The Exponential Curve

Where have I been?

2011-08-05T21:04:00.000-07:00

I've been putting off writing this for a long while, but it's finally time. Since January of 2010, I have gone through some life-changing experiences. I posted already about my sister's death last February; that, plus some difficult health issues (mine and other people's) have made for an extremely trying time. While many things about me have remained the same, overall, I feel like a different person now.

For the past eleven years, I have worked full-throttle at DCP, at a minimum of 50 hours per week, but often more like 60 or 70. I poured all of my energy into improving my curriculum and instruction, providing extra support to my students and their families, and basically eating, sleeping, and breathing school. Doing this was entirely my decision, though the realities we face at the school (students' low skill levels, ever increasing demands for success on testing, fewer and fewer spaces in the CSU system, and continually decreasing state funding that whittles away our program) create an environment in which the passionate teacher feels that working like this is necessary. I've lasted longer than any other teacher there (I was awarded the Lobo of the Decade award!) but I realized earlier this year that I needed to make some changes and run my life differently. I love teaching, and I did not want to leave the classroom, but I needed a new environment where I could finally have some balance.

This past year has not been all bad. In fact, my girlfriend and I decided to get married last fall, and we ended up having the ceremony on July 3rd in Sunnyvale. It was a beautiful wedding - we had a traditional Jewish wedding ceremony as the foundation, but we personalized it to make it fully egalitarian (as opposed to the traditional man's acquisition of the woman), and lots of our friends helped out to make the day amazing. The wedding canopy was designed by two of our friends (one is the art teacher at DCP) - they presented us with an amazing quilted canopy covered in pictures of our family members.

The cake was created in our own kitchen by my wife's friend. The kitchen was a disaster at the end, but the cake came out really well. It fit right in with our rainbow theme. And the cake topper was created by another friend of ours - based off of some pictures of Jen and me!

Besides the Chuppah and the cake, we had lots of other friends help with running all the details of the day. We had a fantastic time, and the whole weekend flew by incredibly quickly. We just got back from our honeymoon in Mendocino - neither of us had been there before, and we really loved it. Very peaceful and beautiful, just what we needed.

My wife Jen is a teacher also, and she started work at a school in San Bruno (near the San Francisco airport) last year. She loves it there, and they had a position open for this year, which I was hired for. I'll be teaching IB Math Studies, a CAHSEE prep class, and Intensive Algebra 1. I think this is a school environment in which I can do a good job for my students, but still be able to pursue outside interests, exercise, and take care of my health. I've met most of the math department there, and they are all nice people, so I am looking forward to starting this new stage in my career.

I don't yet know what I'll do with this blog. I need to find out if there are any school or district policies against blogging. I don't intend to blog under a pseudonym. Which reminds me - instead of my wife taking my last name, we combined our two names (Greene and Wekselbaum), and we've both legally changed our names. So you can now call me Dan Wekselgreene.

In any case, I will leave this blog up indefinitely, since I know people are still finding it and downloading resources. I enjoyed being part of the virtual math blogging community, and I've pretty much lost contact with everyone. Hopefully, I will be able to post more in the future, and also find my way back into the fold (though not too far in - balance is the new name of the game!). Good luck to everyone with the new school year, and may you also seek and find balance, so that you can stay strong both in and out of your classroom.

Some fun(ish) worksheets

2010-04-04T22:22:00.000-07:00

I'm going to try to get my box.net materials updated over this coming week. In the meantime, here are a couple of decent worksheets that you may find helpful.

First, I made one to practice graphing standard form - I just ripped off Mr. K's idea. Thanks! And some of my students actually liked the joke (I googled Laffy Taffy jokes).

For tomorrow, students will be graphing systems of inequalities, so I decided to create a little Ohio Jones adventure (Indiana's lesser known brother). Here is the full lesson and just the activity in pdf form.

(UPDATE: Here is the follow-up lesson in word form - Ohio Jones and the Pyramid of Power. Here is the follow-up lesson in pdf form if you're having trouble seeing the word doc).

Here is what the maze should look like after being solved:

Difficult News

2010-03-20T10:51:00.000-07:00

I haven't posted for a while now, and I wanted to let people know why. My younger sister passed away suddenly about five weeks ago. She would have been 30 next Friday. I'm back at work now, but it is difficult just to get through the days. I love my job, but high stress work is not the best thing when going through something like this, and it's been hard just to get the minimum done. This has also caused me to rethink my priorities and how I lead my life, where I spend my time and energy. I do plan on continuing this blog, and I will start posting ideas and lesson materials again, but I don't know with what regularity right now. Thank you all for your support and understanding.

Algebra 1: Systems of Equations

2010-02-10T20:45:00.000-08:00

We are finally getting to move beyond basic graphing and finding equations of lines. It was a long slog, but the skills tests show that the majority of my students are starting to get the hang of it. I always look forward to the systems of equations unit, because it is a chance for students to synthesize what they have been learning all year - and, in a situated context, no less. My plan this year is to deepen the emphasis on representational fluency and summarizing, to help build all of those neural bridges we want the students to have. We started the unit Monday, and I was really blown away by my classes today - all of a sudden, I have students doing algebra! I had them solving systems in pairs, using mini-whiteboards, where one does the graphical solution and the other does the algebraic solution, and then they compare their answers. They did a great job, and it wasn't until this activity that many students realized the answers should be the same. I got a couple of those hilarious, indignant "you should have told us!" comments. Next week is winter break, which doesn't come a moment too soon; however, I'm worried about how much will be lost over the seven days that nobody is asking them about starting points or rates of change. No matter, it's worth it to have a rest. Here are a couple of examples of what we're doing, and the links to the lesson materials thus far.

Lesson 1 (Intro to Systems of Equations) doc / GeoGebra files / Keynote / Powerpoint
Lesson 2 (Solving y = mx + b Systems) doc / Keynote / Powerpoint
Lesson 3 (Practice Solving Systems) doc

Language and Retention of Math Concepts

2010-02-07T19:46:00.000-08:00

I've been thinking lately that one of the reasons my students have such difficulty with long-term retention of mathematical concepts is due to the small number of times I ask them to thoroughly summarize what they have learned. They do lots of problems, but the language of the problems often does not enter into their brains. As we learned in Orwell's 1984, without language, there is no thought. So I am going to start providing more explicit opportunities for the students to summarize and discuss what we are doing in class.

Comic Strips (Unit 5, Lesson 9: doc / keynote / powerpoint)
Quite a few students are still struggling with graphing lines. They know the general process, but don't pay attention to the details - is the slope positive or negative; if a term is missing, is it the slope or the y-intercept, and how does that change the graph? So, I had all students draw comic strips to summarize the process in these different cases. I like how this went, but I definitely did not provide them with enough time to do all I asked. Here are a few good examples. The first didn't scan that well, but he did an awesome job.

Think-Pair-Share (Unit 5, Lesson 11: doc / keynote / powerpoint)
This is a tool that our humanities classes tend to use a lot. I got some advice from them, and will be trying these periodically during the next couple of units. We did one so far, and it went reasonably well for a first try. Students need a lot of practice both writing down their ideas and sharing them out. Here is the handout I gave (it was used immediately after doing a Do Now problem of the type described).

Both Flattering and Disturbing

2010-02-02T19:26:00.000-08:00

Students in our Numeracy support class have been working on plotting points to help them with graphing in Algebra 1. The students just finished a connect-the-dot cartoon graph assignment. One of my students apparently decided to dedicate her drawing to me. Like my Speedos?

Algebra 1: Representations of Linear Equations

2010-01-10T18:19:00.000-08:00

Increasing my students' representational fluency has been something I've been working on for a while. Our second semester started last Monday, and to start my Algebra 1 students off easy, I had them do a four-fold poster of a linear relationship to review what we did last semester: situation, equation, table, and graph. They did the work fine overall, but quite a few students had more troubling questions than I had expected (i.e. "how do you make a table?"). I guess it just shows that we have to keep going through these different representations and their connections again and again.

We have started the new unit - working with linear equations - in which students have to write the equation of a line given its slope and a point, or two points, or a point and a parallel line. In the past, I have done this only algebraically (except for the initial explanation of concepts); this time around, the students will have to practice the problems both algebraically and graphically. And, more importantly, the skills tests will require them to show mastery with both methods. Let's build those connections!

Here are the first few lessons in the unit.

Lesson 1 (Representations of Linear Functions)
Lesson 2 (Graphing Practice)
Lesson 3 (Write the Equation of a Line)
Lesson 3: Keynote / Powerpoint

And some snippets from the worksheets to illustrate what I am talking about:

Introducing Linear Inequalities

2010-01-07T20:08:00.000-08:00

To show that a line is a representation of an infinite number of points, I like to give my algebra 1 classes an equation, like y = 2x + 3, and then give each student a couple of different ordered pairs - some that are solutions and some that aren't. I have them each work out their points, and then go to the board to plot an open or closed circle, depending. Once all the students sit back down, we look for patterns and see that all the closed circles fell on a straight line. Discuss, and voila.

This extends nicely to linear inequalities (and systems of equations and inequalities). On Tuesday, my algebra 2 students were reviewing linear inequalities so I did this activity with them. I really like it, because it is engaging, and it helps build a mental picture that they can rely on later on when they are struggling through graphing problems on their own. My students often get stuck on the "pick a test point" part of the process; but now, I ask them if they would have plotted a closed or open circle based on their result, and to think about what the picture on the board looked like. This usually helps them see which side of the boundary line to shade, and to be able to explain why.

Here is what the board looks like after students plotted their points:

Then, we looked for patterns. Usually, a student will come to the board and draw some sort of line after getting frustrated with trying to explain it in words. Then I reveal the shading:

And there are usually some audible "ahhs" and such. Another great benefit of this is that the string of open circles on the boundary helps students see what the dotted line is all about, and why changing the inequality to include an equals sign would create a solid line - a string of closed circles.

Here is my lesson that goes with this. And the keynote.

Algebra 1: Graphing Lines Practice

2010-01-07T19:51:00.000-08:00

I just used this worksheet from Mr. K for the first time the other day. I thought it had a pretty cool setup, but I didn't realize just how effective it would be until I used it in my first class. The "solve the joke" aspect of it helps draw them in, but the hidden beauty is in its self-checking properties. Since each line must pass through exactly one number and one letter, a line that doesn't do this must be graphed incorrectly. Students started realizing this and would go back and find mistakes without having to check with an answer key. The only bad part (sorry to say) is that they had absolutely no idea what the answer was supposed to mean (see earlier post).

I made up a "balloon pop" homework to go with this that was inspired by Green Globs. I wish I had the tech access for my students play that game.

Algebra 1: Skills List - Spring Semester

2010-01-04T22:22:00.000-08:00

I spent a good deal of time right before break trying to figure out exactly how far I can push my students for the second semester of Algebra 1. These skill items will be broken down into chunks for the skills tests, and MC-ized for the benchmarks and final exam. I regret how many concepts I had to leave out due to time pressures; and still, the list seems daunting and endless.

If you're interested, this is what my students will be doing over the coming months.

doc / pdf

ELL Joke Worksheets?

2009-12-17T16:45:00.000-08:00

Kate wrote a great post about the value of a well-structured worksheet last month.

I agree that there are huge benefits of having a unified task, with some type of self-checking or affirmation. And a little fun and/or creativity doesn't hurt. Joke worksheets do that pretty well. However, my students (who are generally not native English speakers) hardly ever get the joke. They tackle the sheet with excitement, but there is usually that little moment of disappointment at the end when they don't get the punchline. Instead, of course, of the expected groan and eye-roll that accompanies a quality pun.

"What do you get when you mix prune juice with holy water?"
"A religious movement"

After two minutes of explanation, that loses some of its original zing.

