Update and Upcoming

I haven't posted much recently for a few reasons:

1) We administered the CAHSEE (exit exam) last week, which ate up a bunch of class time. Hopefully, our students will do as well this year as in the past. Last year, we had 88% of sophomores pass the math section on their first try.

2) Perplex City. Ok, I have a problem. :)

3) The last 3 lessons in my rational functions unit are BORING. We just practice adding, subtracting, multiplying, and dividing, and solving rational equations. Good mathematics, but I don't have any clever ideas on how to teach it, so it's just me modeling the method and the students practicing. Nothing wrong with that per se, but nothing much to be said about it either. On Tuesday, we'll have a review lesson before the unit test which is on Thursday, and the students will spend most of the class working on Showdown cards created for this unit.

To make up for the recent lows, I have a couple of cool things coming up which I'll preview here and then write more about later (after they've been, you know, actually created).

1) The Financial Literacy project I wrote about earlier is now coming to fruition. I met last week with the College Readiness teacher and we hashed out the outline for the project. It will look something like this:

• Freshmen will earn weekly income by performing their "job" - i.e. doing homework, being ready for class, etc. They can earn "lobobucks" in all their freshmen classes (assuming we can get all the teachers on board!). Each Friday, students will deposit their lobobucks with their college readiness teacher, and on Monday, they will receive an account statement.


• They will also receive a weekly bill for expenses. For example: "rent" = their chair in class, "utilities" = worksheets and materials they are given. They must use their money to pay their bills. We are considering consequences - i.e., if you don't pay rent, you have to sit on the floor...


• To make things more interesting, students will be required to sign up for a credit card. That is where my Algebra 2 class comes in. On Friday, we begin our exponentials and logs unit. I will teach them about interest rates and credit cards, and they will design their own cards and rate plans for the freshmen to sign up. They can use their credit cards to buy extras (though they come at a steep price!) such as free dress, bathroom passes, homework passes, listening to music during tutorial, and a double lunch period. There will be credit limits to prevent out of control spending too.


• To complicate things further, when students get detentions, the interest rate on their credit card will increase!


• Each Friday, my students will get a log of purchases made by the freshmen, and any payments that have been made. They will take into account any rate hikes, and will then generate a new balance and create a bill, which will be presented to the freshman on the following Monday along with their income and expenses.


• The freshmen that are able to stay in budget (or maybe hit some sort of savings goal) will earn a big prize at the end of the unit (like a pizza party and trip to the imax)


• My students, approaching things from the opposite angle, will be competing to see who can get their "clients" most in debt. What better way to understand how things really work?

2) The "STAR Search" Treasure Hunt. (Can you help me with a better name??)

To help energize my students and prepare them for the STAR test in May, I will create a treasure hunt for them, beginning with a puzzle. I ordered some blank, printable jigsaw puzzles from this site, and I will create a picture/clue that will launch students into the hunt. Each day, they will spend the first 15 minutes of class working on released STAR questions in teams. For each question they get right, they will earn a puzzle piece. By the time the test is here, they should have completed most of the puzzle. Once they do, and they figure out the clue (which leads to a teacher), that teacher will give the group their next puzzle, which will lead to the next, and so on. Each puzzle will require the students to review some Algebra 2 concept, and will also incorporate some sort of fun puzzle, and will lead to another staff member. Ultimately, there will be a prize for the group that gets there first.

The front side of each puzzle will have the same picture. The backs, however, will be different, and will be part of a puzzle that the class will need to solve together, with a class reward as the prize. I don't think any of my students read this, but, just in case they do, I won't post any more details here. After the hunt, I'll post up what we did. For now, just send me an email if you want to hear more, or if you have good ideas for puzzles and clues I can use.

Next Lesson: The Hidden Dangers of Simplifying

(aka Holey Functions, Batman!)

