Regional Integrated Geometry Curriculum
UNIT: Transformational Geometry Time frame for unit: 10 days
TOPIC: Reflections, rotations, & translations Time frame for topic: 5 days
Prior Knowledge8.G.7 Describe and identify transformations in the plane, using proper Geometry function notation (rotations, reflections, translations, and dilations)
8.G.8 Draw the image of a figure under rotations of 90 and 180 degrees
8.G.9 Draw the image of a figure under a reflection over a given line
8.G.10 Draw the image of a figure under a translation
8.G.11 Draw the image of a figure under a dilation
8.G.12 Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation
Content Strands
Note to teacher: Use proper function notation.
Ex:
G.G.55 Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections
G.G.54 Define, investigate, justify, and apply isometries in the Geometry plane (rotations, reflections, translations, glide reflections)
G.G.56 Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism
G.G.57 Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections)
G.G.61 Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90º and 180º, reflections over the lines , , and , and dilations centered at the origin
Concepts
· Reflections
o Line reflections (x-axis, y-axis, x=a, y=b, , )
o Point reflections
o Symmetry
o Orientation, preservation of distance, preservation of angle measure, parallelism, perpendicularity, collinearity, midpoint
o Opposite (indirect) isometry
· Rotations
o About the origin of and , clockwise and counterclockwise
o Orientation, preservation of distance, preservation of angle measure, parallelism, perpendicularity, collinearity, midpoint
o Point reflection, half-turn
o Direct isometry
· Translations
o Orientation, preservation of distance, preservation of angle measure, parallelism, perpendicularity, collinearity, midpoint
o Direct isometry
Essential Questions
What are the similarities and differences between direct and opposite isometries?
Process/skills
G.PS.2 Observe and explain patterns to formulate generalizations and conjectures
G.RP.3 Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion
G.CM.11 Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams
G.CN.1 Understand and make connections among multiple representations of the same mathematical idea
G.R.3 Use representation as a tool for exploring and understanding mathematical ideas
Vocabulary
Collinearity
Image
Invariant
Isometry (direct vs. opposite/indirect)
Mapping / Midpoint
Orientation
Parallelism
Perpendicularity
Pre-image / Prime Notation
Reflection
Rotation
Symmetry
Translation
Suggested assessments
§ Pre-assessment
§ Regents/State exams
§ Projects
§ Participation
§ Q-A Responses
§ Oral responses
§ Conversations / § On-spot checks of class work
§ Homework
§ Observations of students’ work/behaviors
§ Students’ explanations / § Draw a picture
§ Tests and Quizzes
§ T/F
§ Multiple Choice
§ Written Response
§ Ticket out the door
See Blackboard for specific Assessments*
Resources
http://www.emsc.nysed.gov/3-8/MathGeomSample.pdf
Regional Integrated Geometry Curriculum
UNIT : Transformational Geometry Time frame for unit: 10 days
TOPIC: Dilations and Compositions Time frame for topic:4 days
Prior Knowledge8.G.7 Describe and identify transformations in the plane, using proper Geometry function notation (rotations, reflections, translations, and dilations)
8.G.11 Draw the image of a figure under a dilation
8.G.12 Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation
Content Strands
G.G.58 Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries)
G.G.59 Investigate, justify, and apply the properties that remain invariant under similarities
G.G.60 Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism
G.G.61 Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90º and 180º, reflections over the lines , , and , and dilations centered at the origin
Concepts
· Dilations
o Similarity (Corresponding sides are proportional, corresponding angles are congruent)
o Factor of dilation(positive/negative, enlargement/reduction)
· Composition of Transformations (proper notation required)
o Glide Reflection
§ Orientation, preservation of distance, preservation of angle measure, parallelism, perpendicularity, collinearity, midpoint
o 2 or more of a reflection, rotation, translation, or dilation
Essential Questions
Which of the following properties, if any, are invariant under every similarity: area, angle congruence, collinearity, distance, orientation, parallelism? Provide a counterexample for any property that is not invariant under every similarity.
Is the composition of a dilation and an isometry commutative?
Process/skills
G.PS.2 Observe and explain patterns to formulate generalizations and conjectures
G.RP.3 Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion
G.CM.11 Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams
G.CN.1 Understand and make connections among multiple representations of the same mathematical idea
G.R.3 Use representation as a tool for exploring and understanding mathematical ideas
Vocabulary
Collinearity
Composition
Dilation
Glide Reflection
Image
Invariant / Isometry (direct vs. opposite/indirect)
Midpoint
Orientation
Parallelism
Perpendicularity / Pre-image
Reflection
Rotation
Similarity
Symmetry
Translation
Suggested assessments
§ Pre-assessment
§ Regents/State exams
§ Projects
§ Participation
§ Q-A Responses
§ Oral responses
§ Conversations / § On-spot checks of class work
§ Homework
§ Observations of students’ work/behaviors
§ Students’ explanations / § Draw a picture
§ Tests and Quizzes
§ T/F
§ Multiple Choice
§ Written Response
§ Ticket out the door
See Blackboard for specific Assessments*
Resources