Thursday, October 16, 2014

Who Am I? And Section 5-4: Properties of Kites (Day 50)

In yesterday's post, I began revealing some personal information about myself. I mentioned that I am 33 years old, and my weight is -- did I actually state my weight? Well, recently I went to the doctor who weighed me, and I measured between 190 and 200 pounds. (In honor of last week's Metric Week, let me also give my mass as between 85 and 90 kilograms.)

But as much as the students are fascinated by this information, readers of this blog aren't? Instead, you readers want to know such information as, am I a teacher? Where and whom do I teach?

Well, let me give answer that question. Today I was hired to work as a substitute teacher right here in Southern California. That's right -- for the first fifty school days of this blog's existence, I wasn't working in a school at all. Much of what I posted here is based on my knowledge of the Common Core Geometry Standards, as well as what I observed in the classroom in the past. I have completed my California Clear Credential in Single Subject math, but times are tough for newly credentialed teachers. I'm hoping to sub this year and try again to be hired as a full teacher in time for next year.

Now I know that math teachers who read blogs aren't interested in lesson plans and worksheets -- they can get those out of textbooks and other sources. No -- actually, readers of math teacher blogs want to see how teachers interact with students, and how well lessons go with students. But as long as I had no students, there was no way to gauge how well my lessons were written.

So now things are going to change. I'm hoping that I'll be able to sub for a geometry teacher, see how the students react to the new Common Core Geometry lessons, and report the results on this blog.

This means several noticeable changes on this blog. Recall that I've been numbering the school days -- today is Day 50. Technically, this was based on a hypothetical Early Start calendar in which I labeled the last day before winter break as Day 90 -- the end of the semester -- and counted backwards. But now that I'm working for an actual district, I will use the actual calendar for the district.

School in my district started about a week later than Day 1 on this blog. And so to make up for it, I will take an entire week off from posting. I already mentioned that today's lesson, Section 5-4 of the U of Chicago text, is a natural dividing point for the chapter (because I want to state a Parallel Postulate before covering Section 5-5). So I'll post Section 5-4 today, then a short activity tomorrow (Day 51), and then I won't post any school days the following week. This schedule makes out Day 52 to be Monday, October 27th, which aligns better with my district calendar. Counting backwards, the school year in my district began on Tuesday, August 12th, which I should count as Day 0 because only freshmen attended school that day. Day 1 for all students was the next day, which was Wednesday, August 13th. (Oh, and by the way, my district actually took Columbus Day off!)

Starting on Day 52, I will be posting in the late afternoon or early evening -- that is, after school. This way, if I sub for a geometry teacher, I can post on this blog how the students reacted. I plan on beginning with the Parallel Postulate, but that will change depending on what I teach that day. I don't necessarily like jumping around in the text, but it will be much valuable to you readers to let the actual classroom, not a textbook, determine the order of the lessons posted to this blog.

Finally, as a substitute teacher, I already know that students are generally less cooperative for subs than they are for full-time teachers. I've seen students who, the instant they see a sub in the class, give excuses such as "Nobody's going to do this assignment!" Notice it's not I won't do the work, but nobody will -- the implication being that for a sub to do anything other than give the students a free period is highly unreasonable.

To fight this, my strategy will be to give a lesson similar to yesterday's activity. The first question I ask the students to answer will be "What is my age?" This will immediately disarm the student who was planning on saying "Nobody's going to do this assignment!" -- especially when some groups are already shouting out guesses! Only after giving the first two questions will I ask any mathematical question -- even if the full-time teacher has left an assignment, I plan on asking my two questions first before anything from the assignment. Of course, I'll keep tally of the points. I want the students to be behind in points and eager to answer the next question -- and then they find out that they have to complete the teacher's assignment in order to earn the points! The winning students or groups will be the students whose names I leave the regular teacher as having performed and behaved well that day.

And now let's get on with today's lesson. Section 5-4 of the U of Chicago text covers kites. The kite is a relatively new quadrilateral classification. Not only did Euclid never define kite, but many texts made no mention of kites -- including my class geometry textbook from 20 years ago. Nowadays most texts define kite, but some include kites only in bonus questions, not in the main text.

