Flatland

This week we learned all about geometric figures. Plane, point, line etc. When I was in AP Lit senior year of high school, I had the *joy* of being assigned a book called Flatland, which is literally about 2D geometric figures, living on a plane. Now every time I hear about points, lines and polygons I have horrible flashbacks to that book. Hence, the title of this post. Anyway, back to the real reason for this post. I remember back in middle school first learning about geometric figures, and doing it all on the computer, it was actually quite fun. This time around it was still online and still quite fun. One of the most interesting concepts to play with is angles. Especially when you get to or more involved. An angle is made up of two rays that stem from the same endpoint (not to be confused with an angel, which a spiritual being believed to act as an attendant, agent, or messenger of God, conventionally represented in human form with wings and a long robe). The most common angles are acute angles > 90, right angles which are 90, and obtuse angles which are < 90 and A ray is a part of a line segment in which one end has an endpoint, and the other end goes on for infinity (I always remember drawing arrows on the infinity end to demonstrate that fact.) I think for every student, the moment when you realize how angles react and compliment each other, theres a moment of logical wonder. One of the best representations of how the angles interact is when you introduce vertical angles. These types of angles are created by the intersection of two lines. When you add all the angles together, you get 180. The angles opposite each other have the same degree value. Its great to demonstrate these by having one of the lines rotated around, and watching how, despite the fact the angle values change, they still stay equal. It's always fun when you get to discover math by "playing" with it like that. Here is my vertical angle example from class:

Notice how angles ABD and CBE are opposite yet equal. The same goes for angles ABC and DBE. Also notice how angles ABD and ABC add up to 180, as well as ABE and DBE. 

Probability

This week's lesson was on probability. For example, the probability of spinning a 10 in the game of life, the amount of green skittles in a bag, rolling a 6 on a die, flipping a coin, etc. Something I found really interesting was what is called a probability tree. It's a way of representing the different outcomes of two or more "events' (ex. flipping a coin than rolling a die) and writing it out to show the probability of each set. Going with the example of a coin and a die, you start with the two outcomes, or branches, of the first event, here it would be either heads or tails, each with a 1/2 probability of being flipped. From each of those braches, you branch out the outcomes of the rolle of the dice, in this case we're only using one die, so the heads would have six braches, 1,2,3,4,5 and 6, and the tails would as well. Each of those outcomes would have a 1/6 chance in occurring. To figure out the probability of lets say getting heads and rolling a 4, you would multiply the two probabilities together, 1/2 x 1/6 = 1/12.  The tree would look something like this: