Monthly Archives: August 2013

Vectors, Proof, and Various Other Rabbit Holes

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I’m still waiting for my lab supplies to come in so I continued my unit on vector addition. Since students did so good last time with the graphical addition of vectors and the topographic map activity we did, I decided that we could move on to adding vectors using components, trig, Pythagorus, etc. This was a tough topic to teach primarily due to the differing mathematical background of my students — some are currently enrolled in college trig and others have yet to take geometry.

However, when we were discussing how to solve triangles something magical happened.

As I was walking a couple of students through the process of solving a triangle, one of my highly math-literate students asked HOW the first person came up with that. Several students actually began to discuss the question — some more serious than others, but nevertheless I was thrilled. He might have been slightly inspired by an off-topic discussion we had had earlier about the origins of the word “hypotenuse”.

A million thoughts were running through my head about how we could turn this into an awesome learning experience, but I never could quite figure out the best response. So I did what any under-paid state employee would do, played a video. .

Some enjoyed the video, others didn’t, but after it we got into some pretty deep discussions. I showed them a few of the proofs I think are cool (Cantor’s Diagonal ArgumentThe Square Root of 2 is IrrationalInfinitude of Primes, etc.); sadly, the one they found the most intriguing was .999… = 1 which is somewhat interesting, just not as much as the others, in my opinion. They also had some fun with QED, so much so that I had to get the letters put up on the wall. It has become somewhat of a class motto. I was reminded of this XKCD comic but kept that to myself.

I was so inspired by this that I wanted my geometry class to jump in on the fun as well. So I went home and did some research and ultimately created this for my geometry classes (NOTE: I just googled “Easy Sudoku Puzzle jpgs” for the puzzle shown and never worked it since the real goal of the activity is for students to practice proving things. I later realized that the puzzle is impossible because there are already two 1’s in one of the columns. I just told students to move the one on the bottom over a spot and a couple of them managed to work it fine) It did alright with the class my upper level kids, but most of my lower level kids didn’t even know the rules of Sudoku. All-in-all I like the activity for an upper-level class of math or science, but the lower level kids will need some serious scaffolding.

##setbacks ##rainydayactivity

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Day 13 – Classroom Arrangement

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PHYSICS & GEOMETRY

Nothing new in either class today; just followed Day 12’s stuff with the other half of my students. So I wanted to take a chance to write about my classroom layout. I was chosen to be the teacher who is in the lab full-time and I wanted to talk about the implications of that.

First of all, I love how the arrangement of lab tables and stools easily lends itself to group work. I can’t express how great it is to not have to hear the screeching of metal desks across tile floors every time I let students work in groups.  Although, the octagonal design of the tables with the faucets on either side is somewhat awkward, because it really only allows two students to really sit ‘comfortably’ at each station. Also, the location of the drawers on the desk is really awkward too; not sure what the incentive was to put those there.

Next, I love how the grouping allows me to grade less. I played the whole Lecture-Worksheet-Homework-Test-Repeat game last year and realized how useless it really is. When grades are attached to every assignment, students start gaming the system and their grade shifts from true assessment towards a ‘following instructions’ coefficient. I’d love to, one day, switch entirely to standards-based grading, but being a new teacher I’ve got too much other stuff that I am trying to get use to this year.

However, there are some cons to the classroom set up like this. Primarily that the stools are a pain in the butt, literally. I can’t even set in those for more than about ten minutes before I have to stand up, so I know the students can’t enjoy it.

Before I state the second con, let me first admit that I am definitely not a proponent of the indoctrination of students that usually occurs when they are placed in long vertical rows facing the board. But for me, it is almost impossible to get students to settle down for the few times that we have to actually do individual work. Since we don’t use all of the tables in my class I can sometimes achieve this by spreading them out in smaller groups across every table, but that introduces another battle of having to introduce students to a new seating arrangement each class. I suppose the teachers who have honed their classroom management skills more could do this, but admittedly that is my weakness. I am having trouble, at this point in my career, finding the best way of balancing the unfortunate necessary amount of direct instruction in the modeling curriculum with the much preferred collaborative aspect.

