More Thoughts on Geometry Proofs

What does a math teacher think about on the plane while she's traveling to see her dad in another state? Teaching, of course. I'm still working out how I want to start teaching proofs in geometry this coming year, and I had 2 extra ideas that I haven't quite fleshed out yet.

One idea came from a news article I saw the other day. It was something to the effect of school officials allowing nurses to hand out condoms to elementary school kids. The original article JUST stated that they're going to start making them available for K-5, and if my memory serves me correctly, that was ALL the article stated. Well, of course, it sounds so ridiculous and my 1st thought is, "crazy! why?", etc. Then I had to step back and think, what could be the real story behind this, and I came up with a few possibilities.

I discussed it with my friends, and sure enough, a few days later the WHOLE story came out. The school district was making condoms available to any student of ANY grade that asked for them, and I'm guessing that in their minds they were mostly thinking about high school kids. Then maybe the newspaper people got the story and JUST reported that the condoms were now available to elementary kids.

Anyway. What does this have to do with geometry? Here's my thinking. One reason to teach proof in geometry is to get kids to think logically and to think of all (groups of) events or statements that can lead to other events/statements. In this way, they won't be "victims" of people trying to sell them one version of a situation. So I thought this story would be a perfect example of just such a situation.

Somehow I'll begin the proof unit discussing why we learn them and just show them the 1st news story. Then I'll have them get with a partner or in a group and think of what events may have lead up to this article or that may have justified such a move by the school board. Then show them the 2nd article. Then maybe I'll find a provocative picture (in the sense of it may look like something bad or good is happening at 1st glance), and then have them think of things that may lead to this picture (or result from the picture scenario). Anyway, like I said, it's not fleshed out yet, but I think this will be a good start.

My 2nd idea has to do with Excel programming. I don't know if I can pull it off, but it seems like you can do some cool stuff with Excel if you learn a bit of VBA programming, and I'd like to create some sort of interactive proof-filler-outer that will be a quiz for them. If it's easy, then I can pop out several and have them online for the kids to practice with. I'm guessing that's a big if.

Anyhow, vacation with dad is over, and now I have a glorious week to do mathy stuff ... before my next vacation trip. WooHoo! Maui here I come (me and my fish-white belly). I'm guessing I'll look like a pale white tourist, but guess what? That's because I AM a tourist. Maybe I'll just have to drink a lot to be okay with it :).

Meeting Potential Students Outside of School

I started taking Lindy Hop Lessons a couple of months ago because I saw some people doing it, and it looked SO fun. The nice thing about it is that you don't have to be a couple to take lessons, and even if you come in as a couple, you rotate partners frequently in class, so it's not an issue.

Since the "leaders" (guys) and the "followers" (girls) are learning the same steps at the same time, and so we all know what step we'll be doing next, it's hard for me as a follower sometimes to "listen" to the pressure of the hand of the leader to determine what to do next, since I know what we're "supposed" to do, so I just do it (did that make ANY sense)??? Anyway, as with everything, there are good leaders and not-yet-perfected leaders (just as I'm a so-so follower). So, I always like when I rotate to a couple of particular guys because they have that leader thing down. They've got the hand pressure and rotation, and so I know to turn or move back or forward or whatever.

Last class, I was with a great leader, and we were learning a new step where I had to "turn outside". He had to lead this by doing a certain thing with his hand, which I had heard from ANOTHER dance class described as, "just like you're drinking a beer", since he sort of starts with his hand held high up and then tilts it in the same direction as if he's drinking a beer. So I turned to my "great leader" guy and felt I had to share this with him, and after the teacher showed the move, I softly said, "ah, just like you're drinking a beer." My partner looked at me and whispered, "I'm 16." Awkward. I amended it to "milk" and then had the scary thought, oh! he could be my son ... or student ...

