A Preview: Images

Let's include a few images of the theorems mentioned in the previous post. But before I do so, let me make a few comments to clarify some of what I wrote in my last post.

I wanted to rewrite the proof that if a line is parallel to its reflecting line (its mirror), then its parallel to its image as well, so that it is a direct proof. Some may wonder whether this Line Parallel to Mirror Theorem is even worth giving in a geometry class. Going back to the Common Core Standards:

CCSS.MATH.CONTENT.HSG.SRT.A.1.A
A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

So Common Core asks students to know the conditions when a dilation maps a line to a parallel line. Also, Wu proves the conditions when a rotation (of 180 degrees) maps a line to a parallel line. And now I give the conditions when a reflection maps a line to a parallel line. So we see that it's indeed proper to give the Line Parallel to Mirror Theorem.

And in all three cases, the proofs are similar. Taking our cue from Wu, we first find the conditions under which the transformation maps a line to itself. Such a line, by the way, is often called an invariant line of that transformation. Then, to prove that a given line, which we'll call L following Wu, under the right conditions is parallel to its image L', we first choose a point P on L'. Then there must be a point Q on L such that Q' is P, and Q and P lie on an invariant line l of the transformation. Then by the Line Intersection Theorem, L and l intersect at exactly one point -- and that point is Q, not P. Since P is already on l, it can't lie on L. But P is any arbitrary point on L'. So L and L' can't have any points in common -- they are parallel.

Now here are the images:

Notice that this is still a work in progress. I'm not sure whether all of these would be appropriate for a high school class, based on the order in which I will present the material. In particular, the Perpendicular to Parallels Theorem is clearly a generalization of the Corresponding Angles Theorem, the difference being that the latter requires rotations while the former requires only reflections -- and I plan on presenting reflections before rotations. But I also plan on presenting rotations before Playfair's Postulate, so by the time we get to a Playfair proof, we might as well use the rotation version.

This ends the preview. I've decided that I will officially cover the material over the course of an academic year, the same way it would be done in the classroom.

But lately I've noticed that there are two school years commonly followed in this country. The traditional calendar has school start after Labor Day in September. This results in the first semester ending around late January or so. The early-start calendar starts in August, so that an entire semester can be covered before Christmas -- therefore avoiding having winter break separate the finals from most of the semester.

So far, I'll follow an early-start calendar. So I'll begin with Chapter 1 in August at some point.

A Preview: Perpendicular and Parallel Lines

In this post I want to preview specific examples of the geometry that I will provide. As I mention earlier, it will be based on my U of Chicago text, rearranged to satisfy Common Core.

Here is a key example: in Chapter 3, the U of Chicago text covers angles, proofs, and then parallel and perpendicular lines. Let's assume that by this point, we've already covered angles (which I repeat are covered awfully late in the text) and we're ready for perpendicular lines. In the text, Section 3-5 begins by defining perpendicular:

U of Chicago:
"Two segments, rays, or lines are called perpendicular if and only if the lines containing them form a 90-degree angle."

The text then proceeds to prove two theorems relating perpendicular and parallel lines:

Two Perpendiculars Theorem:
"If two coplanar lines m and n are each perpendicular to the same line l, then they are parallel to each other."

Perpendicular to Parallels Theorem:
"In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other."

The text gives an informal proof of these two theorems as being special cases of the Corresponding Angles Postulate and its converse (itself a postulate in this text) in which the corresponding angles happen to be right angles. But this is unacceptable for our purposes. We wish to have only one postulate for parallel lines, the Parallel Postulate of Playfair, and as it turns out, one direction can be proved without any Parallel Postulate. It turns out that the Two Perpendiculars Theorem is provable without Playfair, while the Perpendicular to Parallels Theorem requires that postulate.

Our proof will be very simple. It will depend on reflections -- but once again, Common Core Geometry is all about reflections (and rotations, etc.). This is why I am rearranging the text order so that reflections are covered first. Let's recall our definition of reflection:

"For a point P not on a line m, the reflection image of P over line m is the point Q if and only if m is the perpendicular bisector of segment PQ. For a point P on m, the reflection image of P over line m is P itself."

Of course, we'll have to define "perpendicular" (and "perpendicular bisector") before we can define reflection, but we won't prove any theorems about perpendicular lines until after we've discussed the properties of reflections.

Notice that Section 3-6 covers the construction of perpendicular lines. We refer back to the Common Core Standards:

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."

And we see "constructing perpendicular lines" specifically mentioned in the standard. So we definitely want to do exactly that right here on this blog. The problem is that many classrooms lack the materials mentioned on that list -- especially a compass. Teachers often omit the sections of the text requiring straightedge-and-compass constructions.

The text gives the following steps to construct the perpendicular bisector of a segment AB:

Step 1: Circle A containing B (Compass rule)
Step 2: Circle B containing A (Compass rule)
Step 3: Circles A and B intersect at new points C and D (Point rule)
Step 4: Line CD (Straightedge rule)

Notice that while many texts merely require that the compass setting be any length more than half of AB, this text has the setting be exactly the length of AB.