So my question is if anyone has or knows about these kinds of worksheets developed for ELL students? I'm kind of doubting that there are any, but it never hurts to ask. I think I will probably end up creating some next semester, with jokes solicited from my students. Then I can publish the DCP Spanglish Algebra Joke Book.

Algebra 1: Situation Graphing

2009-12-15T19:58:00.000-08:00

I learned a few years back that jumping right into graphing slope-intercept equations never worked. This is one of those concepts that, before I became a math teacher, I never would have guessed would be so hard for students to master. Start at the y-intercept, use the rate of change to plot the next point, and you're done - right? Yeah, not really. So after a couple of years of teaching, reteaching, re-reteaching, and tearing my hair out, I decided to try some other things. Eventually, I realized that a ton of scaffolding of the concept of slope was needed, along with firmly rooting linear functions in situated contexts.

One of the constant problem areas is deciding which way to draw the line for a negative slope. To graph something like y = -(2/3)x + 5, students would often move down 2 and left 3. My old attempts at correcting this focused only on the mathematical explanation: -(2/3) = -2/3 = 2/-3. So, you either go down 2 and right 3, or up 2 and left 3. If you go down 2 and left 3, that means -2/-3, which is 2/3. This is a perfectly reasonable way to explain it, but it didn't really provide much of a lifeline to my lower-skilled students, as it hinges on mastery of the division rules of signs, as well as remembering that a fraction also represents a division problem.

The other common problem was for students to correctly identify the starting point number, but to plot it on the x-axis instead of the y-axis.

The way I run the unit now is to provide numerous opportunities to graph and describe situations, both with and without numbers, in just the first quadrant of the coordinate plane. Distance, income, height, and so on. The quantity being measured is always on the vertical axis, and the horizontal axis always represents time. When we eventually generalize to y = mx + b equations on the full coordinate plane, I use the situated contexts as memory anchors. If a student doesn't remember where to start, I say something like, "Where do we show that the Hare got a two foot head start? On the feet axis or on the seconds axis?" In these situations, a positive rate of change always means "moving up" and a negative rate of change always means "moving down", while time is always passing to the right. This is a much more helpful way for my students to think about how to graph their decontextualized lines. Suddenly, there is a reason for the direction the line is moving in, instead of just a sign rule.

Another benefit to this approach is that my students are now a lot more flexible with the form of the equations. My situated equations typically are in the form y = b + mx, which seems like a more natural connection to the preferred method for graphing. Once they grasp that the number without the variable is always the starting point, then they can handle both y = b + mx and y = mx + b relatively interchangeably. Also, it really helps them to understand the difference between equations like y = 2 and y = 2x. The first shows a starting point of 2, with zero rate of change. What does it look like on a graph if someone is not moving, but time is still passing? Exactly - a straight line! (I'm still working on that one - even my highest skilled students still say straight when they mean horizontal. My "all lines are straight" response doesn't usually clarify the way I'd like it to.) And in the second, the rate of change is 2. Ahh, it's like a graph of someone running 2 feet per second... but where did he start from? Zero? Where is that?

This approach takes a ton more time, of course, but I can't see any way around it for my students. I hope that I am providing them with a long-lasting ability to understand and graph linear functions. The semester is ending this week (final exams start tomorrow!), so the test will be to see how much they recall in January, when we move into the next unit. We'll be doing those oh-so-fun problems where you give them a point and a slope (or a parallel line and a point, or two points) and they have to give you the equation of the line. I'm going to experiment with doing every problem both graphically and algebraically (both in instruction and on assessments) to see if the focus on multiple representations helps them master these problems. I'll post more about that in late January (after I develop and teach it!).

My Slope and Graphing Linear Functions unit (Unit 4) is pretty much up-to-date in my box.com widget in the side bar. Here are a couple of examples (in pdf format) of the kinds of activities that they were doing. The Word and Keynote files are all in my box. I'd love to get feedback on any of this.

Lesson 10: Practice graphing with tables
Lesson 11: Learning to graph without tables

Review game: Trashketball

2009-12-14T18:19:00.001-08:00

I know that many teachers out there play some form of Trashketball, so this isn't really groundbreaking. However, I always have problems with these kinds of review games. Structuring them so that the higher-skilled students don't dominate or pressure the other students can be quite difficult. Or, looking at it the other way, there are plenty of lower-skilled students who are happy to sit back and let others on their team get the work done for them.

I developed Tic Tac Toe Battle Royale a couple years ago which addresses some of these concerns pretty well. But you can only do the same game so many times. My experiments with Trashketball in the past haven't been that successful, and so I thought about how I could improve it to work more effectively in my class. This is what I came up with:

Break students into groups of 3 or 4 - for me, this yields no more than 6 groups in my Algebra 1 classes. Give each group a letter, and each person in the group a number. Write these in a grid on the board. (If there is an unfilled spot in a group, that spot becomes a wild card - any person can take that number.)
For each round, create 6 separate problems that all target the same concept, but that are slightly different. This prevents the copying problem found in board races.
Hand out a template for doing the work on. My freshmen need an organizer for everything. "Get out a sheet of paper" just doesn't fly.
Show the 6 versions of the problem, giving the class enough time to get it done.
Call for silence. Block the projector. Randomly (or not) call a number between 1 and 4. The student in each group with that number comes to the board - all 6 at once. Have the board sectioned off so they know where to write. They are allowed to bring their own graphic organizer up with them, but no one on the team may offer help at this point. The idea here, of course, is that students must make sure that all group members have done the work. Students who tend to slack off have to at least write down the work that others in their group are doing. Not ideal, but it's better than spacing out.
Have the trashketball basket set up. As students complete their work on the board, tell them if they are right or not (make sure to have answer keys ready!). Right answers get a point, and they get to take a shot for a bonus point. There is less waiting around time this way - some students will still be writing their problems as others are already lining up to shoot.
Record the scores and move on. Winning team gets a whatever.

How it looks:

I did this for the first time today, and was amazed by how well they did. There were only 2 students in the class that I couldn't get totally engaged. The rest did all their work, were excited to take their shots, and so on. It takes longer to make this activity due to the multiple problems, but it was really worth it. Man, do they love tossing paper balls into the recycle bin.

I know it kind of breaks my respect class norm, but it really warms my heart to hear a kid (who I can usually barely get to sit down, and who really wanted to win) say to his teammate who hadn't done his work on the board carefully: "Fool! I told you it was negative eleven!"

Trashketball Problems (Keynote) (Powerpoint)
Answer Template (Word)

Math Department Photo: 2009-2010

2009-11-23T20:21:00.000-08:00

Michelle Longosz, our former photo teacher, always takes amazing department photos for us each year. Here is the radical math department.

XKCD based lesson: The Coordinate Plane

2009-11-15T15:12:00.001-08:00

Ever since I first saw this xkcd cartoon, I wanted to use it in a lesson. I finally put that together this year. I used the cartoon as a way to help convey the idea that points on a coordinate plane are a way to easily visualize the relationship between two different variables. The purpose of the numbers is simply to quantify those relationships, if such a quantification is necessary. I then had students make their own graphs for homework, with variables of their choice. If I had more time to spare, it would have been nice to do this in class (and the outcome would have been better, I think).

This lesson (Unit 4, Lesson 5) and others can be found in my box.com widget to the left. I recently updated Algebra 1, Units 3 and 4.

Here are some examples of the students' work.

Story of 1

2009-10-31T12:06:00.000-07:00

I was clued in to the existence of The Story of 1 a couple of weeks ago from my twitter PLN. I had my sub show it to my algebra 1 classes when I was out of town, and it seemed to go well. Then, one of my colleagues was sick this week and did the same lesson. Her sub said that the students were really engaged with the movie. I couldn't find a question guide on-line for it (though I didn't search all that long), so I made one up.

Algebra 1: Solving Equations Puzzle

2009-10-27T22:35:00.000-07:00

Here is a puzzle activity for reviewing equation solving. I found that it worked better when I made an answer mat for students to put their pieces onto (I indicated a couple of pieces on the mat to help them align the rest of their pieces).

Here are two files in Pages and Word that you can work from to make your own.

Edit:
A comment from David Wees in a previous post with a similar puzzle I did for quadratics:

Yeah your puzzle is cool. So cool that I've created a random generator in Adobe Flex.

See my algebra puzzle generator.

Awesome!

Edit 2:
There is an app called Formulator Tarsia that will do this, but it only works for Windows (which I don't have access to) so I haven't tried it out. Give it a try!

Putting students in control of their learning

2009-10-27T21:36:00.000-07:00

In the last couple of years, I've worked to really clarify exactly what skills I expect my students to learn. The assessment system makes it crystal clear what skills students know and don't know. And then I realized: Oh wait - it's only crystal clear to me. Students focus on their test scores, and come in to retake and improve tests, but they really don't think about what mathematical content they need to develop - only what test number they need to retake. I still have a few students who insist on retaking skills tests even though they haven't done any work to learn the skills that they got wrong the first time. Even when this fails to produce the results they want, they still resist actually working with me to learn the skill.

I think that helping students really understand what the individual skills consist of, and what their personal ability level is on each skill, is really the next step. I want students to understand the connection between their level of numeracy and their success in mastering algebraic concepts. I also want students to make connections between their behaviors in class and their growth (or lack of growth) in the lesson's objectives. Finally, I want to provide students with greater differentiation so that all students can both feel challenged and successful.

So, I put all of that together into a new plan for beginning and ending class. Students will start class with a 10 minute Do Now that has three parts. Part 1 is a Numeracy Skill Builder that targets a specific elementary math concept that is either key to the specific lesson, or something that students have been struggling with. Part 2 consists of one or two algebra concepts that are the lesson objectives. These are broken into basic, proficient, and advanced levels. The proficient level is the form in which the concept will be tested on a skills test. Students are told to solve only one problem in each concept, at the level they feel most comfortable at. Part 3 is a multiple choice test prep question. The purpose of this is obvious, as we need to get students ready for state tests, ACTs, placement tests, and so on.

Students have 10 minutes to complete these problems individually and silently. No helping is permitted here (in general), because the purpose is for students to really get a sense of what they know at the beginning of class on their own. At the end of the 10 minutes, I show the answers so students can see how they did, but we don't spend time actually reviewing these specific problems. I quickly collect the papers.

We have the lesson. Ok.

Now, in the last 5 - 7 minutes, I hand back the papers. On the back, students complete the Exit Slip / Reflection. They are supposed to go back to the Do Now problems, pick one algebra concept, and try a higher level problem. The idea is for them to see how much they can improve in an objective over the course of the class period. So, even if they are only able to accomplish the basic level (when they couldn't before), they can see growth in themselves and feel good about that. Students who already could do the advanced concepts at the beginning of the class have a shot at doing a harder challenge problem, so that they too can push their thinking (my advanced students really like this).

I just started doing this today, so I don't have too much to report about it yet. It seems to have gone well, though it took longer than the 10 minutes because I needed to explain the process a few times until they all got what I was talking about. As it becomes part of the routine, I'll know more about what impact it is really having.

Here is the first one we did, in pdf and word formats.

I'd love to get any feedback on any part of this.

Edit
We decided to make the reflection portion into a progress tracker, instead of copying it individually on the back of each Do Now. This log will be kept in a binder in the class. This will allow students to see how they did in previous classes as they are filling out the current reflection. It will also be a very useful document for discussions during grade conferences.

Distributive Property and Multiplying Binomials

2009-09-21T23:20:00.000-07:00

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.

Algebra 1: Solving Equations

2009-09-12T15:56:00.000-07:00

I am beginning the planning stages of our unit on solving equations in Algebra 1. In my past experiences, some students pick this up very quickly, no matter how you teach it, while other students struggle mightily. I want to try some alternate approaches this year, to really reach those students who have not been able to learn this skill in the past. I remembered an order of operations approach that I read about in the NCTM magazine a few years back. I can't recall the name of the article, but a little google searching found me this document that is even better than what I remembered.

