kitchen table math, the sequel: help desk

## Thursday, January 4, 2007

### help desk I worked an "overlapping triangle" problem in Saxon Algebra 2 today that threw me for a loop.

I finally got the correct answer, but I don't understand the solution in the solution manual.

(Image from Understanding sine at Homeschoolmath.net. I can't add labels to the illustration, unfortunately.)

Look at the bottom side of the left-most triangle, the one with two overlapping triangles.

Assume that the red segment measures 6 cm, the green segment 4 cm.

I've been taught that you would find the scale factor using this equation:

6 x SF = 10

However, the solution manual shows:

6 x SF = 4

I started checking various right triangle problems to see whether you can find a correct scale factor this way....and I'm stumped.

Just based in the triangles I've looked at, the 6 x SF = 4 formula for the bottom red & green segments also holds true for the corresponding 6 and H3 segments of the hypotenuse:

6 x SF = H3

Obviously it does not hold true for the ratio between the two vertical sides labeled 2.6 and 3.9.

What's going on?

++++++++

oh wow!

The homeschool.net page explains it!

hmmm . . .

I think Saxon blew it here. This was too big a leap for me inside a problem set.

Of course, Saxon isn't supposed to be a self-teaching book.

It may be time for me to take a class.

An actual class with an actual teacher.

Though I have to say, attempting to teach myself math I've never seen before is kind of cool.

My dad told me some relative of his taught himself calculus out of a book.

I like that idea.

+++++++

Someone needs to write a sci-fi novel about homeschoolers preserving knowledge for the future.

Which reminds me, I'm still worried about the solution manual for Moise and Downs.

Once it's gone, then what?

Unknown said...

If you write in to Saxon, they will correct the error in the Answer Key. In fact, if you call in, you can have a discussion with an editor there. They are always glad to correct mistakes. (And, seriously, editors, don't get much contact with the outside world, so you may find yourself in an engaging conversation.)

You're looking at dilations, I think. Interesting that understanding (a.k.a., not procedure) here leads you right away to the error.

With the right understanding, you could do this kind of stuff in your sleep.

dank said...

Wait a minute; I'm confused.

Mr. Person, I think Catherine concluded that the scale factor numbers worked out all right. So, when she says she thinks Saxon "blew it," I believe she means that the text didn't present the material well enough to lead her to the solution--not that the solution is in error. Do I have that right?

I certainly do agree, with you though, Mr. Person: The presentation of this problem is very consistent with the topic of dilations.

Dilations is a topic I don't recall at all from my days as a student. All I remember is the similar triangles approach. I really enjoyed teaching dilations this year, though. When using shapes other than triangles and center points that are not vertices of the figure in question, it's neat to draw the lines from the pre-image points to the image points and see how everything comes together at the center. It's kind of a perspective drawing.

Catherine Johnson said...

I believe she means that the text didn't present the material well enough to lead her to the solution--not that the solution is in error.

yes, right

What is a "dilation"?

He hasn't used the term dilation thus far.

The concept he's been teaching is scale factor used to determine the length of a side in a similar triangle.

Suddenly he switched to using the term "scale factor" to mean finding an equivalent segment of a side, if that's the right term, which it probably isn't.

I didn't generalize from one use of scale factor & ratios to another.

At certain points he's said that his purpose in assigning certain problems that belong to geometry is to practice algebra. (I'll look up some of those statements & post.)

I don't understand the concept well, although I have to say that geometry doesn't seem to lend itself to "understanding" the way arithmetic does.

So much seems to depend on proofs; the whole field of geometry seems "abstract" in a way.

Is that wrong?

Catherine Johnson said...

I find that the further I go in math - and we're talking high school math, of course - the less I "understand" it.

I put "understand" in quotes to indicate a feeling of understanding, a feeling I often had while relearning arithmetic of a rich connection between the subject I was relearning and real things in real life.

That was terrific; I loved it. Probably my favorite moment was the day I read Saxon explaining that cash registers are based on the commutative property. (I was a check-out girl in high school and college.)

I don't have that feeling often anymore. I think I'm beginning to see why real mathematicians object so strenuously to making math into a concrete, real world, "thingified" subject.

