**Math Topics**- Common Core
- Initiatives
- Methodology
- Resources
- Projects
- Manipulatives
- Software

**Learning Support**- Standardized Test Prep
- Technology Integration
- Assisting Readers

**Professionalism**- Associations
- Journals
- News
- Professional Development
- Education Standards
- Education Research
- CT4ME Publications

Select the cluster for resources on this page:

**Standards**:

- 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.
- 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).
- CO-A.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
- CO-A.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
- 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.

**Technology-enhanced investigations:**

Mathwords.com: Math Dictionary: Key vocabulary for this domain. Use with CO-A, CO-B, CO-C and CO-D:

LearnZillion: Lesson set: Define geometric terms precisely: Four video lessons: Definitions and representations are distinct, understand the importance of precise definitions, why point and line are undefinable; and define geometric terms precisely. Aligns with CO-A.1.

Wisc-Online: The Circle: In this animated and interactive object, learners examine the definitions and formulas for radius, diameter, circumference, and area. Students also solve practice problems involving the circumference and area of a circle. Use with CO-A.1.

Math.com: Quadrilaterals: Students learn about the five most common quadrilaterals: parallelogram, square, rectangle, rhombus, trapezoid. Use with CO-A.3.

Mathwords.com: Index for Geometry: Terms from plane and solid geometry. This interactive math dictionary has words, terms, formulas, pictures, diagrams, tables, and examples. Use with CO-A.

Mathbits.com: Use with CO-A.2.

- Transformations: This is a lesson on using the graphing calculator (TI 83+/84+) to show reflection, translation, and dilation of a geometric figure.
- Transformations-Rotations: This is a lesson on using the graphing calculator (TI 83+/84+) to show rotation of a geometric figure.

Mathematical Association of America Mathematical Sciences Digital Library: Loci Resources: Using Geogebra and Exploring Geometric Transformations in a Dynamic Environment: There are eight activities, which address reflections, rotations, translations, glide-reflections, and reflections with parallel lines, intersecting lines, and concurrent lines.

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 9 Quadratic Relations and Conic Sections for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for related sections to use within the eWorkbook. Use with CO-A.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources. Description is from the video summary. The Pythagorean Theorem: Geometry's Most Elegant Theorem: "This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications." Aligns with CO-A.1 and SRT-C.6, SRT-C.8.

National Library of Virtual Manipulatives: Transformations

- Reflection
- Rotation
- Translation
- Composition: Use reflection, rotation, and translation

NCTM Illuminations:

- Frieze Patterns: Students experiment with the seven classes of Frieze patterns with this activity. They can explore the transformations that constitute each of the seven categories.
- Transformations and Frieze Patterns: This unit introduces students to symmetry and rotation in figures and patterns. Students learn to recognize and classify symmetry in decorative figures and frieze patterns, and also get the chance to create and classify their own figures and patterns using JavaSketchpad applets.
- Symmetries I: This is the first i-Math in a four-part series of i-Maths entitled Symmetries and Their Properties. Learners investigate rotational symmetry. Fix a center, turn, and you have a rotation.
- Symmetries II: This is the second i-Math in a four-part series of i-Maths entitled Symmetries and Their Properties. Learners investigate reflection, mirror, or bilateral symmetry.
- Symmetries III: In this lesson learners investigate how translations work and what happens when two or more translations are applied one after the other. Those who are familiar with vectors can use them in this context to define a translation in the plane.
- Symmetries IV: In this lesson learners investigate the glide reflection—a symmetry transformation that is made up of two other symmetry transformations, a translation and a reflection.

