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Sunday, April 20, 2014

GEOMETRY ---- BASIC MATHEMATICS-----THE UNITED REPUBLIC OF TANZANIA.

GEOMETRY-- BASIC MATHEMATICS.

Geometry is all about shapes and their properties.

If you like playing with objects, or like drawing, then geometry is for you!

Geometry can be divided into:

	Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper

	Solid Geometry is about three dimensional objects like cubes, prisms, cylinders and spheres.

Hint: Try drawing some of the shapes and angles as you learn ... it helps.

Point, Line, Plane and Solid

A Point has no dimensions, only position
A Line is one-dimensional
A Plane is two dimensional (2D)
A Solid is three-dimensional (3D)

Plane Geometry

Plane Geometry is all about shapes on a flat surface (like on an endless piece of paper).

Perimeter

General Drawing Tool

Polygons

A Polygon is a 2-dimensional shape made of straight lines. Triangles and Rectangles are polygons.
Here are some more:

Pentagon

Pentagram

Hexagon

The Circle

Circle Theorems (Advanced Topic)

Symbols

There are many special symbols used in Geometry. Here is a short reference for you:

Geometric Symbols

Congruent and Similar

Angles

Types of Angles

Degrees (Angle) Radians
Congruent Angles Parallel Lines and Pairs of Angles Transversal A Triangle Has 180°
Supplementary Angles Complementary Angles

Angles Around a Point Angles on a Straight Line
Interior Angles Exterior Angles
Interior Angles of Polygons Exterior Angles of Polygons

Using Drafting Tools

Geometric Constructions

Using the Protractor Using the Drafting Triangle and Ruler Using a Ruler and Compass

Transformations and Symmetry

Transformations:

Symmetry:

Symmetry Artist

Activity: Symmetry of Shapes Activity: Make a Mandala Activity: Coloring (The Four Color Theorem)

Coordinates

Trigonometry

Trigonometry is a special subject of its own, so you might like to visit:

Solid Geometry

Solid Geometry is the geometry of three-dimensional space - the kind of space we live in ...

... let us start with some of the simplest shapes:

Common 3D Shapes

Polyhedra and Non-Polyhedra

There are two main types of solids, "Polyhedra", and "Non-Polyhedra":

Polyhedra : (they must have flat faces)

	Cubes and Cuboids (Volume of a Cuboid)
	Platonic Solids
	Prisms
	Pyramids

Non-Polyhedra: (if any surface is not flat)

	Sphere		Torus
	Cylinder		Cone

CALCULUS ------ BY. MWL. JAPHET MASATU

CALCULUS

The word Calculus comes from Latin meaning "small stone",
Because it is like understanding something by looking at small pieces.

Differential Calculus cuts something into small pieces to find how it changes.

Integral Calculus joins (integrates) the small pieces together to find how much there is.

Read Introduction to Calculus or "how fast right now?"

Limits

Limits are all about approaching. Sometimes you can't work something out directly, but you can see what it should be as you get closer and closer!

Introduction to Limits
Limits and Infinity
Evaluating Limits
Limits (Formal Definition)

Continuous Functions

Derivatives (Differential Calculus)

The Derivative is the "rate of change" or slope of a function.

Introduction to Derivatives
Slope of a Function at a Point (Interactive)
Derivatives as dy/dx
Derivative Plotter
Derivative Rules
Second Derivative
Differentiable
Finding Maxima and Minima using Derivatives
Concave Upwards and Downwards and Inflection Points
Taylor Series (uses derivatives)

Integration (Integral Calculus)

Integration can be used to find areas, volumes, central points and many useful things.

Introduction to Integration
Integration Rules
Integration by Parts
Integration by Substitution
Definite Integrals

If you want more Calculus topics covered, let me know which ones.

ALGEBRA 2 ------ BY. MWL. JAPHET MASATU.

ALGEBRA 2.

OK. So what are you going to learn here?
You will learn about Numbers, Polynomials, Inequalities, Sequences and Sums, many types of Functions, and how to solve them.
You will also gain a deeper insight into Mathematics, get to practice using your new skills with lots of examples and questions, and generally improve your mind.
With your new skills you will be able to put together mathematical models so you can find good quality solutions to many tricky real world situations.

Near the end of most pages is a "Your Turn" section ... do these! You need to balance your reading with doing. Answering questions helps you sort things out in your mind. And don't guess the answer: use pen and paper and try your best before seeing the solution.

Language

So what is this thing called Mathematics? And how do you go about learning it?

	Welcome to Mathematics
	Learning Mathematics
	The Language of Mathematics
	Symbols in Algebra

Sets

Next, we need to think about mathematics in terms of sets.
Introduction to Sets

Numbers

Now you know what a set is, let us look at different sets of numbers that you will be using:

	The Evolution of Numbers

	Prime and Composite Numbers
	Fundamental Theorem of Arithmetic

	Whole Numbers and Integers
	Rational Numbers
	Using Rational Numbers
	Irrational Numbers
	0.999... = 1
	Real Numbers
	Imaginary Numbers
	Complex Numbers
	The Complex Plane

	Common Number Sets

Inequalities

"Equal To" is nice but not always available. Maybe you only know that something is less than, or greater than. So let us learn about inequalities.

	Introduction to Inequalities	a≥b
	Properties of Inequalities
	Intervals

Exponents

You will be using exponents a lot, so get to know them well.

	Exponents
	Using Exponents in Algebra
	Squares and Square Roots
	Squares and Square Roots in Algebra
	nth Root
	Fractional Exponents
	Laws of Exponents
	Exponents of Negative Numbers

Polynomials

Polynomials were some of the first things ever studied in Algebra. They are simple, yet powerful in their ability to model real world situations.

	What is a Polynomial?
	Adding And Subtracting Polynomials
	Multiplying Polynomials
	Dividing Polynomials
	Polynomials - Long Division

	Degree (of an Expression)
	Special Binomial Products
	Difference of Two Cubes

	Factoring in Algebra
	Solving Polynomials
	Roots of Polynomials: Sums and Products

	Rational Expressions
	Using Rational Expressions

	Fundamental Theorem of Algebra
	Remainder Theorem and Factor Theorem
	General Form of a Polynomial
Graphing Polynomials
	How Polynomials Behave
	Polynomials: The Rule of Signs
	Polynomials: Bounds on Zeros

Equations

And, of course, you need to know about equations ... and how to solve them.

	Equations and Formulas
	Solving Equations
	Simplify
	Zero Product Property
	Implication and Iff
	Theorems, Corollaries, Lemmas

Graphs

Graphs can save you! They are a great way to see what is going on and can help you solve things. But you need to be careful as they may not always give you the full story.

	Cartesian Coordinates
	Pythagoras' Theorem
	Distance Between 2 Points
	Graph of an Equation
	Finding Intercepts From an Equation
	Symmetry in Equations

Linear Equations

They are just equations for lines. But they come in many forms.

	Equation of a Straight Line
	Linear Equations
	Point-Slope Equation of a Line
	General Form of Equation of a Line
	Equation of a Line from 2 Points
	Midpoint of a Line Segment
	Parallel and Perpendicular Lines

Functions

A function just relates an input to an output. But from that simple foundation many useful things can be built.

	What is a Function?
	Domain, Range and Codomain

	Evaluating Functions

	Increasing and Decreasing Functions
	Maxima and Minima of Functions
	Even and Odd Functions
	Set-Builder Notation
	Common Functions Reference: Square Function Square Root Function Cube Function Reciprocal Function Absolute Value Function Floor and Ceiling Function


	Function Transformations
	Equation Grapher
	Operations with Functions
	Composition of Functions
	Inverse Functions

Equations of Second Degree

"Second degree" just means the variable has an exponent of 2, like x². It is the next major step after linear equations (where the exponent is 1, like x).

	Quadratic Equations
	Factoring Quadratics
	Completing the Square
	Derivation of Quadratic Formula
	Graphing Quadratic Equations
	Circle Equations

Solving

You already have experience in solving, but now you can learn more!

	Mathematical Models
	Approximate Solutions
	Intermediate Value Theorem
	Solving Radical Equations
	Change of Variables
	Algebra Mistakes

Solving Inequalities

We learned about inequalities above, now let's learn how to solve them.

	Solving Inequalities
	Graphing Linear Inequalities
	Solving Quadratic Inequalities
	Solving Rational Inequalities
	Absolute Value in Algebra

Exponents and Logarithms

You know about exponents ... well logarithms just go the other way. And together they can be very powerful.

	Working with Exponents and Logarithms
	Exponential Function
	Logarithmic Function
	Exponential Growth and Decay

Systems of Linear Equations

What happens when you have two or more linear equations that work together? They can be solved! It isn't too complicated, but can take quite a few calculations.

	Systems of Linear Equations

	Matrices
	How to Multiply Matrices
	Determinant of a Matrix
	Inverse of a Matrix:
	Using Elementary Row Operations (Gauss-Jordan) Using Minors, Cofactors and Adjugate
	Matrix Calculator

	Solving Systems of Linear Equations Using Matrices

	Systems of Linear and Quadratic Equations