Although: there are times when Saxon seems to imply that all of Algebra 2 is preliminary, that later on, in higher math and/or engineering, you come back to concrete applications.

Catherine Johnson said...

I think the first concept I encountered that strongly challenged my "feeling of understanding" was the division of a fraction by a fraction.

I spent quite a lot of time trying to achieve a feeling of understanding and failed.

Ultimately I was "hit" by understanding when a homeschool site (probably the same one!) explained multiplying-by-the-reciprocal by writing a fraction-divided-by-a-fraction problem as a complex fraction (i.e. the dividend in the numerator and the divisor in the denominator).

When you simplify the complex fraction by multiplying both the numerator and the denominator by the reciprocal of the denominator you end up with the classic multiply-by-the-denominator procedure for dividing two fractions.

Voila!

That may have been one of the first moments I "crossed-over" from conrete representations and understandings to abstractions.

Saxon's valiant attempt to explain multiplying by the reciprocal conceptually in 6/5 never "took" for me, even though I went over it many times.

I'll have to go back over it again.

Catherine Johnson said...

I only skimmed the homeschool.net website last night (and have to get going this morning) - but at this point I don't understand why this approach to the problem works.

I assume it's based on something I do know but that I've failed to see the connection.

Unknown said...

Now I'm confused. For the leftmost triangle, the scale factor is 1.5 (or 2/3, depending on which way you're going--small to big or big to small). [So, actually, it should be 6 x SF = 9.]

A dilation is another kind of transformation--in there with reflections (flips), translations (slides), and rotations (turns). Dilations are either enlargements or reductions.

It's not surprising that they don't use the term; similar figures and scale factors are often discussed separately.

The scale factor is the common ratio of corresponding side lengths in similar figures. Basically, it's the end side length over the beginning side length. For the leftmost triangle in your example, the scale factor (of enlargement) is H3 / 6, which MUST be equal to 3.9 / 2.6 if the triangles are similar. In other words, all the corresponding side lengths form equal ratios when the figures are similar.

That's about as clear as mud.

SteveH said...

Perhaps the problem is thinking that trig has to be so very complicated. It isn't.

This is my attempt to simplify the subject.

First, forget "scale factor" and "segment". They are confusing terms.

1. For trig, always use right triangles. If you don't have a right triangle, make it into two right triangles.

2. A triangle consists of three sides.

3. The long side (opposite to the right angle) is called the hypotenuse.

4. The two shorter sides are called legs. The hypotenuse is not called a leg.

5. The two acute angles that are not the right angle will be called "alpha" and "theta". Mathematicians like to use Greek symbols for angles, so you will have to get used to it. If you don't know what the angles are, then alpha and theta are the variable names for the unknown angles, just like you use 'X' and 'Y' for variables in algebra.

6. It doesn't matter which acute angle you call theta and which you call alpha, as long as you are consistent. Acutally, you could call the angles "bart" and "homer" if you want, but you might have trouble getting it published in a technical journal.

7. The sum of the angles of a right triangle equals 180 degrees. This means that alpla + theta = 90. This is important, but not right now.

8. For alpha or theta, the "adjacent" leg is the non-hypotenuse leg that defines the angle. The "opposite" leg is the other non-hypotenuse leg which is on the other or opposite side of the triangle.

9. Many word problems talk about angles and lengths, not triangles. You have to draw a picture and define some right triangles yourself. If a word problem talks about angles, then angles mean trig. Trig means right triangles.

10. How do you define these triangles? The goal is to define right triangles where you know two pieces of data (lengths or angles), not including the right angle. You could know the lengths of two sides, or you could know one angle (not the right angle) and the length of one side. If you know two pieces of information about a right triangle, then you can calculate all of the rest - using trig.

This is the key. How do you go from two pieces of data to find all of the lengths and angles.

Back to sine and cosine.

11. For similar right triangles(the alpha and theta angles are the same for each triangle), there are three ratios that are equal. These ratios can be defined for either of the two acute angles (alpha and theta). Rather than use S1, S2, and S3, I will use the terminology I defined above.

