Scientific Notation

This section covers the conversion of numbers to scientific notation format.

Definition: Scientific notation expresses numbers in the form a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.

Highlight: Scientific notation and standard form are equivalent terms.

Example:

• 2,000,000 = 2 × 10⁶
• 0.0000002 = 2 × 10⁻⁷

Completing the Square

A detailed guide to completing the square steps and methodology.

Formula: The general form is $x + a$² + b

Example: Converting x² + 10x + 7 to completed square form:

1. Half the coefficient of x (10 ÷ 2 = 5)
2. Square this number (5² = 25)
3. Subtract from constant (7 - 25 = -18)
4. Final form: $x + 5$² - 18

Significant Figures

An explanation of significant figures in numerical representation.

Definition: Significant figures are digits that carry meaningful value in a number.

Example:

• 53,879 to 1 significant figure = 50,000
• 0.005089 to 2 significant figures = 0.0051

Highlight: Every digit except leading and trailing zeros is significant.

Algebraic Fractions

Detailed coverage of simplifying algebraic fractions.

Definition: Algebraic fractions can be simplified by canceling common factors in numerator and denominator.

Example: Simplifying complex fractions like $x-2$/6x = $x+1$/2x

Adding and Subtracting Algebraic Fractions

Comprehensive guide to operations with algebraic fractions.

Highlight: The process follows the same principles as regular fraction operations.

Example: Detailed solutions for combining fractions with different denominators.

Expanding Brackets

Guide to expanding algebraic expressions.

Rule: Multiply each term inside the bracket by the term outside.

Example:

• 7$3a + 4b - 5c$ = 21a + 28b - 35c
• 2$5x - 4$ - 3$2x - 1$ = 10x - 8 - 6x + 3 = 4x - 5

Arcs and Sectors

Comprehensive coverage of circular geometry calculations.

Formula:

• Arc Length = (θ/360°) × πd
• Sector Area = (θ/360°) × πr²

Example: Calculating arc length for a 14° sector with diameter 8cm.

Vocabulary:

• θ represents the angle
• r represents the radius
• d represents the diameter

Standard Deviation

This section introduces the fundamental concepts of variance and standard deviation.

The standard deviation formula is presented as: S.D = √$(Σ(x-x̄)²)/(n-1)$

Definition: Standard deviation measures the spread of data points from their mean value.

Example: A detailed calculation using school absentee data (32, 29, 33, 33, 38) demonstrates the step-by-step process of finding standard deviation.

Vocabulary:

• Σ represents the sum
• x̄ represents the mean
• n represents the number of values