Our students in Numeracy already work with bar modeling to solve word problems, so this seems like a natural extension to solving equations. I like this approach because it helps focus on the idea that the variable is a given quantity that must be determined, instead of focusing on the steps that isolate the variable. It also might help with those difficult to master "converting verbal sentences to algebraic equations" problems. Here are a few examples of how this might look. I know the diagrams are a bit confusing at first, but I think they would make more sense to students as they watch them get created and do them by themselves.

I also like the other representation discussed in the article. This is the original order of operations process that I had been searching for. I like this because it gives a very clear framework for solving equations - reversing the order of operations.

When you look at each stage, you can draw equal signs between the boxes. These would be equivalent to the intermediate statements in the traditional "do the same thing to both sides" approach.

So for the unit, I am thinking that we would spend two or three lessons on bar models to build the concept of what we are actually trying to do (find the value of the unknown amount). Then, spend a couple lessons on the order of operations representation to build an understanding of the process for isolating the variable. Finally, transition to the traditional approach, which is clearly the fastest and cleanest way to solve an equation of the three. This would take more time, of course, but the hope is that it would build a more enduring understanding.

Has anyone tried these methods with their students?

Edit:
I wonder now if it would make more sense to start in with modeling sentence/word problems with the bar model method, and not start by saying that we are "solving equations". That way, more students would be engaged with the material, and we could eventually use the bar models to develop the equations.

This way, the unit doesn't start with the problem "solve (3/5)x = 45", which will stop most kids dead in their tracks, but maybe with something like "It took Sandra 45 minutes to finish 3/5 of her homework. How long will it take her to finish it all?", which kids might have more of an entry to. After we solve it, we can then discuss how to represent it as an equation.

Edit 2:
I also need to think about how to incorporate the balance idea and preserving equality... Kids don't always know what the equal sign really means. Maybe in the transition time from the box method to the traditional method?

Edit 3:
(Written on 10/27 - at the end of the unit)
On reflection, the problem was not having enough time to really devote to the two alternative methods. Both did show a lot of promise, but we weren't able to really practice either enough for it to really stick with students. The bar model method really worked to help students set up and solve word problems, so I think I will stick with that next year. Give it some more time so that it really sinks in and can be used to get a deeper understanding of fractional coefficients. I will probably save the GERMDAS method for individual tutoring with students who are not having success with the traditional balance method. Less fights... more differentiation.

All of these lessons have been added to the box widget on the left.

Algebra 1: Skills List

2009-09-07T12:26:00.000-07:00

My goal for this weekend was to complete a rough draft of all the skill items that will be assessed on the first semester final exam. These items are assessed in chunks on the weekly skills tests, and in larger chunks on the 6-week benchmark exams. After each benchmark exam, the plan is to spend a lesson or two on targeted reteaching - any ideas that people have on how to make this effective would be very much appreciated.

I've finished the list, and am interested to hear what other Algebra 1 teachers think about the scope and detail of the items. What would you add? Take away?

My current experience with Algeblocks

2009-09-06T07:18:00.001-07:00

This student finished all of the simplifying expression problems on paper (correctly), and then used the algeblocks for the following:

But can I blame him? Why would a student who can already do a procedure well bother trying to figure out a slower, less portable method?

But then, the students around him who really do need to use manipulatives to help them understand the difference between 2x and x^2 don't want to use the blocks either. They feel stupid and they want to do it the faster way too.

This seems impossible without fully differentiating instruction in the classroom. Which also seems, you know, impossible.

We have our Numeracy class, in which students spend time working on ALEKS. The Numeracy teacher this year is doing small group pull-out during that time to work on specific skill building. I am starting to think we should use the manipulatives in that setting only.

Ok, I'll tweet.

2009-09-05T20:45:00.000-07:00

I've resisted it until now, but it seems like a lot of good stuff is happening in the twitter world. No point in shutting myself out.

Algebra 2: Error Analysis

2009-09-05T12:32:00.000-07:00

My algebra 2 students needed more practice solving equations with rational exponents. I was trying to think of an interesting and yet still effective way for them to keep practicing, and then I thought about how the errors they make tend to be the same again and again. That reminded me that I hadn't done an error analysis activity in a long time - and just like that, the lesson was created. I assigned each table pair one problem, where they had to find the mistake, explain it, and do the work correctly. Then each pair was called to present their findings to the class; the class then worked out the problem and if they all agreed that they had found the correct solution, I allowed them to move on.

Students were generally good at finding the mistakes I had made. Will this activity help keep them from making the same mistakes in the future? We'll find out next week.

Here are the twelve problems plus homework.

Year ten at DCP, off to a strong start! My students this year rock.

2009-08-31T21:14:00.000-07:00

Ok, so it was more than a month. But at least I'm finally writing again. I don't think I will be able to post about every lesson like I did last year, but I am still going to put my files in the box.net account for you all to look at, use, critique, etc. This year, I am reworking our Algebra 1 curriculum, so I'll be trying some new things, and hopefully be getting lots of ideas from all you other teacher bloggers. If you are reading this, and you are not blogging your ideas or posting your work online, I highly encourage you to give it a go. You will get a lot out of it, and you'll give a lot to the community.

We just finished our first unit on evaluating expressions, including order of operations and working with square roots. Up next will be simplifying expressions, and we're going to be using Algeblocks to model combining like terms and the distributive property.

Here is a link to the Unit 1 files. Unit 2 hasn't quite been written yet...

Please send me feedback, especially criticism. I get lots of thanks from people for posting my work, but rarely does anyone tell me what they think would make things better. It's my tenth year, but I still have a long way to go until I'll be satisfied with my work. I need your help!

Back in a month

2009-06-20T13:21:00.000-07:00

I'm going out of town for a few weeks, and with any luck, I won't be thinking about school. Posting for the new school year will start up again probably at the end of July or beginning of August. I'll be teaching Algebra 1 and Algebra 2, and will be redeveloping lots of Algebra 1 materials and posting them like I've done with Algebra 2 this past year. So come back and have a look around - I'll be looking forward to your critiques and comments.

Numeracy Curriculum

2009-05-14T20:30:00.000-07:00

I finally got around to posting my old Numeracy curriculum on my box. This is by no means a final draft of what I think our 9th grade basic math support class should look like, but it is where I left it last year. I didn't decide to start using Keynote presentations until the third unit, so that's why there aren't any Keynote files in the first couple of units. This year, I am not teaching this class, so I haven't had the opportunity to keep developing these lessons. I'm sure I will get around to reworking this stuff at some point. For now, I'll throw it out there for people to look at, borrow, critique, steal, and so forth. I hope someone finds it helpful.

Algebra 2: Quadratics

2009-05-11T21:03:00.000-07:00

We are nearing the end of the quadratics unit, which got chopped up by the STAR test and all the associated hoopla. I don't have too much to say about it right now, except that I think it needs a good deal of revision for next year. Feel free to peruse and comment. Lesson 5 is missing because that was just a midterm review day.

Instead of linking all of the files individually, here is the folder:

I've also updated the skills tests from units 5 - 7 in the Box.

STAR Review

2009-04-13T21:43:00.000-07:00

When we get back to school next week, we have a week of classes, and then we have the STAR tests. Instead of trying to jam in a few more concepts, we're just going to review what we've already learned, in game form. I've got "Who wants to be a millionaire?", "Tic Tac Toe Battle Royale", and "Big-L Bingo" ready to go, as well as a triage lesson (when you look at a problem, should you Guess it? Try it? Kill it?) I hope that this will yield an overall positive result. And when the test is complete, my students won't have to see another multiple choice question for the rest of the year.

Quick update

2009-04-05T10:35:00.001-07:00

I haven't had much energy lately to post, but I've still been updating my box.net account. We ended unit 5 with polynomial division and we are starting unit 6 (quadratics) with completing the square. We are having our midterm before break this Friday. Over break, I'll post in more detail about some of the recent lessons. For now, feel free to download stuff from the box widget on my blog. As always, feedback on what you find there is much appreciated.

Algebra 1: Introduction to Inequalities

2009-03-14T16:26:00.001-07:00

I'm not planning our Algebra 1 classes this year, so I have not been producing much for it. But I did put together a scaffolded introduction to inequalities. The objectives are for students to:

Compare numbers using a number line (i.e. "<" means "to the left of")
Understand the difference between open and closed circles
Graph the solutions of a statement like "x < 3"
Understand graphically why adding/subtracting by any number or multiplying/dividing by a positive number does not change the relative position of two numbers, while multiplying/dividing by a negative number does. In other words, students should understand when and why to "flip the inequality sign" when solving inequalities.
Solve and graph linear inequalities

Here is the file.

Algebra 2: Reducing Polynomial Fractions

2009-03-13T23:06:00.000-07:00

I started this lesson with some theatrics. I asked them to simplify the fraction shown in the picture, and of course they all wanted to cancel the terms (as expected). I let them do it, and then changed the pretty pink heart into the fiery eruption you see here. I told them that those red slashes are like daggers through a math teacher's heart. I also told them that, when they go to college, I never ever want them to make the mistake of canceling out terms. Cancel factors, not terms! We spent a lot of time talking about the difference between factors and terms, and why this rule is true. We talked about why you can't add 5 and 5x, but you can cancel the 5's in 5/5x. I think this was time well spent, because this canceling problem is a persistent weed. From there, we practiced factoring and canceling. Pretty straightforward. In the following lesson, we multiplied and reduced products of polynomial fractions. There really were no new skills to learn, so after modeling one problem, I had them do independent practice work.

And now, I am caught up on postings!

Lesson 11 (Reducing Polynomial Fractions) doc / keynote / quicktime

Lesson 12 (Multiplying Polynomial Fractions) doc

Algebra 2: Factoring Difference of Squares

2009-03-13T22:51:00.000-07:00

Continuing with the lessons, we learned to factor difference of squares expressions. I used a geometric approach to help make sense out of the pattern, and it has really helped some students figure out how to more easily factor the nasty ones like 25x^2 - 16y^4. A quick sketch of the squares, labeled with their side lengths, has proven quite useful.

Lesson 9 (Difference of Squares) doc / keynote / quicktime

Lesson 10 (Review and Practice) doc / keynote / quicktime

Algebra 2: Factoring, and More Factoring

2009-03-13T22:26:00.000-07:00

It's been a while since I posted. The last week of February was our Junior Trip, in which we take all of our junior class on a 4-day-long trip around California to visit various CSU campuses. It's an incredibly important part of our program, because it is the time when our juniors really start to imagine themselves as college students. The tours, the student panels, seeing the dorms and classrooms, the admissions directors, and the DCP alumni all bring things into sharper focus for the 11th graders. We moved the trip earlier this year (it used to be in April) because kids come back inspired and ready to make positive changes, and so we wanted them to have more time to improve their grades before the end of the semester. It's also a great time for students and staff to bond and get to know each other in different ways. Needless to say, a 4-day, 3-night field trip with 80 high schoolers is tiring. We're all pretty much recovered now, and it's been back to business as usual. Time to catch up on some lesson postings.

In Algebra 2, we're nearing the end of the polynomials and factoring unit. I've been focusing on basic factoring techniques (look for the GCF first, then either use trinomial factoring or difference of squares, if possible). I'm still deciding whether to throw sum/difference of cubes into the mix this time around. I decided to bring simplifying and multiplying rational expressions into this unit (instead of waiting for the rationals unit) because it seemed like a good way to have them get more practice with factoring without repeating the same exact problems again and again. Plus, these questions are prominently featured on the STAR test.

One thing that has been helping students deal with factoring out the GCF is teaching them to write the prime factorization of each term in the polynomial, every time (including a -1 factor when there is a minus sign). Though it takes longer, this is pretty much a foolproof way of factoring out the GCF - many students have a lot of difficulty with the "what's the largest expression that divides into both" method.