Shodor Interactivate:

- Translations, Reflections, Rotations: This lesson introduces learners to those concepts.
- Transmographer: Learners can explore translations, reflections, and rotations in the Cartesian coordinate system by transforming squares, triangles and parallelograms. Parameters: Shape, x or y translation, x or y reflection, angle of rotation.
- Transmographer 2: Learners build their own polygon and transform it in the Cartesian coordinate system. Experiment with reflections across any line, rotations about any point, and translations in any direction. Parameters: Shape, x or y translation, x or y reflection, angle of rotation.
- 3D Transmographer: Use the virtual manipulative to experiment with translations, reflections, rotations of polygons on the coordinate plane. Learners can build their own polygons and also revolve them around any line to yield a 3D solid of revolution.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 1 manipulative for CO-A.1, 29 manipulatives for CO-A.2, 14 manipulatives for CO-A.3, 43 manipulatives for CO-A.4, 105 manipulatives for CO-A.5. Among those:

- Reference angles: Aligns with CO-A.1.
- Composing rotations: Aligns with CO-A.2, CO-A.4, CO-B.6.
- Linear transformations of a polygon: Aligns with CO-A.2, CO-A.4, CO-A.5.
- Matrix transformations "F": Aligns with CO-A.2, CO-A.4, CO-A.5.
- Matrix transformation: applied to a circle: Aligns with CO-A.2, CO-A.4, CO-A.5, CO-B.6.
- Classifying quadrilaterals: Aligns with CO-A.3, CO-C.11.
- Rotation about a point in a plane: Aligns with CO-A.3, CO-A.4, CO-A.5.
- 2D rotation using matrices: Aligns with CO-A.3, CO-A.4, CO-A.5.
- Reflection in two dimensions: Aligns with CO-A.3, CO-A.4, CO-A.5.
- Rotation in two dimensions: Aligns with CO-A.2, CO-A.4, CO-A.5, CO-B.6.
- Solve the Loculus of Archimedes by forming a square from given pieces using translations, reflection, and rotation. Aligns with CO-A.2, CO-A.4, CO-A.5, CO-B.6.
- Rotation as product of two reflections: Aligns with CO-A.5.
- Reflection matrix in 2D: Aligns with CO-A.5.
- Quadrilateral slide: Aligns with CO-A.5.
- Understanding 2D reflection: Aligns with CO-A.5.
- Understanding 2D translation: Aligns with CO-A.5.
- Understanding 2D rotation: Aligns with CO-A.5.
- Parallelogram slide: Aligns with CO-A.5.

YouTube video lessons: Use the following with CO-A.1:

- Geometry: Some precise definitions
- Introduction to Geometry (1.1): Define Point, Line, Plane, Segment, Ray, Angle, Collinear, Coplanar.
- Point Line Plane Examples (1.1)

**Multiple Choice:**

CK-12:
Dilations in the Coordinate Plane: Practice questions with review
content and video. Review content includes a *Types of
Transformations Study Guide*, which is an overview of reflection,
rotation, dilation, and translation. Aligns with CO-A.2 and SRT-A.1.

Glencoe Online Learning Center, Geometry (2010) self-check quizzes:

- Ch. 9, Lesson 1: Reflections
- Ch. 9, Lesson 2: Translations
- Ch. 9, Lesson 3: Rotations

Glencoe Online Learning Center, Geometry (2010) standardized test practice: Ch. 9: Transformations and Symmetry

Khan Academy: Practice questions with videos.

- Geometry: Introduction to Euclidean Geometry: Geometric definitions: Aligns with CO-A.1.
- Geometry: Transformations: Aligns with CO-A.2, CO-A.3, CO-A.4, CO-A.5.

MathBitsNotebook: Geometry:

- Segments, Angles and Triangles includes lessons, then practice problems on segments, segments in triangles, midsegments, angles, angles in triangles, isosceles triangles, triangle inequality, pre-proof reasoning (this latter is constructed response), and practice with proofs (this latter is constructed response). Use with CO-A.1, and CO-C.9, CO-C.10.
- Parallels and Perpendiculars includes lessons, then practice problems on angles and parallel lines, perpendicular lines, and proofs involving parallel and perpendicular lines (this latter is constructed response). Use with CO-A.1 and CO-C.9.
- Transformations includes lessons, then practice problems on transformations as functions, symmetry, reflections, translations, rotations, rigid transformations, and compositions (sequences) of transformations. Use with CO-A.2, CO-A.3, CO-A.4, CO-A.5, and CO-B.6.

