Geometry Level Distinctions

Learning Objectives
The student will … / Level 2 / Level 3 / Level 4 / Level 5
1. Develop definitions of geometric terms using visual representations and written descriptions.
Concepts/Reasoning:
  1. Connect visual diagrams with written descriptions
  2. Distinguish between examples and counterexamples
(Counterexamples are important for inductive reasoning)
  1. Identify the characteristics of a good definition
(Lays a foundation for the properties aspect of proof)
4. Relate and distinguish geometric terms and models shifting the emphasis from definitions of whole figures and directing students’ attention to components of figures. / 1. Connect visual diagrams with written descriptions of points, lines (parallel & perpendicular), planes, angles, triangles and quadrilaterals.
Each diagram provides specific characteristics with clarity and no extraneous markings.
2. Distinguish between examples and teacher given counterexamples.
Simpler, somewhat complete supportive statements, minimum formalization, but must have reason-have to make sense of it.
3. Model the characteristics of a good definition
Primarily recognition and identification, emphasis on a match between definition and diagram, single categories only for distinction, and delay distinctions in quadrilaterals until later. / 1. Connect visual diagrams with written descriptions.
Diagrams and written descriptions have greater complexity and are developed in relatively shorter times than Level 2.
Each diagram provides specific characteristics with clarity and some extraneous markings.
2. Distinguish between examples and group determined counterexamples
Simpler, predominantly complete supportive statements, developing formalization, but must have reason-have to make sense of it.
3. Apply the characteristics of a good definition
More work on discriminating different quadrilaterals, more definition writing, take the differences and create diagrams. / 1. Connect visual diagrams with written descriptions
Diagrams and written descriptions have greater complexity and are developed in relatively shorter times than Level 3.
Students must recognize how diagrams are related and generalize these. Explanation or justification are required for statement, setting a foundation for proof
2. Distinguish between examples and counterexamples
3. Develop and use the characteristics of a good definition
Regular and irregular polygons, sets more intersecting, and multiple categories of characteristics / 1. Connect visual diagrams with written descriptions
Diagrams and written descriptions vary by complexity in relatively shorter times.
Students must recognize how diagrams are related and generalize these.Explanation or justification are required for statement, setting a foundation for proof
2. Create examples and counterexamples
3. Develop and use the characteristics of a good definition
Regular and irregular polygons, sets more intersecting, and multiple categories of characteristics

Geometry Level Distinctions

Learning Objectives
The student will … / Level 2 / Level 3 / Level 4 / Level 5
2. Use inductive reasoning to identify patterns and solve problems.
Use deductive reasoning to justify conclusions.
Concepts/Reasoning:
  1. Make conjectures (Recognize the importance of the inductive process in conjecture formulation)
  2. Determine if a conjecture is true (Give deductive arguments for the truth of conjectures)
  3. Generalize number or picture patterns
  4. Write a converse of a statement and determine if it is true.
  5. Write a deductive argument
/ 1. Teacher guided development of writing conjectures.
2. Determine if a conjecture is true. (Give deductive arguments referring to the specifics of interpreting a diagram for the truth of conjectures)
3. To recognize there is more than one way, teacher phrases questions and students argue multiple views to develop flexible reasoning.
4. Write a converse of a statement and determine if it is true. Use more familiar, real-world examples and move to geometric concepts that the students have mastered such as complementary and supplementary angles.
5. Write a deductive argument. The arguments will have the essence of the conjectures with the teacher restating the vocabulary. Problems will center on angle relationships. / 1. Teacher supported development of conjectures.
2. Determine if a conjecture is true. There will be a quicker release to student centered work with emphasis on error analysis. (eg. Distinguishing angles with parallel lines and transversals.)
3. In small groups with some teacher intervention, develop flexible reasoning using and comparing multiple views.
4. Write a converse of a statement and determine if it is true. Use converses on the geometric concepts on the Parallel Line Conjecture.
5. Write a deductive argument. The argument will have the essence of the conjecture and the students will be expected to use more of the vocabulary. / 1. Cooperatively write and apply conjectures.
2. Determine if a conjecture is true.
Assume a level of readiness in interpreting a diagram.
3. Look for all arguments to be sequential w/ a connected quality in the written paragraph.
4. Write a converse of a statement and determine if it is true. Use converses from most geometric situations.
5. Write a deductive argument. The argument will be based on the conjectures, using the correct vocabulary. / 1. Cooperatively write and apply conjectures.
2. Determine if a conjecture is true
Assume readiness in interpreting a diagram.
3. Tap curiosity and motivation students bring to tasks. Arguments need to include more awareness and use of tangential relationships or characteristics. Use larger knowledge base and apply appropriately in proof.
4. Write a converse of a statement and determine if it is true. Use converses from any geometric situations.
5. Write a deductive argument. The argument will be based on the conjectures, using the correct vocabulary.