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Day 12

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GEOMETRY

Did the same thing as yesterday in geometry with my other group of students; good results without any extrinsic motivation from me. I’m really impressed with this group of students.

PHYSICS

A little backstory before I go into the picture of the day, as mentioned in my About Me section, I teach at a small school in rural area of Mississippi. Physics has only been taught at my school once in the past decade, to my knowledge. Because of this my lab is pretty empty when it comes to physics supplies. However, my administration and school district have done a great job backing me up and putting the students’ interests first and are in the process of ordering a great deal of stuff.

I’ve also mentioned before that this year I’m following the modeling curriculum for physics. In the curriculum the first unit, centered around constant velocity, the introductory lab involves students finding the velocity of a slow moving buggy. I really feel like this approach opposed to the traditional “here is the formula… know it”. So I decided that I would try to move things around in the curriculum to stall until my inventory arrived.

To do that I decided to go ahead and teach my students about vectors and scalars. After giving students a few bare-bones notes about the two concepts, we got into graphical addition of vectors. I had the privilege of getting to know the former Mississippi Teacher of the Year, Dr. Paul Cuicchi. The way he taught vectors is the story behind my picture of the day. During the unit on vectors and scalars Cuicchi would give each of his students a topographic map, protractor, ruler, and a set of directions and then let them loose. The students went on a make-shift scavenger hunt through the map. Students seem to pick up on this unit pretty quickly, even when taught traditionally, but I still feel like students enjoy flexing their muscles and doing something like this as opposed to the canned examples at the end of the chapter.

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Day 11: Testing

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GEOMETRY

So over the past couple of classes I have been noticing that my students were struggling with some of the earlier material, so I decided to do a review unit today, covering everything that we have done thus far. However, I didn’t want to waste precious class time droning on at the board with information that they have already seen and heard once. So I decided to let them do the review themselves. I called it Review Roulette. I gave each of my six tables a different stack of sheets — each with a different topic that we have covered thus far. I then put 10 minutes on the timer and told them to get to work. Of course I walked around and assisted with any questions that arose during the time period. At the conclusion of the time period, they shifted to the next table and started fresh on a new topic. This took up the whole day, but I feel like it was necessary and beneficial.

My first two classes needed no incentive to work their butts off for me, but my last class is still caught up in the LWHTR-cycle (lecture, worksheet, homework, test, repeat) and don’t get motivated very easily. So I had to use the grade against them, which I hate to do, but I have yet to find a way to seriously get them interested. I told them they had to have 100 correct answers by the end of class (there was probably 300 questions altogether, so 100 shouldn’t have been too big of a deal). They moaned, they groaned, but they worked — at least for the first four sheets. After that I could feel their attention span slipping away; primarily because many of them had the 100 done before that point. I think next time I need to supplement it with a little bit of discussion.

PHYSICS

Physics today took their first assessment. The test covered graphical methods and data analysis. My average on the test ended up being around a 73, but I suspect that many of the students who did poorly did so because they aren’t quite back in the groove of school yet. I do allow retakes, but do not require them in hopes that students will become more intrinsically motivated.

I also gave the students the chance to give themselves feedback on their quiz. Many students were completely stumped with what to do on this.

“We’re just supposed to grade our test”

“No, you are just giving yourself feedback for the future. I grade the test whether you write or not.”

I still got a dozen or so tests with just the answer written on them. I’m really trying to get them to transform their model of school from a place of work to a place of learning, but it’s tough. The bonus question gave them the chance to grade me. I highly recommend this to other teachers; perhaps not for points, but in general, I received great feedback.

Day 11

This. anytime the author uses “physics” replace with whatever subject you teach.

Day 10

I got off a few days somewhere along the line, but we’re good now. We just lost a few days to the dark ages. I also want to start posting a few pictures of student work from now on to supplement my visual learners.

