I made a review lesson for my Algebra 2 students on these topics, to make sure they are really ready before we start performing operations on complex numbers.
Some instruction, some board races, and there you go. Hope you like it.
Lesson
Keynote
Quicktime
I also updated my Algebra 1 box with unit 2 files and the first four lessons of unit 3.
Ok, so I guess it should really be titled Algebra 1, not 2. But my students always need to review/relearn this topic. We'll go easy for the first lesson - only problems where the GCF = 1 and where the leading coefficient is 1. I made a puzzle for them to put together so that it is more fun than just doing a worksheet. I did something like this in the past with my honors class (but with much harder polynomial equations) and they really enjoyed it. That puzzle, once assembled, instructed them to do push-ups to get some candy. This one only requires that they tell me a joke - I'll add them to my arsenal if they're any good.
Lesson 5 (Factoring Trinomials 1)
Puzzle (doc / pdf)
You may also be interested in the puzzle-based Treasure Hunt I did a couple years ago in Algebra 2.
We just finished our first week of the second semester. The previous two weeks have been our Intersession period, where all students and teachers do totally different classes. This year, I did an algebra review class, helped organize our junior "boot camp" to help get them more ready for the college application process, and taught an anime class.
But now it's back to normal school, algebra 2, and time to start learning about polynomials. The first lesson was not that exciting, as we spent a lot of time learning all the needed vocabulary. But we also did learn about end behavior of polynomial functions, both graphically and algebraically. The next two lessons were more interesting, as we looked at the zero factor property from a graphical perspective, and then we learned how to sketch a polynomial function when given its linear factors graphically. This is scaffolding for the number line model lesson that will happen on Monday, which will allow students to solve factored form polynomial inequalities like (x - 3)(2x + 5) < 0. This isn't in the algebra 2 standards, but this kind of analysis will push them to understand functions more deeply, so I think it is worth the time.
In lesson 3, students worked as a class to generate sketches of product functions by multiplying the linear factors. They really caught on, and were able to easily get through the first problem pictured in this post. It was great to watch them work together as a class so well. The goal is that, on Monday, they will be able to understand and solve the second problem in this post.
Here are the files:
Lesson 1 (Classifying Polynomials / End Behavior) word / keynote / quicktime
Lesson 2 (Zero Factor Property) word / keynote / quicktime
Lesson 3 (Curve Sketching - Graphically) word / keynote / quicktime
Lesson 4 (Curve Sketching - Analytically) word / keynote / quicktime
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.
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.
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.
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.
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.
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.
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