Airplane Method for Factoring

I mentioned the "airplane" method for factoring in a recent post. Someone asked me what that was, so I thought I'd share.

I have seen a lot of methods for factoring a quadratic with a leading coefficient. Out of the ones I've tried, this is my favorite. The analogy to an airplane is a bit of a stretch, but students seem to remember it pretty well. So I'll take it.

I should also mention that, before I show this to students, I always spend some time letting them work on these by trial and error. I figure a process like this is worthless if they don't actually understand what they are doing. Once I feel like students understand the concept but they are still struggling to get every problem to work, I show them this. We treat it like a shortcut, and boy do they appreciate it!

Here is an example:

First, my students know they will need two binomials, so I start with two sets of parenthesis. Then I put the leading coefficient in each parenthesis. Hopefully, the students have a problem with this. We talk about why it is a problem, and I promise them that we will get rid of the extra 2 before we're all done.

Then, multiply a and c. (See the airplane wings? Use your imagination.)

Look for two numbers with product ac and sum b. (Propeller?  I know this is really a stretch.)

Put those numbers in the parenthesis.

Divide the extra 2. (The landing? Maybe.) It is pretty cheesy, but when students are having trouble I can say something like "you forgot the landing", and they know what I mean.

Done.

For something like this, you may need to divide both binomials. I point out how dividing by 3 and by 2 is the same as dividing by 6. We just choose the division that will keep integers.

Happy factoring!

Figuring Out Factoring

I have been thinking a lot about factoring lately. My algebra 2 students really struggle with it, and we have only factored quadratics (no sum/difference of cubes or grouping yet). I am worried because our first unit after winter break is rational expressions/equations. This unit is challenging when you CAN factor well, and almost impossible if you can't.

I am sure it will be necessary to take a few days to work on factoring before we start rationals. I need to review quadratics, teach cubes and grouping, and put them all together. I am working on how to structure that.

I have been doing some reading and searching for ideas. It seems that almost every method out there requires students to memorize some sort of process. (Except for this, and I am not sure my students are ready for that). As far as algorithms go, I am happy with what I've been using.

There are two issues that I want to address:

1. Helping students make a connection between what they SEE (trinomial, difference of squares, four terms, cubes, etc.) and what they DO (write two binomials using your preferred method, grouping, use a formula, or whatever).

2. Helping students get through problems where more than one method is required. (Like factoring out a GCF and then factoring the trinomial, or using grouping and then recognizing that one of the binomials is a difference of squares).

Here is a flow-chart I sketched out this morning:

I am picturing this on the white board, with an answer box next to it. You could 'drop' a polynomial into the top box and it comes out factored. If there was a GCF, you could put it in the answer box and then examine the remaining part and possibly drop it into another box. You could then take the pieces that come out of the second box and either add them to the answer box or drop them into another box. This all seems a little elementary, but I want it to be very obvious and very clear . . .

I am still refining this and hope to use it for the beginning of second semester. I will write about the final product. In the mean time, suggestions welcome. :)

Fraction Exponents. Easy.

Have you ever found yourself teaching a certain thing a certain way for years, and then one day you think about changing your explanation just a teeny tiny bit? And the new way makes infinite more sense to students, and the thing that used to be impossibly hard is now easy? And then you wonder what took you so long to find that more easy/obvious way of explaining something?

That happened to me today with fractions as exponents.

I won't bother to mention how I used to teach it. It was bad. Very bad.

Today, I started by showing them this, and they all shouted several people went "x squared!".

Then I asked them HOW they knew it was x squared. Somebody pointed out that three x-squareds multiply to equal x^6. Someone else said that you divide 6 by 3.

I made a big deal about writing x^(6/3) before writing the answer as x squared. And we did several more of these including some square roots, so they could see that you divide by two.

Then I showed them this, and gave some time to think about it and write down what they thought the answer should be.

Almost every single student wrote down x^(2/3)! And there were angels singing.

Then we worked on going backwards, which was no biggie at all.  Given x^(1/2), students could easily rewrite as square root of x and so on.

And then I didn't know what to do, because it used to take me a full class period to teach that. And some students would still be sitting there going, "Huh?". But today they just got it in, like, five minutes.

I realize that there is nothing earth-shattering about this method. The thing is, I've taught it this way before. Only I didn't lead with this part. I ended with this part. All I did today was change the order.

Oh, I love these moments of finding the tiniest little change that makes a huge difference.

The Question Reveals . . .

I saw this quote somewhere a while ago, I don't remember where:

"It is better to solve one problem five ways than to solve five problems one way."

Recently, for bell work, I asked my students to give a response to that quote. I was kind of proud of myself, because it might be the first time my bell work has been something other than a problem to solve. The lesson for the day was solving quadratic inequalities both algebraically and graphically (they have done both, but I wanted students to see them side-by-side), so it seemed to fit.

The responses were limited to just a few thoughts:

You can use one method to check another method.
Sometimes one of the methods may not work.
You might forget one method so you could use another.

I am not sure what I was looking for. Maybe I was hoping that someone would think about how solving a problem multiple ways helps you better understand the concepts and how they all fit together?

What I think is interesting, is how much these answers reveal about where my students are at in their understanding of math. To my students, math is still a bunch of procedures to remember and repeat.

And here's the thing:  I think I am still mostly teaching that way.

Sorting Out Quadratic Methods

Now that my students can solve quadratics in five different ways, I wanted them to weigh the pros and cons of each method. I wanted them to be able to look at a quadratic equation and choose an efficient method for solving. Maybe it is just me, but watching someone pull out the quadratic formula when the equation can be factored kinda makes me cringe. I also wanted to review all the methods at the same time.

