Geometry

The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school CCSS. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

High School Pacing Guide SY 2022-2023

Module 1: Congruence, Proof, and Constructions

(34 Hours)

Overview

Module 1 embodies critical changes in Geometry as outlined by the Common Core. The heart of the module is the study of transformations and the role transformations play in defining congruence. The topic of transformations is introduced in a primarily experiential manner in Grade 8 and is formalized in Grade 10 with the use of precise language. The need for clear use of language is emphasized through vocabulary, the process of writing steps to perform constructions, and ultimately as part of the proof-writing process.

Standards

G-CO.A.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-CO.A.2 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-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO.A.5 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-CO.C.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.

G-CO.C.10 Prove2 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.C.11 Prove2 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.

Unit Vocab

Isometry (An isometry of the plane is a transformation of the plane that is distance-preserving.)

Module 1: Congruence, Proof, Constructions - All materials here

Topic A: Basic Constructions

Standards

G-CO.A.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-CO.D.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-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle

Overview

Students begin this module with Topic A, Constructions. Major constructions include an equilateral triangle, an angle bisector, and a perpendicular bisector. Students synthesize their knowledge of geometric terms with the use of new tools and simultaneously practice precise use of language and efficient communication when they write the steps that accompany each construction (G.CO.1).

Lessons

Topic B: Unknown Angles

Standards

G-CO.C.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.

Overview

Constructions segue into Topic B, Unknown Angles, which consists of unknown angle problems and proofs. These exercises consolidate students’ prior body of geometric facts and prime students’ reasoning abilities as they begin to justify each step for a solution to a problem. Students began the proof writing process in Grade 8 when they developed informal arguments to establish select geometric facts (8.G.5).

Lessons

Topic C: Transformations/ Rigid Motions

Standards

G-CO.A.2 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-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.A.5 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-CO.B.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.B.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.

Overview

Topic C, Transformations, builds on students’ intuitive understanding developed in Grade 8. With the help of manipulatives, students observed how reflections, translations, and rotations behave individually and in sequence (8.G.1, 8.G.2). In Grade 10, this experience is formalized by clear definitions (G.CO.4) and more in-depth exploration (G.CO.3, G.CO.5).

The concrete establishment of rigid motions also allows proofs of facts formerly accepted to be true (G.CO.9). Similarly, students’ Grade 8 concept of congruence transitions from a hands-on understanding (8.G.2) to a precise, formally notated understanding of congruence (G.CO.6). With a solid understanding of how transformations form the basis of congruence, students next examine triangle congruence criteria. Part of this examination includes the use of rigid motions to prove how triangle congruence criteria such as SAS actually work (G.CO.7, G.CO.8).

Lessons

Topic D: Congruence

Standards

G-CO.B.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.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Overview

In Topic D, Proving Properties of Geometric Figures, students use what they have learned in Topics A through C to prove properties—those that have been accepted as true and those that are new—of parallelograms and triangles (G.CO.10, G.CO.11).

Lessons

Topic E: Proving Properties of Geometric Figures

Standards

G-CO.C.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.

G-CO.C.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.C.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.

Overview

In Topic E, Proving Properties of Geometric Figures, students use what they have learned in Topics A through D to prove properties—those that have been accepted as true and those that are new—of parallelograms and triangles (G.CO.C.10, G.CO.C.11).

Lessons

Topic F: Advanced Constructions

Standards

G‐CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Overview

Topic F is a review that highlights how geometric assumptions underpin the facts established thereafter. Students are presented with the challenging but interesting construction of a nine-point circle.

Lessons

Topic G: Axiomatic Systems

Standards

G-CO.A.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-CO.A.2 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-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Overview

In Topic G, students review material covered throughout the module. Additionally, students discuss the structure of geometry as an axiomatic system.

Lessons

Module 2: Similarity, Proof, and Trigonometry

(34 Hours)

Overview

G-SRT.A.1 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.

