Friday, April 11, 2014

Parametric Equations Project - All the Mathy Details

So, I decided to do a parametric equations animation project in Precalc this year. 

I decided to structure this in phases. Day 1, everyone had to design some motion in their head and get it "checked off" by me as feasible. The basic requirements are that they need to have a wheel moving across space somehow, and they will eventually add a second point that rotates on the edge of this wheel. Day 1 went smoothly. Some of the kids had quite ambitious designs, so we came up with similar but simpler designs as backup options, in case there isn't enough time to do the whole shebang. They got on laptops and played around with the "Warm Up" portion of the packet, that gets them thinking about how to get a point MOVING in the plane as a function of t. I also asked them to try A = (cos(t), sin(t)) as a definition of a point, to see that it circles around (0, 0). They were delighted, but they understood why that works.

On Day 2, their goal was to get the center of the circle, point A, moving in the way that they had designed. So, just defining ONE point using parametric equations. I conferenced with each kid individually to help them get started, because some of their designs are quite tricky. Some kids figured it out entirely on their own. Once they figure it out, I ask them to use parametric equations to define a second point B, that will stay always a few units to the right of point A. 

On Day 3, I thought this was a good check-in point, so I pulled the class together to talk about one person's equations for point A. It was something simple, like A = (2 + t, 4 + t). We talked about what the 2 and the 4 mean, and what the +t does to the coordinate values as t increases. We also talked about how you can get the point to move towards Quadrant III by changing the parametric equations, so that kids start to associate rates with parts of the equation. On Day 3, they created a circle from points A and B (using a circle tool built-in to GeoGebra -- we hadn't learned circular equations yet, and I just thought it was too many new concepts to throw into this project at once, especially because the circular equation is going to be parametrized as well). This is why we want B to remain "r" units to the right of point A; it makes it easy to then create the circle graphically, and it also helps to reinforce algebraically how do you translate the x coordinate without changing the y coordinate? Also on Day 3, I talked about how you would create a third point C = (cos(t), sin(t)) to simply rotate. But, how do we make it rotate around a bigger circle to match yours? How do we then modify it to rotate around a stagnant point such as (2, 3)? How do we then modify it to rotate around a dynamic point (2 + t, 4 + t)?  If your kids are more adventurous than mine, they may be able to figure it all out on their own. Mine needed help with setting up the third point, but afterwards they were able to explain why this is done so.

On Days 4, 5, 6, kids were working on their written explanations of their 6 parametric equations; there are 2 parametric equations for point A (x(t) and y(t)), 2 parametric equations for point B, and 2 parametric equations for point C. They needed to write down explanations for every part of each equation, and explain how that part relates visually to the motion generated. They could also only explain the equations orally during their presentation, but then they would max out at 6/10 points for Communication, instead of potentially getting 10/10 points for both presenting orally and in writing. The explanations took them a long time to write, because I wanted them to do very detailed analysis involving all of these words: period, amplitude, midline, radius, independent variable, dependent variable, parametric equation. They also needed to present ONE interesting graph from among their 6 equations. If their graphs were not sufficiently interesting, they needed to go back and modify their design so that it would generate an interesting graph for our class discussion. Sometimes, this meant modifying the periods so that points A and C do not have the same period of movement.

Anyhow, here are SOME of my student samples. You can right-click on the t values and see the motion designed by my students! I'll talk about them here in more detail, so you can see what a wonderful variety of math concepts came out of this.

The translating wheel controlled by t1 was the first analysis we saw. The wheel simply translates in a linear path across the coordinate plane. The kid showed a graph of two functions that look like this:

Even though his design was relatively simple, he successfully explained with super clarity that one curve above shows the x-position of the rotating point C over time. It's cyclical, with an internal sub-period of 2pi because the equation is x(t) = 3cos(-t) + t, but its midline is no longer horizontal but a diagonal line with a positive slope, because the x-value of the center of the circle is increasing over time, thereby bringing C along with it to generally increase in x-value. He also explained that the second curve shows a downwards trend despite its cyclical nature, because generally the wheel is decreasing in position, so the y-coordinate of the rotating point will generally decrease over time despite the rotation around the circle.

For additional clarity, he graphed the lines of y = x and y = 10 - x on top of the curves, to show us that if we look at the graph of y = 3cos(-x) + x, that curve would have a visible diagonal midline at y = x, and if we see y = 3sin(-x) + 10 - x, that similarly must have a visible diagonal midline at y = 10 - x.