First, I gave them this sheet. It has the bell work and the practice problems.

For bell work, students worked out an example of each method in the first column as a review. Then we had a class discussion about the strengths and weaknesses of each method. We talked about how factoring may be the shortest method, but you can only use it if the quadratic isn't prime. And so on. We also talked about why you might choose one method over another. (Like how complete the square is so much nicer when the coefficient of x-squared is one and the coefficient of x is even.)

Next, I gave them this set of 16 cards* and a piece of card stock divided into four sections. I told them to try to put exactly four quadratic equations in each section. They needed to choose carefully, because next they will use that method to solve that problem. I just walked around and coached them a bit as they worked, and mostly I heard some good discussions going on. A few groups struggled, mostly because they had trouble figuring out how to tell if the equation could be factored or not. That is one problem I was hoping to correct with this activity, and it was pretty easy to identify who needed some help with that.

Then, students worked out the practice problems using the method they chose.

The next day I used a similar set of four quadratic equations for a quiz. Students could solve using any method they chose, but they could get bonus points for using each method only once. Students did fabulously with this. Yay!

*I also put the 16 problems on the practice sheet. Whenever I do an activity like this, I try to create the sheet so that someone who was absent could so something similar outside of class.

Happy Halloween To Me

A former student met me in the hallway today. She heard we had just learned the quadratic formula in algebra 2, and she took it upon herself to choreograph a dance to go with the song I teach. With illustrations.

She offered to come to class and perform it for my current students . . .

. . . but only if I joined in, which I did. Except for the slap the butt part.

Sometimes, high school students are just adorable.

Today's Million Dollar Question

I have been struggling a bit this year with getting a student or two to show steps/process/work/setups (or whatever you call it when you say that it isn't okay to give a lonely answer with no justification). The silver lining is that this struggle forces me to think about WHY students should show work.  Here are a few reasons I have:

Showing steps . . .
Puts the focus on the process, rather than the solution.
Communicates your solution to others.
Makes it possible for you (or someone helping you) to locate your mistakes.
Slows you down, so fewer careless mistakes happen.
Gives evidence your answer is right.
Demonstrates your understanding.
Helps reduce cheating. (Some might still copy, but at least they must copy the work too.)
Finally, in my class an answer bank is given. Showing steps keeps practice from becoming nothing more than a matching game.

I am also asking myself some questions like . . . WHY do some students struggle with showing work?

Maybe because . . .
It takes too much time.
They can do it mentally.
They don't know how to show work.
They are bored.
They don't believe in its value.
They are cheating.

Writing this, I realized that when a student is repeatedly refusing to show their process, I tend to go straight to negative assumptions. I assume they are being stubborn and uncooperative, or that they must be cheating.

I am going to try to put the whole issue in a more positive light and see where the students are coming from. Maybe these students think that showing steps is just for the teacher's sake, and has no benefit to them personally.

Or, maybe they genuinely don't know how to express how they got the answer.

It also has me thinking about the types of questions I am asking. If someone can calculate the answer mentally, maybe the question wasn't challenging enough?

How do you motivate students to show their thoughts?

Bell Work Bliss Gone Bad

I am testing out a new bell work procedure this year, like so:

1.  Student picks up the bell work when she walks in the room. It's right by the door.
2.  Student works out bell work and raises a hand to check.
3.  I check the bell work and give out a green star. Student receives a green pen of her own.
4.  Student checks in with partner to assist as needed and gives the partner a green star.
5.  Student checks in with people sitting nearby to see if green stars/assistance are needed.
6.  I continue passing out green stars and green pens until pretty soon, the green pens and green stars have branched out through the whole room and bell work is done.

The first three or four weeks of this method were completely blissful. Students were self-starting.  Students who understood the problem were coaching the struggling students. I was super proud of myself for putting the responsibility of bell work completion into the hands of the students. Yay for me doing less while students do more!

Weeks 4-6 started going downhill a bit. There is a little lag time between the first and the last students who finish. Students started finishing and chatting. Loudly. Then they started chatting before they got started.

Now, at the 8th week I am frustrated and wondering where it all went wrong. What was different about the first few weeks compared to now? Is the honeymoon over? Or did something else change?

I realized that during the first few weeks I had an extra step at the end of the bell work . . . I directed students toward the few problems I'd put on that day's practice assignment that were review. These were problems they could work on right away without any instruction. So they were doing the bell work, coaching and starring their neighbors, and then moving on to those extra problems. Since it was part of the day's assignment, they were motivated to keep moving along and get that done.

The moral of the story is that if I want to continue to do bell work this way, I need to supply my students with something to do after they finish all the steps. Not busy work. Something worth the time. Something productive that will fill the short gap.

I am thinking I might start using some type of guided questions where they could go ahead and start thinking about the day's lesson, or some ACT practice questions, or maybe just continue with review like before. Maybe some extension or challenge problems for them to think about?

I am not sure which way I will go, but writing this post helped me identify the problem so that now I can figure out a solution.

Yay for blogging!

8 weeks, 8 pencils . . .

I've tried a variety of responses to students who show up to class pencil-less:

1.  Refusing to give them a pencil, so they can learn to be responsible.
(Doesn't work, and I feel like a jerk.)
2.  Providing them with a pencil.
(Pencil is usually gone for good.)
3.  Trading them for something valuable.
(Works, but it is inconvenient.)

This year, I just put out a cup of pencils for students to use as needed.

8 weeks later, I still have all the pencils I put out at the beginning of the year.

It turns out, a plastic spoon is a very affective anti-theft device.

Then I came up with a use for the forks.

(Green pens for checking bell work.)

We figured out right away that the spoon-pencil has a major design flaw:

You can't use the eraser.