Lesson 6 (Factoring the GCF and Trinomials) doc / keynote / quicktime

Lesson 7 (we used Algeblocks to get a better understanding of factoring trinomials) doc

Lesson 8 (Factoring Trinomials by Grouping) doc / keynote / quicktime

Algebra 2: Factoring Trinomials (Part 1)

2009-02-10T21:53:00.000-08:00

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.

Algebra 2: Polynomials and Factoring

2009-02-07T09:29:00.000-08:00

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

Do skills tests work?

2009-01-17T11:51:00.000-08:00

As I've written about before, a large part of the students' grades this year in Algebra 2 are based off of the skills tests. The method I'm using is based of off Dan's, but I've modified it quite a bit. I'll save reflecting on the details of the method, and what should be kept/changed for the end of the year. I'm still getting a feel for the process, and what I've been doing has worked well enough that I don't want to significantly alter it until next year.

The crux of the method is that students are primarily assessed on smaller bits of information, more frequently. They are also encouraged to try and try again at the same concepts until they master them. Since students learn at different rates, and have different things going on in their lives that may prevent them from learning at a certain point in time, they can relearn and retake the skills tests whenever they want, before the end of the semester.

Instead of assessing each skill individually, I've been grouping them into clusters of 4 or 5 related skills. If a student gets, for example, the first 3 out of 5 correct, the score is 3/5. If they retake it, and get the last 4 right (but miss the first this time), I'll raise the score to 4/5, not 5/5 - even though the first one was "mastered" the first time around. This promotes lots of retaking, which is what I want, since my students really need to practice and practice in order to retain concepts.

It took students a while to understand how this system works, but as they figure it out, they love it, because it gives them a chance to really improve their grade when they fall behind. I've had a handful of students bring their grades up from Fs to Cs or Bs just in the last two to three weeks before finals, where this never would have been possible before.

My big fear, of course, is that this style of "micro-testing" would lead to artificially high grades, and that students' retention of material would not pan out. I've been eagerly anticipating the results of the final exam to get some relevant data. The final consisted of 50 questions that were compiled from the skills tests, though of course with different values. First off, here is the distribution of grades on the final exam:

Though this may not look like something to cheer about, for a DCP final exam, this is actually quite good. The average score was a 70 and the median was a 72. But, I was more interested in thinking about the relationship between students' skills test percent and the final exam percent. If the system works as it is meant to, the skills test score should strongly predict the final exam score. The next graph shows a scatterplot of this relationship.

The purple dotted line shows what a y = x relationship would look like, and clearly (as I expected) there are more dots below the line than above - indicating students who performed better on the skills tests than on the final. But how much of a difference was there? I added in the best-fit line, and though it deviates from the purple line, it actually strikes me as not that bad. It's clear that all but a handful of students who failed the skills tests (i.e. didn't do well the first time, and didn't bother retaking them) also failed the final exam. While these students concern me greatly in terms of the task we have in motivating and educating our target students, they actually support the idea that the skills test scores are predictive of the final exam score.

The section of most concern to me is that in the red box. These are the students who had a passing score on the skills tests, but failed the final exam. Are there enough students in that section to show that the system doesn't work? I'm not really sure. Of the students who passed the skills tests, many more of them passed the final exam than did not, and I find this encouraging. And, the 24 dots in the red box all did better than 50% on the final, which means they didn't have catastrophic failure (which is not that uncommon on our final exams). But, they didn't show what we typically consider "adequate" retention, since they didn't get at least 70% of the questions right.

I'm posting this because I would like feedback and impressions from other teachers. What does the data say to you? And for those of you using a concept quiz/skills test method, what kinds of results are you seeing?

Sold out or bought in?

2009-01-07T20:28:00.000-08:00

We're back from break, and it's time to gear up for finals. Since DCP is a California public school, my course is standards-based. I use the standards as a guideline for what to teach, but of course I must pick and choose, modify, add, and subtract in order to meet my students' needs and get them ready for higher level classes. Though it's not fun for anyone, the STAR test must be faced head-on, and I want my students to show that they really are learning math (even if it is hard to see on a day-to-day basis). To that end, I am giving a fully multiple-choice final exam. I copied the language and even the formatting of the STAR test. I feel (somewhat) justified in doing this, since none of the quizzes or cumulative exams have had any multiple choice on them. And, if they don't practice the all-or-nothing multiple choice format, they will do much worse on the STAR test (and the ACT, and the ELM, and the CAHSEE, etc.).

Most DCP students simply don't study. We do our best to teach them, but it takes a long time for students to first believe that studying helps, and then to learn how to do it effectively. On our first day back, I gave the students a practice final exam without any warning. They were not thrilled with it, but they accepted it and actually put in real effort. My purpose was to show them what their score will likely be on the final if they don't study at all. It was time well spent, because before giving them back today, I asked students to write down what percent they think they got on the test. Almost every student guessed way higher than their actual scores, and many were quite shocked. Hopefully, this will help students make wiser decisions regarding studying between now and finals (which start next Wednesday).

Here is the practice final, if you are interested.

Algebra 2: Lines and Systems

2008-12-09T20:50:00.000-08:00

I'm trying to cram in a unit on systems of equations and inequalities before break. It's hard, since so many students are still not totally comfortable with graphing linear functions. But we're making progress. We're up to solving 3x3 systems with linear combination, and most of them have got the idea. These problems are huge, and are probably the longest routine problems my students have ever done. This is cool, because it makes them feel smart and accomplished when they get one right. Unfortunately, a single arithmetic or copying error (which happen all the time) can crumble the whole thing, and then the frustration is back again, eating away at their self-confidence. I'd like to get to graphing systems of linear inequalities before break. When we get back, we need to review for the final, but I'd really like to do some work with linear programming problems first. Here are the files from the last few lessons.

Lesson 1 (Linear Functions) / Keynote / Quicktime
Lesson 2 (2x2 Linear Combination) / Keynote / Quicktime
Lesson 3 (2x2 Word Problems) / Keynote / Quicktime
Lesson 4 (3x3 Systems)
Lesson 5 (Systems Practice)

Algebra 2: Parent Functions

2008-12-02T22:23:00.000-08:00

We ended the functions unit before Thanksgiving. I'm not giving a comprehensive test until just before winter break, and I think that is good, so they can have more time for it to sink in. The new unit is on systems of equations and inequalities, but I'll post about that later on, when I have more time. Here are the last files of the unit.

Lesson 15 (Practice and Skills Test)
Lesson 16 (Translating Parent Functions)
Lesson 16 Keynote
Keynote Quicktime

Algebra 2: Horizontal Shift and Review/STAR Problems

2008-11-20T21:11:00.000-08:00

Translation and transformation have continued to prove extremely difficult for my classes. Even my strongest students have been struggling. I'm still trying to work out what is making it so hard to understand (if anyone has insight on this, I'd really love to hear it). I think they are starting to get the hand of it, but for mastery, we'd need at least another full week, and that is time we just don't have - especially for something that is only tangentially in the standards.

I did incorporate the idea of texting in lesson 14, to introduce what I'm calling "translation notation". We're not talking about vectors or anything like that, but I wanted to give them an efficient way to describe the translations and calculate with them. The kids thought it was really funny; I did play it up, calling it "math chisme" (gossip) and pretending I was texting it under my sweatshirt to my friend. You wouldn't want to type out that whole sentence, right?

Anyway, here are the files from this week.

Lesson 12 (Horizontal Shift) Keynote Quicktime
Lesson 13 (Translation and Transformation Practice) No Keynote for this one
Lesson 14 (More Translation and Transformation) Keynote Quicktime

Algebra 2: Vertical Translation and Transformation

2008-11-15T09:36:00.000-08:00

I've been really good about my timing all year, until this lesson... I wasn't able to finish it in any of my classes. We almost got to the end of the Keynote, and didn't have any time for independent practice. But that's why I don't really create more than one lesson at a time - so I can adapt as needed. Well, that and it takes a huge amount of time, and keeping afloat is what it's all about. I'm still not sure why this lesson took so long; some students were tearing through the class notes, figuring it out on their own and finishing before we even go there. And some students were struggling to keep up. I know it's always kind of like that, but we are working with a very visual representation right now, and it has shifted some of the dynamics of the classes.

Coming soon will be horizontal shift, but not horizontal stretch. I don't want to overload them, and the standards in Algebra 2 really only require that students be able to graph things like f(x) = a(x - h)^2 + k, or to say how one vertex form parabola got shifted to another one. They can learn horizontal stretch in pre-calculus with the trig functions. At least this will give them a good foundation for the tedious work of grinding through f(x) = -2sin(3x-pi/2)+5.

Lesson 11 (Vertical Shift / Stretch)

Lesson 11 Keynote
Keynote Quicktime

Algebra 2: Graphical Analysis Practice

2008-11-09T10:58:00.000-08:00

We have Veterans' Day off on Tuesday, but we still have school on Monday. Would have been nice to get a four-day weekend. How many absences do you think we might have tomorrow? I decided to do a review lesson, both because my students are really struggling with graphical analysis stuff, and because I don't want to move ahead with potentially many students gone. Hopefully that doesn't happen. But there are at least 4 teachers who are taking a personal day, so...

I found a site with some good resources on understanding domain and range graphically, and have included some of those animations in this lesson's Keynote.

Lesson 10 (Graphical Analysis Practice)
Lesson 10 Keynote
Keynote Quicktime

Algebra 2: Graphical Analysis

2008-11-06T22:34:00.000-08:00

Here are the latest files... more work with domain and range (which continues to stump some students) in interval notation form, and my favorite, solving equations and inequalities graphically. These are very challenging concepts for students, even though they don't seem like they would be, compared to some of the other material. But Keynote really shines through for clearly showing how this works.

Lesson 8 (Domain and Range)
Lesson 8 Keynote
Keynote Quicktime

Lesson 9 (Analyzing Graphs)
Lesson 9 Keynote
Keynote Quicktime

Brilliant!

2008-11-06T22:31:00.000-08:00

Students were asking why we have to learn interval notation. I was going on about ease of communication and writing things more simply, but I wasn't getting anywhere until one student piped in with this gem: "Oh, it's just like texting". As soon as she said that, the rest of the class produced a collective "ohh...". Why didn't I think of that? I used it in the following class, and it worked well.

Algebra 2: Interval Notation

2008-11-03T21:53:00.000-08:00

I went back and forth on whether or not to spend time on this, and in the end I decided to go with it. It will be helpful to students who go on to pre-calc and beyond, and it gives us a good opportunity to review solving linear inequalities and to keep working on finding domain and range of graphs. Plus, it's good to have a lesson every once and a while that is pretty easy for students to master right away - someone said today, "This is the easiest thing we've learned in like 50 years!".

Lesson 7 (Interval Notation)
Lesson 7 Keynote
Keynote Quicktime

Algebra 2: Operations on Functions

2008-11-02T08:03:00.000-08:00

I know my posts have grown dull of late. Or maybe just functional. Hopefully the files remain helpful to you all. Enjoy that extra daylight savings hour!

Lesson 5 (Operations on Functions)
Lesson 5 Keynote
Keynote Quicktime

Lesson 6 (Operations on Functions in Multiple Representations)
No Keynote for this one... it was an extended class activity that we did on halloween; I handed out a chocolate kiss to each student in the group each time the whole group finished a round. For my costume, I was La Calavera Matematica de Michoacán.

