Chapter 6 Quiz (Day 61)

You may be wondering why I'm posting a quiz so soon after the last test. Well, the reason has to do with the school district whose academic calendar I'm following on this blog.

You see, I don't know about other states, but here in California, not only is Veteran's Day a holiday on which all schools are closed, but also it must be observed on its actual calendar date -- November 11th -- no matter what day of the week that happens to be. As it turns out, this year, November 11th falls on a Tuesday -- which is obviously not a convenient day because it sets up Monday as a lone day -- a day isolated from the rest of the week.

The two districts at which I sub treat this holiday differently. One of them indeed has the lone day on Monday, and then no school until Wednesday. The other decided to declare Monday a holiday so that there could be a four-day weekend rather than a lone day. (Some districts simply have a professional development day on Monday, while others declare Monday to be "Admissions Day," which is silly because California became the 31st state in September 1850.)

As it turns out, the official calendar used on this blog (the one for which today is Day 61) happens to be from the district with the four-day weekend. Therefore, Day 62 of this blog will not be until Wednesday, November 12th. And it's too much to expect students to remember such an important concept as transformations over the long weekend. And so I post a quiz just before the holiday.

Another thing that I want to point out is that Monday, November 10th really is a holiday. No, it's not Admissions Day, but Pi Day!

Hold on a minute! You probably thought that Pi Day was on March 14th -- and the date on which this blog was launched was Pi Approximation Day, July 22nd. So how can November 10th be yet another Pi Day?

Well, November 10th is the 314th day of the year. And so some people have declared the day to be a third Pi Day:

http://mathforum.org/kb/message.jspa?messageID=7605691

I like the idea of a third Pi Day, based on the ordinal date (where January 1 = 1, November 10 = 314, and December 31 = 365). As the author at the above link pointed out, the three Pi Days are nearly equally spaced throughout the year. So I can celebrate Pi Day every fourth month.

I wouldn't mention the third Pi Day in a classroom, unless I was at a school that was on a 4x4 block schedule, where a student may take math first semester and then have absolutely no math class at all
in the second semester (when the original Pi Day occurs). The only chance a student has to celebrate Pi Day would be the November Pi Day. (Likewise, the second Pi Day -- July 22nd -- may be convenient for a summer school class.)

The only problem would be at a school where Veteran's Day must be November 11th. With the third Pi Day falling the day before a holiday and often part of a long weekend, more often than not there will be no school on November 10th. If that day falls on a Saturday or Sunday, then there will obviously be no school. Nor can there be school on Friday, November 10th, since then Veteran's Day would be on a Saturday and all Saturday holidays get pushed back to Friday. Finally, if November 10th falls on a Monday, as it does this year, there's a four-day weekend. So more often than not, there's no school on third Pi Day. (Notice that if it's a leap year, the 314th day will fall on the 9th -- a day on which there's more likely to be school than the 10th. But I have absolutely no idea what schools do when Veteran's Day falls on a Wednesday. I'll find out next year!)

Hmm, I wonder what I'd do if I taught an integrated math class -- where the students may be studying geometry around November 10th but algebra around March 14th. If possible, I'd try to schedule geometry near the original Pi Day of March 14th, so algebra would be before this.

Both November 10th and March 14th suffer from falling near the ends of trimesters or quarters (depending on whether the school started in August or after Labor Day). Classes may be too busy with trimester or quarter tests to have any sort of Pi Day party.

At home, I like to celebrate and eat pie for all three Pi Days. The pie that I choose is the pie most associated with the season in which that Pi Day occurs. On Monday, I will eat either pumpkin or sweet potato pie due to its proximity to Thanksgiving. On Pi Approximation Day, I ate apple pie, since it occurs right after the Fourth of July, a date as American as apple pie. And for the original Pi Day in March, I eat cherry pie -- the National Cherry Blossom Festival usually occurs between a week and a month after Pi Day.

