Thursday, November 16, 2017

Lesson 6-4: Miniature Golf and Billiards (Day 64)

This is what Theoni Pappas writes on page 16 of her Magic of Mathematics:

"Ever wonder about the f-stop number of a camera? Where did it get its name? How is it determined?"

This is the only page of the subsection "Mathematics & the Camera." Yes, the U of Chicago text mentions cameras in Lesson 15-4, but nothing about f-stop numbers. Well, Pappas tells us what the f-stop number means in the very next sentence:

"'f' stands for the mathematical term factor."

Here are some more excerpts from this page:

"Photographers use what is known as the f-number system to relate focal length and aperture. For example,

f4 = 80 mm lens/20 mm aperture.
f16 = 80 mm lens/5 mm aperture.

Working with f-stop numbers and shutter speeds, you can manually decide how much of the photograph you want in focus."

The only picture on this page is that of a camera lens, with its f-stop numbers listed. As Pappas points out, the size of the lens remains constant, and so it's the aperture, or opening, that changes.

I remember seeing f-stop numbers on Square One TV. In the Mathnet episode "View From the Rear Terrace," Kate Monday adjusts the f-stop number on her camera as she spies on the villain from her upstairs window. Apparently, most f-stop numbers are powers of 2, like 4 and 16 above. I also recall noticing the f-stop numbers 1.4 and 2.8 between 1 and 4 -- I suspect the intention is for the f-stop numbers to form a geometric series, and so 1.4 approximates sqrt(2).

Chapter 4 of George Szpiro's Poincare's Prize is called "An Oscar for the Best Script." Here's how the chapter begins:

"Even though Poincare was an outstanding engineer, his first love remained mathematics. The curriculum at the Ecole Polytechnique, with its classes in mathematics and physics, catered to his tastes."

In this chapter, Szpiro writes about Poincare's return to the mathematical world. His first is to earn his doctorate, but there are a few problems with his dissertation:

"When he presented another paper on differential equations to Jean Darboux, who later became secretary of the Academie des Sciences and would pronounce the eulogy at Poincare's funeral, the Sorbonne professor extolled the essay's virtues but at the same time strongly criticized its rigor."

Of course, Poincare fixes the errors and is awarded his doctorate. For him, 1881 becomes a big year:

"Poincare was appointed to the chair of mathematical physics and probability at the Sorbonne. Simultaneously, he obtained a teaching position at his alma mater, the Ecole Polytechnique. In the same year he married Louise Poulain d'Andecy. The couple had three daughters and a son."

At this time, Poincare works to solve a huge problem in astronomy regarding planetary motion. The problem goes all the way back to 17th century German mathematician Johannes Kepler:

"The observations that he had received, or rather stolen, from his predecessor at Prague, the Imperial Mathematician Tycho Brahe, were the most exact data at the time. But even so, they contained minute errors."

The reasons for the errors is that the known equations solve only the two-body problem, where the motions of only two bodies (such as the sun and a planet) are considered. But there are many more bodies in the solar system, and each one is affected by the gravitational pull of all the others.

The Oscar mentioned in this title is actually a king:

"When he was fifty-five years old, King Oscar II of Sweden and Norway, a great supporter of science, was wondering how to celebrate his sixtieth birthday, which would fall on January 21, 1889. Gosta Mittag-Leffler, a Swedish mathematician [...] proposed the establishment of a prize for an essay about the n-body (here n means any number, including 3) problem as an appropriate way to celebrate the king's birthday."

Let's learn more about Oscar's prize:

"The monarch charged Mittag-Leffler with the minutiae of organizing the prize. He was to work out the rules, choose the jury, announce the prize, and finally select the winner."

And so it was determined that the contest would consist of three questions. One mathematician who doesn't like the questions is Leopold Kronecker -- a known spoilsport. (Indeed, Kronecker is one of the first to challenge Georg Cantor's set theory.) Szpiro writes:

"Kronecker immediately criticized first one, then another of the questions. But his criticisms were recognized as the misgivings of a grouch and not taken seriously."

