## 180 Aims for Geometry (File)

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**Common Core Geometry**–*Suggested Aims**Unit 1: Tools of Geometry*- How do we precisely define the
**basic building blocks of geometry**? (Points, Lines, Planes, Segments, Rays, Postulates, Axioms, Collinear vs. Non-collinear, Co-planar, Parallel Lines, Perpendicular Lines, Angles, Circles) (2 days) - How do we find and compare
**length of****segments**? (Segment bisectors, Congruent Segments, Midpoints, Addition Postulate) - How do we find and compare the
**measures of angles**? (Vertex; Acute, Right, Obtuse, and Straight Angles; Angle bisectors) - How do we identify
**special angle pairs**and use their relationships to find angle measures? (Adjacent, Vertical, Complementary, Supplementary) - How do we make formal
**geometric constructions**with a variety of tools and methods? (Ruler, Compass, Straightedge, Perpendicular bisector) (2 days) - How do we find the
**midpoint**of a segment? - How do we find the
**distance between two points**? (Distance Formula with connection to Pythagorean Theorem) - How do we define and classify
**polygons**? (Review) - How do we find the
**perimeter and area of simple/basic shapes**? - How do we find the
**circumference and area of circles**? - How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 1? (Include a review session, a state test-aligned summative assessment, and/or a performance task( (3 days)

*Unit 2: Reasoning and Proof*- How do we use
**inductive reasoning**to make conjectures? (Include counterexamples) - How do we recognize
**conditional statements**and their parts? (Include hypothesis, conclusion, truth values, truth tables, conjunction, disjunction,**converses, inverses, contrapositives and conditionals,**equivalent statements) (3 days) - How do we write
**biconditionals**and recognize good definitions? - How do use
**deductive**/logical**reasoning**to arrive at a given conclusion? (Include the Law of Detachment and the Law of Syllogism) - How do we connect reasoning in Algebra and Geometry? An introduction to
**simple proofs**. (Include the addition, subtraction, multiplication, division, reflexive, symmetric, transitive, substitution, and distributive**properties**) (2 days) - How do we
**prove and apply theorems about angles**? - How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 2? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

*Unit 3: Parallel and Perpendicular Lines*- How do we identify relationships between figures in space? (Include
**parallel lines**that are both coplanar and non-coplanar,**skew lines**, and**parallel planes**) - How do we identify angles formed by a transversal? (Include
**alternate interior, alternate exterior, corresponding, and same-side interior angles**) - How do we use the properties of
**parallel lines cut by a transversal**to find angle measures? - How can we determine if two lines are parallel? (Converses of theorems learned in previous lesson)
- How do we
**apply geometric methods to solve design problems**by relating parallel and perpendicular lines? - How do we find the
**measures of the angles of a triangle**? - How do we
**use parallel lines to prove theorems about triangles**? - How do we construct parallel and perpendicular lines using a compass?
- How do we
**graph linear functions**and**write**their corresponding**linear equations**? (Review. Include definition and formula for**slopes, slope-intercept form,**and**the point-slope form of lines**) (2 days) - How do we relate the
**slopes of parallel lines**to solve geometric problems? - How do we relate the
**slopes of perpendicular lines**to solve geometric problems? - How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 3? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

*Unit 4: Congruent Triangles*- How do we recognize
**congruent figures and their corresponding parts**? - How do we prove that two triangles are congruent by using congruence criteria for triangles and the
**Side-Side-Side (SSS) postulate**? - How do we prove that two triangles are congruent by using congruence criteria for triangles and the
**Side-Angle-Side (SAS) postulate**? - How do we prove that two triangles are congruent by using congruence criteria for triangles and the
**Angle-Side-Angle (ASA) Postulate**? - How do we use triangle congruence and triangle parts of corresponding triangles to
**prove that parts of two triangles are congruent**? - How do we use and apply the properties of
**isosceles triangles**? - How do we use and apply the properties of
**equilateral triangles**? - How do we apply
**congruence in right triangles**by using the Hypotenuse-Leg Theorem? (Relate to SSS Case) - How do we identify
**congruent overlapping triangles**? - How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 4? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

*Unit 5: Relationships Within Triangles*- How do we use the properties of
**midsegments**to solve problems? - How do we use the properties of
**perpendicular bisectors**to solve problems? - How do we use the properties of
**angle bisectors**to solve problems? - How do we construct a circumscribed circle using the
**circumcenter of a triangle**, i.e., the point of concurrency of the perpendicular bisectors of a triangle? - How do we construct an inscribed circle using the
**incenter of a triangle**, i.e., the point of concurrency of the angle bisectors of a triangle? - How do we identify the properties of medians in a triangle, as well as the
**centroid of the triangle**? - How do we identify the properties of altitudes in a triangle, as well as the
**orthocenter of the triangle**? - How do we use indirect reasoning and negations to write
**indirect proofs**? - How do we use and prove
**inequalities involving angles and sides of triangles**? - How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 5? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

*Unit 6: Polygons and Quadrilaterals*- How do we find
**the sum of the interior and the exterior angles of a given polygon**? - How do we use the relationships among the sides, angles, and diagonals of
**parallelograms**? (Include proofs) - How do we prove that a quadrilateral is a parallelogram?
- How do we use the relationships among the sides, angles, and diagonals of
**rhombi**? (Include proofs) - How do we prove that a quadrilateral/parallelogram is a rhombus?
- How do we use the relationships among the sides, angles, and diagonals of
**rectangles**? (Include proofs) - How do we prove that a quadrilateral/parallelogram is a rectangle?
- How do we use the relationships among the sides, angles, and diagonals of
**squares**? (Include proofs) - How do we prove that a quadrilateral/parallelogram is a square?
- How do we use the relationships among the sides and angles of
**trapezoids**? (Include proofs) - How do we prove that a quadrilateral/parallelogram is a trapezoid?
- How do we use the relationships among the sides, angles, and diagonals of
**kites**? (Include proofs) - How do we prove that a quadrilateral/parallelogram is a kite?
- How do we classify polygons in the
**coordinate plane**? (2 days) - How do prove theorems using figures in the coordinate plane? (2 days)
- How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 6? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

