Download Unit 2 Lines and Transformations Geometry

Unit 2 Lines and Transformations
Geometry
Unit Goals – Stage 1
Number of Days: 29 days
10/10/16 – 11/18/16
Unit Description: The students use skills learned in Unit 1 to justify key relationships among lines and angles. In Unit 2, students investigate intersecting
lines and their resultant angles. Parallel and/or perpendicular lines are identified using the coordinate plane. Students never lose sight of constructing a
logical, and supported, argument as a critical area in Geometry. Adding to their list of proof techniques, students use rigid and non-rigid transformations to
define similar or congruent figures. Dynamic software plays an important role, allowing for student explorations and teacher demonstrations.
Materials: Patty paper, straight edges, compasses, dynamic software (Desmos, Geogebra)
Standards for Mathematical
Practice
SMP 1 Make sense of
problems and
persevere in solving
them.
SMP 2 Reason abstractly and
quantitatively.
SMP 3 Construct viable
arguments and critique
the reasoning of others.
SMP 4 Model with
mathematics.
SMP 5 Use appropriate tools
strategically.
SMP 6 Attend to precision.
SMP 7 Look for and make use
of structure.
SMP 8 Look for and express
regularity in repeated
reasoning.
Transfer Goals
Students will be able to independently use their learning to…
• Make sense of never-before-seen problems and persevere in solving them.
• Construct viable arguments and critique the reasoning of others.
Making Meaning
UNDERSTANDINGS
ESSENTIAL QUESTIONS
Students will understand that…
Students will keep considering…
• Accurate definitions of essential geometric terms are
• How are properties of geometric figures related to their
essential for precise communication.
measurable attributes?
• The converse of a true statement is not necessarily
• Why is it necessary to prove a conjecture before
also true.
considering it a theorem?
• Coordinates can be used to prove relationships
• How does logical reasoning facilitate understanding
between lines and angles.
geometric relationships?
• A conjecture becomes a theorem only after being
• Why is logical reasoning a skill that is important in life
validated by a well-reasoned proof.
outside of school? Where do you see examples of this?
• Rigid transformations preserve a figure’s size and
• What are the big understandings behind “slope”?
shape or distance and angle measure and, thus, may • Based on the definition of parallel lines as two lines that
be used to prove the congruence of two figures.
don’t intersect, are parallel lines always an equal distance
apart? (non-Euclidean geometry)
• A sequence of rigid motions and dilations may be
used to prove the similarity of two figures.
• Can a transformation always be achieved in more than
one way?
• The order in which a transformation is performed may
not
be
commutative.
Standards for Mathematical
Acquisition
Content Clusters Addressed
KNOWLEDGE
SKILLS
[s] G-CO.A Experiment with
Students will be skilled at and/or be able to…
transformations Students will know…
•
Precise
definitions
for
point,
line,
•
Identify relationships between lines and relationships between anglein the plane.
plane, distance, angle, circle,
pairs.
[m] G-CO.B Understand
perpendicular line, parallel line, and • Identify the relationships between parallel lines cut by a transversal and
congruence in
line segment.
the pairs of angles formed by these lines.
terms of rigid
• Formulas for: slope, distance,
• Build a logical progression of reasons to construct a viable argument.
motions.
Pythagorean Theorem.
• Write the converse of a given theorem.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
1
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Unit 2 Lines and Transformations
[m] G-CO.C
Prove
geometric
theorems.
[s] G-CO.D Make geometric
constructions.
[m] G-SRT.A Understand
similarity in
terms of
similarity
transformations
[m] G-GPE.B Use
coordinates to
prove simple
geometric
theorems
algebraically.
[m] G-MG.A Apply
geometric
concepts in
modeling
situations.
•
•
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•
•
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Definitions of rotations, reflections,
and translations in terms of angles,
circles, and line segments.
Rigid and non-rigid transformations.
The composition of rigid motions is
also a rigid motion.
The line of reflection is the
perpendicular bisector of the
segment connecting a point on the
preimage and its corresponding
point on the image, when the point
is not on the line of reflection.
Coordinate rules for a point rotated
about the origin and dilation with a
given scale factor k.