A grouch -- well, this Kronecker reminds me of another Oscar, the one who lives on Sesame Street.

Anyway, over the next three years, a dozen mathematicians submit entries for the contest. As for the winner -- well, there's no real suspense who the winner is, considering whose biography we've been reading in the past two chapters. As the author puts it:

"With great trepidation, the envelope that contained the name was opened in front of a hushed crowd of invited guests: 'And the winner of the Oscar for the best script is ... La La Land [oops, wrong envelope -- dw] Henri Poincare, for "The problem of the three bodies."'"

Okay, that was a bad joke on my part. (I should have said Hidden Figures, not La La Land, since the movie, like Poincare's paper, is related to astronomy.) But as Szpiro explains, the winner's name really is in an envelope, since the papers are submitted anonymously to avoid possible biases.

The author describes Poincare's Oscar-winning paper:

"He proved rigorously that no analytical solutions (i.e., no elegant formulas) exist that would describe the position of the bodies at all times. The surprising implication of this result is that the positions of the planets in our solar system cannot be predicted with total precision."

(By the way, I'm sorry for mentioning Hidden Figures again, but Katherine Johnson also encounters the impossibility of finding an exact analytical solution when one of the bodies is a capsule. This is why she must resort to finding a numerical solution via Euler's method.)

Szpiro continues:

"That could have been the end of the story and all participants could have gone on with their lives. But the troubles were only just about to start."

As it turns out, Poincare's paper contains a fatal flaw. Szpiro explains that it has something to do with the convergence of a series of numbers:

"A series of numbers is an infinite number sequence, such as 1 + 2 + 3 +.... It is said to converge if its entries add up to a finite sum. For example, 1 + 1/2 + 1/4 + 1/8 +... = 2.0, hence the series converges. If the sum tends toward infinity as the number of terms increases, the series is said to diverge."

Szpiro fails to note a coincidence here -- the sum of a series is exactly what Poincare fails to prove in his high school bac exam, and here it bites him again. The problem is when we have a slow-growing yet divergent series. The usual example given, of course, is the harmonic series -- which we've already discussed in reading Ogilvy's book.

"Thus, even if the terms of the series decrease so rapidly that the computation of a planet's orbit can be computed to a precision of several digits behind the decimal point for very, very long time periods -- we are speaking many millions and billions of years here -- convergence of the series must be proven mathematically even as the number of the series's terms tends to infinity."

Part of Poincare's problem is that he fails to explain his work properly. Now maybe you can see why the Common Core requires students to explain their work:

"As he had done in his childhood, when his speech was often unintelligible because ideas came to him faster than he could pronounce them, in his mathematical work too he did not always spend sufficient time formulating the details."

The mathematician tries to halt publication of his erroneous proof, but his efforts are in vain:

"On December 4, three days after Poincare posted the letter, Mittag-Leffler informed him by return mail that prepublication copies of the faulty paper had already been sent out to a few select personalities."

At this point panic ensues -- if word gets out that the prize-winning proof contains an error, the reputations of all people involved, including the king, would shatter into dust:

"Hoping against hope that Poincare would somehow be able to rectify things, Mittag-Leffler took pains to cover up all tracks of the error."

But one of the prize judges, Karl Weierstrauss, eventually hears rumors about the error. And of course, our resident "grouch" wants to know more about a possible flaw:

"And Kronecker was quite willing to listen. So when Weierstrauss was asked by his colleagues about a possible error in Poincare's prize essay, he found himself in an extremely uncomfortable position."

And so Weierstrauss writes a series of complaint letters:

"More to the point, nobody could demand from a juror that he guarantee the truth of every statement in such an extensive manuscript, especially when many computations were not actually performed but only hinted at. The latter was a thinly disguised swipe at Poincare's irritating habit of omitting whole sections of a proof because they seemed clear to him."

