The "Higher" Polynomials

Here is a lesson I love because of graphing calculators. My goals for my students were to be able to look at a graph of a polynomial and know its degree and find an equation given various points on the polynomial. I also wanted them to be able to look at an equation and be able to give a quick sketch of the graph with the intercepts and shape correct.

At the start of class before I mentioned anything about shapes of non parabola polynomials, I handed out graphing calculators and we all got the same window and turned the grid on. In Y=, I had them type in y=(x-2)(x+1)(x-1) (or something similar previously checked out to make sure it fits in the window). I told them not to graph it but to think: what are the x-int, y-int, and make a conjecture about the shape. THEN they could graph and see if they were right. We then had a discussion to link the intercepts with the equation.

Then they typed in y=2(x-2)(x+1)(x-1). I asked them to think about how this might affect the graph and intercepts and shape and THEN graph and confirm their reasoning. We did one more, and then I discussed the shape and degree.

Then same process with y=(x+2)(x-1)^2. They had to think and do the same analysis as above. We then did y=(x-1)(x+2)^2 and such.

Then the fun part: I typed in an equation and just showed them the graph, and they had to match it. We did that a few times. I made sure to change the constant in front sometimes and to make it "upside down" and to have various powers. Then I asked them to find an equation for a 5th degree polynomial with 2 "bounces" off the x-axis going in the same direction (both below the axis or both above). Then for the ones that finished that early: opposite bounce direction.

Then we took notes, and I believe they had a good sense of polynomial graphs.

First Time Teaching Concepts

On my broken record mode (you know, those things that music used to come out of and were bigger than 12" diameter?): I'm loving teaching algebra 1 this year since it's so fun to show them all the cool stuff they'll see over and over again - FOILing, lines, exponents, quadratics...


But it's also daily unsettling because I don't know what they'll find difficult or what will need extra practice or how the best way to explain something will be. Today I told them we were going to do "big kid math" (they were going to start the FOIL process without me saying FOIL just yet). To build up to it I needed to teach them like terms and adding like terms of expressions such as (5x^2)(y^3) or (3xy^7), etc.


I racked my brain (okay I just googled "wracked"/"racked" ....) for a way to present this since I was tired of worksheets and notes. I decided to fold up a white paper to have 20 squares (4x5) and opened it up and put various monomials in each making sure that there were sets of 2 to 3 like terms usually. Then I did it with another piece of paper making sure to put some sort of bubbly border on each square (on one paper only). Then I copied (from 1 to 2) the papers onto various pretty colored papers. The kids then chose their color and cut out the 20 pieces (yay, less work for me) and they had the 2 sets of 20 monomials.


I chose a side (border or no border) and randomly picked a card for the overhead, say (7x^2)(y^6), and told them to find it and cull it from the herd. Then they were to find like terms in their cards (without explaining what like terms were). I wanted to see what they knew. I saw various mistakes as I walked around the room, and it was interesting. So I went up front again and said, "this is like making chocolate milk. You have 2 parts milk and 6 parts chocolate (refering to the exponents). You want to find similar "drinks", and those are like terms". That seemed to work with everyone, and we practiced some more.


Then I had them find a set of like terms, say the cards were 3xy^2, -5xy^2, and 8xy^2. I asked them to add them up to see what they'd get. Instead of 6xy^2, a lot of students got 6xy^6. I was expecting the mistake of (6x^3)(y^6), but I guess the "silent 1 power makes it stay 1!".

Anyhow then we built up the skills for that: 3x + 5x, 10x^2 + 8x^2, etc. They still had the silent 1 problem: 3x+5x = 8x, but 10x^2 + 8x^2 = 18x^4. Then what seemed to work for most (all?) kids was the EXCELLENT box again. Think of the variable parts as boxes filled with x or x^2 or such, and how many boxes do you have and when you add them what happens.

All in all, I think it went well. I also want to share an excellent resource for algebra worksheets. It's been a godsend for extra practice problems all neatly laid out.

The Joys of Spring Break

Happiness is catching up on sleep to start out your break ... and also having lunch with friends and reading books and doing crafts and baking. I hope all teachers everywhere have a rejuvenating break.

I happened across a bookstore yesterday (and by happened I mean I zeroed in on it and ran inside), and found a book another teacher had recommended to me, "The Courage to Teach", by Parker Palmer. I've started to read it, and I see what all her fuss was about.