Now another mathematical text that I saw this week occurred when I was a substitute in a special education class. For some reason, on the bookshelf was an old, but vaguely familiar text -- it was School Mathematics II, published by Addison-Wesley. As it turns out, Part I was the old book that I was telling you about back in Chapter 5, the one that contained an exclusive definition of rhombus. I tried to show up my fifth-grade teacher with that definition, remember? But this was the first time that I had ever seen Part II of this text. (Part I is for seventh graders, and Part II is for eighth graders.)

I looked at the text, and checked out the glossary. Sure enough, rhombus is defined as:

"rhombus: An equilateral convex quadrilateral which is not a square." (emphasis mine)

But notice that in Section 9-2, where rhombus is defined, a rhombus is still a parallelogram. Then I looked at the date of the text. It was 1967. And so we see that some time between 1967 and a mere quarter-century later, 1992 -- the year I made a fool of myself in fifth grade -- the exclusive definition of rhombus has been dropped in favor of the inclusive definition. And now we're approaching another quarter-century later, and if PARCC is influential enough, the exclusive definition of trapezoid will have been dropped in favor of the inclusive definition.

In both Addison-Wesley texts, there are 13 chapters, and every third chapter is geometry. And each geometry chapter has been divided into "Nonmetric Geometry" and "Metric Geometry." In the former there is no measurement, only in the latter. Most modern geometry texts don't respect the difference between nonmetric and metric geometry. For the most part, Chapter 9 of the U of Chicago text is nonmetric while Chapter 10 is metric -- but there are a few measurements made in Chapter 9 (but the heavy hitting volume formulas are indeed in Chapter 10).

My philosophy is that most of the first semester of geometry should be nonmetric. Indeed, most of what I've covered so far on this blog correspond to the nonmetric parts of Addison-Wesley's Chapters 3, 6, and 9, with only the simplest metric sections included.

Here is an overview of the geometry chapters from Addision-Wesley:

• Chapter 3 covers the basics -- points, lines, and planes. It therefore corresponds to Chapter 1 of the U of Chicago text. The metric section covers length, area (of rectangles), and volume (of rectangular prisms). So it's much more basic than Glencoe's Chapter 1.
• Chapter 6 covers congruent angles and triangles. Notice that congruent angles appear in the nonmetric section -- meaning that it must be defined without measuring the angles. This is done by measuring the "opening" of the triangle -- placing points on each side of the each angle and determining whether the segments joining them are congruent. This essentially means that SSS is true almost by definition. (Of course, the Common Core definition of congruence could be considered nonmetric as well!) Triangle area appears in the metric section.
• Chapter 9 covers the Parallel Postulate and its Consequences for parallelograms. Its metric section covers volume of prisms and pyramids, similar triangles, the Pythagorean Theorem, and even some basic trig.
• Chapter 12 is all about circles. Dr. Franklin Mason's Chapter 9, which includes both nonmetric and metric (as in pi) aspects of circles, is a close equivalent.

So the geometry in eighth grade Addison-Wesley corresponds roughly to the eighth grade standards for both Singapore and Common Core.

Interestingly enough, both the Addison-Wesley (in its "Think" sections) and the Common Core packet that I found earlier this week contain the same brainteaser problem -- a problem where several adults and two children need to cross a lake, but the canoe can hold at most two children or one adult at a time.

Here are the answers to the quiz that I posted -- one last quiz for the end of the trimester or quarter, depending on your calendar:

1. Drawing/construction

2. The transformation is a rotation. Since the lines appear to be nearly perpendicular, the magnitude of the rotation will be nearly twice that, or 180 degrees, centered where the two lines cross.

3. HI

4. Angle A

5. A rotation with center where l and n cross, magnitude 180.

6. A translation in the direction of n, twice the distance from l to m.

7. translation

8. rotation

9. 5

10. (a). Notice that this is actually a dilation -- a preview of second semester. Also, I decided to use what is technically function notation, T(x, y). I decided that the only real reason that the U of Chicago introduces the N(S) notation for cardinality (number of elements in a set, previous question) is to prepare the students for function notation, so I might as well use it here. There's only one other place where I see n(A) used for number of elements in set A -- the Singapore Secondary Two standards!


Have a nice Veteran's Day weekend! See you on Wednesday for Day 62!