Download Plainfield Public Schools Mathematics Rigorous Curriculum Design

Plainfield Public Schools
Mathematics
Rigorous Curriculum Design
Unit Planning Organizer
Grade/Course Geometry
Unit of Study Unit 2 Similarity and Proof
Pacing
7 weeks including 2 weeks for reteaching or enrichment
Standards for Mathematical Practice
Mathematical Practices that apply to the unit:
MP1.
MP2.
MP3.
MP4.
MP5.
MP6.
MP7.
MP8.
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.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Unit Standards
G.SRT.1Verify 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.
G.SRT.2 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.
G.SRT.3 Use the properties of similarity transformations to establish the AA
criterion for two triangles to be similar.
G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one
side of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
G.CO.10 Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are congruent;
the segment joining midpoints of two sides of a triangle is parallel to the third side
and half the length; the medians of a triangle meet at a point.
G.C.1 Prove that all circles are similar.
“Unwrapped” Skills
(students need to be able to do)
“Unwrapped” Concepts
DOK
(students need to know)
Levels
FOCUS STANDARD:
G.SRT.2 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.
use
definition of similarity
2
explain
transformations and similarity for triangles
2
FOCUS STANDARD:
G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for
two triangles to be similar.
Use
properties of similarity transformations
2
Establish ( prove)
AA criterion for two triangles to be similar
3
FOCUS STANDARD:
G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a
triangle divides the other two proportionally, and conversely; the Pythagorean Theorem
proved using triangle similarity.
G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of
a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment
joining midpoints of two sides of a triangle is parallel to the third side and half the length;
the medians of a triangle meet at a point.
Prove
a line parallel to one side of a triangle divides the
other two proportionally
the Pythagorean Theorem
measures of interior angles of a triangle sum to
180°
base angles of isosceles triangles are congruent
the segment joining midpoints of two sides of a
triangle is parallel to the third side and half the
length
4
4
4
4
4
4
medians of a triangle meet at a point
“Unwrapped” Skills
(students need to be able to do)
“Unwrapped” Concepts
(students need to know)
Additional Standard:
DOK
Levels
G.C.1 Prove that all circles are similar.
Prove
circles are similar
4
Unit Vocabulary Terms
“Unwrapped” Focus Standards Concepts
Supporting Standards Concepts /Other Unit-Specific Terms
Conditional Statement (all forms)
Two Column Proof
Postulate
Similar Polygon
Scale Factor
Corresponding Angles
Conjecture
Inductive Reasoning
Counter Example
Complimentary Angles
Supplementary Angles
Geometric Mean
Proportion
Ratio
Linear Pair
Vertical Angles
Essential Questions
If two figures are similar, how do you find the length
of a missing side?
Corresponding Big Ideas
Understanding Geometric Relationships In
Diagrams.
How do you identify a similarity transformation in a
plane?
Similar geometric figures can be created by
transformations. All transformations create similar
geometric figures. Dilations, in particular create
figures that are similar, but may not be congruent.
Similar geometric figures have angles that are
congruent and segments that are proportional in
length. Congruence is also similarity. It is just a
more specifically defined similarity where the ratio
of lengths is 1:1.
How can the side of a triangle be partitioned into
segments of a given ratio? How can this
information be used to solve problems involving
similar triangles?
How do you write a geometric proof?
How can similarity and congruence be used to
solve problems and/or prove statements about or
properties of triangles?
The processes of proving include a variety of
activities, such as developing conjectures,
considering the general case, exploring with
examples, looking for structural similarities across
cases, and searching for counterexamples. A proof
can have many different valid representational
forms, including narrative, picture, diagram, twocolumn presentation, or algebraic form.
Engaging Learning Experiences
Engaging Scenario
Student challenge Students design a star for the front of a building based on the criteria of
donor.
Current situation: Star must fit in a designated area. Students analyze space and determine
congruent triangles. The space decreases by 50% and students must redesigns and
maintains the original ratios .
Student role: Students design their projects and grade others.
Intended audience: Donator
Product: Design of two stars.
Performance Task Synopses
Task 1:
 Students determine the shape and coordinates of the original star

Complete the table comparing sides to determine if there are congruent triangles.
(The trick is the congruent triangles that make up the non used space)

Students calculate the 50 % of the space to use
Performance Task
In Detail
Focus Standards:
Which standard(s) (priority/supporting) will the task address? G.SRT.2,
G.SRT.2 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.
G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a
triangle divides the other two proportionally, and conversely; the Pythagorean Theorem
proved using triangle similarity.
What essential Question(s) and corresponding Big Idea(s) will this task target?
Essential Questions
How can the side of a triangle be partitioned into segments of a given ratio? How can this
information be used to solve problems involving similar triangles?
How can similarity and congruence be used to solve problems and/or prove statements
about or properties of triangles?
Big Ideas :
Similar geometric figures can be created by transformations. All transformations create
similar geometric figures. Dilations, in particular create figures that are similar, but may not
be congruent.
The processes of proving include a variety of activities, such as developing conjectures,
considering the general case, exploring with examples, looking for structural similarities
across cases, and searching for counterexamples. A proof can have many different valid
representational forms, including narrative, picture, diagram, two-column presentation, or
algebraic form.
Which “unwrapped specific concepts and skills will this task target?
Skill : use, explain, prove
Concept
 the definition of similarity to decide if transformations are similar using similarity
transformations the meaning of similarity for triangles

Prove theorems about triangles. Theorems include: a line parallel to one side of a
triangle divides the other two proportionally, and conversely; the Pythagorean
Theorem proved using triangle similarity
How can I differentiate the application and/or evidence to meet the varying needs
of my students?
 Give a personal cue to begin work
 Give work in smaller units
 Provide immediate reinforces and feedback
 Make sure the appropriate books and materials are available
 Introduce the assignment in sequential steps
 Check for student understanding of instructions
 Check on progress often in the first few minutes of work
 Provide time suggestions for each task
 Provide a checklist for long detailed tasks
 Use technological resources

Engaging Learning Experiences
Engaging Scenario
Advanced or Exemplary
 All “Goal” criteria plus:
Goal
 Complete the original picture
 complete table for original picture
 complete second picture
 complete table for second picture
 determine theorems based on patterns
Progressing
 Meets _2_ of the “Goal” criteria
Beginning
 Meets fewer than _2 of the “Goal” criteria
 Task to be repeated after re-teaching
 Comments:
Interdisciplinary Connections and
Related Focus Standards
9.1 21st-Century Life & Career Skills All
students will demonstrate the creative,
critical thinking, collaboration, and
problem-solving skills needed to function
successfully as both global citizens and
workers in diverse ethnic and
organizational cultures.
CCSS.ELA-Literacy.RL.9-10.1 Cite strong
and thorough textual evidence to support
analysis of what the text says explicitly as
well as inferences drawn from the text.
21st Century Learning Skills
Specific to Task
those that apply for each task:
❑ Teamwork and Collaboration
❑ Initiative and Leadership
❑ Curiosity and Imagination
❑ Innovation and Creativity
❑ Critical thinking and Problem Solving
❑ Flexibility and Adaptability
❑ Effective Oral and Written Communication
❑ Accessing and Analyzing Information
❑ Other