GEOMETRY

I was really excited about today’s lesson now that we finally made it through the muck that is the undefined terms and midpoint formula (it’s really hard to make this stuff and awesome) and now we’re onto angles and lines. Most of my students are familiar with the basic information about angles–180 degrees in a straight line, 360 degrees in a circle, acute, obtuse, reflex, yadda, yadda, yadda. So I wanted to have them work on a task that required the students to use this information, but without telling them that. So I pulled out an old physics lab that I have never got a chance to actually use in physics class.

hinged mirror

The activity goes like this, you take two plane mirrors and ‘hinge’ them together with tape so that you can adjust the angle between the two mirrors, as seen in the image. Place an object in the middle of them and then count the number of reflections you see. I ask them to fill out a little table to help them organize the data and then to try to find the pattern that was emerging.

Of course not everyone has two plane mirrors for each student to perform this investigation so I got the next best thing, old CDs. The CDs work just as good as mirrors except for the hole in the middle which sometimes blocks out some of the image. Another helpful suggestion is to glue each to a piece of cardboard, so that the mirrors stand on their own.

The activity launches good discussions between students and ultimately provides a great platform to launch into the discussion of angles, circles, straight lines, etc. Another quick note, have a couple of mirrors already set up and demonstrate the procedure for your students. I made the mistake the first time of just verbally issuing instructions and when I got done no one had a clue.

After they finished the activity and we had a short discussion formalizing our findings and some definitions, I gave the groups an assignment that I got from Work Done concerning parallel lines that beautifully introduces vertical and interior angles.

PHYSICS

In physics we broke out the spaghetti again. Since we are still working on the scientific method–specifically using data to make predictions and linear regression–I wanted to give students another hands-on activity as opposed to working on another worksheet. So I showed them my setup of a spaghetti noodle precariously positioned between two supports with a plastic cup hanging from the middle of the noodle by a string (a nod to the modeling curriculum). I then asked them what are some things we could measure if we started adding small masses to the cup. After a couple of off-the-wall ideas were thrown in, we settled on how much mass would it take to break the noodle–and off they went.

The groups started precariously adding masses to the cup. Crashes resounded throughout the room during the first half hour of class, each time followed by students quickly getting up to scribble numbers into their lab notebooks before repeating the whole process again. After all of the data were taken I had a list of questions for students to answer regarding their data, primarily asking them to make predictions from their equation.

  • How much mass could 10 noodles hold? 20 noodles? 50 noodles?
  • How many noodles would be necessary to hold 5 kg?
  • What are the units of the slope of the line?

Some of the students felt very confident with the math necessary to do this–it is the fourth time we have done it in class–but a few were still lost and were even more upset to find out about the upcoming quiz. I was very impressed with the fact that the students who had figured out the process were quick to get up and help out their neighbors, I love that collaboration and have been doing my best to foster a student-first class.

I was also surprised to see the level of interest from the students in the task. It is amazing to me that students find some experiments totally cool and others, just, not. Even though they might be very similar to each other.

Day 7

Geometry

In geometry today we just did some quick definitions, discussed the difference between postulates and theorems, and then practiced with our first two postulates (the ruler postulate and the segment addition postulate). Nothing too fancy here, I wish I could find a way to spice this part of the curriculum up a little bit. Have to work on this for next year.

Physics

Today we spent more time working on data analysis and linear regression. As a quick side note: I encourage every teacher to sit down and look at their curriculum one day and decide what it is that you want students to take away from your class, because they won’t take it all away, probably not even 60% of the material. Whatever you choose should be relevant to many facets of your students’ lives. The only way to ensure that students will leave your class with these skills is by constantly spiraling backwards throughout the course. Repeated exposure is the key. For physics I want students to leave my class with the ability to properly analyze data and to make scientific conclusions; so we are spending a lot of time there (and will continue to).

Today we began class answering questions that students had concerning linear regression that I discussed last time. After it seemed that students were starting to get the hang of it, I decided to do an impromptu laboratory exercise, where they verified (many were discovering instead of verifying) the formulas for circumference of a circle, area of a circle, and area of a square.

It’s still not quite clear for all of the students, but many are beginning to catch on and with plenty of opportunities throughout the semester to catch on, I’m confident that all of them will walk away from my class with the necessary abilities.