I'm also resting easier because I know my plans for next year. Daily, still, I'm frustrated by what's going on at our school, but now it only solidifies my resolve not to be in this situation in the future. In January I sent out my resume, and now it looks like I've secured a great job for next year. It's in the same district, which is a plus, but it's at a more sane-seeming school. All the people I interviewed with seemed like great people to work with, and they had intelligent things to say about teaching math. We clicked.

I'm also excited because I'm going to NCTM in D.C. this year. This will allow me to get great ideas for next year. It looks like I'll be teaching algebra 1 and geometry to begin with. I haven't taught geometry for a few years (after 7 years of teaching it), so I'm eager for fresh ideas. The school I'm starting at is just building up their high school component by adding a grade each year. That's also exciting to be in on the beginning of new traditions.

I also found out last December that I passed my National Board Certification, so, Yay. All in all I'm thinking, that if it hadn't been such a rough year, I wouldn't have looked elsewhere for a job, and maybe I wouldn't have happened onto a better situation (if that's what it turns out to be).

Whew!

Hectic times, low on sleep, high on things to think about. I like that even after 12 years of teaching I still have epiphanies. This last unit in AB calculus was derivatives and integrals involving e^x. In the past I've always grouped the derivatives and integrals on one day and was always amazed that my set of kids just confused everything and didn't know what to do and when to "go forward" and when to "go backward". I also used to just give them a brief intro to the basics and then start tossing challenging problems at them that I thought made them think, but apparently just confused them more. I also used to cut and paste from various sources and cobble together a presentation.

This year, I separated them into one day each. I also slowly and carefully picked my examples to build skills that gradually became harder with about 3 problems per skill .... thinking that I could semi walk them through the first one of each set and then they could practice on the next 2 before we built up difficulty. Also, I grouped the skills together, instead of mixing them up all higgeldy piggeldy. I also kept stressing: "The derivative of e to the box, is e to the box times the derivative of box". This seems to work better for my kids than "e to the u" which is just one more alphabet letter. I physically draw "e" with a blank box in the exponent.

This seems to have been MUCH more successful than in the past. Of course now I say to myself, "self .... DUH! What were you thinking??"

The Running Theme of the Week

A coworker e-mailed all of us a copy of this article on "Grading by entitlement", and it seemed to coincide with several occurrences this week on the same theme.

As I was proctoring the TAKS exam, the students finished in time to sit and chat, and I was casually eavesdropping. One conversation went, "oh did you have so-and-so for a math teacher? Well, on a test if you just call her over and say you don't understand a question, she'll basically work it all for you. That's how I passed the class."

As I was helping tutor my precalculus kids on Thursday for their conics exam on Friday, a student was there for help for the first time in this unit. ... A student who hasn't done any of the conics homework ... A student who was absent for one of the days and missed the treatment of ellipses. ... A student who has pulled this for all 4 of the 6 weeks and always thinks he can go for the "hail Mary pass" at the last week to "pass". ... A student who did this last 6 weeks and ended up with a 68% (not passing in Texas). So. During the tutoring (in which he basically wanted me to do all the work), he grumbled that I wouldn't even give him 2 extra points last 6 weeks even though he tried REALLY hard and came in every day for the last 2 weeks and passed the exams and he should be rewarded for effort. AAAAARRRRRGGGHHH. Don't even get me started on how I was berating him on Thursday and how he still didn't get it and how he still doesn't think he needs to do the hard work to understand the concepts.

As my algebra 1 students were taking a test on solving systems of inequalities, one student early in the test put his head down. I walked over to him and gently but firmly told him not to do that if he wasn't finished. He proceeded to work a little but then his head was down again. I let him be. Another weak student ... same behavior. Towards the end when I wanted to pick up all the tests and teach a new topic, a 3rd student was still working hard, and so I put him outside the class to finish. The other 2 head-downers I went to pick up their tests asking, "are you done?" Their response was, "I don't know what to do (on the test)." I believe they were wanting me to come and guide them through the process. I said, "okay, then you're done. Turn it in." I picked up the tests. Then they saw I was putting the 3rd student in the hall to finish. The 1st head-downer then piped up with, "I want to continue." I said, "but you said you didn't know what you're doing." He said, "I want to try." Okay, so I put them in the hall. At the end, the I-don't-know-what-I'm-doing kid turns in a completely finished test (whereas before 2 out of the 6 questions were finished). Hmph.

Oh my. Okay, but I want to end with a more uplifting article another teacher passed around on learning called, "Try and Fail".