Algebra 2: Solving Basic Exponential Equations

On Thursday/Friday, students learned how to apply logarithms to solve basic exponential equations in the form ab^x+c=d. They did pretty well with it, although, as expected, when I threw in a power equation at the end (like ax^b+c=d) everyone took the log of both sides and then got stuck. Analysis is something my students are notoriously poor at, and teaching students how to analyze is notoriously difficult. In the next lesson, we will review roots and logs, and the focus will be on how to tell when you should use one or the other.

Here are the files:
Lesson 12 (Solving Exponential Equations)
Lesson 12 Keynote
Keynote Quicktime

Algebra 2: Intro to Logarithms

Tomorrow, I will introduce the students to logarithms. I decided to start them early in the year for a couple of reasons. Our first unit is on the Real Number System, along with the operations that can be done on real numbers that we don't study in Algebra 1: nth-roots and rational exponents, absolute value, and logarithms. Secondly, students have lots of trouble mastering the log properties. We typically teach it all at once; my thinking is that front-loading what logarithms are, and how to convert back and forth between logs and exponential form, will make it easier to teach log properties later in the year. There are quite a few log problems on the STAR test, so I'm hoping that this is one standard in which we can make some real growth.

I wrote about using the Big L notation a while back. We used it a little bit last year, and I have anecdotal evidence that it improved students' learning. This year, I am going to go full-on with the Big L, and only practice converting from regular log notation as we approach the STAR test. Just to summarize why I am using Big L:
1) Clearer notation - symbolic instead of a "word"
2) Easier to compare/contrast to radicals
3) Helps students understand that log is an operation, not a number or variable
4) Makes it easier to read and remember log properties

There was a bit of discussion on this on the previous post, but it kind of fizzled out. I'm hoping to get more feedback on this from you all, especially if anyone else decides to try it out.

Here are the files for the next lesson:

Lesson 11 (Intro to Logs)
Lesson 11 Keynote
Keynote Quicktime

Algebra 2: Solving Absolute Value Equations

On Monday, students will learn how to solve absolute value equations. Their next skills test will be during the following class. So far, I think the mastery assessment plan based on skills tests is going well. Of about 130 algebra 2 students, I've had probably 40 - 50 retake their first skills test. Most of the retakers improved their scores, though a few didn't. We might have to examine the merits of studying and getting help before retaking.

This is going especially well due to the presence of the newly formed advisory program. One aspect of this is that all teachers are now asked to keep their online grade books up-to-date (which didn't happen in the past), so that advisors can show their advisees their grades on a weekly basis. Because skills tests in my class are 50% of the final grade, and we've only had one so far, students who had As (from doing all the homework) but scored 4/6 or lower on the test dropped down to Fs. No matter how many times we try to explain how volatile grades are at the beginning of the marking period, students can only see the letter, not the process. In this case, though, it works in my favor, as students see that retaking the test and getting even a 5/6 brings them back to passing, and a 6/6 takes them to an A.

Here are the files for Monday:

Lesson 10 (Solving Absolute Value Equations)
Lesson 10 Keynote
Keynote Quicktime

Update:
Oops.. there was a typo. Keynote files now fixed.

I am now fully qualified! (for the next 5 years)

After three years of emergency credentials, two years of intern credentials, and two years with a preliminary credential + the joy of BTSA:

Dear Daniel Greene

You have met all of the necessary requirements to receive a recommendation for the following document.

Credential: CL--5YR CLEAR EXPIRES EVERY 5 YEARS--: Single Subject Teaching Credential
Issuance: 07/31/2008

Pop the corks! I feel like my lesson plans are about to get a whole lot mathier.

Algebra 2: Intro to Absolute Value

No words of wisdom today. Tired. Here are the files for tomorrow's lesson on the introduction to absolute values. The day after will be solving absolute value equations. I'm skipping absolute value inequalities this year.

Lesson 9 (Intro to absolute values)
Lesson 9 Keynote
Keynote Quicktime

Algebra 2: Rational Power Equations

Apparently, Tic Tac Toe Battle Royale works better with older students. In all of my classes, students got very into the game, despite the fact that there was no candy prize being offered (and only a couple of students even asked). The game was very noisy, but it was the noise of work and competition. I could tell, because each time I put up a new question, it got very quiet at first, and then the noise would build as they started discussing answers - and practicing psychological warfare on their opponents! When the noise reached a certain level, I knew it was time to show the answer. As soon as the answer was on the screen, there was much celebrating (or "aww, man!"), and then we moved quickly to the next problem. I highly recommend this as a review/practice activity.

In the next lesson, students will solve equations in the form ax^(m/n) + b = c. I wasn't going to do this originally, but I thought it would be a good way to reinforce all of the skills we've been working on up to now, set in the context of doing new and harder material.

After this, we will do a couple of lessons on absolute value and solving absolute value equations. Then, we'll move into logarithms. Yes, logarithms already. More on that soon.

