Wednesday, September 24, 2014

Activity: Tic-Tac-Toe (Day 34)

Here are a few more notes on constructions. First of all, we notice that the text writes three rules for using a straightedge and compass. I want to draw attention to the Compass rule:

Compass rule:
A compass can draw a circle with center at a point A and containing a second point B.
Also, a compass can be lifted keeping the same radius.
[Emphasis the U of Chicago's]

Notice the line that the text italicizes. Also, a compass can be lifted keeping the same radius. The point is that back in the days of Euclid, that part was not included! Apparently, for Euclid, a compass was collapsible, so that once lifted, it can't be used to transfer distances. It can only be used to draw a circle with known points at the center and on the circle (that is, whose center and point on the circle adhere to the Point rule).

Interestingly enough, most of the constructions in the text already adhere to this stricter rule. Euclid's Proposition I, of course, adheres to the Collapsible Compass rule. So do the perpendicular bisector and other related perpendicular constructions. My construction of a parallel to a line l through a point  P not on the line can converted to one with a collapsible compass, as follows:

Step 1. Circle P intersects l in points A and B
Step 2. Subroutine: m, the perpendicular bisector of AB
Step 3: Subroutine: k, the perpendicular to m through point P on m

This works because the radius of circle P can be any length, provided it is sufficiently large enough to intersect l in two points, so we aren't transferring distance. As usual, by definition of circle, PA = PB, so P lies on the perpendicular bisector m of AB. The subroutine in step 3 is already partly done since we already have a circle P intersecting m in two points -- call them C and D. Then step 3 merely entails constructing the perpendicular bisector of CD.

But the constructions that do entail transferring distance are the copy constructions. Indeed, the very simple construction for copying a segment requires transferring distance. Yet there is a way to copy a segment with a collapsible compass. We already saw Euclid's First Theorem -- now let's take a look at Euclid's Second Theorem:

This construction is quite complex. We don't expect high school students to master these collapsible compass constructions -- especially since they can use compasses that aren't so restricted. The italicized part of the Compass rule tells us that our compasses are not of the collapsible type -- yet interestingly enough, the constructions in the U of Chicago text are collapsible.

David Joyce tells us that he expects proofs for the simple constructions -- which presumably includes the segment (and angle) copy construction. I wonder whether he's expecting a proof similar to Euclid's Proposition II -- especially since the construction where the compass is not of the collapsible type is so trivial to prove.

Now for today's activity -- I want to have the students construct something that involves parallel and perpendicular lines. So let's construct a tic-tac-toe board. Actually, this symbol has more names than any other symbol that I can think of: number sign, pound sign, sharp sign, octothorpe, and hashtag.

Here's an algorithm that can construct this symbol. The four points A, B, C, and D will be the four vertices of the center square:

Step 1: Circle A containing B
Step 2: Circle B containing A
Step 3: Line AB, intersecting Circles A and B at points X and Y respectively
Step 4: Subroutine: k, the perpendicular bisector of XB
Step 5: Subroutine: l, the perpendicular bisector of AY
Step 6: Lines k and l intersect Circles A and B at points C and D respectively
Step 7: Line CD

Of course, students might come up with other ways to construct the symbol of many names. Once again, there's no need for me to post the worksheet.

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