Saturday, April 30, 2011

Toy Wheel as Intro to Circular Relationships

I built a little toy wheel out of recycled cardboard box, wood glue, some staples, and a paper clip axle. I think it's going to be helpful when we discuss the relationships among rotations through a fraction of a circle, degrees of central angles, and arc lengths.

I also made the wheel 3-D (with thickness of about 1 inch) so that we will be able to easily wrap some yarn around it to verify our conjectures about the relationships between arc lengths greater than the circumference and the number of revolutions and the degrees traveled.

Excited! Love tactile toys! (Even simple ones that don't look flashy and that I build out of empty cardboard boxes after kicking out a family of spiders previously residing in the box.)

Friday, April 29, 2011

Magazine Project Topics

So far, here are the topics my kids have chosen for their Geometry magazine articles:
1 group is doing an article on fiberoptic cables -- their history and also how they are an application of the concept of reflection.

1 group is doing an article on geometric paradoxes. Even though I was picturing them doing paradoxes such as how the Koch fractal is a finite area with an infinite perimeter, for now they're focusing on visual illusions that show that when you re-arrange puzzle pieces to make the same shape, there is a gap in the new "congruent" shape. They're going to use examples from the web, but add their own explanations for why the "illusion" works.

1 group is researching perspective drawings -- what different types (1-point, 2-point, 3-point, etc.) there are, how they differ, and where they are most commonly used.

1 group is researching tessellations -- what makes a shape tessellate and what natural objects (such as beehives) tessellate.

1 group is doing an article on triangulation vs. trilateration -- what are the differences, and what are some modern applications of each?

1 group is focusing on "simple" (ie. right-triangle) trigonometry and its common applications. I recommended looking into the vector stuff in physics, since they seem interested in knowing why all engineers need to learn trig.

1 group is doing an article on sinusoidal waves -- their math discussions will include basic features of a wave function (amplitude, frequency) and how they are represented numerically as transformations to sine or cosine. Their extension will be focused on light waves and its modern applications.

2 groups are working on the applications of fractals. (We recently saw a video on fractals, so they are very interested in researching it further. I told them that they could do the same topic, as long as their actual article content does not overlap.)

1 kid is planning out a two-page comic strip story about the life of Pythagoras.

1 group is doing Pythagorean Theorem -- not sure what their deal is, since they still need to find a narrower focus, but I think eventually they might settle on discussing the various visual proofs of Pythagorean Theorem. (They were looking at some of that stuff on the web and thought it was "so cool"!)

1 group is doing symmetry. I tried recommending them to take a look at the concept of beauty with symmetry, but last I checked they were looking at flags..

1 group is doing how architects use Geometry and software to design buildings. One of their dads is actually an architect and has knowledge / access to AutoCAD. Could be promising.

1 group chose to write about geometric spirals. I was thinking more along the lines of them researching Spirals of Theodorus and other such spirals made of triangles, but they seem to have found some other ones.

1 group (who waited till last second to pick a topic) really hated all of the remaining topics on the list. So, I recommended them looking into spherical geometry. Now they seem to think it's pretty neat, since it debunks a lot of their "duh" ideas of Euclidean geometry.

Anyway -- I'm excited! :) I'm fully expecting their first drafts due next Friday to need a lot of work, but I think it's promising that they find their research topics interesting. The only thing I am a little nervous about is the timing. We have roughly 1 month before finals, and I don't want my volunteer editor and webmaster to still be assembling the magazine and the website while their other teachers crank up the heat before finals. We're going to have to be REALLY on top of the deadlines in order to avoid that!!

Monday, April 11, 2011

Exemplary Proof

This is an exemplary work sample from one of my students -- a regular Geometry student! -- from the recent Midsegment Proof project. I'll admit -- most regular kids did not have this level of work. And neither did many of my honors kids. But, what an accomplishment for this girl. Her algebra work was remarkably perfect down to every last step. I was scrutinizing every project, down to how big the square root signs look. Her algebra work was, well, flawless.

(Depending on your browser, you may have to click the loaded image a couple of times to see the text-readable zoom level. Usually I don't care if you're seeing the full zoom, but this girl's work is too perfect for y'all to be looking at a thumbnail!)

In the end, even if not all kids were able to achieve her level of stellar understanding, I still think it was a worthwhile goal for all kids to shoot for. I'd do it again. :)

Sunday, April 10, 2011

Geometry Magazine Assignment

I am going to have my honors Geometry kids make a single-issue Geometry magazine! I decided that of all the new things I want to try out during the rest of this year, this project is going to be The One that's my baby. I'm doing it with the honors kids for two reasons: 1. There aren't that many of them and they are all reasonably motivated, so it's going to make logistics management easier for me, 2. I would love for my regular kids to do the same, except I think there'd be so much overlap in content that, realistically, the readers are not going to want to read through so many math articles at once.

Here is the instruction file I've pulled together. I am going to give it to the kids this week so that they can start thinking about it over Spring Break. (Some of the extra motivated ones might even try to do the whole thing over Spring Break.) I estimate that it'll take about 3 to 4 weeks outside of class for us to pull this thing together. (Along the way I'll give them a little bit of class time here and there to work on it together as I see fit.)