G-SRT.A.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.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

G-SRT.B.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.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures

G-SRT.C.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.C.7 Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Standards

Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2. To be able to discuss similarity, students must first have a clear understanding of how dilations behave. This is done in two parts, by studying how dilations yield scale drawings and reasoning why the properties of dilations must be true. Once dilations are clearly established, similarity transformations are defined and length and angle relationships are examined, yielding triangle similarity criteria. An in-depth look at similarity within right triangles follows, and finally the module ends with a study of right triangle trigonometry.

Unit Vocab

Cosine (Let θ be the angle measure of an acute angle of the right triangle. The cosine of θ of a right triangle is the value of the ratio of the length of the adjacent side (denoted adj) to the length of the hypotenuse (denoted hyp). As a formula, cos⁡θ=adj/hyp.)

Dilation (For r>0, a dilation with center C and scale factor r is a transformation D_(C,r) of the plane defined as follows:

For the center C, D_(C,r) (C)=C, and
For any other point P, D_(C,r) (P) is the point Q on the ray (CP) ⃗ so

that CQ=r∙CP.)

Sides of a Right Triangle (The hypotenuse of a right triangle is the side opposite the right angle; the other two sides of the right triangle are called the legs. Let θ be the angle measure of an acute angle of the right triangle. The opposite side is the leg opposite that angle. The adjacent side is the leg that is contained in one of the two rays of that angle (the hypotenuse is contained in the other ray of the angle).)

Similar (Two figures in a plane are similar if there exists a similarity transformation taking one figure onto the other figure. A congruence is a similarity with scale factor 1. It can be shown that a similarity with scale factor 1 is a congruence.)

Similarity Transformation (A similarity transformation (or similarity) is a composition of a finite number of dilations or basic rigid motions. The scale factor of a similarity transformation is the product of the scale factors of the dilations in the composition; if there are no dilations in the composition, the scale factor is defined to be 1. A similarity is an example of a transformation.)

Sine (Let θ be the angle measure of an acute angle of the right triangle. The sine of θ of a right triangle is the value of the ratio of the length of the opposite side (denoted opp) to the length of the hypotenuse (denoted hyp). As a formula, sin⁡θ=opp/hyp.)

Tangent (Let θ be the angle measure of an acute angle of the right triangle. The tangent of θ of a right triangle is the value of the ratio of the length of the opposite side (denoted opp) to the length of the adjacent side (denoted adj). As a formula, tan⁡θ=opp/adj.)

Note that in Algebra II, sine, cosine, and tangent are thought of as functions whose domains are subsets of the real numbers; they are not considered as values of ratios. Thus, in Algebra II, the values of these functions for a given θ are notated as sin⁡〖(θ)〗, cos⁡(θ), and tan⁡(θ) using function notation (i.e., parentheses are included).

Module 2: Similarity, Proof, and Trig - All Materials Here

Topic A: Geometry: Scale Drawings

Standards

G-SRT.A.1 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.

G-SRT.B.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-MG.A.3 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).

Overview

Students revisit scale drawings, discovering two methods of how to create them using dilations. Comparing the two methods yield Triangle Side Splitter Theorem/Dilation Theorem.

Lessons

Topic B: Dilations

Standards

G-SRT.A.1 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.

G-SRT.B.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

Overview

Topic B establishes a firm understanding of how dilations behave. Students prove that a dilation maps a line to itself or to a parallel line and, furthermore, dilations map segments to segments, lines to lines, rays to rays, circles to circles, and an angle to an angle of equal measure. The lessons on proving these properties, Lessons 7–9, require students to build arguments based on the structure of the figure in question and a handful of related facts that can be applied to the situation. Students apply their understanding of dilations to divide a line segment into equal pieces and explore and compare dilations from different centers.

Lessons

Topic C: Similarity and Dilations

Standards

G-SRT.A.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.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures.

G-MG.A.1 Using geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Overview

Students learn what it means for two figures to be similar in general, and then focus on triangles and what criteria predict that two triangles will be similar. Length relationships within and between figures is studied closely and foreshadows work in Topic D. The topic closes with a look at how similarity has been used in real world application.