Even though his design was simple, this student's presentation (which happened on Day 4) was a really good example for the rest of the kids, in terms of what level of detail they should incorporate and how to think about their equations. I think his classmates were really grateful that he went first to present, while they were still working on preparing their presentations and explanations, because he gave a really good blueprint to follow.

The rest are out of order for our presentations, but the "bounce" controlled by t2 generated an awesome graph from its rotating point's x-coordinate:
Notice here that for a while (actually, precisely pi seconds) each period, the point stays at x(t) = 0! That's amazing. When we went back to the animation, we noticed that that is true. Half of the time, the rotating point remains on the x-axis -- something that you simply wouldn't notice if you weren't looking at the graph.

When the student presented this, we discussed it in the context of their algebraic equation for that point. x(t) = 2cos(t) + 2|cos(t)|. We talked about why the absolute-value was necessary as part of her point "A" (center of circle) to create the bounce, but also why that cancels out 2cos(t) for the parts of the period when cos(t) < 0. Lovely! Things that normally, my Precalc kids wouldn't see, arising naturally out of one of their creations.

The planetary motion of t3 was interesting, because here the student was very deliberate about how long their periods were to be. One of the sub-periods in their design was 10, another was 20, and another was 30. The kid figured out that this meant that after 60 seconds, the whole system resets, which is visible inside the graph. (You can as well see the shorter internal periods.) I liked this presentation, because it helps to bring about the idea of periods within periods, and when does the system (if ever) reset?







I'll skip discussing t4 except to say that, how interesting it is that even though the motion here appears vastly different from the translating wheel in t1, their horizontal component movements are the same, and therefore their graphs for that component are also the same.

The "falling leaf" pattern controlled by t5 was massively interesting. The student only needed me to ask him  guiding questions through the baby steps of first setting it up such that it's falling straight down (something like (4, 5 - t), then additionally falling left-and-right, something like (cos(t), 5 - t). After that, he was ready to make it entirely his own. His final parametric equations for the falling center looked like this: (t^1.1 cos(t), 20 - 1.2^t). The t^1.1 part, he explained, means that his amplitude of falling left-and-right increases as t increases. He also explained to the class that the 1.2^t means that as t increases, the leaf falls exponentially faster and faster in the vertical direction. This he came up with entirely on his own, by playing around with possibilities of varying the equation! That's amazing. He said that a falling leaf should fall more and more sideways, and faster and faster vertically.

The rotating point for this wheel has a y(t) graph that looks like this (see below). The equation is y(t) = 2sin(t + 2) + 20 - 1.2^t. This student explained that the period of rotation of 2sin(t + 2) is still 2pi, but the "falling leaf" circle starts to drop so fast vertically, that the rotating period doesn't really matter after a little while. After about 10 seconds, the effect of the rotation on the height of that point isn't really visible compared to the overall exponential movement of the wheel.

The parabolic motion controlled by t6 is interesting, because I had to work with the kid to remind him how to do parabolic analysis. He wanted a parabola that would turn around after going up for a while, so we wanted a parabola whose vertex has a positive t value. The student still remembered -b/(2a) for the axis of symmetry equation, so I helped him make the connection that if he wanted a downwards parabola (which he remembered that means a < 0), then b > 0 would cause a positive t-value for the vertex. He then played around with it and decided that a = -3 and b = 15 were good values to use. To help him separate the ideas of t and x(t), I asked him to figure out how to make his wheel go LEFT as it rises and falls. He did well with this, and he was also able to explain why the height of the rotating point gets overwhelmed by the general parabolic change of height. He even explained that normally, t = -15/-6 = 2.5 seconds would be the max height, but here since the rotation changes the shape of the graph we're examining, the max will occur around 2.5 seconds but not then exactly.

Lastly, the wheel that resembles a swinging pendulum was pretty complex to create. I had to help the student create the up-down motion after they created the left-right motion, because the up-down motion has a period that is half of the left-right motion, and also their phase shifts should match up.  (For every time that the pendulum goes all the way right and back to the left, that's one full horizontal period ending at t = 2pi. But, in that time its vertical motion has already completed two full periods, hitting the max at t = pi and t = 2pi.) That's a bit tricky for the kids, but I liked the reminder that the pendulum motion has related but different periods for vertical and horizontal components, since it was something that we had briefly discussed a long time ago.

