Math Transformations: Types and Definition

Transformations in Geometry

• A transformation in math is a mapping of a preimage to an image of the same shape or function.
• There are two categories of transformations: rigid and non-rigid transformations.
• Rigid transformations:
  • Do not change the size or shape of the preimage.
  • Include translations, rotations, and reflections.
• Non-rigid transformations:
  • Can change the size and shape of the preimage.
  • Include shears and dilations.

Types of Transformations

• Translation:
  • A rigid transformation that changes the location of the preimage.
  • Also called a slide.
• Rotation:
  • A rigid transformation that rotates the preimage around a fixed point.
  • Also called a turn.
• Reflection:
  • A rigid transformation that flips the preimage across a line.
  • Also called a flip.
• Dilation:
  • A non-rigid transformation that changes the size of the preimage.
  • Also called an expansion or compression.

Reflections

• A reflection is a type of geometric transformation that flips an object across a line.
• The line of reflection is perpendicular to the preimage and image.
• To find the reflection of an object:
  • Draw a coordinate plane.
  • Extend a line segment from each point in the preimage to the line of reflection.
  • Extend the line in the same direction by the same distance.

Rotations

• A rotation is a rigid transformation that turns an object around a fixed point.
• There are clockwise and counterclockwise rotations of 90, 180, and 270 degrees around the origin.
• Rotation rules:
  • 90 degree rotation: (x, y) → (-y, x)
  • 180 degree rotation: (x, y) → (-x, -y)
  • 270 degree rotation: (x, y) → (y, -x)

Dilations

• A dilation is a transformation that changes the size of a figure using a center of dilation.
• There are three ways a scale factor can affect a preimage:
  • Multiply by a value greater than 1 to stretch away from the center.
  • Multiply by a value less than 1 to shrink towards the center.
  • Multiply by a value equal to 1 to stay the same size.

Similarity and Congruence

• Similar shapes have the same shape and proportional corresponding sides.
• Congruent shapes have the same shape and size.
• Similar triangles have proportional corresponding sides and equal corresponding angles.
• Congruent triangles have equal corresponding sides and equal corresponding angles.

Triangles

• A triangle is a shape with three vertices, three sides, and three interior angles.
• Types of triangles:
  • Scalene triangles: no sides of equal length.
  • Isosceles triangles: two sides of equal length.
  • Equilateral triangles: all sides of equal length.
  • Acute triangles: all angles acute (less than 90 degrees).
  • Obtuse triangles: one angle obtuse (greater than 90 degrees).
  • Right triangles: one right angle (90 degrees).

Tessellations

• A tessellation is a repeating pattern of shapes that fit together without overlapping.
• Regular tessellations consist of one repeated polygon.
• Semi-regular tessellations consist of two or more types of repeated polygons.

Pythagorean Theorem

• The Pythagorean theorem states that a^2 + b^2 = c^2 in a right triangle.
• The theorem is useful for finding the length of the hypotenuse or a leg in a right triangle.

Angle Relationships

• Angles are formed by two rays drawn from a shared endpoint.
• Angle relationships:
  • Complementary angles: add to 90 degrees.
  • Supplementary angles: add to 180 degrees.
  • Adjacent angles: share a common vertex and side.
  • Linear pair: two angles that add to 180 degrees and share a common vertex and side.
  • Vertical angles: two angles formed by intersecting lines.

Three-Dimensional Shapes

• A polyhedron is a three-dimensional solid with flat faces.
• A regular polyhedron (Platonic solid) has all faces that are regular polygons that are congruent to each other.
• Convex polyhedra have no vertices or edges that go "into" the polyhedron.

Cones

• A cone is a three-dimensional object with a circular base that narrows to a tip or apex.
• Dimensions:
  • Height (h): distance from apex to a point at a right angle to the base.
  • Radius (r): distance from center of base to perimeter.
  • Slant height (s): distance from perimeter of base to apex.
  • Area of base (b): area of circular base in square units.

Probability

• Probability is the likelihood that an event will happen, ranging from 0 to 1.
• Types of probability:
  • Simple probability: probability of one event.
  • Sequential probability: probability of two or more events.
• Events can be dependent or independent.### Probability in Real-Life Situations
• Probability is widely used in everyday life to predict outcomes, such as weather forecasts and game-winning chances.

Theoretical Probability

• Theoretical probability is a method to express the likelihood of an event occurring.
• It is calculated by dividing the number of favorable outcomes by the total possible outcomes.
• The result is a ratio that can be expressed as a fraction and a probability value between 0 and 1.
• The probability value can be easily converted into a percentage.

When to Use Theoretical Probability

• Theoretical probability is appropriate when:
  • The researcher has intimate knowledge of the subject.
  • The researcher can determine all possible and all favorable outcomes.
  • Direct experimentation is not possible.

Theoretical vs. Experimental Probability

• Theoretical probability relies on logic and thorough knowledge of the subject.
• Experimental probability uses tests and experiments to determine probability.
• Both are branches of mathematics called probability, which addresses the likelihood of an event occurring under certain conditions.

Measuring Probability

• The scale of measuring probability ranges from 0 to 1.
• 0 represents an impossible event.
• 0.25 represents an unlikely event.
• 0.5 represents an even chance of an event occurring.
• 0.75 represents a likely event.
• 1 represents a certain event.

Experimental Probability Formula

• The formula for calculating experimental probability after trials is: [insert formula].