MathsIsFun.com: Transformations: Learners use virtual manipulatives to experiment with rotations, reflections, and translations of a triangle, rectangle, or L-shape, setting parameters for each experiment. Following each section are multiple choice questions. Aligns with CO-A.5.

McDougal Littell ClassZone, Geometry, 2011: Use with CO-A:

- Ch. 9.1 Quiz: Translate Figures and Use Vectors
- Ch. 9.3 Quiz: Perform Reflections
- Ch. 9.4 Quiz: Perform Rotations
- Ch. 9.5 Quiz: Apply Compositions of Transformations
- Ch. 9.6 Quiz: Identify Symmetry
- Ch. 9.7 Quiz: Identify and Perform Dilations

OpenEd:

- Congruence - definitions: 5 questions combine multiple choice and free response. Includes related review resources. Aligns with CO-A.1.
- Congruence - rotations and reflections: 5 questions combine multiple choice and free response. Includes related review resources. Aligns with CO-A.3.
- Congruence - develop rigid motions: 5 questions combine multiple choice and free response. Includes related review resources. Aligns with CO-A.4.

**Constructed-response:**

Khan Academy: Practice questions with videos.

- Qualitatively defining rigid transformation: Aligns with CO-A.2, CO-A.4.
- Perform translations: Aligns with CO-A.5.
- Perform rotations: Aligns with CO-A.5.
- Perform reflections: Aligns with CO-A.5.
- Transforming polygons: Aligns with CO-A.5.
- Draw the image of a reflection: Aligns with CO-A.5.
- Advanced reflections: Aligns with CO-A.5.

OpenEd:

- Representing transformations - congruence: 6 questions combine free response and multiple choice. Includes related review resources. Aligns with CO-A.2.
- Congruence - transformations: 5 questions combine free response and multiple choice. Includes related review resources. Aligns with CO-A.5.

**Performance tasks:**

Illustrative Mathematics:

- Defining Parallel Lines: Aligns with CO-A.1.
- Defining Perpendicular Lines: Aligns with CO-A.1.
- Reflected Triangles: Aligns with CO-A.5 and CO-D.12.
- Showing a Triangle Congruence: The general case: Aligns with CO-A.5.
- Showing a Triangle Congruence: The particular case: Aligns with CO-A.5.

Mathematics Vision Project, Secondary 1 Student Edition: Module 6: Congruence, Construction, and Proof: This module contains 14 classroom tasks. Geometry standards addressed in Module 5 include CO-A.1, CO-A.2, CO-A.3, CO-A.4, CO-A.5 and CO-B.6, CO-B.7, CO-B.8, and CO-D.12, CO-D.13, and GPE-B.5. Choose the following within the module:

- Task 1: Leaping Lizards! aligns with both standards CO-A.1, CO-A.2, CO-A.4, CO-A.5;
- Task 2: Is It Right? aligns with CO-A.1.
- Task 3: Leap Frog aligns with CO-A.4, CO-A.5.
- Task 4: Is It Leap Year? aligns with both standards CO-A.1 and CO-A.4.
- Task 5: Symmetries of Quadrilaterals; and Task 6: Symmetries of Regular Polygons align with CO-A.3.
- Task 7: Quadrilaterals-Beyond Definition aligns with both standards CO-A.3, CO-A.4.
- Task 8: Can You Get There from Here? aligns with CO-A.5.

**Standards**:

- 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.
- 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.
- 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.

**Technology-enhanced investigations:**

Math Open Reference: Congruent Triangles: Investigate the theorems for congruence.

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 4 Quadratic Functions and Factoring for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for related sections to use within the eWorkbook. Use with CO-B.

NCTM Illuminations: Congruence Theorems: Use this virtual manipulative to investigate congruence by manipulating the sides and angles of a triangle.