The scale factor of a dilation is the
ratio of the corresponding sides of
the image and preimage.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
Geometry
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2
Construct a line through a given point that is parallel to a given line.
Construct a line perpendicular to a line m through a point P, which is not
on line m.
Given a line segment, construct a perpendicular bisector.
Use coordinates to prove simple geometric theorems algebraically
including the slope criteria for parallel and perpendicular lines, distance
formula and the Pythagorean Theorem.
Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure.
Given two figures, use the definition of congruence in terms of rigid
motions to decide if they are congruent.
Represent transformations using coordinate and vector notation.
Identify lines of symmetry and rotational symmetry.
Describe and perform similarity transformations.
Solve real-life problems involving transformations.
Construct a dilation with a given scale factor k.
Posted 9/13/16
Unit 2 Lines and Transformations
Geometry
Assessed Grade Level Standards
Standards for Mathematical Practice
SMP 1
SMP 2
SMP 3
SMP 4
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
SMP 5
SMP 6
SMP 7
SMP 8
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Standards for Mathematical Content
[s] G-CO.A
G.CO.1
G-CO.2
G-CO.3
G-CO.4
G-CO.5
[m] G-CO.B
G-CO.6
Experiment with transformations in the plane.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance
along a line, and distance around a circular arc.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points
in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g.,
translation versus horizontal stretch).
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure onto another.
Understand congruence in terms of rigid motions. [Build on rigid motions as a familiar starting point for development of concept of geometric
proof]
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two
figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
[m] G-CO.C
G-CO.9
Prove geometric theorems. [Focus on validity of underlying reasoning while using variety of ways of writing proofs]
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate
interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those
equidistant from the segment’s endpoints.
[s] G-CO.D
Make geometric constructions.
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper
folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point
not on the line.
G-CO.12
[m] G-SRT.A
G-SRT.1
G-SRT.2
Understand similarity in terms of similarity transformations.
Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all
corresponding pairs of sides.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
3
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Unit 2 Lines and Transformations
[m] G-GPE.B
G-GPE.5
G-GPE.6
[m] G-MG.A
G-MG.3
Geometry
Use coordinates to prove simple geometric theorems algebraically. [Include distance formula; relate to Pythagorean theorem]
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Apply geometric concepts in modeling situations.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working
with typographic grid systems based on ratios).
Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
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Unit 2 Lines and Transformations
Geometry
Evidence of Learning – Stage 2
Assessment Evidence
Unit Assessment
Students will complete selected response and constructed response items to indicate level of mastery/understanding of the unit standards as outlined in
this guide.
[s] G-CO.A
• Students will describe transformations as functions that take points in the plane as inputs and give other points as outputs.
• Students will compare transformations that preserve shape and size to those that do not.
• Students will describe the rotations and/or reflections that carry an image onto itself.
• Students will draw a transformed figure given a series of rotations, reflections and/or translations.
• Students will specify a sequence of transformations that carry a given figure onto itself.
• Students will determine the scale factor for a given preimage and dilated image.
• Students will construct an image from a given preimage and scale factor.
[m] G-CO.B
• Students will use geometric descriptions of rigid motions to transform figures.
• Students will use the definition of congruence in terms of rigid motions to decide if two figures are congruent.
• Students will predict the effect of a given rigid motion on a given figure.
[m] G-CO.C
• Students will prove theorems about lines and angles such as: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior
angles are congruent; and corresponding angles are congruent.
[s]
•
G-CO.D
Students will make formal constructions using straightedge and compass or patty paper.
[m] G-SRT.A
• Students will verify that a line not passing through the center of a dilation to a parallel line, leaves a line passing through the center unchanged.
• Students will verify that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
• Students will use the definition of similarity in terms of similarity transformations to decide if two figures are similar.
• Students will explain the meaning of similarity of triangles as the equality of all corresponding pairs of angles and the proportionality of all
corresponding pairs of sides.
[m] G-GPE.B
• Students will prove the slope criteria for parallel and perpendicular lines.
• Students will use parallel and perpendicular lines to solve geometric problems.
