Next Lesson: Functions and Relations

Today's lesson was simply a chance for the students to begin working with their graphing calculators. I have a bunch of squirrely boys in the class, and they take to the TI-83+ like it was a Game Boy. It's amazing - I can show one of them how to do something once, and 5 minutes later half the class will be experts in it. I can give the same students directions about organizing their binders five times, verbally and in print, and there will still be tons of questions. It just goes to show how important motivation is in the learning process. One of my students keeps asking me where to find the games - after all, that's the whole point of getting one of these things!

Tomorrow, we begin the second unit, which focuses on functions. It's not going to be the most exciting lesson, as it will be primarily direct instruction.

The class starts with an introduction to functions, with the metaphor of a lens. I have two vertical number lines on a handout, with a picture of a lens (which I made in Google Sketchup) in the middle. The directions ask them to draw a stick figure on the left (the original image), with the feet at 0, the knees at 1, the hips at 2, the shoulders at 3, and the top of the head at 4. The lens is a "y = x^2" function, and they must map the projection onto the second number line by drawing guiding arrows from the inputs to the matching outputs. Later, when we do composition, I will use a double lens model to show how the projected image can then be used as the input of the second lens.

After that, we'll look at the definition of relations and functions, and I'll present them with functions in various forms: tables, arrow mappings, equations, graphs, and sets of ordered pairs. It's all about multirepresentational fluency, baby! We'll end that section with the vertical line test, and then learn about function notation.

I realized I screwed up on the homework I assigned from today's calculator lesson - I asked them to sketch a graph of f(x) = cos x. In my tutorial, I saw kids typing into their TIs: Y1 = F(x)cos(x). I forgot that I hadn't taught them yet what f(x) means. I'll just say what the freshmen say when they make a mistake: "just kidding!"

Unit 1 results are in!

I'm happy and relieved to say that our first test went pretty well. It was the right length - all students finished, but needed to work most of the period. I gave them a 7 page test (it was single-sided, and the last page was extra credit problems), and it was guiltily satisfying to hear their gasps of fear when I dropped them on the table in front of them.

I graded them quickly, because I was so anxious, and the results were nice:
Mean = 76%
Median = 79%

Here is a stem-leaf plot of the scores:

10| 0 3
9| 0 1 2 3 8
8| 0 1 2 4 8 9
7| 1 1 3 5 8
6| 1 2 4
5| 0 9 9
4| 8
3| 7
2|
1|


There was no single section that students missed across the board. I found lots of little mistakes that were common, and so I need to make sure to work those in as we move forward. I am trying to make more of an effort this year to constantly spiral in previous topics so they don't get lost.

Breakfast of Champions

Apparently, I found my new shirt to be a lot funnier than the students did...

Next Lesson: Study Groups

Ok, it's actually the last lesson I did, but I didn't have any time to post, what with Back to School Night and other stuff going on this week.

Tuesday is the first unit test for my Algebra 2 class. The first unit test is when reality sinks in for everyone... and we'll see if any of the students decide that they want to switch out of the honors class. I think a few might, but most of them - even if they are struggling - are really showing a lot of determination, which is impressive. I even have a few students who were in Numeracy last year as freshmen (meaning that they came to high school with math skills below a 7th grade level), and now they are in my honors class. I try not to play favorites, but I just have such a soft spot for them. Last year I was helping them learn how to add and multiply fractions, and by the end of this year they will be graphing complicated rational functions and solving 3 x 3 systems of equations. Now that's what you call ganas!

Anyway, one of the skills I want my students to learn is how to effectively use a study group to prepare for a test. Like everything else, this does not come naturally to them, and it needs to be scaffolded. So, after the normal homework review and Do Now practice problems, we did the following:

1) I gave them a handout detailing how they should organize their binders. They have a table of contents at the front, in which they enter each lesson's topic. Each of these entries corresponds to a page of notes, which should be in order at the beginning of the tab (each unit has its own tab). Following the notes, they keep all important handouts relating to the unit, and then all of their homework assignments (each assignment should have the original worksheet, their solutions, and any corrections all staples together). At the end of each unit, I ask them to go through the handouts and recycle (or at least remove from the binder) anything that they probably won't go back to and use anymore. I am trying to get them to learn how to evaluate the importance of materials (i.e. "hmm, this page of important looking notes I should probably keep, while this 10-minute in class worksheet we did can probably go...").

2) I gave them another handout with a structure for how to work in a study group. Here's the text:

How to Study in Groups for a Math Test!

1) Identify the sections that are on the test.
a. From a review sheet
b. From your notes, quizzes, and homeworks

2) Locate sample problems for each section.
a. From old quizzes/tests
b. From notes
c. From homework
(in that order!)

3) Figure out which sections are most difficult for you as an individual.

4) Figure out which sections you want to start on as a group.