The text states that we cannot prove that this algorithm works -- at least not yet. Notice that this is exactly what Joyce criticizes about Chapter 1 of the Prentice-Hall text. He prefers that no construction be given without a proof that it works. Well, here's a proof that this works:

We observe that this construction (two circles centered at A and B, passing through the other point, and intersecting at C and D) is exactly the proof given in Section 4-4 -- which is stated to be the first theorem in Euclid's Elements. And, as Proposition 1 in Euclid tells us, this makes triangle ABC (as well as ABD) equilateral. Now, as equilateral triangles are isosceles, a theorem in Section 5-1 tells us that the perpendicular bisector of the base of both triangles (AB) are also their medians, so that it passes through the other vertices (C and D). QED

Now the proofs in Section 5-1 about isosceles triangles are proved using reflections. So it helps to have Section 4-1 and its definition of reflection. In that section there is also a construction on how to find the reflection image of a point over a line -- but once again, the proof that it works depends on the isosceles triangles of Section 5-1.

Notice what's going on here. We need to jump all the way to section 5-1 just to be able to prove that the constructions in Chapters 3-4 work -- and that's only to satisfy Joyce's preference that proofs accompany constructions. And those constructions are the ones which require straightedge and compass -- the compass that many students won't have in the first place.

But there's a way out of this. Perhaps we can generate reflection images without either constructing them or proving that the constructions work. Let's look at that Common Core Standard again:

"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 ..."

Ah ha! We can just use paper folding! So to reflect a point P about a line l, we simply fold the paper over line l, and the point where P lands is the image of P, which we'll call P' (P prime). And to find the perpendicular bisector of AB, we just fold the paper so that point A lands on top of point B, and the crease line is the perpendicular bisector. We don't actually prove any of this -- all of this is just to get the students to generate reflections and perpendicular bisectors without being bogged down with proofs that they work, or even needing compasses to construct them.

Once we have this, we can now create some simple proofs. All proofs involving reflections ultimately go back to the Reflection Postulate, which we'll now state in full:

Reflection Postulate (U of Chicago):
Under a reflection:
a. There is a 1-1 correspondence between points and their images. This means that each preimage has exactly one image and each image comes from exactly one preimage.
b. If three points are collinear, then their images are collinear. Reflections preserve collinearity. The image of a line is a line.
c. If B is between A and C, then the image of B is between the images of A and C. Reflections preserve betweenness. The image of a line segment is a line segment.
d. The distance between two preimages equals the distance between their images. Reflections preserve distance.
e. The image of an angle is an angle of the same measure. Reflections preserve angle measure.

Wu gives many of these same properties, except for rotations rather than reflections. The U of Chicago uses the mnemonic A-B-C-D: Reflections preserve Angle measure, Betweenness, Collinearity, and Distance.

Using this, let's prove a theorem. Section 3-6 gives a construction for the following, but we shall prove this using not the construction, but using reflections:

Theorem:
Through a point P not on a line l, there is exactly one line through P perpendicular to l. (Notice that this says perpendicular, not parallel.)

Proof:
We are to prove this using reflections, but how? Well, we have a point P and a line l, so why don't we try reflecting the point P over the line l? Then there exists a unique point P' (by part a of the Reflection Postulate) such that l is the perpendicular bisector of PP'. Then voila! Line PP' is exactly the line that we seek -- a line through P perpendicular to l. Its existence and uniqueness are guaranteed because through two points (P and P') there is exactly one line. QED

And believe it or not, this gives us a quick and dirty proof of the Two Perpendiculars Theorem!

Two Perpendiculars Theorem:
If two coplanar lines m and n are each perpendicular to the same line l, then they are parallel to each other.

Proof:
Suppose that m and n weren't parallel -- that is, suppose that they intersect at some point P. Then through P there are two lines perpendicular to the line l (namely m and n) when by the previous theorem, there's only supposed to be one such line, a contradiction. Therefore the lines m and n must be parallel. QED

The only problem with this proof is that it's an indirect proof, which many books seek to avoid until much later -- the U of Chicago text doesn't cover indirect proofs until Chapter 13. I notice that the U of Chicago text gives an interesting definition of parallel:

Definition:
"Two coplanar lines m and n are parallel lines if and only if they have no points in common, or they are identical" (emphasis mine).

So this definition actually allows a line to be parallel to itself! Although not common in most texts, this alternate definition allows the above indirect proof to become a direct proof:

Proof:
We have that either m and n intersect at some point P, or they don't intersect. In the latter case, we are already done. In the former case, we have both m and n as lines through P perpendicular to l, and by the previous theorem, there is only one such line. So m and n are the same line. Therefore m and n have either no point in common, or every point in common -- and by our definition, both of these implies that m and n are parallel. QED

For completion, let me give the proof of the Perpendicular to Parallels Theorem. As we noted earlier, this proof will require Playfair's Parallel Postulate. It also contains an indirect part:

Perpendicular to Parallels Theorem:
In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Proof:
Let's say we're given lines k, l, and m, with k parallel to m and l perpendicular to m. Also, the point where l and m intersect, let's call it A, and the intersection of k and l we'll call B. So our goal is to prove that k and l are perpendicular.