In the end, publication of the faulty proof is finally cancelled:

"Now it was up to Poincare. Work had to be done that went far beyond a simple revision of the essay. The new manuscript was truly pathbreaking. It provided the first traces of chaos theory, which would become so popular only a century later. The possibility of chaotic movement of bodies was just what he had overlooked in the erroneous first version of the essay."

Of course, Pappas mentions chaos theory a couple of times in her book. She hints at it yesterday in the discussion of complex systems (and it doesn't get much more complex than the solar system), and there's a further discussion later in the book (which we read earlier this year -- see my May 8th and 9th posts for more info).

As it turns out, Poincare's fix works, and his contemporaries never see evidence of his original paper filled with errors. Unfortunately, he must spend his entire prize money on removing the mistakes (though the title of Oscar winner is never revoked):

"Only many years later, when historians of science searched through the archives of the Mittag-Leffler Institute and compared the original version with the published paper, was the true magnitude of the differences discovered."

Szpiro tells us that after this fiasco, the Oscar prize -- intended to be given out every four years -- is never awarded again. But a few years later, while the science-loving Oscar is still king of Sweden and Norway, another prestigious award is established -- the Nobel Prizes. (Hmmm, I wonder whether the Poincare problem is a reason that there is no Nobel Prize in mathematics.)

Half a century later, Russian mathematician Andrei Kolmogorov lectures on the n-body problem:

"The subject was the thorny question of what happens to periodic orbits of bodies when small perturbations disturb their course."

Even though Kolomogorov gives a partial solution, it isn't complete:

"The stability of our solar system, which consists of nine planets, not just three, is far from guaranteed by science."

But what happens to Poincare? Szpiro explains:

"Poincare was led away from the manipulation of numbers and formulas and toward the visualization of curves and flows. The subdiscipline he thus pioneered was to become algebraic topology. It will concern us throughout the remainder of this book."

Notice that so far we haven't actually mentioned the Poincare Conjecture yet. As we now see, it is a conjecture in algebraic topology.

The author tells us about the mathematician's final years, after he suffers prostrate problems:

"Back in Paris, Poincare soon resumed his old schedule and everything was normal for another four years. But on July 9th, 1912, he had to undergo surgery."

And, as we found out yesterday, he dies eight days later at the age of 58. Szpiro writes of his legacy:

"The university in his hometown of Nancy, for example, is named after him, and so is one of the town's best-known lycees [high schools -- thank you, my French teacher!]."

Szpiro ends the chapter with a note about Poincare's famous relatives:

"While we're at it, there is also the Avenue du Recteur Poincare [in Paris], in memory of Henri's cousin Lucien, an important university administrator, and in the thirteenth arrondissement an avenue carries the name of his brother-in-law, Emile Boutroux."

Lesson 6-4 of the U of Chicago text is called "Miniature Golf and Billiards." In the modern Third Edition, we must backtrack to Lesson 4-3 to play miniature golf.

This is what I wrote two years ago about today's lesson:

Today we proceed with the next lesson in the text. Lesson 6-4 of the U of Chicago text is all about applying reflections to games such as miniature golf and billiards. I don't need to make any changes to the lesson, so I can just keep what I wrote last year for this lesson almost intact:

One of my favorite TV programs is The Simpsons -- I've been watching it for decades. One of its earliest episodes, having aired almost exactly 27 years ago, was called "Dead Putting Society." In this episode, Bart Simpson is preparing for a miniature golf competition. His sister Lisa shows him how he can use geometry to help him make a difficult shot. After saying this, Bart proclaims, "You've actually found a practical use for geometry."

Lesson 6-4 of the U of Chicago text discusses miniature golf and billiards. Just as Bart learns in this above video, one can use geometry to determine where to aim.