I'm excited! I'm hoping these kids will do a good job, so that they can be my ambassadors of math love to the rest of campus. :) The way I envision it, every article would contain a brief break down of what we had learned in class related to the concept, and then go into an extension part -- either a historical connection (ie. life of Pythagoras or how people used triangulation to find the distance of X star or Y planet), or a modern connection to what cutting-edge technology now uses that Geometry concept (ie. GPSs or medical analysis of cancer as relating to fractals). You know - something a little bit beyond the classroom!

Wish me luck!!

Tuesday, April 5, 2011

Ruminations

Last Saturday I chaperoned another Habitat for Humanity trip. This time we dug gutters in the ground the entire day. The ground was hard and the gutters had to run 30 cm wide and 60 cm deep each, around 3 sides of the house. This is a tropical country; it was humid and the sun shone brightly starting at about 11am. It was warm, and some of the neighbors who weren't doing the digging even had their shirts off. The kids, amazingly, never complained. I was so impressed, considering these were mandatory service hours these seniors had to do in order to graduate.

I dug alongside the kids the whole day, obviously. We used pick axes and shovels to break up the soil. It was hard work. At some point, I reflected out loud, "This makes me very thankful that I earn a living using my mind." Another colleague of mine replied: "When we were in high school, my dad made me and my sister go out and get manual labor part-time jobs, so that we can truly appreciate earning a living with our minds instead of our body." --It's so true. In my entire life I have never had to exert force to earn a living. The hardest job I had was cleaning bathrooms at fast-food restaurants. Not so hard compared with digging with pick axes in the sun, I'd say.

Do your kids know that it's fortunate to have a job where their mind is prized over their physical labor?

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Today was Holocaust Remembrance Day at school. The entire 9th grade was taken out of classes and doing presentations for their research projects, a la History Day style. This year in particular, I was so impressed by their creative employment of manipulatives and activities at every interactive exhibit. It helped me remember some of the facts that I never otherwise would have remembered. I wonder if it's a reflection of their teachers' teaching methodologies, that allows these kids to think outside of the box?

Afterwards, when I sent all the kids an email reminding them what to bring to class tomorrow, I briefly commented on how proud I was of them today and why they should learn about/reflect upon the horrors of the Holocaust. I think it is important that they know that all teachers, not just their history and English teachers, think of this as an important issue that will hopefully shape their character to understand the importance of abolishing hate.

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Some of you on Twitter have probably read this, but recently I found out via Facebook that one of my former students who had lost his parent recently is now apparently a gang-banger, throwing around gang signs in every Facebook photo he takes with his "bros" in the project buildings or on the streets where they hang out. I cried when I found out, and then I cried again when I was telling this to someone I know. I had taught this kid for two years and always spent extra time to pep-talk him outside of class to keep trying. I had always had a soft spot for this kind-hearted kid who struggled in school. I felt utterly heart-broken when I found out, because I know this would never have happened if his parent were still alive. I sent him a message on Facebook and I don't know if he'll ever get it or ever respond. I just want him to know that I've always thought of him as a special kid and that I still think about him and hope his rough patch will pass.

But in reality, I am worried and heart-broken. It's my 5th year teaching. This is the year when I get to see my former 8th-graders go off to college, and also to see some of them throwing gang signs on every photo on Facebook and having status updates about not going to school and smoking and drinking at home instead. The difference between the kids who grew up together is staggering. I know it's just the reality, but the recognition of that fact does not make me feel any less sad.

Monday, April 4, 2011

Hope

Sometimes I have to tell myself that what I'm doing is changing kids' relationship with math little by little. It's especially discouraging when we return to the same topics later on during the same year and kids seem to not be able to do the same problems they used to be able to do. But, recently an experience with my 11th-graders served to give me back a little bit of that glimmer of hope.

Upon my introduction of "weekend homework," various of my 11th-graders came to ask me for help after school this week on computing arithmetic series -- a Ch. 1 concept. They had trouble finding the sum by looking at a Sigma notation, even after they had listed out a few terms and visually identified the linear pattern. I asked them, "How do you add the numbers 1 through 100?" And to my surprise, every kid that came to ask for help was able to say to me (without additional cues), "You put together 1 and 100, and then 2 and 99 -- oh, I get it. So it's first plus last, times the number of pairs." After that short exchange, some of them were able to then walk off and do all those problems and the associated word problems. Others needed me to walk them through just one example, and then they were all set with the Sigma notation problems (and only needed some additional hints on the word problems, some of which were admittedly a bit tricky).

It gave me hope that somewhere, something is sticking.

PS. I have to say that I really have a soft spot for my juniors. Of all the kids I teach at this school, they remind me the most of my kids from the Bronx. So full of silliness (and sometimes a streak of badness that makes me laugh).

Sunday, April 3, 2011

What We're Doing with Proofs

Since some of you asked, here are some proofs I gave / plan on giving to my 9th-grade kiddies. We've been doing proofs using a top-down approach for a little over a week now. In terms of learning modes, I switch off between circulating and looking at their proofs while they work on them in groups during class, going over/highlighting key parts of some proofs on the board after they've all more or less tried / "completed" them, and collecting proofs to fakely grade them at home so that I could provide more detailed feedback.