Let n be the perpendicular bisector of segment AB. So both m and n are perpendicular to l, and so by the Two Perpendiculars Theorem, m and n are parallel. Now we want to reflect m about line n, to produce a new line, m'. But where is m'? We already know where the image of one point on m is -- namely A, of course. The image of A is exactly B since n is the perpendicular bisector of segment AB (definition of reflection). Since A lies on m, its reflection image B lies on m'. This also implies that the reflection image of l is exactly l itself, since A and B lie on l.

Now can m and m' have any points in common? Suppose they have a point C in common -- that is, C lies on both m and m'. Since C lies on m, its reflection image C'  lies on m', and since C' also lies on m, its own reflection image must lie on m'. But the reflection image of C' is just C itself -- the U of Chicago text calls the Flip-Flop Theorem (since n is the perpendicular bisector of segment CC' and trivially C'C as well). So there are two lines passing through both C and C' (namely m and m') when there is only supposed to be one such line, a contradiction. So m and m' have no points in common -- that is, they are parallel.

Recall that B lies on m'. So m' is a line parallel to m passing through B. By Playfair, there is exactly one such line -- and we already know of such a line, namely k. So the reflection image of m over line n is exactly k. By the Reflection Postulate part e, reflections preserve angles, and since l and m are perpendicular (a 90-degree angle), their reflection images l and k meet at the same angle, so they must be perpendicular. QED

This proof is admittedly long. It may be shortened if we've already proved the following Lemma:

Lemma:
If a line is perpendicular to its reflecting line, its image is identical to itself.
If a line is parallel to its reflecting line, its image is parallel to itself.

Notice that Woo proved a similar lemma for 180-degree rotations -- a line passing through the center of rotation is mapped to itself, and a line not passing to the center is mapped to a line parallel to itself -- but he didn't prove this lemma for reflections. I'd say that the Lemma is worth proving, but it still requires an indirect proof unless we use the trick of defining every line as parallel to itself.

Source 4: Jennifer Silverman

My final source, for now, is Jennifer Silverman, an independent math consultant:

http://www.jensilvermath.com/2013/07/17/a-new-way-to-build-a-geometric-system/

On her page, Silverman states seven steps to building a geometric system based on the Common Core Standards. At first, we notice that Silverman's Step 4 is clearly based on rotations, and so this is not much different from Wu's presentation. But Steps 5-6 jump out at me:

5. Lines equidistant at 2 places are equidistant everywhere.
6. Equidistant lines are defined as parallel.

This is an interesting definition of parallel. In fact, I once saw another website (which I've now lost and may no longer exist) which showed an animated image of two parallel lines cut by a transversal and a translation, along the direction of the transversal, which made corresponding angles coincide (as opposed to the rotation making alternate interior angles coincide). Even though Silverman's webpage is based on rotations, I wondered whether there could be another geometric system based on parallel lines, corresponding angles, and translations, using her Steps 5-7.

But there's a problem with Silverman's definition of "parallel." To see why, we must go all the way back to Euclid.

Anyone remotely familiar with Euclid is aware of his Fifth Postulate, the one on parallels. It is independent of the other four postulates -- that is, no matter how hard we try, we can't prove it as a theorem (unlike many of the other so-called "postulates" in modern texts). We now know that there are several non-Euclidean geometries. In particular, we have:

Hyperbolic Geometry: many parallels to a given line through a point not on that line
Euclidean Geometry: exactly one parallel to a given line through a point not on that line
Elliptic Geometry: no parallel to a given line through a point not on that line

Now, as it turns out, some of the results involving parallel lines require the Parallel Postulate to prove, while others don't. In particular, the statement that if two lines are parallel, then alternate interior (or corresponding) angles are congruent require the Parallel Postulate, but the converse does not require it. We see this with Euclid himself, where Propositions 27-28 don't invoke his Fifth Postulate, while Proposition 29 does require the Fifth.

Most modern texts don't distinguish between the former and the latter, but authors like Dr. M and Wu do make the distinction. Mathematicians prefer to use as few postulates as possible when proving theorems -- this is considered more elegant. Indeed, Euclid himself does this -- he proves his first 28 Propositions using only the first four Postulates, then only beginning with Proposition 29 does he invoke the Fifth Postulate. So I should try to prove "if corresponding angles are congruent, then the lines are parallel" without using the Parallel Postulate.

Now here's the problem: the statement "lines equidistant at 2 places are equidistant everywhere" (Silverman's Step 5) requires the Parallel Postulate. It's only true in Euclidean geometry -- indeed, only in Euclidean geometry do there even exist lines that are equidistant everywhere. And so Silverman's Step 6, "equidistant lines are defined as parallel," is not equivalent to the standard definition of parallel unless one assumes the Parallel Postulate. As it turns out, this isn't a problem in elliptic geometry, where there are neither parallel nor equidistant lines (so technically speaking, lines are parallel iff they are equidistant is vacuously true). The problem is in hyperbolic geometry, where there are infinitely many lines parallel to a given line through a point not on the line, but none of those lines are everywhere equidistant from the given line.

