Plus a few of them even had time at the end and chose to spend it playing with Desmos. And they loved this! (or at least they told me they did and they were engaged and I didn’t hear any of the normal “I can’t learn unless you teach me.”) AND they made a lot of the discoveries I wanted them to, some of them more formalized than others, but all of them on the right path. Now this class hates discovery learning with the force of a thousand suns. About 3/4 of them got through during the 50 minute class, with the rest having to finish up the last row for homework. The students worked in pairs in the computer lab and except for mention to a few of them that you can use the sliders instead of typing in each equation (I left off “ like it says in the directions” so I wouldn’t have to put a quarter in the sarcasm jar), I didn’t say anything to guide them. Here is the word file and the desmos file.
My coteacher and I decided that my first draft was asking a bit too much (like the original lesson did), then I found this great worksheet and decided to base my lesson on that, using Desmos instead of the graphing calculator because DUH. I knew I wanted to use desmos and I knew I wanted them to make generalizations. Well, at least a better lesson than what was offered (which was “graph these 5 equations at the same time on the TI 83, come up with generalizations like ‘if |a|<|b| then r can be negative which forms an inner loop’, then forget what you just generalized and use a t-table to graph these.”) Plus I’m helping to write the PreAP curriculum guide so I wanted to make sure it was a good lesson. Then our school decided to teach Calculus BC this year and–oops!–turns out that skill is kind of an important thing. While I did enjoy seeing where the petals and loops came from, we only got through 6 graphs and we didn’t really delve into recognizing the different equations and being able to do a quick sketch. Last year, we did some converting back and forth between polar and rectangular equations and then spent a day on graphing polar equations using t-tables. Cardioids can be created with the equations r = a(1 ± cosθ) and r = a(1 ± sinθ).This week my PreAP Precal classes delved into polar coordinates and graphs. You get a heart shaped figure as a result with a single cusp where the point returns to the beginning.
Above I graphed the rose r=3cos(6θ) see how it has 12 petals? If I were to have graphed sine instead of cosine, my graph still would have had 12 petals, but it would be tilted a little because sine and cosine are phase shifts of one another.Ĭardioids are technically created by rolling a fixed point on a circle around another circle of the same radius (seen below). Therefore, the 'n' variable determines the number of petals your flower will have if n is odd, the flower will have n petals, but if n is even, it will have 2n petals. The variable 'n' represents the b term 2π/b determines the period (distance between repetitions of the graph). Just as when graphing on the Cartesian plane, the variable 'a' determines the size (amplitude) of the curve. Polar roses can be created using the equations r = a cos( n θ) or r = a sin( n θ). With this in mind, we can make polar roses, cardioids, and lemniscates. This opens up a world of possibilities because coterminal angles (π/2 and 5π/2, for example) will have the same value of cosine or sine, and instead of going on forever in waves, they will be bounded by those values of r and theta.
The polar coordinate system looks like a series of concentric circles representing r centered at the origin. If we graph on the polar axis, we graph in terms of r and theta (the angle measure). If you plot y=sinx or y=cosx on the Cartesian (x,y) plane, you get the familiar waves that are anchored on the x-axis. When we talk about sine, cosine, and tangent functions and their reciprocals, we are talking about what are called periodic functions, meaning that they repeat values at regular intervals. So for anyone with a little precalc background, let's talk polar roses and their cousins. So I've been teaching her some topics that she hasn't yet gotten to in class (and I'm nervous that with all that's going on, she WON'T get to at all), and one of these topics has been things involving trig and polar equations of all sorts. One of my students is prepping for the SAT, and that prep includes prep for the Math SAT II. However, with all the doom and gloom we have in the news, and because while we've been self isolating in our houses, spring has arrived, I figured I'd change gears today and head back to math activities. My original plan was to publish another long form rant, this time about the murder of Thomas Becket in 1170 (I have a soft spot for everything related to Henry II of England).