The length of an "adjacent" leg I will call 'A'

The length of an "opposite" leg I will call 'O'

The length of the hypotenuse I will call 'H'

Ratio 1: A/H is the same for all similar right triangles = cosine of the angle

Ratio 2: O/H is the same for all similar right triangles = sine of the angle

Ratio 3: O/A is the same for all similar right triangles = tangent of the angle

I like this terminology because each acute angle of a right triangle has its own sine, cosine, and tangent values, and for each acute angle, there is no confusion over what is 'A', 'O', and 'H'. No matter how you draw your triangle or how it's oriented on a drawing, there is no confusion over "adjacent", "opposite", and hypotenuse once you focus on one of the acute angles.

12. This is important. Each acute angle of a right triangle has its own sine, cosine, and tangent based on the ratios above.

How does this work in practice? If you have a right triangle and know two things from the list of 'A', 'O', 'H', alpha, and theta, you can find the other values.

If you know 'H' and the 'A' for the angle alpha, but you don't know alpha, then

cos(alpha) = A/H. [You calculate A/H and then use the cos^-1 key on the calculator.] When I started, I had to calculate A/H with long division and go to the the back of the book and interpolate to find the angle. We had interpolation races in class.

If you know only 'A' and 'O' for angle alpha, then:

tan(alpha) = O/A

If you know the angle alpha and the length of the hypotenuse, then the length of 'A' is found with

Cos(alpha) = A/H

Students need to be able to define right triangles where two pieces of information are known. They have to quickly (for either acute angle) write down the trig equation that allows them to find the rest of the data. I have found that knowing which ratios belong to sine, cosine, and tangent and using 'A', 'O', and 'H' help an awful lot.

A good practice would be to show a series of right triangles that have only two pieces of information given. They student has to find the rest.

KDeRosa said...

There you go, an entire semester worth of trig boiled down to one class. The rest is just practice.

SteveH said...

"I spent quite a lot of time trying to achieve a feeling of understanding and failed."

There are different kinds of understanding. The modern fuzzy kind of understanding seems to be a sort of Zen-like understanding that somehow magically happens via extended pattern recognition or discovery without any prior knowledge. It's very hard to define and achieve directly.

True mathematical understanding comes from axioms, definitions, proofs, and lots of practice. The purpose of this kind of mathematical understanding is to allow you to do things without any additional understanding. Does this sound strange?

A better way to put it is that mathematical understanding leads to a Zen-like understanding, not the other way around. You can use calculus to analyze equations and uncover their secrets. Math is a tool for understanding - if you follow the rules.

But then, what does it mean to "understand" the rules? By using other rules. That's what math is all about. This is quite different from rote understanding. Of course, everyone wants to understand things on a gut level, not just a definition basis, but a gut-level understanding does not necessarily lead to a mathematical understanding. I see this in Everyday Math. They attack understanding from a gut-level or common sense approach, but never quite get to a proper mathematical definition approach. They go from simple descriptive examples to rote methods and bypass the formal mathematical definitions. They think that's enough.

A Zen-like understanding will come in time, but it has to be a by-product of a formal understanding of the definitions and rules of math. Practice is the true path to mathematical enlightenment. Anonymous said...

"Practice is the true path to mathematical enlightenment."

This is what Keith Devlin, NPR's math guy, says:

"Expertise does not come from understanding, it comes from practice. The part of our brain that provides conscious understanding did not evolve to control and direct our detailed actions, it evolved to make sense of them --- after the fact. (The benefit of that sense making is that we can make use of our understanding to guide future action at a higher, more strategic level).

...

Understanding follows experience.

[But M]astery of skills without understanding is shallow, brittle and subject to rapid decay.... Understanding mathematical concepts is crucially important to mastering math. But because of the way the human brain works, that understanding can arise only as a consequence of practice .... lots of it."

I don't know if he's right about evolution, but his explanation makes sense to me. It suggests that the Everyday Math approach is highly unnatural. And his theory probably makes a lot of sense with higher level math which is so far removed from concrete, everyday experience.

RDW said...

Your explanation of trig was very clear.... so clear, that even I could understand it. That kind of big picture overview is also very important in any math class. That, followed by practice.
(It's also missing in Everyday Math.)

I don't remember having trig explained to me that clearly in high school, but maybe I just wasn't paying attention.