One way to avoid this is to combine Steps 5 and 6: "lines equidistant at 2 places are parallel" (where "parallel" means the standard definition -- using this as a new definition remains invalid). As it turns out, this is provable without the Parallel Postulate. Its proof involves a shape called a "Saccheri quadrilateral" -- a figure with opposite sides parallel (so it's a parallelogram) and two right angles, yet isn't necessarily a rectangle (unless one adds the Parallel Postulate back).

Indeed, let's return to translations -- what I was originally trying to accomplish here. We look at the definition of translation as given by the U of Chicago text -- notice the similarly between this definition and that of rotation given yesterday:

U of Chicago:
"A translation (or slide) is the composite of two reflections over parallel lines."

This definition is invalid unless we assume the Parallel Postulate. Indeed, Wu's definition of "translation" also mentions the Parallel Postulate. So without the postulate, we're having trouble even defining "translation"!

Notice that in elliptic geometry, there are no parallel lines and hence no translations. In hyperbolic geometry, there is a definition of "translation" that's similar to the U of Chicago definition. There, a translation is the composite of two reflections over ultraparallel lines. As it turns out, most pairs of parallel lines are "ultraparallel" in the sense that through a point not on a given line, there are infinitely many ultraparallel lines but only two parallel lines that aren't ultraparallel. If the reflecting lines are parallel but not ultraparallel, the composite is called a "horolation." So in hyperbolic geometry, there are five different isometries (horolation, translation, rotation, reflection, and glide reflection) whereas in Euclidean geometry there are only four. (In elliptic there are only three.)

And now we see a problem -- are we really, in a high school course, going to start discussing ultraparallel lines and Saccheri quadrilaterals for the sole purpose of proving "if corresponding angles are congruent then the lines are parallel" without using the Parallel Postulate? ...especially when we'd soon introduce the Parallel Postulate anyway to prove the converse (and ultraparallel lines become parallel lines, and Saccheri quadrilaterals become rectangles)? ...especially when we can use rotations instead (and rotations, being the composite of reflections in intersecting lines, have nothing to do with parallel lines and so nothing to do with the Parallel Postulate)?

Of course we won't. And so I'll stick to using the Wu rotation method -- after all, the student I was tutoring quickly understood the rotation proof, and the goal of teaching is to present the material so that the students understand it.

It's a shame, though. One thing I like about the translation method is that it unites the two definitions of "corresponding angle" commonly used in geometry classes -- two lines cut by a transversal form corresponding angles, and congruent triangles have corresponding angles (and sides) that are congruent (CPCTC). Here, "corresponding angles" would be angles that are the preimage and image of some isometry, whether it be a translation for the parallel lines or a translation (or reflection, etc.) for the congruent triangles. But unfortunately, we can't use this method in a high school course.

Now I want to avoid being a hypocrite. I just criticized Silverman for using a definition of "parallel" that isn't equivalent to the standard definition unless one assumes the Parallel Postulate, then turn around and endorse definitions of "translation" that require a Parallel Postulate. And the definition of "translation" given by the U of Chicago in terms of composites of reflections is similar to the one that hyperbolic geometers use -- just make the reflecting lines "ultraparallel."

But it's much more important to avoid confusing high school students with concepts such as "ultraparallel" lines. And besides, notice that I will move rotations up to Chapter 4, but keep translations in Chapter 6. In the intervening Chapter 5, the U of Chicago text gives a brief description of non-Euclidean geometry. This will be a good place to mention the Parallel Postulate, in the form first proposed by the 18th century Scottish mathematician John Playfair (which is why both Joyce and Dr. M call it "Playfair's postulate"): "given a line and a point not on it, at most one line parallel to the given line can be drawn through the point." Notice the phrase "at most," which is used to rule out hyperbolic geometry. As it turns out, elliptic geometry is actually ruled out by other postulates besides Euclid's Fifth Postulate. Then defining translations as the composite of two reflections in parallel lines is OK because by that point we've already assumed Playfair.

In my next post, I will try to provide a few visual previews of what the geometry on this blog will actually look like.

Source 3: Hung-Hsi Wu

My next source is Hung-Hsi Wu, a professor at Berkeley:

http://math.berkeley.edu/~wu/CCSS-Geometry.pdf

Like Dr. M, Wu is also developing a Common Core course, although he is also working on the eighth grade geometry portion as much as the high school class -- after all, the peculiarities of Common Core Geometry begin in 8th grade.

