Given a preimage and image, describe the transformation and represent the transformation algebraically using coordinates.
Examples
Example: Given a triangle whose vertices have the coordinates (-3,4), (2,1.7) and (-0.4,-3). If this triangle is reflected across the y-axis the transformation can be described using coordinates as (x,y)→(-x,y) resulting in the image whose vertices have the coordinates (3,4), (-2,1.7) and (0.4,-3).Clarifications
Clarification 1: Instruction includes the connection of transformations to functions that take points in the plane as inputs and give other points in the plane as outputs.Clarification 2: Transformations include translations, dilations, rotations and reflections described using words or using coordinates.
Clarification 3: Within the Geometry course, rotations are limited to 90°, 180° and 270° counterclockwise or clockwise about the center of rotation, and the centers of rotations and dilations are limited to the origin or a point on the figure.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Geometric Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Coordinate Plane
- Dilation
- Origin
- Reflection
- Rigid Transformation
- Rotation
- Scale Factor
- Translation
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 8, students developed an understanding of single transformations. In Algebra 1, students extended this knowledge of transformations to transforming functions using tables, graphs and equations. In Geometry, students will understand transformations as functions and will transform figures, using two or more transformations, using words and coordinates. In later courses, types of transformations will be expanded to include stretches that can transform functions and conic sections and conversions between rectangular and polar coordinates.- Instruction includes describing transformations using words and coordinates. Common transformations are provided below.
- Rotations can be described algebraically as the following:
90° counterclockwise about the origin (, ) → (−, )
90° clockwise about the origin (, ) → (, −)
180° counterclockwise about the origin (, ) → (−, −)
180° clockwise about the origin (, ) → (−, −)
270° counterclockwise about the origin (, ) → (, −)
270° clockwise about the origin (, ) → (−, ) - Reflections can be described algebraically as the following:
Over the -axis (, ) → (, −)
Over the -axis (, ) → (−, )
Over the line = (, ) → (, )
Over the line = − (, ) → (−, −) - Dilations
Dilation by a factor of a, where a is a real number (, ) → (, )
- Rotations can be described algebraically as the following:
- Translations
- Horizontal translation by units, where is a real number (, ) → ( + , ) Vertical translation by units, where is a real number (, ) → (, + ) Horizontal translation by units, where is a real number, and vertical translation by units, where is a real number (, )
- Instruction includes examining the effect of transforming coordinates by adding, subtracting, or multiplying the - and -coordinates with real-number values to make the connection between functions and transformations.
- For example, ΔPQR, with vertices P(−1, 4), Q(3, 4) and R(1, 7) can be transformed using the coordinates representation (, ) → (, − 1).
- For example, if the vertices of ΔABC are (4, −2), (4, 5) and (3, 3), respectively, and the vertices of ΔA′B′C′ are (8, −4), (8, 10) and (6, 6), respectively, the coordinate representation can be determined.
- Instruction includes the use of hands-on manipulatives and geometric software for students to explore transformations.
- Instruction includes using a variety of ways to describe a transformation using coordinates. (MTR.2.1)
- For example, the same translation can be described using words as 2 units to the right and 4 units down, using coordinates as (, ) → ( + 2, − 4) or as , = ( + 2, − 4), or .
- For example, a reflection over the -axis can be represented as (, ) → (, −) or as -axis (, ) = (, −).
- For example, a 90°rotation counterclockwise about the origin can be represented as (, ) → (−, ) or as 0,90o (, ) = (−, ) where is the origin.
- The discussion of translations can be extended to include vectors when describing translations.
- For example, if a point is translated 3 units to the left and 4 units up the translation vector is . The vector summarizes the horizontal and vertical shifts.( + , + )
Common Misconceptions or Errors
- Students may believe the orientation of a figure would be conserved in a rotation in the same way that the orientation of a car, or gondola, is preserved when rotating on a Ferris wheel).
- Student may not be able to visualize some transformations like rotations. To address this, instruction includes using folding paper (e.g., patty paper) or interactive geometric software to allow students hands-on experiences and flexibility in exploration.
Instructional Tasks
Instructional Task 1 (MTR.3.1)- Use the graph to the below to answer the following questions.
- Part A. Describe the transformation that maps ABCD to A'B'C'D'.
- Part B. Represent the transformation described in Part A algebraically.
- Part C. Algebraically represent the transformation needed to map A"B"C"D" onto ABCD.
- Part D. Describe the transformation that maps A"B"C"D" onto A"'B"'C""D"".
- Part E. How is the transformation described in Part D related to the transformation needed to map A"'B"'C""D"" onto A"B"C"D"
Instructional Task 2 (MTR.5.1)
- Part A. Ask students to plot A, B and C and A', B' and C' on the coordinate plane. What do you notice?
- Part B. How can you describe the transformation using words? Explore the patterns among the coordinates of the points of the preimages and the images.
- Part C. How can you describe the transformation using coordinate notation??
Instructional Items
Instructional Item 1- A triangle whose vertices are located at (, −1), (−4,− ) and (3,1) is shifted to the right 2 units.
- Part A. What are the coordinates of the triangle after the translation?
- Part B. Describe the transformation that would map the preimage to the image algebraically.
Related Courses
This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
Related Access Points
Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.2.AP.1a: Given a preimage and image, identify the transformation.
MA.912.GR.2.AP.1b: Select the algebraic coordinates that represent the transformation.
Related Resources
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Formative Assessments
Lesson Plans
MFAS Formative Assessments
Demonstrating Translations:
Students are asked to translate a quadrilateral according to a given vector.
Transformations And Functions:
Students are given examples of three transformations and are asked if each is a function.
Student Resources
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Parent Resources
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