ALEX Classroom Resource

  

Geometry Module 1, Topic C: Transformations/Rigid Motions

  Classroom Resource Information  

Title:

Geometry Module 1, Topic C: Transformations/Rigid Motions

URL:

https://www.engageny.org/resource/geometry-module-1-topic-c-overview

Content Source:

EngageNY
Type: Lesson/Unit Plan

Overview:

Module 1, Topic C, Transformations, builds on students’ intuitive understanding developed in Grade 8. With the help of manipulatives, students observed how reflections, translations, and rotations behave individually and in sequence (8.G.1, 8.G.2). In Grade 10, this experience is formalized by clear definitions (G.CO.4) and more in-depth exploration (G.CO.3, G.CO.5). The concrete establishment of rigid motions also allows proofs of facts formerly accepted to be true (G.CO.9). Similarly, students’ Grade 8 concept of congruence transitions from a hands-on understanding (8.G.2) to a precise, formally notated understanding of congruence (G.CO.6). With a solid understanding of how transformations form the basis of congruence, students next examine triangle congruence criteria. Part of this examination includes the use of rigid motions to prove how triangle congruence criteria such as SAS actually work (G.CO.7G.CO.8).

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
4 ) Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [G-CO4]


Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.4- Given a geometric figure of a reflection or a translation of that figure, identify if the geometric figure is a reflection or translation.


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
21. Represent transformations and compositions of transformations in the plane (coordinate and otherwise) using tools such as tracing paper and geometry software.

a. Describe transformations and compositions of transformations as functions that take points in the plane as inputs and give other points as outputs, using informal and formal notation.

b. Compare transformations which preserve distance and angle measure to those that do not.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a variety of transformations (translations, rotations, reflections, and dilations),
  • Represent the transformations and compositions of transformations in the plane using a variety of methods (e.g., technology, transparencies, semi-transparent mirrors (MIRAs), patty paper, compass).
  • Describe transformations and compositions of transformations functions that take points in the plane as inputs and give other points as outputs, explain why this satisfies the definition of a function, and adapt function notation to that of a mapping [e.g., f(x,y) → f(x+a, y+b)].
  • Compare transformations that preserve distance and angle to those that do not.
Teacher Vocabulary:
  • Transformation
  • Reflection
  • Translation
  • Rotation
  • Dilation
  • Isometry
  • Composition
  • Horizontal stretch
  • Vertical stretch
  • Horizontal shrink
  • Vertical shrink
  • Clockwise
  • Counterclockwise
  • Symmetry
  • Preimage
  • Image
Knowledge:
Students know:
  • Characteristics of transformations (translations, rotations, reflections, and dilations).
  • Methods for representing transformations.
  • Characteristics of functions.
  • Conventions of functions with mapping notation.
Skills:
Students are able to:
  • Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane with and without technology.
  • Communicate the results of performing transformations on objects and their corresponding coordinates in the coordinate plane, including when the transformation preserves distance and angle.
  • Use the language and notation of functions as mappings to describe transformations.
Understanding:
Students understand that:
  • Mapping one point to another through a series of transformations can be recorded as a function.
  • Some transformations (translations, rotations, and reflections) preserve distance and angle measure, and the image is then congruent to the pre-image, while dilations preserve angle but not distance, and the pre-image is similar to the image.
  • Distortions, such as only a horizontal stretch, preserve neither.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.21.1: Define distance, angle, input, output, plane, translation, reflection, rotation, and dilation.
GEO.21.2: Compare transformation that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
GEO.21.3: Describe transformations as functions that take points in a plane as inputs and give other points as outputs.
GEO.21.4: Represent transformation in the plane.
GEO.21.5: Generate an input output table.
GEO.21.6: Compare the distance and angles of the figures from the pre-image to the image.
GEO.21.7: Measure distance and angle(s) of an image.