Wu begins by giving some basic postulates, then moves on to transformations. He covers rotations first and gives some of their properties. Then Wu gives his first theorem, Theorem 1 (under high school, not 8th grade):

Wu:
"Theorem 1. Let L be a line and O be a point not lying on L. Let R be the 180-degree
rotation around O. Then R maps L to a line parallel to L itself"

We are considering rotations about a point O, and we want to know what happens when we rotate lines about that point O 180 degrees (since rotations require a center and a degree). The answer is that it depends on whether the line passes through O or not. A line passing through O ends up being rotated to itself -- this is essentially because a line is a straight angle of 180 degrees. Less obvious is what happens if the line doesn't pass through O. If L is the line that we are trying to rotate and O doesn't lie on L, where is the image of L? We begin by choosing any point on line L, and Wu labels this point Q. Now we want to rotate Q 180 degrees about O, and Wu refers to this new point as P. So where is P -- can it lie on line L? We consider the line that passes through points Q and O, and Wu labels this line with a lowercase l. Like all lines passing through O, l is rotated to itself, and so P, the rotation image of Q, lies on l. This proves that P can't lie on L, for if it did, there would be two lines passing through P and Q (namely L and l) when there is only supposed to be one line through those points. (And L and l can't be the same line, since O lies on l yet not on L.) But point Q is arbitrary -- we've shown that every point Q on the line L is rotated to a point P that is not on L. Therefore L and its rotation image can't have any points in common -- and lines that have no points in common are called "parallel." QED. (Technically, parallel lines are lines in a plane that have no common points -- but "plane" is assumed here because the rotations, reflections, and translations that we're performing are transformations of the plane.)

Now this Theorem 1 can be used to prove other statements -- for example, if two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. This is Theorem 12 in Wu. To prove this, we observe that the two lines and the transversal form a "Z" shape, and rotating this "Z" 180 degrees gives the same "Z" shape. (Wu does this more formally.) This rotation maps the transversal to itself and one of the lines to the other -- and since lines are parallel to their rotation images by Theorem 1, the original lines are parallel.

So we have two ways to prove the Alternate Interior Angles Theorem -- using the Exterior Angle Inequality following Dr. M (and Euclid), and using rotations following Wu. As a math tutor, I decided to present both the Exterior Angle Postulate and Wu's Theorem 1 to a geometry student who is reading a traditional text (not any of the texts mentioned on this blog so far) that defined transformations in its final chapter. The student is taking a fast-paced summer course in which all the assessments are multiple-choice (as there is no time to grade a free-response test), and so there are no proofs on the tests. Yet when I asked him to prove the Alternate Interior Angles Theorem, he saw the required 180-degree rotation for Wu right away. Perhaps this was because not doing many proofs in class, he was especially unfamiliar with the indirect proof that was required in the derivation from the Exterior Angle Postulate. At any rate, if an actual student found the rotation proof easier to understand, then this is how I should teach it. And besides -- Common Core Geometry is all about rotations (and other transformations), and so I should use rotations as much as possible in a Common Core class (as opposed to the derivation from the Exterior Angle Postulate, which doesn't use transformations at all).

What I will do on this blog is present the material from my U of Chicago text, which is already very similar to Common Core Geometry, and fill in the gaps using information from Wu. The first seven chapters in my U of Chicago text are as follows:

1. Points and Lines
2. Definitions and If-Then Statements
3. Angles and Lines
4. Reflections
5. Polygons
6. Transformations and Congruence
7. Triangle Congruence

Notice that while reflections are given in Chapter 4, the other transformations aren't given until two chapters later. I understand that there is a newer version of the U of Chicago text -- mine is dated 1991 -- in which all the transformations are covered in Chapter 4. But the 1991 text is the one in my hands, and so I will work from this one.

As for the other chapters, the first two chapters are similar to a traditional text. Notice that parallel lines are covered very poorly in this text -- corresponding angles are mentioned in Chapter 3, alternate interior angles in Chapter 5, and same-side interior angles not at all.

Here's how I'll rearrange the chapters so that they fit Common Core:

1. Points and Lines
(I'll focus on sections 1-6 onward.)

2. Definitions and If-Then Statements
(I'll move angles up to this chapter. Chapter 3 is awfully late to first present angles -- most traditional texts cover angles in Chapter 1!)

3. Reflections
(I might as well get to reflections right away. Since reflections are defined using perpendicular lines, some of the information on perpendicular lines fit here.)

4. Rotations
(Here's where I'll include information from Wu's page, including his Theorems 1 and 12.)

5. Polygons
(This chapter is mostly intact.)

6. Transformations and Congruence
(This chapter is mostly intact, except that rotations have been moved up.)

7. Triangle Congruence

Notice that U of Chicago presents reflections first, while Wu gives rotations first. The definitions of reflection and rotation in U of Chicago requires that reflections have priority:

U of Chicago:
"For a point P not on a line m, the reflection image of P over line m is the point Q if and only if m is the perpendicular bisector of segment PQ. For a point P on m, the reflection image of P over line m is P itself."

"A rotation is the composite of two reflections over intersecting lines."

Wu uses a similar definition for reflections, but his definition of rotations is a more traditional one in terms of the center and angles and so is independent of reflections. The problem is that for Wu, reflections are not well-defined unless we prove that there is exactly one point Q such that the reflecting line is the perpendicular bisector of segment PQ -- and his proofs ultimately derive from Theorem 1 and rotations. This isn't a problem for U of Chicago because it merely assumes, via the Reflection Postulate, that reflections are well-defined:

Reflection Postulate (U of Chicago):
Under a reflection:
a. There is a 1-1 correspondence between points and their images. This means that each preimage has exactly one image and each image comes from exactly one preimage.