Spring (or Decade) Cleaning

I'm not the best of housekeepers, and that's an understatement. Sure, the dishes are done and laundry done and trash put away, but there is dusting and vacuuming and clutter cleanup that only seems to get done when company is coming over - which is about twice a year or so. Ew, put that way, I'm gross.

Anyway, since my inclination is to sit around all summer and just read (this or this or this) or nap or do puzzles, I thought I'd give the house a once over this once.

Tons of old books will be donated to the library, usable clothes to charity, and our own personal landfill of electronics to Goodwill. We've just gotten in this complacent state of clutter all over the house, and I guess we don't see it any more. But now that I've cleaned 3 rooms (so far) of excess and unnecessary things we don't even use anymore, it looks so peaceful and inviting instead of this underlying current of unease and discontentedness.

Whew! I should have people over to celebrate.

Adding & Sutracting Integers

I talked with a math teacher friend today about how she teaches adding and subtracting integers to her 7th graders. She mentioned the following type of method, and I decided to put it to pictures. I'm going to try it next year with my high school kids who STILL mix things up. Maybe this will do the trick.

What's My Grade??

Probably like most teachers/schools, our grades are weighted for the grading period. Last year it was 45% homework and 55% tests/quizzes. Students always seem in such wide-eyed wonder as to how various pieces affect their grades. I think they're under the total points worth mentality, and this doesn't always translate into good grades.

Also, during the 1st day of school, I always have them do some math activity to get them thinking while I'm doing all my administrative stuff (roll call, seating chart, ...). So, VOILA, I decided next year to combine my 2 tasks into one. I have a draft of the "calculate your grade" assignment that I will hand out to all my students on the 1st day. Any suggestions on additions or revisions? I kind of hate to leave it at 3 pages since then I have a blank back page. Ideas?

Homework...

I've been thinking more lately about how I "do" homework. I'm still in the camp that wants to assign a completion grade only, so that they don't get penalized for making mistakes while they're learning. I see how many problems they attempt, and how many they grade in class as per my instructions, and I give them a grade accordingly. I still like to provide some kind of answer bank, so that they get immediate feedback and are more apt to persevere through a problem if it's wrong. And yet, I still understand that for some kids, this completion grade leads to laziness and lack of an honest effort and potentially to not grasping topics.

I have a thought rumbling around in my head of something I may want to add to the grade. I want there to be an EXTRA question group on EVERY homework that MUST be answered in order to get full credit (any credit?). Here it is:

1. What is a problem you had most trouble with?

2. What specifically is wrong, examples:
* I don't know how to start
* I don't understand some words
* I get stuck at the point ______
* I'm messing up with my calculations somewhere
* I ______

3. Write out which of the following you did:
* I looked through the book and found extra examples
* I went to our online book to see if there was a tutorial on this problem
* I went on Ms. Cookie's website to see if she had links that may help
* I looked through my notes to see if we did something like this in class
* I called/texted/e-mailed a friend and got help
* I did ______ and it helped
* I did nothing and waited until the next day in class to ask questions

4. Describe what progress you made after you did this, examples of what could happen:
* I didn't do anything, so I made no progress
* I solved the problem
* I understood more about what part confused me
* I made progress in the problem but still did not solve it (describe progress)

Maybe I'll print this out on pretty paper and have them tape it in their composition books as a reference. In this way, it'll cause the kids to reflect nightly (or the period before-ly) on their homework. My lists will hopefully also serve the purpose of prompting them to find other sources to persevere before giving up. And giving a grade for this portion will maybe be the incentive to do the reflection.

Which brings me to my next change next year. I'm a recent CONVERT of composition notebooks. Before I just let the kids buy whatever 3-ring binder or spiral notebook or whatever they wanted. At my not-so-new school they had a tradition of requiring composition notebooks for other classes, and the kids would tape pictures to the front that represented whatever class it was for, and the teacher would put packing tape over the cover to preserve it. Now all the notes are in the book and not easily lost and if there were handouts, the kids taped them to the appropriate dated page. I didn't do it this year, but ... Love it! I believe I'll require them to get the graph papered composition notebook. Then POOF! instant graph paper on demand.