Algebra 2: Composite Functions

2008-10-25T16:19:00.000-07:00

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

2008-10-25T15:44:00.000-07:00

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

2008-10-22T20:16:00.000-07:00

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

2008-10-19T20:17:00.000-07:00

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

2008-10-19T19:57:00.000-07:00

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

2008-10-15T19:18:00.000-07:00

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

2008-10-12T20:25:00.000-07:00

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

2008-10-12T20:13:00.000-07:00

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

2008-10-07T22:01:00.000-07:00

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

2008-10-02T22:02:00.000-07:00

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

2008-10-02T20:44:00.000-07:00

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

Algebra 2: Solving Basic Exponential Equations

2008-09-27T14:33:00.000-07:00

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

2008-09-22T21:56:00.000-07:00

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

2008-09-21T10:36:00.000-07:00

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)

2008-09-16T21:11:00.000-07:00

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

2008-09-15T22:43:00.001-07:00

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

2008-09-14T18:22:00.000-07:00

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

2008-09-10T23:27:00.000-07:00

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

2008-09-08T21:05:00.000-07:00

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

2008-09-06T18:04:00.000-07:00

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

2008-09-03T23:26:00.000-07:00

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?

2008-09-01T11:06:00.000-07:00

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.

Algebra 2: nth-roots

2008-08-31T21:58:00.000-07:00

My first unit this year is on real numbers. In the first two lessons, we learned the types of real numbers and how they are related. In this lesson, students learn what nth-roots are (they are only familiar with square roots from algebra 1). They also learn how to evaluate, estimate (including plotting on a number line), and simplify nth-roots.

Speaking of number lines, they can be a pain to make. Here is a tutorial on how to make number lines and coordinate planes in MathType. Who knew?

We have a lot of resources to work with at DCP, but one thing we are lacking in is digital projectors. We have four for the whole staff, which is adequate if they are only used for occasional movies and demonstrations. However, I started using one last year for daily lessons, and I liked it so much that I kind of reserved it every single day. It worked out ok, but I felt bad hogging a quarter of the resource for myself. So I decided to get one of my own this year. I just bought it from Amazon, and it seems like a great deal. Here is the link if you are interested. If you know anything bad about this projector, please don't tell me, since it's already being shipped!

Here are the handouts and the Keynote:
Lesson 3
Keynote (zipped)
Keynote Quictime

Update:
The projector arrived, and it is fantastic. I got the box 15 minutes before class, opened it up, plugged it in, and was good to go. It is bright enough to use with the lights on, and the colors look exactly like what's on the screen! (Neither of these things are true with our media cart projectors). And now I can take it home on the weekends and watch movies on the wall, instead of peering at the laptop screen (we don't have a TV, so...)

College Habits

2008-08-29T21:52:00.000-07:00

I stole this idea from the teachers at Impact Academy in Oakland. I learned about it two days before classes started, and pretty much immediately decided to implement it.

The idea is that you establish a few "college habits" that students should be working toward, and then they self-assess at the end of each class on how well they demonstrated those habits.

Here are the habits I am using:

Punctuality: When the bell rings you are in your seat, binder and pencil/pen out, checking homework.
Materials: Every day, you need your binder, planner, pen, and pencil.
Supporting Others: Always support other students’ learning. Help others and ask for help when you need it. Never distract other students.
Focus: Focus on doing your best work. Don’t let your mind wander. Your body language should show active engagement.

For habits 1 and 2, students can earn either 1 or 0 points. For 3 and 4, they can earn 1, .5, or 0 points. I save the last 3 - 4 minutes of the class to do this: students think about how they did, in each category, write it in their logs, and total their scores.

The key - which at first seemed a little off-putting - is that they then say their score out loud, one by one, for the teacher to record. The point is that students are being held publicly accountable for their behaviors in class, and have to face up to it when they have been harmful to the learning environment. I made a couple of rules for this. First, no students are allowed to comment in any way on other students' scores; it is silent during this time except for the person reporting their score. Second, even if I don't agree with the score, the student has final say over their score, and it does factor into their grade (albeit, only a small percentage of the whole).

I've seen my Algebra 2 classes twice each so far, and this has gone surprisingly well. I wasn't sure how they would handle it, but they are doing a good job. For example, today I had to send a student out of the room for being distracting. (We have a cool new reflection process for this sort of thing, which I'll write about later.) She came back in, finished the class in a much better mood, and at the end, gave herself a 1 out of 4 for the day. And I overhead another girl asking her group if they thought she was focused today.

I hope that this structured reflection on behaviors, tied in with some public accountability and the ultimate control over a small piece of their grade will help build a strong classroom culture this year.

Algebra 2: Real Number System

2008-08-28T19:38:00.000-07:00

I've taught the first lesson of Algebra 2, and it went well. I love the energy that 9th graders have, but I do have to admit that it is pleasant not to have to spend the first few weeks breaking the kids in from scratch. My first unit is on the real number system. The skills that I plan on covering are here.

In the past, students have had great difficulty dealing with the Venn Diagram of the number system. They always got tripped up with the fact that all whole numbers are integers, but not all integers are whole numbers (for example). So I decided to make my first lesson of the year about classification with Venn Diagrams. The second lesson extends that understanding to the number system. I'm hoping that this scaffolding will make the number system much easier to understand and remember.

Here are the lessons if you want to see what I am doing. (I plan to post all of my lessons as they go, so stay tuned if you are interested. And, any and all feedback on lessons would be greatly appreciated, especially if you try out something of mine in your class).

Lesson 1 (classification)
Lesson 2 (real number system)

Island Maps complex instruction skillbuilder I used in lesson 1

DCP Alviso

2008-08-26T21:46:00.000-07:00

Today, DCP launches our second campus, located in Alviso. It is a 6th-12th grade school; right now, there are only 6th and 7th graders. Here is a link to the story shown on ABC news.

Algebra 1: Intro to square roots

2008-08-23T09:44:00.001-07:00

Students often have trouble seeing a square root as a number when the radicand is not a perfect square. The point of this activity is to help students develop this understanding by using a geometric metaphor.

square root intro.doc

Group work skill builders

2008-08-23T09:28:00.000-07:00

We are doing a major overhaul of our Algebra 1 and 2 program this year, and one of the elements we will be working on is increasing the use of complex instruction.

To be successful at this type of learning, students must be able to work effectively in groups - which is something that does not come naturally to most students. The designers of complex instruction have a set of skill builder activities that can be used to teach students how to accomplish group tasks. If you plan on doing any sort of group work, you should definitely check out the link and read through the activities.

Back to work

2008-08-07T23:57:00.000-07:00

It's been a lovely summer, but now it is time to get back to work. I will be teaching Algebra 2 this year (4 sections) and Algebra 1 (1 section). So, most of my lesson posts are going to be about Algebra 2.

I have been working with H. on creating a skills mastery assessment system. She has posted a lot about it, so I won't bother repeating it. You can catch up here.

I just found out about box.net and I will be posting my files there as I go this year. There is a handy widget on the blog now (if you are reading this by RSS), and you can even subscribe to that by RSS. Feel free to look at and use anything I post, but of course nothing may be published/sold/turned in to your ed school instructor without my permission.

I describe how my system will work in one of the files, but here it is if you don't want to bother downloading it:

Each skill test will be given at least twice – the scores are added together. The purpose of this is to promote retention of concepts. If students receive a perfect score on the second administration of the test, the first score will be raised to a perfect as well. Students are able to retake these tests as many times as they want before the end of the semester, though they can’t take the same test more than once per day. The questions on the skills tests are all single topic items that reflect typical STAR questions.

Homework will be graded. To make this possible, students will be called on to grade their own work. This will require trust and buy-in on the students’ part – but, the weight is limited to 10% to limit temptations for cheating. Each class, students will be shown the answers to the problems, and five to ten problems will be selected for grading. Each problem is either right or wrong, no partial credit. However, students may do corrections to the incorrect problems and turn them in the following class to earn their points back.

Thus, a full 60% of the points are “recoverable”, and students who put in sufficient effort should be able to earn the full amount of points. A student who earns all of the recoverable points needs only a 25% average on comprehensive tests and the final in order to earn a 70% in the class. This way, the class is passable by lower ability students who are willing to work hard, and these students will hopefully be better prepared to take the STAR test. Students who want to earn a B or an A must also do well on the comprehensive tests; this allows me to create tests that are more challenging and focus on analysis and synthesis problems.

In the Unit Skills Lists, skills that are marked CE will be tested only on a comprehensive exam. These are skills that are not required by the benchmarks/released questions, but are key to progressing to higher level math.

This system is still in its development phase, so any feedback you may have would be much appreciated.

The Numeracy Project - Live

2008-06-21T22:21:00.000-07:00

This is our Spirit Week 2007 gig. We rocked the house!
Voted best math teacher band in downtown San Jose!
Check out my singing debut in the second clip (get ready to turn down the volume...)

Another year finished...

2008-06-10T20:24:00.000-07:00

Classes are done, finals are being graded, and graduation is on Friday.

Instead of writing about how I'm feeling right now, I think this email I got from a student pretty much sums it up.

Algebra 2 Exhibition

2008-05-20T22:38:00.000-07:00

We just completed a project in Algebra 2. Each student designed a picture made up only of lines and shaded regions. They had to then determine the equations of the boundary lines and the systems of inequalities that described each region. The drawings and equations/inequalities were all displayed today; to finish the assignment, each student had to pick someone else's project, copy down their inequalities, graph it, and see if they got the same picture.

Graphing systems of inequalities is always a challenging topic for my students; this project underscored just how difficult the topic is, as most students needed repeated explanations and examples, just to figure out what they were supposed to do. Though it took longer than expected (and time budgeted), it seems to have been worth the time. Ideally, this should be followed up by a linear programming unit, but we don't have the time this year. Here are some pictures:

And, a bonus picture from a few weeks back... each year we have a spirit week in April, and part of it is class competitions. Here is our media center all decked out for the event.

A winning review activity

2008-05-14T17:58:00.000-07:00

My students always complain that we don't play enough games in class. I know they love games, but most of the games I've seen are quite ineffective. "Showdown" is one of my favorite review activities for my older (more mature, more motivated) students in Algebra 2, but it doesn't work so well for my freshmen in numeracy.

To many of my students, tic tac toe is a riveting activity to be played surreptitiously during a dull lesson, or after a test. I thought I would capitalize on that, and so I present to you:

The students are broken into pairs that collaborate against other pairs. Each group of 4 is given a game board with several tic tac toe grids on it. One pair picks X and the other gets O. Turns alternate with each problem. When the problem is shown, both teams should work on it. If it is X's turn, if they are right, they get their square. If they are wrong, and the O's are right, the O's get to steal a square. (Students took a while to get this - at first they all thought it was unfair). When a game is won, the winning team gets a point. At the end of the activity, whichever teams won more games get a prize.

The benefits:
- All students are engaged on every problem. Even if it's not their turn, they can steal if the other team is wrong.
- Students have a partner to collaborate with, so weaker students are not put on the spot and can learn during the activity.
- Pairs monitor each other for cheating - they can only get the square if they've shown their work.
- Tic tac toe is the funnest game on the planet. Apparently.
- Generic mechanical pencils in fun colors come 30 to a pack for $5. Great prizes! Mini candy bars work too.

Enjoy! Let me know if you play it and it works (or doesn't work!) for you.

Big changes

2008-04-23T20:38:00.000-07:00

The 10% cut in school funding in CA has really thrown the school for a loop, and we've spent the last couple of months figuring out how to respond in a way that is positive for both families and staff. It's been very difficult, but it has also helped spur us on to re-look at some of our practices and hopefully make some positive changes. I'll write more about the actual changes once they get finalized - everything is still in the discussion phase, although we are beginning to make some real headway.

Note to bay area math teachers (or those wanting to relocate to the 408): we are hiring! Check out the website or email me if you want more info.

