Download Unit 2 - Long Beach Unified School District

Unit 2 Transformational Geometry
Math 7 ACC
Unit Goals – Stage 1
Number of Days: 24 days
10/17/16 – 11/18/16
Unit Description: Students study congruency and similarity by experimenting with dilations, rotations, reflections, and translations of geometric figures.
During this unit, students are introduced to similar figures. Students will explore the relationships between interior and exterior angles of triangles and
angles formed by parallel lines that are cut by a transversal.
Materials: patty paper (tracing paper), calculators, pattern blocks, Geoboard dot paper*, protractors*
(* Paper tool available at LBUSD Curriculum Intranet  Instructional Tools  Middle School  Mathematical Tools)
Standards for Mathematical
Transfer Goals
Practice
Students will be able to independently use their learning to…
SMP 1 Make sense of problems and
• Make sense of never-before-seen problems and persevere in solving them.
persevere in solving them.
• Construct viable arguments and critique the reasoning of others.
SMP 2 Reason abstractly and
Making Meaning
quantitatively.
UNDERSTANDINGS
ESSENTIAL QUESTIONS
SMP 3 Construct viable arguments
Students will understand that…
Students will keep considering…
and critique the reasoning of
• A two-dimensional figure is congruent to another if the
• How do transformations affect lines, line
others.
second figure can be obtained from the first figure by a
segments, angles, parallel lines, and twoSMP 4 Model with mathematics.
sequence of rotations, reflections, and translations.
dimensional figures?
SMP 5 Use appropriate tools
• Similar figures have the same shape, but not
• How can you use transformations to determine
strategically.
necessarily the same size. A two-dimensional figure is
if two figures are congruent or similar?
SMP 6 Attend to precision.
similar to another if the second figure can be obtained
• How can you find missing angle measures
SMP 7 Look for and make use of
from the first figure by a sequence of rotations,
when parallel lines are intersected by a
structure.
reflections, translations, and dilations.
transversal?
SMP 8 Look for and express
Acquisition
regularity in repeated
KNOWLEDGE
SKILLS
reasoning.
Students will know…
Students will be skilled at and/or be able to…
• The definition of academic vocabulary
• Verify the properties of rotations, reflections, and
Standards for Mathematical
words, such as adjacent angles, alternate
translations experimentally.
Content Clusters Addressed
exterior angles, alternate interior angles,
• Describe the effect of dilations, translations, rotations, and
[a] 7.G.B Solve real‐life and
complementary angles, dilation, image, line
reflections on two-dimensional figures using coordinates.
mathematical problems
of reflection, remote interior angle, rotation, • Describe sequences of rotations, reflections, translations,
involving angle measure,
same-side interior angles, similar,
and dilations that can verify whether two-dimensional
area, surface area, and
supplementary angles, transformation,
figures are similar or congruent to one another.
volume.
translation, transversal, and vertical angles. • Use facts about supplementary, complementary, vertical,
[m] 8.G.A Understand congruence
• Transformations include translations,
and adjacent angles in multi-step problems to write and
and similarity using
reflections, rotations, and dilations
solve equations for unknown angles in a figure.
physical models,
•
Complementary
angles
add
up
to
90°
and
•
State facts about the angles created when parallel lines
transparencies, or
supplementary
angles
add
up
to
180°.
are cut by a transversal.
geometry software.
• Vertical angles are congruent.
• Informally argue facts about the measures of interior and
exterior angles of triangles and the angle-angle criterion
• The sum of the measures of the three
for similarity of triangles.
interior angles of a triangle is 180o.
LONG BEACH UNIFIED SCHOOL DISTRICT
1
Posted 10/4/16
2016-2017
Unit 2 Transformational Geometry
Math 7 ACC
Assessed Grade Level Standards
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.
Standards for Mathematical Content
[a] 7.G.B
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.G.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple
equations for an unknown angle in a figure.
[m]
8.G.A
8.G.1
8.G.2
8.G.3
8.G.4
8.G.5
Understand congruence and similarity using physical models, transparencies, or geometry software.
Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of
rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations,
reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity
between them.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel
lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same
triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
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Posted 10/4/16
Unit 2 Transformational Geometry
Math 7 ACC
Evidence of Learning – Stage 2
Assessment Evidence
Unit Assessment
Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency.
Concepts and skills that may be assessed in Claim1:
7.G.B
•
The student solves real-life and mathematical problems involving angle measure including problems requiring writing and solving equations.
8.G.A
•
The student verifies that rigid transformations preserve distance and angle measures.