5) Begin working on individual problems. Use a clean sheet of binder paper to do this – don’t write on top of your old problem. There are several ways you can structure this.
a. Each person works on the same problem first, and then everyone compares answers.
b. One person who knows how to do a problem shows the rest of the group.
c. Each person in the group works at their own pace, and asks for help as needed.

6) When you are stuck, do the following things in order:
a. Check your notes.
b. Ask someone in your group.
c. Ask your friendly teacher.

3) Then, because this was their first time, I gave them the sections that would be on the test (i.e. step 1). I told them that by the third or fourth unit test, I would expect them to be able to do this part on their own. I asked them to rank each section (0 = I don't remember this at all; 1 = I need someone to help me on this; 2 = I just need to practice this by myself; 3 = I am ready to do this on the test), and then to discuss their ranks as a group and come up with a plan of attack.

4) After that, groups had about 30 minutes left to begin their studying. This does feel like a lot of time to use up on not actually reviewing problems - I think it will have a long term benefit if I stick with it, but I might have to be prepared for some short term hits. Most of the groups took it seriously and they all had different styles to how they worked on the problems, and what problems they chose to focus on.

5) For homework, I gave them their practice test problems. I decided to not let them have it until the end of class, because they would have just done those problems and avoided looking through their old quizzes and notes.

So that's that. Wish them luck! I'll post the outcome of the test next week sometime. In case you're interested, and still reading, here are the topics that are on this test. (It's kind of a hodgepodge of review and new topics... I haven't figured out a way to do this unit better yet. I can't wait to get into unit 2, because it is all new and it has a more consistent structure.)

Unit 1 Test Sections
1) The real number system (types of numbers and properties of real numbers)

2) Solving equations with rational coefficients (fractions and decimals in front of the variables)

3) Converting decimals to fractions (including repeating decimals like 0.4222...)

4) Working with parallel and perpendicular lines

5) Systems of equations (linear combination, word problems)

6) Isolating a variable (even complicated ones, and when the variable is in the denominator)

7) Solving and graphing compound inequalities (AND and OR inequalities)

8) Using interval notation (going between number lines, inequalities, and interval notation)

9) Absolute value equations

10) Absolute value inequalities

Next Lesson: Absolute Value Inequalities

The last lesson seemed to go fine. Half of the class time was devoted to doing our computerized math grade level diagnostic testing, so I didn't really get a chance to see if they mastered the objective. I'll have to see how their homework comes in.

In tomorrow's lesson, we will begin with an error-checking Do Now. Students are given a series of problems (in this case, mainly focused on converting between number lines, interval notation, and inequality notation) that are already worked out, and must determine whether or not they are correct. If the problems are incorrect, students must identify, explain, and correct the error. I'm trying to do these kinds of activities more frequently, to build students abilities to reflect on mathematical solutions and identify common sources of error.

After this, we will extend the 5-step model used for solving absolute value equations to inequalities. Everything is basically the same - the hardest parts for students tends to be drawing a number line that represents the inequality, and then converting that number line into a compound inequality. I'm predicting that the work done in the Do Now will be sufficient scaffolding for this to be less of a challenge (I've definitely learned from mistakes made last year!).

The class will then have about 20 - 25 minutes to practice these problems individually or in pairs. I really hope I don't cut in to this time too much. For my students, this type of practice time in class, immediately following direct instruction, seems to be indispensable. But it is all to easy to go on and on during direct instruction, trying to get every last one of the students to understand what you're saying. This is never going to happen, obviously, but there is something that seems to compel me when I'm in front of the class to keep going, answer every single question, and make sure every student is ready to work. I've been working with a timer this year for the first time, and that seems to help. I do tend to say things to the timer when it gives me a warning beep ("alright, alright, I'm hurrying!") and the students look at me funny.

Anyway, handouts to be posted on ILoveMath.org.

Next Lesson: Absolute Value Equations

For Tuesday's lesson, we will be looking at the meaning of absolute value, and how to solve absolute value equations. I decided to go with the traditional idea of treating |x| as the distance between x and 0, and using that as the base for solving equations such as 2|x - 5| - 8 = 4. The method we will use is:
1) isolate the absolute value
2) write the meaning in words (just for now, until it sinks in)
3) draw a number line
4) use the number line to write 2 equations and solve them
5) plot the final solutions on a new number line

This is different than what we are doing in the regular (not honors) algebra 2 classes. There, the method is pretty much the same, but we are teaching them the fundamental concept that |x - a| is the distance between x and a. They are being taught to rewrite any absolute value equation in that form first. I think it is helping them solve the basic equations and inequalities, and it is definitely helping them with the "tolerance" word problems. That's ok for the regular class, because that satisfies the standards for absolute value, and they won't be doing anything else with it for the rest of the year.

