Lesson 8: Rigid Transformations on the Coordinate Plane

Core Objective: The primary goal is to develop the ability to describe a sequence of transformations that maps a preimage onto its image on the coordinate plane.
Standards Alignment: This lesson specifically addresses Standard GS.MGSR.2.1, which focuses on measurement, geometry, and spatial reasoning.
Alignment Context: The material is designed for South Carolina Geometry curriculum requirements, emphasizing the use of the coordinate plane to formalize rigid transformations.

Distance Determination:
- When calculating the distance between two specific points (such as Point $A$ and Point $B$ ), one effective method involves creating a right triangle.
- The length of the hypotenuse of this right triangle represents the distance between the two points.
- In a specific scenario where Point $A$ and Point $B$ are separated by specific horizontal and vertical displacements, the distance is calculated as $5\,units$ .
- Alternative Method (Distance Formula): While right triangles provide a visual method, students may suggest the distance formula. This formula is intrinsically connected to the Pythagorean Theorem through the coordinate system.
Transformation Language:
- Casual descriptions like "Move $3\,units$ down and $4\,units$ right" must be translated into formal geometric language.
- Translation: A rigid motion that moves every point of a figure the same distance in the same direction.
- Directed Line Segments: These are used to specify the exact direction and distance of a translation. A translation can be described as being "along" a directed line segment from one coordinate to another.

Rotation and Translation Sequence:
- Example Procedure: Rotate a figure $90^\circ$ clockwise using center $(0,0)$ .
- Following the rotation, translate the figure according to the directed line segment from $(0,0)$ to $(3,2)$ .
- The result of such a multi-step sequence is typically labeled with a new letter, such as $R$ .
Reflection and Rotation Sequence:
- Example Procedure: Reflect a figure across the $y$ -axis.
- Following the reflection, rotate the figure $90^\circ$ counterclockwise using center $(0,0)$ .
- The result of this sequence is labeled $L$ .
Effect on Coordinates:
- Reflecting across the $y$ -axis results in the $x$ -coordinate changing its sign (becoming its opposite) while the $y$ -coordinate remains the same.
- If a point is $(x, y)$ , after a reflection across the $y$ -axis, the new point becomes $(-x, y)$ .

Case 1: Triangle Mapping ( $\triangle ABC$ to $\triangle DEF$ )
- $\triangle DEF$ is the image of $\triangle ABC$ after a sequence of rigid motions.
- One possible sequence: $\triangle ABC$ is rotated $90^\circ$ counterclockwise around the center $(0,0)$ . It is then translated according to the directed line segment from $(0,0)$ to $(0,-2)$ .
Case 2: Quadrilateral Mapping ( $ABCD$ to $KMNP$ )
- $KMNP$ is the image of $ABCD$ after a sequence of transformations.
- One possible sequence: Quadrilateral $ABCD$ is reflected over the $y$ -axis and then translated left $2\,units$ and up $4\,units$ .
Case 3: Triangle Mapping ( $\triangle XCH$ to $\triangle NGM$ )
- $\triangle NGM$ is the image of $\triangle XCH$ after a sequence of transformations.
- One possible sequence: $\triangle XCH$ is reflected over the $y$ -axis, rotated $90^\circ$ clockwise, and then translated right $2\,units$ .

Coordinate Plane Integration: While sequences of transformations can be explored off the coordinate plane, using the plane allows for the use of ordered pairs to describe precise locations and movements.
Descriptive Precision: When describing transformations, it is necessary to include specific details:
- For Rotations: Specify the angle of rotation and the center of rotation (e.g., $(0,0)$ ).
- For Translations: Specify the directed line segment or the horizontal and vertical shifts.
- For Reflections: Specify the line of reflection (e.g., the $y$ -axis or a line such as $x = -2$ ).
Synthesis Questions:
- What is the specific angle and center of rotation used?
- What is the directed line segment defined for the translation?
- How can a translation be written out using different notations?

Problem 1: Complex Sequence
- Task: Reflect triangle $ABC$ over the line $x = -2$ , then translate it by the directed line segment from $(0,0)$ to $(3,1)$ .
- Solution Coordinates: The resulting points are labeled $A''$ , $B''$ , and $C''$ .
Problem 2: Coordinate Analysis
- Preimage Coordinates: Triangle $ABC$ has coordinates $A$ , $B$ , and $C$ .
- Image Coordinates: Following a sequence of transformations, the triangle becomes $A'B'C'$ with coordinates $A'$ , $B'$ , and $C'$ defined.
- Possible Sequence: Translate triangle $ABC$ along the directed line segment from $(3,0)$ to $(1,2)$ , and then reflect the translated image across the $x$ -axis.
Problem 3: Midpoint Rotations
- Task: Rotate a figure $180^\circ$ around the midpoint of segment $BC$ .
- Result: A $180^\circ$ rotation around a midpoint essentially flips the segment upon itself, maintaining the orientation relative to that center point.

MLR8: Discussion Supports: These routines are used to support students in explaining their thinking and justifying their sequences of transformations.
Language Call-outs: Specific phrases like "reflect across the $y$ -axis" and "rotate clockwise" are highlighted to help students produce mathematical reasoning and identify the results of their partner's transformations.
Tools for Success: For students who struggle with visualizing transformations, tools such as tracing paper and physical manipulatives (like transparency sheets) are highly recommended. These tools help bridge the gap between abstract coordinate movements and concrete geometric changes.
Feedback Loops: Students are encouraged to meet with multiple partners to share sequences and receive feedback on specific details, such as verifying the centers of rotation or the accuracy of directed line segments.