"Using your visualization skills and folding techniques, determine a way to make one straight cut to separate the checkerboard into 2 * 2 squares, such as this one."
This is the second page of the subsection "Checkerboard Mania." Once again, the first page of this section was blocked by the weekend.
But we don't need to see the first page -- nor, for that matter, the pictures on this page -- in order to understand the problem. The checkerboard is a standard 8 * 8, and we are being asked to fold it so that a single cut will divide it into sixteen 2 * 2 squares. Pappas describes it as "taking a checkerboard apart with one fell swoop."
As usual, I'll post the solution tomorrow. This problem isn't easy at all -- the folds and the cut to make is extremely clever.
Lesson 5-2 of the U of Chicago text is called "Types of Quadrilaterals." In the modern Third Edition of the text, quadrilaterals appear in Lesson 6-4.
This is what I wrote two years ago about today's lesson:
Lesson 5-2 of the U of Chicago text covers the various types of quadrilaterals. There are no theorems in this section, but just definitions. The concept of definition is important to the study of geometry, and in no lesson so far are definitions more prominent than in this lesson.
The lesson begins by defining parallelogram, rhombus, rectangle, and square. There's nothing wrong with any of those definitions. But then we reach a controversial definition -- that of trapezoid:
Definition:
A quadrilateral is a trapezoid if and only if it has at least one pair of parallel sides.
(emphasis mine)
Just as with the definition of parallel back in Lesson 1-7, we have two extra words that distinguish this from a traditional definition of trapezoid -- "at least." In other textbooks, no parallelogram is a trapezoid, but in the U of Chicago text, every parallelogram is a trapezoid!
To understand what's going on here, let's go back to the first geometer who defined some of the terms in the quadrilateral hierarchy -- of course, I'm talking about Euclid:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI22.html
Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
Of course, the modern term for "oblong" is rectangle, and a "rhomboid" is now a parallelogram. The word "trapezia" is actually plural of "trapezium." In British English, a "trapezium" is what we Americans would call a trapezoid, but to Euclid, any quadrilateral that is not a parallelogram (or below on the quadrilateral hierarchy) is a "trapezium." But the important part here is that to Euclid, a square, for example, is neither a rectangle (oblong) nor a rhombus. He makes sure to say that a rectangle (oblong) is "not equilateral," and that a rhombus is "not right-angled." And of course, neither a rectangle nor a rhombus is a parallelogram (rhomboid).
These are called "exclusive" definitions. For Euclid, there was no quadrilateral hierarchy -- each class of quadrilaterals was disjoint from the others. But since the days of Euclid, more and more geometry texts have slowly added more "inclusive" definitions.
One of the first inclusive definitions I've seen was the definition of rectangle. It was mentioned on an episode of Square One TV, when a Pacman parody character named Mathman was supposed to eat rectangles, and then ate a square because "every square's a rectangle." I would provide a YouTube link, but I haven't found the link in years and it doesn't come up in a search. (I even remember someone posting in the comments that just as for me, his first encounter of the inclusive definition of rectangle was through watching that clip when it first aired so many years ago!)
But many of my family members were also teachers, and one relative gave me an old textbook that still mentioned some exclusive definitions. In particular, it declared that a square isn't a rhombus. A little later, my fifth grade teacher then taught the inclusive definition of rhombus. I then blurted out that a square isn't a rhombus, then actually brought the old text to school to prove it! She replied, "Wow!" but then, if I recalled correctly, told me that this definition was old, and that by the new definition, a square is a rhombus. And so all modern texts classify the square as both a rectangle and a rhombus, and that all of these are considered parallelograms.
So we see that there is a tendency for definitions to grow more inclusive as time goes on. (We see this happening in politics as well -- for example, the definition of marriage. But I digress.) And so we see the next natural step is for the parallelogram to be considered a trapezoid.
One of the first advocates I saw for an inclusive definition of trapezoid is the famous Princeton mathematician John H. Conway. He is best known for inventing the mathematical Game of Life, which has its own website:
http://www.conwaylife.com/
But Conway also specializes in other fields of mathematics, such as geometry and group theory (which is, in some ways, the study of symmetry). Twelve years ago, he posted the following information about why he prefers inclusive definitions:
http://mathforum.org/kb/message.jspa?messageID=1081135
The preference for exclusive definitions arises, I think, from
what I call "the descriptive use". Of course, one wouldn't DESCRIBE
a square table as "rectangular", since that would wantonly use
a longer term to convey less information. So in descriptive uses,
there's a natural presumption that a table called "rectangular"
won't in fact be square - in other words, a natural presumption
that the terms will be used exclusively.
But the descriptive use is unimportant to geometry, where the
really important thing is the truth of theorems. This means that we
should use a term "A" to include "B" if all the identities that
hold for all "A"s will also hold for all "B"s (in the way that
the trapezoid area theorem holds for all parallelograms, for
instance).
You might worry about the consistency of switching to the
inclusive use while other people continue with the exclusive one.
But there can be no consistency with people who are inconsistent!