SteveH said...

"I don't remember having trig explained to me that clearly in high school, but maybe I just wasn't paying attention."

It wasn't explained to me like this when I had "traditional" trig in high school. This approach kind of evolved over the years while I tackled geometry problems. Add to this some very basic ideas of vectors, dot products, and cross products and you have the beginnings of a very powerful geometry toolbox.

I wrote it off the top of my head so I wasn't as careful as I should have been. One of the pieces of information about the triangle has to be a length. If you know one of the acute angles, you can figure out the other, but you can't determine any of the lengths. So, when you create your triangles from your word problems, the length of one of the sides of the triangle must be known.

After a little bit of practice, you can look at a problem showing an angle and hypotenuse and quickly write down that the length of the adjacent side is H*cosine(angle). It's very important for students to "see" these triangles and relationships very quickly.

Unknown said...

"So much seems to depend on proofs; the whole field of geometry seems "abstract" in a way.

Is that wrong?"

No, not wrong. One of the major purposes of geometry is the proofs -- that is, learning basic logic.

SteveH said...

"A dilation is another kind of transformation--in there with reflections (flips), translations (slides), and rotations (turns). Dilations are either enlargements or reductions."

Since when did "they" start calling it "dilation"? Does that sound more sexy than "scaling". In all of my computer graphics texts, (including the 2000+ pages of the Graphics Gems series), there is not one word about "dilation". Is this a "Sketchpad" term? Of course, dilation just refers to getting wider or larger; scaling means getting larger or smaller.

Newman/Sproull calls it a scaling transformation. Foley/vanDam calls it scaling. Rogers/Adams calls it scaling. Not one word of dilation in Mortenson.

I did a search on "math and dilation" and found most all references related to education. I even saw a description for dilation related to the Regents Math A exam!

I don't know why, but this really struck a nerve with me. Do "educators" like to make up their own terms? Are they completely ignorant of the technical literature or do they like to substitute their own cutesy terms?

I suspect that "dilation" is an old math term that is finding renewed popularity in the education community, but I have never heard of the term in the last 30+ years of working with computer graphics and geometric modeling.

Unknown said...

There are examples here and here.

Publishers match language and meaning to state standards. New York uses the term "dilation" in the 8th grade, and I believe they use the term to cover both enlargements and reductions:

8.G.7 Describe and identify transformations in the plane, using proper Geometry function notation (rotations, reflections, translations, and dilations)

8.G.11 Draw the image of a figure under a dilation

8.G.12 Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation

SteveH said...

From Wolfram Mathworld:

"A dilation corresponds to an expansion plus a translation."

"Expansion is an affine transformation (sometimes called an enlargement or dilation) in which the scale is increased."

[translation is] "A transformation consisting of a constant offset with no rotation or distortion."

A dilation means an increase in size, not a decrease. There is a separate transform (contraction) that does the decrease in size.

" ... and I believe they use the term to cover both enlargements and reductions:"

Which is wrong by definition.

From www.regentsprep.org

"A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. The description of a dilation includes the scale factor and the center of the dilation."

This is wrong if they allow smaller sizes.

This is another "educational" definition of dilation I found.

"Definition: A geometric transformation that changes a figure's size, but its shape, orientation, and location stay the same
Context: The animator performed a geometric dilation in order to create the illusion that the figure was getting larger without changing its shape, orientation, or location."

This is even more wrong. There is a change of location. The translation could be zero, but that doesn't mean that the object won't move. Confusing? Not if you really understand what is going on. The problem is that for any object, what doesn't move? If you resize something, it moves. Moves relative to what? It's center? What is the center of an object?

When you scale an object, it has to be relative to a scaling origin, what they call the center of dilation. The center does not move, but the center could be anywhere near or far from the object. The center of scaling is not (necessarily) the center of the object. The [X,Y,Z] coordinates of the object always move unless they are located at the center point. When you apply the transformation to all of the points that define an object, the only point that won't move is the one (if any) located at the center of dilation.

