Geometry – 12 (hrs)
Nov 06, 2020 20201124 12:42Geometry – 12 (hrs)
Geometry – 12 (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
 Copy and Bisect an Angle Copy
 Construct a Perpendicular Bisector Copy
 Points of Concurrencies Copy
 Solve for Unknown Angles—Angles and Lines at a Point Copy
 Solve for Unknown Angles—Transversals Copy
 Solve for Unknown Angles—Angles in a Triangle Copy
 Unknown Angle Proofs—Writing Proofs Copy
 Unknown Angle Proofs—Proofs with Constructions Copy
 Rotations, Reflections, and Symmetry Copy
 Translations Copy
 Characterize Points on a Perpendicular Bisector Copy
 Construct and Apply a Sequence of Rigid Motions Copy
 Applications of Congruence in Terms of Rigid Motions Copy

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

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

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

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