The graph that was generated from the horizontal position of the rotating point C looked like this:
It was another good reminder of how to analyze the sub-periods of a function formula such as f(x) = 2cos(6x) + 5cos(x) by looking at the formula (and cross-referencing it to the graph).

Overall, this project was definitely challenging, but I thought that it really stretched their understanding of the forms of functions. It was also a great way to tie in math to something real. I told them that for making animated movies, people obviously don't manually put in every coordinate of the object for every fraction of a second. A lot of it is controlled by parametrized movements and automated calculations as a function of t!

Thursday, April 10, 2014

Precalculus Parametric Projects - a Sneak Peek

Here is another parametric equations playground, this time with some of my students' work!

http://www.geogebratube.org/student/mX8LkY8Om 

I plan to write soon about why I liked this assignment and how I used these particular ones to stretch our class's understanding of functional forms. Stay tuned...

Saturday, April 5, 2014

Algebra 2 Picture of Functions Project

This is definitely not a new idea on the web, but this is my first time doing a functions pictures project with my students! I am really excited to see the final outcome. What a GREAT way to teach restricted domain, technology / math notation, piecewise graphs, solving for y in a formula, and to review function transformations!

Here are my students' projects from Algebra 2. Stay tuned for my Calculus students' projects (which include an integral component). The basic requirements in Algebra 2 were to use quadratic, linear, and absolute-value functions. Some of the kids took the liberty to learn about sideways parabolas (which had to be turned into square root functions in order for them to limit its domain), circles and ellipses, and one student learned about using matrices to solve for coefficients, to help him find cubic functions to fit his points.

I think a writing component is essential for this particular project, when done in Algebra 2. Even though they were able to create the functions with only a minor amount of help from me and each other, it was when they had to write down their explanation of the impact of every parameter on the graph, that they really learned. They learned the correct math vocabulary, the specificity of the language, and for some kids who needed extra conceptual reinforcement, this was a great time for them to slow down and to make sure that they articulated once more the horizontal and vertical transformations and vertical scaling. I find now that I can grab any kid from my Algebra 2 classes, sit them down in front of any of the basic functional forms, and they can correctly analyze the parameters instead of mixing them up, as they kept doing earlier this year. Great stuff.

WIN. This project is really a MUST in Algebra 2, and (judging from my Calculus students' great enthusiasm for it) worth repeating at a later age, with greater detail and more complex requirements. It also fit beautifully with our annual Arts Fest, which will happen next week. I'm putting these bad boys on display, along with the Calculus designs (to come in a separate post).

Thank you, teachers of the interwebs, for your generous sharing of ideas!!!

Wednesday, April 2, 2014

My Turtle Adventure

This is off-topic, but here is my recent personal pet project: I made a Teenage Mutant Ninja Turtle costume for Geoff and me, in preparation for a local Comic-Con!

I had seen some photos of someone else's home-made TMNT costume, and figured that I could probably do the same. My coworker donated a small container of papier mache glue right before February break, when she had heard me talking about my plan to stay around town and to possibly work (idly) on this potential costume over the break.

So, here it is! I actually did it!!! I took some photos along the way, and even though the photos I had found online were pretty detailed and I had done a lot of research in addition to looking at those photos, I encountered quite a few snags along the way that were unexpected and required some minor problem-solving. Initially, I almost gave up. (I realized during the process that I 1. had never taken a proper art class before, 2. had never done papier mache!! Go big or go home, I guess.) But, I am glad I stuck it out.

So, here's a little documentation of my adventure. My husband was amazed, because seriously, I have a very short attention span and he was expecting me to never have followed through on this.

Step 1: Proof of concept. Building a papier mache base from used newspaper, and putting 3 or 4 layers of papier mache on top. The base was actually really not easy to make, since it had to already form the basic shape of the turtle head and had to be big enough to fit over our heads. The photographed instructions didn't really say how to make the base look correct, so you have to just rely on your artistic instincts. I had to take measurements of our heads and to do a little estimation as well. It helped that, online, I saw that once you make a base, you can cover it in masking tape, and then massage Vaseline all over the masking tape. This makes it easier to rip out the insides when you're done, since the papier mache won't stick to Vaseline. You also have to wait until the papier mache is completely dry to take out the insides. I wasn't that patient, but it helped that our central heat was on during the February break, and I just left it in front of the heating vent overnight. (Be careful of fire safety though.)


Step 2: Once the base was dry, I created lips and eyelids using modeling clay. I taped over the clay details with masking tape, and papier mached another few layers on top.