Prepositional Nightmare: Anywhere a Cat Can Go

2008-03-19T21:40:00.000-07:00

One of the problems with block scheduling is that, when you lose a day of school, it throws your whole system off. Due to community day on Thursday, and spring break starting on Friday, periods 1 - 4 met twice this week, but 5-6 only met once. So, it was time for a slush lesson. Sorry, I mean "enrichment". I find these hard to do well, because if it is something worthwhile - such that you can justify spending 80 minutes of time with periods 1 through 4 - then you want the other periods to see it to. And if it isn't worthwhile, then why not just have a pizza party or something? But you'll never catch me throwing away a lesson like that. There's just no time to waste.

So I decided to experiment with correcting a linguistic problem that bothers me, but is not necessarily mission critical. That is the reversal of terms when saying division and subtraction problems out loud, confusing divided by with divided into, and my personal favorite, "subtract 7 to both sides". I know that part of the problem here is the somewhat arbitrary nature of prepositions, and I've been told that fluency with prepositions is one of the last things to develop when a person is learning a new language, and can take many years of practice. When students make these mistakes in class, I tend to repeat their words back to them, using the correct language, but not making a big deal out of it. My thinking here was that I could do a lesson on it, and then, when they make those mistakes in the future, I can just say "remember the correct way to say that?" and jog their memory, instead of launching cold into an explanation again and again.

I did the lesson. Nothing fancy - just some explanation, some practice, a little board wars (which I typically shun, but it's a slush lesson, so what the hey) and some delectable Easter candy prizes. Yesterday, the students were pretty non-enthused about working on prepositions (shocking, I know), and board wars was so-so, although there were quite a few kids who were very motivated to win the giant bunny lollipops. Today, I had some pig- and ducky-shaped candies to give away, and I think I struck gold, because the minute I showed them to my class, they freaked out and got super-focused. I don't really like bribery, but I think it's probably ok to break form on the day before vacation.

In any case, we were well into the first round of board wars when the phone rang. When I picked up the receiver, I heard some students say "Mr. Greene, we're in Algebra class right now and we have a question." I was pretty confused, until their teacher came on the line. He had them on speaker phone, and said, "My students are telling me that I'm not speaking like a mathematician." (Speak Like a Mathematician is the phrase I use with them for all matters linguistic.) They were all laughing in the background. I finally got what was going on, and said, "Hold on, let me put you on speaker phone here." When I did that, his class erupted in a cheer, which my class could hear, and they were shouting hellos back and forth (although nobody knew who was in each class). They quieted down, and I had them ask me the question - it seems that their teacher said "subtract by 7", and not only did they notice the mistake, they had enough confidence in themselves to call him on it. So I settled it for them, all the kids shouted goodbye to each other, and we went on to an excellent board wars competition.

Later, when talking about it with the other teacher, he told me that he had actually read a problem that said "reduced by 7", but the students swore he said "subtracted by 7" and he decided to play it up for them and call me since he knew I'd been working on it with them. Moments like that are really cool (and potentially powerful), and they can't really be planned out. I love when the last class before a break is a really good one.

p.s.
Does anyone else remember the phrase "anywhere a cat can go"? I still remember it from 7th grade French.

p.p.s
Funny cat videos. My classes loved these for the physical humor. But if you've ever had a cat, you'll see that the cartoonist captures their behavior really well. Enjoy!

Physics is Phun

2008-03-16T22:07:00.000-07:00

I don't have the proper hardware to experiment with this, but it looks extremely cool. Check it out: Phun - 2d Physics Sandbox.

Don't tell, but I learned something on YouTube

2008-03-16T10:26:00.000-07:00

I've been using Keynote this semester as an experiment, to see how it could work in my Numeracy class. So far, it's gone pretty well - especially after I bought a remote mouse so I could control it from anywhere in the room. Combined with the mini-whiteboards, it's been a really efficient way of getting students to do work. After presenting a concept, I can have them practice a few problems right away by showing the next slide, and having them work on their boards. There is no time wasted passing out worksheets. Also, I can make sure all students are focusing on a specific set of problems (versus on a worksheet, where they tend to start jumping around right away, based on what seems easiest). Then, I can show work/answers on the slide without having to pull out a transparency.

Since I've got the projector reserved and set up now, I can easily insert fun and interesting images, sounds, and video clips. I've recorded myself and other teachers singing little ditties (like the infamous "Don't add across"). I've started scouring YouTube for interesting stuff... though the ratio of total crap to interesting stuff is quite high, I've found a couple of gems. I even unearthed my old calculus professor from college, who recorded a "top ten algebra mistakes hit parade" as well as "all of calculus in 20 minutes".

So I'm in my fraction adding unit now, and we've been working with fraction circles to understand adding. Now, we're taking a break from that to do some work on prime factorization, reducing fractions by canceling common prime factors, and finding LCM. Once they get all this mastered, we can go back to adding fractions using common denominators. I hope they don't forget it all over spring break... I've always found it difficult to teach factors and multiples, and GCF and LCM because students confuse these concepts very easily. Part of the problem is their difficulty with the language of division. Just about every student I have says "divide 6 by 40" when they mean 40÷6. If I ask "does 3 go into 12?", they'll say yes. But they'll also say yes if I ask "does 12 go into 3?". (Aside: I think I'm going to devote an entire lesson to this issue - along with the whole "subtracted from"/"subtracted to" issue.)

In any case, I YouTubed LCM and GCF to see if there was anything interesting out there. I was surprised to find a method for finding both LCM and GCF at the same time using Venn Diagrams that I'd never seen before. It's mathematically equivalent to looking at the prime factorizations and picking the right factors, but it provides a nice structure for students to remember which is which. So I designed a lesson to practice finding factors and multiples, and then using this model to find LCM and GCF. It went quite well. I don't know how much will be retained over the weekend, but we'll practice more on Monday/Tuesday because I want them to have LCM down solid. Here are two of my slides, and then the original video I got the idea from.

Math & Art: Big Numbers

2008-02-23T22:24:00.000-08:00

Check out this site. Really cool images.

"Depicts one million plastic cups, the number used on airline flights in the US every six hours."

"Detail at actual print size:"

What's the percentage of "adders-across" in Numeracy?

2008-02-23T19:23:00.000-08:00

In the past, I've given diagnostics before a unit so as to be able to compare pre- and post-instruction scores. Now, in the spirit of differentiation, I'm going to go one step further.

The next unit is about adding and subtracting fractions and mixed numbers. On my diagnostic, I wanted to see what percent of the students are still "adders-across" (#25 down: snakes that are bad at math). That would be 68/80, or 85%. The remaining 12 students could all do the basic algorithms, but most stumbled on the more complicated mixed number subtraction problem.

So here's the plan. In each class, I will assign one of the non-adders-across (NAA) to an adder-across (AA), tasking the NAA to help the AA learn over the coming lessons. If I see that they remain on task during practice time, the NAA will not have to take the quizzes, earning an automatic 100% on them. This seems reasonable, since they have already shown me they know the skill. Additionally, if the AA passes the quizzes (i.e. becomes an NAA!) then the NAA helper will earn some oh-so-coveted extra credit points. This way, the NAA has strong incentive to help, but there is no penalty if the AA doesn't make enough improvement.

Since almost no students showed mastery of the mixed number subtraction problems, every one will need to take that quiz when we get to it.

Now, the only thing that remains is to pair up the NAAs with the AAs effectively. I need to factor in personality, motivation, and so forth. Also, this experiment really highlights the imbalance between classes, even though we try to avoid any tracking (a constant difficulty in a small school). Here are the numbers of NAAs by period... Period 1: 5, Period 2: 4, Period 4: 2, Period 6: 1.

My mini-whiteboard love-hate relationship... Can you help?

2008-02-22T18:24:00.000-08:00

I've been using mini-whiteboards daily in my numeracy classes all year. Students use them most of the time, except when I have a worksheet for them to do (and even then, they tend to use them for scratch work).

Positives:

I can see, from anywhere in the room, what students are doing, and if they are on task.
Students enjoy writing on their whiteboards more than on paper.
Students don't have to waste paper for scratch work (this is especially helpful for those students who have still not mastered the art of bringing school supplies to class).
And I don't have to make worksheets for every single task either.
It makes collaboration easier during pair/group work tasks.
It's great for quick checks of understanding - put a problem up, students do it on their boards, and then immediately lift them up for inspection.

Negatives:

We burn through markers like nobody's business, and the ones that are low-odor cost about a buck a piece. I've tried the cheaper ones, but they run out really fast, or have fumes that cause much complaining of headaches.
Tables and hands tend to get really messy (for some students more than others...) Our beautiful white laptops are getting covered in whiteboard marker smudges.
"Mr. Greene, can I please go wash my hands???"
Some students not able to respect materials, destroying markers by pounding in their tips, or writing with them on paper till they run out.
Some students unable to stop drawing beautiful works of art when I am presenting material. Or maybe this is a positive because I can see that they are off-task, whereas if they were doing plain old paper-and-pencil doodling, I might not notice?

I was wondering if anyone had any ideas to help with the logistical issues of mess and expense? Remember the Magna Doodle?

4.58 x 10,000 = 4.580000

2008-02-21T17:28:00.000-08:00

Most of my numeracy students remember that helpful rule from middle school: "Multiplying by 10 means adding a zero", and so we get results like the title of this post. This is one of those fundamental place-value problems, the type of thing that betrays just how little some students really get about the number system. It's taken about two weeks of practice to get them comfortable with the idea of shifting the decimal place left and right (and remembering which way to shift it, depending on the operation).

We are also currently struggling with the issue of the missing decimal point... when there is no point shown in a number, where is it really? Some of my students still think that you put the point at the front of the number. Why do they think this? I'm not sure. Before break, we spent a whole lesson on what the decimal point means, and it seemed to go well. Since we've been back in the second semester, the question of where the missing decimal point goes has been asked and answered many times each class period. They are getting better at comparisons: if I ask them to compare 473 and .473, or .4 and .39, or .4 and .04, they are usually getting it right. And yet, when faced with the problem 473 ÷ 100,000, some students seem to forget it all and start with the decimal at the front of the 473 (or sometimes between the 4 and the 7), forgetting that this changes the value of the number.

No wonder scientific notation is such a bear to teach in Algebra... To reinforce both concepts, I've been teaching scientific notation (with positive exponents only) in this unit, and it's finally starting to work. From the start, my students could tell me that 10^6 was the number 1 followed by 6 zeros, but they couldn't see the relationship between the problems 9.02 x 10^6 (which was totally confusing) and 9.02 x 1,000,000 (which is finally becoming easy). Converting a number into scientific notation is starting to make more sense to them now, since I've finally figured out another flaw in some of the students' understanding: they don't really get the significance of the equals sign. I would show over and over why 302,000,000 = 3.02 x 10^8, and some kids just weren't catching on. But then, when I asked them what they would get if they multiplied 3.02 x 10^8, they were surprised to see that it was 302,000,000. I would get lots of "ohhhs" as they realized that the two parts of the equation had to be the same, and that you could multiply to check your answer. The main problem I still have is getting them to remember that the first part must be between 1 and 10. But at least we're making progress! Though we have been learning dividing by powers of 10 at the same time, I don't want to introduce scientific notation with negative exponents now (since they have never seen negative exponents before). I want to give this time to sink in, and maybe come back to it later in the year.

We have the rest of this week off for winter break; when we start next week, I think it's time to move on from this percent and decimal concepts unit and start in on fraction operations. 1/2 + 2/3 = 3/5, here we come! (One of my favorite things to show numeracy students is why this equation doesn't make sense.)

Welcome

2008-02-18T17:19:00.000-08:00

Most of our students are English language learners, but most have Spanish as their native language. As of a few weeks ago, we have a new student who is a refugee from Myanmar - she showed up in my SSR period and in my Numeracy class. Not only is language a huge barrier, there is also her difficult past. Working in her favor, however, is a massively strong desire to learn.

An article came out in today's paper which gave us all more insight.

Here is the text of the article (if the link is bad).

