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

Don't worry, I'm not dead...

My Numeracy students are now about a month into their ALEKS experience. I started all students out on the third grade standards level (the lowest ALEKS goes), and on average, my students scored around 50% mastery on their initial diagnostics. At this point, some of the students have completed level 3 and are onto level 4, and many others are close to completing the level. There are the stragglers too, of course. I'll do more detailed stats later on. The goal I've set with students is that they should try to complete 3 entire levels by the end of the year (i.e. 3 years of growth in math ability). I was skeptical at first, but after seeing how the students interact with the program, I have much more hope. ALEKS is not a creative, fun, snazzy program. Essentially, students get a sample problem to try. If they don't know how to do it, they read an explanation and try again. When they get a certain type of question right 3 or 4 times in a row, without asking for help, the concept is added to their pie chart. Periodically, they are re-assessed by the program, and concepts they no longer know are pulled back out of their pie chart.

I have been impressed by how self-reliant the students are being. They are managing to read the explanations and figure out the problems on their own. Some students are really getting into it, and are bragging to each other about how much of their pie they have completed. They have also figured out that getting a problem wrong, or clicking on the "explain" button causes the program to require more correct problems to add the concept to the pie. For that reason, they are actually trying harder to get the problem right the first time. The immediate feedback has been very helpful for the students. My favorite moments are now at the end of class; sometimes, when I tell students they need to log off, a few will be like "oh wait, let me just get this one last problem so I can add it to my pie".

Right now, I am just assessing them on time spent on ALEKS - not on the actual amount of progress being made; it seems to be effective enough, and the whole point is to allow students to work at their own pace. We'll see if I need to modify that policy in the future.

On a different note, we have been working on bar modeling to solve word problems every class for 15-20 minutes. I assigned the first problem set as homework last week, and I graded them this weekend. They were quite bad. It's always a bad feeling when you realize your students are a lot farther behind than you thought. I've pushed ahead into more complicated problems, but I just realized that many students are still having trouble with the basics. That's ok.. we'll just cycle back to the beginning and have another go at it.

In Algebra 2, we've started in with the basic idea of logarithms, using the Big L notation I wrote about in an earlier post. I think it is working well. We have been focusing on the similarities between roots and logs: in a root, the index tells you the exponent, and you are looking for the base. In a log, the subscript tells you the base, and you are looking for the exponent. Last year, many students had trouble in power expressions determining when to use a log or a root; I think they will have a much better understanding of it this year.

The Big "L"

There are some students who, no matter what, can’t seem to comprehend what a logarithm (when treated like an operation) is doing. I see students that:

1) Cancel the log.

2) Multiply by log.



3) Ask where the 2 went when log2(8) is simplified to 3.

These mistakes indicate that “log” is being perceived as some sort of quantity to be manipulated, not as an operation. This may be due to the fact that “log” is the first time students are exposed to an operation that is represented as a word instead of as a symbol or other numerical notation. Texts apparently assume that this is a natural transition, not even worth mentioning, but it’s pretty clear that it is not as obvious as one might think.

To help students see what is going on, I’ve tried expressing other operations in a similar manner and drawing parallels. For example, take a look at roots and powers:

Logarithm does not have a symbol; our initial idea was to therefore rewrite exponentiation in terms of the “word operation" exp. We then explained that logarithms are the inverse of exponentiation, and that they undo each other, just like addition and subtraction, multiplication and division, and powers and roots.

This seems to have worked moderately well in terms of getting students to be able to evaluate and solve the log problems that they encounter on the STAR tests. However, I don’t think it’s really helped them to understand what a logarithm is, and their ability to apply the concept flexibly is quite limited.

I’m wondering now if going the other direction would have been better. Instead of rewriting exponentiation as a “word operation", we could have invented a symbolic representation for logarithms – say, a big L. (Not to be confused, of course, with the L formed by thumb and pointer finger, raised to the forehead!).

Inverse operations could then be modeled like this:

When I ask my students what “the third root of 8” means, they are pretty good about saying something like “what number to the third power gives you 8”.

When I ask them what “the log base 2 of 8” means, they rarely can say “2 to what power gives you 8”. I wonder if using a symbolic representation of logs will allow this meaning to be clearer. After all, when you think of a log in this way, it’s not really that much more confusing than a root.

