Next Lesson: Polynomial Inequalities and End Behavior

Students took to the number line models pretty well in the last lesson. I think it can be really eye-opening to see how lines, parabolas, cubics, and so on are all related in a very simple and elegant way. Some of the students started recognizing the alternating positive/negative pattern that occurs when all the x-intercepts are real and there are no repeated roots. In this next lesson, we will look at polynomial functions (still in factored form) that have repeated roots, such as f(x)=(x-2)^2(x+1).

The students will then take notes on what end-behavior means, and what it looks like for a polynomial function. We will use quasi-limit notation like: "as x→+∞, f(x)→-∞". There will also be direct instruction on how to solve factored-form polynomial inequalities. But I think this will go relatively quickly, as they have already practiced solving quadratic and absolute value inequalities graphically (and writing the solutions in interval notation), and because they are understanding the number line models so well. Scaffolding = Success. (This equation reminds me of the team name of the two smart but disruptive boys I put together in one of my other classes - "Sexy + Math = Johnny + Ben" [names changed to protect the dorky].)

After the lecture portion, students will work individually or in groups on the practice work. The problems ask them to make number line models of polynomials to sketch a reasonable graph, solve inequalities, and describe end-behavior. The last page is a mini-exploration on how to find the end-behavior of a polynomial in standard form. I'm not sure if there will be time to adequately understand that, but we'll give it a go. It's one of those concepts that seems really easy (i.e. just plug in a big positive or negative number into the leading term, and see if your result will be positive or negative), but last year, most students took a long time figuring out how to do those problems.

The homework will be more similar problems to review for a quiz on Friday. I have also recommitted to assessing students on their note-taking and organization, as they tend not to do things unless they are assessed (long-term goals are still not immediately accessible to many sophomores). The once-per-unit binder checks I did last semester were not very effective: organized students didn't need me to check on them, and they just got free points; disorganized students would try to get it all together the day before the test (when I checked binders), and they never could do it. This semester, I am going to check the notes and table of contents for the week each Friday as they take a weekly quiz. These short-term objectives should help some of the more disorganized students stay on top of things.

Next Lesson: Polynomials and the Number Line Model

Intersession is complete, and we're back to regular school (i.e. normal crazy instead of crazy crazy).

In the next Algebra 2 lesson, we will start with the basic understanding of what a polynomial is, and how to categorize them by degree and number of terms. Students also need to know how to write a polynomial in standard form and identify the leading coefficient. Nothing too exciting or creative here, just some definitions to get out of the way.

The second half of the class should be a little more interesting. Based on the work in the previous class, students should be able to look at a set of linear graphs and determine where the x-intercepts of the product function will be. From there, we will learn how to generate number line models. Students will plot the x-intercepts on the number line, and then write, in interval notation, the intervals that are defined. Students will look at the lines to determine the sign of the product function in each interval, putting a "+" above or a "-" below the number line in each interval. They will then sketch what a reasonable graph might look like based on this, and then check their work by graphing on the calculator.

After practicing this, we will move on to doing the same thing, but given a factored form polynomial function instead of the liner graphs.

If all goes well, we can move into working with functions that have repeated roots in the next class, along with solving factored form polynomial inequalities.

Next Lesson: Creating a cubic function from 3 linear functions

Intersession is nearing completion, and I am finally getting back on track with my curriculum. My Algebra 2 students had their Quadratics and Complex functions test today, which I hope to grade later tonight, but we'll see...

Tomorrow, we have a short class, and it works out well, because we're going to do a little exploration type activity. On the homework that is due tomorrow, I gave them a graph with two linear functions on it; there are guiding questions that help them graph the sum of the functions both graphically and algebraically, and then to compare their results. Then, the second part asks them to repeat the process, but finding the product instead of the sum. This is a neat way of visually understanding why the product of two linear factors yields a parabola, and why the zeroes of the parabola are at the zeroes of the lines. So tomorrow, students will repeat this activity, this time graphing the product of three lines to generate a cubic function. We will focus on the roots of this product function, and how the roots split the x-axis into intervals, and how you can easily determine the sign of the product function within any given interval.

I hope that this will lay a good foundation for later on when we use number lines to sketch polynomial functions and solve inequalities in factored form.

I am posting the worksheets for this on ILoveMath. If you use it, let me know what you think.

Intersession Update

I haven't posted for a while, in part due to my new Perplexcity obsession. But today seems like a good day to write.