Utah State University, National Library of Virtual Manipulatives: investigate methods for determining congruent triangles. Explore SSS, SAS, ASA, SSA. Note SSA does not guarantee congruent triangles.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 32 manipulatives for CO-B.6. Among those:

- Solve the Loculus of Archimedes by forming a square from given pieces using translations, reflection, and rotation. Aligns with CO-A.2, CO-A.4, CO-A.5, CO-B.6.
- Matrix transformation: applied to a circle: Aligns with CO-A.2, CO-A.4, CO-A.5, CO-B.6.
- Rotation in two dimensions: Aligns with CO-A.2, CO-A.4, CO-A.5, CO-B.6.
- Composing rotations: Aligns with CO-A.2, CO-A.4, CO-B.6.
- Symmetries of a prism: Aligns with CO-B.6.

**Multiple-choice:**

Glencoe Online Learning Center, Geometry (2010) self-check quizzes: Use with CO-B.7:

- Ch. 4, Lesson 3: Congruent Triangles
- Ch. 4, Lesson 4: Proving Congruence: SSS, SAS
- Ch. 4, Lesson 5: Proving Congruence: ASA, AAS

Khan Academy: Practice questions with videos.

- Congruent triangles 1: Determine congruent triangles using SSS, SAS, ASA, AAS. Aligns with CO-B.7.

MathBitsNotebook: Geometry:

- Transformations includes lessons, then practice problems on transformations as functions, symmetry, reflections, translations, rotations, rigid transformations, and compositions (sequences) of transformations. Use with CO-A.2, CO-A.3, CO-A.4, CO-A.5, and CO-B.6.
- Congruent Triangles includes lessons, then practice problems on rigid motions and congruence, numerical practice with congruence, recognizing congruent triangles, completing congruent triangle proofs (this latter is constructed response), congruent triangles (multiple choice), and congruent triangle proofs (this latter is constructed response with proofs - rigid motion, direct, double, indirect, flow). Use with CO-B.7 and CO-B.8.

McDougal Littell ClassZone, Geometry, 2011: Use with CO-B.6 and CO-B.7:

- Ch. 4.2 Quiz: Apply Congruence and Triangles
- Ch. 4.3 Quiz: Prove Triangles Congruent by SSS
- Ch. 4.4 Quiz: Prove Triangles Congruent by SAS and HL (Hypotenuse-Leg)
- Ch. 4.5 Quiz: Prove Triangles Congruent by ASA and AAS
- Ch. 4.6 Quiz: Use Congruent Triangles
- Ch. 4.7 Quiz: Use Isosceles and Equilateral Triangles
- Ch. 4.8 Quiz: Perform Congruence Transformations. Aligns with CO-B.6.

OpenEd: Triangle congruence (ASA, SAS, SSS): 5 questions combine multiple choice and free response. Includes related review resources. Aligns with CO-B.8.

**Constructed-response:**

AlgebraLab.org: Congruent Triangles: Lesson and fill in the blank practice problems, including word problems. Be sure to show the related AlgebraLab documents for additional practice problems.

Khan Academy: Practice questions with videos.

- Defining congruence through rigid transformations: Aligns with CO-B.6, CO-B.7.
- Explore rigid transformations and congruence: Aligns with CO-B.6, CO-B.7.
- Congruent triangles 2: Find angles in congruent triangles. Aligns with CO-B.6, CO-B.7.

**Performance tasks:**

Illustrative Mathematics:

- Reflections and Isosceles Triangles: Aligns with CO-B.
- Properties of Congruent Triangles: Aligns with CO-B.7.

Mathematics Assessment Project: Standards: High School: Geometry: http://map.mathshell.org/stds.php?standardid=1367 Task 452: Analyzing Congruence Proofs

Mathematics Vision Project, Secondary 1 Student Edition: Module 6: Congruence, Construction, and Proof: This module contains 14 classroom tasks. Geometry standards addressed in Module 5 include CO-A.1, CO-A.2, CO-A.3, CO-A.4, CO-A.5 and CO-B.6, CO-B.7, CO-B.8, and CO-D.12, CO-D.13, and GPE-B.5. Task 5: Symmetries of Quadrilaterals; Task 6: Symmetries of Regular Polygons; and Task 7: Quadrilaterals-Beyond Definition align with standard CO-B.6. Task 9: Congruent Triangles aligns with standards CO-B.6, CO-B.7, CO-B.8.