Prior Knowledge Skills:
  • Define rotation, reflection, and translation.
  • Recognize translations (slides), rotations (turns), and reflections (flips).
  • Distinguish between lines and line segments.
  • Demonstrate how to measure length.
  • Demonstrate how to use a protractor to measure angles.
  • Identify parallel lines.
  • Define square root, cube root, inverse, perfect square, perfect cube, and irrational number.
  • Define square root, expressions, and approximations.
  • Identify perfect squares and square roots.
  • Demonstrate how to locate points on a vertical or horizontal number line.
  • Define ordered pairs.
  • Show how to plot points on a Cartesian plane.
  • Locate the origin on the coordinate plane.
  • Identify the length between vertices on a coordinate plane.
  • Recall how to read a graph or table.
  • Draw and label a coordinate plane.
  • Plot independent (input) and dependent (output) values on a coordinate plane.
  • Plot pairs of integers and/or rational numbers on a coordinate plane.
  • Arrange integers and/or rational numbers on a horizontal or vertical number line.
  • Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
  • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
  • Calculate the distances between points having the same first or second coordinate using absolute value.
  • Define number line.
  • Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.
  • Calculate missing input and/or output values in a table.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives.


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
22. Explore rotations, reflections, and translations using graph paper, tracing paper, and geometry software.

a. Given a geometric figure and a rotation, reflection, or translation, draw the image of the transformed figure using graph paper, tracing paper, or geometry software.

b. Specify a sequence of rotations, reflections, or translations that will carry a given figure onto another.

c. Draw figures with different types of symmetries and describe their attributes.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a geometric figure,
  • Explore rotations, reflections, and translations using graph paper, tracing paper, and geometry software.
  • Produce the image of the figure under a rotation, reflection, or translation using graph paper, tracing paper, or geometry software.
  • Describe and justify the sequence of transformations that will carry a given figure onto another.
  • Draw figures such as rectangles, parallelograms, trapezoids, or regular polygons.
  • Identify which figures that have rotations or reflections that carry the figure onto itself.
  • Perform and communicate rotations and reflections that map the object to itself.
  • Distinguish these transformations from those which do not carry the object back to itself.
  • Describe the relationship of these findings to symmetry.
Teacher Vocabulary:
  • Transformation
  • Reflection
  • Translation
  • Rotation
  • Dilation
  • Isometry
  • Composition
  • horizontal stretch
  • vertical stretch
  • horizontal shrink
  • vertical shrink
  • Clockwise
  • Counterclockwise
  • Symmetry
  • Trapezoid
  • Square
  • Rectangle
  • Regular polygon
  • parallelogram
  • Mapping
  • preimage
  • Image
Knowledge:
Students know:
  • Characteristics of transformations (translations, rotations, reflections, and dilations).
  • Techniques for producing images under transformations using graph paper, tracing paper, or geometry software.
  • Characteristics of rectangles, parallelograms, trapezoids, and regular polygons.
Skills:
Students are able to:
  • Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane with and without technology.
  • Communicate the results of performing transformations on objects and their corresponding coordinates in the coordinate plane.
Understanding:
Students understand that:
  • Mapping one point to another through a series of transformations can be recorded as a function.
  • Since translations, rotations and reflections preserve distance and angle measure, the image is then congruent.
  • The same transformation may be produced using a variety of tools, but the geometric sequence of steps that describe the transformation is consistent.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.22.1: Define rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GEO.22.2: Describe the effects of rotations, reflection, and translations on two dimensional figures using coordinates.
GEO.22.3: Illustrate figures transformed by a rotation, reflection or translation.
GEO.22.4: Describe the process of transforming a given figure.
GEO.22.5: Graph a figure on a coordinate plane.

