Algebra 2: Composite Functions

Last week we started work on composite functions. Using different representations was a very effective way to scaffold the idea of having the output of one function be the input of another function. Instead of students getting lost in algebra, using table and graph representations first let them clarify what was actually going on. Then, I added in the equation representation, and most students were able to figure out what to do before I even explained it. On Monday, we will continue with composite functions, where the goal is not to evaluate something like f(g(2)) but to simplify something like f(g(x)) when equations are given for f and g.

Lesson 3 (Composite Functions) - Part 1
Lesson 3 Keynote
Keynote Quicktime

Lesson 4 (Composite Functions) - Part 2
Lesson 4 Keynote
Keynote Quicktime

Newsflash: Students Don't Study

The results from my first Algebra 2 comprehensive test were (predictably) bad. Though they were even worse than I was anticipating. The test included a reflection that asked students, among other things, if they felt that they were well prepared for the test. Most of them were honest and said that they didn't really study. I gave students the option to create and use a 1-page study sheet for the test; less than half of them bothered to do this. It's an ongoing battle trying to get students to see and believe that there is a connection between their actions and the grades that they receive. Many students wrote that they thought they would be able to pass the test without studying. I hope that this is a wake-up call for them. I know that they want to succeed in the class - I have to do a better job of teaching them how to study and convincing them that studying actually has a purpose.

Algebra 2: Evaluating Functions

We're beginning our trek into the deep waters of representational fluency - my favorite part of the algebra 2 curriculum. If my students only retain one thing from my course, I'd like it to be the ability to move back and forth between equations, graphs, and tables with (relative) ease. So, today we evaluate functions in all these forms. Tomorrow, we do the same, but toss in composite functions. I think the Keynote animations are really powerful here - especially when working with graphs. Though they know which axis is which, students tend to get all turned around when trying to read graphs this way. I think the vertical lines and moving points help them see what they are looking for.

Lesson 2 (Evaluating Functions)
Lesson 2 Keynote
Keynote Quicktime

Department Photos

It's been a yearly tradition for some time now for the DCP staff to take creative department photos. Our former photo teacher, Michelle Longosz, comes in, and we do a full on photo shoot. Math decided to go with a Brady Bunch theme this year. Here are math and science, and you can see them all on our website.

Algebra 2: Intro to Functions

I have graded only 1 out of 4 classes worth of midterms... can't put them off too much longer. That period averaged a 61%. I was hoping it would be higher, but given the nature of the cumulative exams, I guess that is pretty good. It really helped identify the kids that have no idea what is going on, or are not retaining anything, in a way that the skills tests don't.

Tomorrow, we move on to unit 3, which is on functions. The first lesson is on the various representations of relations (table, arrow map, graph, equation, set of ordered pairs), what domain and range is, and how to determine if a relation is a function. I put together a Keynote that I think is pretty good, though it took way too long to build. I hope it is useful to someone besides me.

Lesson 1 (Intro to Functions)
Lesson 1 Keynote
Keynote Quicktime

Web-based scientific calculator

There is a new online scientific calculator that is worth checking out. It allows you to type in text instead of keying in numbers, and it also does unit conversions. It also allows you to set up variable expressions, and plug in different values.

Algebra 2: Complex Number System and Review

The skills tests I give target micro skills, in the form that they are presented on the state tests. I do my best to make sure that each question only tests one algebra skill at a time. The system has been working out well in terms of teaching and assessing those individual skills, but I still wanted to leave room for assessing students' abilities to synthesize and analyze. So I am giving periodic comprehensive exams that attempt to do that. To the right is an example of what I mean.

We are approaching the first of these, which covers material from units 1 and 2. So the last lesson and tomorrow's lesson mainly focus on review. The only newish material is to expand our understanding of the real number system to include imaginary and complex numbers. I've got some Showdown going on, some review packet action, some "how do you study for a math test" work, and so on. Here are the files.

Lesson 5 (Practice with Complex Numbers)
Lesson 6 (Complex Number System and Review)
Lesson 6 Keynote
Keynote Quicktime

Understanding Weighted Grades

Many of my students still don't get the idea of weighted grading (and, let's face it, neither do some of the staff members). I want them to understand that the skills tests are the biggest part of their grade, and thus very important, but that the other parts of their grade are important too. So, in preparation for the upcoming midterm, I made a little visual presentation to help them see how it all fits together. I think it helped. This show is dedicated to all the Renees, Toms, and Michelles out there.

Weighted Grading Keynote
Quicktime

Algebra 2: Operations on Complex Numbers

This week, we've been working on adding, subtracting, multiplying, and dividing complex numbers. The students who are good at polynomial operations from algebra 1 love this unit, because it is so easy. But there are many students who never really mastered this in algebra 1, so they are more frustrated, but it is a good chance for them to review and finally get this fundamental algebra skill. Here is an example of the main problem that students have:
(3 + 2i)(4 - 5i)
(3 + 2i) - (4 - 5i)
The number of students who do these two problems the same way, even after focusing on this specific distinction multiple times, is kind of staggering. They are all FOIL happy!

Well, little by little, bit by bit, we'll make progress, as always. And then they'll forget it, and we'll start again. One day it'll hold, I just have to believe.

Here are the files from Monday and Tuesday.

Lesson 3: Adding, Subtracting, and Multiplying Complex Numbers
Lesson 3 Keynote
Keynote Quicktime

Lesson 4: Dividing Complex Numbers
Lesson 4 Keynote
Keynote Quicktime

Algebra 2: Equations with Complex Solutions

Today, students learned how to simplify radicals with negative radicands, and how to solve equations with complex solutions. "No Real Solution" is no longer an acceptable answer to a problem. Nothing fancy, just some key examples and practice time.

The skills tests project is still going well. I had about 30 kids in after school today who were retaking one or more skills tests. The tests are short, and each one only takes me a few seconds to grade (since there is no partial credit). Students love seeing their grade go from a C or F to an A or B, just like that. Giving them frequent chances to master small sets of knowledge is proving way more effective than giving few chances to master large sets of knowledge. Not that this should be so surprising.

The end of the grading period is this week, and I have much higher grades now than I ever have had before. I won't really know how this all works out until final exams come around, and STAR Test results come in (next summer!), but so far, things are looking good.

Here are the files from today's lesson:
Lesson 2 (Complex Solutions)
Lesson 2 Keynote
Keynote Quicktime

Algebra 2: Intro to Complex Numbers

We've finished the first unit, which was on real numbers. Unit two focuses on complex numbers and their operations, and solving basic quadratics with complex solutions.

I went through the main ideas with them about what imaginary numbers are, how the imaginary number line works, and multiplying imaginary numbers. I used the "multiplying by i = 90 degree rotation" idea that I wrote about before, and with the use of Keynote animations, it was even more effective. I really suggest using this method to teach about powers of i.

Here are the files:
Lesson 1 (Intro to Complex Numbers)
Lesson 1 Keynote
Keynote Quicktime