ALGEBRA 1----- BASIC MATHEMATICS ---- THE UNITED REPUBLIC OF TANZANIA.

ALGEBRA  1----BASIC   MATHEMATICS.

Algebra is great fun - you get to solve puzzles!

With computer games you play by running, jumping or finding secret things. Well, with Algebra you play with letters, numbers and symbols, and you also get to find secret things!

And once you learn some of the "tricks", it becomes a fun challenge to work out how to use your skills in solving each "puzzle".

The Basics

• Introduction to Algebra
• Balance when Adding and Subtracting (animation)
• Introduction to Algebra - Multiplication
• Order of Operations - BODMAS, or PEMDAS
• Substitution
• Equations and Formulas
• Inequalities and Solving Inequalities
• Basic Algebra Definitions

Exponents

• What is an Exponent?
• Negative Exponents
• Reciprocal in Algebra
• Square Roots, Cube Roots, and nth Roots
• Surds
• Simplifying Square Roots
• Fractional Exponents
• Laws of Exponents
• Using Exponents in Algebra
• Multiply and Divide Variables with Exponents

Simplifying

• Expanding (removing parentheses)
• Multiplying Negatives
• Associative, Commutative and Distributive Laws
• Cross Multiply
• Directly Proportional and Inversely Proportional
• Fractions in Algebra

Factoring

• Factoring - Introduction

Logarithms

• What Is A Logarithm

Polynomials

• What is a Polynomial?
• Adding And Subtracting Polynomials
• Multiplying Polynomials
• Polynomial Long Multiplication
• Rational Expressions
• Dividing Polynomials
• Polynomial Long Division
• Conjugate
• Rationalizing The Denominator
• Special Binomial Products

Linear Equations

• Equation of a Straight Line
• Explore the Straight Line Graph
• Cartesian Coordinates

Quadratic Equations

• Quadratic Equations
• Quadratic Equation Solver
• Factoring Quadratics
• Completing the Square

Functions

• Definition of a Function

Sequences and Series

• Sequences and Series
• Sequences - Finding a Rule

VECTORS ------ FORM FOUR BY. MWL. JAPHET MASATU

VECTORS.

This is a vector:

A vector has magnitude (how long it is) and direction:

The length of the line shows its magnitude and the arrowhead points in the direction.

You can add two vectors by simply joining them head-to-tail:

And it doesn't matter which order you add them, you get the same result:

Example: A plane is flying along, pointing North, but there is a wind coming from the North-West.

The two vectors (the velocity caused by the propeller, and the velocity of the wind) result in a slightly slower ground speed heading a little East of North.
If you watched the plane from the ground it would seem to be slipping sideways a little.

Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seem to fly sideways. Vectors help explain that.

Subtracting

You can also subtract one vector from another:

• first you reverse the direction of the vector you want to subtract,
• then add them as usual:

Other Notation

A vector can also be written as the letters of its head and tail with an arrow above, like this:

Calculations

Now ... how do we do the calculations?
The most common way is to break up a vector into x and y pieces, like this:

The vector a is broken up into
the two vectors a_x and a_y

Adding Vectors

And here is how to add two vectors after breaking them into x and y parts:

The vector (8,13) and the vector (26,7) add up to the vector (34,20)

Example: add the vectors a = (8,13) and b = (26,7)
c = a + b
c = (8,13) + (26,7) = (8+26,13+7) = (34,20)

Subtracting Vectors

Remember: to subtract, first reverse the vector you want to subtract, then add.

Example: subtract k = (4,5) from v = (12,2)
a = v + −k
a = (12,2) + −(4,5) = (12,2) + (−4,−5) = (12−4,2−5) = (8,−3)

Magnitude of a Vector

The magnitude of a vector is shown by two vertical bars on either side of the vector:

|a|

OR it can be written with double vertical bars (so as not to confuse it with absolute value):

||a||

You can use Pythagoras' theorem to calculate it:

|a| = √( x² + y² )

Example: what is the magnitude of the vector b = (6,8) ?
|b| = √( 6² + 8² ) = √( 36+64 ) = √100 = 10

A vector with magnitude 1 is called a Unit Vector.