Prior Knowledge Skills:
  • Recognize dilations.
  • Recognize translations.
  • Recognize rotations.
  • Recognize reflections.
  • Define rotation, reflection, and translation.
  • Recognize translations (slides), rotations (turns), and reflections (flips).
  • Distinguish between lines and line segments.
  • Identify parallel lines.
  • Demonstrate how to locate points on a vertical or horizontal number line.
  • Define ordered pairs.
  • Show how to plot points on a Cartesian plane.
  • Locate the origin on the coordinate plane.
  • Identify the length between vertices on a coordinate plane.
  • Recall how to read a graph or table.
  • Draw and label a coordinate plane.
  • Plot independent (input) and dependent (output) values on a coordinate plane.
  • Plot pairs of integers and/or rational numbers on a coordinate plane.
  • Arrange integers and/or rational numbers on a horizontal or vertical number line.
  • Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
  • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
  • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
  • Calculate the distances between points having the same first or second coordinate using absolute value.
  • Define number line.
  • Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives.


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
23. Develop definitions of rotation, reflection, and translation in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Use geometric terminology (angles, circles, perpendicular lines, parallel lines, and line segments) to describe the series of steps necessary to produce a rotation, reflection, or translation.
  • Use these descriptions to communicate precise definitions of rotation, reflection, and translation.
Teacher Vocabulary:
  • Transformation
  • Reflection
  • Translation
  • Rotation
  • Dilation
  • Isometry
  • Composition
  • Clockwise
  • Counterclockwise
  • Preimage
  • Image
Knowledge:
Students know:
  • Characteristics of transformations (translations, rotations, reflections, and dilations).
  • -Properties of a mathematical definition, i.e., the smallest amount of information and properties that are enough to determine the concept. (Note: may not include all information related to concept).
Skills:
Students are able to:
  • Accurately perform rotations, reflections, and translations on objects with and without technology.
  • Communicate the results of performing transformations on objects.
  • Use known and developed definitions and logical connections to develop new definitions.
Understanding:
Students understand that:
  • Geometric definitions are developed from a few undefined notions by a logical sequence of connections that lead to a precise definition.
  • A precise definition should allow for the inclusion of all examples of the concept and require the exclusion of all non-examples.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.23.1: Define rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GEO.23.2: Describe the effects of rotations, reflection, and translations on two dimensional figures using coordinates.
GEO.23.3: Describe the effects of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GEO.23.4: Describe the process of transforming a given figure.
GEO.23.5: Illustrate figures transformed by a rotation, reflection or translation.
GEO.23.6: Recognize the type of transformation from a pre-image to an image.

Prior Knowledge Skills:
  • Recognize dilations.
  • Recognize translations.
  • Recognize rotations.
  • Recognize reflections.
  • Analyze an image and its dilation to determine if the two figures are similar.
  • Define dilation.
  • Recall how to find scale factor.
  • Give examples of scale drawings.
  • Recognize translations.
  • Recognize reflections.
  • Recognize rotations.
  • Identify parallel lines.
  • Compare translations to reflections.
  • Compare reflections to rotations.
  • Compare rotations to translations.
  • Define diameter, radius, circumference, area of a circle, and formula.
  • Identify and label parts of a circle.
  • Recognize the attributes of a circle.
  • Define rotation, reflection, and translation.
  • Recognize translations (slides), rotations (turns), and reflections (flips).
  • Distinguish between lines and line segments.
  • Identify parallel lines.
  • Define square root, cube root, inverse, perfect square, perfect cube, and irrational number.
  • Define square root, expressions, and approximations.
  • Demonstrate how to locate points on a vertical or horizontal number line.
  • Define ordered pairs.
  • Show how to plot points on a Cartesian plane.
  • Locate the origin on the coordinate plane.
  • Identify the length between vertices on a coordinate plane.
  • Recall how to read a graph or table.
  • Draw and label a coordinate plane.
  • Plot independent (input) and dependent (output) values on a coordinate plane.
  • Plot pairs of integers and/or rational numbers on a coordinate plane.
  • Arrange integers and/or rational numbers on a horizontal or vertical number line.
  • Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
  • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
  • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
  • Calculate the distances between points having the same first or second coordinate using absolute value.
  • Define number line.
  • Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.
  • Calculate missing input and/or output values in a table.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives.


Tags: compass, congruence, geometry, reflections, rigid motions, rotations, straightedge, transformations, translations, triangles
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There are ten lessons in this topic.

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Author: Hannah Bradley