• Students will find the point on a directed line segment between two given points that partitions the segment into a given ratio.
[m] G-MG.A
• Students will use geometric shapes, measures and properties to solve real-world problems.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
5
Posted 9/13/16
Unit 2 Lines and Transformations
Geometry
For selected content, students will need to…
• Solve complex problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.
• Clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others.
Other Evidence
Formative Assessment Opportunities
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Opening Tasks
Informal teacher observations
Checking for understanding using active participation strategies
Exit slips/summaries
Modeling Lessons (SMP 4)
Tasks
Formative Assessment Lessons (FAL)
Quizzes / Chapter Tests
Big Ideas Math Performance Tasks
SBAC Interim Assessment Blocks
Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website – “M” Mathematics – Curriculum Documents
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
6
Posted 9/13/16
Unit 2 Lines and Transformations
Geometry
Learning Plan – Stage 3
Days
Learning Target
1-2
days
I will explore the art
of symmetry,
reflection and
rotation by
participating in
Fantastic Folded
Name.
I will identify
relationships
between two lines
by…
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math Geometry
Expectations
(Activities and Lessons)
Opening Task- Fantastic Folded Name
The opening task requires students to use reflection
and rotation to create an “art project” featuring their
name displayed eight times around the center of a
square. After the art is finished, there are suggested
questions for the teacher to ask to preview some of the
unit’s content.
• Knowing precise definitions of terms which are based
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•
3-5
days
I will prove
relationships
between lines and
angles by…
4-5
days
on the undefined notions of point, line, and distance
along a line. (SMP 6)
Experimenting with coplanar and non-coplanar lines
to discover the relationships between the given lines.
Identifying lines and planes, and pairs of angles
formed by transversals.
Proving theorems about parallel lines. (SMP 3)
Using lines and angles to solve real-world problems.
Answering questions such as…
o What tools can you use to illustrate geometric
concepts?
o What are the intersection possibilities for two
lines? Three non-coplanar lines? Two planes? A
line and a plane?
o If two lines do not intersect, are they parallel?
o Are corresponding angles always congruent?
• Formally constructing parallel lines. (SMP 5)
• Writing the converse of conditional statements and
deciding whether each is true or false.
• Proving theorems about parallel lines. (SMP 3)
• Using the transitive property to prove that lines are
parallel.
• Finding the distance from a point to a line.
• Formally constructing perpendicular lines. (SMP 5)
• Proving theorems about perpendicular lines. (SMP 3)
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
7
• Section 3.1
• STEM Video: Squaring a
Tree House
• Section 3.2
Curriculum Intranet
Procedural Skills and
Fluency:
• Fantastic Folded Name
Conceptual Understanding:
• Illustrative Mathematics:
Defining Parallel Lines
Task
• Illustrative Mathematics:
Defining Perpendicular
Lines Task
• Which One Doesn’t
Belong - Lines
Applications:
• STEM Performance Task:
Squaring a Treehouse
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Section 3.3
Section 3.4
Section 3.5
Conceptual Understanding:
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Illustrative Mathematics:
Parallel Lines in the
Coordinate Plane Task
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Illustrative Mathematics:
When Are Two Lines
Perpendicular Task
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GeoGebra: Transversals
and Parallel Lines
Demonstration
Posted 9/13/16
Unit 2 Lines and Transformations
Geometry
Learning Plan – Stage 3
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math Geometry
Days
Learning Target
Expectations
(Activities and Lessons)
• Solving real-world problems using the properties of
parallel and perpendicular lines.
• Using slope to identify parallel and perpendicular
lines.
• Writing equations for parallel and perpendicular lines.
• Using slope to find the distance from a point to a line.
• Answering questions such as…
o When two lines are intersected by a transversal,
will there always be 4 acute and 4 obtuse angles?
o Given a conditional statement, how do you write
its converse?
o What evidence is needed to know that constructed
lines are parallel?
o Under what conditions will two lines be
perpendicular?
o Why are cities often designed on a grid with
streets running parallel and perpendicular to each
other?
o How would you prove that horizontal lines are
perpendicular to vertical lines?