I say "fakely grade" because I told the kids that these grades are not going into their grades but merely serve as feedback for them and for me, to see how they're doing on a "test- or quiz-level expectation." The 9th-graders are LOVING the fake grades, and take the fake grades way more seriously than I would have thought. I guess it's something along the lines of Shawn's observation that teenagers just want you to hear/see what they can do. They don't want you to actually judge them.

Anyway, since it was a bit of work to pull these proof exercises together, and I'm liking them all so far, I thought I'd share them.

The first thing I did was to give the kids a reading from Proofs for Dummies that gives them a super kid-friendly intro to what attitude they should take toward constructing proofs. (I'm not implying that my kids are dummies, obviously. This was the most kid-friendly article about proof-writing out there, period.) Following the reading, we looked at one example proof and highlighted reasons vs. statements to start to "see" what a proof structure looks like. Then they did one proof -- showing that when you draw two overlapping triangles with parallel sides, then the resulting triangles will be similar (by the AAA Similarity Theorem*). That stuff is all here: Proof Packet #1.

*The AAA Similarity Theorem is missing from Packet #2, which I handed out at the same time as Packet #1 in order to allow kids to reference the common theorems right from the start. I had the kids add it by hand.

I noticed while I was making the answer key to Proof Packet #1 that the proof for Midsegment Theorem is actually really algebraic and difficult, so I told the kids to skip it in Proof Packet #1 and to go on to Proof Packet #2 to try their hands at the easier triangle congruence proofs first. They worked on some of it in class, in groups, and the last triangle congruence proof they took home and did on their own. I collected that last proof of Proof Packet #2, gave it a fake grade, and went over it step-by-step upon returning it to the kids. (You should also note that the list of theorems in the front of the packet are not complete. Kids were able to use them to cover the majority of cases, but in some cases I still needed to have them add missing theorems to the list. That's why doing the answer key yourself beforehand is a good idea.)

After Proof Packet #2, I decided that all kids are now ready to attempt the algebraically hairy Midsegment Theorem proof. So we went back to Proof Packet #1, spent 2 days on that in class, and I had the kids bring me back fresh copies the next day to submit as a project. --Sorry but I won't get around to grading the projects till later today, so no student samples... yet. :)

When the algebraic proof "project" was all submitted, I handed out Proof #3 packet and had the kids read all of the circle definitions and basic circle theorems before we discussed each of them as a class. I gave them illustrated definitions on the board when I went over them, and made sure that every kid understood each part. I also had them add Thales' Theorem and Isosceles Triangle Theorem to their list of reference theorems, since I had forgotten to include those the first time and they're needed to complete some of the proofs. The kids then began working on the circle proofs in groups in class. That's where we left off last week, and I plan on picking up from there on Monday.

So far, I'm really pleased with how it's going. I think the kids are doing fabulously with all the increasingly difficult proofs. For both my regular and honors kids, after this third set of proofs (which is mostly circle proofs, plus one Triangle Angle Sum proof), I plan to switch gears and have kids work on some algebra problems to practice applying what they now know as common properties of chords and angles inside a circle. These easy circle proofs I pulled from here, by the way. Thank you Oswego School District!

Then, for the regular kids, we'll move on and do a series of simple algebraic proofs (Thanks, Guillermo!), and the honors kids will first do a series of advanced circle proofs as "borrowed" from Mr. Tytler of Rochester, NY, before moving on to those same algebraic proofs as the regular kids.

I'm excited. I can't prove it, but I think we're doing good stuff. I'm sure when I teach proofs again next year (or whenever) I'll want to revamp the whole thing again, but for now, I'm really loving the layers of logical thinking my kids are showing. It's miles beyond what I could pull out of my kids last year!

Friday, April 1, 2011

April Fool's Day

A really funny thing happened today. To paint a quick picture, I'm one of those teachers that start teaching 30 seconds before the tardy bell rings and teaches all the way through the dismissal bell. I always plan about 70 minutes' worth of material for a 50-minute class (or 90 minutes worth of material for a 75-minute class). When the tardy bell rings, the Do Now is already up on the board, as well as the Aim. I can't bear the thought of losing class time to even write things on the board at the beginning of a class, because I always feel like we are running out of learning time -- everyday, every week, and every year.

Today near the end of one of my morning prep periods, I was famished, so I decided to walk down to the cafeteria to buy a snack. While I was there, I heard the bell ring and I asked the cashier if that bell signaled that it was the start of the morning recess ("break"), since each day runs on a rotating schedule. He said in Spanish, "Oh no. That's still 10 minutes away." Figuring that the bells were probably off, I sat down and leisurely ate my pupusa and then got up to leisurely walk back to my classroom. I am a lightning-fast eater, so the whole thing took about 10 minutes. I stopped to chat with my department head in the hallway about the proof project I'm doing with the 9th-graders. I told him that I would love to show it to him at some point.