Sets, Data and Data Collection

• Set represents a group of items.
• Elements are the individual items contained within a set.
• Cardinality signifies the quantity of elements in a set.
• Subset refers to a smaller group derived from within a larger set.
• Null Set is a set containing no elements, also known as an empty set.

Set Operations

• Union (A ∪ B) combines all elements from both sets into one.
• Intersection (A ∩ B) identifies and extracts only the elements common to both sets.
• Complement (A’) includes all elements that are NOT in the specified set.
• Venn Diagrams visually represent the relationships and overlaps between sets.

Collecting and Organizing Data

• Data represents information that has been gathered, including numbers and words.
• Qualitative data consists of descriptive words expressing qualities like colors and opinions.
• Quantitative data involves numerical figures, such as age and test scores.

Data Collection Methods

• Survey involves collecting data by asking participants a series of questions.
• Observation involves collecting data by watching and recording events as they unfold.
• Interview involves directly talking to individuals and asking questions to gather data.
• Counting & Measuring involves collecting data using numerical values, like height and temperature.
• Sampling involves selecting a smaller group to represent the characteristics of a larger population.
• Random Sampling involves choosing individuals entirely by chance.
• Non-Random Sampling involves selecting particular individuals based on specific criteria.
• Organizing Data involves using tables to group and arrange numerical data to identify patterns.

Graphing Data

• Bar Graph compares different items using rectangular bars of varying lengths.
• Pie Chart represents the parts of a whole using slices of a circular pie.
• Line Graph displays changes over a period of time using connected data points.
• Stem-and-Leaf Plot organizes numerical data in ascending order for easy visualization.
• Graphs enable quick and efficient data comprehension.

Integers and Their Subsets

• Integers encompass all whole numbers, including positive, negative, and zero.
• Positive Numbers are integers greater than zero (e.g., 1, 2, 3...).
• Negative Numbers are integers less than zero (e.g., -1, -2, -3...).

Real-world Applications of Integers

• Temperature uses integers to represent hot (positive) and cold (negative) conditions.
• Money uses integers to represent savings (positive) and debt (negative).
• Vertical Movement uses integers to represent going up or down (stairs, elevators, etc.).
• Number Line visually represents integers and their relationships in a linear fashion.

Adding and Subtracting Integers

• When adding integers with the same sign, add their absolute values and keep the original sign.
• When adding integers with different signs, subtract their absolute values and keep the sign of the larger absolute value.
• Subtraction of integers can be performed by changing the minus sign to a plus sign and reversing the sign of the second number.

Methods to Add/Subtract

• Integer Chips use colored counters to represent positive (yellow) and negative (red) values.
• Number Line uses movements to the right for addition and to the left for subtraction.

GEMDAS (Order of Operations)

• GEMDAS dictates the sequence of operations in mathematical expressions.
• G represents grouping symbols (e.g. parentheses), which are evaluated first.
• E represents exponents, which are evaluated after grouping symbols.
• MD represents multiplication and division, which are performed from left to right.
• AS represents addition and subtraction, which are performed from left to right.

Example problem

• Problem: (-6 + 2) × 3 - 5²
• Step 1: Parentheses (-6 + 2 = -4).
• Step 2: Exponents (5² = 25).
• Step 3: Multiply (-4 × 3 = -12).
• Step 4: Subtract (-12 - 25 = -37).
• Following the correct order of operations ensures accurate solutions.