In July I got a phone call from a number with an Austin area code saying I was named as a finalist in the state of Texas for the 2015 PAEMST competition. Completing the application took the better part of 3 months, and the video taped lesson was quite the ordeal, but clearly we managed to capture something good.

Last week at the Texas State Board of Education meeting for the first quarter of 2016, myself and the other nominees were recognized.

No, the meeting doesn't take place inside the state capitol, unfortunately. Though naturally, Texas-shaped trophies make the best trophies. Being involved in all of this has been very interesting, and a lot of people want to shake your hand. The real winners are the students I chose to video tape, the people who helped plan/shoot the video, and all of you teachers on the internet who helped shape the procedures and teaching style I was able to put on display. I just hope the national committee announces their choice before these kids graduate college.

For the curious, I submitted a video of my kids working through Polar Coordinates, specifically the opening exploration activity. The long essays discussed all the introductory activities and the follow up: sidewalk chalk.

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AuthorJonathan Claydon

A couple years ago I had an idea for a display piece for vectors. Take some straws, create some vectors in each of the four quadrants, cut the straws to length, and demonstrate the right triangle nature of a vector. It was ok. But there was a lot of struggle. The wrong kind of struggle. The "I'm going to do this because you said so but I may not be learning anything" kind of struggle. It happened two years in a row, so I figured it was a time for a change.

Since I shifted my focus on vectors to a bigger, broader category like Images, I thought that would be the starting point for this display piece. Creating images has yielded all the aspects you'd want to cover with vectors: operations, scaling, magnitude, and direction. And edge cases like vectors with 0i and 0j come up naturally, I don't have to force it or attempt to single it out as one of those capital letter Named Issues.

Instructions

given: colored paper, markers, rulers, full sheet of grid paper

  • construct a 7-sided (at a minimum) shape
  • identify the vectors that define the sides of the shape
  • calculate the length of each vector
  • scale the shape by a factor of 2 or 3
  • identify the vectors that identify the sides of the scaled shape
  • prove that the N+1 side of the shape is equivalent to the first N vectors added together
  • calculate the "true" angles for each of the sides
  • compare the side lengths of your original shape to the scaled shape
  • compare the angles of your original shape to the scaled shape

Lots of good work here. It can be a little tedious once they start digging into the calculations, but the creativity on display here was great.

The great discussion point came when it was time to calculate angles. At first some confusion, do I calculate my "true" angle (basically a +180 or +360 correction if you're pointed at the II, III, or IV quadrant) based on the physical location of the vector or the orientation of that vector?

Many kids saw through the simplistic nature of the two questions. Of course the lengths will double if I scaled the whole thing by 2, duh! Why would the angles change? Everything's pointed the same way, duh!

I enjoyed this version of the project a lot better. My kids continue to amaze me with how well they handle tasks with some structure but not a script. More to come on this.

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AuthorJonathan Claydon

I came up with a clever test question last year:

I wanted to tie vectors together with solving triangles for a couple reasons. Primarily, to give purpose to calculating the angle between two vectors. If you go the mathematical route using dot products and all that it's just all so...fake. Second, to connect the idea of vector magnitude to representing something in a geometric shape. In the above problem, the idea is to find the magnitude and angle of side b in order to find the angle between a and b. After all that you can find the area of the triangle or anything else about it.

The issue was I came at this idea at the end of our vector discussion and it generated a lot of confusion. Primarily they weren't making the connection that vectors could actually mean something other than be arbitrary arrows. This year I was going to lead with the idea. In the future, I hope this helps me lay the groundwork for changing Pre-Cal from a random list of topics into a concrete set of applications.

Computer Machines

I open with a discussion of computer graphics. All the favorite images they love to like and heart are usually rendered as bitmaps, hard coded pixels in a grid. I find a random .jpg from the internet, open it in Photoshop and zoom:

I do the same with a vector image that's been rendered out as a PDF. No grid of dots. What gives? To keep things simple (and because I don't know all the gritty details) we discuss that image files contain information about relatively defined coordinates which the computer uses to dynamically draw whatever lines they see. If you ever zoom in on a PDF and catch it go fuzzy before smoothing out again, this is kinda sorta what's going on, the computer/phone is redrawing everything on the fly. There was a side note that computers are trained to avoid showing you the pixel grid as much as possible through approximation, ie why bitmaps get blurry when blown up.

