Grade 8 · Geometry · MAT-08-GEO-003
Rotate Figures About the Origin
Represent and solve rotate figures about the origin problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve rotate figures about the origin problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify rotate figures about the origin results in context, including units, constraints, and reasonableness.
Prerequisite check
Make sure the foundation is ready.
Grade 8 uses this prerequisite as active knowledge in a more formal representation.
Learn
Build the idea from meaning, not memorization.
Meaning and structure
Rotate Figures About the Origin belongs to geometry. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent rotate figures about the origin numerically and symbolically, then connect it to the Grade 8 Math Box. Tables, graphs, equations, coordinate models, diagrams, and verbal descriptions must agree.
Verification and interpretation
Check with substitution, an inverse operation, an equivalent representation, geometric invariants, estimation, or context. State whether the result is exact or approximate and preserve relevant units and constraints.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Transformation Lab
- Model rotate figures about the origin in the Math Box.
- Change one input and predict which quantities or invariants should change.
- Explain what the visual, equation, and numerical result say about the same relationship.
Worked examples
Twenty different ways to see the concept work.
Rotate (1,2) 90° counterclockwise.
(-2,1)90° counterclockwise maps (x,y) to (-y,x).
Rotate (2,4) 180°.
(-2,-4)A 180° rotation reverses both signs.
Rotate (3,6) 270° counterclockwise.
(6,-3)270° counterclockwise maps (x,y) to (y,-x).
Rotate (4,8) 90° counterclockwise.
(-8,4)90° counterclockwise maps (x,y) to (-y,x).
Rotate (5,3) 180°.
(-5,-3)A 180° rotation reverses both signs.
Rotate (6,5) 270° counterclockwise.
(5,-6)270° counterclockwise maps (x,y) to (y,-x).
Rotate (7,7) 90° counterclockwise.
(-7,7)90° counterclockwise maps (x,y) to (-y,x).
Rotate (8,2) 180°.
(-8,-2)A 180° rotation reverses both signs.
Rotate (1,4) 270° counterclockwise.
(4,-1)270° counterclockwise maps (x,y) to (y,-x).
Rotate (2,6) 90° counterclockwise.
(-6,2)90° counterclockwise maps (x,y) to (-y,x).
Rotate (3,8) 180°.
(-3,-8)A 180° rotation reverses both signs.
Rotate (4,3) 270° counterclockwise.
(3,-4)270° counterclockwise maps (x,y) to (y,-x).
Rotate (5,5) 90° counterclockwise.
(-5,5)90° counterclockwise maps (x,y) to (-y,x).
Rotate (6,7) 180°.
(-6,-7)A 180° rotation reverses both signs.
Rotate (7,2) 270° counterclockwise.
(2,-7)270° counterclockwise maps (x,y) to (y,-x).
Rotate (8,4) 90° counterclockwise.
(-4,8)90° counterclockwise maps (x,y) to (-y,x).
Rotate (1,6) 180°.
(-1,-6)A 180° rotation reverses both signs.
Rotate (2,8) 270° counterclockwise.
(8,-2)270° counterclockwise maps (x,y) to (y,-x).
Rotate (3,3) 90° counterclockwise.
(-3,3)90° counterclockwise maps (x,y) to (-y,x).
Rotate (4,5) 180°.
(-4,-5)A 180° rotation reverses both signs.
Practice
Guided practice with hints
01Rotate (5,7) 270° counterclockwise.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (7,-5)
02Rotate (6,2) 90° counterclockwise.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (-2,6)
03Rotate (7,4) 180°.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (-7,-4)
04Rotate (8,6) 270° counterclockwise.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (6,-8)
05Rotate (1,8) 90° counterclockwise.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (-8,1)
06Rotate (2,3) 180°.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (-2,-3)
Practice
Independent practice
01Rotate (3,5) 270° counterclockwise.
Answer: (5,-3)
02Rotate (4,7) 90° counterclockwise.
Answer: (-7,4)
03Rotate (5,2) 180°.
Answer: (-5,-2)
04Rotate (6,4) 270° counterclockwise.
Answer: (4,-6)
05Rotate (7,6) 90° counterclockwise.
Answer: (-6,7)
06Rotate (8,8) 180°.
Answer: (-8,-8)
07Rotate (1,3) 270° counterclockwise.
Answer: (3,-1)
08Rotate (2,5) 90° counterclockwise.
Answer: (-5,2)
Common mistakes
Learn to catch the error, not just the answer.
Rigid transformations preserve size; dilations generally do not.
a²+b²=c² applies to right triangles.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Use transformations in design and coordinate geometry and Pythagorean reasoning for distance.
- Model capacities of cylindrical, conical, and spherical objects.
Challenge problems
Rotate (3,7) 180°.
(-3,-7)
Rotate (4,2) 270° counterclockwise.
(2,-4)
Rotate (5,4) 90° counterclockwise.
(-4,5)
Rotate (6,6) 180°.
(-6,-6)
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve rotate figures about the origin problems and justify each result with a valid verification method.
Practice
Mastery check
01Rotate (7,8) 270° counterclockwise.
Answer: (8,-7)
02Rotate (8,3) 90° counterclockwise.
Answer: (-3,8)
03Rotate (1,5) 180°.
Answer: (-1,-5)
04Rotate (2,7) 270° counterclockwise.
Answer: (7,-2)
05Rotate (3,2) 90° counterclockwise.
Answer: (-2,3)
06Rotate (4,4) 180°.
Answer: (-4,-4)
07Rotate (5,6) 270° counterclockwise.
Answer: (6,-5)
08Rotate (6,8) 90° counterclockwise.
Answer: (-8,6)
Terminology
Words to know
- transformation
- A rule mapping a figure to an image.
- congruent
- Having the same size and shape.
- similar
- Having equal corresponding angles and proportional corresponding lengths.
- Pythagorean theorem
- For a right triangle, a²+b²=c².
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Grade 8 instructional guidance, mathematical practices, modeling, and progression toward high-school mathematics.Common Core State Standards for Mathematics — Grade 8
Grade-level expectations for real numbers, equations, functions, geometry, and statistics.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.