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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
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.
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
Not directly related to Geometry, but another minor inconsistency throughout the standards:
‘multi-step’ is sometimes hyphenated and sometimes not.
(I noticed because I was searching for ‘multi-step’ and only found some of the references).
Seems to me that falls under 1.OA.6 not 1.OA.3 seeing as these are generic strategies and not strategies based on properties of operations:
1.OA.6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
It is “a later standard” in the since that it occurs after this standard in the document. However, they are both Grade 1 standards and can therefore be taught in whatever order is deemed appropriate.
I have the same question as Lynda:
“when are students expected to know the formula for the volume of a pyramid?”
“3.NF.3. b. Recognize and generate simple equivalent fractions, e.g., 1/2 =
2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by
using a visual fraction model.”“In Grade 4, instructional time should focus on three critical areas: […]
developing an understanding of fraction
equivalence,”
“Because the equations 28/4=7 and 36/4=9 tell us that
28=4*7 and 36=4*9, this is the fundamental fact in disguise:
(4*7)/(4*9)=7/9
It is possible to over-emphasize the importance of simplifying fractions in this way. There is no mathematical reason why fractions must be written in simplified form, although it may be convenient to do so in some cases.”
http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_nf_35_2011_08_12.pdf?#page=4
See the comments contained within the standards on this issue here:
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf?#page=83
1.MD.1
Order three objects by length; compare the lengths of two objects indirectly by using a third object. a string to represent their forear’s length
From the progressions document:
“Students build these competencies, often more slowly, in the domain of three-dimensional shapes. For example, students may intentionally combine two right triangular prisms to create a right rectangular prism, and recognize that each triangular prism is half of the rectangular prism. 1.G.3”
It seems to me that composing three-dimensional shapes falls under the standard 1.G.2, not 1.G.3 as indicated. I see that there is the concept of two halves making a whole, a concept that a student might transfer from learning 1.G.3, but looking at 1.G.3 reveals that it limits itself to the realm of 2D shapes (circles and rectangles).
I’d like to see 1.G.3 addressed more fully in the Progressions document.
