Download Unit C: Congruency Proofs

Geometry / Unit C: Congruency Proofs
Unit Overview
Math Florida Standards
Content Standards
Students prove theorems—using a variety of formats—and solve problems about triangles,
quadrilaterals, and other polygons. They use triangle congruence as a familiar foundation for the
development of formal proof. Students will prove theorems about lines, angles, triangles, and
parallelograms and use the theorems to solve problems.
MAFS.912.G-CO.3.9
MAFS.912.G-CO.3.10
MAFS.912.G-CO.3.11
Standards for
Mathematical Practice
MAFS.K12.MP.1.1
MAFS.K12.MP.3.1
MAFS.K12.MP.5.1
MAFS.K12.MP.6.1
Textbook Resources
Pearson Prentice Hall Geometry
Sections: 4.1-4.7, 6.9
Videos:
Tasks:
Projects:
?:
Mathematics Formative Assessment System Tasks
The system includes tasks or problems that teachers can implement
with their students, and rubrics that help the teacher interpret
students' responses. Teachers using MFAS ask students to perform
mathematical tasks, explain their reasoning, and justify their solutions.
Rubrics for interpreting and evaluating student responses are included
so that teachers can differentiate instruction based on students'
strategies instead of relying solely on correct or incorrect answers. The
objective is to understand student thinking so that teaching can be
adapted to improve student achievement of mathematical goals
related to the standards. Like all formative assessment, MFAS is a
process rather than a test. Research suggests that well-designed and
implemented formative assessment is an effective strategy for
enhancing student learning.
Mathematics Formative Assessment System Tasks
This a working document that will continue to be revised and improved taking your feedback into consideration.
Other Resources
Classroom Resources/ Manipulatives
Patty paper, AngLegs
GeoGebra
Interactive Digital Demonstrations:
Congruent line segments
Congruent angles
Congruent triangles
SSS, ASA, SAS, AAS, H-L
Why doesn’t AAA prove congruence?
Why doesn’t SSA work?
Video Resources:
Virtual Nerd
You Tube
Khan Academy
Pasco County Schools, 2014-2015
Geometry / Unit C: Congruency Proofs
Unit Scale (Multidimensional) (MDS)
The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for
measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1,
“What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps
students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the
academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and
declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards.
4.0
In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was taught within these
standards. Examples:

3.0
The Student will:



2.0
Utilize the theorems about lines, angles, triangles and parallelograms to construct a multi-step proof.
Prove theorems about lines and angles; use theorems about lines and angles to solve problems. 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. (MAFS.912.G-CO.3.9)
Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a
triangle sum to 180°; triangle inequality theorem; 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. (MAFS.912.G-CO.3.10)
Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms
with congruent diagonals. (MAFS.912.G-CO.3.11)
The student will recognize or recall specific vocabulary, such as:

Vertical angles, alternate interior angles, corresponding angles, perpendicular bisector, equidistant, triangles, triangle sum, base angles,
isosceles, midpoints, midsegment, median of a triangle, centroid, properties of a parallelogram, opposite sides, opposite angles, diagonals
The student will perform basic processes, such as:



1.0
Understand the theorems about lines and angles.
Understand the theorems about triangles.
Understand the theorems about parallelograms.
With help, partial success at 2.0 content but not at score 3.0 content
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Geometry / Unit C: Congruency Proofs
Unpacking the Standard: What do we want students to Know, Understand and Do (KUD):
The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question
#1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what
student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional
scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are
essential for mastery.
Domain: Congruence
Cluster: Experiment with Transformations in the Plane (Major)
Standard: MAFS.912.G-CO.3.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a
parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
Parallelograms have special relationships, including:




opposite sides of a parallelogram are congruent
opposite angles of a parallelogram are congruent
diagonals of a parallelogram bisect each other
rectangles are parallelograms with congruent diagonals
Know
Declarative knowledge: Facts, vocab., information
Identify parts of a parallelogram, including:



Opposite sides
Opposite angles
Diagonals
Do
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Level 3 (Analysis)
Prove theorems about parallelograms
Corresponding parts of congruent polygons are
congruent
Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard?
Congruency, characteristics of a parallelogram
Learning Goals:
Moving Beyond:
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Geometry / Unit C: Congruency Proofs
Sample Uni-Dimensional, Lesson Scale:
The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of
knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown
below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to
further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1,
“Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the
actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level,
not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard.
Standard:
MAFS.912.G-CO.3.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a
parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Learning Progression
Sample Tasks
Score
I can…
 Given a proof, find and defend a different way to prove a
quadrilateral is a parallelogram
4.0
3.5
Given
EFGH is
a
parallelogram with diagonals EG
and HF.
Since the diagonals of a parallelogram bisect each other, FK ≅ HK and
GK ≅ EK. EF≅ GH, because opposite sides of a parallelogram are
congruent. Therefore by the SSS congruence postulate, triangles EFK
and GHK are congruent.
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
Suppose ABCD is a parallelogram, and that M and N are the midpoints of
I can…
Use theorems, postulates, or definitions to prove theorems about
parallelograms, including:
3.0
2.5
Prove this a different way.
segment AB and segment CD, respectively. Prove that MN = AD, and that line
MN is parallel to AD.
 Opposite sides of a parallelogram are congruent
 Opposite angles of a parallelogram are congruent
 Diagonals of a parallelogram bisect each other
 Rectangles are parallelograms with congruent diagonals
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Geometry / Unit C: Congruency Proofs
I can ….
2.0



1.0
I need prompting and/or support to complete 2.0 tasks.
Prove congruency in triangles
Corresponding parts of congruent polygons are congruent
Recognize or recall specific vocabulary, such as, quadrilateral,
parallelogram, rectangle, diagonals, distance formula, midpoint
formula, slope, bisector, congruence properties
Sample High Cognitive Demand Tasks:
These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply
demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them
as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and
common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?”
and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New
Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors).
MAFS.912.G-CO.3.11
MAFS Mathematical Content Standard(s)
Design Question 1; Element 1
MAFS Mathematical Practice(s)
Design Question 1; Element 1
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent,
the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent
diagonals.
1. Make sense of problems and persevere in solving them.
3. Construct viable arguments and critique reasoning of others.
Marzano’s Taxonomy
3.0 (Analysis)
Teacher Notes
Scaffolding: use any of the following questions to encourage thinking.
What is the information given to you in the problem? In the picture?
How will you use that information? What theorems do you know about parallelograms? What mathematical evidence
would support your solution?
Questions to develop mathematical thinking,
possible misconceptions/misunderstandings,
how to differentiate/scaffold instruction,
anticipate student problem solving strategies
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Geometry / Unit C: Congruency Proofs
Task
*These tasks can either be teacher created or
modified from a resource to promote higher
order thinking skills. Please cite the source for
any tasks.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015