I've seen many geometry books that MAKE the exclusive definitions,
but none that manage to USE them consistently for more than a few
theorems.
Indeed, Conway advocated taking it one step forward and actually abolishing the trapezoid and having only the isosceles trapezoid in the hierarchy! After all, there's not much one can say about a trapezoid that's not isosceles -- just look ahead to Lesson 5-5. There's only one theorem listed there about general trapezoids -- the Trapezoid Angle Theorem, and that's really just the Same-Side Interior Angle Consequence Theorem that can be proved without reference to trapezoids at all. All the other theorems in the lesson refer to isosceles trapezoids. In particular, the symmetry theorems in the lesson refer to isosceles trapezoids. (Recall that Conway's specializes in group theory -- which as I wrote above is the study of symmetry.) I suspect that the only reason that we have general trapezoids is that they are the simplest quadrilateral for which an area formula can be given.
This is now another digression from Common Core Geometry, so I'll just provide another link. Notice that here, Conway also proposes a hexagon hierarchy based on symmetry. There's also a pentagon hierarchy, but there are only three types of pentagons -- general, line-symmetric, and regular -- just as there are for triangles. It's easier to make figures with an even number of sides symmetric.
http://mathforum.org/kb/message.jspa?messageID=1074038
Another advocate of inclusive definitions is Mr. Chase, a Maryland high school math teacher. (And no, in yesterday's post and today's I'm referring to two differentMaryland teachers.) I see that he is so passionate about the inclusive definition of trapezoid that he devoted three whole blog posts to why he hates the exclusive definition of trapezoid:
http://mrchasemath.wordpress.com/2011/02/03/why-i-hate-the-definition-of-trapezoids/
http://mrchasemath.wordpress.com/2011/02/18/why-i-hate-the-definition-of-trapezoids-again/
http://mrchasemath.wordpress.com/2013/08/12/why-i-hate-the-definition-of-trapezoids-part-3/
One reason Chase states for using inclusive definitions is that it simplifies proofs:
When proving that a quadrilateral is a trapezoid, one can stop after proving just two sides are parallel. But with the exclusive definition, in order to prove that a quadrilateral is a trapezoid, you would have to prove two sides are parallel AND the other two sides are not parallel.
Regarding some of the others to whom I refer regularly, Dr. Wu uses the inclusive definition:
"A quadrilateral with at least one pair of opposite sides that are parallel is called a trapezoid. A trapezoid with two pairs of parallel opposite sides is called a parallelogram."
while Dr. Mason uses the exclusive definition:
"A trapezoid is by definition a quadrilateral with precisely one pair of parallel sides."
(emphasis Dr. M's)
So which definition should I use for trapezoid? Well, this is a Common Core blog, so the definition favored by Common Core should have priority over all other definitions. The following is a link to the information that will appear on the PARCC End-of-Year Assessment for geometry:
http://www.parcconline.org/sites/parcc/files/ES%20Table%20Geometry%20EOY%20for%20PARCC_Final.pdf
And right there in the column under "Clarifications," it reads:
i) A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”
And that plainly settles it. The PARCC Common Core assessment uses the inclusive definition of trapezoid, and so it's my duty on a Common Core blog to use the Common Core definition. Of course, we notice that this is the definition given by PARCC -- but so far I've seen no information on what definition Smarter Balanced is using. It would be tragic if PARCC were to use one definition and Smarter Balanced the other. But as I can't say anything about Smarter Balanced, I will use the only definition that's known to be on a Common Core test, and that's the inclusive definition. The fact that this definition is already used by the U of Chicago is icing on the cake.
There is one problem with the inclusive definition of trapezoid, and that's when we try to define isosceles trapezoid. The word isosceles suggests that, just as in an isosceles triangle, an isosceles trapezoid has two equal sides -- the sides adjacent to the (parallel) bases. But in a parallelogram, where either pair of opposite sides can be considered the bases -- the sides adjacent to these bases are also equal. This would make every parallelogram an isosceles trapezoid. But this isn't desirable -- an isosceles trapezoid has several properties that parallelograms in general lack. The diagonals of an isosceles trapezoid are equal, but those of a parallelogram in general aren't. But the diagonals of a rectangle are equal. So we'd like to consider rectangles, but not parallelograms in general, to be isosceles trapezoids.
This dilemma is mentioned in the comments at one of the Chase links. It's pointed out that there are two ways out of this mess -- we may either define isosceles trapezoid in terms of symmetry, as Conway does, or we can use the U of Chicago's definition:
"A trapezoid is isosceles if and only if it has a pair of base angles equal in measure."
Some don't like this definition, because it violates linguistic purity -- the word isosceles comes from Greek, and it means "equal legs," not "equal angles." But as it turns out, it's a small price to pay to make the quadrilateral hierarchy and other theorems work out. And besides, any geometer who calls a nine-sided polygon a nonagon should just shut up about linguistic purity!
By the way, Mr. Chase has returned to posting in 2017, at least in January through April. His most recent post is about the biennial National Math Festival, held in Washington, DC.
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