Create Your Own

Since we aren't computers, our ability to render and re-render vector based images is quite slow. We'll need to define our coordinates by hand and then determine the relative horizontal and vertical moves necessary to move from point to point. Here we introduce i-hat and j-hat notation.

Their task here was to define a series of points, connect the dots, and define the vectors that move them around the perimeter. In phase two, we'll talk about calculating the lengths of those lines, how to scale our shapes, and use the grid paper to approximate the area of the shapes. It all slowly builds towards the problem I wanted to tackle in the first place.

What I like about this activity is it shows the absolute nature of a vector. The same line can be defined two different ways depending on direction of travel. Would -3i + 4j have a different length than 3i - 4j ? Most of them say no, because duh, same line. Can we prove that?

Another interesting "hmm..." moment is showing them that in a shape of n + 1, the first n vectors sum to the value of the final one. Why might that be?

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AuthorJonathan Claydon

This is a continuing series of posts about how I approach topics in Pre-Cal They represent a sample of what I get the most questions about which always seem to start "how do you..."

Objective

Students understand the unusual nature of a trigonometric equation. Students understand there is a base set of solutions on the domain 0 to 2pi, but an extension of that domain extends the solution set. Students understand that it is possible to make a statement that summarizes all the terms in the infinite series of solutions. Students understand that the frequency of the trig functions relates to the way solutions repeat along the domain.

Progression

We don't start with algebra. We start with the concept of an inverse trig function. For convenience, I stick with angles on the unit circle. My first request is that given a particular x or y coordinate from the unit circle, can you find all the angles associated with that coordinate? We condense that statement into math notation, sin^-1 x or arcsin x, for example. The idea that multiple answers are possible is interesting, and lays the foundation for the algebra later.

Next, we look at graphs. I hand out a set of trig equations.

To keep things simple I have them graph the even set or odd set on a restricted domain of 0 to 4.5pi, recording every intersection they notice with a sketch of what they saw. This continues a big algebra idea, every equation has a graph that validates the algebra. For more complex algebra, I always like to start with the graph.

For some of these equations there are many intersections, for others there is but one. Some comments on how weird tangent is, and questions on how to type sin^2 x.

Now the interesting question. Why so many answers? What happens if we widen the domain restriction? What if we removed it entirely? Many correctly guess that there are far more solutions than we could ever write down.

But is there something to this? Is there a way to condense a small infinity like this into something a little more manageable? We take a closer look at just what numbers appear as solutions. Are they random or regular? How might we predict the next number?

After the graphing exercise I graph another random trig equation. We focus on the first two positive solutions. I name them Solution Zero. We discuss the frequency of the function in question. I take our Solution Zero and add one repetition of the frequency to those numbers. Amazingly, the next number in the solution sequence comes out. Minds blown.

Eventually we discuss that an infinite set of solutions condenses to Solution Zero + n[frequency] both in radians and degrees. Finding Solution 1000 is an easy task now. It's fun to ask about how we would determine solutions to the left of zero, many determining that n = -1 would do the trick.

Lastly I demonstrate the algebra necessary to find these solutions without a graph, validating the need for an inverse trig function.

We build from there and talk about squared functions and how the scope of Solution Zero is much larger (base 4 instead of 2). All of it a lovely dance back and forth between algebra and graphs, avoiding impractical (and graphically dubious) equations used by textbooks to convince somebody (ANYBODY) that kooky trig identities are a thing real people use.

Assessment

We discuss Trig Equations in the early parts of the second semester. I look for student understanding of both the graphical and algebra aspects of the solution sets. Students validate whether a graph accurately represents a trig equation and determine solutions beyond Solution Zero.

Test 11, Test 12

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AuthorJonathan Claydon

More goofy classroom culture things for you. I enjoy writing tests. When it comes to final exams, it's always fun to throw a few curve balls in there to lighten the mood.