Orphans survive wars, find safety in Bay Area
REFUGEE CHILDREN ADOPTED THROUGH UNITED NATIONS
By Rebecca Rosen Lum
Bay Area News Group
Article Launched: 02/18/2008 01:33:04 AM PST

Kate's smooth brow buckles when she thinks about the soldiers who muscled their way into the house where she lived with her grandmother - plundering belongings, forcing their attentions on her and ordering them to prepare meals.

"The soldiers make me too sad," said Kate, discriminated against as an ethnic minority in Myanmar. "I don't like."

One day Kate, now 16, fled to the home of sympathetic friends in a neighboring town. She learned soon afterward that the soldiers killed her grandmother in retaliation.

After a desperate flight through underground channels of Southeast Asia, Kate has found a lasting safety: She now lives with a family in San Jose. "Baba" and "Mama" are the Rev. Ben and Anne Daniel; she has three siblings.

As rain pounds on the roof of Ben Daniel's church, Kate sits comfortably between her new parents, a delicate girl with shiny black hair and a wide open smile. She has been here little more than a month, but she says this is home.

"Everything OK," she said. "Not tired. Not scared. I happy."

Kate is one of a trickle of refugee orphans finding homes with Bay Area families through a special program of Catholic Charities, one of two agencies that contracts with the U.N. High Commissioner on Refugees to place the children.

In such countries as Liberia, Uganda, Sri Lanka, Myanmar and Nepal, children have been driven out by armed conflict or pressed into service by government militias and rebel groups - as combatants, sex slaves and virtual pack mules.

If an adoption always includes risk and reward, these adoptions offer a double dose of both.

Preparing food is now a source of surprise and delight for Kate. She likes oatmeal with hot sauce. At first, she dissolved in giggles at the sight of Baba popping up a skillet of popcorn on family movie night. (Men don't cook in Myanmar). Now they fix dinner together.

Kate dropped out of school after her fourth year to help her grandmother farm corn and beans. She asked to start school the morning after she arrived: "I want right now," she said, laughing. She studies music with Anne and says she hopes to become a minister, like Ben.

Kate's odyssey hardly seems likely for a child, but it is mirrored throughout countries where war and strife have made homelands unlivable. Many have been persecuted for religion, ethnicity, or political affiliation. They have been separated from their parents or seen them killed. The children escape brutality by guts, wit and luck, walking for miles, hiding in jungles, riding on the backs of sympathetic elders to safety - mainly, in refugee camps.

Five million refugees have fled their homelands, according to Refugees International, a non-profit organization. If one includes those who are trapped in their home country, such as in Darfur, that number balloons to 14 million. They can't go home in many cases because home is no more; their villages have been destroyed.

Tracy Weiss read all she could get her hands on about the conflicts that racked the Eastern coast of Africa after she agreed to adopt three siblings from Monrovia, Liberia.

When she picked them up from Mineta San Jose International Airport, Sadiki, the eldest and tallest, stood in front, "scanning everyone, looking for danger in every direction." His sister Maryama tucked in behind him, holding a bag, the U.N. signal for a refugee arrival. Antimana, called "Ansu," crouched behind his two siblings. They wore donated clothes - Ansu, a 1930s-era man's suit.

"I said, 'Hi. I'm your new mom,' " Weiss remembered. "Ansu was the first to break into a grin."

The trio has been living with Weiss in Los Altos for three years and - Maryama counts on her fingers - six months.

Rebels executed the children's Mandingo father, as well as Sadiki and Ansu's mother. The children and Maryama's mother ran from rebels, living in the bush, moving constantly, sometimes getting separated. They settled for a time in Bo, a village in Sierra Leone. Sadiki - he thinks he was 3 or 4 - made many friends there.

"Then things got bad if you are not a citizen," said Sadiki, now 18. "We had to find a way to stay alive."

Sadiki's earliest memory is of a village in chaos, with people running everywhere to escape the approaching rebels. Alone, he held up his arms in hopes someone would carry him to safety. Someone did.

He thinks the family spent five to seven years on the run.

Chatting one afternoon, Sadiki's new mother asked him if he had any photos from his earliest years.

"Mom," he said evenly. "You are running with a whole stack of things on your head. You step and you fall in the river, everything gets ruined."

They eventually made their way to the Bandajuma refugee camp, where his stepmother died from complications of diabetes.

It took them some time to get used to the idea that they could make the four-block walk through their wooded suburban neighborhood to school without getting mugged, that loud pops were not likely to be gunshots. Weiss had to quickly abort a July Fourth trip to see fireworks in San Francisco when the multiple blasts badly shook the children.

While life here brings a sense of safety, negotiating the social minefield of a new culture can prove dicey.

Language is a separator at the outset. Then come the mutual misconceptions of American kids and the newcomers.

The refugee orphans are surprised to see all Americans aren't wealthy and white. Alternatively, few Americans have had to run for their lives.

"One kid said to me, 'Did you ever fight a lion?' " Sadiki recalled, howling with laughter. "I said, 'Yes, two.' "

Many don't even know where Africa is, Maryama said, and they know much less about the violence that devastated her homeland and scarred her family.

"I can't be angry at them," she said. "They don't know. When they know, they care."

Divide by Zero?

2008-02-09T21:51:00.000-08:00

This is pretty cool. I've never heard of the animated lego genre before, but I guess it's pretty popular. Most of these films are shorts, but some are feature length! Wow..

Caught Being Good

2008-02-06T22:01:00.000-08:00

If you're a Harry Potter fan, you've probably noticed that classroom management at Hogwarts isn't much of an issue. Sure, they get to fly around and do magic all day. And parent involvement seems quite strong. But what else do they have that keeps the young'uns in line and focused on getting to a four-year wizard college? An entirely hassle-free incentive system. I'm talking, of course, of the Hogwarts' House Cup, and the constant cries of "10 points for Gryffindor" and the like.

At DCP, we've only formally developed negative consequence systems (detentions, referrals, contracts, etc.). These work to an extent, but not for all students, and not in all cases. For a while now, I've wanted to get a positive system put into place as well. I thought that this would help increase student buy-in, especially for the freshmen making the transition to DCP and becoming a college-prep student. So, combine this need with our students' love of getting points, use Hogwarts as a model, and presto-hey you've got the "Chalice of Pride"!

I got some other teachers together, and we made a plan for this at the beginning of the year, but we haven't been able to get it off the ground until now (time, time, time...). We originally had a more complicated setup, but the lack of magic wands put a damper on our plans - the system had to be totally easy for the teachers, or it wouldn't fly. So here's how it works: each freshman tutorial class (there are 5) is competing for the Chalice of Pride. Students can earn a Ganas Point (i.e. a "caught being good" ticket) for any behaviors that really demonstrate one of the school values. This is simply handed to the student by the staff member. The student then must put the slip into the clear plastic locking drop-box that is assigned to his or her tutorial (these are attached to the wall in a central location). At the end of each month, we tally up the slips in each box, and the winner gets to display the "Crest of Community" and claim bragging rights. At the end of the year, the tutorial with the most total points earns the "Chalice of Pride" and a field trip (like a day at the beach, or whatever floats their collective boat). We're in day 3 of the system, and I'm starting to see slips in the boxes already. I'm looking at this as a kind of experiment in positive incentive systems - clearly, it can be done more effectively - but we have to start somewhere.

Continuing on with the positivity trend, we've also updated our homework checker system for the new semester. For the past few years, students have had to carry a homework checker with them to each class; it got marked each time they didn't have their homework, and then the checker would be looked at by the tutorial teacher and the parents. Of course, this would cause students to "forget" their checker on days when they did not have their homework. Therefore, we gave detentions to students for not bringing their checkers (to force them to produce them), and this really never led anywhere good. So, we made a simple switch to stamping their checkers when they do have their homework, with some simple rewards attached to getting a certain number of stamps over the 6 weeks. The rewards and reward-levels were created by the student council: 85% = prizes like stickers, candy, etc.; 90% = a free homework pass; 95% = free dress day and double lunch. We also started this on Monday, and so far, it seems to be working well. Students really want the rewards, and they are making sure to get their stamps. Even if they miss a homework, they will still be more likely to produce the checker the following class period so they can get the next stamp (in the past, they knew we would just mark all missing homeworks when they finally brought out the checker, so some students never would).

Who knew how effective stamping and stickering could be in high school?

A Jewish-Italian Hannakristmas

2007-12-25T17:36:00.000-08:00

I'm in Cleveland at my dad's house for the annual event... the homemade gnocchi and sauce were great, the kids got lots of noisy plastic crap to forget about by tomorrow, and the vegan chocolate-peanut butter cake from Mustard Seed cafe was tasty: I had two, ok three, slices.

After desert, board games, and being shot at with nerf guns (which have gotten scarily high-caliber), I was talking with my 10-year-old half-brother about his school. Specifically, I wanted to see how he is doing in math. He is (not surprisingly) unable to articulate exactly what he is doing in math, so I was asking him specific questions to see what his math is like. Last year, I was surprised to find out he knew square roots already - I explained about cube and higher roots, and he picked it up instantly.

I wanted to see what he knew about fractions as a 4th grader at a typical Cleveland-area public school. I asked him if 1/2 or 3/4 was bigger.. way too easy. I asked him if 2/3 or 3/4 was bigger. He got it right, and quickly, but couldn't really explain why. I then asked him if 3/7 or 3/8 was bigger, and he said 3/7 immediately. I asked him to explain how he knew, and he looked at me like I was stupid, saying, "a seventh is bigger than an eighth, so..". I asked if he had worked with mixed numbers, and he hadn't, so I asked him to figure out what 3 1/4 - 1 1/2 is. He couldn't do it in his head, so I told him to get paper and draw a picture. That's all I said - he drew fractions circles correctly, crossed off a whole, the fourth, and then another fourth from a whole, and came up with 1 3/4.

I've only ever taught at DCP, so I don't have much of a frame of reference for knowing if he is above average or not, but this is the kind of thinking that all students must have to be successful in high school math. This is the kind of numeracy ability I want my students to develop; this way, when they get to a new problem, instead of giving up, they can reason it through and at least make progress. I struggle daily to get them to pay attention, to care, to think, to not give up when a problem is hard, and their mathematical progress is painfully slow. In a couple weeks, when we start reviewing for finals, and half the kids don't even remember what an integer is, it will be painful. But I know that, by the end of the year, most of my students will have improved their math abilities in many ways. Never as much as I want, but it will have to do! I just gave our grade-level equivalency test before break, and the median score has improved by 1.1 grade levels (from 5.9 to 7.0) and the average by 1.65 grade levels (from 5.76 to 7.42) since the summer. If I can squeeze that kind of growth or better out of them during the second semester, most will be in pretty good shape for next year.

A case study: freshmen's ability to listen...

2007-12-20T12:26:00.000-08:00

(Background: cell phones are not permitted. If seen/heard, they are confiscated until parents pick them up.)

{Phone rings; I answer}

Me: C, they need you at the front desk with your cell phone, because your sister got hurt and they don't have your mom's cell phone number.

C: (Looking angry) But I don't even have a cell phone!

Me: They don't want to take your phone, they just need your mom's number.

C: But I don't have a phone!

Me: Just go...

(5 minutes later)

C: Mr. Greene, do you know why they wanted me? My sister hurt her leg, and they needed my mom's number!

Me: That's exactly what I told you.

C: You did?

More decimal and percent work

2007-12-19T21:27:00.000-08:00

For the next lesson in Numeracy, I wanted to keep building students' ideas about what percents and decimals are, and how they relate to fractions. I've mentioned before that I am teaching the students to use bar modeling to solve word problems, but I haven't been posting the problems, and some examples would probably be nice. This week, I've started incorporating percents into the problems, which is, of course, throwing the students for a loop. But, they will get it eventually (and some already have), and I think that continually reinforcing the visual connection between percents and fractions is important. So here are the three problems I'm using this week:

Lesson 1

Diego and Dora both took a test in Algebra 1. Diego got 70% of the questions correct, which was 42 points. Dora did very well, and even got the bonus problem right, so she got a 105% on the test. How many points did Dora score on the test?