There are more parts, but part (a) is the only part that matters now. Part (a) is what guarantees that reflections are well-defined.

To conclude this post, let me state that Chapters 1-7 constitute what I would like to be the first semester of the geometry course. Other concepts such as area, volume, and similarity (and thus dilations -- the transformations on which similarity is based) can wait until second semester. I'm thinking of a student who just barely passes Algebra I (earning the lowest grade that the school allows for entry into Geometry). The last thing that such a student wants to see is more algebra, yet concepts like measurement and similarity are obviously algebra-heavy. I'd like to see the first semester of Geometry be algebra-light -- the most calculation required in Chapters 1-7 is to determine whether angles add up to 90, 180, or 360 (the Polygon Angle-Sum Theorem) -- so that the barely-passing Algebra I student has a chance to earn a good grade in at least one semester of the geometry course.

Source 2: Franklin Mason

My next source is Franklin Mason (also known as Dr. M), a high school math teacher in Indiana:

http://ateacheratextandaculture.blogspot.com/

Unlike Joyce, whose webpage was last updated before Common Core, Dr. M actually has the Common Core Standards in mind. His website contains a complete course in Common Core Geometry, complete with Powerpoints and worksheets, along with an actual textbook that is still in the works.

Dr. M focuses on rigor and proofs. In fact, some of Joyce's desires for a rigorous proof-based geometry course are fulfilled in Dr. M. For example, Dr. M states that the intersection of two lines being at most one point is a consequence of an earlier postulate (not a postulate itself). Under Chapter 2 (of Prentice-Hall), Joyce writes:

Joyce:
"In a return to coordinate geometry it is implicitly assumed that a linear equation is the equation of a straight line. A proof would depend on the theory of similar triangles in chapter 10."

This is actually omitted from most traditional texts -- after all, it's so "obvious" that a linear equation is a line that neither algebra nor geometry texts prove this. But Dr. M now actually uses similar triangles to prove this in his final chapter, Chapter 13, on Analytic Geometry.

Regarding circles, Joyce writes:
"The tenth theorem in the chapter [5 of Prentice-Hall] claims the circumference of a circle is pi times the diameter. It is apparent (but not explicit) that pi is defined in this theorem as the ratio of circumference of a circle to its diameter. So the content of the theorem is that all circles have the same ratio of circumference to diameter. This theorem is not proven. It would require the basic geometry that won't come for a couple of chapters yet, and it would require a definition of length of a curve and limiting processes. It would be nice if a statement were included that the proof the the theorem is beyond the scope of the course."

Dr. M actually has a section on least upper bounds in his Chapter 9. These are the "limiting processes" to which Joyce refers. Thus, Dr. M is able to give even a rigorous definition of pi! And of course, he uses translations, rotations, and reflections to derive SAS and the others.

Since Dr. M's website is so thorough, one may wonder what I can provide on this blog that would improve upon it. Well, regarding parallel lines, Joyce writes:

Joyce:
"Chapter 7 is on the theory of parallel lines. It begins by postulating that corresponding angles made by a transversal cutting two parallel lines are equal. One postulate is enough, but for some reason two others are also given: the converse to the first postulate, and Euclid's parallel postulate (actually Playfair's postulate). A proliferation of unnecessary postulates is not a good thing. One postulate should be selected, and the others made into theorems."

Dr. M improves on Prentice-Hall by giving two postulates for parallel lines. One of them is Playfair's Postulate as mentioned by Joyce. The other is called "Exterior Angle Postulate." As it turns out, this is Euclid's Proposition 16:

Euclid, Proposition 16:
"In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles."

Dr. M uses the Exterior Angle Postulate to prove that if two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel (as Euclid in Proposition 27). But Euclid proves Proposition 16 itself. In fact, Dr. M gives the proof right on his blog -- as of today, it's the third post from the top (dated August 3rd of last year). This proof derives from SAS. And so we wonder, why does Dr. M call it a postulate rather than prove it as a theorem?

I think that earlier this year, Dr. M actually did include the Exterior Angle Inequality as a theorem, but he recently changed it to a postulate. Notice that this proof is what I call a low-level proof. It's used in proofs of other results (including both the Alternate Interior Angles Theorem and the Triangle Inequality), but is not truly an interesting result in itself -- after all, the Exterior Angle Equality tells that the exterior angle actually equals the sum of the remote interior angles, not merely that it's greater than either one. So, as many authors do with low-level proofs, Dr. M just makes the Inequality a postulate rather than cause students to worry about its proof.

But there are other ways to prove the Alternate Interior Angles Theorem besides the Exterior Angle Inequality. I will discuss them in my next post.