•
The student describes sequences of rotations, reflections, translations, and dilations that can verify whether two-dimensional figures are similar or
congruent to one another.
•
The student constructs a new figure after the original figure is dilated, rotated, reflected, or translated.
•
The student describes the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Claim 2: Students can solve a range of wellposed problems in pure and applied
mathematics, making productive use of
knowledge and problem-solving strategies.
Standard clusters that may be assessed in
Claim 2:
•
7.G.B
•
8.G.A
Claim 3: The student can clearly and precisely
construct viable arguments to support their own
reasoning and critique the reasoning of others.
Standard clusters that may be assessed in
Claim 3:
•
7.G.A
•
8.G.A
Claim 4: The student can analyze complex,
real-world scenarios and can construct and use
mathematical models to interpret and solve
problems.
Standard clusters that may be assessed in
Claim 4:
•
7.G.B
Other Evidence
Formative Assessment Opportunities
• 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
Intranet Unit Assessment Resources
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
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Posted 10/4/16
Unit 2 Transformational Geometry
Math 7 ACC
Learning Plan – Stage 3
Days
Learning Target
1 day
I will explore
transformational
geometry by
participating in the
Opening Task.
I will investigate
congruency by…
7-8
days
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math Advanced 2
Expectations
(Activities and Lessons)
OPENING TASK – Escher’s Artwork
This Opening Task is a discussion and should take
should take about 20 minutes. Display the website
with Escher’s artwork and facilitate a class discussion
with the discussion questions utilizing Talk Moves.
This task is a gateway into the entire unit of
transformational geometry.
• Experimentally verifying the properties of
translations, reflections, and rotations. (SMP 5)
• Transforming lines, angles, parallel lines, and
two-dimensional figures using translations,
reflections, and rotations to verify congruence.
• Describing the effect of translations, reflections, and
rotations on two-dimensional figures using
coordinates.
• Describing a sequence of transformations that
exhibits congruency, given two congruent figures.
• Explaining that a two-dimensional figure is
congruent to another if the second can be obtained
from the first by a sequence of translations,
reflections, and rotations. (SMP 3)
• Answering questions such as…
o How can you use coordinates to describe the
results of a translation, reflection, or rotation?
o Compare and contrast translations, reflections,
and rotations. Explain why these
transformations generate congruency.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
4
• Section 2.1
(Activity 1;
Examples 1, 2, and 3)
• Section 2.2
(Activities 1 and 4;
Examples 1, 2, and 3)
• Section 2.3
(Activities 1, 2, and 3;
Examples 1, 2, and 3)
• Section 2.4
(Activity 2;
Examples 2, 3, 4 and 5)
Curriculum Intranet
Conceptual Understanding:
• Escher’s Artwork
Discussion Questions
• M.C. Escher’s Artwork
Website
• Article: The Mathematical
Side of M.C. Escher
• What is a Tessellation?
Conceptual Understanding:
• Patty Paper
Transformations Using
Coordinates Activity
• Zombie Apocalypse
Lesson Plans and Game
Procedural Skills and
Fluency:
• Reflecting a Rectangle
Over a Diagonal Task
• Same Size Same Shape?
Task
• Transformations—
Shortest Sequence
• Triangle Congruence with
Coordinates Task
Application:
• Alumni Center
Performance Task
• Desmos: Translations
• Transformations: Aaron’s
Designs Task
• Modeling: How Did They
Make Mrs. Pac-Mac?
(SMP 4)
• Modeling: How Do
Skytypers Write
Messages? (SMP 4)
Posted 10/4/16
Unit 2 Transformational Geometry
Math 7 ACC
Learning Plan – Stage 3
Days
Learning Target
I will investigate
similarity by…
•
•
•
•
•
•
•
4-5
days
2-3
days
I will check my
understanding of
transformations by
participating in the
FAL.
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math Advanced 2
Expectations
(Activities and Lessons)
Extending knowledge of transformations to include
• Section 2.5
dilations.
(Activity 1;
Examples 1, 2, and 3)
Connecting scale factors to dilations.
• Section 2.7
Finding missing measures in similar figures.
(Activity 1;
Describing the effect of dilations on twoExamples 1, 2, 3, and 5)
dimensional figures using coordinates.
• STEM Video: Shadow
Describing a sequence of transformations that
Puppets
exhibits similarity, given two similar figures.
Explaining that a two-dimensional figure is similar to
another if the second can be obtained from the first
by a sequence of dilations, rotations, reflections,
and translations. (SMP 3)
Answering questions such as…
o What is dilation? What is a scale factor?
o Explain how similarity is connected to
proportionality.
o How does the dilation affect the length of line
segments?
o How can you use coordinates to find scale
factors?