Here's what John Conway wrote about the kite -- nearly 20 years ago, right around the time that I was taking my geometry class:

In fact it's not quite true, either, because "kite" is not
a very traditional name - it was obviously inserted because
this was a type of quadrilateral that SHOULD have received a
traditional name, but didn't, until recently.

Why do we include the kite - plainly because it represents
the one type of symmetry not otherwise mentioned. But this
reason suggests we should also EXCLUDE the non-isosceles

David Joyce, meanwhile, doesn't find kites to be necessary at all:

Too much is included in this chapter. The sections on rhombuses, trapezoids, and kites are not important and should be omitted.

Notice that both Conway and Joyce want to exclude trapezoids -- but Joyce is the only writer I know who wants to omit rhombuses. Of course, for Joyce, the emphasis should be on triangles and parallel lines, not quadrilaterals like rhombuses. Also, notice that kites are not specifically mentioned in the Common Core Geometry Standards. But I find that students can identify a kite more readily than a rhombus or trapezoid.

Now every kite contains two special vertices, known as its "ends." As defined by the U of Chicago:

"The common vertices of the equal sides of a kite are the ends of the kite."

Notice that Michael Serra doesn't define ends, but does give a name to the two angles whose vertices are the ends of the kite -- the vertex angles, in analogy with the vertex angle of an isosceles triangle. I see that from a proof standpoint, this makes sense, since the first thing that we do in the proof of our main theorem is divide the kite into two isosceles triangles.

And now here is our main theorem: the Kite Symmetry Theorem. As I mentioned back in the lesson on isosceles triangles, we use symmetry to determine the properties of kites. (A pre-Common Core proof might divide the kite into two triangles to be proved congruent by SSS.) I will post a proof here of the Kite Symmetry Theorem. It is taken directly from the U of Chicago, except that I, as always, add a Given step to the beginning of the proof. Since the U of Chicago's proof has eight steps, mine has nine:

Kite Symmetry Theorem:
The line containing the ends of a kite is a symmetry line for the kite.

Given: ABCD is a kite with ends B and D.
Prove: Line BD is a symmetry line for ABCD.

Statements                                           Reasons
1. ABCD is a kite with ends B and  1. Given
2. AB = BC, AD = DC                         2. Definition of ends of kite (meaning)
3. Tri. ABC and ADC are isosceles      3. Definition of isosceles triangle (sufficient)
4. Let m be the perp. bis. of AC          4. A segment has exactly one perp. bisector
5. A' = C, C' = A                                  5. Definition of reflection (sufficient)
6. m contains B and D                         6. The perp. bis. of the base = angle bis. of the vertex angle
                                                                 (so it contains the vertex)
7. B' = B, D' = D                                 7. Definition of reflection (sufficient)
8. ABCD reflected over m is CBAD     8. Figure Reflection Theorem
9. m (Line BD) is a symmetry line      9. Definition of symmetry line (sufficient)

Notice that more than half of the reasons in this proof are definitions. This underlines how important definitions are to the study of quadrilaterals.

As for the other theorems in this lesson, the Kite Diagonal Theorem follows directly from lines 4 and 6 of the above proof. It makes the symmetry diagonal the perpendicular bisector of the other diagonal and so the diagonals of a kite are perpendicular, and the symmetry diagonal bisects the other one.

Finally, we have the Rhombus Symmetry Theorem. It states that a rhombus has two symmetry lines, as both of the lines containing its diagonals are symmetry lines. This follows directly from the classification of a rhombus as a kite. In texts that define kite exclusively, the theorems "The diagonals of a kite are perpendicular" and "the diagonals of a rhombus are perpendicular" are two separate theorems, often in two separate sections. But here we can easily see why the diagonals of both the kite and the rhombus are perpendicular -- because the rhombus is a kite! Also, since we are defining kite inclusively, we don't need any extra steps in the proof to ensure that our kite isn't a rhombus.

This is our last section for now that doesn't require a Parallel Postulate. After all, a kite doesn't have parallel sides (unless it is a rhombus), and so the Parallel Postulate can't appear in the proofs.

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