Lessons

Topic D: Applying Similarity to Right Triangles

Standards

G-SRT.B.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

Overview

The focus in Topic D is similarity within right triangles. Students examine how an altitude drawn from the vertex of a right triangle to the hypotenuse creates two similar sub-triangles. Students work with adding, subtracting, multiplying, and dividing radical expressions. Finally, students prove the Pythagorean Theorem using similarity.

Lessons

Topic E: Trigonometry

Standards

G-SRT.C.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.C.7 Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems

Overview

Students link their understanding of similarity and relationships within similar right triangles formally to trigonometry. In addition to the terms sine, cosine, and tangent, students study the relationship between sine and cosine, how to prove the Pythagorean Theorem using trigonometry, and how to apply the trigonometric ratios to solve right triangle problems.

Lessons

Module 3: Extending to Three Dimensions

(13 Hours)

Overview

Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. They reason abstractly and quantitatively to model problems using volume formulas.

Standards

G-GMD.A.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.A.3 Use volume formulas for cylinders, pyramids, cones and spheres to solve problems.

G-GMD.B.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.

G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g. modeling a tree trunk or a human torso as a cylinder). G-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G-MG.A.3 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)

Unit Vocab

Cavalieri’s Principle (Given two solids that are included between two parallel planes, if every plane parallel to the two planes intersects both solids in cross-sections of equal area, then the volumes of the two solids are equal.)

Cone (Let 𝐵𝐵 be a region in a plane 𝐸𝐸, and 𝑉𝑉 be a point not in 𝐸𝐸. The cone with base 𝐵𝐵 and vertex 𝑉𝑉 is the union of all segments 𝑉𝑉𝑉𝑉 for all points 𝑃𝑃 in 𝐵𝐵. If the base is a polygonal region, then the cone is usually called a pyramid.)

General Cylinder (Let 𝐸𝐸 and 𝐸𝐸′ be two parallel planes, let 𝐵𝐵 be a region in the plane 𝐸𝐸, and let 𝐿𝐿 be a line which intersects 𝐸𝐸 and 𝐸𝐸′ but not 𝐵𝐵. At each point 𝑃𝑃 of 𝐵𝐵, consider the segment 𝑃𝑃𝑃𝑃′ parallel to 𝐿𝐿, joining 𝑃𝑃 to a point 𝑃𝑃′ of the plane 𝐸𝐸′. The union of all these segments is called a cylinder with base 𝐵𝐵.)

Inscribed Polygon (A polygon is inscribed in a circle if all of the vertices of the polygon lie on the circle.)

Intersection (The intersection of 𝐴𝐴 and 𝐵𝐵 is the set of all objects that are elements of 𝐴𝐴 and also elements of 𝐵𝐵. The intersection is denoted 𝐴𝐴 ∩ 𝐵𝐵.)

Rectangular Pyramid (Given a rectangular region 𝐵𝐵 in a plane 𝐸𝐸, and a point 𝑉𝑉 not in 𝐸𝐸, the rectangular pyramid with base 𝐵𝐵 and vertex 𝑉𝑉 is the union of all segments 𝑉𝑉𝑉𝑉 for points 𝑃𝑃 in 𝐵𝐵. )

Right Rectangular Prism (Let 𝐸𝐸 and 𝐸𝐸′ be two parallel planes. Let 𝐵𝐵 be a rectangular region in the plane 𝐸𝐸. At each point 𝑃𝑃 of 𝐵𝐵, consider the segment 𝑃𝑃𝑃𝑃′ perpendicular to 𝐸𝐸, joining 𝑃𝑃 to a point 𝑃𝑃′ of the plane 𝐸𝐸′. The union of all these segments is called a right rectangular prism.)

Solid Sphere or Ball (Given a point 𝐶𝐶 in the three-dimensional space and a number 𝑟𝑟 > 0, the solid sphere (or ball) with center 𝐶𝐶 and radius 𝑟𝑟 is the set of all points in space whose distance from point 𝐶𝐶 is less than or equal to 𝑟𝑟.)

Sphere (Given a point 𝐶𝐶 in the three-dimensional space and a number 𝑟𝑟 > 0, the sphere with center 𝐶𝐶 and radius 𝑟𝑟 is the set of all points in space that are distance 𝑟𝑟 from the point 𝐶𝐶.)

