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.
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
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?
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.
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.
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.
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.
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!".
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 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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:
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.
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.
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!
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...)
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.
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).
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.
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.
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.
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.
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...)
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.
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.
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.
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!
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.
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.
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?
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.)
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."
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..
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?
I am a math teacher in the Bay Area, California. Previously, I taught for eleven years at Downtown College Prep charter high school in San Jose, where the target student is low-skilled and of the first generation to go to college. Currently, I am teaching at Capuchino high school in San Bruno.
On this blog, I will post strategies that I am trying, and I would love to see strategies that others are using to help bring students up to grade level and teach them real math skills.
If you want more information on any of my lessons, feel free to email me at yochanan AT gmail.com. Also, if you have ideas to share, please comment.
Note: if you don't have Gill Sans, try changing the font to Tahoma. These files were all created in Mac OS X.
I work in Keynote, and export the presentations to Quicktime for those who don't have Keynote to check them out. If you want something in Powerpoint, let me know, and I can usually convert it reasonably well.
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Algebra 2 Files
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Numeracy Files
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