The key is reflections -- one of the important transformations in Common Core Geometry. It is often said that when a ball bounces off a wall, the angle of incidence equals the angle of reflection. The text describes where to aim a golf ball G so that it bounces off of a wall and reaches the hole H:

"In this situation, a good strategy is to bounce (carom) the ball off a board, as shown [in the text]. To find where to aim the ball, reflect the hole H over line AB. If you shoot for image H', the ball will bounce off AB at P and go toward the hole."

We can write a two-column proof to show why the angle of incidence -- the angle at which the ball approaches the board, which is BPG -- equals the angle of reflection APH:

Given: H' is the reflection of H over line AB.
Prove: Angle APH = Angle BPG

Statements                            Reasons
1. H refl. over line AB is H'   1. Given
2. Angle APH = Angle APH' 2. Reflections preserve angle measure
3. Angle APH' = Angle BPG 3. Vertical Angle Theorem
4. Angle APH = Angle BPG  4. Transitive Property of Equality

Notice that for this proof, I've skipped a few steps. Technically, we should write that the reflection images of both A and P are the points themselves, since they lie on the mirror (Definition of Reflection), and so angle APH' is the image of APH (Figure Reflection Theorem). But I'm tired of writing that over and over again -- how much less, then, will the students want to write that.

Of course, this only works if the ball caroms only once. The U of Chicago text describes the game of billiards, where the player is required to bounce the ball off of three cushions. The text writes:

"Pictured [in text] is a table with cushions wxy, and z, the cue ball C, and another ball B. Suppose you want to shoot C off x, then y, then z, and finally hit B. Reflect the target B successfully over the sides in reverse order: first z, then y, then x. Shoot in the direction of B"' [...]

"Notice what happens with the shot. [...] On the way toward B"', it bounces off side x in the direction of B". On the way toward B", it bounces off y in the direction of B. Finally it hits z, and is reflected to B. [End of quote]"

In the video clip above, four caroms are required for the golf ball shot by Bart to find the hole. But unfortunately, the path of the ball as drawn on the show is impossible. To see why, let's label the direction from the starting triangle to the hole "North," and all the walls appear to meet at right angles, so they are all oriented in the north-south or east-west directions. Bart begins with the ball slightly to the right side of the starting triangle, so the initial direction of the ball is northwest. After hitting the first east-west wall, the ball is now traveling southwest. But then, after hitting the second east-west wall, the ball should be traveling northwest again. Indeed, we can use the Alternate Interior Angles Consequence and Test Theorems to prove that the path of the ball after hitting two walls should be parallel to the original direction of the ball. Yet the show depicts the ball as travelling due north after hitting two walls.

In fact, we can show that the only correct path to make the ball arrive in the hole involves hitting a north-south wall -- most likely the wall to the far west (in front of "Do not sit on statuary"). Only by hitting a north-south wall can the direction change from anything-west to anything-east, which is necessary for the ball to approach the hole. Lisa's advice to her brother is sound, but the way that it is animated is geometrically impossible.

For this worksheet, I decided to reproduce Bart's golf course, but tilt two of the walls in order for the path of the ball to be geometrically correct. I used equilateral triangle paper to create this page [2017 update: yes, the isometric graph paper I used for another assignment last year], so that the students will be able to figure out the paths without needing a ruler or protractor, for classrooms in which these are not available. Just as for Bart, four caroms are needed to get from G to H.

Working backwards on this worksheet, we can determine the path from A to H by reflecting H to the point H', then aiming from A to H'. But to determine a path from B to H using two caroms, we can't reflect H to first H' and then H", because H" would be well off the page. It may be better to aim from B to A', the reflection of A in the necessary wall.

Notice that if there are two walls and the paper is large enough, it may be actually possible to perform both of the necessary reflections. If the two walls meet at right angles, it is fairly easy to perform both reflections -- because the composite of those two reflections is a rotation. So to find the direction to aim at, we take the target point and rotate it twice 90, or 180, degrees around the point where the two walls meet. And if the ball is bouncing off of two parallel walls, then the composite of the two reflections is a translation, so we can just translate the target twice the distance between the two walls.