Step 3: Since my husband's head is somewhat bigger than mine, I was shooting originally to make a mask to fit his head first. That didn't work. I had underestimated, since his head wouldn't fit through the neck hole, even though it would have fit the rest of the mask. So, I erred on the side of making the second mask HUGE.


Step 4: I painted them over with Gesso (primer for crafts), and then acrylic paint. The Gesso was great, since it really smoothed out the rough edges of the papier mache. I noticed that the masks were starting to get a little deformed from the weight and also from moisture, so I propped them up on wine bottles to help keep their shape while drying.

Step 5: Putting the fabric masks on was pretty tricky, as I had expected. They needed to cover the eyes precisely, which meant cutting out the eye holes beforehand, but not too much, and gluing them down without smearing glue everywhere. I glued the masks down with Elmer's all-purpose glue, but they were sticking up in some places so I had to rip them off and to re-glue a new strip. Even then, they were still a little bubbly/uneven because of all the details around the eyes, so I had to twist them in order to make them stay down. Elmer's glue is a little messy for this job. Glue gun probably would have been better, but at the time I wasn't sure how well glue gun would work on papier mache, or whether the heat might destroy the material.

With some research, the shells I made out of disposable turkey baking trays. I was lucky that when I went to the supermarket, they already had oval-shaped baking trays! I hammered them down to make the edges smoother, before painting them green and outlining the patterns using a paint marker. For minimal work, I was quite happy with them.


Step 6: For the front side of the shells, I used leftover cardboard and pieces of modeling clay to create some texture, and then I masking taped the whole thing down before painting it in yellow. I attached the front to the back side with leftover fabric, and moisture-sealed everything a dozen times. This is a photo of my husband hanging out on our rooftop on a Saturday, while I sealed the paint.

Step 7: I covered the eyes and mouth with wire frame, and used paint markers to fill in the details. The wire frame was a really great move, because the masks would have otherwise been really stuffy and hot! They were very breathable in the end. I did eventually invest in some glue and a mini glue gun. I used only the low-temperature setting when gluing everything together, but I used lots of glue to make sure there was no random wire pieces sticking out inside. (My husband was concerned about safety.) 

Step 8: We also ordered a green-man bodysuit from Amazon, as well as cheap sais and nunchucks. I also made a belt for each of us with our turtle initials on them, and I velcroed them to the front and back of the turtle shells. A detail is that velcro is strong, so I ended up sewing (by hand) the velcro to parts of the cardboard, since glue gun wasn't always reliable.

Here is us with a kid at Comic-Con! It was actually a lot of fun. People took a lot of photos of us, because they were either TMNT enthusiasts or home-made costume enthusiasts, or both! It was really fun to walk around Comic Con in our costumes.



My next pet project will be taking sewing lessons, because eventually I hope to be able to make my own sun dress. This time, Geoff doesn't know whether to believe my resolve, or not. haha

Thursday, March 20, 2014

Exponential Growth and Sustainability

I recently got interested in the idea of exponential growth in the "real world," and a little bit of research got me thinking about the possibility of framing a math project surrounding the idea of exponential growth and (environmental or social) sustainability implications.

For example, one of the things that I spoke to my Algebra 2 students about is the idea that as our world population grows exponentially, so does our utilization of non-renewable resources, our CO2 emission rates, and our general waste emission. With technology also exploding exponentially (Moore's Law), our world economic output also increases exponentially. I dug up some statistics that the world economy grows, on average, 4% a year, which means that in one person's lifetime (less than 60 years), it'll multiple 10 folds. In a century, it'll multiply 50 folds. That's insane. Even if a part of this "growth" is due to inflation, the increase in our economic output is still growing substantially over time. The earth's finite resources cannot support even a steady linear growth, let alone a sustained exponential growth. The exponential pattern we experience now is simply not environmentally sustainable, and it points to the hypocrisy of our governments who simultaneously support environmental concerns and continued economic expansion.

Similarly, I did some digging on inflation rates, causes of inflation, and rises in public four-year university tuition as an example of inflation. The trend is exponentially increasing at a staggering rate, which comes back to a social justice issue, on the front of social sustainability. I told the students that the in-state tuition of UW is already twice as expensive now as when I attended university in California 10 years ago, and I was an example of a "regular" person who barely made it through college at those former tuition rates, working part-time and on financial aid. The more I think about it, the more I think that there is a prime opportunity here for an end-of-year project linking math and either social or environmental implications. Students can research data sets of their interest, create mathematical models, and then create a PSA to educate the school community about the social and environmental implications of such trends.

