1st Quarter Math for Grade 8

1st Quarter Math Lessons for Grade 8

1. Real Numbers

Types of Numbers:
- Natural Numbers: {1, 2, 3, ...}
- Whole Numbers: {0, 1, 2, ...}
- Integers: {..., -2, -1, 0, 1, 2, ...}
- Rational Numbers: Fractions, terminating or repeating decimals.
- Irrational Numbers: Non-repeating, non-terminating decimals (e.g., √2, π).

2. Exponents and Powers

Exponents: Indicates how many times a number (base) is multiplied by itself.
Laws of Exponents:
- Product of Powers: a^m × a^n = a^(m+n)
- Quotient of Powers: a^m / a^n = a^(m-n)
- Power of a Power: (a^m)^n = a^(m*n)
- Zero Exponent: a^0 = 1 (a ≠ 0)

3. Algebraic Expressions

Terms: The parts of an expression separated by + or -.
Coefficients: The numerical factor in a term.
Like Terms: Terms with the same variable part.
Combining Like Terms: Simplifying expressions by adding/subtracting like terms.

4. Solving Equations

One-step Equations: Solving by performing the inverse operation (e.g., x + 5 = 12 → x = 12 - 5).
Two-step Equations: Involves two operations (e.g., 2x + 3 = 11 → 2x = 8 → x = 4).
Multi-step Equations: Requires combining like terms and using inverse operations.

5. Functions

Definition: A relation where each input has exactly one output.
Function Notation: f(x) represents the output of function f when the input is x.
Evaluating Functions: Substitute the input value into the function.

6. Linear Equations

Standard Form: Ax + By = C, where A, B, and C are integers.
Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
Graphing: Plotting points based on the equation or slope/intercept.

7. Systems of Equations

Definition: A set of equations with the same variables.
Solving Methods:
- Graphing: Finding the intersection point.
- Substitution: Solving one equation for a variable, then substituting.
- Elimination: Adding or subtracting equations to eliminate a variable.

8. Inequalities

Definition: A mathematical statement that compares two expressions (e.g., x > 5).
Solving Inequalities:
- Similar to equations but reverse the inequality sign when multiplying/dividing by a negative number.
Graphing: Use a number line; open circle for < or >, closed circle for ≤ or ≥.

9. Data and Statistics

Mean: Average of a set of numbers.
Median: Middle value when data is ordered.
Mode: Most frequently occurring value.
Range: Difference between the highest and lowest values.

10. Probability

Definition: The likelihood of an event happening.
Formula: P(E) = Number of favorable outcomes / Total number of outcomes.
Basic Events: Simple events that cannot be broken down any further.

These notes provide an overview of essential concepts covered in the 1st quarter of grade 8 math, helping students grasp foundational topics in algebra and statistics.

Real Numbers

Natural Numbers: Set of positive integers starting from 1 (e.g., {1, 2, 3,...}).
Whole Numbers: Includes all natural numbers plus zero (e.g., {0, 1, 2,...}).
Integers: Set of whole numbers and their negative counterparts (e.g., {..., -2, -1, 0, 1, 2,...}).
Rational Numbers: Can be expressed as fractions, includes terminating and repeating decimals.
Irrational Numbers: Cannot be expressed as fractions; includes non-repeating and non-terminating decimals like √2 and π.

Exponents and Powers

Exponents represent repeated multiplication of a base.
Laws of Exponents:
- Product of Powers: Combine exponents when multiplying (a^m × a^n = a^(m+n)).
- Quotient of Powers: Subtract exponents when dividing (a^m / a^n = a^(m-n)).
- Power of a Power: Multiply exponents when raising a power to another power ((a^m)^n = a^(m*n)).
- Zero Exponent: Any non-zero base raised to the zero power equals one (a^0 = 1).

Algebraic Expressions

Terms are the individual components of an expression, separated by addition or subtraction.
Coefficients are the numerical part of a term (e.g., in 3x, 3 is the coefficient).
Like Terms share the same variable component, enabling simplification.
Combining Like Terms simplifies expressions by adding or subtracting terms with the same variable.

Solving Equations

One-step Equations involve a single operation to isolate the variable (e.g., x + 5 = 12).
Two-step Equations require two operations (e.g., 2x + 3 = 11).
Multi-step Equations need combining like terms and using inverse operations for solution.

Functions

A function relates each input to one specific output.
Function Notation: f(x) denotes the output of function f when x is the input.
Evaluating Functions involves substituting a specific input value into the function.

Linear Equations

Standard Form: Represented as Ax + By = C, where A, B, and C are integers.
Slope-Intercept Form: Expressed as y = mx + b, where m is the slope and b is the y-intercept.
Graphing involves plotting points derived from the equation or using slope and y-intercept.

Systems of Equations

Systems consist of multiple equations sharing variables.
Solving Methods include:
- Graphing: Finding the point where equations intersect.
- Substitution: Isolating one variable in an equation and substituting into another.
- Elimination: Adding or subtracting equations to remove a variable.

Inequalities

An inequality compares two expressions, represented with symbols like >, <, ≥, or ≤.
Solving Inequalities is akin to solving equations, with caution to reverse the inequality sign when multiplying/dividing by negative numbers.
Graphing Inequalities involves the number line, with open circles indicating strict inequalities (< or >) and closed circles indicating inclusive inequalities (≤ or ≥).

Data and Statistics

Mean: Calculated by summing values and dividing by the total count.
Median: The middle number in a sorted set; if even count, the average of the two middle numbers.
Mode: The value that occurs most frequently in a data set.
Range: The difference computed by subtracting the lowest value from the highest.

Probability

Probability measures the likelihood of an event occurring, expressed as a value between 0 and 1.
Probability Formula: P(E) = Number of favorable outcomes / Total number of outcomes.
Basic Events are the simplest outcomes in a probability scenario, cannot be divided further.