So, then I really got carried away . . .

I would recommend attaching the eraser to the handle part instead . . . 

My Favorite Partner Practice

I am a fan of anything cooperative, as long as it is structured in such a way that everyone participates. I spent a lot of time last year seeking out new cooperative practice ideas and trying them out.

I am an even bigger fan of partner cooperation. There is something about a group of three or four that makes me feel like someone is probably getting lost in the interaction. When you are working with a partner, you are directly accountable to one other person. I even have my students seated as partners.

I learned my favorite partner practice from another teacher in my grad program a few years ago. I found out later that it is a Kagan cooperative learning structure called boss/secretary.

Student 1, Boss:  This student's job is to watch and tell. Student tells how to solve the problem, and describes each step with enough detail that the partner can get it written down. This student should be talking and watching, but not writing.

Student 2, Secretary:  This student's job is to listen and write. Student writes down the solution as the partner describes it. This student should be writing but not talking. The exception is if the partner gets stuck or makes a mistake. Then the student can coach and assist with the solving.

What I like about boss/secretary:  It can be done at any time at the drop of the hat. It can also last for any length of time. I will often stop in the middle of a lesson and check for understanding this way. I will just throw out two problems and tell the person sitting on the right (or wearing the most green, or the biggest feet, or the oldest . . .) to be solver of the first problem and then switch roles. I also just used it for a unit review that lasted a whole class period. We used white boards and I signed off on each section as it was completed.

What I don't like:  Students have to stick to their roles in order for it to be a meaningful interaction. I have seen students who are responsible for listening take charge of the problem, or students who are in charge of talking to take a passive role. It is helpful to have students switch roles with every problem.

I Need Adventure (also my trial run of Solve, Crumple, Toss)

Yesterday I tried another goodie from f(t) . . . Solve, Crumple, Toss. Just in case you've never read that post, make sure you read the part about how you should use "6-8 problems with somewhat lengthy solutions". I either skipped or disregarded that part of the description, and it was the downfall of the day.

My students are working on writing linear equations in standard form, given different types of information. I just printed up twelve problems, six per page. Students cut them apart, worked them out (one at a time) and brought them to me to check. If all was dandy, then they got to take a shot for points.  If not, I sent them on their way to keep trying and encouraged partners to check each other's wrong answers and help find the errors. I had no time to coach and help find errors myself, because I had a line of students waiting for me to check their answers. I expected them to finish all twelve problems and make twelve shots. Fine as long as I have a student shooting, like, every eleven seconds. Note to self: Fewer, more complex problems next time!!!

The student response ranged from a few who acted totally tortured because they had to get up out of their seat twelve times, all the way to enthusiastic participation by many. This was fun and productive, and there will definitely be a next time.

In related news, I realized how much more I enjoy my job when I am trying something new. I have been teaching the same classes in the same school for 6 years now, so it is super easy to just pull out what I did last year and do it again. I don't want to take that approach to teaching, so I am always looking for ways to improve on what I have already done. Still, I think my best lessons are the ones that I create from scratch with a fresh perspective.

So today I am just thinking about the payoff for extra time spent trying something new. Sometimes these things work, and sometimes they don't. Still, I am happiest when I am being adventurous.

I Stopped Answering Questions

Not really . . . but I did stop using 10-15 minutes at the beginning of class to discuss and answer questions on the previous day's assignment.

Initially, it was an experiment. I felt like students weren't really invested in this time, and that most of them were falling into one of four categories:

1.  The procrastinators:  These students were using this time to finish the assignment.
2.  The ones who lacked perseverance:  These students would encounter a challenging problem and then stop working on it (or not even attempt it in the first place) because they could just "ask about it in class".
3.  The ones who were really engaged:  Most days, it felt like maybe 2 kids.
4.  The ones who were bored:  These students had the assignment finished and were ready to move on to  a new lesson.

So I stopped spending time on questions. (My students have answers, so they can check for correctness as they practice). And this is what happened:

1.  Most of the procrastinators found a time to finish the assignment before class.
2.  Many more students persevered through challenging problems because they didn't have the crutch of asking about it later.
3.  Many with legitimate questions would drop by before school to ask. Most of our students arrive 30 minutes before first bell, so this works well at our school.
4.  Most everyone started finishing the assignment outside of class.

These outcomes alone were enough for me to turn my experiment into a permanent routine, but there was another benefit that I wasn't expecting . . . I suddenly had an extra 10 - 15 minutes in every class period. What can you do with an extra 10 - 15 minutes?! Here's how I use the extra time:

I use a few bell work problems every day (I am testing out a new bell work strategy, more on that later) to review and check for understanding. If there are any major misconceptions, I can usually identify and address them during this time.

While teaching a new lesson, I have a lot more time for practice and checking for understanding. I still do a lot of talking, but I also do a lot of pausing while students try this or that and check with me (or a partner). I have time to work in several mini-formative checks, and address common misconceptions. The result is fewer issues on the practice/assignment, which in turn further reduces the need for the question/discussion time at the beginning of class the next day.

Sometimes I still wonder if I should bring back the question time, structured differently to eliminate the problems I was having. I haven't done this because I don't miss it. And neither do my students. I realized today that in 2 or 3 years, I haven't had a single student complain about why I don't answer questions at the beginning of class.

Missing the Point

Recently, I wrote about my system for assigning practice. I give students an answer bank with every assignment so that they can find mistakes and revise their work.

I know why I do it this way. The purpose of these assignments is practice, so I want to give them feedback while they are working. I want students to keep practicing until they are doing it correctly. I want them to learn from their mistakes.