Vector vs Scalar

When using vectors we call an ordinary number a "scalar".

Scalar: just a number (like 7 or −0.32) ... definitely not a vector.

A vector is often written in bold,

so c is a vector, it has magnitude and direction
but c is just a value, like 3 or 12.4

Example: kb is actually the scalar k times the vector b.

Multiplying a Vector by a Scalar

When you multiply a vector by a scalar it is called "scaling" a vector, because you change how big or small the vector is.

Example: multiply the vector m = (7,3) by the scalar 3

a = 3m = (3×7,3×3) = (21,9)

It still points in the same direction, but is 3 times longer

(And now you know why numbers are called "scalars", because they "scale" the vector up or down.)

Multiplying a Vector by a Vector (Dot Product and Cross Product)

How do you multiply two vectors together? There is more than one way! The scalar or Dot Product (the result is a scalar). The vector or Cross Product (the result is a vector). (Read those pages for more details.)

More Than 2 Dimensions

The vectors we have been looking at have been 2 dimensional, but vectors work perfectly well in 3 or more dimensions:

Example: add the vectors a = (3,7,4) and b = (2,9,11)
c = a + b
c = (3,7,4) + (2,9,11) = (3+2,7+9,4+11) = (5,16,15)

Example: subtract (1,2,3,4) from (3,3,3,3)

(3,3,3,3) + −(1,2,3,4)
= (3,3,3,3) + (−1,−2,−3,−4)
= (3−1,3−2,3−3,3−4)
= (2,1,0,−1)

Example: what is the magnitude of the vector w = (1,-2,3) ?

|w| = √( 1² + (-2)² + 3² ) = √( 1+4+9 ) = √14

Magnitude and Direction

You may know a vector's magnitude and direction, but want its x and y lengths (or vice versa):

	<=>
Vector a in Polar Coordinates		Vector a in Cartesian Coordinates

You can read how to convert them at Polar and Cartesian Coordinates, but here is a quick summary:

From Polar Coordinates (r,θ) to Cartesian Coordinates (x,y)	From Cartesian Coordinates (x,y) to Polar Coordinates (r,θ)
x = r × cos( θ ) y = r × sin( θ )	r = √ ( x2 + y2 ) θ = tan-1 ( y / x )

An Example Sam and Alex are pulling a box. Sam pulls with 200 Newtons of force at 60° Alex pulls with 120 Newtons of force at 45° as shown What is the combined force, and its direction?

Let us add the two vectors head to tail:

Now, convert from polar to Cartesian (to 2 decimals):
Sam's Vector:

• x = r × cos( θ ) = 200 × cos(60°) = 200 × 0.5 = 100
• y = r × sin( θ ) = 200 × sin(60°) = 200 × 0.8660 = 173.21

Alex's Vector:

• x = r × cos( θ ) = 120 × cos(-45°) = 120 × 0.7071 = 84.85
• y = r × sin( θ ) = 120 × sin(-45°) = 120 × -0.7071 = −84.85

Now we have:

Now it is easy to add them:

(100, 173.21) + (84.85, −84.85) = (184.85, 88.36)

We can convert that to polar for a final answer:

• r = √ ( x² + y² ) = √ ( 184.85² + 88.36² ) = 204.88
• θ = tan^-1 ( y / x ) = tan^-1 ( 88.36 / 184.85 ) = 25.5°

And we have this (rounded) result:		And it looks like this for Sam and Alex:

They might get a better result if they were shoulder-to-shoulder!

BASICS OF COMPUTER -----BY. MWL. JAPHET MASATU.

BASICS   OF   COMPUTER.  INTRODUCTION: 1. Historical Development of Computers 2. Computer Hardware 3. Types of Computers 4. The Organization of Computer System 5. Architecture of a Computer System 6. Information Flow With in a Computer 7. Input and Output Devices 8. Types of Printer 9. I/O Devices (Input and Output devices) 10. Memory of Computer 11. Types of Memories 12. Features of a Computer 13. Binary Coding System 14. BinaryFractions 15. Octal andHexadecimal Number Systems 16. Character Representation 17. Software Components 18. Programming Languages 19. Assembler, Compilers and Interpreters 20. Types of Software 21. Information Organization and File Management 22. Operating Systems MS-DOS 23. Operating System Windows 24. Parts of Window Operating Systems 25. Working with Documents in Windows OS 26. Computer Applications 27. Word Processing 28. Data Communication 29. E-mail 30. Networks 31. Internet

SURVEYING & CHAIN SURVEY / TAPE SURVEY ---- BY. MWL. JAPHET MASATU

Watershed Management Including Surveying  INTRODUCTION: 1. Introduction to Surveying 2. Object, Use & Principles of Surveying 3. Difference between Plane and Geodetic Surveying 4. Instruments used in Surveying 5. Instruments for Setting out Right Angles 6. Method of Measuring Distance 7. Ranging out Survey Line 8. Chain Survey 9. Cross Staff Survey 10. Plane Table Surveying 11. Methods of Plane Tabling 12. Computation of Area 13. Introduction to Levelling 14. Dumpy Level 15. Classification and Principle of Leveling 16. Study of Contour 17. Introduction to Runoff 18. Factors Affecting runoff 19. Direct Runoff and Time of Concentration 20. Runoff Coefficient 21. Runoff Measuring Devices 22. General Requirement for the Setting and Operation of Weirs 23. Parshall Flumes 24. Orifices 25. Principles of Soil and Water Conservation 26. Types of Soil & Water Erosion 27. Factors Affecting Soil & Water Erosion 28. Measures for Soil and Water Conservations 29. Terracing 30. Contour and Graded Bunding 31. Types of Pumps 32. Centrifugal Pumps, Vertical Turbine Pumps and Submersible Pumps 33. Concepts of Watershed Management 34. Classification of Watershed 35. Watershed Management

CHAIN / TAPE SURVEY ----- BY. MWL. JAPHET MASATU.

CHAIN SURVEY  OR   TAPE  SURVEY

Chain survey or  Tape  Survey is the simplest method of surveying. In this survey only measurements are taken in the field, and the rest work, such as plotting calculation etc. are done in the office. This is most suitable adapted to small plane areas with very few details. If carefully done, it gives quite accurate results. The necessary requirements for field work are chain, tape, ranging rod, arrows and some time cross staff.

Survey Station:

Survey stations are of two kinds

1. Main Stations
2. Subsidiary or tie

Main Stations:

Main stations are the end of the lines, which command the boundaries of the survey, and the lines joining the main stations re called the main survey line or the chain lines.

Subsidiary or the tie stations:

Subsidiary or the tie stations are the point selected on the main survey lines, where it is necessary to locate the interior detail such as fences, hedges, building etc.

Tie or subsidiary lines:

A tie line joints two fixed points on the main survey lines. It helps to checking the accuracy of surveying and to locate the interior details. The position of each tie line should be close to some features, such as paths, building etc.

Base Lines:

It is main and longest line, which passes approximately through the centre of the field. All the other measurements to show the details of the work are taken with respect of this line.

Check Line:

A check line also termed as a proof line is a line joining the apex of a triangle to some fixed points on any two sides of a triangle. A check line is measured to check the accuracy of the framework. The length of a check line, as measured on the ground should agree with its length on the plan.

Offsets:

These are the lateral measurements from the base line to fix the positions of the different objects of the work with respect to base line. These are generally set at right angle offsets. It can also be drawn with the help of a tape. There are two kinds of offsets:

1) Perpendicular offsets, and
2) Oblique offsets.
 
The measurements are taken at right angle to the survey line called perpendicular or right angled offsets.

The measurements which are not made at right angles to the survey line are called oblique offsets or tie line offsets.

Procedure in chain survey:

1. Reconnaissance:

The preliminary inspection of the area to be surveyed is called reconnaissance. The surveyor inspects the area to be surveyed, survey or prepares index sketch or key plan.