Quite a few students got the hidden message, and they did it faster than I would have expected. Mostly, it went like this:

Student: "Mr. Greene, I got the answer."
(I look at student's paper and see only a string of numbers.)
Me: "What does it say?"
Student: "What do you mean 'say'? You can't read numbers!"
(I look at student meaningfully. Single or double arched eyebrow, with a slight off-center forward head-tilt. Admit it - you're doing it right now!)
(Student looks at the puzzle again.) "Oh, wait!" (Student excitedly grabs pencil and gets back to work.)

Anyway, we took a quiz at the end of the class, and they did fine, so I hope we are ready to move into simplification in tomorrow's lesson.

For the warm up, students will review finding vertical asymptotes and end behavior functions, and they will do this when given a graph only, or when given a function. (Go Representational Fluency!)

Then, I will have them analyze f(x) = (x^2-x-6)/(x+2). (Sorry, I haven't spent the time to learn LaTex yet...) They will assume there is a vertical asymptote at x = -2, and won't they be surprised when they see the graph on the TI! This will lead in to the discussion of 0/0 and holes in functions, and so forth. We'll do a couple practice problems, where students need to simplify (clearly writing the domain of the simplified form of the function) and graph (clearly indicating any holes). If there is more time left, I have a few practice problems for them to do on their own.

Wish us luck!

Rational Review

Today, I did a lecture on the end behavior of rational functions. We used polynomial division to rewrite the rational function, and then figured out what terms would approach 0, leaving us with a lovely end-behavior function.

Students are starting to get overwhelmed - though they know the differences between x-intercepts, y-intercepts, and vertical asymptotes, when called on to figure them all out for a problem, they tend to mix things up. So I need to stop and take a day for review. So, I present:

Another Perplex City inspired creation for tomorrow's lesson. Enjoy!

Can you decipher the message? Leave a comment.

Next Lesson: Rational Functions

The unit 5 test is over, vacation is over, and I'm ready to get back on track with posting.

In the last lesson, we started the Rational Functions unit (as I described in a previous post). As a warmup, I had students do some division work to explore what happens to a quotient as the divisor approaches zero. They did this visually (i.e. fitting smaller and smaller boxes into a fixed space) and numerically (filling in tables of values).

After they were clear on the effects of dividing by a number approaching 0, I gave them a graph with two linear functions on it, and asked them to work in teams to find the quotient function. They had to look at each value of x, estimate the y-values of the two lines, divide, and then plot a point for the quotient function. It doesn't sound like this would take too long, but I knew from experience that it would take at least a half hour (and it did!). But the division warmup did really help a lot, and my main goal was for them to really understand why a vertical asymptote occurs.

We then moved into some direct instruction where we reviewed the difference between 0/4 and 4/0, I introduced them to hyperbolas (the shape of the graph generated when you divide two linear functions... conic section definitions will have to wait), and we looked at vertical asymptotes and x-intercepts, and where they occur. Students have a lot of trouble with fractions (duh!) and this translates to confusion when trying to deal with rational functions. I hope that continued reminders about what happens when you divide by 0 will help them remember. Finally, I taught them the "as x approaches 2 from the left/right" type notation, with the minus/plus sign as superscript.

We did some example problems, and that was that. I came up with a good way of testing their understanding in the homework: I gave a graph of a hyperbola with two linear functions A and B, and asked them to determine which line was the numerator and which was the denominator.


In tomorrow's lesson, students will continue to practice these ideas, and I will introduce them to Rational Functions as a concept. We will solidify their understanding of x-intercepts, y-intercepts, and vertical asymptotes, and we will discuss the domain of rational functions. I will throw in some factoring, but nothing yet that simplifies (holes will be discussed a few lessons later on).

Cool Puzzles

Check out this puzzle, just one of many you can find at the new version of Perplex City. Come on, check it out (you'll thank me later!). Root for yochanan1 on the leaderboard!

"Pictured is an amazing geometric figure: a rectangle partitioned into ten different squares... each square a different size (all have whole number lengths). Knowing only that the side length of the small, white square is 3 units, can you determine what the side length of the yellow square must be?"

