Geometry Curriculum Map

Tools of Geometry (Points, Lines, and Planes)Unit(Chapter 1) / 1st Nine Weeks
Essential Questions / Topics

What are the building blocks of geometry?
How can you describe the attributes of a segment or angle?

Points, Lines, and Planes
Measuring Segments
Measuring Angles
Exploring Angle Pairs
Midpoint and Distance in the Coordinate Plane

Instruction and Strategies / Assessments

Sorting and Matching activity
Always, Sometimes, Never activity

Points, Lines, Planes Quiz
Angles Quiz
Points, Lines, Plane Test

How can you make a conjecture and prove that it is true?

Patterns and Inductive Reasoning
Conditional Statements
Biconditionals and Definitions
Deductive Reasoning
Reasoning in Algebra and Geometry
Proving Angles Congruent

Instruction and Strategies / Assessments

Pass-around proof activity
Proof sorting and matching activity

Intro to Proofs Quiz
Proofs Quiz
Proofs Test

Prepares for Standards:
#9 (G-CO9)
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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
#10 (G-CO10)
Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180º, 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, and the medians of a triangle meet at a point.
#11 (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.
Parallel Lines Unit - (Chapter 3) / 1st Nine Weeks
Essential Questions / Topics

How do you prove that two lines are parallel or perpendicular?
What is the sum of the measures of the angles of a triangle?
How do you write an equation of a line in the coordinate plane?

Lines and Angles
Properties of Parallel Lines
Proving Lines Parallel
Parallel and Perpendicular Lines
Parallel Lines and Triangles
Constructing Parallel and Perpendicular Lines
Equations of Lines in the Coordinate Plane
Slopes of Parallel and Perpendicular Lines

Instruction and Strategies / Assessments

Sorting and Matching activity
Always, Sometimes, Never activity

Properties of Parallel Lines Quiz
Proofs and Angles of a Triangle Quiz
Parallel and Perpendicular Lines Test

Standards
#1 (G-CO1)
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.
#9 (G-CO9)
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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
#12 (G-CO12)
Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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.
#31 (G-GPE5)
Prove the slope criteria for parallel and perpendicular lines, and use 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).
Transformations Unit - (Chapter 9) / 2nd Nine Weeks
Essential Questions / Topics

How can you change a figure’s position without changing its size and shape?
How can you change a figure’s size without changing its shape?
How can you represent a transformation in the coordinate plane?
How do you recognize congruence and similarity in figures?

Translations
Reflections
Rotations
Compositions of Isometries
Congruent Transformations
Dilations
Similarity Transformations

Instruction and Strategies / Assessments

Human Transformations Activity
Patty Paper
Transformations Story Project

Transformations Quiz
Transformations Test

Standards
#2 (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).
#3 (G-CO3)
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
#4 (G-CO4)
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
#5 (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.
#6 (G-CO6)
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.
Congruent Triangles Unit – (Chapter 4) / 2nd Nine Weeks
Essential Questions / Topics

How do you identify corresponding parts of congruent triangles?
How do you show that two triangles are congruent?
How can you tell whether a triangle is isosceles or equilateral?

Congruent Figures
Triangle Congruence by SSS and SAS
Triangle Congruence by ASA and AAS
Using Corresponding Parts of Congruent Triangles
Isosceles and Equilateral Triangles
Congruence in Right Triangles
Congruence in Overlapping Triangles

Instruction and Strategies / Assessments

Geometer’s Sketchpad
Patty Paper

Triangle Congruence Quiz
Triangle Congruence Test

Standards
#7 (G-CO7)
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.
#8 (G-CO8)
Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions.
#18 (G-SRT5)
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Relationships Within Triangles Unit – (Chapter 5) / 2nd Nine Weeks
Essential Questions / Topics

How do you use coordinate geometry to find relationships within triangles?
How do you solve problems that involve measurements of triangles?
How do you write indirect proofs?

Midsegments of Triangles
Perpendicular and Angle Bisectors
Bisectors in Triangles
Medians and Altitudes
Indirect Proof
Inequalities in One Triangle
Inequalities in Two Triangles

Instruction and Strategies / Assessments

Paper folding

Quiz 1
Quiz 2
Test

Standards
#10 (G-CO10)
Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180º, 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, and the medians of a triangle meet at a point.
#17 (G-SRT4)
Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.
#26 (G-C3)
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Constructions Unit – (Chapter 5) / 2nd Nine Weeks
Essential Questions / Topics (Constructions):

What properties do you use to construct the tools of Geometry?
What is the most efficient way to construct the tools of Geometry?

Copying a segment
Copying an angle
Bisecting a segment
Bisecting an angle
Perpendicular lines
Perpendicular bisector of a segment
Parallel line through a given point not on the line
Equilateral triangle
Square
Regular hexagon inscribed in a circle

Instruction and Strategies / Assessments

Use of compasses and straightedges
Instructional videos
Graphic organizers
Construction Card Project

Intro to Constructions Quiz
Constructions Quiz
Constructions Card Project

Standards
#12 (G-CO12)
#13 (G-CO13)
Polygons and Quadrilaterals Unit – (Chapter 6) / 3rd Nine Weeks
Essential Questions / Topics

How can you find the sum of the measures of polygon angles?
How can you classify quadrilaterals?
How can you use coordinate geometry to prove general relationships?