We have only been in school for 8 days, but I have spent a lot of time trying to communicate this to students. I keep getting responses that tell me students don't really get it, like these:

1.  Extreme excitement, because having the answers feels like legalized cheating.
2.  Confusion (usually from the high achieving students), because they don't think it is fair that everyone has an equal chance of getting all the answers right.
3.  Resistance, from students who feel like it is pointless to show their process since they already know the answer.

Practice is a process, dear students! I will assess you soon, I promise.

I am not sure how I can help my precious Algebra 2 students to understand.

It seems obvious (but not all inclusive) to use some sort of sports analogy:  If you are learning a new football play and you totally mess it up, the coach doesn't mark a D- in the grade book and call it a day, does he? No, he doesn't. He sends you back out there to try it again until you get it right.

That is all I have for now . . . I am going to keep working on it.

Sticker Survey

I have always done some kind of survey-type thing at the beginning of the year. My lofty goal is always to collect the data and learn valuable information about my students. Sadly, most years the surveys get filed away. I get busy with the planning of lessons and the grading of quizzes and such, and I never look at them.

This year I decided to put the survey questions on card stock and post them around the room. I gave each student a strip of stickers.

I think these two are interesting, placed right next to each other. (The grade I usually get in math is . . . , and My goal for a grade in math is . . . ) I assured them their responses would be anonymous.

This was kind of a last minute idea, so I had to work with what stickers I could find. I think next year I will try to find something real distinctive for each hour, so I can easily compare the data for a single class.

Anyhoo, it was a fun first day activity and it was definitely more useful than my old survey.

My Thoughts On Homework

Lately I have been thinking about how I assign, collect, and give points (or not) for homework. (For the record, when I say ‘homework’ I really mean ‘practice’. I want my students to practice every day. Sometimes they practice at home.)

I know a lot of bloggers have had success with not giving points for homework, but I am not ready to go there yet. I tried not giving points once during my first year of teaching and it was a disaster. Then again, lots of things during my first year of teaching were a disaster. But if giving students a score on a paper helps them to reflect on the quality of their efforts as they practice math, then I’m okay with using points.

So this is what I do . . . In bold is the thing I am trying to accomplish, and after that is how I attempt to make it happen.

1.     The perfect system emphasizes quality practice. Students need to reflect and make corrections as they are working:  I spend a lot of time teaching students how to practice. I want them to work out a problem, and then check the answer and find any mistakes and revise their work as needed. I hate to say just “show your work”. I do emphasize the importance of justifying your solution so that you can communicate to others how you found it, and prove it is correct. I make up a page of problems where some are perfect, others are missing work/justification, others have a wrong answer, and some have a right answer but the work/justification is incorrect or incomplete. Then students work with a partner to critique and discuss the quality of the practice. This takes time, but it is worth it.

2.     A good system gives students feedback while they are working, whether at home or at school. Ideally, they can find out if their answer is right or wrong without being told the actual correct answer: I put the answers to the problems in random order in the margin of the assignment. When students finish a problem, they find the answer in the margin and cross it out.  The only drawback here is that they can use the process of elimination to know the answer to the problems at the bottom of the page. Still, it works pretty well. I am thinking of tweaking this a little this year using the sum of a couple answers (sort of like Kate’s Add ‘em up). Instead of writing all the answers in the margin, I think I will try something like “the sum of #1 and #2 is _____”.  Then they will know if they need to fix their work without giving away any answers.

3.     If points are given, the points should reflect the quality of the practice vs. the number of answers that are correct on the first try:  A problem counts for points if it has correct work (or justification of some kind) leading to the correct answer, regardless of how many tries it takes you to get there. I don’t really even think of it as a “homework” grade. I want the point value to help students think about how well they are practicing. Hmm, maybe I will start calling it the “quality of practice” grade. Or something like that . . . I will have to think of something more catchy.

4.    If points are given, the system minimizes teacher time spent grading and recording:  My students spend so much time learning what good practice looks like that they know whether a problem they have finished qualifies. It has correct work leading to the correct answer and it counts, or it doesn’t. So students take the number of problems that qualify as good practice divided by the total number of problems times 5 (because I want a practice assignment with 20 short problems to have the same value as a practice assignment with 4 or 5 lengthy ones). I will even put that formula at the top of the paper to make it simple. Round the number to the nearest tenth and hand it in. Teacher records that number.

I am pretty happy with this system, but I still have a few problems. Sometimes students put a score on their paper that isn’t accurate (pretty easy to catch). Sometimes students rely too heavily on clues from the answers in the margin (maybe my little tweak will help that). Sometimes students copy their friend’s homework in the hallway before school (but at least they have to copy the work, too).

What's your homework system?

Common Core Response: Panic?

Last fall, when I heard about my state's adoption of the Common Core State Standards, my response was panic. But it wasn't the "Oh no, we have new standards" kind of panic. It was the "Oh no, we're gonna have a new assessment" kind of panic. Yes, it is true. I suffer from a fear of high stakes testing.

I set out to learn as much as possible about the new standards. I attended a couple of webinars, I enrolled in an online class, and I attended a two-day conference. I spent a lot of time during the month of June reading documents, writing papers, and participating in discussions. I felt like I was learning quite a bit, and I resolved to blog my thoughts.

Each time that I sat down to write about CCSS, I had nothing to write. I was completely blank. Weird. But I did notice that the panic was gone, so that is a good thing. The best I can do now is a bulleted list of random thoughts . . . which seem to have led me away from the panic I was feeling:

1.  The CCSS are not just about WHAT we teach, but also about HOW we teach. There are now 8 practice standards to guide our teaching methods. Students need to learn how to think, not just follow a set of steps. That is something I need to work on anyway.