Geometry Proofs

As I still have momentum from the previous year, I thought I'd get some ideas down for next year. I believe I'm teaching 2 levels of geometry (in addition to the PLTW IED course). I know how hard it is for kids to get up to speed with the flow proofs we teach, so I thought I'd scaffold it more this year, and potentially start with the following idea:



After a few (2?) of these, I'd try to incorporate a math story; then take away the "skeleton" at the bottom of the page for another story (but keep the statements), so they had to create the flow. Then start taking away more and more and finally bringing it to just a geometry problem statement. I'll have to flesh it out more, but that's the seed of my idea.

I like the flow proof style instead of the 2-column proof because there is a visual connection between what follows from what. I also like teaching proofs because it's a challenge for the kids and stretches their brains. It's also the first time (only time?) they see what it means to actually prove something instead of just give a convincing argument.

How to make a nice graph for tests and such

Someone requested a tutorial for making a usable graph for your worksheets, so here it is. (okay, the person nicely agreed to a tutorial after I was pushy and suggested one). Here's the 3 page download of my instructions.

And here's what it looks like:

A Surprising Finals Question (Apparently)

I gave the following 2 sets of questions on my algebra 1 final (click on it to enlarge):



For 13 & 14, they were given a line and were to draw a parallel line going through (4,1) then give me the equation in point-slope form. For these 2 questions, people generally did fine.

For 15 & 16, I had the same original line drawn, and they were supposed to draw the perpendicular line through (4,1) and then give the equation in point-slope form. This is the set of problems they had problems with.

I thought I was giving them a gentle teacher nudge by having them draw the picture before they wrote the equation. The original slope is -2/3. Holy Moly. So many students thought NOTHING of the fact that the line they drew was NOT perpendicular looking. They INSISTED the slope of the perpendicular line had to be 2/3, and by Golly, that's the line they drew. 90 degrees? eh!

Note to self, when I teach algebra 1 again, I have to do more of this type of problem before we get to finals. I never did do the drawing thing in conjunction with just the "what's the slope/equation/etc of the perpendicular line to ..." type of question. Apparently, I should have.

On a positive note, I'll be teaching them geometry next year, so guess what we'll be "reviewing".

Surveying the Students

We're supposed to survey our students at the end of the year to get feedback on how class went. I've seen in the blogging world that others do this, and I've done it periodically in the past on my own. At school I asked if there was a standard document, and there wasn't so I made up the following and have since passed it out to 3 of my 4 "real" math classes.

Before I let them quietly work on it for about 10 minutes, I had a discussion about how in life it's a useful skill to be able to give constructive feedback. I said that they'd be bosses or employees or students or teachers or whatever, and they wanted to help "improve" or "praise" someone else's work effectively, and we talked about good and bad ways of doing that. Anyway, I liked their comments. Some things that were good for me to know:

* "Don't just always joke around with 1 or 2 students; make sure you joke around with everyone to make them feel special.". This was good for me to hear. Usually, I just engage with kids that engage with me, and then I feel comfortable continuing that "razzing" throughout the year. I can see how it would feel exclusionary to others instead of, ha ha, I'm a part of this if only as an observer.

* "I like that you just take homework for a completion grade. That way I don't feel pressured when I'm trying to learn." Hmmmmm, that was my intent, but it has its flip side for some students and some years when they slack off because they think it doesn't count.

* "I like when you provided answer banks on the homework. That gave me instant feedback." me, too.

* and various other comments that reaffirmed what I thought didn't go well or did.

Here is the download of this sheet: (p.s. thank you to the commenter who taught me about Screen Print. I'm ALL about Screen Print these days :))

Quadratic Formula Dialog Update

Well, the kids are starting to come in to take their "test" of the dialog I mentioned a few posts ago. They're doing that AND multitasking by whining in various ways about why they have to do it. I good-naturedly hold my ground and explain to them why (the dialog has lines that address 3 key sources of mistakes for them; the memorization and repetitive stress of remembering it will allow it to potentially stick in their heads).

