I did it! This past week, Week 26, my half-year mark, I hit my goal of finishing the unit Circles and the entire course High School Geometry. Up to this point, I’m happy with my progress and my overall commitment and self-dedication but, looking ahead, to hit my 2020 goal of at least getting into the calculus courses by the end of the year, I think I’m going to need to speed up my rate of progress by ~25-35%. Since I started, I’ve approximately gone from doing ~5 hours/week on KA to ~6-8 hours/week. It’s likely I’m going to need to ramp up the amount of time I’m working on this to ~10 hours/week if I want to get to calculus by 2021.

Finishing up the unit Circles this week was more difficult than I anticipated (which always seems to be the case). I was introduced to the equations that are used to graph a circle which I found still find very confusing. At points throughout the week, I was being shown things that were being presented as if they had already been covered in depth in former units yet they were completely new to me. Though I struggled, here’s what I managed to figure out about the equations of a circle:

• Standard Equation
  • (x – _)^2 + (y – _)^2 = r^2
    • Ex. for a circle that has a center of (4, -3) and a radius of 2 the equation would be:
    • (x – 4)^2 + (y – +3)^2 = 2^2
  • When going through the videos/exercises, the denotation used in place of the circles center (x, y) coordinates were h and k.
• Expanded Equation
  • x^2 + y^2 + hx + ky + r = 0
    • Ex. for a circle that has a center of (-9, -7) and a radius of 5 the equation would be:
    • x^2 + y^2 + 18x + 14y + 105 = 0
    • To get from the expanded equation to the standard form, you need to group the x- and y-terms together and complete the square of each. For the above example it looks like:
      • x^2 + y^2 + 18x + 14y + 105 = 0
      • (x^2 + 18x) + (y^2 + 14y) = -105
      • (x^2 + 18x + 81) + (y^2 + 14y + 49) = -105 + 81 + 49
      • (x + 9)^2 + (y + 7)^2 = 25, therefore
      • (x – (-9))^2 + (y – (-7))^2 = 5^2
      • (-9, -7) = Circle’s center and r = 5

The main part of what confused me when learning about circle equations was how to get the center (x, y) coordinates from the standard equation. In the expanded equation example shown above, getting the circles center (x, y) coordinates occurs when going from Step 4 to Step 5. In Step 4, the values associated with the x- and y-terms are +9 and +7 respectively. For some reason, you’re supposed to reverse the signs and put them into essentially the same equation going to Step 5. Those values associated with the x- and y- terms in Step 5 ((-9) and (-7)) are the coordinates for the circles center. To be honest, I still don’t fully understand how or why this works which is disappointing and frustrating to me. Unless I missed something or am forgetting something, I feel like the KA videos skipped over how the equations for circles work.

After getting through Circles, the last thing I did this week was the course test which I scored 93% on (28/30). The first question I got wrong was one that (surprise, surprise) I didn’t fully comprehend. The question had a diagram of what was essentially four triangles and asked me to choose one of the given answers that explained why a reflection would prove that two of the four triangles would line up on top of each other and where the reflection would need be done in order to make that happen. The question was very awkwardly worded and difficult to understand so I wasn’t too upset when I got it wrong. I got the other question wrong for similar reasons in that I really didn’t understand what it was asking me. It essentially asked, if you took a 2D slice of a 3D cone from any and all theoretical directions, what is the maximum number of straight sides and curved sides could you create from any theoretical 2D slice. Again, it was worded very awkwardly and didn’t make much sense to me so I was ok with getting it wrong. I was happy, however, that when I was given questions on trig, the area/perimeter/volume of shapes, etc. (things that I had learned 1-2 months ago), I was able to remember the formulas and got the rest of the 28 questions correct.

This coming week I’ll finally be starting the course Algebra 2 (750/12,300 M.P.), beginning with the unit Polynomial Arithmetic (400/1200 M.P.). It’s funny, I’m actually excited and eager to get started and learn more about algebra. I found the course High School Geometry interesting at times, but I enjoyed learning about algebra in the previous course Algebra 1 more. Algebra to me is like a code/puzzle that you have to work through to come to the correct answer. I want to learn more about it so I can work through more difficult, and therefore more satisfying, “codes”/”puzzles”. Plus, learning more about how algebra works also makes me feel like a god damn genius.

WOOO!! MATH!!