8th Grade Math: Algebraic Expressions

• Algebraic Expressions, Linear Equations, Geometry, Statistics and Probability, and Percentage and Ratio concepts are covered in Grade 8 Mathematics.

Algebraic Expressions

• An algebraic expression is a combination of variables, constants, and algebraic operations.
• Variables are symbols (usually letters) that represent unknown values.
• Constants are fixed numerical values.
• Terms are parts of an expression separated by + or - signs.
• Coefficients are the numerical part of a term.
• Like terms have the same variables raised to the same powers and can be combined by adding or subtracting their coefficients.
• Simplifying algebraic expressions involves combining like terms.
• Expanding expressions uses the distributive property: a(b + c) = ab + ac.
• Factoring is the reverse of expanding, expressing an expression as a product of its factors.
• Identities are equations that are true for all values of the variables.
• Common identities include:
  • (a + b)² = a² + 2ab + b²
  • (a - b)² = a² - 2ab + b²
  • (a + b)(a - b) = a² - b²
  • (x + a)(x + b) = x² + (a + b)x + ab

Linear Equations in One Variable

• A linear equation in one variable can be written in the form ax + b = 0, where a and b are constants and x is the variable.
• Solving a linear equation involves finding the value of the variable that makes the equation true.
• Steps for solving linear equations:
  • Simplify both sides of the equation by combining like terms and removing parentheses.
  • Use inverse operations to isolate the variable on one side of the equation.
  • Add or subtract the same value from both sides.
  • Multiply or divide both sides by the same non-zero value.
• Transposing terms involves moving a term from one side of the equation to the other while changing its sign.
• Applications of linear equations include solving word problems involving unknown quantities.

Geometry Concepts

• Quadrilaterals are polygons with four sides.
• Types of quadrilaterals:
  • Trapezium: One pair of parallel sides.
  • Parallelogram: Two pairs of parallel sides.
  • Rectangle: Parallelogram with all angles equal to 90 degrees.
  • Square: Rectangle with all sides equal.
  • Rhombus: Parallelogram with all sides equal.
  • Kite: Two pairs of adjacent sides are equal.
• Properties of parallelograms:
  • Opposite sides are equal and parallel.
  • Opposite angles are equal.
  • Diagonals bisect each other.
• Special parallelograms (rectangles, squares, and rhombuses) have additional properties.
• Sum of the angles of a quadrilateral is 360 degrees.
• 3D Shapes: Cube, Cuboid, Cylinder and Cone.
• Euler's formula relates the number of faces (F), vertices (V), and edges (E) of a polyhedron: F + V - E = 2.

Statistics

• Data is a collection of facts or information.
• Data can be organized and presented in various ways, including:
  • Bar graphs: Use bars to represent data.
  • Histograms: Similar to bar graphs but used for continuous data, with no gaps between bars.
  • Pie charts: Divide a circle into sectors to represent proportions of data.
  • Line graphs: Use lines to show trends in data over time.
• Measures of central tendency:
  • Mean: The average of all data values (sum of values divided by the number of values).
  • Median: The middle value when the data is arranged in order.
  • Mode: The value that appears most frequently in the data.
• Range: The difference between the highest and lowest values in the data.
• Probability is a measure of the likelihood that an event will occur.
• Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes).
• Probability is expressed as a fraction, decimal, or percentage between 0 and 1 (or 0% and 100%).
• Experiments involve random outcomes.
• Events are specific outcomes of an experiment.

Percentage and Ratio

• Percentage means "out of one hundred."
• To convert a fraction or decimal to a percentage, multiply by 100.
• To convert a percentage to a fraction or decimal, divide by 100.
• Percentage increase = [(New Value - Original Value) / Original Value] * 100.
• Percentage decrease = [(Original Value - New Value) / Original Value] * 100.
• Ratio is a comparison of two quantities.
• Ratios can be written in the form a:b or as a fraction a/b.
• A proportion is an equation stating that two ratios are equal: a/b = c/d.
• To solve proportions, cross-multiply: ad = bc.
• Unitary Method: Finding the value of one unit and then using it to find the value of multiple units.
• Direct Proportion: When two quantities increase or decrease together.
• Inverse Proportion: When one quantity increases and the other decreases.