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Defining Rotations


Alignments to Content Standards: G-CO.A.4

Task

Consider the following possible definitions for rotation of the plane by an angle $a$ about the point $P$:

  1. If $Q$ is a point in the plane, then we send $Q$ to the point $R$ so that the measure of $\angle RPQ$ is $a$.
  2. If $Q$ is a point in the plane, then we send to $Q$ to $R$, where $|QP| = |RP|$ and the measure of $\angle RPQ$ is $a$.
  3. If $Q$ is a point in the plane, then we send to $Q$ to $R$, where $|QP| = |RP|$ and the measure of $\angle QPR$ is $a$.
  4. If $Q$ is a point in the plane and $C$ is the circle with center $P$ containing $Q$, then we send $Q$ to the point $R$ on $C$ so that the measure of $\angle QPR$ is $a$.

Which, if any, of these definitions are valid (that is, do they make sense and have the desired effect) for all points $P$ and $Q$?

IM Commentary

The goal of this task is to encourage students to be precise in their use of language when making mathematical definitions. After working on this problem, they should be prepared and encouraged to give a proper definition of a rotation by an angle $a$ about a point $P$. The rotation by an angle $a$ about $P$ maps $P$ to itself; for any point $Q \neq P$ it sends $Q$ to $R$, where $|QP| = |RP|$ and the measure of $\angle QPR$ is $a$.

Providing precise definitions for the different rigid motions of the plane is challenging. This comes in part from the fact that we have a natural visual intuition for what it means to rotate, translate, or reflect the plane. As a result, it is easy to rely on this intuition and not communicate the correct information with the definition. This task looks at some reasonable possibilities for defining rotations: each definition has at least one problem but it takes patience to exhibit cases where they fall short. Parts (c) and (d) are essentially correct and just require an additional note stating the (clear) fact that the center $P$ of rotation does not move when we apply the rotation.

Rather than asking students whether any of the definitions are valid, the teacher may prefer to tell students that none are valid and ask them determine why. Also, instead of these definitions, the teacher may wish to take proposals from (groups of) students. Although the definitions of the different transformations inevitably come early on in work with rigid motions, this task should not be done until students have some level of comfort with this material. Ideally they have already gained this in their previous courses but it is not a bad idea to postpone this discussion until later on in the course: this is because without a good stock of examples and experience with reasoning, it is difficult to identify what is good or bad with a particular definition.

Work on this task exemplifies MP6, ''Attend to Precision.'' Indeed the subtitle of this mathematical practice is ''Communicate Precisely'' which requires using language carefully and, in particular, defining concepts crisply and precisely.

Solution

  1. There are several problems with this definition. Most notably, it talks about ''the'' point $R$ so that $m(\angle RPQ) = a$ but there are multiple such points and so if we are not careful we end up changing the distance from $P$ to $Q$ after rotating. Here is an example of a point $R$ which satisfies this definition but $R$ is not the rotation of $Q$ about $P$ by an angle of $a$:

    Rotationdef1_a0dceb306558201e5b46a89dab5ea921

    This definition also suffers from the flaws of the definitions in parts (b) and (c) indicated below.

  2. This definition captures one of the essential aspects of rotations, namely that these maps do not alter distances between points. One problem with this definition is that the rotation goes from $\overrightarrow{PQ}$ to $\overrightarrow{PR}$ and therefore it is not $\angle RPQ$ but $\angle QPR$ whose measure is $a$. In other words, the number $a$ has a sign, positive for counterclockwise and negative for clockwise. This definition ''reverses'' the sign. Below is a picture:

    Rotationdef2_1768713594d0852f9024586f77b5d3d3

    The second definition has an additional flaw as well as indicated in part (c).

  3. This definition gets the direction of the rotation correct and the fact that it preserves distances. The only flaw here is a very subtle one: it does not tell us where to send the point $P$, the center of the rotation. Intuitively we know that $P$ does not move when we perform the rotation but this needs to be specified in the definition. In this definition, there is no angle formed when $Q = P$ and so the definition does not tell us what to do with $P$.

  4. This definition comes with the same picture and has the same content as (c):

    Rotationdef3_4a2e4b9946cb18e5405029112cf4e586

    Here too, something needs to be said about what we do with the point $P$. In particular the point $P$ is sent to itself. In this case, it does make sense to talk about a circle of radius 0 with center $P$: this would just be the point $P$. The angle in question, however, would be $PPP$ and this does not make sense.

Howard Phillips says:

over 2 years

Technically c) is OK, as if we are considering what happens to P then the point Q and the point P are coincident. So the circle with centre P through point Q has zero radius, all its points are coincident with P, including R. So Q goes to R but neither of them go anywhere and the angle bit is now irrelevant. Generally the CCSS is poor at dealing with special or degenerate cases of anything, which is a shame, as the general intentions are quite "mathematical". Simple example: a square is classified as a rectangle, but no mention is made of it being a special case. Poor little kiddies, they get so confused.

Cam says:

over 2 years

Hi Howard,

Thanks again for your comments. Three points:

1) I'm not sure why you think kids won't see this in CCSS. Grades K-3 have geometry standards rife with such comparisons, including many relationships between quadrilateral classes. And regardless of the language of the standards themselves, curriculum adopted which adheres to the standards would certainly address these points.

2) There's a sharp difference between degeneration and special cases. While we as mathematicians can appreciate that "a point is a degenerate circle", this perspective takes time to appreciate, and would be inappropriate to mandate for introductory HS geometry.

3) Your claim that (c) works out because the angle is irrelevant does not change the fact that the definition is ambiguous as to what to do to P. If you had curriculum that specifically introduced the notion that the angle between any point and itself could be taken to be any real number (whether or not this is a wise or sensible definition), then perhaps this definition could be workable.

kassymov says:

almost 4 years

Regarding part b. This is the only reference to negative angles I can think of in the context of CC Geometry (obviously, it's something students will when advancing with trigonometry). Should the explanation be restated, in possible, in terms of clockwise-counterclockwise? Otherwise, I can see how a lot of things will have to be made unnecessarily complicated. For example, the sum of the angle measures of a triangle is either 180 or -180, or, alternatively, it's the sum of the absolute measures of the angles.