| Title: |
Unit 5: Congruence, Proof, and Construction |
| Author & Email: |
Debbie Seldomridge dseldomr@access.k12.wv.us
and Cynthia Burke caburke@access.k12.wv.us |
| Grade Level: |
High School Math I |
Unit Overview:
|
Throughout the unit, students explore rigid motions isometries (translations, reflections, and rotations) in order to develop an understanding about what it means for two objects to be congruent. They establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. Students solve problems about triangles, quadrilaterals and other polygons. They apply reasoning to complete geometric constructions and explain why they work. |
Unit Calendar
| http://wveis.k12.wv.us/teach21/cso/upload/UP74CAL.doc |
Next Generation Content Standards and Objectives:
|
Objectives Directly Taught or Learned
Through Inquiry/Discovery | Evidence of Student Mastery of Content | | M.1HS.CPC.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 | Draw Along Activity
| | M.1HS.CPC.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). | Geometric Transformation Workout Transformation Golf
Translations in the Plane (Part 2)
Reflections in the Plane with GeoGebra
Rotations in the Plane with GeoGebra
Assessment of Learning
Algebraic Transformations | | M.1HS.CPC.3 given a rectangle, parallelogram, trapezoid or regular polygon, describe the rotations and reflections that carry it onto itself. | Reflections in the Plane with GeoGebra
Rotations in the Plane with GeoGebra
Assessment of Learning
Algebraic Transformations | | M.1HS.CPC.4 develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments. | Geometric Transformation Workout
Transformation Golf
Translations in the Plane (Part 2)
Reflections in the Plane with GeoGebra
Rotations in the Plane with GeoGebra
Assessment of Learning
Algebraic Transformations | | M.1HS.CPC.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. | Geometric Transformation Workout
Transformation Golf
Translations in the Plane (Part 2)
Reflections in the Plane with GeoGebra
Rotations in the Plane with GeoGebra
Assessment of Learning
Algebraic Transformations | | M.1HS.CPC.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. | Geometric Transformation Workout
Transformation Golf
Translations in the Plane (Part 2)
Reflections in the Plane with GeoGebra
Rotations in the Plane with GeoGebra
Assessment of Learning
Algebraic Transformations | | M.1HS.CPC.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. | Triangle Congruence Constructions Students construct triangles given three sides, or two angles and the included side, or two sides and the included angle, or two angles and a non-included side. They compare their constructions to verify that they have constructed congruent triangles. Logo Construction In a problem-situation students inscribe an equilateral triangle, a square, and a hexagon in a circle. They use dynamic geometric software to explore methods of combining formal constructions techniques and continue to formalize and defend how these constructions result in the desired objects. Students inscribe an equilateral triangle, a square, and a hexagon using formal construction techniques. | | M.1HS.CPC.8 explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. | Draw Along Activity
Students create illustrations of two triangles that are congruent by SSS, SAS, and ASA and include appropriate notation, as they demonstrate an understanding of triangle congruence conjectures. Logo Construction In a problem-situation students inscribe an equilateral triangle, a square, and a hexagon in a circle. They use dynamic geometric software to explore methods of combining formal constructions techniques and continue to formalize and defend how these constructions result in the desired objects. Students inscribe an equilateral triangle, a square, and a hexagon using formal construction techniques. | | M.1HS.CPC.9 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. | Triangle Congruence Constructions Students construct triangles given three sides, or two angles and the included side, or two sides and the included angle, or two angles and a non-included side. They compare their constructions to verify that they have constructed congruent triangles. Constructions Students create geometric constructions that include copying and bisecting angles and line segments. Challenge Problem
Students construct a copy of a triangle that is both a vertical and a horizontal reflection of the original. Triangle Construction
Students construct a triangle given three sides. More Constructions
Students create geometric constructions that include constructing perpendicular lines, constructing the perpendicular bisector of a line segment, and constructing a line parallel to a given line through a point not on the line. Students apply their understanding of reflections by reflecting a triangle over a given line. Logo Construction In a problem-situation students inscribe an equilateral triangle, a square, and a hexagon in a circle. They use dynamic geometric software to explore methods of combining formal constructions techniques and continue to formalize and defend how these constructions result in the desired objects. Students inscribe an equilateral triangle, a square, and a hexagon using formal construction techniques. | | M.1HS8.CPC.10
construct an equilateral triangle, a square and a regular hexagon inscribed in a circle. | Logo Construction In a problem-situation students inscribe an equilateral triangle, a square, and a hexagon in a circle. They use dynamic geometric software to explore methods of combining formal constructions techniques and continue to formalize and defend how these constructions result in the desired objects. Students inscribe an equilateral triangle, a square, and a hexagon using formal construction techniques. | |
Mathematical Practices::
