Sitting down to write this post, I don’t have the good news I was hoping to have. I only got through the unit Equations this week but it certainly wasn’t due to a lack of effort. Looking back through my notebook, I wrote out 22 pages (front and back) worth of notes this week which is most likely my record. I spent close to 15 hours working through videos and exercises so overall I feel ok about not making as much progress through KA as I had hoped since I put in a solid effort.

I don’t have any noteworthy updates regarding the Coronavirus. Since last week the number of total cases has risen essentially as predicted. It’s hard to say but it seems to me that the “curve” may be flattening which is a big relief though it’s pretty clear we’re far from being out of the woods, so to speak. I will have a better idea of whether the curve is actually flattening by the end of next week.

It seems like people are starting to really struggle to continue to self-isolate. My impression from social media is that people are starting to socialize again. My feeling is that when people initially started social isolating they didn’t realize what it actually meant and how it would feel.  Having gone through times where I felt isolated and alone for months on end, becoming depressed and verging on suicidal, it bothers me that some people can’t handle being isolated for the 4 weeks it’s been. I realize it’s a contemptuous perspective, but I feel like the pain people are feeling from being on their own for this relatively short amount of time is very little compared to the pain I felt at my most isolated and makes me think they’re weak. I realize it’s impossible to fairly compare people’s mental/emotional pain but, nonetheless, that’s how I feel.

Getting back to KA, the first thing I learned (or perhaps relearned) this week was about inverse functions. An inverse function (similar to an inverse fraction) is essentially the mirror image of its original function flipped across a line of symmetry that runs through the point (0, 0) and has a slope of 1/1. Similarly to how the inverse of the fraction 3/4 would be 4/3, the inverse coordinate from a function that went through the point (2, 5) would be the coordinate (5, 2). In order to find an inverse functions equation you follow these steps:

• Original function:
  • h(x) = y = (-2x – 1)/(x + 5)
• To find the inverse function (written as h^-1(x)), you take the original equation, swap the x and y variables and then solve for y:
  • OG function — h(x) = y = (-2x – 1)/(x + 5)
    • Swap the x and y variables:
  • Inverse function — h^-1(x) = x = (-2y – 1)/(y + 5)
  • Solve for y:
    • x = (-2y – 1)/(y + 5)
    • x(y + 5) = -2y – 1
    • xy + 5x = -2y – 1
    • 2y + xy = -5x – 1
    • y(x + 2) = -5x – 1
    • y = (-5x – 1)/(x + 2) = h^-1(x)

I next worked through videos and questions that had me solving equations which contained square and cube roots. Solving these types equations which contained roots was not too difficult, however I struggled to understand what are known as extraneous solutions. Extraneous solutions are solutions that can be thought of as ‘answers’ (i.e. values for variables) that don’t ‘work’ when substituted back into the original equation. You most often get an extraneous solution when you take the square root of one side of an equation because this results in two possible answers, the negative and positive integer. For example, the square root of 36 could be either 6 or -6. Often, putting the negative number back into the original equation will not result in a correct/equal equation and is therefore considered the extraneous solution. The phrasing of these questions was incredibly confusing to me and very frustrating to get through.

About halfway through the week I got stuck not remembering how to factor quadratics with a leading coefficient greater than 1. For the first time, I had to go back into KA and review/relearn how to do something which I was initially pretty frustrated and disappointed about having to do. Eventually I came to the realization that, though it felt like I was going backwards, it’s imperative to have a foundational understanding of factoring in order to move forward. An example question of factoring quadratics with a leading coefficient greater than 1 would be:

• a(x)^2 – b(x) – c = 6x^2 – 5x – 4
  • Step 1 – Multiply a and c
    • 6 * -4 = -24
  • Step 2 – Find two numbers whose product equals the product of a and c but when added together equal b.
    • -8 * 3 = -24 = a * c
    • -8 + 3 = -5 = b
  • Step 3 – Use the two numbers in Step 2 in place of b in the original equation and factor.
    • 6x^2 – 5x – 4
      • = 6x^2 + 3x -8x – 4
      • = 3x(2x + 1) -4(2x + 1)
      • = (2x + 1)(3x – 4)

The final section of this unit worked through interpreting functions when seeing them in graphical form. Relative to extraneous solutions, this part was a breeze. In this section I used a website called Desmos which is essentially a graphing calculator. It turns out graphing calculators are dope and make things a lot easier. The exercises I worked through gave me questions that asked me to input the functions into Desmos and, based on the resulting graph, find the solutions (i.e. figure out the coordinates of the points where the functions intersect). This was simple but it gave me a much better intuitive understanding for what functions look like and made functions as a whole much easier to understand.

One thing I got better at throughout this week was writing my notes out more legibly. As insignificant as this may sound, I often make simple mistakes because I can’t read my writing properly. Taking more time to write my notes and making them more clear made things much easier going from step to step. Looking back through my notebook, I also think my notes look more beautiful (which is a weird thing to say) which I find very satisfying and something I find oddly important, though I can’t put my finger on exactly why I think it’s important.  

In the past two weeks I’ve gone through sections which, relative to past material, I found very difficult to understand. I’m worried that the rate of difficulty going from unit to unit seems to be going up exponentially. When I first started KA I assumed that as long as I fully understood any particular unit the following unit should be relatively easy to understand. In my mind, I thought of each successive unit as being a “+1” in difficulty from the unit before. If I was to go from arithmetic to calculus, however, that would be something like a “+40” jump in difficulty and would be much too difficult to understand. Going through these past two weeks, however, I feel like certain parts of what I’ve learned have been a “+3” jump in difficulty from the previous unit and makes me worried that I’m getting into material that I’m simply not intelligent enough to understand. An alternative thought I have is that KA has just done a poor job at explaining some of these sections.

It’s looking less likely that I’m going to get through this course by the end of April. I’ll have to get through the last 3 units and finish the course test in the next 11 days. The units aren’t small either. I’m going to try my hardest to make it happen but I won’t be too upset if it doesn’t. I’m more concerned with fully understanding what I’m learning than I am about getting through KA in a certain time frame. That being said, I believe I’ve pulled off a KA hat-trick twice (i.e. finishing 3 units in 1 week) so it may still be doable. I’m really hoping I don’t run into any “+3” sections otherwise a hat-trick probably won’t hap-pen.