The Polygon Angle-Sum Theorems
Properties of Parallelograms
Proving that a Quadrilateral is a Parallelogram
Properties of Rhombuses, Rectangles, and Squares
Conditions for Rhombuses, Rectangles, and Squares
Trapezoids and Kites
Polygons in the Coordinate Plane
Applying Coordinate Geometry
Proofs Using Coordinate Geometry

Instruction and Strategies / Assessments

Geometer’s Sketchpad
Graphic Organizer
Properties Chart
Venn Diagram

Properties of Parallelograms Quiz
Special Parallelograms Quiz
Polygons and Quadrilaterals Test

Standards
#11 (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.
Similarity Unit – (Chapter 7) / 3rd Nine Weeks
Essential Questions / Topics

How do you use proportions to find side lengths in similar polygons?
How do you show two triangles are similar?
How do you identify corresponding parts of similar triangles?

Ratios and Proportions
Similar Polygons
Proving Triangles Similar
Similarity in Right Triangles
Proportions in Triangles

Instruction and Strategies / Assessments

Graphic Organizer
Scale Drawings
Patty Paper

Dilations and Similarity Quiz
Properties of Proportions and Similar Polygons Quiz
Ratio, Proportions, and Similar Polygons Quiz
Similarity Test

Standards
#14 (G-SRT1)
Verify experimentally the properties of dilations given by a center and a scale factor.

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.
The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

#15 (G-SRT2)
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.
#16 (G-SRT3)
Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar.
#18 (G-SRT5)
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Right Triangles and Trigonometry Unit – (Chapter 8) / 3rd Nine Weeks
Essential Questions / Topics

How do you find a side length of angle measure in a right triangle?
How do trigonometric ratios relate to similar right triangles?

The Pythagorean Theorem and Its Converse
Special Right Triangles
Trigonometry
Angles of Elevation and Depression
Law of Sines
Law of Cosines

Instruction and Strategies / Assessments

Graphic Organizer
Clinometers
Calculating Height of Real Objects
Trig Project

Trig Project
Special Right Triangles Quiz
Trigonometric Ratios Quiz
Right Triangles and Trig Test

Standards
#17 (G-SRT4)
Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.
#19 (G-SRT6)
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.
#20 (G-SRT7)
Explain and use the relationship between the sine and cosine of complementary angles.
#21 (G-SRT8)
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
#22 (G-SRT10)
(+) Prove the Law of Sines and the Law of Cosines and use them to solve problems.
#23 (G-SRT11)
(+) 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).
Area Unit – (Chapter 10) / 4th Nine weeks
Essential Questions / Topics

How do you find the area of a polygon or find the circumference and area of a circle?
How do perimeters and areas of similar polygons compare?

Areas of Parallelograms and Triangles
Areas of Trapezoids, Rhombuses, and Kites
Areas of Regular Polygons
Perimeters and Areas of Similar Figures
Trigonometry and Area
Circles and Arcs
Areas of Circles and Sectors
Geometric Probability

Instruction and Strategies / Assessments

Paper cutting discovery
Graphic organizer

Intro to Area Quiz
Area of Regular Polygons Quiz
Area Test

Standards
#33 (G-GPE7)
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*
#34 (G-GPE(AL)
Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics.
#42 9S-MD6)
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
#43 (S-MD7)
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
Surface Area and Volume Unit – (Chapter 11) / 4th Nine Weeks
Essential Questions / Topics

How can you determine the intersection of a solid and a plane?
How do you find the surface area and volume of a solid?
How do the surface areas and volumes of similar solids compare?

Space Figures and Cross Sections
Surface Areas of Prisms and Cylinders
Surface Areas of Pyramids and Cones
Volumes of Prisms and Cylinders
Volumes of Pyramids and Cones
Surface Areas and Volumes of Spheres
Areas and Volumes of Similar Solids

Instruction and Strategies / Assessments

Orange peeling Activity
3D hands-on objects

Surface Area Quiz
Volume Quiz
Surface Area and Volume Test

Standards
#35 (G-GMD1)
Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
#36 (G-GMD3)
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
#37 (G-GMD(AL)
Determine the relationship between surface areas of similar figures and volumes of similar figures.
#38 (G-GMD4
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
#39 (G-MG1)
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*
#40 (G-MG2)
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).*
#41 (G-MG3)
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios).*
Circles Unit – (Chapter 12) / 4th Nine Weeks
Essential Questions / Topics

How can you prove relationships between angles and arcs in a circle?
When lines intersect a circle, or within a circle, how do you find the measures of resulting angles, arcs, and segments?
How do you find the equation of a circle in the coordinate plane?

Tangent Lines
Chords and Arcs
Inscribed Angles
Angle Measures and Segment Lengths
Circles in the Coordinate Plane
Locus: A Set of Points

Instruction and Strategies / Assessments

Group teaching
Compasses
Graphic organizer

Circles Quiz
Circles Test

Standards
#1 (G-CO1)
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.
#13 (G-CO13)
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
#24 (G-C1)
Prove that all circles are similar.
#25 (G-C2)
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.
#26 (G-C3)
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
#27 (G-C4)
(+) Construct a tangent line from a point outside a given circle to the circle.
#28 (G-C5)
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.
#29 (G-GPE1)
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
#35 (G-GMD1)
Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.