Period 1: Slam Poetry

We had a guest presenter come today to run the class - Mark Pinate. He's a great performance poet who also teachers classes. He performed a couple pieces for the kids and then did a workshop with them where they wrote some pretty good "I Am" poems. Tomorrow we'll revise their work, and Thursday he'll come back to help them work on their delivery.

Period 2: Algebra 2 Honors

We continued to practice completing the square. This year I am using the traditional method, as opposed to a new method I read about and tried last year. It was a good idea, but the kids found it very confusing. Then, I showed them how to prove the quadratic formula. They followed it the whole way and were asking questions when appropriate. At the end, I pointed out how much math they really needed to know to be able to understand the proof, and we talked about how far they've come. They spent the rest of the class working in pairs and groups on practice problems, and they were all working really hard. At the end, I overhead one of the students say, "I feel like I really accomplished something today." That may sound like an ordinary statement, but not one that you really hear coming from a student all that often.

Period 3: English 1 Review

We had a great start to the class: I had a fresh new seating chart, now that I know the kids, and they started off on their prompt decoding quietly and with a lot of focus (for 9th graders, anyway). We got through the agree/disagree activity reasonably well (where they had to move to one side of the room or the other depending on if they agreed or not with a statement pertaining to their new essay). Then, we were transitioning into reading the articles for evidence when a couple of girls wouldn't stop talking to each other. I went over to refocus them, but then I realized that they were arguing. They weren't listening to me and started to get a bit nasty, so I tried my severe voice: "Ok, that's enough. Stop." But then they started to really swear and stand up, and I got worried. I started to lead one out of the room to cool off (my hand lightly pushing her back), but as I was doing so, the other girl's friend began shouting taunts, and the girl I was taking out turned around and lunged back in. I had to drag her out by her arm to prevent her from attacking the other girls. I got her to the office, got back to the room with the principal, and had her take the other two girls. I tried to go on with the lesson for a bit, but I guess I had a lot of adrenaline going (man, it's scary to watch kids go into that red zone) and I was shaking a little, so another teacher stepped in to run the class for about 10 minutes while I calmed down and told the admin what had happened.

Now I ask you, what other job has this kind of daily excitement and variety?

Intersession

At DCP, between fall and spring semesters, we have a period of time called intersession. It originally started as a time for students failing math and English to take a two-week mandatory review course, with a chance to retake the final at the end (getting an 85% would change the F to a C-). There are four periods a day, so the rest of the time is filled with fun and/or academic classes invented by the staff. Since then, we have dropped both the mandatory requirement and the chance to pass by retaking finals. We now sell the review classes as "a chance to catch up on the skills you missed, so that you can start strong in the spring semester". Plus, there is no homework. Surprisingly, most of the students who should sign up for these classes do sign up. We normally have a rotating 6-period block schedule, but these two weeks, we meet all 4 periods (rather 3, as most of us have a prep period) everyday. No grades, no homework, just a focus on learning and fun.

Here is my schedule:

1st: Slam Poetry
I think performance poetry is pretty cool to listen to - when it's good. When it's not good, it can cause quite a cringe factor. I'm going to help the students develop their creative writing abilities, as well as getting them more comfortable standing up, speaking, and showing emotions in front of a group. What's my qualification for this? None! That's why intersession is so fun; i'll learn along with the students. Today, we were working on emotions. I gave groups a quote from that timeless classic "Mean Girls", and they had to brainstorm a variety of emotions, and then say the quote steeped in one of those emotions. The rest of the group had to guess which emotion was being, well, emoted. The best moment came right at the end of class when one of the students (whose emotion was "seductive") got up on the table, lounging one one elbow, and did a PG-rated (thankfully) seduction scene. The hardest part with this kind of thing is getting students to feel comfortable enough with each other to be themselves, and today was a pretty good start. Now, I need to look up more theater games and good writing prompts...

2nd: Algebra 2 Honors
I have required my class to meet through intersession . If they want to accelerate, they have to make some sacrifices after all. For them, the new semester starts today, along with homework, quizzes, and tests. They're not thrilled, but they're being good sports about it so far.

3rd: English 1
Yes, I am teaching one section of the English 1 review class for the freshmen (thought it's planned by one of the English 1 teachers). It's a small class of 15, and we will be working on the foundational stuff that they still need help with. It's a nice change from teaching math, but I don't think I could handle it long term. Initially, I thought I would be an English teacher, but after doing some subbing in English 1 classes, I realized it wasn't for me! I love reading and talking about literature, but that doesn't happen with our students until much later on. At the beginning, it's all topic sentences, capitalization, and subject-verb agreement.