Lesson 2

By the end of tutorial, Mariana completed 45% of her homework. She spent 54 minutes working (the rest of the time, she was giggling with Gricelda). If she works at the same speed at home, how much longer will it take her to finish all of her homework?

Lesson 3

At the school dance, 70% of the students were girls, and the rest were boys. Ms. Vasquez wondered why there were so few boys there – she counted only 36 boys. How many students were at the dance in total?

We only do one problem like this per lesson (3 lessons per week - block schedule), because it really takes 15 - 20 minutes for the whole process (more when the students are unfocused) to play out. What's nice about these problems and this method is that it naturally connects percents to the work students have been doing for months drawing whole number bars (first) and then fraction bars. Right now, lots of students are still struggling, but I think it's more due to the proximity of vacation affecting their ability to care about math than a conceptual problem. We'll pick up with this after break as we review for finals, and I think it'll go better.

For this lesson, after finishing the problem solving portion, we did another class activity. I gave each group a set of 10 post-it notes with various decimals and percents, all between a pair of consecutive whole numbers. They had to stick their post-it notes to the board (where I had blue-taped up a long number line) drawing arrows with marker to indicate more precisely where the number should go. This only took about 5 minutes or so, and then I had everyone sit back down so we could evaluate how we did. I told them that they would earn 2 team points (whoopie!) for each number in the right place, and 1 bonus point (what can I say? Freshmen love their points!) if they could find a mistake in another team's positioning. We went through team by team, and I asked the class to point out any mistakes. The mistakes that were pointed out lead to additional discussions and modeling, until students seemed satisfied that everything was in the right place.

After doing this with my first period, I wasn't sure if the activity had been all that useful. I asked students if they found it useful (many did) and to share something that they had learned. This lead to some good questions and observations - the key one being that 2.45 is less than 2.5 (I still had the papers on the board from the previous lesson to refer back to) because of the value of each number; similarly, several realized that 2.45 is not more than 2.5 even though it has more numbers in it. Other students said that it helped them see better how a percent and a decimal could both be plotted together on the same line (yes!). So this made me feel better about the whole thing, and I didn't modify the lesson for the other periods.

Now that we've used manipulatives and done a couple of class activities, in tomorrow's lesson, students will be doing a worksheet I put together to get some solid independent practice time. They will start by drawing paper pieces (as in the last lesson) to represent a variety of decimals, fractions, and percents. Then, given a number in one form, they will have to write it in the other two forms. Finally, they have a bunch of problems where they must compare a pair of numbers - written in any form - to see which is larger.

The decimal point's job... where's the one?

2007-12-14T20:20:00.000-08:00

This has been a long and stressful week, but it ended well yesterday. The biology classes were on a field trip, so my Algebra 2 class only had about 5 kids in it. I didn't bother trying to teach anything new - I just helped them with the homework, and then we spent the rest of the time looking at different problems they needed help with, and talking about stuff that was on their minds... some "remember when", since I taught most of them as Freshmen - they remember amazingly well things that I said over two years ago, as long as they are not math related, apparently... we talked through some of their fears about going to college... we reviewed fractions... I wish there were more times available to just sit and chat with students.

In my Numeracy classes, the goal of the day's lesson was to learn how the base-10 positional system works, with a focus on the decimal side of things. Specifically, I wanted them to be able to represent fractions and percents as decimals, and to understand how each decimal place relates to a specific base-10 fraction. This lesson went very poorly on Thursday with my first two classes, due to management issues (i.e. they were behaving terribly). Is there ever a point in time when you can get through a week without feeling like a first-year teacher again? Anyway, I didn't rewrite the lesson for my Friday classes, because I was confident that it was actually a good lesson, and worthwhile.. I just tightened it up, and made sure to stay on top of things behaviorally, and it really paid off. The discussions in both periods (and these are my two weaker classes) were quite rich and productive. On my board, I had taped up papers as in the graphic above, and told students that each smaller piece had been made by cutting the previous one up into 10 pieces. I asked how big each was, and they easily saw that each paper was 1/10 of the one to the left. I then asked which paper represented one whole. Some students thought it was the 10 big pieces, some thought it was the single full-size piece, some thought it was one of the smaller ones. They discussed it, and as I kept asking more students what they thought, in each class, several students started saying that any one of the pieces could be one whole. They convinced each other without much of my prompting - it was like magic! I then pulled out my magic Decimal Point cutout, and told them that the decimal point has one and only one job - to determine which shape is worth one whole. I taped the point up on the board (making the full sheet of paper the one whole for this lesson), and we then proceeded to label all the place values relative to one whole. All of a sudden, why the tenths are called "tenths" started to make sense for some students.

I showed what the fraction 71/100 would look like by taping up 7 strips and 1 tiny square in the right columns, and then translated this to the decimal 0.71. We did a couple more examples (I called students up to try some), and then they began asking about adding zeros at the end. Some thought it was ok, and others thought it would change the value. So I had them debate for a while if 0.6 and 0.60 were the same or not. Eventually, they were convinced that they were the same. So, to push them, I asked why 6 and 60 are not the same, which allowed me to show them that the 0's only change the value when they push a digit into a different place value. For each example, I also went back to the definition of percent (a fraction out of 100) to show that the tiny squares must therefore be considered 1% each. This made it easy to see why 0.36 would be the same as 36%, as they could see the 36 little squares on the board (comprised of 3 strips and 6 squares). It also made it easy to explain why 0.237 is the same as 23.7% - you have 23 little squares, and 7/10 of a little square.

I saw quite a few light bulb moments over the course of this lesson. I need to find a way to get my first two periods back on board now... We'll review this on Monday; I hope that, by the end of next week, students will all be able to compare and order decimals, and easily move back and forth between decimal, fraction, and percent representations. That would be quite an accomplishment, and a good way to go into break.

Fidelity in Math

2007-11-30T18:54:00.000-08:00

In Algebra 2, the topic today was an overview of functions. Some students were having difficulty understanding the "each input has exactly one output" condition, and the previous example (percent score --> letter grade) just wasn't cutting it.

The follow-up example was much better. The domain was a set of boys' names, the range was a set of girls names, and the mapping was "dating". And, of course, one of the boys had an arrow pointing to three different girls. We discussed why this was not a function, and one student said, "So to be a function, they have to be faithful!". Exactly! I took her up on this, and had them add to their notes: Functions are Faithful! This instantly made sense to them, and this language carried forth through the rest of the lesson. I then added another boy pointing to one of the girls that was already in the list, and asked if everyone was still faithful. They said no, and we clarified things; our new "taken-as-shared" idea was that only the boys (the inputs) have to be faithful for it to be a function. (I mentioned that if all the girls were faithful too, then it is called a one-to-one function, and we'd look at that later.)

It was really amazing - even when we did examples involving decontextualized numbers, they were still very comfortable using the analogy: i.e., that set of ordered pairs is not a function because the 4 is being unfaithful! It even made the vertical line test a breeze to teach.

It's always nice to find something new to add to the bag of tricks.

Happy Tofurkey Day

2007-11-21T11:40:00.000-08:00

If you've read my blog in the past, you may have noticed I haven't been writing much lately. I've had some bad health problems over the past few months, which have made it difficult for me to get anything but the essentials done. It's finally under control, and I am feeling relatively human again. I am thankful for strong drugs and medical advances!

So, my lack of energy along with the typical November doldrums have made my freshman Numeracy classes less effective and positive than I'd like. I hope that this mini-break will give us all a chance to recover a bit, and come back to end the semester strong. Here are some of the major things I need to work on:

1) Multiplication Tables.
I decided not to focus on teaching the tables this year to the whole class, as it is a waste of time to the 1/2 to 2/3 of the kids that know them. And, I'm not sure how to do it really effectively for those that are in high school and still don't know them. ALEKS doesn't deal with multiplication tables.

I've given students who need them 12 x 12 tables to keep in their binders and look at as they do things like reducing fractions. I need to find a way that they can actually work on improving, and I think this will be different for each student. Some combination of flash cards, games, incentives, and quizzes will be needed. But how to work this seamlessly into the class? Hmm... Needs more thought..

2) Classroom Culture / Readiness
I've posted about this in the past, and my readiness checker was working very well. But, recently, class hasn't been getting started very efficiently. The students recognize this, and freely admit that, now that they are well into school and have made friends, there is a lot more temptation to hang out between classes and avoid getting into their classroom or seat in the room until the last possible second.

I don't sweat it if we miss out on the first minute or two of class, but more seriously and annoyingly, school supplies seem to be growing ever more sparse (kids don't have pencils or binder paper, and act shocked when I ask them to take these things out!). It's like they buy some new stuff in September, and when they're out, that's it for the year. Also, binders have gotten to be a mess (i.e. holy terror) again - and that's for the kids who still have binders.

I need to come up with an incentive system to get this all back on track - and a way that forces me to stay on top of it.

3) Problem Sets - Singapore Bar Models
I overshot greatly with my initial stabs at assigning students problem sets. My goal was to get them started on longer-term planning, while assessing their ability to use the bar model method to solve word problems. My initial idea was to assign ten problems on Monday, due the following Monday. I would grade and return them, and they would have till the following Monday to do revisions to increase their scores. Sounds reasonable? Well, it was still way too much for them to handle. The second time around, I made mandatory progress checks during the week, to help keep them on track. I got more turned in the second time around, but they were still not very good. And, there is a big problem with cheating. Unfortunately, students don't really understand all the time what cheating is, and it's hard to get them to see it. I want them to work together to help each other; many of them think that copying someone's answers who is "helping you" isn't cheating. This is going to require a lot more coaching. Here are my initial ideas for how to run problem set #3:
- Reduce the number of problems from 10 to 6.
- Continue with the progress checks, for regular homework credit.
- The day the assignment is due, give a one- or two-question quiz with selected problems from the assignment, but with numbers changed. This will help me see which kids actually understand the work, and which kids copied.

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

2007-10-21T18:11:00.000-07:00

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.

Our Department Picture: Math Rox!

2007-09-24T15:58:00.000-07:00

Are you ready for this?

2007-09-17T20:34:00.000-07:00

I quickly mentioned the Readiness Checker idea a few posts ago, as a new idea I got from another teacher. I want to revisit it here, now that school has been under way for a few weeks. I have to say, it is simply excellent. In the past, I've done a variety of readiness checks, with various degrees of success. This is working better than any other method I've tried.

So here's how it works. To be ready, students must, *by the time the bell rings* do the following:

Take out binder

Take out homework (and Readiness Checker)

Have a pencil sharp and ready

Put backpack in back of the room

Begin working on the Do Now

If all of those things are completed, the student earns a sticker or stamp on their Readiness Checker. When the checker is filled in (I currently have 9 spaces on mine, but will probably extend it to 12 for the next round), it turns into a "get out of homework" pass. This has the benefit of putting a nice positive incentive on being ready for class, no negative consequence, and it is not directly tied to points in the grade.

It has been working like a charm in my 9th grade Numeracy classes. Most of the time, I have at least 3/4 of the students earn a sticker (often more), which lets class start quietly, focused, and on track. Of course, there are off days (like Friday afternoons), but overall this has been a fantastic new innovation. If you teach students that have difficulty getting started, I highly recommend a system like this. A couple of our 12th grade English teachers are doing this too (I was surprised, but they say the students love it.) Best of all, there is no added management on your part - if the student loses their checker, give them a nice new blank one. (So far, though homework sometimes "gets lost", I haven't had a single student lose their Readiness Checker. What a surprise! :)