Download Geometry Curriculum

Geometry Curriculum
The Standards for Mathematical Practice (SMP)
The SMP should be taught in connection with the math content.
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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
Geometry Quarter 1
Unit 1: Tools of Geometry
G-CO. 1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Points, Lines, and Planes
• Understand basic terms of geometry
• Understand basic postulates of geometry
Segments, Rays, Parallel Lines, and Planes
• Identify segments and rays
• Recognize parallel lines
Measuring Segments and Angles
• Find lengths of segments and of angles
G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a
segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines,
including the perpendicular bisector of a line segment; and constructing a line parallel to a given line
through a point not on the line.
Basic Constructions
o Construct congruent segments and congruent angles
o Construct a straightedge to bisect segments and angles
G-CO. 1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment
in a given ratio.
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The Coordinate Plane
• Find the distance between two points in the coordinate plane using distance formula and
Pythagoreans Theorem
• Simplify radicals and perform radical operations (add, subtract, multiply, divide)
• Find the coordinates of the midpoint of a segment in the coordinate plane
G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,
using the distance formula.
G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit
arguments.
Perimeter, Circumference, and Area
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Incorporate application problems and composite figures
Find perimeters of rectangles and squares
Find circumference of circles
Find areas of rectangles, squares, and circles
Introduction to Reasoning and Proof
Conditional Statements
• Recognize conditional statements
• Write converses of conditional statements
Reasoning in Algebra
• Focus on the justification of each step
• Connect reasoning in algebra and geometry
G-CO.9 Prove theorems about lines and angles. 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.
Proving Angles Congruent
• Identify angle pairs
• To prove and apply theorems about angles
Unit 2: Parallel and Perpendicular Lines
G-CO.9 Prove theorems about lines and angles. 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.
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Properties of Parallel Lines
• Identify angles formed by 2 lines and a transversal
• Prove and use properties of parallel lines
Proving Lines Parallel
• Use a transversal in proving lines parallel
• Relate parallel and perpendicular lines
G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric
problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a
given point).
Lines in the Coordinate Plane
• Graph lines given their equations
• Write equations of lines
Slopes of Parallel and Perpendicular Lines
• Relate slope and parallel lines
• Relate slope and perpendicular lines
G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a
segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines,
including the perpendicular bisector of a line segment; and constructing a line parallel to a given line
through a point not on the line.
G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric
problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a
given point).
Constructing Parallel and Perpendicular Lines
• Construct parallel and perpendicular lines
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.
Parallel Lines and the Triangle Angle-Sum Theorem
• Classify triangles and find the measures of their angles
• Use exterior angles of triangles
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-CO11 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.
The Polygon Angle-Sum Theorems
• Classify polygons
• Find the sums of the measures of the interior and exterior angles of polygons
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Geometry Quarter 2
Unit 3: Congruent Triangles
G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a
given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid
motions to decide if they are congruent.
G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are
congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Congruent Figures
• Recognize congruent figures and their corresponding parts
G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a
given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid
motions to decide if they are congruent.
G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are
congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition
of congruence in terms of rigid motions
G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a
segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines,
including the perpendicular bisector of a line segment; and constructing a line parallel to a given line
through a point not on the line.
Triangle Congruence by SSS, SAS, ASA, and AAS
• Prove two triangles congruent using SSS and SAS Postulates
• Prove two triangles congruent using the ASA Postulate and the AAS Theorem
Using Congruent Triangles: CPCTC
• Use triangle congruence and CPCTC to prove that parts of 2 triangles are congruent
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.
Isosceles and Equilateral Triangles
• Use and apply properties of isosceles triangles
G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a
given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid
motions to decide if they are congruent.
G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are
congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Congruence in Right Triangles
• Prove triangles congruent using the HL Theorem
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Using Corresponding Parts of Congruent Triangles
• Identify congruent overlapping triangles
• Prove 2 triangles congruent by first proving two other triangles congruent
Unit 4: Similarity
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.
Similar Polygons
• Do not do golden rectangle/golden ratio
• Identify similar polygons
• Apply similar polygons
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.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships
in geometric figures.
Proving Triangles Similar
• Show students Coordinate Proof of Similar Triangles (do not assess)
• Use and apply AA, SAS, and SSS similarity statements
G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships
in geometric figures.
G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
Similarity in Right Triangles
• Review Pythagorean Theorem
• Find and use relationships in similar right triangles
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-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships
in geometric figures.
Proportions in Triangles
• Use the Side-Splitter and Triangle-Angle-Bisector Theorem
Perimeters and Areas of Similar Figures
• Find the perimeters and areas of similar figures
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Unit 5: Relationships within Triangles
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.
Midsegments in Triangles
• To use properties of midsegments to solve problems
G-CO.9 Prove theorems about lines and angles. 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.
Bisectors in Triangles
• Use properties of perpendicular bisectors and angle bisectors
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-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a
segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines,
including the perpendicular bisector of a line segment; and constructing a line parallel to a given line
through a point not on the line.
G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
Concurrent Lines, Medians, and Altitudes
* Spend at least a few days on this section
• Constructions of the points of concurrency
• Identify properties of perpendicular bisectors and angle bisectors
• Identify properties of medians and altitudes of triangles
Inverses, Contrapositives, and Indirect Reasoning
• Only cover Indirect Reasoning
G-CO.10 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.
Inequalities in Triangles
• Make sure to cover Hinge Theorem (assess)
• Use inequalities involving angles of triangles
• Use inequalities involving sides of triangles
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Geometry Quarter 3
Unit 6: Right Triangle Trigonometry
G-SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the
triangle, leading to definitions of trigonometric ratios for acute angles.