Officially, this lesson is the Guided Notes for Lesson 6-4 of the U of Chicago text. But this lesson naturally lends itself into a group activity -- the teacher can provide additional golf courses for the students to solve, or even allow the students to make up their own via the Bonus Question. It is a nice activity to give right before the week-long Thanksgiving break.

Wednesday, November 15, 2017

Lesson 6-3: Rotations (Day 63)

This is what Theoni Pappas writes on page 15 of her Magic of Mathematics:

"Complexity is an emerging science which may hold answers or at least explanations to such questions as...."

This is the last page of the subsection on complexity and the present. Pappas gives a full list on many questions that have to do with complexity. I'll only give part of her list below:

How is it that

  • the universe emerged out of the void?
  • on January 17, 1994 Los Angeles suffered an earthquake of unexpected magnitude and destruction?
  • Yugoslavia was thrown suddenly into severe internal wars?
  • for no apparent reason the stock market surges upward or plunges downward?
By the way, as a Southern Californian, I lived through the 1994 Northridge earthquake that occurred a few months before Pappas wrote her book. I was a seventh grader at the time. Notice that January 17th that year was Martin Luther King Day, and so all schools were closed for the holiday. I woke up with a jolt early that morning, but otherwise I was too far away from the epicenter to suffer damage.

Let me give excerpts from the rest of this page, so we can hurry up and get into Szpiro's book:

"The underlying common factor of these events is that each represents a very complex system. The factors which act on such a system are ever growing and changing. There seems to be a continual tug of war between order and chaos. It is the means by which the system regains equilibrium by changing and adapting itself to constantly changing factors/circumstances. These scientists and mathematicians fell that today's mathematics, along with other tools and high tech innovations, are capable of creating a complexity framework that can impact major aspects of our global world, especially economics, the environment, and politics."

In short, complex math is used to study complex systems.

Chapter 3 of George Szpiro's Poincare's Prize is called "The Forensic Engineer." It begins at the end, which a newspaper article reporting a mathematician's passing:

"Monsieur Henri Poincare, Professor of Mathematical Astronomy in the University of Paris, Member of the French Academy and the Academie des Sciences, died suddenly this morning of an embolism of the heart. The death of Henri Poincare at the comparatively early age of 58 deprives the world of one of its most eminent mathematicians and thinkers."

This chapter and the next are the story of Poincare, who first formulated the conjecture that would haunt mathematicians for over a century. Szpiro now begins with his birth:

"Jules-Henri Poincare was born in Nancy, a town in the northeast of France, on April 29, 1854. His father, Leon, was a doctor and professor of medicine at the University of Nancy; his uncle, Antoni, inspector of public works and an acknowledged authority on meteorology."

The first half of this chapter is all about the young Henri's education -- and as a teacher, I like to write about the educations of the subjects whose biographies I read. Because he is diagnosed with diphtheria at the age of five, his mother homeschools him for the first three years of his education. But they are worried when it's time for him to start at a public school:

"His very first essay was described as a small masterpiece by his teacher, who ranked the boy first in his class. Henri would keep that rank in all subjects throughout his school years."

But, as Szpiro points out, math isn't always Poincare's best subject:

"His favorite subjects were history and geography, and his aptitude for literary and philosophical essays aroused the interest of his educators. Henri's interest in mathematics only become apparent when he was fourteen years old."

After surviving the Franco-Prussian war, he prepares to take some big tests:

"In August 1871, Poincare sat for the literary baccalaureat, the French high school exams. He did extremely well in the Latin and French essays and excelled in all other subjects."

In high school, I myself took French, and as part of learning the language, we were taught all about French culture, including the big tests that our overseas counterparts must take. Furthermore, some traditionalists believe that the French bac and similar tests are superior to the PARCC, SBAC, and other state tests.