Rational Functions

This post is in response to Lsquared's question in the previous post.

Here is the basic outline for my rational functions unit:

1. 
   Introduce the shape of the hyperbola by giving students two linear functions, and having them divide the y-values for a bunch of different x-values. This shows that the x-intercept occurs when the numerator line has an x-intercept, and the vertical asymptote occurs when the denominator line has an x-intercept. My students also need some understanding about what happens when you divide a constant by a smaller and smaller number, so we do some numerical work with tables here, as well as conceptual understanding of what dividing by a smaller and smaller fraction means.


2. After this, we begin some basic feature analysis of simple hyperbolas. Students should be able to look at a graph and/or equation of a simple hyperbola and be able to quickly determine the domain, range, vertical asymptote, and intercepts. We practice matching graphs to equations and generating graphs by hand.


3. At this point, I want them to start thinking about end behavior, so we bring back polynomial division. We divide the numerator by the denominator, and then talk about what happens as "x gets really big". This is good scaffolding for limits. Last year, I held students accountable for determining all sorts of end-behavior by this method - not just horizontal asymptotes. I'm not sure if this is too much for them at this level or not. They definitely struggled with it last year.


4. Last year, I had students practice with more complicated hyperbolas, that had multiple vertical asymptotes (but no holes yet). I think I will cut this for this year, mainly to save time.


5. Now we look at what happens when you reduce a rational function, and discuss functions that are the same at all but one point, like f(x) = x^2/x and g(x) = x. We discuss holes in graphs, and when they occur. Students have trouble determining when a value for x causes a hole, a vertical asymptote, or an x-intercept, so we spend some time really focusing in on those concepts.


6. After this, I spend a couple of lessons just having them practice analyzing and graphing rational functions, using all that they've learned so far.


7. When I think students have a clear picture of what is happening, we move in to operations. I start them with multiplying and dividing rational functions, as that is much easier to do.


8. In the following lesson, we practice factoring and finding the LCD, so we can add and subtract rational functions.


9. Finally, we have a couple of lessons where we work on solving rational equations. I teach them multiplying by the LCD and cross-multiplication as two main strategies to use. Students had difficulty last year with understanding extraneous solutions, so I'm going to need to think that through better this time around.

That's it. We have a full period review, and then the unit test.

This may change, as I haven't yet begun really planning the unit for this year yet, but I think I will be sticking with the same basic outline. Lsquared - I am definitely interested in hearing how you teach rationals. Anything I can steal?

Next Lesson: Science + Math = Love

Yet another busier-than-normal week has gone by. I am looking forward to our break next week to catch up on some sleep, get some work done, and even relax a little.

Last Friday, I taught the students how to do polynomial division, and it seemed to go ok. I had typed up most of the notes already on their note-taking template, just leaving the examples for them to do, and they were very excited by not having to write as much down. The algorithm is pretty straightforward, and the only student who really had trouble was one who had learned division in a different country, with a different algorithm. A lot of students who learn long division in Mexico use the same DMSB algorithm that we do in the US, except they do the multiplication and subtraction steps in their head and just write down the difference. But this student had a totally different format (the division sign is written upside-down, the numbers go beneath, etc.). I'd never seen it before, but after watching him use it to do a division, I got how it worked. I couldn't come up with an analog for polynomial division on the spot, however, so I just tried to work with him on that a little more. Maybe I can offer him some extra credit if he works out a way to base a polynomial division algorithm on his division method...

I decided to skip synthetic division this year since you don't really need it if you can do polynomial division, and I am also skipping the factor and remainder theorems. (I'm not holding them accountable for knowing this stuff, but I am offering it up as an extra credit assignment over the break.) I'd like to push those concepts into our pre-calc curriculum. It's the middle of February and I already feel the STAR test breathing down my neck. I need to get through Rational Functions (which is a long unit - I'm already thinking about what concepts I can trim and save for pre-calc) and well into Exponentials and Logarithms before the test, as it has an absurdly heavy focus on logs.