Subset (A set 𝐴𝐴 is a subset of a set 𝐵𝐵 if every element of 𝐴𝐴 is also an element of 𝐵𝐵.)

Tangent to a Circle (A tangent line to a circle is a line that intersects a circle in one and only one point.)

Union (The union of 𝐴𝐴 and 𝐵𝐵 is the set of all objects that are either elements of 𝐴𝐴 or of 𝐵𝐵 or of both. The union is denoted 𝐴𝐴 ∪ 𝐵𝐵.)

Topic A: Area

Standards

G-GMD.A.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.

Overview

Topic A studies informal limit arguments to find the area of a rectangle with an irrational side length and of a disk (G-GMD.A.1). 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.

Lessons

Topic B: Volume

Standards

G-GMD.A.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.A.3 Use volume formulas for cylinders, pyramids, cones and spheres to solve problems.

G-GMD.B.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.

G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

G-MG.A.3 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).

Overview

Students study the basic properties of two-dimensional and three-dimensional space, noting how ideas shift between the dimensions. They learn that general cylinders are the parent category for prisms, circular cylinders, right cylinders, and oblique cylinders, and study why the cross section of a cylinder is congruent to its base. Next students study the explicit definition of a cone and learn what distinguishes pyramids from general cones, and see how dilations explain why a cross-section taken parallel to the base of a cone is similar to the base.

Students revisit the scaling principle as it applies to volume and then learn Cavalieri’s principle, which describes the relationship between cross-sections of two solids and their respective volumes. This knowledge is all applied to derive the volume formula for cones, and then extended to derive the volume formula for spheres. Module 3 is a natural place to see geometric concepts in modeling situations. Modeling-based problems are found throughout Topic B, and include the modeling of real-world objects, the application of density, the occurrence of physical constraints, and issues regarding cost and profit.

Lessons

Module 4: Connecting Algebra and Geometry through Coordinates

(15 Hours)

Overview

G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G-

GPE.B.5 Prove3 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).

G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Standards

In this module, students explore and experience the utility of analyzing algebra and geometry challenges through the framework of coordinates. The module opens with a modeling challenge, one that reoccurs throughout the lessons, to use coordinate geometry to program the motion of a robot that is bound within a certain polygonal region of the plane—the room in which it sits. 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 will describe the region via systems of algebraic inequalities and work to constrain the robot motion along line segments within the region.

Unit Vocab

Normal Segment to a Line (A line segment with one endpoint on a line and perpendicular to the line is called a normal segment to the line.)

Module 4: Connecting Algebra and Geo: All Materials Here

Topic A: Rectangular and Triangular Regions Defined by Inequalities

Standards

G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Overview

Students impose a coordinate system and describe the movement of the robot in terms of line segments and points. This leads to graphing inequalities and discovering regions in the plane can be defined by a system of algebraic inequalities. Students then program the robot to move on lines cutting through these regions rotating and rotate 90°clockwise or counterclockwise about an endpoint.

Lessons

Topic B: Perpendicular and parallel Lines in the Cartesian Plan

Standards

G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

G-GPE.B.5 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).

Overview

The challenge of programming robot motion along segments parallel or perpendicular to a given segment leads to an analysis of slopes of parallel and perpendicular lines. Students write equations for parallel, perpendicular, and normal lines. Additionally, students will and study the proportionality of segments formed by diagonals of polygons.

Lessons

Topic C: Perimeters and Areas of Polygonal Regions in the Cartesian Plane

Standards

G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Overview

Students sketch the regions, determine points of intersection (vertices), and use the distance formula to calculate perimeter and the “shoelace” formula to determine area of these regions. Students return to the real-world application of programming a robot and extend this work to robots not just confined to straight line motion, but motion bound by regions described by inequalities and defined areas.

Lessons

Topic D: Partitioning and Extending Segments and Parameterization of Lines

Standards

G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point �1, √3� lies on the circle centered at the origin and containing the point (0, 2).

