Do other states even have to teach Locus? I have a feeling it might be one of those annoying NY topics that they like to cram in the curriculum for no good reason.
Anyway, Locus is fun for me to teach since I cracked the code.
Just in case anyone is not on the same page yet, "Locus" comes from the Latin for "place", and we can think of it as a location in space. More specifically, a collection of points that fit some description.
For example, the locus of points equidistant from a given point is...a circle. The locus of points equidistant from two given points is...the perpendicular bisector of the segment connecting those two points. You can have more fun in 3D. The collection of points equidistant from a line is...a cylinder. Etc.
This leads to problems like...Two statues are 10 meters apart and connected by a fence. A tree will be planted equidistant from the statues and also 13 meters from one of the statues. How many possible locations are there for the tree? How far away from the fence will it be? Problems pinpointing the location of buried treasure are also very popular.
OK so back to cracking the code. Our success was limited while I stuck to having the kids draw pictures. We were marginally more successful when graph paper was incorporated. It's much easier to grok "4 units away" and "equidistant" when you can count distances on graph paper.
The breakthrough came from getting the concept off the paper. For a class learning the material for the first time, I acquired different-colored tape. On the floor, I used the tape to make two parallel lines, and one line intersecting them. I also used the tape to make a few small X's on the floor representing points. That took a little prior planning, $10 at the hardware store, plus, the custodians yell at you if you don't get all the tape off the floor.
In the class, we pushed all the desks aside, and as a group the kids were given vocal instructions by me like, "Everyone stand so that you are 4 floor tiles away from the green line." "Everyone stand so that you are 4 floor tiles away from the blue X." "Everyone stand so that you are equidistant from the blue X and green X." After each instruction, we discussed the shape they had arranged themselves in. I proclaimed them "locus experts!" before they had even touched pen to paper. This lesson stuck. I heard stories from other teachers and students ("Today I was a perpendicular bisector!") The rest of the locus unit was much easier for them and me to deal with and scores even improved.
This year I am sadly not teaching Geometry. No Locus unit. However, I am teaching a section of Honors Precalculus. Today I was introducing the parabola as the locus of points equidistant from the focus and directrix, and x^2 = 4py. With a gleam in my eye I told the kids to shove the desks to the side of the room, and got out my tape. After a few warm-up human loci, I threw my keys on the floor, and told them to stand so that they were equidistant from the keys and the wall.
A parabola materialized. Made of children.
Then I scooted the set of keys closer to the wall and said, "Do it again."