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Let’s use the definition of Similarity given in the standards:
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
We’ll start by showing any two circles are the same. Take any two circles, and slap some Cartesian Coordinates on them, such that the first is at the origin. Translate the second circle to the origin, then dilate it until the radii match. Thus the pair of circles is similar.
If any two circles are similar, then all circles are similar by transitivity of similarity. QED
That explanation sounds like circular reasoning to me. How is that a proof? You state what similar means and then say since they are the same shape, they are similar. Are there any axioms involved? Sorry if I’m being thick, but I really don’t get this.
You can’t prove they are similar until you have a definition of similar. The definition is based on transformations and congruence. The axioms are not being invoked directly, but everything done is undergirded by them.
Jim-“Prove all circles are similar” is a specific common core standard that ALL high school students are supposed to be able to do. But your reply you state “slap some Cartesian coordinates on them” and then, in my opinion, just do a bit of handwaving.
Are you saying all Euclidean geometry is now dependent on coordinates?
Most high school students, teachers ,and textbooks do not think about similarity in this way .
Traditionally, Euclidean Geometry did not depend on a coordinate system for meaning. This is new to me. When, and who, made this decision?
Jim-“Prove all circles are similar” is a specific common core standard that ALL high school students are supposed to be able to do. But your reply you state “slap some Cartesian coordinates on them” and then, in my opinion, just do a bit of handwaving.
Are you saying all Euclidean geometry is now dependent on coordinates?
Most high school students, teachers ,and textbooks do not think about similarity in this way .
Traditionally, Euclidean Geometry did not depend on a coordinate system for meaning. This is new to me. When, and who, made this decision?
Dan,
Sherry asked “How do you prove all circles are similar?” (emphasis added).
I gave my answer. In no way did I suggest it is the only proof. One of the beautiful things about math is that there are sometimes many ways to prove the same thing. So no, Euclidean geometry is not dependent on Cartesian coordinates, but coordinate proofs are one of the tools in the toolbox even of the high school geometry student.
Additionally, I was outlining, rather than detailing the proof, so it may have looked like handwaving because I was allowing the reader to fill in the details.
Since it seems more clarity is called for, feel free to see my extended explanation below:
To show: All circles are similar.Similarity of two figures is defined as obtaining the second “from the first by a sequence of rotations, reflections, translations, and dilations.” I intend to show that any circle is similar to the unit circle, and that the unit circle is similar to any circle. Since a combination of two sequences of the above transformations is still a sequence of the above transformations, this would succeed in showing that the two circles are similar. Without Loss Of Generality place a circle in the plane centered at (h,k) with radius r. It can be described by the equation (x-h)^2+(y-k)^2=r^2. Apply the transformation (x,y)->(x-h,y-k) which we’ll call T_1. Then apply the dilation (x,y)->(x/r,y/r) which we’ll call D_1. The sequence T_1, D_1 transforms any circle to the unit circle. Let T_2, D_2 be transformations that likewise take (x-g)^2+(y-j)^2=s^2 to the origin. Because translations and dilations are both invertible, the sequence (D_2)^(-1), (T_2)^(-1) transforms the unit circle into this second arbitrary circle (x-g)^2+(y-j)^2=s^2. So T_1, D_1, (D_2)^(-1), (T_2)^(-1) is a sequence of translations and dilations which allows you to obtain any circle in the plane from any other circle in the plane. QED
This version has less handwaving. It also has a lot more notation and makes the concept of the proof ugly and obscured.
I think the expectation for a high school student is more along the lines of my original argument:
You can take two circles and move one so that they have the same center, then dilate it so they are the same size. QED
Jim-
“Proof is making obvious what was not obvious.” -Rene Descartes
I already know all that you wrote, and could have written it myself. But to a sophomore it’s all irrelevant BS. Why must such a simple idea be made so complicated?
I guess what I’m really asking is why must all students in US high schools be expected to “Prove all circles are similar” when it is obvious?
Re your simple version: “You can take two circles and move one so that they have the same center, then dilate it so they are the same size. QED”
If that’s all there is to do, then why in the world is this trivial idea a specific standard?Jim,
Your “proof” seems to be proof by definition. I am a curriculum developer and have been trying to come up with some activity or activities for students that address this standard.
Maybe the authors of the Common Core can tell us why this is a standard, how you prove it, and what you would do in the classroom to prepare students for an assessment item on this standard. Yes, PARCC plans on assessing this standard. Perhaps Bill McCallum, Jason Zimba, or Phil Daro can tell us why this is a standard and how to prove it. Can you help us out here?Jim basically had it right in his first post, but Dan and Sherry were correct that the coordinates were unnecessary, and that the key point went by very quickly. To my mind the key sentence in Jim’s proof is “then dilate it until the radii match.”
Why is it even possible to dilate until the radii match? Because by definition all the points on the circle are the same distance (the radius) from the center. So that means that if you dilate the smaller circle from the center, all its points will arrive at the larger circle at the same time.
In more detail: Given two circles, translate the first one so that its center coincides with the center of the second circle. If the first circle has radius r and the second circle has radius R, then perform a dilation on the first one from its center with scale factor k = R/r. Since every point on the first circle is a distance r from the center, every point on the dilated circle will be a distance kr = R from the center, so the dilated circle is identical to the second circle.
This would be easier to explain with visual aids, of course.
As for activities to support this, I can imagine having students play around with a dynamic geometry program, and asking them perform similarity transformations that map circles onto each other. At first maybe using the mouse, but then by giving the precise commands: “perform the translation that takes O to O'” and “dilate around O’ with scale factor 1.2”. To find the scale factor they would have to realize it is the ratio of the radii. Then you could ask what it is about a circle that makes this work (it has a constant radius).
I started a forum thread with some curriculum ideas here http://commoncoretools.me/forums/topic/prove-that-all-circles-are-similar-2/#post-1966.
Take a look at this site.
http://learnzillion.com/lessonsets/427-prove-that-all-circles-are-similarConsider the first “proof” in this link but instead, start with a circle radius 2 and a square with side 4.
Then please correct me if I’m wrong, but doesn’t the “similar triangles” proof written out in this link also prove that the 2 unit circle and 4 unit square are similar?Yes, the first proof is just wrong, and also misguided. The similarity of circles follows very directly from the definition of similarity in terms of dilations, as explained above. Trying to go via similar triangles strikes me as extremely irrational. And the review of triangle similarity is out of sync with the standards.
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