the two types of tangent appear in back-to-back sections -- Section 10-7 for tangent of a circle, and Section 10-8 for the trig ratios. There is a brief mention of a unit circle to justify having the trig functions in the same chapters as circles. Spheres also appear in Chapter 10 -- leading to the awkward situation that the surface area and volume of a sphere are known before the surface area and volume of a cube!

Chapter 9 of the AGS is on perimeter and area -- it therefore corresponds to the sections of Chapter 8 of the U of Chicago that appeared on the last test posted to the blog. Chapter 7 is on similarity -- it therefore corresponds to the same numbered chapter in Glencoe and Chapter 12 of the U of Chicago.

But Chapter 8 -- a chapter solely devoted to the Pythagorean Theorem -- doesn't appear in any of the other texts that we've discussed here on the blog. The AGS does the exact opposite of what we did here on the blog. On this blog we rushed through the Pythagorean Theorem, whereas in AGS we slow it down and spend an entire chapter on Pythagoras.

Here are the sections of Chapter 8 of the AGS text:

Section 8-1: Pythagorean Triples
Section 8-2: Pythagorean Triples and a Proof
Section 8-3: Pythagorean Demonstration
Section 8-4: Pythagorean Theorem and Similar Triangles
Section 8-5: Special Triangles
Section 8-6: Pythagorean Proof and Trapezoids
Section 8-7: Distance Formula: Pythagorean Theorem
Section 8-8: Converse of the Pythagorean Theorem
Section 8-9: Algebra Connection: Denominators and Zero

So notice that we begin with Pythagorean Triples -- numbers that satisfy a^2 + b^2 = c^2 -- before saying anything about right triangles. Sections 8-3 and 8-4 both contain area-based proofs of the Pythagorean Theorem -- a bit awkward, since area doesn't appear until Chapter 9 of this text. I believe that the proof mentioned in Section 8-6 is President Garfield's proof. Section 8-9, as well as the last section of every chapter of the AGS text, is an "Algebra Connection" or algebra review.

In some ways, maybe spending extra time on the Pythagorean Theorem is worthwhile. Most texts include it as part of other chapters, such as the area or similarity chapters -- and often separate the Distance Formula by including it in a different chapter. Perhaps my blog represents a compromise -- we introduce it quickly, but we remind students of the theorem whenever they need to remember it.

And one of those times is now. I post these worksheets based on AGS's Sections 8-1 and 8-2 in order to remind students of the Pythagorean Theorem, just ahead of the lessons on pyramids and cones. After all, the radius, vertical height, and slant height of a cone form a right triangle, and so one might need the Pythagorean Theorem to calculate their lengths before finding the surface area or volume of the cone. As I said, the "Chapter 8" at the top of this posted worksheet refers to the AGS text, although one might assume that it refers to the U of Chicago text since the Pythagorean Theorem appears in Section 8-7 of that text and we just covered the rest of Chapter 8 last week.

In most classes, I ended up only discussing Pythagorean Triples from the Section 8-1 worksheet -- which I don't post to this blog -- and not any triangles at all. I played a modified version of the usual game that I give because some of the special ed classes had only four or five students.

Only in third period did I actually reach the triangle worksheet that I posted here. Many of the students were confused when they were asked to solve for a side of the triangle. Part of this is because the first two problems involve integers, and then everything jumps into decimals. Students were allowed to use a calculator, but many of them had only a simple online calculator that lacked a square root function.

Perhaps if I were the one preparing this lesson, I would have included more simple integer questions before jumping into decimals with irrational square roots so quickly, but this was a lesson that was already prepared by the regular teacher. Then again, we can't shield the students from decimals and irrational square roots forever. A few of the problems that appear in Sections 10-2 and 10-7 of the U of Chicago text on pyramids and cones require decimal square roots, and of course we might expect such questions on PARCC or SBAC as well.

And so this concludes my last post before spring break. Once again, I plan on making one or two posts next week, during spring break itself. One of the posts may be a continuation of what I posted in February about the German education system and what would happen if one tried to apply their system to the U.S. And the other may be about Section 9-5 of the text -- a lesson that we'll skip as it's not important to PARCC or SBAC, but it's interesting to me. The Common Core standards devote much time to reflections and other transformations of the plane, and now Section 9-5 discusses what happens with reflections in 3D space. After the break, we will continue with pyramids and cones.

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