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8TH Grade Mathematics [Pre-Algebra]

TABLE OF CONTENTS
  • MAIN OBJECTIVES
  • EQUATIONS AND ENEQUALITIES
    |--Writing expressions
    |--Solving equations
    |--Solving Inequalities
  • EQUATIONS IN GEOMETRY
    |--Length and Perimeter
    |--Area and Volume
    |--Congruent Figures
  • PROBABILITY, STATISTICS, AND GRAPHS
    |--Probability
    |--Statistics
    |--Line graphs and bar graphs
  • GRAPHS OF EQUATIONS AND ENEQUALITIES
    |--The Coordinate Plane
    |--Slope and Intercepts
    |--Graphing Inequalities
  • REFERENCES
    MAIN OBJECTIVES

    Required Textbook: A Beka Pre-Algebra, Work-Text, Pensacola, Florida, current edition.

    [a] Understand the basics of numerical operations, quantitative reasoning and algebraic thinking.
    [i] Numerical operations include an understanding of numbers negative, positive, whole, decimal, and fraction.
    [ii] Quantitative reasoning and algebraic thinking enables you to make generalizations before performing the actual mathematical operations and study relationships among quantities.
    [iii] Algebraic thinking uses a variety of representations; numerical or graphical to model mathematical situations and solve meaningful problems.

    [b] Get the knowledge and skills to understand that a function represents a dependence of one quantity on another and can be described in a variety of ways.
    [i] Describe independent and dependent quantities in functional relationships and retrieve data to determine functional relationships between quantities.
    [ii] Describe functional relationships and write equations or inequalities to answer questions arising from the situations.
    [iii] Represent relationships among quantities using tables, graphs, diagrams, verbal descriptions, equations and inequalities.
    [iv] Identify and sketch the general forms of linear (y = x) and quadratic (y = x2) functions.
    [v] Identify the mathematical domains and ranges, determine reasonable domain and range values for given situations (√x is valid in what domain?).
    [vi] Interpret situations in terms of graphs or creates situations that fit given graphs.
    [vii] Understand how algebra can be used to express generalizations and use of symbols to represent situations. Such symbols represent unknowns and variables. These must sometimes be manipulated to simplify algebraic expressions.
    [viii] Find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary. The simplification process might use the commutative, associative, and distributive properties.

    [c] Understand that linear functions can be represented in different ways and you can determine whether or not given situation can be represented by a linear function.
    [i] Determine the domain and range values for which a linear function makes sense.
    [ii] Understand the meaning of the slope and intercepts of linear functions and interpret the effects of changing their parameters (the effects of changes in m and b on the graph of y = mx + b).
    [iii] Write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept.
    [iv] Determine the intercepts of linear functions from graphs, tables, and algebraic representations.

    [d] Understand that the graphs of quadratic functions and the effects of changing their parameters: the effects of changes in a on the graph of y = ax2 and the effects of changes in c on the graph of y = x2 + c.
    [i] Understand there is more than one way to solve a quadratic equation, solve them using appropriate methods.
    [ii] Use patterns to generate the laws of exponents and apply them in problem- solving situations.
    [iii] Analyze data and represent situations involving exponential growth and decay using tables, graphs, or algebraic methods.

    1 EQUATIONS AND ENEQUALITIES

    1.1 - Writing expressions
    Verbal phrases that suggest addition or subtraction can be translated into numerical or algebraic expressions. The phrase "5 increased by 3" translates into 5 + 3. The phrase "A number increased by 3" translates into n + 3. Tphrase "A number decreased by 4" translates into n - 4.
    5 increased by 3          -->  5 + 3
    A number increased by 3   -->  n + 3
    A number decreased by 4   -->  n - 4
    
    There is more than one way to say the same thing. For example,
    y increased by 3          -->  3 more than y
    A number decreased by 3   -->  3 less than n
    
    For multiplication and division,
    3 times as large          -->  3y or 3x
    One third as large        -->  y/3 
    
    Being able to translate verbal phrases into numerical or algebraic expressions is a fundamental skill in algebra. Translating words into symbols is equivalent to modeling a situation using an equation and variables. Similarly, algebraic equations and inequalities can represent the quantitative relationship between two or more objects.
    Variables have important roles in algebra. Often, they serve as placeholders in equations for which there are unknown quantities. In such cases, finding the specific value of the variable for which an equation is true yields the solution to the problem.
    For example, a problem uses a variable in this way to find the width of a rectangle when the area and length are known. Soppose the area is 96 square units and the length is 8 units,
                       length = 8 units
                   --------------------------
                   |                        |
                   | Area = 96 square units | <-- width = y units
                   |                        |
                   --------------------------
    
    An equation for finding the width of the rectangle is 8y = 96. The variable y does not vary; it is a placeholder representing an unknown number of units. Let's solve for y,
                   8y = 96
                    
                   8y = 96
                   --   --
                   8    8
                    y = 12
    
    Thus the width of the rectangle is 12 units.
    ...


    1.2 - Solving equations

    Suppose we wish to solve the equation
    3x + 15 = x + 25

    The important thing to remember about any equation is that the equals sign represents a balance. What an equals sign says is that whatís on the left-hand side is exactly the same as whatís on the right-hand side. So, if we do anything to one side of the equation we have to do it to the other side. If we do this, the balance is preserved. The first step in solving this equation is to perform operations on both sides so that terms involving x appear on one side only, usually the left. We can subtract x from each side, because this will remove it entirely from the right, and give
    2x+15 = 25

    We can subtract 15 from both sides to give
    2x = 10

    and finally, by dividing both sides by 2 we obtain
    x = 5

    So the solution of the equation is x = 5. This solution should be checked by substitution into the original equation in order to check that both sides are the same. If we do this, the left is 3(5) + 15 = 30. The right is 5 + 25 = 30. So the left equals the right and the solution is correct.
    For example; suppose we wish to solve the equation 2x+3= 6 - (2x - 3).
    We first remove the brackets on the right to give
    2x+3= 6 - 2x + 3

    so that
    2x+3= 9 - 2x

    This now has the same form as the equation in the first example. We can remove terms involving x from the right by adding 2x to each side.
    4x+3= 9

    Now subtract 3 from each side:
    4x = 6

    so that
    x = 6/4

    The answer can be left as is, but you can simplify the fraction
    x = 3/2

    Remember that the equation you will use to answer a question will not always be given, it is up to you to figure it out.
    For example; given this diagram,
              15           8x
           |------|----------------------|
           
           |------------79---------------|
    

    you are asked to write an equation to represent it and solve for the value of x. The total lenght of the lower line is 79, so this number goes on one side of the equal sign,
    ______ = 19

    The two segments of the upper line, put together have the same lenght as the upper line line, so these two expressions go on the other side of the equal sign,
    15 + 8x = 19
                 15 + 8x = 79
                -15           -15      First subtract 15
               ____________
                         8x = 64 
                         8x = 64
                        ___   ___      Then divide by 8
                         8     8
                          x = 8
    
    The answer is x = 8. Now lets find an equation for the number of one square foot tiles as a function of a pool's area (see diagram).


    2 EQUATIONS IN GEOMETRY

    2.1 - Length and Perimeter
    ...

    3 PROBABILITY, STATISTICS, AND GRAPHS

    3.1 - Probability
    ...

    4 GRAPHS OF EQUATIONS AND ENEQUALITIES

    4.1 - The Coordinate Plane
    ...

    REFERENCES

    [1] A Beka Pre-Algebra, Work-Text, Pensacola, Florida.