Am I being picky here? I don't think so. This all has to do with being mathematically correct and careful. It has to do with understanding. How many kids exposed to dilation really know that it is two transformations? How many kids could write down both the scale factor and the translation coordinates? How many kids will be able to create the one transformation that combines both a scaling and a translation, given the center of dilation and the scaling factor? How many will learn that to define a dilation, you really need three(!) basic transformations? How many will know that these three transforms can be concatenated and multiplied into one (more complicated) transformation? How many kids will know that if they become a computer animation specialist at Pixar, the only time they will hear the word "dilation" is when they talk about the scaling transformations that will make a pupil grow larger?

If educators want to preach about "real world" applications, then they better do their homework. I can't imagine the misconceptions "dilation" students will have when they get to a real course in matrix transformations.

Unknown said...

It is interesting that the Regents site defines dilations as covering both expansions (enlargements) and contractions (reductions), yet the more mathy Wolfram deals with dilations only as enlargements, and--as you mentioned--refers to reductions separately.

Two professors of mathematics, Carolyn Goldberg and Thomas Tucker are listed as contributors on the 2005 NY Learning Standards for Mathematics document. It would be nice to know what they think about the differing definitions. I'll put in E-mails to them and see what happens.

Unknown said...

Well, that was fast. This was the response from Professor Tucker:

As far as I know, dilations mean both enlargement and reduction (that way, the inverse of a dilation is also a dilation). Out of curiosity, I did Google search, and the math sites I found all dealt with dilations this way.

dank said...

I am not speaking from a position of rigorous knowledge. My opinion, worth at most two cents, is that the competing definitions of dilation are just semantic differences. You can choose to use whichever convention you like, as long as you are consistent.

The text we use in my geometry class says that a dilation with a scale factor greater than 1 is an enlargement. A dilation with a scale factor less than one is a reduction.

I can live with that. By way of analogy, consider acceleration in physics. If the acceleration acts to slow a moving object, is it okay to call it "deceleration"? Sure. As long as we understand acceleration to be a vector quantity with a magnitude and direction, we know that acceleration can be in the positive or negative direction. We also know that it can act in any direction relative to the motion of the object. So, the term acceleration is really all we need. That doesn't mean we need to outlaw the word deceleration. In some contexts it is more descriptive.

Similarly, we can use different terms for enlargement and reduction, or we can call them both cases of dilations.

SteveH said...

It appears that dilation is whatever one wants it to be. I managed to find a usage in the second edition of Mortenson's Geometric Modeling (1997), but it's not used in his classic first edition (1985). His second edition uses the term interchangeably with a simple scaling transform (no translation). I don't see dilation used in any of my other books, even some classic books on geometry and computer graphics.

The problem with dilation is that it is not a common term, not that it is used for both enlarging and reducing. It is also not just about scaling. It's not a term used by people in the real or academic (beyond K-12) worlds. It's surprising that it was picked up by the K-12 crowd.

"I did Google search, and the math sites I found all dealt with dilations this way."

That's because all of these sites are K-12 education sites.

It's also not a clearly defined term, although most educational examples define it with a scaling factor and a center of dilation. The explanations get a little bit confusing when the center of dilation is not at the origin. They kind of ignore the fact that it is really two transformations; three, if you want to construct it from the data.

For example, in the regentsprep.org examples, if the center of dilation is at the origin, you have to multiply the coordinates by the scale factor. If the center is somewhere else, they tell you to multiply the distances. Two different methods for what should be just one method if you really understand what is going on.

All of this reminds me of the superficial symmetry stuff my son gets in Everyday Math. At the Regents level, no one is talking about matrix transformations and concatenations, so they just require a superficial descriptive understanding of transformations that will do them little good in advanced math classes. These kids will start talking about dilations and nobody will know what they are talking about and they won't be able to explain themselves in mathematical terms.

dank said...

I don't have a problem with the two different methods you mention, Steve. Obviously, when the center is the origin, it is just simpler to multiply the coordinates. But, as you say, it's still a consistent transformation with the one centered at any other point. It's just a simpler method for the easy case.

It's kind of like using Pythagorean triples. If your recognize that the sides of a right triangle are multiples of 3, 4, and 5, you don't need to bother working with the squares. That's a different method for a simpler case. I see no problem in teaching it.

SteveH said...

