Geometry Basics: Shapes, Angles, and Measurement

Geometry

• Properties of 2D and 3D Shapes:
  • Angles: acute, obtuse, right, straight, and reflex
  • Sides: congruent, opposite, and adjacent
  • Vertices: number of vertices in different shapes (e.g., triangle, quadrilateral, hexagon)
• Types of Angles:
  • Complementary angles: sum of two angles is 90°
  • Supplementary angles: sum of two angles is 180°
  • Adjacent angles: share a common vertex and side
• Calculating Perimeter and Area:
  • Perimeter: sum of all side lengths
  • Area: formulas for triangles, quadrilaterals, and other shapes
• Transformations:
  • Reflections: flipping shapes over a line
  • Rotations: turning shapes around a fixed point
  • Translations: sliding shapes without changing size or orientation

Data Analysis

• Types of Data:
  • Qualitative data: descriptive, non-numerical data (e.g., colors, flavors)
  • Quantitative data: numerical data (e.g., heights, scores)
• Data Displays:
  • Bar graphs: comparing categorical data
  • Picture graphs: using symbols to represent data
  • Histograms: showing frequency distributions
• Measuring Central Tendency:
  • Mean: average value of a dataset
  • Median: middle value of a dataset
  • Mode: most frequent value in a dataset
• Interpreting Data:
  • Analyzing graphs and charts to draw conclusions
  • Identifying patterns and trends

Equations

• Simple Equations:
  • One-step equations: solving for a variable using addition, subtraction, multiplication, or division
  • Two-step equations: solving for a variable using a combination of operations
• Equation Properties:
  • Reflexive property: a = a
  • Symmetric property: if a = b, then b = a
  • Transitive property: if a = b and b = c, then a = c
• Solving Equations:
  • Using inverse operations to isolate variables
  • Checking solutions by plugging them back into the equation
• Applications of Equations:
  • Real-world problems: using equations to model and solve problems
  • Word problems: translating words into equations

Geometry

• Properties of 2D and 3D Shapes
  • Angles can be acute (less than 90°), obtuse (greater than 90°), right (90°), straight (180°), or reflex (greater than 180°)
  • Congruent sides have the same length, while opposite sides are opposite each other
  • The number of vertices in a shape determines its name, such as a triangle (3 vertices), quadrilateral (4 vertices), or hexagon (6 vertices)
• Types of Angles
  • Complementary angles add up to 90° (e.g., 30° + 60°)
  • Supplementary angles add up to 180° (e.g., 120° + 60°)
  • Adjacent angles share a common vertex and side, but do not overlap
• Calculating Perimeter and Area
  • The perimeter of a shape is the sum of all its side lengths
  • The area of a triangle is calculated using the formula (base × height) / 2
  • The area of a quadrilateral is calculated using the formula (base × height) or (side × side)
• Transformations
  • Reflections involve flipping a shape over a line, resulting in a mirror image
  • Rotations involve turning a shape around a fixed point by a certain angle
  • Translations involve sliding a shape without changing its size or orientation

Data Analysis

• Types of Data
  • Qualitative data describes characteristics, such as favorite colors or flavors
  • Quantitative data involves numerical values, such as heights or scores
• Data Displays
  • Bar graphs are used to compare categorical data, such as favorite colors
  • Picture graphs use symbols to represent data, such as a chart showing the number of pets
  • Histograms show the frequency distribution of data, such as the number of students in a class
• Measuring Central Tendency
  • The mean is the average value of a dataset, calculated by adding all values and dividing by the number of values
  • The median is the middle value in a dataset, which can be found by arranging the data in order
  • The mode is the most frequent value in a dataset, which can be found by identifying the most common value
• Interpreting Data
  • Analyzing graphs and charts helps to identify patterns and trends in data
  • Identifying patterns and trends allows us to draw conclusions about the data