Geometry Level Distinctions

Learning Objectives
The student will … / Level 2 / Level 3 / Level 4 / Level 5
3. Make conjectures based on investigations using geometric constructions.
Concepts/Reasoning:
  1. Develop conjectures based on the figures constructed.
  2. Using the given constraints, construct triangles and quadrilaterals based on their definitions.
  3. Recognize and construct the points of concurrency from real-world problems.
/ 1. Develop conjectures based on the figures constructed. The constructions are completed on patty paper. The conjectures are written with teacher guidance, questioning and modeling.
2. Using the given constraints, construct on patty paper, triangles based on their definitions.
3. Construction of triangles and quadrilaterals lay a foundation for conjectures they will see later on.
4. Develop accountability for verbally distinguishing similarities and differences.
5. A broader acceptable range for precision in constructions. / 1. Develop conjectures based on the figures constructed. The constructions are completed on patty paper. The conjectures are derived through student-centered activities.
2. Using the given constraints, construct, on patty paper, triangles and rectangles based on their definitions.
3. Students develop and write inferences that are not limited to the guidelines of the investigation.
4. Follow steps without overarching reasoning- making sense in the local example. / 1. Constructions are completed with patty paper and construction tools. Students develop and write the conjectures.
2. Using the given constraints, construct, with patty paper and construction tools, triangles and quadrilaterals based on their definitions.
3. Recognize and construct the points of concurrency from real-world problems.
4. Students develop concept-extending inferences.
5. Preview conjecture, move toward one right answer.
6. Read statements and make sure they make sense, referring back to the diagram for affirmation.
7. Given constraints, will follow literally.
8. Accountability to get to conclusion / 1. Constructions are completed with patty paper and construction tools. Students develop and write the conjectures and other alternatives.
2. Using the given constraints, construct, with patty paper and construction tools, triangles and quadrilaterals based on their definitions.
3. Recognize and construct the points of concurrency from real-world problems.
4. Based on constructions, students create other conjectures. Students will make distinctions given minimum constraints.
5. That which they see in constraints generates curiosity in different directions and initiates additional connections
6. Skip constraints, let students produce variations.

Geometry Level Distinctions

Learning Objectives
The student will … / Level 2 / Level 3 / Level 4 / Level 5
4. Investigate the properties of triangles, analyze relationships between their sides and angles, and articulate the conditions that guarantee that two triangles are congruent.
Concepts/Reasoning:
1. Demonstrate a working knowledge that physical determination is tied to logical determination or implication: (Essential to all deductive reasoning)
  1. Distinguish parts that determine a unique triangle
  2. Identify the information/conditions that determines the congruence of two triangles
2. Recognize and demonstrate justification, organization and communication as essential in proof. (Distinguish sequential and non-sequential steps) / 1. Demonstrate a working knowledge that constructions and diagrams lead to logical determination or implication:
  1. Distinguish parts that determine a unique triangle
  2. Use the conditions that determines the congruence of two triangles
2. Recognize and demonstrate justification of congruency in triangles.
3. Patty paper constructions only / 1. Demonstrate a working knowledge that constructions and diagrams lead to logical determination or implication:
a.Distinguish parts that determine a unique triangle
b.Identify and use the conditions that determines the congruence of two triangles
2. Determine facts they can use to identify congruent triangles.
3. Recognize and demonstrate justification, organization in the form of paragraph or flowchart proofs.
4. Patty paper constructions only / 1. Demonstrate a working knowledge that physical determination is tied to logical determination or implication:
a.Distinguish parts that determine a unique triangle
b.Identify, use and model the information/conditions that determines the congruence of two triangles
2. Sketch counterexamples for true / false questions for false statements about triangle relationships.
3. Recognize and demonstrate justification, organization and communication as essential in proof. / 1. Determine what congruence conjectures are and an expectation of how to apply them.
a.Distinguish parts that determine a unique triangle
b.Identify use and model the information/conditions that determines the congruence of two triangles
2. Determine if triangle relationships are unique.
3. Sketch and explain counterexamples for false statements about triangle relationships.
4. Recognize and demonstrate and model in different forms justification, organization and communication as essential in proof.

