Math

Our current Curriculum includes several courses for the Field of Mathematics:

General Math

The topic of General Math is meant to include everything taught from first grade until the last year before you can take Algebra. You begin with Integers and you learn how to add, subtract, multiply. Soon after learning division you learn about Fractions and Decimals. Fractions provide a lead into Rational Numbers. Square Roots might be covered. These last topics are covered again in higher classes (Algebra and a few higher). Students may additional topics such as working with Graphs or an introduction to Functions. It will be interesting to see how much has changed since 1983-1990.

Algebra – Algebra Topics / Algebra

Algebra teaches the use of symbols to replace values. In Computer Science we say “variable” and it is helpful to compare what you learn in Algebra to what do when writing computer code. A lot of time is spent studying the manipulation of equations. By that we mean, the study of what you can do to an existing equation and after you are done, the two sides will still be equal. Three Axioms of Equality are introduced. Basic rules for Addition and Multiplication, involving 1 and 0 are discussed, and you probably were aware of quite a few of them from your work in General Math.

Geometry

Lines, rays, line segments and angles are introduced in Geometry. Students are taught to draw shapes using a straight edge and a compass. Hopefully at some point you will learn a little bit about Euclid, a Greek mathematician. Students learn basic rules about triangles, parallel lines and intersecting lines, with an emphasis on side lengths and angles. Students learn to look at drawings of lines and see Corresponding Angles. Students learn about Right Triangles and Trigonometric Functions. This might seem odd since Trigonometry is a stand alone subject taught in “Algebra II and Trigonometry”, a course taken after taking Geometry. Most of Trigonometry (or should we say “all”?) is about the Trigonometric Functions.

Trigonometry

Trigonometry is the study of a set of functions called Trigonometric Functions. Three trigonometric functions are introduced and then their inverses are introduced, giving the student six trigonometric functions. For the first three, Sine, Cosine and Tangent (usually abbreviated to sin, cos and tan), Cosine is Sine offset by 90 degrees: cos x = sin (x – 90). Tangent is sine divided by cosine, thus all three functions can be expressed as functions of Sine, and consequently the same is true for the three inverse functions. Work with the Unit Circle shows how the function “sine squared plus cosine squared equals one” plots all the points of a circle. This idea is a tool in several areas of Physics.

Calculus

Calculus is the study of math with infinitesimals. It begins with learning to use Limits when there is an answer we can’t “get” but we can approach it. This let us get the slope of a single point even though you need two points to calculate a slope. It also lets you calculate the area under a curve using rectangles under the curve. The flatness of the top of the rectangles creates an error, the error being area that isn’t counted, but as the rectangles are made thinner and thinner, this error decreases. Differentiation is the study that looks at point slopes and Integration is the study that looks at areas under curves. Both of these are important for the study of Kinematics in Physics.

Linear Algebra

Vectors and Matrices are studied. Matrices can be used for Linear Transformations–the laws of addition and scalar multiplication in Matrix Math correspond to the definition for Linearity: T(a+b)=T(a)+T(b) and T(cA) = cT(A). Students are introduced to Vector Spaces–and it might be helpful to students forthright that in math, almost every space is a Vector Space. Hours have been lost trying to find an explanation for “Space” without “Vector”.

Homework Page

Separate work is in progress to provide homework problems for students studying these topics. The questions aren’t intended to be difficult, but rather, to instill confidence and focus attention on specific things in one topic that will be helpful in a future topic.

Appendix A

Lists from several categories

Algebra
  • a=a – Reflexivity of Equality
  • a=b \rightarrow b=a – Symmetry of Equality
  • a=b \land b=c \rightarrow a=c – Transitivity of Equality
  • a+0=0+a=a – Identity Element of Addition
  • a(1)=(1)a=a – Identity Element of Multiplication
  • a+(-a)=(-a)+a=0 – Inverse Element of Addition
  • a\frac{1}{a}=\frac{1}{a}a=1 – Inverse Element of Multiplication
  • a(0)=0(a)=0 – Multiplication by Zero
Geometry
  • Point := a location, a point has no size.
  • Line := the set of all points such that for any two unique points in the set, the slope for those two points is a constant; a line can be defined from two points.
  • Circle := the set of all points that are equidistant from a center point.
  • Plane := a flat surface that can be defined from three points.
Calculus
  • Differential := a vanishingly small, nonzero value that often has a value constrained to another differential
  • Limit := a calculation which can be made to approach a believed value as close as we wish, by moving an independent variable closer and closer to a value of interest.
  • Differentiation := a calculation that uses a Limit and can calculate the slope for a single point in a function; a set of these calculations is a function called a derivative.