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.