I’d be interested in hearing any thoughts on this. Would a symbol for log be helpful? Confusing?

Staying on top of blogging is like solving a greased Log

Or something. Ok, I haven't posted in a while about what's going on in class. So a summary:

1) The STAR Search(tm) treasure hunt is in effect. The idea of giving away puzzle pieces for each correct answer on the daily 10-minute 5-question warmup has worked really well. Most of the teams are nearly complete with their puzzles, and a few have worked it out already. It was the smiling face of another teacher (photoshopped to make him a bit harder to recognize) with the text "Find Me!" and "Make my noise of disgust." (He has a patented barf-noise he makes whenever PDA is observed in the halls). So a couple teams have done this, and are now working on puzzle #1. I'll post them all eventually, but this one has them reviewing concepts of the real number system from unit 1. There are a bunch of true false questions that they convert to 1s and 0s, and then must research binary to figure out what number is being represented, which will lead them to the next teacher based on a look-up list. An insight into DCP student critical thinking: one student asked me today what to do, because he got the 1s and 0s, but didn't understand how to get any of the numbers. I asked him to read me the title of the puzzle. "There are only 10 people in the world: those who know binary and those who don't." He looked at me. I asked him if there were any words in the title that he didn't understand. He said, "binary". I suggested that finding out what that means might be a good place to start.

2) In Algebra 2, aside from the treasure hunt, I am now desperately trying to get them to grasp logarithms before the STAR test next week. The proximity of the test has forced me to teach the unit totally out of order, and it bums me out. On the test, they only need to be able to evaluate basic logs, change the base, use the log properties, and spot incorrect lines in a log simplification problem. There is nothing about the log function and its domain, the inverse relationship with exponentials, solving real log or exponential equations, and so forth. The order I would have preferred would have been:

1. introduce the concept of exponentials with a "trick" problem like a lottery or the grains of rice on a chessboard type thing
2. develop an understanding of exponential functions - growth and decay, and maybe some translations and transformations
3. present word problems (population growth, interest, depreciation, carbon dating, etc.) and model them with exponential functions
4. use these models to help students realize that we don't have a tool for finding the x when we know the y, and why we need one - springboard to the logarithm operation
5. develop a feel for how logs work, by estimating their value to being between a pair of consecutive integers; compare the log operation to the nth-root operation
   
6. convert back and forth between log and exponential form, and solve basic log and exponential equations
7. learn how to change the base of logs
   
8. go back and use logs to answer the questions in the word problems that we previously couldn't
9. derive and learn the properties of logs, drawing parallels with the properties of exponents
10. use the properties to solve more complex log equations, including discussion of domain and extraneous solutions
11. explore the graphs of log functions
12. use that as a lead in to a brief unit on inverse functions

So instead, I jumped in at 5 and continued with 6, 7, and 9. Then, after the test next week, we'll go back to 1 and move forward.

3) I have a student teacher now, and he is taking over the instruction, as of today. It's really cool to work with someone like that, and help them learn how to become a teacher. But I didn't realize how hard it would be to change my own work habits (I can't just plan where and when I want to), and it's difficult to know how much to do versus how much to let him do, knowing that he's got to try things on his own, yet also wanting to make sure that my students learn the material. He's got a great natural patience and rapport with the students, and once he gets the teachable stuff down, he'll be on fire.

4) We're in Spirit Week now. Yesterday was boy-dress-like-girl-girl-dress-like-boy-day (umm... student council came up with the themes...) and today was dress-like-your-culture-day. Lots of sombreros and mariachis walking around, and apparently "hoochie" and "jeans and t-shirt" are cultures too. I abstained yesterday, but today I wore my bar-mitzvah talit and kipah, which always leads to the expected questions: "You're Jewish? Really? Do you speak Jewish? What's Jewish? etc." Today, a freshman took one look at me and said, "What race are you supposed to be?" Tomorrow is class color day (Go purple! Sophomores! Wise Fools!) and the Numeracy Project will be playing; I'll be doing my world-premiere version of "Slope Is Rise Over Run" (The Animals). That will lead into "y = mx + b" (borrowed from Semisonic), and then the ever popular standard "Sweet Home Alameda".