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