Source 1: David Joyce

Even before the advent of Common Core, there has been much debate about what to include in a geometry text, and ultimately a geometry course. Here are some of my favorite sites discussing geometry and how it should be taught. I will refer to these sites often throughout this blog.

http://aleph0.clarku.edu/~djoyce/java/elements/geotfacw.html

On this page, David Joyce, a professor at Clark University in Massachusetts, gives a review for a certain geometry text. It is not my U of Chicago textbook, but a book published by Prentice-Hall in 1998. I am a math tutor who has taught several students whose classes use this text, so I am familiar with it.

Joyce criticizes the Prentice-Hall text very harshly, and I agree with some of what he writes. In particular, he laments how late the text covers parallel lines (Chapter 7) and congruent triangles (Chapter 8). Most books I’ve seen cover these topic in Chapters 3-4, and so I also wonder why these are mentioned so late in the Prentice-Hall text.

But there are several topics that Joyce argues should be left out of the course (like trig for example). Even though this webpage was written before the Common Core Standards, many state standards (including my home state of California) required trig in the geometry courses. So Prentice-Hall includes topics that Joyce finds extraneous because the states required them

The cornerstone of Joyce’s critique involves logic -- in particular, postulates and theorems. I bet that most geometry students don’t know the difference between a postulate and a theorem. Why, for example, are SSS, SAS, and ASA called “postulates,” while AAS and HL are called “theorems”? The reason is that the first three are assumed, while the last two are proved in the text. But most students don’t learn the proofs of AAS and HL, and most teachers don’t teach these proofs either. “Postulate” and “theorem” end up being a distinction that Joyce and most textbooks make, yet most actual students and teachers in the classroom ignore this distinction.

Indeed, we see how important the difference between “postulate” and “theorem” is to Joyce in his discussion of Chapter 1:

Joyce:
"Chapter 1 introduces postulates on page 14 as accepted statements of facts. The four postulates stated there involve points, lines, and planes. Unfortunately, the first two are redundant. Postulate 1-1 says 'through any two points there is exactly one line,' and postulate 1-2 says 'if two lines intersect, then they intersect in exactly one point.' The second one should not be a postulate, but a theorem, since it easily follows from the first. And what better time to introduce logic than at the beginning of the course. No statement should be taken as a postulate when it can be proved, especially when it can be easily proved."

Think about it for a moment -- suppose Postulate 1-2 were false and two lines (say m and n) could really intersect in two points (say P and Q). Then through two points (P and Q) there would be two lines (m and n) when there's only supposed to be one line through the points, by Postulate 1-1. So this is a contradiction, and so Postulate 1-2 is true. But this constitutes a proof of Postulate 1-2, which would make it a theorem, not a postulate!

So why doesn't Prentice-Hall label this a "theorem"? I suspect it's because this is an indirect proof (or a proof by contradiction), which Prentice-Hall (along with most other texts) cover late. The authors probably didn't want to bring up indirect proofs in Chapter 1. (Interestingly enough, the U of Chicago text does call this a theorem, the "Line Interesection Theorem.")

On this blog, I'll classify proofs by different levels. A high-level proof is what most students think about when they consider proofs in geometry -- given two triangles, with some of the pairs of corresponding sides and angles congruent (either given or easily derived using vertical angles, the Reflexive Property, etc.), prove the triangles congruent using SSS, SAS, etc., then use CPCTC to derive another pair of congruent sides or angles. In other words, they are exercises or problems to do in the homework.

A medium-level proof is the proof of a well-known theorem. Most textbooks give the proofs or leave them as exercises, but students and teachers skip over them. Students typically think of theorems as something to use in high-level proofs, not something to be proved themselves. But since this is a Common Core blog, we must refer to the Common Core Standards, and the standards do require students to prove these theorems:

CCSS.MATH.CONTENT.HSG.CO.C.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

A low-level proof is a proof of a theorem, but it isn't as well known. Most books simply label the statement as a "postulate" rather than a theorem and omit the proof. An example of this is given at the Joyce link:

Joyce:
"Chapter 8 finally begins the basic theory of triangles at page 406, almost two-thirds of the way through the book. This chapter suffers from one of the same problems as the last, namely, too many postulates. The three congruence theorems for triangles, SSS, SAS, and ASA, are all taken as postulates. One is enough. The other two should be theorems."

And I myself had long thought of SSS, SAS, and ASA as postulates, not theorems. Almost every text I'd seen presented these three as postulates, so I was shocked to read this. For explanations on how to prove these, we turn to the author of the world's first geometry text -- the ancient Greek mathematician Euclid, who lived about 2,300 years ago. Joyce gives a link to Euclid's famous text Elements, which contains basically nothing but proofs:

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html

Here, Euclid's Proposition 26 contains proofs of both ASA and AAS. Both of them use Proposition 4 (SAS) and are indirect proofs (which explains why most books don't include these proofs). Also, Proposition 8, which is SSS, is also dependent on Proposition 4. It appears that Joyce suggests having only SAS be a postulate, and both SSS and ASA proved as theorems. Once again, I point out that students think of SSS and ASA as statements to use in proofs as opposed to being proved themselves.

