Math Vocabulary and Concepts Study Notes

Set: A collection of distinct objects, considered as an object in its own right.
Copy Set: A duplicate of a set with the same elements.
Whole Numbers: Non-negative integers including zero: {0, 1, 2, 3, …}.
Natural Numbers: Positive integers starting from 1: {1, 2, 3, …}.
Isolate: To separate a variable on one side of an equation.
Inquesty: Likely a misspelling, possibly intended as "inquiry" or "inquest," referring to an investigation or examination.
Element: An individual object within a set.
Rational Numbers: Numbers that can be expressed as a fraction of two integers (where the denominator is not zero).
Linear Equation: An equation that forms a straight line when graphed, typically expressed in the form $y = mx + b$.
Included: To be part of a set, for example, the inclusion of specific elements.
Evaluate: To calculate the value of an expression.
Formula: A mathematical relationship or rule expressed in symbols.
Subset: A set where all elements are contained within another set.
Simplify: To reduce an expression to its simplest form.
Interval Notation: A way of representing a range of numbers.

Real Numbers: All numbers on the number line, including integers, fractions, and irrational numbers.
Least Common Denominator (LCD): The smallest common multiple of the denominators of two or more fractions.
Integers: Whole numbers that can be positive, negative, or zero.
Expressions: A combination of numbers, variables, and operators (e.g., $x + 2$).
Equations: A statement that asserts the equality of two expressions.
Equivalent Equations: Different equations that yield the same solution.
Commutative Property: A property stating that the order in which two numbers are added or multiplied does not affect the sum or product.
Brackets and Parentheses: Symbols used to group parts of mathematical expressions
Associative Property: States that the way numbers are grouped does not change their sum or product.
Identity: An equation that is true for all values of the variable involved.
Compound Inequality: An inequality that combines two inequalities, usually joined by "and" or "or."
Inverse: A relationship or operation that reverses another.
Inclusive vs. Exclusive: Inclusive means including endpoints/values, while exclusive means omitting endpoints/values.
Inconsistent Equation: An equation that has no solution, often represented by parallel lines.
Distribute: To multiply a term by each term in a parentheses.
Intersection: The set of elements common to two or more sets.

Order of Operations: The rules that determine the order in which calculations should be performed, often remembered by "PEMDAS" (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Conditional Equation: An equation that may only be true under certain conditions.

Resource Manager: Responsible for ensuring that everyone has necessary materials and that ideas are shared.
- Questions to consider:
  - What ideas do we have?
  - Who can answer that question?
  - What questions should we ask the teacher?
  - What supplies do we need for this activity?
Facilitator: Ensures that everyone understands the solutions without leaving anyone out.
- Questions to consider:
  - Does anyone know where to start?
  - What does the question mean?
  - What are we supposed to do?
  - Do we all agree?
  - Does anyone have a different idea?
  - Are we ready to go on?

To find midpoint, use:
$Midpoint (x, y) = \bigg( \frac{x1+x2}{2}, \frac{y1+y2}{2} \bigg)$
To find distance between two points (x₁, y₁) and (x₂, y₂):
Distance = ext{sqrt}igg{(} (x2-x1)^2 + (y2-y1)^2 igg{)}
Writing Linear Equations:
- Find y-intercept: Set $x = 0$ to find (0, y)
- Find x-intercept: Set $y = 0$ to find (x, 0)

To graph:
1. Plot the y-intercept.
2. Use slope to find more points.
- Slope Formula:
 $m = \frac{y2 - y1}{x2 - x1}$
Find missing lengths and slopes
Parallel & Perpendicular Lines:
- Parallel lines: same slope, different intercepts.
- Perpendicular lines: slopes are negative reciprocals (m1 imes m2 = -1)

Graphing Method:
- Graph each linear equation in the system.
- Identify the intersection of the lines as the solution (x, y).
Substitution Method:
- Isolate one variable in one equation.
- Substitute this variable into the other equation to find the missing value.
Elimination Method:
- Eliminate one variable to solve for the remaining variable by adding or subtracting equations.
- Write the solution as (x, y).

The notes detailed above encompass key vocab, graphing principles, methods for solving systems, and practical strategies for collaborative learning. Each section provides comprehensive coverage to well equip students for developing their understanding of algebra and its applications.