CCSS Math: HSG.CO.A.4
Read a dialog where a student and a teacher work towards defining rotations as precisely as possible.
The dialog below is between a teacher and a student. Their goal is to describe rotations in general using precise mathematical language. As you'll see, the student must revise their definition several times to make it more and more precise. Enjoy!
Teacher:
Today we will try to describe what rotations do in a general way.
Suppose we have a rotation by θ\theta degrees about the point PP. How would you describe the effect of this rotation on another point AA?
Student:
What do you mean? How can I know what the rotation does to AA when I don't know anything about it?
Teacher:
It's true that you don't know anything about this specific rotation, but all rotations behave in a similar way. Can you think of any way to describe what the rotation does to AA?
Student:
Hmmmm... Let me think... Well, I guess that AA moves to a different position in relation to PP. For example, if AA was to the right of PP, maybe it's now above PP or something like that. This depends on how big θ\theta is.
Teacher:
Neat. We can describe what you just said as follows:
Suppose the rotation maps AA to the point BB, then the angle between the line segments PA\overline{PA} and PB\overline{PB} is θ\theta.
Student:
Yes, I agree with this definition.
Teacher:
Remember, however, that in mathematics we should be very precise. Is there just one way to create an angle P\angle P that is equal to θ\theta\,?
Student:
Let me see... No, there are two ways to create such an angle: clockwise and counterclockwise.
Teacher:
Right! Rotations are performed counterclockwise, and our definition should recognize that:
A rotation by θ\theta degrees about point PP moves any point AA counterclockwise to a point BB where mAPB=θm\angle APB=\theta.
Of course, if θ\theta is given as a negative measure, the rotation is in the opposite direction, which is clockwise.
Student:
Cool. Are we done?
Teacher:
You tell me. The definition should make it absolutely clear where AA is mapped to. In other words, there should only be one point that matches the description of BB.
Is there only one point that creates a counterclockwise angle that is equal to θ\theta\,?
Student:
I think so... Wait! No! There are many points that create this angle! Any point on the ray coming from PP towards BB has an angle of θ\theta with AA.
Teacher:
Good observation! So, can you think of a way to make our definition better?
Student:
Yes, in addition to the angle being equal to θ\theta, the distance from PP should stay the same. I think you can define this mathematically as PA=PBPA=PB.
Teacher:
Well done! We can summarize all of our work in the following definition:
A rotation by θ\theta degrees about point PP moves any point AA counterclockwise to a point BB where PA=PBPA=PB and mAPB=θm\angle APB=\theta.
Student:
Wow, this is very precise!
Teacher:
Indeed. As a bonus, let me show you another way to define rotations:
A rotation by θ\theta degrees about point PP moves any point AA counterclockwise to a point BB such that both AA and BB are on the same circle centered at PP, and mAPB=θm\angle APB=\theta.
Student:
Yes, this also works because all the points on a circle have the same distance from the center.
Teacher:
That's right! The main difference between the two definitions is that the first uses line segments and the second uses a circle.
Student:
Cool. So is that it?
Teacher:
Yes. I think we've defined rotations as precisely as we can.