From 2010, featuring a dated Lady Gaga reference:

From 2011, featuring a less dated musical reference:

This one drew probably the funniest answer I've gotten to one of these. I'm paraphrasing but the answer went something like "I would drown myself in the ocean so I didn't have to live in a world without him." That particular student had a lot of greatest hits when it came to creative answers. Goofy questions is a time honored tradition around here and this year's was pretty good. It went like this:

43. Draw a kitty in a spaceship. Yes, you read that correctly. DRAW THE KITTY.

And here are 12 cats piloting spaceships, because I know what you want:

A little irreverence in the face of stoic final exam culture is not a bad thing.

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AuthorJonathan Claydon
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You want to see the inside of my cabinet? Sure you do.

1. An oddball space that's hard to use. Other than the boxes, crates, and tubs, I use this space for poster tubes and a boom microphone pole. What few posters I still have kicking around sit in here.

2. The shelf kids always spot when I open up. Here sits cups, puzzles, a couple jengas, Uno, Zombie Dice, and playing cards. There is a Barbie in there, but you can you only spot the foot.

3. The Hotel Snap shelf. Several tubs of linking cubes with more cups and old signs for classes I no longer teacher.

4. Primarily for colored paper. Also featuring a bell, party plates, boxes with unused iPad chargers, batteries and the Varsity Math cash box. Oh, and cups.

5. Video equipment lives here. The shotgun mic, camera, and audio recorder are tucked away in this area, plus tennis balls, mini whiteboards, erasers, poker chips, an Iron Man mask, and oh...more cups.

6. Duct tape, old notebook tubs that are too small (would you believe I had class sizes in the 20s once?), and a box with log war cards, trig war cards, and straws. Probably a cup hidden in here.

7. And lastly, bulkier items. My printer lives down here in the summer. For the school year, we've got a lamp I no longer use, power strip, toner refills, a couple thousand straws, and Promethean accessories I don't use. No cups to speak of, just a random roll of neon orange tape.

When I first inherited this thing, it was half full with a set of Algebra 2 books and the piles and piles and piles and piles of resources that textbook companies thought I needed (spoiler: didn't). One year I decided warehousing textbooks was not the purpose of this thing and finally arranged it in a functional manner.

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AuthorJonathan Claydon

Here's an interesting exercise I thought up as a Day 1 activity for Calculus in the new semester. Having covered the ideas of position, velocity, and acceleration and their more general relationship in f, f prime, and f double prime, I decided to hand out some scenarios with a simple request: would an integral or derivative be helpful?

No actual answers to determine, just tell me the method. Prompted a lot of good discussion. The most common confusion was the phrase "area under the curve" which is a little weird because I had been keying in on the relationship between "area" and "integral" for a while, but it worked out. Having them catch on to a correct method by spotting units is also a handy skill.

It didn't take a lot of class time (12 minutes + discussion) and it was a nice restart activity. I think a second round of these questions is in order.

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AuthorJonathan Claydon

As fall semesters close, there's always those fun final exams to think about. The problem with Calculus is that our seniors are allowed to exempt fall final exams. I can't override it, which is fine, but it influences the kind of final I give. Spend the time to prep something intense and everyone will exempt, wasting the effort. Instead, I put the effort into their third AP Mock Exam of the semester and conduct that a week before final exams start.

So what do you do during the exam period? Enter arts and crafts options.

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While they were working I wanted to be able to spend time discussing AP Exam projections with each student, so I picked projects that with loose requirements that covered material they would know the best. I didn't have time to hop around answering lots of questions.

Students could choose one of the options to complete during the final exam period. We broke out the colored paper and markers and it had to be hanging on the wall by the end. They had about an hour and twenty minutes to complete it.

I overhead a lot of good discussions. And my hunch was right, most of them were super comfortable with this stuff. That blew my mind a little bit given that this time last year we had covered approximately zero of the topics considered in each of the options. Bigger bonus, each project was rather conceptual in nature and required them to develop something out of the blue on their own.

Really digging Calculus this year.

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AuthorJonathan Claydon

Mildly boring, but important nonetheless. I gave homework in Calculus! On a regular basis! And the kids did it nearly every time!