2.  The new standards are more rigorous and more complete than our retired standards. In Kansas, our standards only went through the 10th grade. We've only been testing a limited number of these standards. A more rigorous update was needed, no question.

3.  The thing we want to know about the most (assessment!) is the thing about which we know the least. There are two companies working on the new assessment, and it is in the early stages of development.

4.  We most likely have 3 - 4 years before the new assessment rolls in.

5.  I know I am going to need to do some things differently, but three years is a relatively long time to make the necessary adjustments. I am going to make a list of things to change, and a time line for myself. One. Step. At. A. Time.

6.  I want to believe that if I work towards helping students be better thinkers, and make sure that the standards are included in my curriculum . . . if I focus on good teaching vs. state assessment preparing . . . that everything will be fine . . .

7.  I believe that this is a positive change, even though the transition will have its challenges.

8.  Panic isn't going to help.

Dear Summer, Please Slow Down

I can't believe that four of my precious summer weeks are gone! Here's what I've been doing:

1.  Moving:  My husband has taken a new job. He is leaving his position as principal of a small K-12 school to be an administrator in a larger district. So, we moved to a more centralized location between our schools. My last day of school was May 26th and we moved the very next day.  

2.  Unpacking, Sorting, Organizing, Tossing, Putting Away . . . On the bright side, our new home is located in a town 20 times larger than the old one. It has multiple grocery stores and multiple stop lights. Now we can walk to the park, dentist, hardware store, library, pool, and more. The mail is delivered right to our front door. Not to mention, there is a place down the street with half price ice cream on Tuesdays. We are living large, for sure.

3.  Recovering from a hail storm:  Five days after moving in, we had baseball-sized hail. I have never seen anything like it! We needed a new roof, new gutters, paint on the house . . . Oh, and new cars (both were totaled in the storm). Garages are only helpful in these situations if you park inside of them.

4.  My other full-time job:  Mama to this curly-headed cutie. She is almost 2. Full time job, indeed! She is growing up so fast. If some things on my summer list don't get finished because I spent too much time playing with her, I am not going to feel bad. 

5.  Common Core, Common Core, Common Core:  One of my biggest professional concerns has been our state's adoption of the CCSS. It means there will be a new assessment, when we are pretty comfortable with the current one. It means there will be changes to what I teach and how I teach it, but I am  not sure what kind of changes. I just finished an online course, and I will be attending a two-day conference this week. I plan to post some of the things I am learning. Soon, I hope.

Lost Treasure, Found! (a fun way to start class)

Most of my teaching career has taken place at my current school, but my husband's job took us away for a couple of years. During that time, I taught at Bulldog High.

There was a really great chemistry teacher at Bulldog High, Mr. S. He was a tough teacher, but the students loved him. He opened every class period with an enthusiastic class chant. If the classroom door was open, you could hear it all the way down the hall.

It started like this:
Mr. S:  Good morning Bulldogs!!!
Students:  Good morning Mr. S!!!
Mr. S:  How are you today?
Students:  Super great and getting better!

The chant when on and on for awhile. Over the years, students had suggested additions and it became a tradition in Mr. S's class. The students loved it!

Anyway, I really wanted my own chant for my class -- but that was Mr. S's thing and I didn't want to take away from its uniqueness. When I left that school, I asked Mr. S to write down the entire chant. I thought I would modify it for myself and start using it, since I'd be in a different school. I was super excited to do this!!!

Sadly, I lost my copy of the chant. Then I forgot it existed . . . until today! The hubs and I are packing up for another move. I found a small box, never unpacked, full of personal stuff I'd packed up from my Bulldog classroom. The chant was in that box!! Woo Hoo!

I am already thinking of how I can use it next year.

Here is the entire chemistry cheer, as written by Mr. S:

Mr. S:  Good morning bulldogs!
Students:  Good Morning Mr. S!
Mr. S: How are you today?
Students:  Super great and getting better!
Mr. S:  How far?
Students:  All the way.
Mr. S:  How much?
Students:  All of it.
Mr. S:  How many?
Students:  6.022 times 10 to the 23rd particles per mole
Mr. S:  What's our favorite class?
Students:  We love chemistry!
Mr. S:  What's our motto?
Students:  Can do!
Mr. S:  What's the truth of the matter?
Students:  The impossible takes a little longer.
Mr. S:  What's the platinum rule?
Students:  If you mess it up, clean it up.
Mr. S:  What's our philosophy of life?
Students:  WE NEVER GIVE UP!!!

So fun! Thank you, Mr. S!

Color Coding: For Sketching Piecewise

I tried a new (for me) approach for introducing piecewise functions in my Algebra 2 class, and it went over pretty well.  There is nothing really earth-shattering about this method, but it does involve color-coding -- and that is totally on my list of favorite things!

First of all, I am using only basic parent functions and their transformations that my students are already familiar with (linear, quadratic, and absolute value).  They do not need to plot points, because they already know how to sketch these graphs. I spent some time reviewing these before the lesson.

I start out with the idea of a restricted domain.  Students sketch the function, using the entire coordinate plane . . .

Then we worry about this "if" that comes after the function. Students color the restricted domain, and the corresponding portion of the x-axis.

Then they draw vertical lines to enclose the restricted area and shade it in completely.

And then they erase everything that is not in the restricted area.

We practiced these for a while before moving on to piecewise.

For piecewise, we shaded each restricted domain with a different color. Then shade in the corresponding restricted areas.

By the time we reached this point, most students could draw the graphs within the restricted areas without drawing the entire graph and erasing.

I've taught this lesson before, but pairing it with colored pencils was a first for me. I am pretty happy with the results, especially considering that a bunch of them are *DONE* and have started to shut down for summer.