*Unit 7: Similarity**Review*of ratios, proportions, and solving linear as well as quadratic equations strongly encouraged at the beginning of the unit if need/skill gap is found (2 days)

- How do we identify and apply the properties of
**similar polygons**? (Review of ratios, proportions, and solving linear as well as quadratic strongly encouraged if need/skill gap is found) (2 days) - How do we use the
**Angle-Angle (AA) Postulate**to solve problems involving similar figures and to prove relationships in geometric figures? - How do we use the
**Angle-Angle (AA) Similarity Postulate**to solve problems and to prove relationships in geometric figures? - How do we use the
**Side-Angle-Side (SAS) Similarity Postulate**to solve problems and to prove relationships in geometric figures? - How do we use the
**Side-Side-Side (SSS) Similarity Postulate**to solve problems and to prove relationships in geometric figures? - How do we verify if two figures are similar?
- How do we prove that triangles are similar?
- How do we find and use relationships in
**similar right triangles**? - How do we use the
**Side-Splitter Theorem**and its corollary? - How do we use the
**Triangle-Angle-Bisector Theorem**? - How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 7? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

*Unit 8: Right Triangles and Trigonometry**Review*of simplifying radical expressions strongly encouraged at the beginning of the unit if need/skill gap is found (2 days)

- How do we use the
**Pythagorean Theorem**and its converse to solve and prove right triangles? - How do we apply the Pythagorean Theorem to solve real-life situations?
- How do we use the properties of
**the 45-45-90 special right triangle**to solve problems? - How do we use the properties of
**the 30-60-90 special right triangle**to solve problems? - How do we use the sine, cosine, and tangent
**trigonometric ratios**to determine side lengths and angle measures in right triangles? - How do we apply trig ratios to solve real-world situations?
- How do we use
**angles of elevation and depression**to solve problems involving trig ratios? (Include real-world applications) *Extension:*How do we apply the**Law of Sines**to find unknown measurements – sides and angles – in right as well as non-right triangles? ((Include real-world applications)*Extension:*How do we apply the**Law of Sines**to find unknown measurements – sides and angles – in right as well as non-right triangles? (Include real-world applications)*Extension:*How do we apply the**Law of Cosines**to find unknown measurements – sides and angles – in right as well as non-right triangles? (Include real-world applications)- How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 8? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

*Unit 9: Transformations*- How do we define
**rigid motions**and how do we identify what properties they preserve? - How do we apply
**translations**to images and draw the resulting figure? - How do we apply
**reflections**to images and draw the resulting figure? - How do we apply
**rotations**to images and draw the resulting figure? - How do we apply
**dilations**to images and draw the resulting figure? *Extension*: How do we identify the**types of symmetry**that result when figures are reflected or rotated? (Line Symmetry, Point Symmetry, Rotational Symmetry)- How do we perform
**composition of transformations**? - How do we define and classify
**isometries**? - How do we prove triangle congruence using isometries?
- How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 9? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

*Unit 10: Area*- How do we find the
**area of a triangles, rectangles, and parallelograms**? - How do we find the
**area of trapezoids, rhombi, and kites**? - How do we use the apothem to find the
**area of regular polygons**? - How do we find the
**area of regular polygons using trigonometry**? - How do we find the
**area of irregular shapes**? - How do we find the
**perimeter and area of similar figures**using ratios/scale factors? - How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 10? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

*Unit 11: Surface Area and Volume*- How do find the
**surface area and volume of a sphere**? - How do find the
**surface area and volume of a cylinder**? - How do find the
**surface area and volume of a cone**? - How do find the
**surface area and volume of a rectangular prism**? - How do find the
**surface area and volume of a triangular or different type of prism?** - How do find the
**surface area and volume of a pyramid**? *Extension*: How do we find the**lateral area**of the shapes learned in this unit?- How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 11? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

*Unit 12: Circles*- How do we find the measures of
?*central angles and arcs* - How do we find the area of
**sectors and segments of circles**? *Extension*: How do we compare the circumference and area of circles with the perimeter and area of regular**polygons inscribed in and circumscribed about the circle**?- How do we use the properties of a
**tangent to a circle**? - How do we identify and describe
**relationships among inscribed angles, radii, and chords**? - How do we find the
**measure of inscribed angles, intercepted arcs, and angles formed by a tangent and a chord**? - How do we find the
**measures of angles formed by chords, secants, and tangents**? - How do we find the
**lengths of segments associated with circles**? - How do we write the
**equation of a circle in the coordinate plane**? (Include finding the radius and the center from the equation and vice versa) (2 days) - How do we draw and describe a
**locus**? - How do we
**prepare for the Regents exam by reviewing and applying the main concepts**learned in Unit 12? (Include a review session, a state test-aligned summative assessment, and/or a performance task) (3 days)

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*Additional Note:*The remaining 20 days can be used for Regents prep, review, performance tasks, applications to real-world situation, academic interventions for Algebra skills, and/or extensions of the material.}