Last year was not nearly as successful, though there were bright spots I used to build with this time around. Here were my considerations when setting up a homework system:

  • homework respects students' time, assignments are given Monday, due Friday
  • homework respects students' time, assignments might take 30 minutes to complete, max
  • homework is not meant to be a struggle, material is at least a week old
  • homework is not meant to be a struggle, solutions are posted online
  • homework is not meant to penalize, multiple assignments were lumped together into one grade
  • homework is not meant to penalize, students could complete older ones and receive credit
  • homework has purpose, students are told upfront the reasons I assign it
  • homework has purpose, a focus on old material helps build retention

Providing solutions was an incentive I started last year. It contributed to what little success I had with homework. The second factor was the nature of the assignments. Last year I handed out textbooks and cobbled together problem sets from the book. They weren't perfect and it was hard to find problems that aligned well with what we were really up to. This year I said forget it, no textbooks, and I wrote my own assignments (hw1, hw2 are the assignments, anything else was just classwork). They are short, they are focused, and most importantly, they serve to spiral back. Only old material goes on homework. I use it for retention purposes. My goal is not to make a student struggle with unfamiliar material they just learned, it should reinforce ideas with which they are comfortable.

They took advantage of the solutions too:

The spikes are Thursdays (and Friday mornings), because teenagers. Many of them don't even use it, which is totally cool. Others like playing a game of find the error (I am Supreme Lord of Errors you guys). You might say, well aren't they just going to copy it and pawn it off as original? Sure, maybe. It's usually easy to root those kids out come assessment time. But recall my design goal, homework is not meant to penalize. If you stress the purpose of the assignments, you'll get kids on your side. Respecting their time is huge too. Long, daily assignments are just asking for trouble and cutting corners. Some of my lowest performing students tell me they'd rather turn in nothing than a series of copied answers, because they want to learn. It's crazy. If your homework is a means of punishment, well, you're going to earn every bit of the resulting culture.

Props to my kids for doing a great job so far, we'll see what happens as Senioritis beings. And props to the Calculus people at TMC15 for helping me think through this issue.

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AuthorJonathan Claydon
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Grouping is that tried and true way to make your classroom look cutting edge. No, seriously, ask the average student how often they're encouraged to talk with their classmates about things during a class period. It may depress you. The number of "would you look at that" comments I get about my room layout is equally concerning.

But not you, you are super down for this group thing! Except, you may be concerned about how to handle it. Who should sit with who? What are the "roles" going to be? Where am I going to come up with great group task ideas?

The short answer: don't think about it so hard. My success with grouping has come with not treating it as anything special. Start with desk arrangement. Creeping class sizes have gotten me to what I call my Standard Group Unit: 6 desks with a screen and a supply bucket.

Here's a better look at the front half of the room (featuring 3 Standard Group Units):

Given the density of people in my room, this is the most efficient use of limited floor space. Groups are nothing special, it's just how we sit around here. Your group is your family. Historically, some groups have latched onto that idea quite heavily.

Now that they're sitting in groups all day, every day, how do we leverage this? You might think I conduct a lot of projects with defined roles and detailed rubrics for grading final products. You'd be wrong. The groups exist to take an overwhelming class of 36 and make it six neat little families. Ok, so what the heck, there's got to be more to it? Indeed.

  • Take a problem set of considerable length, assign half to even desks and half to odd
  • Take an exploration activity, work on it with a partner from your group
  • Take an activity with not quite enough work for six people, divide your group in half and complete the activity
  • Take an extensive construction project, work on it with a partner from your group
  • Take a design activity, require each group member to complete the same task, collaborating with but not copying the work of other group members
  • Take a problem set of considerable length, make it into a deck of cards using half size index cards, trade problems within the deck among the group until each group member has solved X number of cards, have that be step one of a game like Log War 
  • Take a problem set (usually requiring a graph or some other illustration) of considerable length and equal difficulty, have each group solve the full set and compile the results as a poster

Running groups this way reduces a lot of the overhead commonly associated with group work. I'm not fretting over who works with who because I figured that out a long time ago making seating charts. I'm not fretting over one group having three thumb twiddlers while the other three do all the work. The groups are small little six person classes within a larger whole. All the situations I outlined are just a small fraction of the big picture. On a daily basis, the groups are just who students sit with. If we're taking notes, it's who they'll turn to if they miss something. If we're doing classwork, it's who they'll turn to with questions. As I wander, I can address the needs of many with only a few stops.

Many kids tell me how much they look forward to their time in my room. Part of it has nothing to do with me, it's all about family time with their group (and maybe the lava pits).

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AuthorJonathan Claydon