Equations

• Simple Equations
  • One-step equations involve solving for a variable using a single operation, such as 2x = 6
  • Two-step equations involve solving for a variable using a combination of operations, such as 2x + 3 = 7
• Equation Properties
  • The reflexive property states that a value is equal to itself, such as a = a
  • The symmetric property states that if a = b, then b = a
  • The transitive property states that if a = b and b = c, then a = c
• Solving Equations
  • Using inverse operations, such as addition and subtraction, helps to isolate variables
  • Checking solutions by plugging them back into the equation ensures that the solution is correct
• Applications of Equations
  • Real-world problems, such as calculating the cost of goods or the area of a room, can be modeled and solved using equations
  • Word problems, such as "Tom has 5 apples and gives 2 to his friend," can be translated into equations and solved

Ratios and Rates

• A ratio is a comparison of two quantities, which can be written in simplest form, such as 2:3 or 2/3.
• Equivalent ratios have the same value, and can be used to compare different quantities.
• Rates are ratios that compare two different quantities, such as km/h, kg/m.

Percent Increase and Decrease

• Percent increase is the percentage by which a quantity increases, calculated using the formula: (new value - original value) / original value × 100.
• Percent decrease is the percentage by which a quantity decreases, calculated using the formula: (original value - new value) / original value × 100.
• Percent increase and decrease can be used to solve problems involving markups, discounts, and sales tax.

Probability

• Probability is a measure of the likelihood of an event occurring, on a scale from 0 (impossible) to 1 (certain).
• Experimental probability is based on repeated trials, while theoretical probability is based on the number of favorable outcomes.
• Probability can be expressed as a fraction, decimal, or percentage.

Data Analysis

• Data can be organized and displayed in various ways, including tables, bar graphs, histograms, and circle graphs.
• Measures of central tendency include the mean (average), median (middle value), and mode (most frequent value).
• Measures of variability include the range (difference between highest and lowest values) and interquartile range (IQR).

Equations

• An equation is a statement that two expressions are equal, and can be solved using various methods, including addition/subtraction, multiplication/division, and balancing.
• Variables can be represented by letters or symbols, and equations can be used to solve problems involving unknowns.

Transformations

• A transformation is a change in the position or size of a shape, and can be classified into four types: translations (slides), reflections (flips), rotations (turns), and enlargements (resizing).
• Transformations can be represented using graphs and coordinates, and can be used to solve problems involving congruent and similar shapes.

Probability

• An experiment is an action or situation that produces a set of outcomes
• An outcome is a specific result of an experiment
• Sample space is the set of all possible outcomes of an experiment
• Theoretical probability is the number of favorable outcomes divided by the total number of possible outcomes
• Experimental probability is the number of times an event occurs divided by the total number of trials
• Probability of an event is always between 0 and 1
• Sum of probabilities of all outcomes in a sample space is 1

Data Analysis

• Qualitative data describes characteristics or attributes (e.g. favorite color, hair color)
• Quantitative data is numerical data (e.g. height, test scores)
• Mean is the average value of a dataset
• Median is the middle value of a dataset when it is in order
• Mode is the value that appears most frequently in a dataset
• Histograms display quantitative data
• Bar graphs display categorical data
• Box plots display quantitative data and show outliers

Equations

• Solving equations involves adding or subtracting the same value to both sides or multiplying or dividing both sides by the same non-zero value
• Linear equations are in the form Ax + By = C, where A, B, and C are constants
• Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept
• Equations can model real-world situations (e.g. cost, distance, area)

Transformations

• Translations involve moving a figure a certain number of units in a specific direction
• Reflections involve flipping a figure over a line or axis
• Rotations involve turning a figure a certain number of degrees around a fixed point
• Translations can be represented as (x, y) → (x + a, y + b), where a and b are the translation values
• Reflections over the y-axis can be represented as (x, y) → (-x, y)
• Reflections over the x-axis can be represented as (x, y) → (x, -y)
• 90° clockwise rotation can be represented as (x, y) → (y, -x)
• 90° counterclockwise rotation can be represented as (x, y) → (-y, x)