Week 18 – Dec. 31st to Jan. 5th

My goal was to get through the course Algebra 1 by end of 2020 and happy to say, goal achieved!

I began the week by getting through the unit Irrational Numbers and then finished the Algebra 1 course challenge on Tuesday (Dec. 31st) morning scoring 97%. The one question I got wrong I could have gotten right if I had spent more time thinking it through. Going through the course challenge and looking back, it’s a nice feeling knowing that I’ve made progress and learning and retaining these ideas/concepts. Though I don’t know exactly what this is all leading up to, I definitely feel a sense of accomplishment.

The unit Irrational Numbers was relatively small, comprising predominantly of videos and only had two practice modules. The notes I took away from the unit are:

  • Rational Numbers
    • Can be described as the “ratio of two numbers”, i.e. any number that can be written as a fraction
    • Ex. 0.375, 1/3, 17, -12, 0.66*repeated
  • Irrational Numbers
    • Any number that cannot be written as the “ratio between two numbers”, i.e. cannot be written as a fraction, i.e. is a number that does not repeat itself and goes on forever.
    • Ex. the square root of two, Pi, etc.
  • Adding and Multiplying combinations:
    • R + R = R
    • R + I = I
    • R * R = R
    • R * I = I
    • I + I = R or I
    • I * I = R or I

After the course challenge, I was happy to start the new course High School Geometry. This week, I only managed to get through the first unit Geometry Foundations which, to be fair, was fairly large at 3000 M.P. Although it was a relatively big unit, it was mainly a refresher on a subjects I had already worked through in previous weeks. I learned that the word “geometry” means “earth” (geo-) “measurement” (metry). Other key terms that were defined included:

  • Point
    • Shown simply as a dot
    • Are often labelled point A, B, or C or X, Y, or Z
    • You cannot “move” on a point (otherwise it would be a line)
    • It has 0 dimensions
  • Line Segment
    • A line between two distinct points (two end points are required for a line segment)
    • Labelled via it’s end points, ex. AB or CD (you draw a little line over the two letters but I don’t know how to do that on here)
    • One dimensional
  • Ray
    • A line with one end point but goes on indefinitely in the opposite direction
    • The end point would be labeled with a letter (ex. A or X) with an arrow on the opposite side to indicate it goes on indefinitely in that direction
    • Written as AD-> (the arrow goes above the letters)
  • Line
    • A line that goes on forever in both directions
    • Can be drawn as a line segment (i.e. there are two distinct points on it) but has arrows on both ends indicating it goes on indefinitely in each direction
    •  Written as BC <-> (again, the line with arrows goes above the letters)
  • Collinear
    • If two or more points fall on the same line segment/ray/line, the points are collinear.
  • Congruent
    • Equal length (ex. a square has four congruent sides)
  • Adjacent
    • Two sides of a shape that meet at the same vertex
  • Supplementary Angles
    • Two angles that add up to 180 degrees
    • Ex. when you draw a cross, any two of the angles that are next to each other (i.e. not across from each other) add up to 180 degrees and are known as supplementary angles.
  • Transversal
    • One line segment/ray/line that goes through two other line segments/rays/lines
  • Corresponding Angles
    • When looking at a transversal, two angles that don’t have the same vertex but have the same measurement.
  • Polygon
    • A two dimensional shape
  • Polyhedron
    • A three dimensional shape that has flat surfaces and straight edges
    • Ex. a cube, pyramid, prism, etc.
  • Convex Polygon
    • A shape without a “dent” in it (this means all the interior angles are less than 180 degrees).
  • Concave Polygon
    • A shape with a “dent” in is (at least one interior angle of the shape is greater than 180 degrees).

I learned that the equation to find out the total sum of the interior angles in a shape is the (number of sides – 2) * 180 = total interior degrees. For example,

  • A triangle = (3 sides – 2) * 180 = 180 degrees
  • A quadrilateral = (4 – 2) * 180 = 360 degrees
  • A pentagon = (5 – 2) * 180 = 540 degrees
  • A hexagon = (6 – 2) * 180 = 720 degrees, etc.

Another way to think of it is the total number of sides of a shape minus two is the number of triangles that can be created inside any given shape. The number of triangles multiplied by 180 (every triangle has 180 degrees inside of it) equals the total degrees of the initial shape.

I was introduced to what’s known as the Triangle Inequality Theorem. This theorem states that any given side of a triangle cannot be smaller than the absolute value of other two sides subtracted from each other and cannot be bigger than the other two sides added together.

The unit spent a good amount of time covering quadrilaterals which are all shapes that have four sides. These include:

  • A square
    • All four corners are 90 degrees
    • All four sides are the same length
  • A rectangle
    • All four corners are 90 degrees
    • Opposite sides are the same length but one set of sides is longer than the other set
  • Trapezoid
    • Can have two 90 degree angles at most
    • Most common definition is four sided shape with exactly two sides being parallel
    • Some people argue that both opposite set of sides can be parallel
  • Parallelogram
    • A tilted rectangle i.e. opposite sides are parallel and no interior angle equals 90 degrees
    • Opposite sides are congruent
    • Some people argue this is also a definition of a trapezoid
  • Rhombus
    • A tilted square, i.e. opposite sides are parallel, all four sides are congruent, and no interior angle is 90 degrees
  • Kite
    • A four sided shape that has two sets of congruent, adjacent sides but opposite sides are not congruent

I learned that the formula to find the area of a trapezoid is (add the length of the two parallel sides)/2 * height. I had to work through a few of my own practice questions to get an intuitive understanding of why this works. I still find it tough to put into words why it works exactly but it does.

I relearned certain formulas for circles and worked through practice modules on these types of questions. The two formulas included were:

  • Pi = circumference/diameter
  • Area = Pi * (radius)^2

The last thing the unit covered was 3D shapes (a.k.a. polyhedrons) and worked on finding their surface areas. To find the surface area, I learned that a Polyhedron Net is the 2D shape created when “unfolding” a polyhedron. It’s possible to create many different polyhedron nets from the same polyhedron.

I’ve once again rejigged my ultimate goal. It being the start of a new year, I’ve decided my new year’s resolution is to finish all the KA calculus courses by the end of the year. I like this goal because it gives me a more realistic time-frame to complete the courses, but having that deadline will encourage me to keep going at a semi-fast pace to finish on time.