Here are the files:

Lesson 8 (solving rational power equations)
Lesson 8 Keynote
Keynote Quicktime

Algebra 2: More Rational Exponents

Time to slow down. I can tell that I am forging ahead a bit too rapidly - I know that students need at least another day to work on rational exponents. This was a constant challenge last year. Some students, no matter how many times we went over it, could never remember what to do when the exponent was a fraction. A negative fractional exponent with a negative base might cause some heads to explode. So, more practice, more time, more scaffolding. Hopefully it will stick a little better this year. But I have to remember to be patient and willing to invest the time to practice. So, next class, before taking the first skills test of the year, we'll do a little Tic Tac Toe Battle Royale to practice. We'll see if the older students like it more or less than the freshmen.

Per H's request, I've started putting the presentations in Quicktime, so you should be able to view it even if you don't have Keynote. I also went back and added links in the previous lesson posts.

Lesson 7 (More rational exponents)
Lesson 7 Keynote
Keynote Quicktime

Algebra 2: Rational Exponents

Mondays are exhausting. I have four 80-minute classes, a 45-minute advisory, a meeting during lunch to plan advisory, and a Leadership Team meeting after school. And then I have to plan for Tuesday. That's why it's 9 pm and I am just getting ready to go home now. Is this sustainable? Um...

So the lesson for tomorrow is on evaluating rational exponents. Lecture, practice, review, repeat. Hard to be creative sometimes... Anyway, here it is. Hopefully it can save you some time some day.

Lesson 6 (rational exponents)
Lesson 6 Keynote
Keynote Quicktime

Algebra 2: Power Equations

I had a great Friday. My Algebra 2 classes went well, though the Keynote took about 10 minutes longer than I wanted. But, the slide where I showed them how you can use the Pythagorean Theorem to locate root 2 on the number line was quite successful (I think). They agreed that we could construct the original square out of a 1' x 1' piece of wood. Then, after we use the compass to map out the length of the diagonal, I showed that we could cut a piece of wood that is exactly root 2 feet long. I'm really trying to drive home the idea that irrational roots are still real amounts, and this slide made their brains hum.

And my single Algebra 1 class is starting to go really well. Over a week, and not a single referral. And, only once did I ask a student to step out of the class to calm down. I've got a TA who helps out by checking and logging homework, and then assisting students during practice time. I taught her a couple years back in Algebra 2 honors, and she is now one of a handful of seniors taking Calculus at a local junior college. Plus, I have another former student senior who has decided to use her free period to come every class and sit with Kate, and Kate is very happy with this arrangement. We learned the first part of the order of operations (aside from parentheses and exponents), and though they all surprisingly had heard of PEMDAS and knew that multiplication and division come before addition and subtraction, only about half knew the "left to right" part of it. So, when we got to that example, a big debate erupted, along with "you wanna bet"s and so forth, but it was all done in a positive way. And when the answer was revealed, the kids who were fighting for the wrong side were gracious about it (though I did make it extra clear that they could have been right too, and mathematicians just had to pick one way to do it). They were my last period of the day, and as a gift to them, when I got home I made a positive phone call home to every kid in the class. It took about an hour or so, but I'm hoping that it will turn out to be a good investment in furthering our class culture. The parents were very grateful to hear from me - even the ones who almost had a heart attack when the math teacher was already calling home. I had to do some quick assurances that "todo esta bien, no hay problema!"

On Monday, we will be solving power equations in Algebra 2. Nothing too fancy, but we will be doing Showdown for the first time - one of my favorite collaborative activities. Here are the files:

Lesson 5 (solving power equations)
Lesson 5 Keynote
Keynote Quicktime

Algebra 2: Intro to power equations

In the last lesson, students learned what n-th roots are. In this lesson, we explore what happens with positive and negative radicands when we have an even or an odd index. Students need to understand why taking an even root of a negative number yields no real solution, and that this is different than an irrational solution that is real. The estimation exercises in the last lesson were meant to start tackling this misconception: i.e., "you can't take the square root of 12 because no number times itself is 12".

In this lesson, students also learn when to include the plus/minus and when not to - which I think they will tend to confuse with the real/not real question. Additionally, we discuss the difference between exact and approximate solutions. I had originally planned on solving equations like 3x^5 - 40 = 152 in this lesson, but I decided that really focusing on the stuff I described above merits a whole lesson.

By the way, I really wowed the students with the new animations in Keynote - flame, sparkle, etc. I know that it's bad design to rely on animations, and I only typically use dissolve (to make text appear more gently), wipe (when I have arrows), and pop (when I want something eye-catching to appear, like a circle around a group of numbers). But I just couldn't resist having a wrong answer burst into flames. A little showmanship really made lesson 3's presentation more fun. I held the remote behind my back, and clicked as I threw a fireball at the screen with my other hand. Quite a few kids were properly amazed at my magical talents. And later in the class, I used sparkle to make the 2 disappear off the radical sign: I flicked the screen with my finger, and it sparkled away. I told them that Tinkerbell took the 2 away.

Ok, here are the files for tomorrow:

Lesson 4 (real or not, into to power equations)
Keynote for Lesson 4
Keynote Quicktime

Does anyone else find this to be hilarious?

Or is it just me?

Anyway, if you don't already have xkcd in your RSS, you should definitely add it right away. And don't start browsing the archives unless you have a couple of hours to spare.