For one thing, notice that Poincare takes the tests in 1871, when he is 17 years old -- in other words, a high school senior. Standardized testing in our own country ends in junior year -- for some reason, seniors don't take tests like PARCC or SBAC. In France, the entire senior year is devoted to taking the bac. In addition, we see that the literary bac -- the equivalent of the ELA exam (or even the verbal section of SAT or ACT) -- contains a Latin section. We also see that Poincare sits his literary exam early in the school year -- in August -- and the math bac is three months later:

"Amazingly, Henri nearly failed the written test. The exam question asked for a proof of a formula to compute the sum of a geometric series. Henri had arrived late for the examination."

Fortunately, Poincare passes his math test on the strength of the oral part of the exam. Of course, American math standardized tests don't have an oral section, and of course high school students wouldn't be ask to prove (rather than merely use) the geometric series sum on a test.

"Following high school, young Frenchmen with exceptional aptitude for mathematics underwent, and still undergo today, a two-year grueling course in mathematics in preparation for entry to the grandes ecoles, the prestigious engineering schools."

Again, there is no equivalent in the American education system. Imagine if students here had to study for two extra years just to get in to an Ivy League school, MIT, or Caltech. During these two years, Poincare does so well that he passes his math courses without taking many notes, and even wins several national math competitions.

"Henri's next challenge was to enter the competition to gain admission to one of France's top engineering schools. Of all the grandes ecoles the most prestigious was, and still is, the Ecole Polytechnique in Paris."

The Polytechnique was the math and science school in France. I've mentioned the famous university twice on the blog in connection with other mathematicians (Evariste Galois and Sophie Germain -- the latter being unable to matriculate there due to her gender, but, as Szpiro points out, women can go there now).

Eventually, Poincare does attend the Polytechnique, and moves on to a special grad school for prospective mining engineers. Before we leave, Szpiro does tell us about the grading scale in France. Officially, there is a 20-point scale, but I already knew that earning a 20 is basically impossible, according to Numberphile:

The Numberphile video implies that earning 19 out of 20 is possible, but according to Szpiro, even 18 isn't achievable -- and scores from 16 to 18 are "reserved for the professor himself." Most of the grades earned by Poincare are between 15 and 16 -- except for Geometry (our subject!) and a closely related subject, drawing.

After he graduates, he works in the coal mines for a year. At this point, Szpiro spends the rest of the chapter telling an fascinating story about an explosion at the mines, where 16 people were killed. And Poincare uses mathematics to solve the mystery of what causes the explosion -- which I choose not to write about here, since we must hurry up to get to Geometry. (Again, you can read Szpiro's book for yourself to get the whole story.) His solution is similar to that of one of our recent logic puzzles -- indeed, the mine is a very complex system, just as we read about in Pappas today,

At the end of the chapter, Poincare leaves the mines and starts thinking about math again, as he works to become a math professor:

"Poincare was a formidable, even heroic engineer. His real calling, however, lay elsewhere."

Lesson 6-3 of the U of Chicago text is called "Rotations." In the modern Third Edition, we must backtrack to Lesson 4-5 to learn about rotations.

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

Yesterday, we discussed translations on the coordinate plane, and so now we move on to rotations. I point out that we learned how to perform translations of the form (xy) -> (x + hy + k) -- which turns out to be every translation in the plane.

But with rotations, we only perform a precious few of them. The only rotations that appear on the PARCC and other Common Core tests are those of magnitude 90, 180, or 270. Yet we've seen a few of these rotations centered at points other than the origin on the PARCC.

We'll begin with rotations that are centered at the origin, though. Just as we used the Two Reflections Theorem for Translations yesterday, today we'll use the Two Reflections Theorem for Rotations. So to perform the rotation of 180 degrees about the origin, we compose two reflections in mirrors that intersect at the origin, at an angle of half of 180, or 90 degrees. The obvious choices for mirrors are the x- and y-axes. We've already proved that the reflection image of (xy) in the x-axis is (x, -y) and the reflection image of (xy) in the y-axis is (-xy). It doesn't matter in which order we compose these as reflections in perpendicular mirrors always commute. So we prove that the rotation image of (xy) centered at the origin and of magnitude 180 degrees is (-x, -y).