New York City School System: High School Geometry: Company Logo: Unit Overview: “This 3-4 week unit uses an investigation of rigid motions and geometric theorems to teach students how to verify congruence of plane figures and use the implications of congruence to solve problems and create proofs about geometric relationships. Students will demonstrate mastery of the content by making sense of the Company Logo Performance Task and persevering in solving the task.” Geometry standards addressed include CO-B.6, CO-B.7, CO-C.9, CO-C.10, and SRT-B.5. Mathematical Practices standards include MP.1, MP.3, and MP.6.

**Standards**:

- 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*. - 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*. - 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*.

**Technology-enhanced investigations:**

LearnZillion: Lesson set: Prove theorems concerning triangles and parallelograms: Six video lessons: Use triangle congruence (SAS), use triangle similarity to prove Triangle Midsection Theorem and Concurrency of Medians Theorem, use parallel lines and triangle congruence theorems to prove properties of parallelograms, including properties of diagonals of parallelograms, use properties of parallelograms and triangles to solve a real-world problem. Aligns with CO-C.10 and CO-C.11.

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 1 Equations and Inequalities, Ch. 2 Linear Equations and Functions, Ch. 3 Linear Systems and Matrices, Ch. 4 Quadratic Functions and Factoring, Ch. 5 Polynomials and Polynomial Functions, and Ch. 8 Rational Functions for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for related sections to use within the eWorkbook. Use with CO-C.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources. Description is from the video summary. The Broken Stick Experiment: Triangles, Random Numbers and Probability: "In this video lesson, during in-class sessions between video segments, students will learn among other things: 1) how to generate random numbers; 2) how to deal with probability; and 3) how to construct and draw portions of the X-Y plane that satisfy linear inequalities." Aligns with CO-C.10.

NCTM Illuminations:

- Angle Sums: Use this virtual manipulative to examine the angles in a triangle, quadrilateral, pentagon, hexagon, heptagon or octagon. The purpose is to find a relationship between the number of sides and the sum of the interior angles of the polygon.
- Pieces of Proof: Lesson: "There is a leap to be made from understanding postulates and theorems in geometry to writing proofs using them. This lesson offers an intermediate step, in which students put together the statements and reasons to build a formal proof."

Saltire Software: Geometry Applet Gallery from contains some excellent applets, which are easy to use and suitable for exploring several geometry theorems. Select from the following:

- Basic Geometry: Nine applets for exploration of angles on a chord, angles on a chord and center, similar triangles, angles between parallel lines, right kites (2 applets), perpendicular bisectors of a right triangle, the incircle, and Pythagorean triples. Students can drag points to explore concepts.
- Triangle Calculators: Four triangle calculators are presented. Students can enter SSS, SAS, or ASA, and see the unique triangle that is formed with the applet. If they enter SSA, they see that the resulting triangle is not unique (two are possible) or no triangle is possibly formed with given information.

Shodor Interactivate: Angles: This lesson addresses acute, obtuse, and right angles as well as relationships between angles formed by parallel lines crossed by a transversal, including vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, same-side interior angles, same-side exterior angles, and corresponding angles.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 1 manipulative for CO-C.9, 15 manipulatives for CO-C.10, 2 manipulatives for CO-C-11. Among those:

- Parallel lines: Angles with a transversal: Aligns with CO-C.9.
- Interior angles of a triangle sum to Pi (radians): a straight line is formed. Aligns with CO-C.10.
- The Sum of the Interior Angles of a Triangle Equals 180 Degrees by Paper Folding: Aligns with CO-C.10.
- Halving a Triangle: The median divides the triangle into two halves of equal area. Aligns with CO-C.10.
- The Medians of a Triangle Divide It into Three Smaller Triangles of Equal Area: Aligns with CO-C.10, CO-C.11.
- The Medial Division of Triangles: The medians of a triangle divide it into six smaller triangles of equal area. Aligns with CO-C.10.
- The Medians of a Triangle Are Concurrent: A Visual Proof: Aligns with CO-C.10.
- The Centroid of a Triangle Divides Each Median in the Ratio 1:2: Aligns with CO-C.10.
- Triangle Inequality: The sum of lengths of two sides of a triangle is at least as large as the length of the third side. Aligns with CO-C.10.
- Euclid, Book 3, Proposition 22: in a cyclic quadrilateral, opposite angles sum to 180 degrees: Aligns with C-A.2, C-A.3, and CO-C.10.
- Angles in a quadrilateral total 360 degrees: Aligns with CO-C.10.
- Classifying quadrilaterals: Aligns with CO-A.3, CO-C.11.

YouTube Videos:

- Theorems for parallel lines and transversals: Use with CO-C.9.
- Triangle Application theorems: Use with CO-C.10.
- Properties of Parallelograms: Use with CO-C.11.

**Multiple-choice:**

Khan Academy: Practice questions with videos.

- Triangle inequality theorem: Aligns with CO-C.10.

McDougal Littell ClassZone, Geometry, 2011:

- Ch. 1.5 Quiz: Describe Angle Pair Relationships: Use with CO-C.9.
- Ch. 2.6 Quiz: Prove Statements About Segments and Angles: Use with CO-C.9.
- Ch. 2.5 Quiz: Prove Angle Pair Relationships: Use with CO-C.9.
- Ch. 3.2 Quiz: Use Parallel Lines and Transversals: Use with CO-C.9.
- Ch. 3.3 Quiz: Prove Lines are Parallel: Use with CO-C.9.
- Ch. 3.6 Quiz: Prove Theorems about Perpendicular Lines: Use with CO-C.9.
- Ch. 5.2 Quiz: Use Perpendicular Bisectors: Use with CO-C.9.
- Ch. 5.2B Quiz: Use Perpendicular Bisectors, More Problems: Use with CO-C.9.
- Ch. 5.3 Quiz: Use Angle Bisectors of Triangles: Use with CO-C.9.
- Ch. 5.4 Quiz:
Use Medians and Altitudes: Use with CO-C.9.

- Ch. 4.1 Quiz: Apply Triangle Sum Properties: Use with CO-C.10.
- Ch. 4.6 Quiz: Use Congruent Triangles. Proofs to show triangles congruent. Use with CO-C.10.
- Ch. 5.1 Quiz:
Midsegment Theorem and Coordinate Proof: For Triangles.
Use with CO-C.10.

- Ch. 8.2 Quiz: Use Properties of Parallelograms: Use with CO-C.11.
- Ch. 8.3 Quiz: Show that a Quadrilateral is a Parallelogram: Use with CO-C.11.
- Ch. 8.4 Quiz: Properties of Rhombuses, Rectangles, and Squares: Use with CO-C.11.

OpenEd:

- Line and angle theorems: 5 questions combine multiple choice and free response. Includes related review resources. Aligns with CO-C.9.
- Triangle theorems: 6 questions combine multiple choice and free response. Includes related review resources. Aligns with CO-C.10.
- Parallelogram theorems: 5 questions combine multiple choice and free response. Includes related review resources. Aligns with CO-C.11.

Shodor Interactivate: Angles: This applet with drop-down choices for responses provides an interactive quiz for learners to practice their knowledge of acute, obtuse, and alternate angles. Also, practice relationships between vertical, adjacent, alternate, same-side, and corresponding angles. Learners can use either one or two transversals crossing parallel lines.

Utah State University, National Library of Virtual Manipulatives: walk through a visual proof of the Pythagorean theorem.

**Constructed-response:**

Khan Academy: Practice questions with videos.

- Line and angle proofs: Aligns with CO-C.9.