1. Dilation is not a common term outside of K-12 math. It is not used out in the "real world". I find it amazing that it has been adopted by the K-12 math crowd. I suspect that it's because it allows scaling relative to any center. This hides a lot of the mathematical details ... and understanding.

2. Dilation (no matter what the scaling factor) is really two transformations in one. Do the students understand this?

3. Do the students know how to define the two transformations, given a scaling factor and center of dilation? How about just writing down one method that will work for an arbitrary center of dilation. Actually, it's easier to define with three transformations. First, translate the points from the center of dilation to the origin. Second, apply the scaling transform to the coordinate points. Third, translate the points back to the center of dilation. Three matrices that can be multiplied together to get one transformation matrix. I suspect that at the Regents level they don't talk about transformation matrices, so what's the goal of teaching these things?

4. If they approach the problem in two different ways depending on the center of dilation, do the kids understand why they can do this? This is not about shortcuts. It has to do with understanding - a mathematical basis for doing the steps. Can they explain why the two different methods are equal? With 3,4 5 as the sides of a right triangle, students know that they can plug the numbers into the formula to show it works. Can they do the same when the center of dilation is not at the origin?

5. Outside of a descriptive understanding of basic transforms, what underlying mathematical concepts have they learned? If they mathematically understand what a dilation is, then they should be able to write down the equations that will map any coordinate point (X,Y) to its new transformed (dilated) position (X',Y'), given the (h,k) center of dilation and the scaling factor 's'.

Like this:

X' = (X-h)*s + h

Y' = (Y-k)*s + k

If the center is at the origin (h=0; k=0), then the formulas reduce to:

X' = X*s

Y' = Y*s

Students should be able to learn this without knowing about matrix transformations. With the center at the origin, it is a pure scaling transformation. When the center is somewhere else, there is a translation. If you multiply out the first equations, you get:

X' = X*s - h*s + h

Y' = Y*s - k*s + k

The scaling is still defined by 's', but the translation is:

h - h*s

for 'X' and

k - k*s

for 'Y'

Two transformations; one scaling and one translation.

These equations can also be derived using the three transformation steps I gave above.

Translate back to the origin

(X-h)

(Y-k)

Scale

(X-h)*s

(Y-h)*s

Translate back to the center of dilation

(X-h)*s + h

(Y-k)*s + k

This is the sort of mathematical understanding I would expect a student to know. Add to that the fact that they will probably never hear the word dilation after they get out of high school.

dank said...

Steve,

This will be my last post on this topic. It seems that I have fallen into one side of an argument here, and I am arguing largely from a basis of ignorance. My only acquaintance with dilations has been the way it’s presented in Prentice-Hall’s Geometry: Tools for a Changing World. I don’t know how widely dilations are taught or used. I also don’t know how much definitions of dilations vary. My only point is that I think the way I have seen it explained, it is a useful transformation and perfectly mathematically reasonable. Who am I to make such a judgment? Nobody, really. When I say that this is my humble opinion, there is no snark involved. Still, I will take the time to explain what I like about dilations. It will probably turn into a long (and, alas, boring) post, but it will cover all my thoughts on the topic. Feel free to correct or rebut me.

One thing that I like about dilations can be illustrated by the original post in this thread on the KTM main page. The original diagram of overlapping triangles can be expressed very cleanly using a dilation. For informal reference, let’s give things some labels. Let’s say the red triangle is the original image and the green triangle is the transformed image. Let’s label the vertices of the red triangle X, Y, and Z, in a clockwise direction, starting with the point to the left. So, the angle at vertex Z is the right angle. From the discussion, it’s not entirely clear to me what the side lengths are supposed to be. So, I’ll pick my own: Let’s say the hypotenuse has length 6. I believe it is fairly common notation to write this as XY = 6. We’ll set the horizontal leg to 5: XZ = 5; and that leaves the vertical leg equal to the square root of 11: YZ ~= 3.3.