Today, we were working on "glue words" such as "since", "because", "if", and "while" that can be used to combine simple sentences into more complex ones. Trying to get away from the "I like soccer. Soccer is fun." pattern of early writers.

4th: Prep
That's right now! Time to get back to work... gotta plan Slam and Algebra for tomorrow.

Perplex City: the new obsession

Perplex City is a game, based out of England, that has been gaining popularity for the last couple of years. It is an "alternate reality game", meaning it can take many different forms (websites, emails, live events, text messages, and even skywriting). The basic story is that there is a place on some other planet called Perplex City where the citizens highly value games and puzzles; their most important artifact, the Receda Cube, has been stolen and brought to Earth where it is now hidden. In trying to search for it, the people of Perplex City have established this game to engage the people of Earth in the hunt. There is apparently a real-life reward of $200,000 for anyone who finds the actual cube. The game does a good job of blurring real-life and its own environment.

The only part of the game I have really played so far are the puzzle cards (which, I think, is the main part). The company (Mind Candy) has released a set of 256 cards, each of which has some sort of puzzle on it. The cards range in difficulty from trivial to nearly impossible (one unsolved one asks you to prove the Riemann Hypothesis), with a huge variety of types of puzzles (math, logic, word, visual, trivia, etc.) On one I had to play a game of minesweeper and another I had to cut out and fit together as a puzzle. These cards are highly addictive, and I'm not quite sure why. As you solve them, you can enter the answers on their website; if you are right, you earn points and move up the leaderboard. They also say that there are little clues burried in the different cards (some of which are done in heat-sensitive or invisible ink), that taken together can help find the cube. Plus, the backs of the cards form a map of Perplex City. Most math teachers like a good puzzle - go ahead and click, you won't be disappointed (except for the fact that you can no longer do anything else in your free time!).

Finals Results

The results of the Algebra 2 Honors final:

10 | 0
9 | 6
8 | 0 1 1 4 4
7 | 1 1 1 4 4 6 6 8
6 | 1 1 5 6 9
5 | 4
4 |
3 |
2 |
1 |

For a final exam, that's fairly solid. Students did worst on the most recent material, which makes sense, because they haven't had enough time to let it really sink in. Next week, we are going to need to review factoring and dealing with the various forms of parabolas. Then we'll learn to complete the square and prove the quadratic formula.

Every student in the class ended up passing with a C- or better (a new first for me!), so that is something we will definitely celebrate next week.

The biggest success: the boys who shook my hand, looked me in the eye, and thanked me for helping them get prepared, both after the review sessions and after getting their tests back. Usually the girls are pretty vocal about their appreciation; the boys tend to wave, put on their iPods/cell phones, and rush out the door (not to generalize, but...) So I kind of smiled on the inside when these boys conveyed their awkward gratitude - it was a milestone along their paths to maturity and they didn't even know it.

Radical Butterflies from Hell

We used to have our fall finals at the end of December, right before the students went on break. In the past couple of years, we have moved them to after break in order to keep the same schedule as the rest of the schools in our district. This has both positive and negative effects, the latter obviously being that two weeks of vacation tends to sap the content knowledge out of even the most diligent students' brains. Most teachers spend the first week back reviewing everything, since finals happen Tuesday through Thursday of the second week (this coming Tuesday is my Algebra 2 final).

To make it a little more fun for my class, I decided to hold the First Annual DCP Algebra 2 Honors Fall Final Review Math Olympics, and I told them that the ceremony for the bronze, silver, and gold winners will be nationally televised (I'm still tring to come up with the prizes. I wonder, if I take them off campus for lunch, is that a prize or a punishment? Please leave a comment if you can think of any good prizes that 10th graders might enjoy!). The students are in teams of three and needed to come up with a team name using a type of number, along with a place to represent in the games. We have, for example, the Imaginary Mathematicians from New York City, the Radical Mexicans from Michoacan, and the Radical Butterflies from Hell; the Real X-tremes from Hawaii are currently in the lead. Each day this week, there was a series of events to review key concepts, and each homework was a set of multiple choice STAR-type problems that counted as additional events.

Today (Saturday), we had a terrific review session. I told students I would be at school at 10:30, and many were there by 10:15. I had most of the class show up (only about 5 or 6 didn't), and they stayed until after 2:00, ordering pizza for lunch. We pushed the tables together into one big conference table, and they all sat around it, going over old tests, doing their homework, working together, and asking me questions. I really need to recognize the whole class at our next school assembly for such high quality effort, work, and behavior. I'm proud of how they're coming together as a class; I hope (now more than ever) that they all pass the final.