Secret to motivating students this time of year, anyone?

Or, to get everyone to at least bring a pencil?

Just THREE!

I have narrowed my list of things to focus on for 2011-2012 to just three. This new list isn't elephant-sized at all. I am feeling much more optimistic, now that I have a plan.

1.  Making videos:  I am still completely intrigued by the concept of the inverted classroom, but I just can't wrap my brain around how that will work for my high school students. And I am not convinced that the payoff is there. I am not trashing the idea forever, but I have decided to postpone it in order to focus on higher priorities. In the mean time, I am going to be diligent about recording all of my direct instruction LIVE. This means I won't be spending any extra time outside of class making videos. I will be able to make these videos immediately available to students who were absent or who want to review the material. As an extra added bonus, I should end the year with a bank of videos for future use -- voila!

2.  Common Core:  One of the biggest concerns for myself and my colleagues is our state's adoption of Common Core standards. We are in a comfort zone with our current standards. We know how to prepare our students for our state assessment, and we are nervous about the transition to a new one. I am hoping to gather as much information as possible over the summer, so that we can start getting ready for the changeover. I am attending a 2-day conference, and I've enrolled in an online class through a local university. I am hoping that my colleagues and I can use our PLC time next year to do concept lists for our classes. I don't plan on making any major changes until then. One step at a time, right?

3.  Physics:  I have decided to let my growth as a physics teacher happen more naturally. I have added some physics blogs to my reader. I have started to tag some videos, demos, and lesson ideas as I find them. In general, I am looking for ways to enhance what I am already doing vs. a complete overhaul.

Speaking of three, that is how many weeks until summer!  13 school days, to be exact.

Gosh Darn It (A Cheesy Pep Talk To Myself)

To say that I am a perfectionist would be a ginormous understatement. Lately, I have been thinking about how I am probably way too hard on myself.


For example, our state assessment results were awesome this year -- 97% of our junior class was proficient! While I was ecstatic about those results, I kept thinking about the one student in my math strategies class whose score actually decreased from the first time he took the test. I felt responsible for that student, and I felt like I had failed with him. Never mind that I did all that I could, and this student made some choices that were out of my control.


Second example, at the end of every school year I am always looking back over the year for what things I want to do differently in the future. This always results in a list that is way too long for one human to complete, because I end up thinking that almost everything I am doing could be a little better.


I don't think that I am alone. I have read plenty of blogs, written by amazing teachers, talking about how they suck and how this lesson or that one was crap. I know it is just their way of saying "I am frustrated, and that lesson didn't go the way I wanted", and they are being honest about what that feels like. 


Obviously, there is nobody out there who has attained perfection. We all know that that we have strengths and weaknesses. We all have lessons that play out in less than perfect fashion.


So, here is a reminder to myself:  I don't suck. Really. Also, I wouldn't allow my students to say that about themselves. So I should try not to say that either, right? I know that there is always someone who is doing something better than me. I learn from that person. I teach because I care about kids, and I want the best for them.  I love the subjects I teach, and I am committed to continually learning. I love my job, and gosh darn it, I am good at what I do!

Improvement Overload: I'm There

I've been reading math blogs for about a year now. At first, it was a treasure trove of resources I never knew existed. I loved reading so many new thoughts and ideas. I tried some new things and a bunch of them went really well. It was fun. I wanted more and more!

But recently, my blogospheric bliss has turned into desire-to-improve overload.

I want to improve my this or that, or figure out a better way to do this other thing, and add more of x, and rearrange the order of how I teach such and such, and start getting ready for my state's new test . . .

I made a list, and the list is haunting me. I try to think about something on the list, and I just hit a wall.

Improvement is what we all strive for, right? But right now I am not feeling like I want to improve. Right now, I want to think that what I am already doing is okay the way that it is.

I am sure that I will get out of this funk, and figure out a balance. I will narrow my list to 2 or 3 things, and I will accept the rest 'as is' for now.

In the mean time, I need a break.

And then I need some motivation.

My Physics Island

I really love teaching physics. I like the fact that it is so easily relatable to everyday life. I tell my students that after you study physics, you will never see the world the same. Most of them believe that by the end of the year.

But there is a catch:  I teach physics in a small school where there is only one section of the class. We don't have a ton of lab materials, and I have no physics colleagues. In order for a small school like mine to have a physics teacher, they have to hire someone who can also teach other stuff. Enter: Me!

I tend to be more math-focused in my approach to physics. And sadly, I end up putting more energy into my classes that are state-assessed.

I don't think I have done a horrible job teaching physics. I mean, I have had my share of students go on to be successful in physics and engineering careers. One former student landed an internship at Cern physics laboratories in Switzerland.

But I do know that I am a better math teacher than a physics teacher. I also know that I have room for improvement. And it is hard to improve on this island with no colleagues.

Anyway, enough whining already. I do have this thing called the internet. Therefore, there is no excuse for me. The resources are out there, I know it. I must find them.

Next year, I am hoping to make better use of internet resources to kick it up a notch. Elephant List #1. I want more labs, but not just labs for the sake of labs. I want labs that produce "aha" moments for students. I want labs that can be set up with everyday materials. I have laptops, so computer simulations would be okay, too.

This search is long overdue.

If anyone out there has some ideas to help me get started, it would be much appreciated!

Conic Card Follow-Up

Recently, I wrote about my affection for Cindy Johnson's Conic Cards. I thought I would mention, like anything else stolen borrowed, I have added a few things to make it my own.

(This probably won't make any sense unless you have read that post.)