Curriculum Intranet
Conceptual Understanding:
• Dilation Exploration
Procedural Skills and
Fluency:
• Are They Similar? Task
• Effects of Dilations on
Lengths, Area, and Angles
Task
Application:
• STEM Performance Task:
Master Puppeteer
• Soccer Complex
Performance Task
• Desmos: Polygraph
Transformations
FORMATIVE ASSESSMENT LESSON
• Representing and Combining Transformations
(SMP 1, 3, 5, 6, 7, 8)
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
5
Posted 10/4/16
Unit 2 Transformational Geometry
Math 7 ACC
Learning Plan – Stage 3
Days
Learning Target
I will explore angle
relationships by…
•
•
•
•
•
2-3
days
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math Advanced 2
Expectations
(Activities and Lessons)
Identifying vertical angles, adjacent angles,
• Section 12.1
supplementary angles, and complementary angles.
(Activities 2 and 3;
Examples 1, 2, and 3)
Writing equations to find unknown angles in
complex figures that combine adjacent, vertical,
• Section 12.2
complementary, and supplementary angles.
(Activities 1 and 2;
Examples 1, 2, and 3)
Finding unknown angle measures when parallel
lines are intersected by a transversal using
• Section 3.1
knowledge of corresponding angles, alternate
(Activities 1 and 2;
interior angles, alternate exterior angles, and
Examples 1, 2, 3, and 4)
same-side interior angles.
Justifying angle relationships through
transformations to show congruency using patty
paper. (SMP 5)
Answering questions such as…
o Explain why two angles cannot be vertical and
adjacent at the same time.
o Ariana claims that two angles are
complementary only if they are adjacent and
make a right angle. Is Ariana correct? Justify
your reasoning.
o How can you use angle pairs to solve problems?
o How can you find missing angle measures when
parallel lines are intersected by a transversal?
o How are corresponding angles, alternate interior
angles, alternate exterior angles, and same-side
interior angles similar and different?
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
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Curriculum Intranet
Conceptual Understanding:
• Angle Relationships
Investigation with Patty
Paper
• Vertical Angles Dynamic
Tool
• Corresponding Angles
Dynamic Tool
Procedural Skills and
Fluency:
• Flipbook: Angle
Relationships
• Rigid Motions and
Congruent Angles Task
• Wanted Poster: Angles
Application:
• Street Intersections Task
• Truss Bridge Performance
Task
Posted 10/4/16
Unit 2 Transformational Geometry
Math 7 ACC
Learning Plan – Stage 3
Days
Learning Target
I will apply
transformations and
angle relationships
to triangles by …
•
•
•
•
3-4
days
I will prepare for the
unit assessment on
transformational
geometry by…
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math Advanced 2
Expectations
(Activities and Lessons)
Visually representing and explaining the Triangle
• Section 3.2
Sum Theorem. (SMP 3, SMP 6)
(Activities 1, 2, 3, and 4;
Examples 1, 2, and 3)
Finding a missing angle measure in a triangle,
including the exterior angles and remote interior
• Section 3.4
angles of triangles.
(Activities 1 and 3;
Examples 1 and 2)
Determining if two triangles are similar by using the
Angle-Angle (AA) Postulate or using proportional
side lengths.
Answering questions such as…
o How can you justify the Triangle Sum Theorem?
o How can you determine when two triangles are
similar?
o What is the Angle-Angle Similarity Postulate?
o How can you determine if corresponding sides
are proportional?
o Are two pairs of congruent angles enough
information to conclude that two triangles are
similar? Explain. (SMP 3)
• Incorporating the Standards for Mathematical
Practice (SMPs) along with the content standards
to review the unit.
1-2
days
1 day
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
•
•
•
•
Ch. 2 Study Help (p. 68)
Ch. 2 Review (p. 91 – 99)
Ch. 3 Study Help (p. 116)
Ch. 3 Review
(p. 133 – 139)
Curriculum Intranet
Conceptual Understanding:
• Triangle Sum Theorem
PowerPoint
• Triangle Sum Theorem
Activity
• Interior Angles of a
Triangle Dynamic Tool
• Similar Triangles Dynamic
Tool
• Triangle Exterior Angle
Dynamic Tool
Procedural Skills and
Fluency:
• What’s My Angle Task
Procedural Skills and
Fluency:
• Intersection Performance
Task
Application:
• Transformations
Performance Task
Unit Assessment
Synergy: 2016-17 Math 7 ACC Unit 2
7
Posted 10/4/16