MathBitsNotebook: Geometry:

- Segments, Angles and Triangles includes lessons, then practice problems on segments, segments in triangles, midsegments, angles, angles in triangles, isosceles triangles, triangle inequality, pre-proof reasoning (this latter is constructed response), and practice with proofs (this latter is constructed response). Use with CO-A.1, and CO-C.9, CO-C.10.
- Parallels and Perpendiculars includes lessons, then practice problems on angles and parallel lines, perpendicular lines, and proofs involving parallel and perpendicular lines (this latter is constructed response). Use with CO-A.1 and CO-C.9.
- Quadrilaterals includes lessons, then practice problems on properties of quadrilaterals (fill-in chart, then a quiz), numerical problems and algebraic problems about quadrilaterals, applications of quadrilaterals, and proofs dealing with quadrilaterals (this latter includes both multiple choice and constructed response). Use with CO-C.11.

**Performance tasks:**

Illustrative Mathematics: Geometry:

- Tangent lines and the radius of a circle: Aligns with CO-C.9.
- Points equidistant from two points in a plane: Aligns with CO-C.9.
- Is this a parallelogram?: Aligns with CO-C.11.
- Parallelograms and Translations: Aligns with CO-C.11.
- Midpoints of the sides of a parallelogram: Aligns with CO-C.11.
- Congruence of parallelograms: Aligns with CO-C.11 and SRT-B.5.

Inside Mathematics: MARS Tasks:

- Quadrilaterals: Students use geometric properties to solve a problem. This task aligns with standards CO-C.9 and CO-C.11.
- Circles in Triangles: Students use geometric properties of circles and triangles to prove that two triangles are congruent. This task aligns with geometry congruence standards CO-C.9 and CO-C.10 and geometry circle standards C-A.2 and C-A.3.

Mathematics Assessment Project: Standards: High School: Geometry: http://map.mathshell.org/stds.php?standardid=1367

- Task 429: Solving Geometry Problems: Floodlights
- Task 212: Evaluating Statements About Length and Area
- Task 419: Proofs of the Pythagorean Theorem

Mathematics Vision Project, Secondary 2 Student Edition:

- Module 5: Geometric Figures: This module contains 8 classroom tasks addressing geometry congruence standards CO-C.9, CO-C.10 and CO-C.11.

New York City School System: High School Geometry: Company Logo: Unit Overview: “This 3-4 week unit uses an investigation of rigid motions and geometric theorems to teach students how to verify congruence of plane figures and use the implications of congruence to solve problems and create proofs about geometric relationships. Students will demonstrate mastery of the content by making sense of the Company Logo Performance Task and persevering in solving the task.” Geometry standards addressed include CO-B.6, CO-B.7, CO-C.9, CO-C.10, and SRT-B.5. Mathematical Practices standards include MP.1, MP.3, and MP.6.

**Standards**:

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

**Technology-enhanced investigations:**

Geogebra: free open-source software

C.a.R. (Compass and Ruler) is a dynamic geometry program simulating compass and ruler constructions on a computer.

Khan Academy: Geometric Constructions, Tutorial videos:

- Constructing bisectors of lines and angles: Aligns with CO-D.12.
- Constructing a line tangent to a circle: Aligns with CO-D.12.
- Constructing regular polygons inscribed in circles: Square, equilateral triangle, hexaxon. Aligns with CO-D.13.

MathsIsFun.com: Constructions: This is an excellent series of 17 animated geometric constructions, which learners would do with a compass, pencil, and straightedge only. They demonstrate how to construct a perpendicular bisector, angle bisector, 30-45-60 degree angles and equilateral triangle, perpendicular line to a line from a point on or off the line, parallel lines, congruent angles, a line cut into "n" pieces. Circle constructions include to find a center, tangent lines from an external point, inscribed and circumscribed circles for triangles, circle touching three points, and a pentagon construction. Aligns with CO-D.12 and CO-D.13.

Math Open Reference: Constructions: The site provides step-by-step animations of constructions with compass and straightedge. The list is comprehensive for lines, line segments, angles, triangles, circles, arcs, ellipses, polygons, and non-Euclidean constructions. Aligns with CO-D.12 and CO-D.13.