1.  Extra time:  I have found that following Cindy's outline, I ended up with 5-10 minutes left at the end of my 50 minute class periods. So I would put up an equation for whatever conic(s) they had learned so far and ask students to discuss with their partners to identify the vertex, center, major/minor axis, or whatever. Then I would randomly pick a color and have the students who worked with that color of cards that day answer the question. Since everyone knows they may be called on, they all talk to their partner and make sure they know the answer.

2.  Giving directions:  I found that it was really hard to give directions while students had cards in their hands. I didn't feel like they were listening because of all the sorting. Then again, I WANT them to be sorting. Instead of fighting it, I wrote out some directions (per Cindy's outline) so that they could go through at their own pace. I stopped them at a few points to write something down, but mostly they were able to figure things out on their own. Here are those sheets:

Parabolas (practice at the end is Cindy's)
Circles (practice at the end is Cindy's)
Ellipses (I combined Day 3 and Day 4 from Cindy's outline)
Sketch Parabolas, Circles, Ellipses
Hyperbolas

3.  Green Globs:  This is some software our school has purchased, and we like it. For conics, students type an equation to try to match a graph that is given. If they are correct, they advance to the next level. If not, they can see the graph of their incorrect guess so that they can figure out what adjustments to make. Games for linear equations are also included.

Thinking about the Flip

Item #2 on my Elephant list:  Experiment with flipped/inverted classroom.

Just a few months ago, I read about the idea of the inverted or "flipped" classroom for the first time. Since then, I have noticed that a bunch of people in the blogosphere are talking about it. I am completely intrigued by the idea. Instead of giving direct instruction and sending kids home to practice, you assign a video as homework and do the practice in class.

One thing I love about this idea is that it doesn't feel like it would be that big of a stretch for my teaching style. I have been recording videos of lessons for several years now, using an interwrite pad:

This device allows me to record my voice and handwriting as I teach. I can record an explanation live, and make it available to students who were absent. I have also left videos for a substitute when I was gone for the day. Once, when I had a terrible sore throat and could barely talk, I played a pre-recorded video in class to save myself from the talking. The thought never occurred to me to record a lecture and assign it as homework. But why wouldn't that work? I want to try it.

Here are a few of my concerns:

1.  I am not sure how to make the videos accessible to all students.

2.  I don't know if I have enough time to record a bunch of videos.

3.  I don't know how to make videos anymore. I know I said I've been making videos like this for years, but now that we are in a new school I have a completely new set of audio/visual gadgets to figure out. My interwrite pad has been replaced with a smart slate. It is supposed to be the same, but for me it hasn't been as user-friendly.

4.  What if a student shows up to class without having completed the homework?

To help process my thoughts, I had a discussion with my current calculus class about the idea. I was just curious what they would think. In general, they weren't thrilled. The main concern they had was that they want to be able to ask questions if they don't understand something in the lecture. They are also worried about accessibility of the videos, as some of them do homework during their break at work, or riding the bus to and from a sports event. I was sort of surprised. I kind of expected them to be more excited about the idea.

One of my students said "What is the point?". He wasn't being rude, he legitimately wondered what was the advantage of flipping. Good question. My answer was that it takes the part of the lesson that students are most likely to need help with, and puts them in the classroom with teacher and peer support while they are doing that.

But now I am starting to question it myself. Why? I don't want to do something just because it is new and interesting and it seems like it would work. Will it truly enhance learning? I think the key is going to be a combination of quality videos AND how I choose to structure the practice time in class. If all I am going to do is say, "Any questions on the video? Okay, here's your practice", then it probably isn't anymore affective than teaching the traditional way.

I still plan on experimenting with the flip in calculus next year. I can see that it might be a harder sell than I originally thought. . .

Cake Day in Calculus

If you teach calculus, you probably can't look at one of these without seeing a volume of rotation.

I presented this as a problem to my calc class, after introducing volumes by the disk and shell method. I gave each student a sheet of cm graph paper, and a slice of cake:

Students drew the volume of rotation and found the outer and inner radii. It wasn't too challenging for them, since the cake slice was just a rectangle. This one was my attempt to do the problem with them:

To check for accuracy, I filled up the original cake pan with water.

The result:  1700 mL.  I calculated 2412.7 mL.  I was really hoping for more accurate results. I am not sure what went wrong other than the sides of the pan were a little slanted, and we treated them like they were vertical. 

Then, to practice the shell method, we used one of these guys:

Here is one student sketching out the cake.

Students tried to use a parabola to model the shape of the cake, which seemed like a good choice. They knew how to use transformations to flip the parabola and translate it to the right location, but they didn't know how to adjust the width. They ended up choosing a random fraction like 1/2 or 1/3 because they knew a fraction in front of x^2 would make it wider.  But the results were no where close to the actual cake volume.

Then again, we were running out of time. It was right before lunch. Everyone was hungry. The room smelled like cake. And everybody wanted to eat cake more than they cared about the exact formula for this parabola, or even if it should have been a parabola.

I like the concept of this activity, but I would like to figure out what adjustments to make to get more accurate results next time. I probably just didn't give my students enough time and/or resources to really figure out the right equation for the cross-section.

In the end, the cake was good. We did some math. And everyone thanked me for having a cake day in calculus. So it wasn't a total disaster.

Crazy for Conic Cards!

I just finished another one of my favorite units: Conic Sections! I used to hate them, but now I love them (and so do my students). And it is all thanks to Cindy Johnson for sharing her conic section cards at an NCTM conference a few years ago. Her cards did more than enhance my unit on conics, they completely revolutionized the way that I teach this particular topic! Students learn by identifying patterns, not laboring over tedious formulas. Learning conics has never been so fun and painless.

