# A Follow Up (regarding Centers of Rotation)

It’s 10:44pm on a Saturday evening, and I have been thinking about math. Whoooo hooooo! I finally got a chance to ponder how I’m going to attack this question that I posed earlier this week. For those too lazy to click:

Imagine you’re a geometry teacher, and you want students to discover a “foolproof method” to coming up with the center of rotation (given an original figure and a rotated figure). You also want students to be able to understand (deeply) and articulate why this method gives you the center of rotation.

Although not perfect, I have whipped some stuff up in the past hour that I hope will get to the heart of this question. Of course in my head, I have class discussions, and we gently get at this. These sheets alone don’t get us there. But if you’re interested, feel free check ’em out.

(Also, since I just whipped them up, there might be some things that need fixing/tweaking…)

There are five of them (all combined).

• Rotations of two points [after this, through discussion, get student to make the connection between the perpendicular bisector as all points equidistant from the endpoints of a segment and the radii of a circle]
• Rotation of a line segment [after this, through discussion, get students to recognize that they are really considering two perpendicular bisectors… we are looking at one perpendicular bisector to find all possible points which will rotate one end of a line segment to the new point and a second perpendicular bisector to find all possible points which will rotate the other end of a line segment to the new point… for both endpoints to simultaneously rotate to their new location, we have to look for the intersection of the perpendicular bisectors!]
• Rotation of three points [after this, make it clear that this is not the same as saying that you can trace a triangle on patty paper in two different places and find a center of rotation that will bring the first triangle to the second triangle… in fact, maybe I should have this as an exercise…]
• Center of rotation practice
• Rotations of a complex figure

(.docx)

PS. If you’re talking about multiple “center of rotation”s, do you say “centers of rotation” or “center of rotations” or “centers of rotations”? It makes me think of culs de sac, which indeed is the plural of cul de sac. Thank you Gilmore Girls.

1. Sam.
Number 2 has two centers
Number 3: It looks as though the centre of rotation is A but it doesn’t say so.

1. Hihi,

It took me a second. But then I realized you were totally right about #2. Muy interesante! If you call AB the first segment and CD the second segment, you have to find a center of rotation that maps A to C and B to D… and also a center of rotation that maps A to D and B to C! It also made me think about the times where there is NO center of rotation. That will be a fun discussion!!!

As for #3, I’m not seeing what you’re saying… there is only one place that will (when you spin the patty paper) have B and C both whoosh over the original location of point A (that place being the intersection of the three perp bisectors of the sides of triangle ABC). Am I thinking wrongly about this?

Sam

1. Hello Sam, Sorry about the delay.
I am confused by your “As for #3”.
The triangle of interest is BCD. You move D to D’ by an unspecified angle. This can be achieved with the center anywhere on the perp bisector of DD’, with corresponding angle.
I don’t get it about the original position of A, in fact, what has A got to do with it?
ps
Rotations can be through any angle, not just 90. The algebra gets a bit messy until one uses a matrix description.

2. Sam — You have to say “centers of rotation” because if you say “center of rotations,” you will die of embarrassment and we cannot have that. K? K.

– Elizabeth (@cheesemonkeysf)

1. Eeep! That’s the best thing because it IS like culs de sac. :)
I also think you way underestimate my tolerance for embarrassment.

1. jwilson828 says:

(or at least your classroom in the future)!