My impression, though, is that teaching it this way somehow would make it harder if you go on to study absolute value functions. For example, I want my honors students, if given a graphical representation of a function (and not necessarily the equation), to be able to graph the absolute value of that function. I think that this requires you to think of the absolute value as something that always returns a positive value (i.e. the magnitude of the distance from 0), and not necessarily the distance between two non-zero numbers. I really believe that this type of representational fluency is necessary for them to be able to understand Calculus. Anyway, if anyone has any reactions or advice, please feel free to leave a comment.

The handout and sample problems I will use for this lesson can be found on ILoveMath.

Slam I Am

I really enjoy teaching math. I like working on math problems myself, and I find reading about the history of mathematics to be fascinating, but I'm no mathematician. When I first got interested in education, I definitely saw myself as more of a humanities person - probably English. It was actually subbing at DCP (before I taught there full time) that convinced me to become a math teacher instead. Our freshmen come in knowing very little about anything academic, to be quite honest, and watching them struggle through writing a simple sentence, I knew that English class (at least for the first 2 or 3 years) would not be about reading great books and having deep class discussion. Their math abilities were in comparable disarray, but it seemed like a more manageable problem - teaching math is much more linear than English. I've since learned that, even when students struggle with basic oral and written communication, it's still possible for the classes and assignments to be quite interesting. I like pushing the students on their numeracy, problem solving, and critical thinking skills, but I definitely miss being able to connect with students on a different level - i.e. hearing their thoughts about life, reading what they write about their families and friends, and so forth. I'm too busy pushing them to the next math standard.

Last year, through a partnership with MACLA, a local Latin American arts and culture venue (if you live in South Bay, I highly recommend you click the link and check it out!), another teacher and I began the DCP Slam Poetry team. MACLA hosts the South Bay Teen Slam League, and they offer free Friday poetry workshops to high school students. We had a few students last year who were regulars, and even competed in one or two slams, but we never had enough momentum to really get off the ground. Yesterday, we began the new season, and we took over about 15 students. The room was packed and the students really had a good time. It's an uncensored workshop and it's hosted by local poets (not musty old teachers). Though the actual "teaching of poetry" part of the workshop is not always the strongest, it gives the students a great venue for saying whatever the hell they want to say. I bring the students there, and I usually stay and listen to what goes on (brainstorms and free-writes, improv activities, read-alouds, etc.), but I don't participate at all. They get into it, and soon I'm not there - they begin to talk like they do outside of school. I like being able to not monitor what they are saying and to just listen. My goal this year is to actually engage in the same writing activities that they do (they always asked me why I wasn't writing, and what could I say?). So we'll see.. if I actually end up writing a good poem maybe I'll post it. I hope that enough students stay interested in it, and that the team will have enough momentum to move forward on its own. It's an amazing thing to see how empowering it is to those who muster the courage to get up on stage and (for lack of a less cliche description) bare their souls. When they finally perform, they come off the stage crackling with energy and life (which is not necessarily the normal state of being for a teenager!)

Next Lesson: Compound Inequalities and Interval Notation

After reviewing the homework and the results of the last quiz (mean: 76%, not great, not terrible), we will dive in to a little guided instruction time.

We'll start with a quick review of inequalities - students will add, subtract, multiply, and divide the inequality 4 < 8 by both 2 and -2 to see which operations cause the inequality symbol to reverse. We'll also review the idea that a < b means that a exists to the left of b - some students have problems with this, especially when the numbers are negative. We'll also see why x < a is the same as a > x, and that when graphing, you can't just draw an arrow in the direction that the inequality symbol faces.

Then, I'll show them how to solve and graph both simple and compound inequalities. I'll do a couple of examples, and then they will also.

Finally, I'll hand out a reading that introduces students to interval notation. They will read it and then complete the practice problems on their own, as I walk around and coach them. This topic is not part of the standards, but it is used extensively in Calculus, and I think Algebra 2 is the right place for students to see it. The handout (with problems) is on ILoveMath.org.

Update:
The lesson went fine, but there was not enough time for the students to complete the interval notation reading, so I'm pushing that to the next class. They were asking good questions about inequalities (i.e. why expressions like 3 < x > 6, or -2 > x > 2 don't make sense). These students seem to have retained a lot from Algebra 1, which is always a good sign!

Next Lesson: Isolating a Variable

In this lesson, students will learn how to solve for a variable that may be in multiple terms, as well as trapped inside of parentheses. I realized I needed to make this a specific objective in Algebra 2 when my Calculus students in previous years didn't know how to isolate dy/dx after taking an implicit derivative. They have also had trouble in science: for example, the Gas Laws in Chemistry throw them for a loop when they have to solve for a variable that's in the denominator. Hopefully, getting in some good practice with this solving technique now will pay off when it is reinforced later on.