Here is the basic idea:  You have a bunch of decks of cards (hopefully, you have a student aide to copy, laminate, and cut them for you). Each deck contains 20 equation cards (5 for each conic), 20 information cards, and 20 graph cards.  They are corresponding so that students can match each equation to its information and graph.  There are also four title cards (with the words Parabola, Circle, Ellipse, and Hyperbola) and eight formula/reference cards (with all the a's, b's, h's, and k's explained).

Each day students learn the characteristics of a new conic. Then they separate, sort, and match the corresponding cards. Each deck is different, so they work with different cards each day. One of my students says to me, "I love this, I wish we could learn all our math with decks of cards".

Each card has a letter, number, or symbol in the corner. There is a key for each deck so you can check for correctness at a glance.

At the end of the investigation (which takes 6-8 days), students can identify conics along with their vertices, opening, center, radius, major/minor axes, and asymptotes. And they can sketch them.

I don't go into any more depth than that at the Algebra 2 level. I think advanced students could do the matching more quickly, and you could follow up with some more in-depth study of all the formulas for the formulas. At our school, I leave that to the Precalc teacher.

I have had some contact with Cindy since NCTM, and recently I asked about her policy for sharing the cards. I have no desire to take credit for Cindy's great idea, I just want to help spread it far and wide so that others can benefit the way my students and I have. She said I could share her email, so here it is. You can send her a note to request the Conic Card files. Thanks a million times, Cindy!

Update August 1, 2014:  Cindy's cards are now available on google drive!

Click Here for Conic Cards

Also, I am linking a follow-up post.

A New Elephant

This is the time of the year when I finally feel like I can breathe a little . . . State assessments are over. The results were great, what was I worried about? (Well, there was one oh-so-frustrating exception that I want to blog about, but shouldn't.) And, I have my plans pretty much laid out for the rest of the year.

With all these things squared away, I really start to reflect on how the year has been. I try to focus on some areas to improve for next year, or just some new things that I would like to try. Then I make a long list, and delete some stuff until it feels halfway reasonable. I will get as much of these done as possible before school is out in seven weeks, do some over the summer, and then work on the rest during the next year.

Here is my list, and also possibly the titles of my next ten posts:

Physics:
1. More meaningful lab experiences.

Calculus:
2. Video lectures and experimenting with inverted classroom.

Algebra 2:
3. Look for ways to go more in-depth with fewer topics (like this year with logs)
4. More effective use of ACT practice questions.
5. Look at order of topics (parent functions/transformations first?).
6. Look at how I teach/review factoring.
7. Make sure all topics are aligned to College Readiness/ACT Standards.
8. Look at homework collection/grading procedures.

All:
9. Learn more about Common Core Standards, recently adopted in Kansas.
10. Work on atmosphere of partner cooperation/peer tutoring.

Despite my efforts to edit, the list is still a bit overwhelming.

This calls for the elephant-eating approach:

One. Bite. At. A. Time.

Crossing My Fingers, Knocking On Wood

Why, oh why, does this make me so nervous?

In Kansas, we only test students once at the high school level. It is up to schools to decide when they feel students are ready to test. At our school, we test freshmen who are in geometry (our most advanced students), and all the sophomores who haven't already tested.

If a student doesn't reach proficient level, we can remediate and have them test again. We use our math strategies class (junior year) for remediation and retesting.

All scores are banked until a particular class's junior year, then count together for that class. We don't know our complete results for a class until after the retesting takes place. That is happening this week.

Our school has received Standard of Excellence for five years in a row. This involves meeting AYP requirements for a percentage of students reaching the proficient level, but it also means that you must have at least 15% of the class in exemplary level and no more than 15% in academic warning.

We feel pressure to continue to achieve that level of performance.

The whole thing turns into an agonizing numbers game:

1.  This year's freshman class is 2 students short of the 15% exemplary. We have already tested the most advanced students, so we'll have to pick up a couple more exemplary from next year's sophomores in geometry.

2.  This year's sophomore class has just the right number of exemplary students, but what if a bunch of new students enroll and we need another 1 or 2 to reach the 15%?

3.  This year's junior class has enough exemplary, but needs 5 more proficient. Those students who did not reach proficient the first try will be retesting this week in their math strategies classes. There are 17 of them, and we're pretty sure there will be more than 5 who make it. So all is probably well for this year, but we'll know for sure by the end of this week.

I hate that we have to do this.

Not the testing. I am okay with that, for the most part. I hate the counting of students and the calculating of percentages, and the worrying that we might be one short of the goal. And the feeling that I am not teaching math as much as I am teaching strategies. And the feeling that you have done all that you can and it might not be enough. I worry too much, I guess.

At this point, I'm just hoping for the best.

Puzzles!

I love love love puzzles! And I forgot all about this fun puzzle (not my original idea, but I cannot remember where it came from) until recently. I made some of these a few years ago, and we just pulled them out for our math strategies classes to review a few of the tested standards for the Kansas 10th grade math assessment.

Here is how it works:  You start with a template that looks like this, or you could make your own by creating a table in word.

Each border between shapes is used for a problem and answer, or two pieces of matching information. This one shows the names of properties and corresponding examples.

I usually tell students which piece goes in the middle, to help them begin.

Then students just have to match the edges like a puzzle.

I've written a few of these, and I have learned there is a ton of potential for varying the difficulty level.  You can write distracting answers along the outer edges to make it more challenging, or not.  You could tell students which piece goes in the middle to help them get started, or not.  You could white out the happy faces and write in different directions so that students don't know which side is "up".  You can repeat answers, or not. I recommend trying to solve the puzzle yourself before you give it to the students, though. The puzzles I wrote ended up having a lot of variation in difficulty without my even realizing.

Here is the template.