Teaching Conics

I'm loving this unit more and more each year. It's a chance for the students to practice their ever-waning algebra skills (hello completing the square, I love you). It's also a chance to do some hands-on stuff. AND it's a chance to see some cool applications.

One year (not this one, because ... well, just because) I saw an application that you could build a pool table in the shape of an ellipse, and then if a ball was at one focus point and you hit it, then in an ideal world, it would pass through the other focus point. I then just had to try it. I had the kids construct an ellipse (lots of string and a large piece of paper. Then I had my patient/loving husband carve this out of some wood and hollow out the inside to be the "pool table" in the shape of the ellipse. I brought it to class and we recreated where the foci were and we tested it out. It worked most of the time and was cool.

This year for ellipses, I just had them create ellipses on paper with a partner and a loop of string and 2 sharp pencils held down for the foci. Then they took their notes on this creation. Every student had 2 ellipses, one on each side of the paper, one with vertical foci, and one with horizontal. It worked well.

I'm having them do a project of searching for practical uses of each conic (circles, parabolas, ellipses, and hyperbolas). Let's see what they wow me with.

The Worst Part of Turning in Grades

Grades were due at 2pm on Monday. The previous Thursday night I was driving out of town to go to a math conference and would be out Friday. I warned the kids and told them that the LATEST they could turn in grades would be 4:16 on Thursday afternoon. I made a joke of it to hammer it home.

"Do not run after my car waving your homework at 4:30"
"Do not slip your work under my door after 4:16 and expect me to get it"
"Do not secretly slip it in my mailbox"
"Do not come to my house, please, this weekend to turn things in."

Mostly it went okay. But then there's always the special cases that I make allowances for without telling anyone else. One student's dad had recently died and she was having a rough time of it. She was also out the end of last week for FFA. She talked to me before Thursday, and I said that she could turn in late work AND work that was way past acceptable to turn in.

One student is struggling socially and familially and scholastically and has been in tears and has a hard time keeping it together. I went to school Monday and found some test corrections on my desk from him. I accepted them. He still didn't pass, but this brought his grade up.

Then there's the students that beg and plead and such after the fact and after they have slacked off ALL 6 weeks (which is 7 weeks this time, but who's counting).

Student #1. Monday morning he shows up. "I see my grades a 50%. Is there anything I can do?". Hmmmm, well, you turned in no work all 6 weeks, you barely did your late work. You didn't take advantage of tutoring or retests or test corrections. "No, there's nothing you can do. The time has passed." ... "Please, please, please, PLEASE. I'll do anything. I'll do quadruple work, I'll turn it in before 2pm (due time), I'll, I'll, I'll." .... "NO!". I mean, quite honestly, it's nothing to me if he turned in more late work. I had time to put it in. But I made the decision that that would not be beneficial to him. What would he have learned in that case. "Oh, I can always slack off and then be super polite and hang dog faced and teachers will let it slide at the end." Still it put a sour taste in my mouth and I felt horrible for the day, but ultimately I know it was the right decision.

Student #2 Basically the same story as student #1, except he came in AFTER school AFTER grades were turned in. "IS there ANYthing I can do? I need to stay eligible for band.". ... NO. This one had the extra added effect of super politeness, "yes ma'am, no ma'am, thank you ma'am", and the dipped head of sorrow. He wouldn't leave. He kept waiting around looking all glum waiting for me to change my mind. No, no, no. It may seem nicer to give in now, but it's not useful to you in the long run.

Argh. Hard decisions. I feel like "mean teacher", but I have to remember that I'm trying to do what's ultimately best for each kid.

Teaching Neatness

I just graded my algebra 1 tests over Solving Systems by Substitution. Oh my. The kids generally knew what they were doing, but their sloppiness got in their way. Some couldn't read their own handwriting and dropped negatives or made 3's into 13's and such. Some wavered all over the place and then misread their work that way.

Today we had a lesson on neatness and how part of your job as a "mathematician" is not just to get the answer but to communicate to others how the problem is done, so they can just follow along by reading your work and you don't have to be there to interpret the "doctor handwriting" as I call it. I implored (ordered) them to write each step and keep it all lined up going down the page and not all higgeldy-piggeldy every which way.

We practiced. We practiced some more. We refreshed our memory on fractions. We refreshed our memory on the fact that "3x/4" means the same thing as "3/4 x". We refreshed our distributing skills of "5 - 3(x - 2)" types of situations.

We discussed how to check our work (plug (x,y) back into BOTH equations).

"But why do I have to check both?"
"It could be right in one but your previous mistake makes it wrong in the other. Check both!"
"But I'm not going to be a mathematician. I'm going to be a doctor."
"Well, after surgery you don't just want to check ... 'did I leave the scalpel in the body? No? Good, sew him up' and meanwhile, you didn't check that you left the saw in the body."

Differentiating

Yesterday, for some strange reason, I actually had more time than my usual 2 seconds to prepare for classes, and I was giving a test in precalculus, and I have a handful of very smart kids that are polite and bored in class because it's going too slow for them.

This all adds up to a differentiated test on vectors and polars. The bulk of the class just had the standard test that asked them to:
find the angle between 2 vectors,
convert a polar point to a rectangular point and visa versa,
plot r = 3 sin theta + 2, etc.

Without saying anything to them, I handed the super-smart kids a different version of the test and kept an eye on them throughout the period to see their reaction. They did fine. It took them the whole allotted time (whereas usually they're done in less than 1/2 the time of the other students). They're aware of this fact, so periodically, one of them would look up as another student handed in his test. I wonder what was going through their minds.

Their questions were more of the variety of (and maybe I could have made them harder, but ...) :

give me 2 vectors that are perpendicular to each other and neither lies on the axes or has equal components,

plot 4 points on the polar plane that when connected form a rectangle, none are on the axes. then give me their coordinates, each in 2 ways.

vector u is <6,> find me a vector in the same direction that is 1 unit long (and we did NOT cover this in our short time with vectors). I did give them a hint by making them answer a "similar triangle" type question right before this one.

Anyway, I'm glad I could "unbore" them briefly, and hopefully I can do this again.

Polar Graphs

I'm so excited. I taught the graphing r= 4 sin(theta), r = 2 cos(theta) + 3 and such today, and I think THIS method will stick more successfully than what I've done in the past.

Two years ago at the NCTM Atlanta Conference, a teacher from North Dakota shared her strategy, and it made so much sense, and this year I adapted it and tried it.

I made up a packet where I have 12 such graphs mapped out on a rectangular coordinate system. I don't even label which ones they are. Right next to these graphs are blank polar coordinate systems. The tick marks (or angle marks) on each are divided the same (into pi/6). This ND teacher stressed to make the connection between "y = f(x)" and "r = f(theta)" and link x to theta and y to r and to keep mentioning it. Then you transfer each point from (x,y) to (r,theta) accordingly, and voila! You have your graph.

On the front page I had 3 similar ones, and after they/we graphed all three, then we refreshed our memory on what the equations were. Then we discussed what the connection was between "amplitude BIGGER/smaller than vertical shift" was, etc.

We got through 4 in class, and they have the rest for homework. I'm thinking it will work, because even I can now remember what the graphs should look like by doing such an analysis (whereas before, I had to refresh my memory each year).