Now our other common rotation magnitude is 90 degrees -- and this time, it will make a difference whether it's clockwise or counterclockwise. The angle between the mirrors will now have to be half of 90, or 45 degrees. There's one mirror to consider that will help us with a 45-degree angle -- the line whose equation is y = x.

We've hinted at several proofs involving reflection over the line y = xLet's look at the quadrilateral whose vertices are (0, 0), (a, 0), (aa), and (0, a). We can show that this figure is a kite.

So now we can apply the properties of a kite -- the Kite Symmetry Theorem. The diagonal of our kite running from (0, 0) to (aa) bisects the angle between the x- and y-axes -- and since we know that the angle between the axes is 90 degrees, the diagonal must form a 45-degree angle with each axis. And reflecting across this symmetry diagonal must map the axes to each other and x = a to y = a.

Recall that at this point, we don't know the equations of lines, so we aren't yet certain that the graph of y = x is even a line (which we'd better figure out before trying to use it as a mirror). But we see that the value of a in the above proof is arbitrary -- it's true for every single real number a (although in case a is negative, we should probably say that the kite has sides of length |a|, not a). Therefore every single point of the graph of y = x lies on the bisector of the angle between the axes -- that is, the graph of y = x is exactly that line. And reflecting in that line maps x = a to y = a and vice versa -- that is, it switches x and y. Therefore the image of (ab) must be (ba).

Now that we know how to reflect in the line y = x, let's use it to perform a 90-degree rotation. It's probably easiest just to start with the reflection in y = x first, so (xy) maps to (yx). As for the second mirror, it depends on whether we want to go clockwise or counterclockwise. To go clockwise, the second mirror must be 45 degrees clockwise of the first mirror, y = x. That is the x-axis, and to reflect in it, we change the sign of the second coordinate. So (yx) reflected in the second mirror is the point (y-x), so mapping (xy) to (y, -x) rotates points 90 degrees clockwise. To go counterclockwise, the second mirror must be 45 degrees counterclockwise of the first mirror, y = x. That is the y-axis, and to reflect in it, we change the sign of the first coordinate. So (yx) reflected in the second mirror is the point (-yx), so mapping (xy) to (-yx) rotates points 90 degrees counterclockwise.

Notice that some of the PARCC questions mention 270-degree rotations -- for example, there was a released question that mentions a 270-degree clockwise rotation. Of course, a 270-degree clockwise rotation is equivalent to a 90-degree counterclockwise rotation, so it maps (xy) to (-yx). If students forget this, they can still take half of 270 degrees to get 135 degrees clockwise, and they can see that 135 degrees clockwise from the line y = x is still the y-axis, just as it would have been if they'd gone 45 degrees counterclockwise instead.

Last year, I created a quick worksheet to help students perform any of the reflections and rotations mentioned in this post. (This was late in the year when we were covering PARCC questions, but now I'm giving this lesson much earlier.) It takes the coordinate plane labeled with the positive x-, negative x-, positive y-, and negative y-axes. Students can then perform the rotations on the axes to see what happens. For example, let's try our 270-degree clockwise rotation. After we rotate the paper 270 degrees clockwise, we see that where the x ought to be, we see -y instead, and where the y ought to be, it's +x. Thus the image must be (-yx).

Okay, so we've taken care of all the rotations centered at the origin, But on the PARCC, there are questions with rotations centered at other points. These questions that I've seen direct the students to take a triangle ABC and rotate it around one of its vertices -- let's say C. Well that makes things a little easier, since then the rotation image of C is C itself. So then there are only two points that we need to find, A' and B'.

It's possible, in principle, to find formulas to determine the image of (xy) under reflections in mirrors parallel to the axes and rotations centered at points other than the origin. We've seen, for example, that the point (xy) reflected in the line x = a is (2a - xy). An interesting question is, where exactly does the 2 in 2a - x come from?

