Tuesday, March 10, 2009

Locus

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."

Matt said...

Bravo! That is pretty sweet.

unapologetic said...

I'm not sure. I know I learned about loci of points (that's the plural, btw) in Maryland, but I was in an unusual program within the local public school system, and I don't know if the normal curriculum included it (with or without the term "locus").

What I do know is that I'm teaching college algebra now, and it's pretty obvious that if Kentucky schools require it none of the students remember it. Not that I have to teach it myself, but it's pretty invaluable in translating word problems into mathematical relations.

And you can tell your students that it goes all the way up. Earlier this semester, I gave my multivariable calculus students a problem about describing the plane of points equidistant from two points in space. They seemed genuinely surprised that you can just write down the formula for distance to the first point and the formula for distance to the second point, and set the two of them equal to each other. Maybe a little locus practice back in high school would have helped.

Kate said...

Thanks for the perspective, unapologetic. I appreciate it and will pass it on. Descriptions of shapes and relationships are always tough - I'll be more appreciative of the practice.

Thanks Matt!

sumidiot said...

Sounds like fun! I should try getting my students up and moving more often in class.

Sarah Cannon said...

You piqued my curiosity, so I flipped back through my standards. Nothing in SD that explicitly says locus and nothing in them jumped out at me.

It makes sense as something to try to incorporate, but I won't add it in until we're good with other material. (Sorry Unapologetic. But we're so far behind in other arenas. If it fits in, I'll try to incorporate to help them at your level.)

unapologetic said...

Sarah, I fully understand that there are more fundamental concepts than this sort of thing. But it's something that I was sort of surprised to find that I can't assume multivariable calculus students know. Please don't think I'm tearing down the job that high school teachers do manage to do just because I've run across one side topic they don't cover well.

Sarah Cannon said...

Unapologetic, totally agreed and very much understood.

jd2718 said...

Kate,

it's most often in precalc (or honors Algebra II), and that would lead it to be missing from most state standards.

New York's standards were written with the accelerated sequence Alg, Geo, Alg II/Trig, Calc in mind... so precalc-ish topics are often either omitted or pushed down.

One year I used a student as the demo... Where are all the places (what is the locus of all points) equidistant from Gerardo's eyes? His nose. What part? Right down the middle. [I motion a bunch of isosceles triangles]. Good. Where else? The middle of his mouth! His chin! [that's the moment were they, and I, and probably you, saw where this was going, and I lost the class to laughter. Not well thought out, jd, not well thought out]

jd2718 said...

Oh, I think I learned about locus in Algebra II, and returned to it in calc 2 years later. Traditional district in Connecticut.

Kate said...

Ah, ok, a little history helps. So, 20 years ago an accelerated curriculum was written, with the intention of omitting a year of precalc, and wasn't designed for every student to take it. When passing a regents exam actually meant you had met a high standard.

Now, there are legacy, extra topics still in the curriculum, which we now expect every student to get through.

That is as good an explanation as any for why our curriculum is a mile wide and an inch deep. Jonathan: you, me, and a pot of coffee could write a better list of high school standards in an afternoon.

Also, poor Gerardo! I'm sure it got their attention though.

Kevin said...

The California math framework mentions locus exactly once, in the section on "mathematical analysis" for grades 8-12:

5.2
Students can take a geometric description of a conic section—for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6—and derive a quadratic
equation representing it.

Kate said...

Hunh. That seems more like an accident of how they worded a conic section standard.

This is what we get from NY in the Geometry core curriculum:

G.G.22
Solve problems using compound loci
G.G.23
Graph and solve compound loci in the coordinate plane