I've had about 8 students do the dialogue, so far. I've also given them "inspiration" by putting a temporary "msg" in the electronic grade book. This is "missing" and has the same effect on their grade as a zero. Whew you should have seen the grades plummet.

Anyway, I've also had some kids in one class tell me they're nervous and can they do the whole dialog by themselves or can they do it with another student while I watch (because apparently I'm intimidating ... who knew). So I made up a FAQ and mass e-mailed it to them:

1. When is it due?
Let's say by this Friday, so that if you don't finish it, I have time to nag you next week (2 days only!)

2. I'm nervous and Ms. Cookie is intimidating, can I do the dialog with someone else and we switch roles?
Yes, and you can get full credit if there are no major mistakes.

3. Can I do the WHOLE dialog by myself playing both parts?
Yes, but Ms. Cookie will have to take off some points (can still earn in the high 90's)

4. Ooh, I don't know the whole thing really well, but I know most of it, so I don't want to come do it.
Well, Ms. Cookie can prompt you .... she'll take off some points depending on how well you do, but it'll be a WAY better grade than a zero.

5. EEK! Nervous.
Well, the lowest grade you'll get is a 70% if Ms. Cookie feels you don't know it THAT well (but most of it), or a 0% if you never come in.
Do the math :)

6. Why do we have to do this dialog?
The dialog has some key points on things students mess up on. Also, when you memorize something and have to do it over and over again, it sticks in your head. Thus, using my professional judgment, it's good for you and your quadratic formula use knowledge (like math vitamins) even though it's stressful. AND you'll be proud of yourself for doing something you thought was not possible.

7. Has anyone taken it yet?
As of 1:30pm Monday, 5/24/10, 8 people have taken and passed it (lowest score 80%).

Best Interpretation of the Dialog: J./M. & G.
First Done: D. W.
Person Who Overcame Their Most Stress So Far: C. J.

Final Review For Algebra Finals. Finally.

I can't believe finals start this week for our freshmen. Let's see, so far this semester they've had the LTF tests (Laying the Foundation), the EOC tests (end of course), the TAKS test ("exit" exam test), and now finals! I know they're excited and will be going through test withdrawals this summer. Maybe I can pay home visits and test them on various things.

Anyhow, here's the review I'll give them this week:

All About Me.....

Every year without fail one or 20 students will ask me what kind of music I listen to. I'm always stuck for an answer because I like a ton of different kind of music depending on the mood I'm in ... maybe that's the way for everyone. Also, I don't listen to music much other than the odd time in the car when I'm not listening to my books on CD (Love you Simply Audiobooks), or when I'm cooking dinner and need to rock out to Amy Winehouse. I don't have an iPod. I run early in the morning when it's still dark, so I'm more listening to the sound of deer that may crash into me or the stray car that may or the other runners going in the opposite direction who, like me, aren't wearing reflective gear. But I digress.

Well, must be a slow learner, since I should know this question is coming up every time I meet/teach new kids. Anyway, that leads me to my idea I just implemented that will allow me to send the students away and answer their questions at the same time. And as a bonus, I have a music play list in case my ears are ever hungry for tunes while I'm on the computer.

I have a website for my students where I post the homework using GoogleCalendar and I have a recap of what we learned that week with links to various helpful websites or practice problems or tutoring sites that are related to the topics. In addition, I have a little click/link on my weebly site that sums up my life in a few paragraphs (another question from the kids). So anyway, on this last link, I just created a short list, that I can add to later, of songs/artists I like that are links to YouTube videos. It looks like this, and it was a ton of fun to make.

Time to Review

I'm having fun looking up all these crazy facts to use for my review sheets. It gives the kids maybe a wee bit more motivation to actually review. I'm also trying something new. For each question, I've referred them back to a helpful page in the book. Maybe that will cure the, "I didn't know how to do it" syndrome, and the "oh! I didn't THINK to look in my notes" malady.