I will use rigid
• Section 4.1
• Performing translations, reflections, rotations,
transformations to
• Section 4.2
dilations and compositions of transformations.
show that two
• Section 4.3
• Using dynamic software to explore properties of
figures are
• STEM Video: Rotational
transformations. (SMP 5)
congruent by…
Doors
• Verifying that a translation preserves the segment
• Section 4.4
length of the preimage for a specific case.
• Representing translations using coordinate and
vector notation. (SMP 2)
5-6
• Understanding that the line of reflection is the
days
perpendicular bisector of the segment connecting a
point on the preimage and its corresponding point on
the image, when the point is not on the line of
reflection. (SMP 6)
• Identifying lines of symmetry and rotational
symmetry.
• Reflecting a preimage more than once over parallel
or intersecting lines. (SMP 6)
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
8
Curriculum Intranet
Procedural Skills and
Fluency:
•
Distance Formula: School
Yourself Geometry
Interactive Tool
Conceptual Understanding:
• Illuminations: Flip-n-Slide:
Exploring Transformations
through Modeling and
Computer Games
Interactive Activity
• Desmos: Translations
Activity
• MAA: Exploring
Geometric
Transformations in a
Dynamic Environment
Procedural Skills and
Fluency:
• Determine Coordinates
for Transformation-
Posted 9/13/16
Unit 2 Lines and Transformations
Geometry
Learning Plan – Stage 3
Days
2-3
days
Learning Target
I will check my
ability to recognize
and visualize
transformations of
2D shapes by
participating in the
FAL.
I will engage with
similarity
transformations
by…
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math Geometry
Expectations
(Activities and Lessons)
• Describing and performing congruence
transformations and similarity transformations.
• Solving real-life problems involving transformations.
• Using theorems about congruence transformations to
identify congruent figures.
• Answering questions such as…
o Have you heard of angle measures greater than
360°? In what context?
o Does the order matter with composite
transformations?
o Can rotations be written as the composition of
several reflections?
o What does it mean for two figures to be
congruent?
Formative Assessment Lesson - Representing and
Combining Transformations
• Identifying and performing dilations.
• Exploring the coordinate rule for dilations.
• Constructing a dilation with a given scale factor and a
point outside the preimage. (SMP 6)
• Solving real-life problems involving scale factors and
dilations.
• Performing and describing similarity transformations.
3-5
• Proving whether two figures are similar by a single
days
dilation or a composition of rigid motions and
dilations. (SMP 3)
• Answering questions such as…
o How can you solve for the scale factor of a
dilation?
o What will the scale factor k tell you about a
dilation?
o How do you find the center of a dilation?
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
9
• Section 4.5
• Section 4.6
Curriculum Intranet
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Shooting Pool Interactive
Tool
Illustrative Mathematics:
Identifying Rotations Task
OpenMiddle:
Transformations –
Shortest Sequence
Applications:
• STEM Performance Task:
Revolving Doors
Conceptual Understanding
• Representing and
Combining
Transformations
Conceptual Understanding:
• Verify experimentally
properties of dilation given
by a center and a scale
factor Videos
• Illustrative Mathematics:
Dilating a Line Task
• Dilations in the
Coordinate Plane Task
• GeoGebra: Dilations
Demonstration
Posted 9/13/16
Unit 2 Lines and Transformations
Geometry
Learning Plan – Stage 3
Days
Learning Target
1-2
days
I will prepare for the
unit assessment on
lines and
transformations by...
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math Geometry
Expectations
(Activities and Lessons)
o When a preimage is transformed to an image, is
there only one set of rigid motions and dilations to
achieve that image? Is the transformation unique?
o If two figures are congruent, are they similar?
o If two figures are similar, are they congruent?
o How can you prove that two squares are similar?
o Hold your hands out in front of you palms up. Is
there a series of transformations that will map one
hand onto the other?
• Incorporating the Standards for Mathematical
Practice (SMPs) along with the content standards to
review the unit.
1-2
days
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
Curriculum Intranet
Unit Assessment (LBUSD Math Intranet, Assessment)
10
Posted 9/13/16