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Clarifications
Clarification 1: Transformations include translations, dilations, rotations and reflections described using words or using coordinates.Clarification 2: Within the Geometry course, figures are limited to triangles and quadrilaterals and rotations are limited to 90°, 180° and 270° counterclockwise or clockwise about the center of rotation.
Clarification 3: Instruction includes the understanding that when a figure is mapped onto itself using a reflection, it occurs over a line of symmetry.
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 the elementary grades, students learned about lines of symmetry. In grade 8, students learned about the effects of translations, rotations, reflections, and dilations on geometric figures. In Geometry, students use their knowledge of translations, dilations, rotations and reflections to identify a sequence or composition of transformations that map a triangle or a quadrilateral onto another congruent or similar figure or onto itself, and they connect reflections to lines of symmetry. In later courses, lines of symmetry are identified as key features in graphs of polynomials and trigonometric functions.- To describe the sequence of transformations, students will need to know how to describe each one of the transformations in the composition using words or using coordinates. In each case, they will specify vertical and horizontal shifts, center and angle of the rotation, clockwise or counterclockwise, line of reflection, center of the dilation and scale factor, when needed. (MTR.3.1)
- Provide multiple opportunities for students to explore mapping a variety of triangles and quadrilaterals onto congruent or similar figures (given the preimage and the image) using both physical exploration (transparencies or patty paper) and virtual exploration when possible. This will allow students to experience multiple compositions of transformations and realize that more than one sequence can be used to map a figure onto another. (MTR.2.1)
- Instruction includes examples where preimages and images partially overlap each other.
- When a sequence includes a dilation, it may be helpful that students identify the dilation first, and then continue to identify any rigid motions that may be needed.
- Students can explore the sequence of a reflection over the -axis followed by a reflection over the -axis (or any sequence of two reflections over axes perpendicular to each other). To help students make the connection between different sequences of transformations, ask “Is there a single transformation that produces the same image as this sequence?” (MTR.5.1)
- To map a figure onto itself, explore the effect of each transformation. Discuss with students the possibilities of using translations or dilations.
- When a reflection maps a figure onto itself, the line of reflection is also a line of symmetry for the figure. Explore the lines of symmetries of isosceles and equilateral triangles, and rectangles, rhombi, squares, isosceles trapezoids and kites.
- When a rotation is used, explore the cases of regular polygons (equilateral triangles and squares) and how to determine the angles of rotation that will map them onto themselves. (MTR.5.1)
- Instruction includes discussing the case of a dilation with a scale factor of 1. Even if this case is considered trivial, it leads the conversation to the relationship between congruence and similarity. If a dilation is a similarity transformation, then it produces an image that is similar to the preimage. But if a dilation with a scale factor of 1 produces an image that is congruent to the preimage, then congruence is a case of similarity. In other words, when two figures are congruent, then they are necessarily similar to each other.
- An extension of this benchmark may be to explore the angle of rotation needed to map a regular polygon of 5 or more sides onto itself.
Common Misconceptions or Errors
- Students may believe there is only one sequence that will lead to the image. Instead, students should explore the fact that multiple sequences will result in the same image.
Instructional Tasks
Instructional Task 1 (MTR.2.1)
- Part A. From the list provided, choose and order transformations that could be used to map ΔABC onto ΔA"'B"'C"'.
- Part B. Describe the transformation that maps ΔA"B"C" onto ΔA"'B"'C'".
Instructional Items
Instructional Item 1- A single rotation mapped quadrilateral ABCD onto quadrilateral A'B'C'D'.
- Part A. What is the center of the rotation?
- Part B. If the rotation is counterclockwise, how many degrees is the rotation?
- Part C. Describe another transformation that maps quadrilateral ABCD onto quadrilateral A'B'C'D'.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorial
Perspectives Video: Professional/Enthusiast
Problem-Solving Tasks
MFAS Formative Assessments
Students are asked to describe the transformations that take one triangle onto another.
Students are asked to use rigid motion to explain why the HL pattern of congruence ensures right triangle congruence.
Students are asked to describe a sequence of rigid motions to demonstrate the congruence of two polygons.
Students are asked to describe a rigid motion to demonstrate that two polygons are congruent.
Students are asked to describe a rigid motion to demonstrate two polygons are congruent.
Students are asked to describe a rigid motion to demonstrate two polygons are congruent.
Students are asked to describe a sequence of transformations to show that two polygons are similar.
Students are asked to describe a sequence of transformations to show that two polygons are similar.
Students are asked to describe a sequence of transformations that demonstrates two polygons are similar.
Students are asked to describe the transformations that take one triangle onto another.
Original Student Tutorials Mathematics - Grades 9-12
Learn to describe a sequence of transformations that will produce similar figures. This interactive tutorial will allow you to practice with rotations, translations, reflections, and dilations.
Student Resources
Original Student Tutorial
Learn to describe a sequence of transformations that will produce similar figures. This interactive tutorial will allow you to practice with rotations, translations, reflections, and dilations.
Type: Original Student Tutorial
Problem-Solving Tasks
The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.
Type: Problem-Solving Task
This task provides a concrete geometric setting in which to study rigid transformations of the plane.
Type: Problem-Solving Task
Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.
Type: Problem-Solving Task
This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''
Type: Problem-Solving Task
The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.
Type: Problem-Solving Task
In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Tasks
The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.
Type: Problem-Solving Task
This task provides a concrete geometric setting in which to study rigid transformations of the plane.
Type: Problem-Solving Task
Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.
Type: Problem-Solving Task
This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''
Type: Problem-Solving Task
The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.
Type: Problem-Solving Task
In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.
Type: Problem-Solving Task