But I digress. This week, we've been working on the properties of exponents, and operations on rational monomial expressions. I have been putting a heavy focus on having students understand why the properties of exponents act as they do - especially when dealing with negative exponents. When kids just learn the rules (add/subtract/multiply the exponents), they constantly make mistakes, putting the result in the wrong place, multiplying instead of adding, and so forth. I've found that this year, so far, they are doing a lot better since I am not talking about the "rules" at all, and instead, having them reason through their work each time. We have explored how a negative exponent works, and the only "rule" I want them to use now is to move the factor from the numerator to the denominator (or vice versa) and make the exponent positive.

I have also been doing a lot of problems like 3^900 / 3^x = 9. These help push a deeper understanding about what is happening when you divide and multiply power expressions.

Tomorrow we will do a scientific notation review (hence the title of today's post), just to make sure they have this down before they move into chemistry next year. The end of the exponents unit seems like a good time to do it - especially now that they better understand what x 10^-5 actually means.

Perplexing Polynomials

This has been a very busy and productive week. Maybe that explains why I haven't been able to post recently...

I just came back from an event at Villa Montalvo, a beautiful place in the mountains around Los Gatos where they host artists in residence. A group of poet/actors called headRush was in residence there and helped a group of our students create a one-act play. It developed from short skits to a larger play, and tonight they performed for their families and for the Montalvo guests on a real stage with lights and sounds. They were extremely nervous but did an excellent job, and there were many misty eyes in the house :)

I had a couple of good lessons this week in Algebra 2. On Tuesday, I taught them about u-substitution and how it can be used to convert expressions into quadratic form. I think this is a good thing to get used to, so these kinds of substitutions won't be as much of a mystery when they get to trig and calculus. Then, I handed out a factoring flow chart that I made. I'm not sure yet how effetive it will be, but now, whenever a student tells me that they don't know what to do next, I tell them to show me where they are on the flow chart. They groan, then open their binder and pull it out. They look at it, and then suddenly know what to do next, without me saying a word. It's magic!

Today, I spent a long time making a puzzle for them to solve, but it was worth it, as it was one of the best lessons I've had in a long time. I was inspired by my recent obsession with Perplex City to create this review activity. I made a 6 x 4 grid, where each square had various equations and/or solutions along the edges, and a letter on the back. The students needed to solve the equations and match them with the solutions in order to assemble the puzzle. Then, they had to turn over the pieces to see the message that was formed. When they responded to the message, they won the prize. Here is an image of the finished puzzle, which I also posted on ILoveMath.

Here is the puzzle in word and pdf form.



I gave students the cut out puzzle pieces, and all I told them was that they needed to figure out how to put the puzzle together, and that I'd know when they were done based on their actions. Most of the students were really confused at first, and wanted me to tell them exactly what to do. But I persisted in not telling them, and after a few minutes, they all figured out how the puzzle worked. I think this was a good move on my part, because the small success of figuring out what to do helped them get more excited about actually doing it.

I gave the students an hour to work on this puzzle (I wasn't sure if it was going to be too much or too little time). When there were 20 minutes left, most of the groups had some clusters of pieces assembled, but no more. I was worried that no one would complete it, but with 10 minutes to go, the first team got all the pieces together. They turned over the pieces and stared at the message for a while. The way they put it together, it was backwards and upside down, and it took them a couple of minutes to understand it. But then a lightbulb went on for them, and the four boys dove to the floor and cranked out their 5 pushups. The rest of the class (who had not yet read the message) looked at them like they were crazy. Another group was about to be finished, but one of the girls puffed in frustration (by accident) and all of their papers went flying. She was mortified; I need to think about laminating these for weight in the future. Then, two more groups got it, and 8 more kids jumped to the floor to do their pushups. A couple landed on top of each other. The prize was a lovely box of valentine chocolate cards that said, "You won my heart!". Of course, there were chocolate Kisses for all as consolation prizes. It was a lot of fun, and I hope someone else can use this activity and enjoy it too.