G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Overview

Students find midpoints of segments and points that divide segments into 3 or more equal and proportional parts and extend this concept prove classical results in geometry. Students are introduced to parametric equations in an optional exercise and derive and apply the distance formula.

Lessons

Module 5: Circles with and without Coordinates

(21 Hours)

Overview

This module brings together the ideas of similarity and congruence and the properties of length, area, and geometric constructions studied throughout the year. It also includes the specific properties of triangles, special quadrilaterals, parallel lines and transversals, and rigid motions established and built upon throughout this mathematical story. This module's focus is on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page.

Standards

G-C.A.1 Prove that all circles are similar.

G-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Include3 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.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove2 properties of angles for a quadrilateral inscribed in a circle.

G-C.B.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-GPE.A.1 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.

G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Unit Vocab

Arc Length (The length of an arc is the circular distance around the arc.)

Central Angle (A central angle of a circle is an angle whose vertex is the center of a circle.)

Chord (Given a circle 𝐶𝐶, let 𝑃𝑃 and 𝑄𝑄 be points on 𝐶𝐶. The 𝑃Q line is called a chord of 𝐶𝐶.)

Cyclic Quadrilateral (A quadrilateral inscribed in a circle is called a cyclic quadrilateral.)

Inscribed Angle (An inscribed angle is an angle whose vertex is on a circle, and each side of the angle intersects the circle in another point.)

Inscribed Polygon (A polygon is inscribed in a circle if all vertices of the polygon lie on the circle.)

Secant Line (A secant line to a circle is a line that intersects a circle in exactly two points.)

Sector (Let 𝐴B be an arc of a circle. The sector of a circle with arc 𝐴B is the union of all radii of the circle that have an endpoint in arc 𝐴B. The arc 𝐴B is called the arc of the sector, and the length of any radius of the circle is called the radius of the sector.)

Tangent Line (A tangent line to a circle is a line in the same plane that intersects the circle in one and only one point. This point is called the point of tangency.)

Module 5: Circles with and w/o Coordinates: All materials here

Topic A: Central and Inscribed Angles

Standards

G-C.A.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.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Overview

Topic A leads students first to Thales' theorem (an angle drawn from a diameter of a circle to a point on the circle is sure to be a right angle), then to possible converses of Thales' theorem, and finally to the general inscribed-central angle theorem. Students use this result to solve unknown angle problems. Through this work, students construct triangles and rectangles inscribed in circles and study their properties.

Lessons

Topic B: Arcs and Sectors

Standards

G-C.A.1 Prove that all circles are similar.

G-C.A.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.B.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.

Overview

Topic B defines the measure of an arc and establishes results relating chord lengths and the measures of the arcs they subtend. Students build on their knowledge of circles from Module 2 and prove that all circles are similar. Students develop a formula for arc length in addition to a formula for the area of a sector and practice their skills solving unknown area problems.

Lessons

Topic C: Secant and Tangents

Standards

G-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords.

G-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Overview

In Topic C, students explore geometric relations in diagrams of two secant lines, or a secant and tangent line (possibly even two tangent lines), meeting a point inside or outside of a circle. They establish the secant angle theorems and tangent-secant angle theorems. By drawing auxiliary lines, students also notice similar triangles and thereby discovered relationships between lengths of line segments appearing in these diagrams.

Lessons

Topic D: Equations for Circles and Their Tangents

Standards

G-GPE.A.1 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.

G-GPE.A.4 Use coordinates to prove simple geometric theorems algebraically.

Overview

Topic D brings in coordinate geometry to establish the equation of a circle. Students solve problems to find the equations of specific tangent lines or the coordinates of specific points of contact. They also express circles via analytic equations.

Lessons

Topic E: Cyclic Quadrilaterals and Ptolemy's Theorem

Standards

G-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Overview

The module concludes with Topic E focusing on the properties of quadrilaterals inscribed in circles and establishing Ptolemy's theorem. This result codifies the Pythagorean theorem, curious facts about triangles, properties of the regular pentagon, and trigonometric relationships. It serves as a final unifying flourish for students' year-long study of geometry.

Lessons