In today's eighth grade math classes, after having completed Lesson 2-5 yesterday, the class suddenly jumped back to Lesson 2-1, on Solving Equations with Rational Coefficients. Well, I guess I can't blame the teacher for jumping around the lessons in Chapter 2 since, after all, I jump around entire chapters in the U of Chicago text on the blog. As usual with fraction problems, the questions the students struggled with the most were the ones required both simplification (reducing) and conversion between mixed and improper fractions.
The Math/Computing class was still working on Graphing Using Intercepts. The teacher decided that even though they were using the intercepts to graph, she'd have the students find the slopes of their lines anyway. This confused the students -- for example, the first graph was x + y = 5. They had already seen slope-intercept form (from Lesson 3-4) and so thought that since the coefficient of x was 1, the slope of the line x + y = 5 had to be 1. I tried to show them that since they had already graphed the line using intercepts, they could see visually that the slope was clearly "nice and negative" -- after all, Sarah Hagan's "Slope Dude" was still posted on the wall, and I believe this might have helped some of the students figure out why the slope had to be -1. (Thanks again, Ms. Hagan!)
In case you are curious, here's all of both chapters in detail:
2-1. Solve Equations with Rational Coefficients
2-2. Solve Two-Step Equations
2-3. Write Two-Step Equations
2-4. Solve Equations with Variables on Each Side
2-5. Solve Multi-Step Equations
3-1. Constant Rate of Change
3-3. Equations in y = mx Form
3-4. Slope-Intercept Form
3-5. Graph a Line Using Intercepts
3-6. Write Linear Equations
3-7. Solve Systems of Equations by Graphing
3-8. Solve Systems of Equations Algebraically
These last few weeks, I've made so many changes to my worksheets that I haven't been quoting my old posts much lately. But today I'm going back to quoting my post from last year on today's lesson -- one, because I'm not making that many changes to the old worksheet, and two, because it is still my most popular blog post. It has received more views that any other entry on this blog. And so I quote it, but with one small change that I'm making to one of the definitions:
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:
A quadrilateral is a trapezoid if and only if it has at least one pair of parallel sides.
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:
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:
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:
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
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
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.
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 different Maryland 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:
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:
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!
But there is a similar intersection problem that's too advanced to be given here, and where the answer differs depending on what geometry one is using. The intersection of the set of all parallelograms and the set of all isosceles trapezoids is, in Euclidean geometry, the set of all rectangles. (One way to prove this is to note that isosceles trapezoids have equal diagonals, and -- though this isn't proved in the U of Chicago -- parallelograms with equal diagonals are rectangles.) Yet in hyperbolic geometry, there are no rectangles, but there exists figures that are both parallelograms and isosceles trapezoids -- in particular, the Saccheri quadrilateral is both. A hyperbolic geometer may miss the fact that a Saccheri quadrilateral is an isosceles trapezoid because he/she is using the exclusive definition, where a parallelogram can't be a trapezoid. But in some ways, a Saccheri quadrilateral is more like an Euclidean isosceles trapezoid than a Euclidean parallelogram, since the Saccheri and the isosceles trapezoid share the same type of symmetry line that the general parallelogram lacks.
OK, so last year, I wrote about Saccheri quadrilaterals as a sort of curiosity -- we are teaching Euclidean geometry, where "Saccheri quadrilateral" simply means "rectangle."
But this year, I'm making yet another change to my curriculum -- we are now going to learn about the properties of translations, which we'll then use to derive the Parallel Tests and Consequences. And we found out that unless we're careful, the proofs here will end up being circular.
I mentioned last spring that the way out is to use Saccheri quadrilaterals. We use the properties of Saccheri quadrilaterals to development translations, and then we can use translations to derive the theorems for parallels.
Now here's the thing -- it's terrible pedagogy to start using and defining the phrase "Saccheri quadrilateral" in a high school Geometry class. Why, after all, should we start talking about "Saccheri quadrilaterals," only to prove eventually that "Saccheri quadrilateral" just means "rectangle" -- especially for the sole purpose of making a proof non-circular?
Since we want our proofs to remain neutral as long as possible, we can't just say "rectangle" instead of Saccheri quadrilateral. But if we use the inclusive definition of trapezoid, then we can prove in neutral geometry that all Saccheri quadrilaterals are isosceles trapezoids. And so "isosceles trapezoid," not "rectangle," can be our replacement phrase for "Saccheri quadrilateral."
This is why we must be so careful with how exactly we define "isosceles trapezoid." We know that the natural definition -- an isosceles trapezoid is just a trapezoid that is isosceles (that is, with equal legs) -- is inconsistent with the inclusive trapezoid because it would then make every parallelogram an isosceles trapezoid. The U of Chicago text sacrifices the word "isosceles," instead defining it as a trapezoid with equal base angles.
But there's another choice -- we can sacrifice the word "trapezoid" instead. That is, we define an isosceles trapezoid as a quadrilateral with equal legs and equal base angles. As it turns out, this definition is very convenient, because now a Saccheri quadrilateral is an isosceles trapezoid almost by definition -- it's simply an isosceles trapezoid whose (equal) base angles happen to be right angles.
And just as with the U of Chicago definition, where we can ultimately prove that an isosceles trapezoid really is isosceles (that is, with equal legs), with our definition, we can ultimately prove that an isosceles trapezoid really is a trapezoid (that is, with parallel bases). In fact, the key to our development of translations will be the proof that an isosceles trapezoid is a trapezoid.
So the only change we need to make to last year's lesson is the definition of isosceles trapezoid. But the worksheet asks students to fill in the definition, and so I don't need to change it at all.
By the way, Mr. Chase is no longer an active blogger as his blog has no 2015 posts. (The last entry is dated Christmas Eve 2014.)