Thoughts? Have you done such a project and have resources that I can look at? I want to find a balance of leaving it completely open for them to choose something they care about, and providing some general framework to ensure it's going to be productive and meaningful. But, I've been thinking a lot about this! Brainstorming slowly for the end of the year...

Saturday, March 15, 2014

Parametric Playground

I'm playing around with the idea of letting kids design and create an animation via parametric equations in GeoGebra. To play around with the idea myself, I tried to come up with some possible motions. Basically, I think that any motion that the kids can dream up, as long as it's a physically sensible pattern, we can create an animation via parametric equation.

Check these examples out: http://www.geogebratube.org/student/mnl0rUuxl . In this, I played around with a bouncing ball, a rotating circle, a rolling wheel, a dampened bouncing ball, and a ball that flies through air with downwards acceleration. In each case, there is an additional rotating point that stays with the circle as it moves around.

If they can dream it, they can create it! My thought is that they would design something, create it in GeoGebra via parametric equations, explain every part of their parametric equation, and plot x(t) and y(t) functions in terms of t (not by hand by via technology), and analyze some critical points along the graph.

Exciiiting!! I love projects like this, because as a baseline, it's reinforcing everyone's understanding of parametric equations, but the upper end is limitless to allow the creative and mathematically confident students to challenge the limits of their knowledge. For example, in order to create the dampened bouncing ball, I had to use the form y1(t) = a/(t + 1)*|sin(bt)| + c just to get the center of the ball to move/bounce in a dampened way, which made the height of the rotating point around the circle more complex:
y2(t) = csin(dt) + a/(t + 1)*|sin(bt)| + c .... In creating even something that looks simple, I incorporated rational, absolute-value, and sine functions. I would be very happy if some of my kids approached this level of complexity in their own projects.

Thursday, March 13, 2014

Creative Problem-Solving on the Rollercoaster Project

This is going to be a fuzzy-wuzzy post maybe. But, I wanted to write down how much I loved reading through the math work of my Calculus students, who completed those wonderful rollercoasters!

In the end, they took a variety of approaches:

* Some first chose the boundaries x = k, and then wrote down the boundary conditions f'(k)  and f(k). They then took a generic form of the next adjacent function g, shifted it over to make g(x-k), and then differentiated g(x-k) and set g'(k - k) = f'(k), and g(k - k) = f(k) to solve for constraints on the remaining parameters in the g equation. In other words, they first chose boundaries, then did transformations, then got the derivatives to match via a standard algebraic approach (which we eventually, at the end of the project, went through as a class in preparation for their quiz).

* Some other students were clever. They first played with functions centered around x = 0, for example f(x) = ax^3, or f(x) = ae^x. They did this because it was easy to manipulate just "a" and the x value to get a numerical derivative and general shape that they wanted. For example, if they wanted a downwards parabola that connects with a derivative of 3, they might first get y = -x^2, and then figure that at x = -1, y' = -2(-1), so y' = 2. So, they figured that if they change "a" to be -1.5, then y' = 3. Bam, they got a general shape and a derivative value to match what they wanted at the boundary. And then all they had to do was to transform
y = -1.5x^2 over and up to the boundary, which is an easy task.

* Some other students chose their boundaries LAST. They first placed the pieces of functions down loosely, then took derivatives of connected equations and set the derivative equations equal f'(x) = g'(x). In their graphing calcs they solved for the x value where this occurs, and used that as the boundary x value. After that, they just shifted the g function up or down to meet the other function in height as well. 

* Yet some other students used the principles of turning points to help them connect pieces. They also used horizontal symmetry around a vertex to predict steepness at a future part of a curve, etc.

* Many groups had trouble with ending their rollercoasters with the same height AND derivative value as in the very beginning. To help them make their lives easier, I recommended that they use the vertex form y = a(x - h)^2 + k on both ends, setting k to be the eventual height they wish to reach. They then had to put in an (x, y) value from the other boundary, and solve for a and h as a system. Even in doing this, there were some clever kids who did some clever substitution in order to make it easier to solve a rational system, while other kids turned it into a quadratic-linear system and solved graphically. Loved - it!

I felt really inspired by all of their individuality on this project. Even though it took a few more classes than I would have liked, I felt that the learning -- and moreover, the OWNERSHIP -- made it totally worth it in the end. Yeah!!!