Tomorrow: polynomial long division! Hmm... I don't think it will be quite as fun, but not every class can be pushups and Kisses, I guess.

The Difference Between Good and Evil

Though I finally finished my credential work over a year ago (I was in the system for a long time.. emergency, intern, preliminary, and now BTSA), reading stuff like this:

Masters program update:
[√ ] lack of content organization
[√ ] belaboring of small points
[√ ] disjointed course work
[ ] elevation of jargon over skill
[√ ] permanent residence in the Baltic Avenue of Bloom's Hierarchy
[ ] general awkwardness

makes me think back fondly on those days (I was at the same institution as TMAO).

On our refrigerator, we have a sample lesson plan handed out to us by the instructor in the "Teaching Second Language Learners" class. Note: this was held out as an example of how to write a lesson plan for language learners, not an example of how not to write a plan. Also keep in mind that some people had trouble passing this class. Ok, without further ado, I give you, dear readers, the unedited "Difference Between Good and Evil" lesson plan.
____________________________________________________________________
Title: The Difference Between Good and Evil
Subject:English (All Levels)
Name: ****** ******

Theme/Performance Standards
The students will examine some of the ways we determine the difference between good and evil. This information will be used to evaluate literary characters throughout the year. In addition, the students will learn how to write a one page, one paragraph paper.

The assignment will take two days and the students will be able to:
- state one difference between good and evil.
- compose a paragraph explaining that difference.

Preparation
The only material needed is a white board and a marker.

Building Background
The students will be asked for their opinions on this topic and vocabulary will be introduced as necessary.

Instruction
In round robin fashion, each student will be asked about the difference between good and evil. Responses will be placed in clusters on the board.

The process will be repeated several times and each idea will be clarified and examples will be given. If necessary, important vocabulary will be translated into English by other students in the class.

(This next bit is my favorite part:) The next part is the writing of one paragraph on one of the ideas. As the students write, the teacher will circulate around the room and help by asking clarifying questions that are slightly above the student's zone of proximal development. (ROTFLMAO) Students then read their papers to the class. The final assignment is rewritten after the teacher grades it.

Review/Evaluation
The paragraph determines if the student understands the assignment. Students that have problems will receive individual help. The final papers will be hung on the wall.

______________________________________________________________

I'm going to end this post the same way so many of my students end their presentations:

so, and, yeah.

Next Lesson: Sum & Difference of Cubes

This lesson will start with the new weekly quiz + notes check. If students' are unorganized or have incomplete notes, they will lose points. I just did this in my reglar algebra 2 class this morning, and only a handful of people got full credit on the notes check. While that is discouraging, it also shows how critical it is that they get continually assessed on short bursts of organization (they only needed to show me a binder with 2 days worth of notes, and an index with 2 entries).

After this, there will be some basic direct instruction on factoring polynomials with the difference and sum of cubes patterns (which I'm not exactly clear why this is worthy of a state standard, but there you go..). The only interesting part of this is that I will use the visual model shown in this post to help them see where the pattern comes from. You can think of a^3 - b^3 as the big cube's volume minus the small cube's volume. This volume is equal to the volumes of shapes I + II + III. You can easily get expressions for their volumes, and then factor out the common (a - b) factor from each term to derive the (a - b)(a^2 + ab + b^2) pattern.

That segues nicely into the next piece, which is factoring cubic polynomial by grouping. Students already learned this method for factoring quadratics, so this piece should go pretty smoothly.

I hope to get through the instruction piece fast enough to allow a good chunk of time for students to just practice using these techniques. They definitely have troubles applying the patterns to expressions like 125x^3 - 64y^3, or even worse, when you need to factor out a common monomial first. I'm trying to coach them to always write the problem in the form ( )^3 + ( )^3 first, so they can clearly see what the values for a and b are.