Okay, first stop: review of lines and slopes. Also, does ANYONE know how to stop there from being a 2nd page in a word document if you create a table at the bottom of the page. My version seems to want to start a new page, and I can't get rid of it. I remember I fixed it in the past, and I know I can browse and probably find the answer myself, but ....

Ahhhh Finals Time

There's no better way to spend your Sunday than getting finals and final reviews and final review study ideas written at the last minute. Good Times. Of course now that I know you can watch Netflix movies (or seasons of TV shows) on line, maybe I need little breaks in between. You know, to refresh my brain and be more productive.

Anyway, here's what I'm handing my algebra students tomorrow. Hopefully, they'll look at all the topics and be motivated to study (after they want to curl up in a little ball and rock back and forth). Also, hopefully when they do the calculations of their grades on the 2nd page, that will be another incentive to work hard.

Gush Alert...

I am a lucky teacher. There is such a gi-normous difference between my teaching experience last year at my old school and my time this year at my current school. Sure I had a rough 1st semester getting on board with all the new ways of doing things and all the higher expectations and all the students sort of eying you to see what type of person/teacher you are before they decide to like you and play along with learning. Sure I've had many times this year (in the last couple of weeks especially) when I'm asking myself, they want me to do WHAT now in addition to the other 5 bazillion things I'm doing??? Sure I've been tossed into the pool and told to swim and figure things out and then jump back out, dry off, and assess how things went and make notes for the next time we do something new we haven't done before.

But.

I'm loving it. Maybe it's analogous to life experiences in that you ultimately are more satisfied with something you've had to work hard for. Maybe you're more proud of yourself when you've agreed to try new things and not come out any worse for wear. Maybe I just work with phenomenal people that push me every day to be a better teacher/person/employee. In any case. Gush. Gush. Gush. Okay, hope you're through being queasy.

And 3 more off-this-topic observations I've been mulling around in my head.

1. A while ago I blogged about a student cheater. I don't know if I mentioned it, but this student was caught cheating on another test. This was last semester. I never mentioned anything to her, and things settled down, and I'm in general very present during tests and walk around and check up on students. Anyway. Can I just tell you I'm glad I didn't severely damage my relations with this kid by saying something that would stick in both of our minds. She has turned out to be a joy and is one of my favorites now. She makes me laugh constantly, and I'm always happy to see her. Another instance where things are not black and white, and are ever-shifting.

2. A shorter while ago I blogged about a student who had attendance issues and was frustrating me with her lack of math progress. Well, this is a poised kid who excels at other things, and I've had opportunities to see her in action and was impressed and told her so. So I guess this is an instance to remind me that I'm not just teaching math robots who only exist to come to my class and churn out math; they are well-rounded and just like the rest of us are trying to juggle many things and growing up (keep reminding myself that they're only 14-15 years old) and learning how to do things and do them successfully.

3. There are the types of people who feel they have to ask permission for everything (not talking etiquette here), and there are the types of people who just push on ahead and do things with the mindset that no one's the boss of them. Example, some of our teachers went to a meeting the other week where there were people from various schools. This one teacher was asking for help and suggestions on how to set up field trips and transportation and logistics of visiting various colleges. She seemed to be flustered by the whole idea. Our teachers just looked at each other because we had just done that with a set of our kids. Hey! we just contacted people and transportation and figured out the logistics on our own. Just get it done. Most likely your common sense or perseverance will allow you to figure things out. And if not, you'll learn from it and do better next time.

Shifting of Parabola Functions

I don't know how your school year is ending up, but at our school it's CRAZY CRAZY CRAZY. Practically every day the schedule is altered in some way, and we're all just practicing our deep breaths and our focus of just thinking of one wacky day at a time. Because of this I wanted a lesson that discussed shifting of graphs so that the kids could work independently in case I saw different classes for different amounts of time. Now I'm thinking I should do this more often. I really know I should, but other things get in the way.