2. Marking Station:

Surveyor fixes up the required no stations at places from where maximum possible stations are possible.

3. Then he selects the way for passing the main line, which should be horizontal and clean as possible and should pass approximately through the centre of work.

4. Then ranging roads are fixed on the stations.

5. After fixing the stations, chaining could be started.

6. Make ranging wherever necessary.

7. Measure the change and offset.

8. Enter in the field the book.

STATISTICS FORM FORM TWO & THREE ----- BY. MWL. JAPHET MASATU.

NUMERACY   SKILLS

TABLE  OF   CONTENTS:

1. Using averages
   1. Introduction
   2. Part one: The Mean
   3. Part two: The Median
   4. Part three: The Mode
2. Working with percentages
   1. Introduction
   2. Using percentages to compare information
   3. Calculating percentage change
   4. Calculating percentage increase or reductions
   5. Summary
3. Measures of variability: the range, inter-quartile range and standard deviation
   1. Introduction
   2. The Range
   3. The Inter-quartile Range
   4. Calculating the Inter-quartile range using Excel
   5. The Standard Deviation
   6. Population and sample standard deviations
   7. Formulae for the standard deviation
   8. Calculating the standard deviation using Excel
   9. Summary
4. Presenting numerical data
   1. Introduction
   2. Including numbers in the main body of text
   3. Presenting numbers in tables
   4. When to use tables
   5. Table design
   6. Examples of poor and better practice in the presentation of data in tables
5. Graphs
6. Types of Graph
   1. Bar charts
   2. Histograms
   3. Pie charts
   4. Line graphs
   5. Scatter plots
   6. Good graph design
   7. Components of a graph
   8. Chart area
   9. Reference to source of information
7. Other design considerations
   1. Use of colour and shading
   2. 3D effects
   3. Summary
8. Bar charts
   1. What is a bar chart?
   2. What types of data can be displayed using a bar chart?
   3. Different types of bar charts
9. Histograms
   1. What are histograms?
   2. Grouping data in a histogram
10. Pie charts
    1. What is a pie chart?
    2. What data can be presented using a pie chart?
    3. Design issues
    4. Comparing pie charts
    5. 3D effects and exploding pie charts

THE STANDARD DEVIATION ----- BY. MWL. JAPHET MASATU

THE  STANDARD   DEVIATION ---- BY.   MWL.   JAPHET   MASATU.

The standard deviation is a measure that summarises the amount by which every value within a dataset varies from the mean. Effectively it indicates how tightly the values in the dataset are bunched around the mean value. It is the most robust and widely used measure of dispersion since, unlike the range and inter-quartile range, it takes into account every variable in the dataset. When the values in a dataset are pretty tightly bunched together the standard deviation is small. When the values are spread apart the standard deviation will be relatively large. The standard deviation is usually presented in conjunction with the mean and is measured in the same units.
In many datasets the values deviate from the mean value due to chance and such datasets are said to display a normal distribution. In a dataset with a normal distribution most of the values are clustered around the mean while relatively few values tend to be extremely high or extremely low. Many natural phenomena display a normal distribution.
For datasets that have a normal distribution the standard deviation can be used to determine the proportion of values that lie within a particular range of the mean value. For such distributions it is always the case that 68% of values are less than one standard deviation (1SD) away from the mean value, that 95% of values are less than two standard deviations (2SD) away from the mean and that 99% of values are less than three standard deviations (3SD) away from the mean. Figure 3 shows this concept in diagrammatical form.

If the mean of a dataset is 25 and its standard deviation is 1.6, then

1. 68% of the values in the dataset will lie between MEAN-1SD (25-1.6=23.4) and MEAN+1SD (25+1.6=26.6)
2. 99% of the values will lie between MEAN-3SD (25-4.8=20.2) and MEAN+3SD (25+4.8=29.8).

If the dataset had the same mean of 25 but a larger standard deviation (for example, 2.3) it would indicate that the values were more dispersed. The frequency distribution for a dispersed dataset would still show a normal distribution but when plotted on a graph the shape of the curve will be flatter as in figure 4.