Now, let’s consider the green triangle. Again we will start at the left vertex and go clockwise, labeling the vertices X’, Y’ and Z’. Let’s further say that X’Y’ = 12, X’Z’ = 10, and Y’Z’ = 6.6. Since X’ is exactly the same point as X, we can say that the green triangle is the product of a dilation (enlargement) with a scaling factor of 2 and a center at X. Presenting the green triangle that way describes it completely. It doesn’t matter what the coordinates of any of the points are. X could be the origin or some point in quadrant IV. The way we have expressed the dilation tells us exactly how to get from XYZ to X’Y’Z’. Now, if we wish to pay attention to the coordinates, the dilation covers them just fine. For any point P, we know that XP’ = XP * (scaling factor) and that the line XP’ (I apologize for not being able to put the two-headed arrow over the vertices) is the same as line XP. We can use the distance formula in the coordinate plane to compute XP and XP’ if we like.

Now, I may be unaware of a common convention, but I think there is less precision in saying that the green triangle is similar to the red one, with a similarity ratio of 2. You see, that doesn’t tell us that X = X’. Of course, it also doesn’t limit us: the similar triangle could be rotated or reflected or translated and still be similar. With the dilation, we know exactly where the transformed image is and how it is oriented. I like that. We could also position the green triangle so that Z’ = Z. I guess we would still say that it is similar to the red one with a similarity ratio of 2. But we’d also have to specify that Z’ = Z, and the orientation had not changed. To me it is more succinct—and in my humble opinion, more elegant—to say that the green triangle is the product of a dilation with scale factor 2, centered at Z. With that description, I know exactly where the green triangle is.

Do people in the real world use dilations? I don’t know. I’ll take your word for it that they don’t. I don’t get exposure to a lot of the real world. I can think of a real world example that I would like to have explained in terms of dilations—but it is the product of my own persistent ignorance. When I set up a copier machine to enlarge by a factor of 150%, I’m never sure just where to place my original. My problem is that I don’t know where the center of the dilation is. If the enlargement is a dilation centered at the upper right hand corner of the paper, then I would know exactly where to place what I wanted enlarged to get the image I wanted. By the same token, the dilation could be centered at the center of the 8.5 x 11 page (at coordinates (4.25, 5.5) from the corner, if you will). Now, it may be that nobody in the copier industry or the printed media business uses the term dilation. I just wish they did. Then they would have a succinct way of describing exactly what happens when I program in an enlargement for my copy.

I did a little experiment in MS Powerpoint. I drew some arbitrary rectangle. By default, the sides were horizontal and vertical, and that’s fine with me. I then did a copy-and-paste to place an identical rectangle over the first. I right clicked on the second image and selected “Format AutoShape.” From the resulting page, I chose the “size” tab. There, I clicked to lock the aspect ratio, and scaled the image by 200%. I then observed the resulting image and compared it to my original image. To me, the clearest way to describe what happened is to say that there was a dilation centered at the upper left vertex with a scale factor of 2. I have no idea what the coordinates of any of the points on those rectangles were. Still, I can fully describe the transformation in these terms. That’s excellent to me. And don’t take my dismissal of the coordinates the wrong way. If I knew the coordinates, I could tell you exactly the coordinates of the transformed image, too.

I did a second Powerpoint experiment. I drew a right triangle oriented like the red one in the original post. Then I scaled it in the same way. Interestingly, the software did not use any vertex of the original image as the center of the dilation. I think this is because there was no upper left corner in this case. What the software did was to seemingly impose a rectangular window around the figure, then use the upper left corner of that window as the center of the dilation. So, the transformed image had no points in common with the original image. Still, I think the simplest way to describe the transformation is to say that it was a dilation with a scale factor of 2 about a center whose x-coordinate is the x-coordinate of point X and whose y-coordinate is the y-coordinate of point Y. The dilation terminology just seems like an elegant way to describe it. Now the software engineers who wrote Powerpoint may never have designed it to be a dilation, but that term describes exactly what it is. I have no idea whether programmers in Java or graphics-oriented languages refer to these things as dilations. If they don’t, then that’s an argument that dilations is an unknown term in the real world. But if they use the term “flip,” does that mean that “reflection” doesn’t pass real world muster?