To find out, we notice that if we were reflecting in the y-axis (which is parallel to x = a), then the point (xy) is mapped to (-xy). Now that extra 2a term looks just like a horizontal translation of exactly 2a units.

So somehow, our reflection in the line x = a appears to be the composite of a reflection in the y-axis and a horizontal translation. (This is not a glide reflection between the mirror is perpendicular to the direction of translation -- we found out last year that such a composite yields a simple reflection, not a glide reflection.)

Using symbols, let's call the composite transformation T. It is the composite of a y-axis reflection, which we'll call r_y, and a horizontal translation of 2a units, H_2a:

T = H_2a o r_y

But the horizontal translation is itself the composite of two reflections. The two mirrors here must be vertical mirrors spaced exactly half of 2a, or a units apart. We might as well let the two mirrors be the y-axis itself and the line x = a.

T = H_2a o r_y
    = r_(x = a) o r_y o r_y
    = r_(x = a) o I
    = r_(x = a)

which is exactly what we wanted -- a reflection in the line x = a.

Likewise, we see that the reflection in the line y = b maps (xy) to (x, 2b - y). The composite of both reflections is a 180-degree rotation about the point (ab), which maps (xy) to (2a - x, 2b - y) -- and that's also the composite of a 180-degree rotation about the origin and yet another translation.

Now 90-degree rotations about points other than the origin are even trickier, because now we'd have to reflect about mirrors with equations like y = x + b -- and we don't even know that's a line yet. The algebra involved in this reflection gets very messy.

Of course, if we try to visualize the rotation, another composite transformation jumps at us. To perform a 90-degree rotation (either clockwise or counterclockwise) about the point (ab), it appears that we can first perform the translation that maps (ab) to (0, 0), then perform the rotation centered at the origin, and finally translate (0, 0) back to (ab).

This seems to work, but is there any reason why it should? Let's use symbols again -- in order to remember what the symbols stand for, we let "rot" stand for the rotation and "trans" stand for the translation mapping (0, 0) to (ab). Then trans^-1 can stand for the inverse translation -- the one mapping (ab) to (0, 0). This gives us:

T = trans o rot o trans^-1

This composite has a name in classes like linear algebra and above -- conjugation. That is, we are conjugating the rotation by the translation.

We now want to rewrite both the translation and origin-rotation with two mirrors each. And as usual, we want to choose the mirrors strategically so that some of the reflections cancel out. For the translation, we'll let k be the line joining the points (0, 0) and (ab). Then l will be the line perpendicular to k passing through the origin, m will be the line perpendicular to k passing through the midpoint of (0, 0) and (ab), and n will be the line perpendicular to k passing through (ab). Then the rotation can be written as r_l o r_k, and the translation can be written as r_m o r_l. Notice that the inverse translation can be written r_l o r_-- but it can also be written as r_m o r_n (as either l and m, or m and n, are the correct distance apart). So we write it:

T = trans o rot o trans^-1
    = r_m o r_l o r_l o r_k o r_m o r_n
    = r_m o I o r_k o r_m o r_n
    = r_m o r_k o r_m o r_n
    = r_k o r_m o r_m o r_n (as reflections in perpendicular mirrors commute)
    = r_k o I o r_n
    = r_k o r_n

which is the composite of reflections in perpendicular mirrors intersecting at (ab). And so T is in fact the rotation centered at (ab), which is what we were expecting.

Once again, though, this is not the sort of symbolic manipulation I'd want my students to see. But then, what should we expect students to do when faced with a PARCC question where they have to rotate around a point other than the origin?

Most likely, this is something that can wait until we discuss the Slope Formula -- especially since it's this rotation that leads to the slopes of perpendicular lines. For now, one can consider such rotations only informally -- after all, the PARCC questions usually include graphs, so students might be able to perform the rotations just by counting units on the graph, rather than using an algebraic formula or manipulating mirrors and symbols.