Geometry – 5 (hrs)
Sep 24, 2020 20201124 13:02Geometry – 5 (hrs)
Geometry – 5 (hrs)
Study of transformations and the role transformations play in defining congruence. The study of scale drawings, specifically the way they are constructed under the ratio and parallel methods, gives us the language to examine dilations.
It also focuses on properties of area that arise from unions, intersections, and scaling. These topics prepare for understanding limit arguments for volumes of solids. Topic B begins with a lesson where students experimentally discover properties of threedimensional space that are necessary to describe threedimensional solids such as cylinders and prisms, cones and pyramids, and spheres. Crosssections of these solids are studied and are classified as similar or congruent. A dissection is used to show the volume formula for a right triangular prism after which limit arguments give the volume formula for a general right cylinder.
Students explore and experience the utility of analyzing algebra and geometry challenges through the framework of coordinates. To set the stage for complex work in analytic geometry (computing coordinates of points of intersection of lines and line segments or the coordinates of points that divide given segments in specific length ratios, and so on), students describe the region via systems of algebraic inequalities and work to constrain the robot motion along line segments within the region.
This focuses on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page. If the lines are perpendicular, and one passes through the center of the circle, then the relationship encompasses the perpendicular bisectors of chords in a circle and the association between a tangent line and a radius drawn to the point of contact. If the lines meet at a point on the circle, then the relationship involves inscribed angles. If the lines meet at the center of the circle, then the relationship involves central angles. If the lines meet at a different point inside the circle or at a point outside the circle, then the relationship includes the secant angle theorems and tangent angle theorems.

Congruence, Proof, and Constructions
 Construct an Equilateral Triangle
 Copy and Bisect an Angle
 Construct a Perpendicular Bisector
 Points of Concurrencies
 Solve for Unknown Angles—Angles and Lines at a Point
 Solve for Unknown Angles—Transversals
 Solve for Unknown Angles—Angles in a Triangle
 Unknown Angle Proofs—Writing Proofs
 Unknown Angle Proofs—Proofs with Constructions
 Rotations, Reflections, and Symmetry
 Translations
 Characterize Points on a Perpendicular Bisector
 Construct and Apply a Sequence of Rigid Motions
 Applications of Congruence in Terms of Rigid Motions

Similarity, Proof, and Trigonometry
 Scale Drawings
 Making Scale Drawings Using the Ratio Method
 Making Scale Drawings Using the Parallel Method
 Comparing the Ratio Method with the Parallel Method
 Scale Factors
 Dilations as Transformations of the Plane
 How Do Dilations Map Lines, Rays, and Circles?
 How Do Dilations Map Angles?
 Dividing the King’s Foot into 12 Equal Pieces
 Dilations from Different Centers
 Properties of Similarity Transformations
 The AngleAngle (AA) Criterion for Two Triangles to Be Similar
 BetweenFigure and WithinFigure Ratios
 The SideAngleSide (SAS) and SideSideSide (SSS) Criteria for Two Triangles to Be Similar
 Similarity and the Angle Bisector Theorem
 Families of Parallel Lines and the Circumference of the Earth

Extending to Three Dimensions
 Properties of Area
 The Scaling Principle for Area
 Proving the Area of a Disk
 ThreeDimensional Space
 General Prisms and Cylinders and Their CrossSections
 General Pyramids and Cones and Their CrossSections
 Definition and Properties of Volume
 Scaling Principle for Volumes
 Scaling Principle for Volumes
 The Volume of Prisms and Cylinders and Cavalieri’s Principle
 The Volume Formula of a Pyramid and Cone
 The Volume Formula of a Sphere

Connecting Algebra and Geometry Through Coordinates
 Finding Systems of Inequalities That Describe Triangular and Rectangular Regions
 Lines That Pass Through Regions
 Criterion for Perpendicularity
 Segments That Meet at Right Angles
 Equations for Lines Using Normal Segments
 Parallel and Perpendicular Lines
 Perimeter and Area of Triangles in the Cartesian Plane
 Perimeter and Area of Polygonal Regions in the Cartesian Plane
 Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities
 Dividing Segments Proportionately
 Analytic Proofs of Theorems Previously Proved by Synthetic Means
 The Distance from a Point to a Line

Circles With and Without Coordinates
 Thales’ Theorem
 Circles, Chords, Diameters, and Their Relationships
 Rectangles Inscribed in Circles
 Experiments with Inscribed Angles
 Inscribed Angle Theorem and Its Applications
 Unknown Angle Problems with Inscribed Angles in Circles
 The Angle Measure of an Arc
 Arcs and Chords
 Arc Length and Areas of Sectors
 Unknown Length and Area Problems
 Properties of Tangents
 Tangent Segments
 The Inscribed Angle Alternate—A Tangent Angle
 Secant Lines; Secant Lines That Meet Inside a Circle
 Secant Angle Theorem, Exterior Case
 Similar Triangles in CircleSecant (or CircleSecantTangent) Diagrams
 Equations for Circles and Their Tangents