Geometry Level Distinctions

Learning Objectives
The student will … / Level 2 / Level 3 / Level 4 / Level 5
5 Investigate, analyze and articulate the properties of quadrilaterals, and the relationships between their sides and angles.
Concepts/Reasoning
  1. Draw a concept map relating different kinds of quadrilaterals
  2. Recognize that properties of one category are inherited by all subcategories
  3. Use the symmetries of various quadrilaterals to identify properties
  4. Identify and distinguish relationships between polygons using all, some, or no (or always, sometimes, never)
/ 1. Recognize, identify, and label the quadrilaterals by properties.
2. Recognize that properties of one category are inherited by all subcategories.
3. Use constructions incidentally to determine various congruent parts of quadrilaterals
(tools limited to patty paper)
4. Identify relationships between quadrilaterals using all, some, or no.
5. Investigate properties of specific quadrilaterals, then generalize from examples. / 1Group quadrilaterals based on common properties.
2Recognize the properties of one category are inherited by all subcategories.
3Use constructions incidentally across various tools (patty paper, and compass and straight edge) to determine congruent parts of quadrilaterals
4Identify and distinguish relationships between quadrilaterals using all, some, or no.
5Informal proofs using properties of one quadrilateral to relate to another. /
  1. Investigate and develop categorical relationships between quadrilaterals.
  1. Use the properties of one category of quadrilaterals to establish the additional properties of a subcategory.
  1. Use constructions incidentally across various tools (limited use of patty paper, and predominant use of compass and straight edge) to determine congruent parts of quadrilaterals
  1. Distinguish relationships between quadrilaterals using all, some, or no (or always, sometimes, never)
/
  1. Investigate and develop categorical hierarchyrelating different kinds of quadrilaterals
  1. Use the properties of one category of quadrilaterals to establish the additional properties of a subcategory
  1. Use the symmetries of various quadrilaterals to identify properties
  1. Distinguish and specify relationships between quadrilaterals using all, some, or no (or always, sometimes, never)

Geometry Level Distinctions

Learning Objectives
The student will … / Level 2 / Level 3 / Level 4 / Level 5
6 Use geometry tools to explore, recognize and articulate relationships among angles and line segments, in and around circles.
Concepts/Reasoning:
  1. Connect basic vocabulary of circles with visual representations.
  2. Use points of tangency to recognize
a)the relationship between radius and tangent
b)the congruency of tangent segments to a circle from a common point.
  1. Discover and recognize articulate properties of central angles, inscribed angles, chords, and arcs of circles.
  2. Understand pi as the ratio between circumference and diameter and its arc length implications
/
  1. Connect basic vocabulary of circles with visual representations.
  1. Use points of tangency to recognize
a)the relationship between radius and tangent
b)the congruency of tangent segments to a circle from a common point.
  1. Discover, recognize, and articulate properties of central angles, inscribed angles, chords, and arcs of circles.
  1. Identify pi as the relationship between the circumference of a circle and its diameter
/
  1. Connect basic vocabulary of circles with visual representations.
  1. Use points of tangency to recognize and demonstrate
a)the relationship between radius and tangent
b)the congruency of tangent segments to a circle from a common point.
  1. Discover, recognize, articulate, and use properties of central angles, inscribed angles, chords, and arcs of circles.
  1. Demonstratepi as the relationship between the circumference of a circle and its diameter
/
  1. Connect basic vocabulary of circles with visual representations.
  1. Use points of tangency to develop and apply
a)the relationship between radius and tangent
b)the congruency of tangent segments to a circle from a common point.
  1. Develop, communicate and apply properties of central angles, inscribed angles, chords, and arcs of circles.
  1. Apply pi as the relationship between the circumference of a circle and its diameter and its arc length implications.
/
  1. Create visual representations of basic circle vocabulary
  1. Use points of tangency to develop and apply
a)the relationship between radius and tangent
b)the congruency of tangent segments to a circle from a common point.
  1. Develop, communicate, and applythrough proof properties of central angles, inscribed angles, chords, and arcs of circles.
  2. Apply and model pi as the relationship between the circumference and diameter and its arc length implications