Ironically, Proposition 4 also has a proof in Euclid:

Euclid, Proposition 4:
Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF respectively, namely AB equal to DE and AC equal to DF, and the angle BAC equal to the angle EDF.
I say that the base BC also equals the base EF, the triangle ABC equals the triangle DEF, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ABC equals the angle DEF, and the angle ACB equals the angle DFE.
 If the triangle ABC is superposed on the triangle DEF, and if the point A is placed on the point D and the straight line AB on DE, then the point B also coincides with E, because AB equals DE.
Again, AB coinciding with DE, the straight line AC also coincides with DF, because the angle BAC equals the angle EDF. Hence the point C also coincides with the point F, because AC again equals DF.
But B also coincides with E, hence the base BC coincides with the base EF and equals it.
Thus the whole triangle ABC coincides with the whole triangle DEF and equals it.
And the remaining angles also coincide with the remaining angles and equal them, the angle ABC equals the angle DEF, and the angle ACB equals the angle DFE.
Therefore if two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.

Euclid called this "The Principle of Superposition." But, as it turns out, this is exactly the method that Common Core uses to prove SAS! Common Core uses translations, rotations, and reflections, and notice that we have a translation (A slides to where D is) and then a rotation (keeping A fixed on D while turning so that B ends up where E is). But Euclid does make one oversight here -- the angles BAC and EDF are indeed congruent, but the angles may be on opposite sides of AB and DE. It's possible that a reflection is needed (so that we flip one of the angles so that it's on the same side as the other). Finally, the triangles must coincide (one right on top of the other) and so must be congruent!

Some critics decry Common Core as changing the way that math has been taught for years, but as it turns out, Common Core Geometry (at least regarding SAS) actually makes geometry more, not less, like the way it's been taught since Euclid. This is also how the U of Chicago text presents triangle congruence, and so SSS, SAS, and ASA are all theorems!

In conclusion, let's see what Joyce has to say about transformational geometry, the cornerstone of the Common Core method:

Joyce:
"Chapter 3 is about isometries of the plane. The entire chapter is entirely devoid of logic. How are the theorems proved? 'The Work Together illustrates the two properties summarized in the theorems below. Theorem 3-1: A composition of reflections in two parallel lines is a translation. ...' Moving a bunch of paper figures around in a "work together" does not constitute a justification of a theorem.
"The theorems can be proven once a little actual geometry is presented, but that's not done until the last half of the book. A little honesty is needed here. Why not tell them that the proofs will be postponed until a later chapter? Or that we just don't have time to do the proofs for this chapter. Even better: don't label statements as theorems (like many other unproved statements in the chapter). Like the theorems in chapter 2, those in chapter 3 cannot be proved until after elementary geometry is developed.
"In summary, either this chapter should be inserted in the proper place in the course, or else tossed out entirely. Since there's a lot to learn in geometry, it would be best to toss it out."

When Joyce writes that the theorems of Chapter 3 can't be proved until after geometry is developed, he means after the parallel and congruence results of Chapters 7-8. But Common Core flips this around. Instead of using the traditional parallel and congruence results to prove theorems about reflections and translations, in Common Core we use theorems about reflections and rotations to prove the traditional parallel and congruence results! Joyce suggests that we toss this material out, but Common Core requires that we keep it and treasure it.

And those proofs -- using transformations to prove traditional results -- are the focus of this blog.

Happy Pi Approximation Day!

Today is July 22nd, or 22/7 in the international date format, and 22/7 is approximately equal to pi, a constant famous for its use in geometry. As such, today is Pi Approximation Day, one of two days of the year devoted to that number -- the other being 3/14, Pi Day. Today is therefore a perfect day to launch this blog, since it is all about geometry.

I am David Walker, an aspiring math teacher. A major challenge facing all of us math teachers is the transition to the Common Core State Standards -- especially in geometry, since they are so different from the way the subject is traditionally taught. When I was working on my preliminary credential, I showed a fellow geometry teacher a textbook that mentions reflections, translations, and rotations, and she was completely confused. So I can only imagine how much more the students will be confused by this new way of learning geometry if we teachers can't even figure it out!

That book was published by Scott, Foresman and designed by the University of Chicago School Mathematics Project. It was not the textbook that I'd used to learn geometry, for I had moved to another district in time for Algebra II and saw the Geometry students carrying around this bright pink textbook. But our school discarded them in favor of texts more aligned with the California State Standards, and I ended up purchasing a copy at the used bookstore at the local library for $2. It is ironic, then, that this old U of Chicago text is actually better aligned with the Common Core Standards while the books in the classrooms are still set to the California State Standards.

Throughout this blog, I will present Common Core Geometry using my the U of Chicago text as a major guiding force. But it won't be my only guide. Along the way, I will mention a few other teachers and their own visions of what our subject will look like in the near future. I write this blog mainly to help myself know what to teach -- since when I'm in the classroom, I find that often I, even more than the students, am the one who needs guidance. If anyone else, be it a student or another teacher, gains a greater appreciation of Common Core Geometry by reading this blog, it'll be icing on the cake.

Welcome to Geometry, Common Core Style!