We will start by looking at x + xy = 1, and finding the value of y for different values of x. Students will likely conclude that it would be easier to isolate y first. We'll then look at some specific formulas (Gas Laws, temperature conversion) and how to isolate for any given variable. The key to isolating a variable:
1) use the distributive property (if the variable is trapped inside parentheses).
2) get all the terms with the variable on one side of the equation.
3) get all the terms without the variable on the other side.
4) if necessary, factor the variable out, and then divide what remains.

The type of problem I want them to be able to do: Isolate y in (3 + 7x)y – 6(xy + 5) = 15

Following the direct instruction portion, students will have independent practice time with a related homework. These will be posted on ILoveMath.

Update:
The lesson went ok, but I realized I made the same mistake as last year... I need to teach students how to deal with the division form of the distributive property first, so they can understand why certain steps are invalid. For example, given the equation xy = 2y + 7, students will tend to "divide both sides by y" to get x = 2 + 7. I didn't think of this because none of the problems I gave require a division step like that - but I forgot to take into account that they would mistakenly do it. Woops. I think that will come up in the next lesson.

Next Lesson: 2 x 2 systems practice

The students liked learning the linear combination method. I like to play up things like that for them - "Now that you're in Algebra 2, you're ready to learn this new technique..." Of course, I screwed up my first example, which was supposed to have a simple integer solution - I copied the wrong number down in the problem and we ended with thirteenths or something, and then I decided to push ahead anyways (cause everyone loves fraction practice!).. So after thoroughly losing them, I started over with the other examples, and they caught on pretty quickly.

Next class:
- Students will practice solving linear equations with rational coefficients. I am using the "partner problems" idea that I saw on ILoveMath.org - you put the problems in two columns, where the matching problems are different but have the same answer. This way, students work on their own problem, but can easily check if they are getting it right by comparing their results to their partner.

- Then, I will do a brief direct instruction piece where we work through solving a 2 x 2 system word problem together following the key steps (identify the variables, write the equations, solve, answer in a sentence) on the overhead.

- Finally, students will practice with more word problems in the Problem Relay format. Each group selects a runner. I sit at the front with a stack of problems for each group. The runners get the first problem and the teams work. They send the runner up when they are finished to have the problems checked. If they are right, they move on to the next problem. If they are wrong, they get a penalty (either points or time lost) - and I use my own discretion to determine what kind of help to give them. At the end, the team with the most correctly solved problems will earn some mythical extra homework points.

Update:
The students loved the relay game. Almost everyone was working full-tilt for 30 minutes straight. One of my students has Asperger's and he was chosen to be a problem runner for his team. He took it a little too seriously - the first time his team finished a problem, he launched himself at me full speed, and I literally had to fend him off so that he didn't crash into me. I coached him a little on how to properly bring the problem up; the next time, he ran the same way, but stopped himself in time, and only knocked a bunch of papers off the table. He's a funny kid - the only student at our school who is on a "no reading" contract, and who gets his books taken away (like others lose their cell phones) because he won't ever stop reading, even in class. Anyway, we'll have to do this activity again in a few weeks.

Next Lesson: Linear Combination

I am trying to review key algebra 1 concepts as well as move forward with the new stuff. We do not teach linear combination in algebra 1, because it takes all of our effort to just get across the idea of solving a system graphically and algebraically (through substitution), and how the graph is related to the algebra.

In my next lesson, I will start by reviewing the substitution method a bit more, with an example that causes lots of fractions to form. Then I will introduce the idea of linear combination through 3 progressively harder examples (i.e. needing to multiply neither, then one, then both equations for it to work out).

Finally, students will work in pairs on basic practice problems. The only innovation I have here over last year is that I am requiring students to explicitly answer these questions before beginning to solve each system:
1) Is there anything you can do to simplify either equation (i.e. can you divide out the GCF, or multiply by the LCD to clear the denominators?)
2) Which variable will you eliminate?
3) Which equation (if any) will you multiply, and by what factor?

So no need to post anything on ILoveMath . But I'm hoping that forcing students to answers those questions will help them clarify the process in their minds.

I Love Math!

I discovered the "I Love Math" site (see link to the left) a couple months ago. I've decided to start posting my stuff on there. If you're a math teacher, I really recommend that you check out the site, download stuff, read the message boards (though there still seems to be some problems registering for them), and upload files that you create. Think about how much time you could save if that one activity you were going to create was already there! And for me, I find seeing the way other people approach teaching a topic to be very informative. Sometimes, just seeing a new way to pose a problem can really get me thinking.

So feel free to look at my files, and send any feedback my way. I decided to not post everything from previous years, but to just post my Algebra 2 materials as I revise them this year. As I update my lessons, I'll post on this blog what I'm doing in class, and what files I've submitted to the site.

If there is a specific topic that you want to see how I teach, or see if I have any files to support it, just leave a comment or send me an email.