Common Core High School Geometry Teaching and Learning Resources

Domain: CO: Congruence

• CO-A: Experiment with transformations in the plane

• CO-B: Understand congruence in terms of rigid motions

• CO-C: Prove geometric theorems

• CO-D: Make geometric constructionss

CO-A: Experiment with transformations in the plane

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:

 Angle Alternate Interior Angles Congruent Rectangle Altitude Corresponding Vertical Angles Square Angle Bisector Transformation Median of a Triangle Point Circle Triangle Congruence Diagonal of a Polygon Rhombus Arc of a Circle Parallelogram Dilation Trapezoid Line Perpendicular Rotation Equilateral Triangle Parallel Lines Line Segment Reflection Isosceles Triangle Transversal Midpoint Translation Regular Polygon Alternate Exterior Angles Perpendicular Bisector Theorem

Imagine Learning Classroom (formerly LearnZillion) subscription needed: 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.

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: Lessons and practice problems include symmetry, reflections, translations, rotations, and dilations of a geometric figure.
• Geometry: Transformations: Lessons on using the TI-84+ family of calculators to perform transformations (reflect, translate, rotate, dilate).

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.

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 (Java required for online use)

Shodor Interactivate:

• Angles formed by parallel lines and one or two transversals.  This activity allows users to practice angle vocabulary.
• 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.

Walter Fendt's Sum of Angles in a Triangle, Virtual Manipulative

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:

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

Khan Academy: Practice questions with videos.

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.

Constructed-response:

Khan Academy: Performing transformations: unit on translations, rotations, reflections, dilations. Practice questions with videos.

Illustrative Mathematics: Geometry:

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 8: Can You Get There from Here? aligns with CO-A.5.

NCTM's Reasoning and Sense Making Task Library: Taking a Spin includes the task overview, teacher notes for its use in the classroom, and student activity sheet.  Aligns with CO-A.3, CO-A.4 and mathematical practice standards 3 and 8.

CO-B: Understand congruence in terms of rigid motions

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.

National Library of Virtual Manipulatives: investigate methods for determining congruent triangles (Java required). 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:

Multiple Choice:

Khan Academy: Practice questions with videos.

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.

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.

Math.com: Congruent Figures: Lesson and fill-in the blank matching problems.

Illustrative Mathematics: Geometry:

Mathematics Assessment Project: High School: Geometry: 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.

NCTM's Reasoning and Sense Making Task Library: Bank Shot includes the task overview, teacher notes for its use in the classroom, and student activity sheet.  Aligns with CO-B.6, CO-C.9, SRT-B.5, and mathematical practice standards 1, 2, 3, and 7.

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.

CO-C: Prove geometric theorems

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:

Imagine Learning Classroom (formerly LearnZillion) subscription needed: 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.

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.

Saltire Software: Geometry Applet Gallery 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:

Multiple Choice:

Khan Academy: Practice questions with videos.

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.

National Library of Virtual Manipulatives: walk through a visual proof of the Pythagorean theorem.

Constructed-response:

AlgebraLab.org:  Lesson and fill in the blank practice problems.  Be sure to show the related AlgebraLab documents for additional practice problems on each topic.

Khan Academy: Practice questions with videos.

Math.com: Angles and Intersecting Lines: Lesson and fill-in the blank matching problems.

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.

Illustrative Mathematics: Geometry:

• 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: High School: Geometry:

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.

NCTM's Reasoning and Sense Making Task Library: Bank Shot includes the task overview, teacher notes for its use in the classroom, and student activity sheet.  Aligns with CO-B.6, CO-C.9, SRT-B.5, and mathematical practice standards 1, 2, 3, and 7.

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.

CO-D: Make geometric constructions

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:

Desmos: Geometry Tool

GeoGebra: free open-source software

Khan Academy: Geometric Constructions unit includes tutorial videos on the following:

• 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 Table of Contents for the Constructions 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.

Robo Compass is an online app for geometric constructions, transformations, patterns and tesselations.  It works "exactly as you do using a physical straightedge, compass and protractor."  Documentation is included.

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:

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.

Constructed-response:

Illustrative Mathematics: Geometry:

Mathematics Assessment Project: High School: Geometry:  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.