Anyway, I like how this worked out. I told the kids I wouldn't help them and they had to work in their groups to figure things out while I walked around. I told them they could ONLY work on the 1st page and we'd talk before they went on. I walked around and cleared up misconceptions. We got through about 3 pages in a wacky 75 minute class after doing homework checks and such.

Here's the 1st & 4th pages: (and here is the whole packet of 4 pages)

Adding & Subtracting Radicals

I teach 3 different classes of algebra 1 preAP. My poor 1st class gets all my mistakes and things I test out. My 2nd class gets the refinement, but still not "the best" I'm going to offer this round. By my 3rd class, I've had time to mull over ideas and see what works and doesn't work and anticipate their problems. So... Here's the 1st page of what my 3rd class is going to get on Monday when we learn adding and subtracting radicals. You don't even want to SEE what I subjected my 1st period to. (here's the whole 2 pages).

Circles and Tangents and Horizons

We're on the "tangents to circles" section in geometry, and I'd assigned a standard problem of "the satellite is ____ miles from the earth ... how far is it from the horizon?" Too many kids drew goofy pictures and didn't get the concept, so I started class on Friday with the following task.

1. We had a discussion about how when you're at the (ocean) beach, and you can see the horizon, maybe you wonder how far away it is .... any guesses? And do you think tall people and short people will see the same distance away?

2. Then I had them draw the "circle" earth, and we discussed drawing them standing on the earth (as a math stick figure with a big eyeball dot on top) and made sure they were standing up straight (an extension to the radius of the earth).

3. Then I asked them to just PLACE their ruler as if it was their line of sight to the horizon and walked around to make sure they knew what they were doing and corrected some misconceptions.

4. After they drew the line of sight to the horizon, they had "AH!"s of the right triangle possibility of finding the answer.

5. I asked them what information they knew or could find out, and we looked up the radius of the earth in miles on the Internet, and we recalled the number of feet in a mile.

6. Then before I set them on their way, I asked that if I was 5'4", how many feet tall am I and show me on your calculator as I walk around. I heard good conversations and them correcting each other. Then I got another kid to tell her height, and we did the same thing to make sure everyone got it.

7. Then I set them on their way to find how far each of them individually with THEIR OWN HEIGHT could see to the horizon. As I walked around, I saw too many kids doing everything in feet (because "I don't know HOW TO DO IT in miles")... they converted the radius of the earth (about 3960 miles) to feet!!! We had a discussion about how these are too large of numbers.

8. I stopped the class temporarily and did a mini-lesson. Let's say I have 53", how many feet is that. What about 38 feet, how many yards is that. Okay, let's say I have 5.25 feet, how many miles is that? They caught on. Then we had to have another mini-lesson on using our store button on the calculator instead of rounding early or writing EVERY BLESSES DIGIT out on paper and then retyping it in.

9. THEN when they'd calculated the length from their eyeball to the horizon, we asked the shortest kid what their answer was, and the tallest kid their answer. And we had to fix some kids' work and computation until they did it correctly.

10. Interesting span of possibilities of how far we all could see .... which I won't divulge in case you want to try it.

Graphing Parabolas

I had really good luck with this sheet when we were practicing graphing parabolas today. I apparently haven't been doing enough graphing in class because I still got a handful of questions during class, "how do I find points on the graph". Scary. Anyway, I'm not too worried because with other topics earlier in the year that they couldn't do, now they're great at them (like terms, solving for x, ...). We only got through this front side because of all the other extra administrative stuff we needed to take care of.

What I like about having a preprinted sheet is that I can stop them at each step and have a discussion: why is it called the axis of symmetry? how do you know what x value to plug in for the vertex? what kind of shape is the axis of symmetry (a point or a line or other?), how do you know how to find the y-intercept? why? does every parabola have a y-intercept? (that brought surprising answers when I asked for thumbs up for yes and down for no) ....