Transformations in Geometry

• A transformation in math is a mapping of a preimage to an image of the same shape or function.
• There are two categories of transformations: rigid and non-rigid transformations.
• Rigid transformations:
  • Do not change the size or shape of the preimage.
  • Include translations, rotations, and reflections.
• Non-rigid transformations:
  • Can change the size and shape of the preimage.
  • Include shears and dilations.

Types of Transformations

• Translation:
  • A rigid transformation that changes the location of the preimage.
  • Also called a slide.
• Rotation:
  • A rigid transformation that rotates the preimage around a fixed point.
  • Also called a turn.
• Reflection:
  • A rigid transformation that flips the preimage across a line.
  • Also called a flip.
• Dilation:
  • A non-rigid transformation that changes the size of the preimage.
  • Also called an expansion or compression.

Reflections

• A reflection is a type of geometric transformation that flips an object across a line.
• The line of reflection is perpendicular to the preimage and image.
• To find the reflection of an object:
  • Draw a coordinate plane.
  • Extend a line segment from each point in the preimage to the line of reflection.
  • Extend the line in the same direction by the same distance.

Rotations

• A rotation is a rigid transformation that turns an object around a fixed point.
• There are clockwise and counterclockwise rotations of 90, 180, and 270 degrees around the origin.
• Rotation rules:
  • 90 degree rotation: (x, y) → (-y, x)
  • 180 degree rotation: (x, y) → (-x, -y)
  • 270 degree rotation: (x, y) → (y, -x)

Dilations

• A dilation is a transformation that changes the size of a figure using a center of dilation.
• There are three ways a scale factor can affect a preimage:
  • Multiply by a value greater than 1 to stretch away from the center.
  • Multiply by a value less than 1 to shrink towards the center.
  • Multiply by a value equal to 1 to stay the same size.

Similarity and Congruence

• Similar shapes have the same shape and proportional corresponding sides.
• Congruent shapes have the same shape and size.
• Similar triangles have proportional corresponding sides and equal corresponding angles.
• Congruent triangles have equal corresponding sides and equal corresponding angles.

Triangles

• A triangle is a shape with three vertices, three sides, and three interior angles.
• Types of triangles:
  • Scalene triangles: no sides of equal length.
  • Isosceles triangles: two sides of equal length.
  • Equilateral triangles: all sides of equal length.
  • Acute triangles: all angles acute (less than 90 degrees).
  • Obtuse triangles: one angle obtuse (greater than 90 degrees).
  • Right triangles: one right angle (90 degrees).

Tessellations

• A tessellation is a repeating pattern of shapes that fit together without overlapping.
• Regular tessellations consist of one repeated polygon.
• Semi-regular tessellations consist of two or more types of repeated polygons.

Pythagorean Theorem

• The Pythagorean theorem states that a^2 + b^2 = c^2 in a right triangle.
• The theorem is useful for finding the length of the hypotenuse or a leg in a right triangle.

Angle Relationships

• Angles are formed by two rays drawn from a shared endpoint.
• Angle relationships:
  • Complementary angles: add to 90 degrees.
  • Supplementary angles: add to 180 degrees.
  • Adjacent angles: share a common vertex and side.
  • Linear pair: two angles that add to 180 degrees and share a common vertex and side.
  • Vertical angles: two angles formed by intersecting lines.

Three-Dimensional Shapes

• A polyhedron is a three-dimensional solid with flat faces.
• A regular polyhedron (Platonic solid) has all faces that are regular polygons that are congruent to each other.
• Convex polyhedra have no vertices or edges that go "into" the polyhedron.

Cones

• A cone is a three-dimensional object with a circular base that narrows to a tip or apex.
• Dimensions:
  • Height (h): distance from apex to a point at a right angle to the base.
  • Radius (r): distance from center of base to perimeter.
  • Slant height (s): distance from perimeter of base to apex.
  • Area of base (b): area of circular base in square units.

Probability

• Probability is the likelihood that an event will happen, ranging from 0 to 1.
• Types of probability:
  • Simple probability: probability of one event.
  • Sequential probability: probability of two or more events.
• Events can be dependent or independent.### Probability in Real-Life Situations
• Probability is widely used in everyday life to predict outcomes, such as weather forecasts and game-winning chances.

Theoretical Probability

• Theoretical probability is a method to express the likelihood of an event occurring.
• It is calculated by dividing the number of favorable outcomes by the total possible outcomes.
• The result is a ratio that can be expressed as a fraction and a probability value between 0 and 1.
• The probability value can be easily converted into a percentage.

When to Use Theoretical Probability

• Theoretical probability is appropriate when:
  • The researcher has intimate knowledge of the subject.
  • The researcher can determine all possible and all favorable outcomes.
  • Direct experimentation is not possible.

Theoretical vs. Experimental Probability

• Theoretical probability relies on logic and thorough knowledge of the subject.
• Experimental probability uses tests and experiments to determine probability.
• Both are branches of mathematics called probability, which addresses the likelihood of an event occurring under certain conditions.

Measuring Probability

• The scale of measuring probability ranges from 0 to 1.
• 0 represents an impossible event.
• 0.25 represents an unlikely event.
• 0.5 represents an even chance of an event occurring.
• 0.75 represents a likely event.
• 1 represents a certain event.

Experimental Probability Formula

• The formula for calculating experimental probability after trials is: [insert formula].