I did it. This week I achieved my goal of finishing the units Quadratics and Polynomials and Equations and Geometry, AND finished the Algebraic Basics course challenge. In Quadratics and Polynomials, I started getting a much better understanding of how polynomials and quadratics work, and understanding how to factor and distribute them. I found out some important equations and “rules” to remember include:

• (x^2 – a) = (x + a)(x – a) (a.k.a.  “Difference of Squares”),
• (a + b)^2 = a^2 + 2ab + b^2 (equation used to factor a “Perfect Square Trinomial”),
• (a – b)^2 = a^2 – 2ab + b^2 (again, an equation used to factor a “Perfect Square Trinomial”),
• (Ax + B)^2 = (Ax + B)(Ax + B) = (A^2x^2 + 2ABx + B^2)
  • Ex. 16x^2 + 24x + 9 = (4x)^2 + 2(4)(3)x + 3^2
• When factoring a trinomial to a binomial, 1) look for products to factor out, 2) check to see if it is a perfect square trinomial, and 3) then look for 2 numbers that multiply together to get a and b and add together to get c.
  • Ex. 2x^2 – 40x +200 = 2((A)x^2 (C)–20x + (B)100) = 2(x^2 – 10) + (-10x +100) = 2(x(x – 10) + -10(x – 10) = 2(x – 10)(x – 10) = 2(x – 10)^2

I realized that “factoring” a trinomial means turning it into 2 binomials and “distributing” 2 binomials means multiplying them together to get a trinomial.

The following unit, Equations and Geometry, started out by working through questions with angles. I learned the denotation symbols for the terms Perpendicular, Ray (although I’m still not 100% clear on the definition of Ray) Angle, and Measurement (m). Apart from measurement, I don’t know how (or if it’s even possible) to show the symbols for the other terms here. One thing that seems a bit odd to me is that the letter used to denote Measurement (“m”) is the same letter used to denote slope (also “m”). It makes me wonder if they’re related in some way that makes it reasonable to use the same letter.

The unit then went on to work through questions using the Pythagorean Theorem (a^2 + b^2 = c^2). This theorem states that if you measure the two sides of a right triangle that meet at the 90 degree angle, square them, and add them together, it will equal the third side (a.k.a. the hypotenuse) squared.

Next, I learned why the three corners of a triangle always add up to 180 degrees and, knowing this, worked through questions to find out the missing value of one angle of a triangle. I learned that:

• A “Congruent” triangle has the exact same dimensions as another triangle and is denoted with the symbol ~ with an equal sign below it,
• A “Similar” triangle is a triangle that is the same shape as another (i.e. all angles are the same) except all three sides are scaled up or down by the same factor (i.e. it’s the same shape just bigger or smaller) and is denoted with the symbol ~, and
• (k) is the variable most often used for “scale factor” in these types of questions.

To determine if a triangle is Similar to another, I learned three methods to use:

1. Angle, Angle (A.A.)
   • If two angles are the same in two different triangles, they’re similar,
2. Side, Side, Side (S.S.S.)
   • If the ratio of all three sides in two different triangles are the same (i.e. all sides are scaled up or down by the same factor), the triangles are similar, and
3. Side, Angle, Side (S.A.S.)
   • If two sides of two triangles have the same scale and the angle between the sides in both triangles is the same, the triangles are similar.

Lastly, I completed the course challenge and got 28/30 questions correct. The two questions I missed both had to do with inequalities. I got the first question wrong because I didn’t fully comprehend what the question was asking. Had I understood it, I’m certain I would have got it correct. The second question showed me an inequality and four graphs to represent it asking me to choose the graph that fit the inequality. I got the type of graph right but chose one with a solid line (i.e. less-than-or-equal-to) when I should have chosen the dotted line (i.e. less-than).

I’m happy to be starting the next course Algebra 1. The first unit of the course is already done (woo!) and the second unit is called Solving Equations and Inequalities which, based on the last course challenge, will be useful for me to do as a refresher. This course is bigger than Algebra Basics but it is already 28% completed (again, woo!). My hope is that I can finish it before December. That said, considering the grind I had to go through this week to finish the two units and the course challenge, I’m not going to commit to that timeline as a goal just yet.