National Security Agency (NSA):

- Pythagorean Puzzle: In this self-guided investigation, students use Geometer’s Sketchpad to construct a right triangle and discover a geometric proof of the Pythagorean Theorem. Students test the geometric proof with acute and obtuse triangles.
- Concurrent Events: This lesson enhances student knowledge of how to use Geometer's Sketchpad to explore geometric concepts (e.g. the points of concurrency in a triangle). It includes the construction of line segments, triangles, circles, perpendicular bisectors of line segments, angle bisectors, altitudes of triangles, and medians of triangles.

Math Open Reference: Constructions: Students explore using a compass and straightedge to perform constructions with lines, angles, triangles, circles, arcs, ellipses, and polygons. Each exploration shows visually how to do the construction step by step.

NCTM Illuminations: Interactive Geometry Dictionary: Lines in Geometry: More than just the definitions, learners use applets with each definition to explore properties of lines, segments, rays; perpendicular lines; medians; perpendicular bisectors; angle bisectors; the Euler line; and the Simpson line. They can see steps in performing constructions.

Whistler Alley Mathematics: Geometry Construction Reference: This includes illustrations and steps for geometric constructions.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 20 manipulatives for CO-D.12, 3 manipulatives for CO-D.13. Among those:

- Distance of a point to a segment: Aligns with CO-D.12.
- Triangle side length inequality: Construct a triangle with three given sides. Aligns with CO-D.12.
- Angle bisectors of two intersecting lines: Aligns with CO-D.12.
- Angle bisectors in a triangle: Aligns with CO-D.12.
- The perpendicular bisectors of a triangle: Aligns with CO-D.12.
- Triangle: scalene, isosceles, equilateral: Aligns with CO-D.12.
- Mascheroni's construction of the center of a circle: Aligns with CO-D.12.
- Constructing a regular pentagon with a ruler and compass: Aligns with CO-D.12.
- Indestructible perpendicular lines: Aligns with CO-D.12.
- Bisection of line segment: Aligns with CO-D.12.
- Largest triangle inscribed in a circle: the equilateral triangle. Aligns with CO-D.13.
- Largest isosceles triangle inscribed in a circle: the equilateral triangle (via exploration). Aligns with CO-D.13.
- Maximum rectangle inscribed in a circle: the square. Aligns with CO-D.13.

**Multiple-choice:**

MathBitsNotebook: Geometry: Constructions and Concurrencies includes lessons on how to construct with compass and straightedge and other tools (e.g., a reflective edge, string, paper-folding, and dynamic geometric software), points of concurrency, and locus theorems. Practice problem are on basic and more constructions, using constructions, constructions and reflections, points on concurrency, and locus. Use with CO-C.10, CO-D.12 and CO-D.13.

OpenEd:

- Geometric constructions: 4 questions combine multiple choice and free response. Includes related review resources. Aligns with CO-D.12.
- Constructions of regular shapes: 4 questions combine multiple choice and free response. Includes related review resources. Aligns with CO-D.13.

**Constructed-response:**

**Performance tasks:**

Illustrative Mathematics:

- Angle bisection and midpoints of line segments: Aligns with CO-D.12.
- Construction of perpendicular bisector: Aligns with CO-D.12.
- Reflected Triangles: Aligns with CO-A.5 and CO-D.12.
- Inscribing a square in a circle: This task aligns with standard CO-D.13.

Mathematics Assessment Project: Standards: High School: Geometry: http://map.mathshell.org/stds.php?standardid=1367 Task 403: Inscribing and Circumscribing Right Triangles

Mathematics Vision Project, Secondary 1 Student Edition: Module 6: Congruence, Construction, and Proof: This module contains 14 classroom tasks. Geometry standards addressed in Module 5 include CO-A.1, CO-A.2, CO-A.3, CO-A.4, CO-A.5 and CO-B.6, CO-B.7, CO-B.8, and CO-D.12, CO-D.13, and GPE-B.5. Task 11: Under Construction; Task 12: More Under Construction; Task 13: Justifying Construction; and Task 14: Construction Blueprints align with standards CO-D.12 and CO-D.13.

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