Geometry Level Distinctions

Learning Objectives
The student will … / Level 2 / Level 3 / Level 4 / Level 5
7.Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic object, and, conversely, determine whether and how one object can be transformed to another by a transformation or a sequence of transformations.
Concepts:
  1. Use geometric transformations to define symmetry and isometry.
  2. Recognize that properties (including parallelism, angle measurement, distance and area) are preserved by all isometries.
  3. Describe or demonstrate how to compose transformations to make other transformations.
  4. Classify and identify monohedral, regular, and semi regular tessellations.
  1. Use the symmetries of various quadrilaterals to identify properties
/ Initially perceive transformations of polygons to recognize congruence of the shape rather than parts.
1Use geometric transformations to define reflectional symmetry and isometry.
2Recognize that distance is preserved by all isometries. Use coordinate plane to focus on location
3Initial teacher demonstration to compose transformations over parallel lines to make a translation.
4Represent transformations in the plane; describe transformations as being the movement of a point in a coordinate plane to another point that’s its image
5Use a point as a means of moving a figure Take a pentagon use one of its vertices to model rigid transformation. / Initially focus oncongruence using symmetries to reflect figures and build images.
Emphasis on spatial elements as opposed to component parts.
1Use geometric transformations to define symmetry and isometry.
2Recognize that angle measurement and distance are preserved by all isometries.Use coordinate plane to focus on distance and location
3Students demonstrate how to compose transformations over parallel lines and intersecting lines to make translations and rotations using geometry tools (Mira, patty paper)
4Notation of prime to underscore corresponding parts / Use transformations to construct congruent polygons
Demonstrate fluidity between corresponding partsof image and pre-image.
1Use constructions of geometric transformations to define symmetry and isometry.Use coordinate plane to focus on distance and location. Line of symmetry can be off the axes.
2Recognize that parallelism, angle measurement, and distance are preserved by all isometries.
3Describe a glide reflection and demonstrate how to compose transformations over parallel lines and intersecting lines to make translations and rotations using geometry tools (Mira, patty paper, compass/straight edge) / Flexibly move between polygons and their correspondent parts under various transformations.
4Use constructions of geometric transformations to define symmetry and isometry. Use coordinate plane to focus on distance and location. Line of symmetry can be off the axes.
1Recognize that parallelism, angle measurement, distance and area are preserved by all isometries.
2Describe or demonstrate how to compose transformations over parallel lines and intersecting linesto make translations and rotations or glide reflections using geometry tools (Mira, patty paper, compass/straight edge)

Geometry Level Distinctions

Learning Objectives
The student will … / Level 2 / Level 3 / Level 4 / Level 5
8. Use a variety of strategies to estimate and determine perimeter and area of plane figures and surface area of 3D figures.
Concepts:
(Inter-related rather than sequential)
  1. Derive formulas and methods for finding and relating areas.
Recognize, apply and describe using the sub concepts of area: unit iteration, additivity and invariance.
  1. Determine a generalizable way to find the area of any polygon.
  2. Determine a generalizable way to approximate the area of an irregular figure.
/ 1Derive formulas for rectangle, parallelogram, triangle and trapezoid (visual) and methods for finding and relating areas, using the sub concepts of area: unit iteration, additivity and invariance.
2Use decomposition limited to previously specified figures to find the area of any polygon or irregular figure.
3Derive area of regular polygons and circles
4Find the area of shaded regions
5Through teacher –guided instruction, determine area for parallelograms, triangles, and trapezoids / 1Derive formulas for rectangle, parallelogram, triangle, trapezoid, Kite(visual) and Regular Polygons and methods for finding and relating areas, recognize, apply and describe using the sub concepts of area: unit iteration, additivity and invariance.
2Emphasis on reasoning through the use of composition, decomposition, and transformation of parts, recognition and use of congruent parts, and the correspondence of these parts.
3Variation based on the types and examples of polygons, circles and composite figures
4Determine areas of sectors, annulus of a circle, and segments of circles / 1Derive formulas and methods for finding and relating areas of rectangle, parallelogram, triangle, trapezoid, kite,regular polygons, circles, and circle regions (visual)
2Recognize, apply and describe formulas and methods using the sub concepts of area: unit iteration, additivity and invariance.
3Emphasis on reasoning through the use of composition, decomposition, and transformation of parts, recognition and use of congruent parts, and the correspondence of these parts.
4Variation based on the types and examples of polygons, circles and composite figures
5Determine areas of sectors, annulus of a circle, and segments of circles / 1Derive formulas and methods for finding and relating areas of rectangle, parallelogram, triangle, trapezoid, kite, regular polygons, circles, and circle regions (visual). Sequence and justify methods for finding and relating areas,
2Recognize, apply and describe formulas and methods using the sub concepts of area: unit iteration, additivity and invariance.
3Emphasis on reasoning through the use of composition, decomposition, and transformation of parts, recognition and use of congruent parts, and the correspondence of these parts.
4Variation based on the types and examples of polygons, circles and composite figures.
5Base derivation on prior knowledge of geometric relationships

Geometry Level Distinctions