I’m not sure I followed your discussion of the matrix representation of dilations. I’m sure you’re right, but I just didn’t follow. When my class discussed dilations we did it more in terms of the coordinate plane, the distance formula, slopes, and intersections (at the center). The matrix notation was mentioned as a notational option. The transformation was not motivated from a transform matrix basis. As an aside, I’ll mention that in my past engineering work in speech coding, we represented the math for our processing of speech samples in matrix form. We did that because it was a convenient notation for describing our work in papers. We didn’t start by thinking of 40 consecutive samples of speech as a 40-dimensional space. Still, once we had expressed our processing in matrix form, it did lead us to take advantage of some matrix structures and manipulations. I look upon dilations the same way. I don’t see them as based in matrices. Once you represent it in matrices, though, you have the potential of exploiting benefits of using matrices. I don’t see how that fuzzes up the math at all.

Finally, our discussion of dilations did not present it as two different methods, where one used the origin as the center and one did not. If anything, the two cases we illustrated were where the center was a vertex of the object and not. But in either case, we found the position of the image point (P’) by finding the distance from the center to P and applying the scaling factor. It was only one transformation. The fact is that things do get simpler if the center is at the origin. If I apply a dilation of scale factor 3 centered at (0,0) to point P(2,5), it’s really easy to compute that P’ has coordinates (6,15). To me, that simplification is not somehow cheating the math; it grows out of understanding slope and distance. To me, it’s a shortcut just like recognizing and applying a Pythagorean triple.

Finally (Sorry, again, readers. I warned you I’d be wordy.), I think I misunderstand the objection that a dilation is really three transformations in one. That may not be what you were saying. I didn’t fully understand. Assuming that what you meant is that we don’t need to define some “new” transform when its function can be accomplished by three others that we already have, I don’t get it. In conditional logic, we know how to form the converse, inverse, and contrapositive of a conditional statement. Is the contrapositive a bad thing to define because we can form it by applying both the converse and the inverse? That’s probably a bad analogy. I think I missed the point.

Anyway, I will summarize by repeating that I like the dilation as a transformation. The way I know it to be described, it is succinct and mathematically sound. I think it is a useful term in describing things like photocopier enlargements and reductions or perspective drawings. I know of no example of the term being used in the real world, but now that I know the term, I think I could understand things better if the term were commonly used.

SteveH said...

"Anyway, I will summarize by repeating that I like the dilation as a transformation. The way I know it to be described, it is succinct and mathematically sound."

Respectfully,

I never said it was unsound, but nobody out in the real world calls it dilation.

It is also a common function in computer graphics libraries - pass in the scaling factor and a center for the scaling, and get back the 4 X 4 transformation matrix used to multiply the coordinates that define an object. I wrote a routine to do this for my graphics library.

The question is what does the course expect the students to know, just a descriptive understanding of the transform, or some underlying mathematical understanding? Are they learning about the general idea of transformations, or are they just getting a flavor for the subject?

Perhaps most importantly, if students are required to apply the transform on [X,Y] coordinates, they need to know some of the math that backs up their calculations. In other words, can they write down and understand the formulas I wrote above? They might be able to figure it out in their heads when they look at a picture, but can they translate that into the equations? It doesn't matter whether they use equations or 2 or 3 transformations, there should be some mathematical definition.

This also applies to rotation, where you have a center for the rotation. However, they don't give that a separate name. There is still the confusion over simple transforms (rotation about the origin), versus combined transforms (rotation plus translation).

It would be better to teach students that a simple scaling or rotation transform ALWAYS uses the origin. Since this is not very useful, you need to use additional translation transforms to deal with scaling or rotation at different locations. A number of the dilation links I saw seemed to ignore or gloss over these details.

Unknown said...

A final E-mail response to the dilation question I threw out there. This one's from Dr. Stephen West:

Professor Carolyn Goldberg forwarded your email regarding the definition of dilation to me to answer.

In the current mathematics vernacular, the term dilation does include both enlargements and reductions (depending on the size of factor of similitude).

The most common definition of a dilation is: A transformation of the plane such that if O is a fixed point, k is a non-zero real number, and P' is the image of point P, then O, P and P' are collinear and OP'/OP = k. If k > 1 then O-P-P'; if 0 < k < 1 then )-P'-P and if k < 0 then P'-O-P. (Note if k = 1 then P' = P.)

The term contraction, although very descriptive, is not commonly used. I personally prefer the term homothety.