Lesson 3-6: Constructing Perpendiculars (Day 36)

Today I subbed in a high school French class. It's in my old district, and so today is actually Day 36 there. (In my new district, today is Day 28.)

There's no need for "A Day in the Life" here. But as usual, I will say a little about the classes. There are three sections of French I (mostly freshmen, of course) and two sections of combined French III (juniors) and AP French (seniors). The freshmen are copying vocabulary words, while the older students have some work to do out of their respective texts.

I'm unable to resist singing "Nine, Nine, Nine" in the French language -- but only in the freshman classes. Even though the lesson isn't about French numbers, I've already set up that song in French.

I did consider converting another song to French. One song I was in the mood to sing was "No Drens." But that song has a huge problem -- how exactly do I translate it? "Dren," after all, is a word that I made up.

Well, I coined "dren" from the word "nerd." This isn't a word that most French classes will bother to teach, so I looked up Google Translate. It interpreted the word "nerd" as ballot. (It's strange that it looks just like the English word for "something to cast votes on.") Anyway, to be consistent, just as I reversed "nerd" to "dren," we reverse ballot to "tollab." To make it sound French, the final b must be silent, so it would sound like "toe-la."

But this assignment causes problems when I try to establish a good/bad list or determine whether the song incentive has been earned. The assignment is to copy each of the 37 words in French, then in English, then in French five more times. But some students wrote all 37 words in French, then all 37 in English, etc., while others did one word at a time (French, English, French * 5, next word). Perhaps what I should have said was "fill in any 100 boxes to avoid the bad list, 200 boxes for good list), since there were 37 * 7 = 259 boxes in all.

By the way, I'm glad I sang a math song, since chanter (to sing) and les mathematiques (math) are two of the vocab words. Another word the students struggle with is manger (to eat) -- they wanted to write "manager" instead.

Today on her Mathematics Calendar 2019, Theoni Pappas wrote:

If one circle's radius is three times as long as another's, then how many times greater is the larger circle's circumference?

Well, the Fundamental Theorem of Similarity (Lesson 12-6 of the U of Chicago text) tells us that similarity transformations increase all linear dimensions proportionally. If you need to, you can write it out:

R = 3r

2pi R = 3(2pi r) (multiplying both sides by 2pi)

The area would have increased by a factor of 3^2 or nine. But the desired circumference increases only by a factor of three -- and of course, today's date is the third. (It's nice to have an error-free Pappas question after what happened yesterday.)

Lesson 3-6 of the U of Chicago text is "Constructing Perpendiculars." (It appears as Lesson 3-9 in the modern edition of the text.)

This is what I wrote last year on today's lesson:

Section 3-6 of the U of Chicago text deals with constructions. So, we're finally here. The students will need a straightedge and compass to complete this lesson.

Here's a good point to ask ourselves, which constructions do we want to include here? The text itself focuses on the constructions involving perpendicular lines. Well, let's check the Common Core Standards, our ultimate source for what to include:

CCSS.MATH.CONTENT.HSG.CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

CCSS.MATH.CONTENT.HSG.CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

But let's also go back to what David Joyce writes about constructions:

The book [Prentice-Hall 1998 -- dw] does not properly treat constructions. Constructions can be either postulates or theorems, depending on whether they're assumed or proved. For instance, postulate 1-1 above is actually a construction. On pages 40 through 42 four constructions are given: 1) to cut a line segment equal to a given line segment, 2) to construct an angle equal to a given angle, 3) to construct a perpendicular bisector of a line segment, and 4) to bisect an angle. Later in the book, these constructions are used to prove theorems, yet they are not proved here, nor are they proved later in the book. There is no indication whether they are to be taken as postulates (they should not, since they can be proved), or as theorems. At the very least, it should be stated that they are theorems which will be proved later.

David Joyce, after all, emphasizes that at the very least, constructions should be proved. He writes here that they can be proved later -- but of course, he prefers that theorems not be stated until they can be proved.

So which of the theorems in the Common Core list can be proved so far? Let's look back at that list one by one:

Copying a segment: This should be trivial to construct and prove. The student simply uses the straightedge to draw a line, marks a point O on it, and opens up the compass to the length of the given line segment AB to mark the second point P. The proof that these segments AB and OP have the same length simply follows from the definition of straightedge and compass. It's a bit surprising that the U of Chicago text doesn't begin with this as the first construction, as this one should be easy for the students.

Copying an angle: This is the one that we can't prove yet. The usual construction requires SSS to prove. This is one reason why Joyce would prefer that Chapter 8 of his text occur before these constructions in his Chapter 1.

Bisecting a segment; constructing perpendicular lines, including the perpendicular bisector of a line segment: This is the focus of Section 3-6 of the U of Chicago text. Notice that as soon as we've constructed the perpendicular bisector, we've already done the other two constructions (bisecting the segment and drawing its perpendicular). And so, as soon as we prove the perpendicular bisector construction, we are done.

Given: Circle A contains B, Circle B contains A, Circles A and B intersect at C and D.

Prove: Line CD is the perpendicular bisector of AB

Proof (in paragraph form -- can be converted to two columns later):

Just as in the proof of Euclid's first theorem (Section 4-4), since both B and C lie on circle A, AB = AC by the definition of circle, and since both A and C lie on circle B, AB = BC. Then by the Transitive Property of Equality, AC = BC -- that is, C is equidistant from A and B. So, by the Converse of the Perpendicular Bisector Theorem, C lies on the perpendicular bisector of AB. In the same way, we can prove that D also lies on the perpendicular bisector of AB. And through the two points C and D there is exactly one line -- and that line is the perpendicular bisector of AB. QED

In many texts, it's pointed out that the compass opening for the two circles need not be exactly the same as AB. All that's necessary is for the opening to be greater than half of AB -- that guarantees that the two circles intersect in two points. 

The text states that the midpoint of AB has been constructed as a "bonus" -- so we've bisected the segment, as requested. All we need now is to construct perpendicular lines -- and that's exactly what the text does in the next example, construct a perpendicular to line AP through point P on the line, using our perpendicular bisector algorithm as a subroutine.