G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
Sine, Cosine, and Tangent Ratios
• Show students Secant, Cosecant, and Cotangent (assess)
• Show Trig Identities as well (sin² + cos² = 1) (do not assess)
• Use sine, cosine, and tangent ratios to determine side lengths in triangles
G-SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the
triangle, leading to definitions of trigonometric ratios for acute angles.
G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
Angles of Elevation and Depression
• Focus on application problems of sine, cosine, and tangent
G-SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems.
G-SRT.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown
measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Law of Sines and Cosines:
• Use Law of Sines and Cosines to solve right triangles
G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
G-SRT.9 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from
a vertex perpendicular to the opposite side.
Trigonometry and Area
• Show students how to find the area of polygons using triangles. Can use the idea of
the apothem but do not use the formula.
• Find the area of a triangle and a regular polygon using trigonometry
Vectors
• Describe vectors and use vectors to solve problems
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Unit 7: Quadrilaterals
Classifying Quadrilaterals
• Define and classify special types of quadrilaterals
G-CO.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.
Properties of Parallelograms
• Use relationships among sides and angles of parallelograms
• Use relationships involving diagonals of parallelograms or transversals
Proving That a Quadrilateral is a Parallelogram
• Incorporate problems using system of equations
• Determine whether a quadrilateral is a parallelogram
Special Parallelograms
• Use properties of diagonals of rhombuses and rectangles
• Determine whether a parallelogram is a rhombus or a rectangle
Trapezoids and Kites
• Verify and use properties of trapezoids and kites
G-GPE.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
Placing Figures in the Coordinate Plane
• Name coordinates of special figures by using their properties
Proofs Using Coordinate Geometry
• Show Coordinate Geometry of Similar Polygons
• Prove theorems using figures in the coordinate plane
Unit 8: Area
Areas of Parallelograms and Triangles
• Find the area of a parallelogram
• Find the area of a triangle
The Pythagorean Theorem and its Converse
• Use the Pythagorean Theorem and its Converse
• Focus on problems with simplifying radicals
Special Right Triangles
• Use properties of 45°-45°-90° and 30°-60°-90° triangles
Areas of Trapezoids, Rhombuses, and Kites
• Find the area of a trapezoid, rhombus, and kite
Areas of Regular Polygons
• Find the area of a regular polygon
G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
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G-C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the
relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right
angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
G-C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to
the radius, and define the radian measure of the angle as the constant of proportionality; derive the
formula for the area of a sector.
G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit
arguments.
Circles and Arcs
• Find the measures of central angles and arcs
• Find the circumference and arc length
G-C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to
the radius, and define the radian measure of the angle as the constant of proportionality; derive the
formula for the area of a sector.
Areas of Circles and Sectors
• Find the areas of circles, sectors, and segments of circles
Geometry Quarter 4
Unit 9: Surface Area
Space Figures and Nets
• Recognize nets of space figures
G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and
identify three-dimensional objects generated by rotations of two-dimensional objects.
Surface Areas of Prisms and Cylinders
• Find the surface area of a prism
• Find the surface area of a cylinder
Surface Areas of Pyramids and Cones
• Symmetry of Geometric Solids needs to be covered
• Find the surface area of a pyramid and a cone
G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit
arguments.
G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
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Volumes of Prisms and Cylinders
• Find the volume of a prism and cylinder
Volumes of Pyramids and Cones
• Find the volume of a pyramid and cone
G-GMD.2 Explain volume formulas and use them to solve problems.
G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Surface Areas and Volumes of Spheres
• Find the volume of a sphere
• Find the surface area of a sphere
G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Surface Area and Volumes of Composite
• Find the surface area of different composite solids (application problems)
Areas and Volumes of Similar Solids
• Find the relationships between the ratios of the areas and volumes of similar solids
Unit 11: Circles
G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a
segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines,
including the perpendicular bisector of a line segment; and constructing a line parallel to a given line
through a point not on the line.
G-C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the
relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right
angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
G-C.4 Construct a tangent line from a point outside a given circle to the circle.
Tangent Lines
• Include constructions of tangent lines
• Use the relationship between a radius and a tangent
• Use the relationship between 2 tangents from 1 point
G-C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the
relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right
angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Chords and Arcs
• Use congruent chords, arcs, and central angles
• Recognize properties of lines through the center of a circle
G-C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the
relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right
angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
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G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
Inscribed Angles
• Find the measure of an inscribed angle
• Find the measure of an angle formed by a tangent and a chord
G-C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to
the radius, and define the radian measure of the angle as the constant of proportionality; derive the
formula for the area of a sector.
Angle Measures and Segment Lengths
• Find the measures of angles formed by chords, secants, and tangents
• Find the length of segments associated with circles
Circles in the Coordinate Plane
• Make sure to define chords and tangents of spheres
• Write an equation of a circle
• Find the center and radius of a circle
Unit 12: Transformations
G-CO2 Represent transformations in the plane using, e.g., transparencies and geometry software;
describe transformations as functions that take points in the plane as inputs and give other points as
outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation
versus horizontal stretch).
G-CO3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
G-CO4 Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
G-CO5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure
using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that
will carry a given figure onto another.
G-CO12 Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a
segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines,
including the perpendicular bisector of a line segment; and constructing a line parallel to a given line
through a point not on the line.
Reflections
• Identify isometries
• Find reflection images of figures
Translations
• Describe translations
• Find translation images
Rotations
• Draw and identify rotation images of figures
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G-SRT1 Verify 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.)
Dilations (12.7):
• Locate dilation images of figures
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