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| Mathematical Practices | Evidence of Student Engagement in Mathematical Practices | | MP1. Make sense of problems and persevere in solving them. | Students determine both the minimum number of pairs of corresponding parts and the specific pairs of corresponding parts which must be congruent to ensure that triangles are congruent. Students begin by investigating the most basic cases and working towards the triangle congruence postulates. Students create examples and counterexamples as they develop and test triangle congruence conjectures. | | MP2. Reason abstractly and quantitatively. | Students determine both the minimum number of pairs of corresponding parts and the specific pairs of corresponding parts which must be congruent to ensure that triangles are congruent. Students create examples and counterexamples as they develop and test triangle congruence conjectures. | | MP3. Construct viable arguments and critique the reasoning of others. | Students are presented with examples and non-examples of terms to be defined. They determine the necessary attributes of the terms to be defined and compose definitions. They test the accuracy of their definitions by attempting to find counterexamples. Students determine both the minimum number of pairs of corresponding parts and the specific pairs of corresponding parts which must be congruent to ensure that triangles are congruent. Students begin by investigating the most basic cases and working towards the triangle congruence postulates. Students create examples and counterexamples as they develop and test triangle congruence conjectures. | | MP4. Model with mathematics. | Students determine both the minimum number of pairs of corresponding parts and the specific pairs of corresponding parts which must be congruent to ensure that triangles are congruent. Students begin by investigating the most basic cases and working towards the triangle congruence postulates. Students create examples and counterexamples as they develop and test triangle congruence conjectures. Students investigate creating a triangle given segments of any three lengths; they determine that it is only possible to create a triangle if sum of the lengths of any two sides of the triangle is greater than the length of the third side. | | MP5. Use appropriate tools strategically. | Students use a variety of tools, including transparencies and geometry software, in representing transformations in the plane. They make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). | | MP6. Attend to precision. | Students examine examples and non-examples of the terms and develop precise definitions for geometric terms. Students examine and analyze the created definitions looking for counterexamples. | | MP7. Look for and make use of structure. | Students use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. | | MP8. Look for and express regularity in repeated reasoning. | Students use the criteria for triangle congruence (ASA, SAS, and SSS) to justify formal construction techniques. | |
Focus/Driving Question:
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Why is it important to carefully define words? How does a mapping apply to the entire geometric shape? What algebraic rules can be generated to describe transformations? What properties are preserved by reflections? What properties are preserved by rotations? What properties are preserved by translations? How can triangles be shown to be congruent without verifying that each pair of corresponding angles and each pair of corresponding sides is congruent? How can rigid motions be used to explain geometric constructions? |
Student will Know:
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Terminology and notation associated with geometric objects Properties of isometries Construction methods to duplicate a segment, an angle, and a polygon Construction methods for perpendicular bisectors, midpoints, a perpendicular through a point to a line, a perpendicular through a point on a line, angle bisectors, and parallel lines Triangle congruence conjectures (SSS, SAS, ASA) |
Student will Do:
|
Identify and create translations, rotations, and reflections of figures in the plane Identify the characteristics of reflectional and rotational symmetry Apply concepts of reflectional, rotational, and translational symmetry Create algebraic representations of translations, reflections, and rotations Identify and create isometries that result from the composition of other isometries Distinguish among constructions, sketches, and drawings of geometric figures Use construction tools to duplicate a segment, an angle, and a polygon Construct perpendicular bisectors, midpoints, a perpendicular through a point to a line, a perpendicular through a point on a line, angle bisectors, and parallel lines Triangle congruence conjectures (SSS, SAS, ASA) |
Materials/Resources/Websites:
|
Lesson 1:
Chart paper and markers
Counterexamples
Examples and Non-Examples Lesson 2:
Reference materials for definitions and symbolic representation
Chart paper and markers
State or local map (optional)
Terms and Definitions
Geometry Glossary
Glossary Key Landmark Key Lesson 3:
Transformations on a Geoboard and Answer Key
Geometric Transformation Workout
Geometric Transformation Workout Answer Key
Powerpoint of Transformation Foldable
Additional Websites:
http://web.mnstate.edu/peil/geometry/c3transform/JavaSketch/0Intro.htm http://www.mathsisfun.com/geometry/transformations.html http://teachers.henrico.k12.va.us/math/GeoGebra_Site/ Lesson 4: Transparency paper Permanent markers
Powerpoint on M.C. Escher
Transformations in the Plane (Instructions Page)
Transformation Graph Paper
Transformations in the Plane (Notes Page) and Answer Key
Transformation Golf and Answer Key
http://www.mathsonline.co.uk/nonmembers/gamesroom/transform/golftrans.html Lesson 5:
Examples of Translation
Translations in the Plane (Part 1 and Part 2) and Answer Key
Translations in the Plane with GeoGebra and Answer Key
Additional Websites:
http://www.brightstorm.com/math/geometry/transformations/translations/
http://zunal.com/webquest.php?w=53929 Lesson 6:
Examples from the website: http://zunal.com/process.php?w=80350
Reflections in the Plane (Part 1 and Part 2) and Answer Key
Investigation of Lines of Symmetry (Answers are included on PowerPoint)
Reflections in the Plane with GeoGebra and Answer Key
Assessment of Learning Question
Additional Websites:
Annenberg Lesson on Symmetry
http://www.learner.org/courses/learningmath/geometry/session7/index.html
Illuminations Lesson on Symmetry
http://illuminations.nctm.org/LessonDetail.aspx?ID=L556
Lesson on Reflection
http://www.brightstorm.com/math/geometry/transformations/reflections/
Lesson on Reflectional Symmetry
http://www.brightstorm.com/math/geometry/transformations/reflectional-symmetry/
techsteps.com
Activity Library: “Reflections” and “Simple Reflections”
GeoGebra http://www.geogebra.org/cms/ Lesson 7:
Examples from the website: http://zunal.com/process.php?w=80350
Rotations in the Plane (Part 1 and Part 2) and Answer Key
Rotations in the Plane with GeoGebra and Answer Key
Assessment of Learning Question
Texas Instrument Lesson: What’s the Spin?
http://education.ti.com/calculators/downloads/US/Activities/Detail?id=1286 Lesson 8:
Dilations (Part 1 and Part 2)
Transformation Review Graphic Organizer
Algebraic Transformations and Answer Key
Additional Websites to explore dilations:
http://www.quadrivium.info/GGB/SimTri.html
http://www.quadrivium.info/GGB/SimQuad.html
http://www.quadrivium.info/GGB/SimPenta.html
Lesson 9:
Sets of six straws of lengths 2, 3, 4, 5, 5, 6 inches (It is suggested that different colors be used for each different length of straw.)
Glue
Construction paper
Markers
Patty paper Angle Diagrams Isometries
Triangle Congruence Conjecture Results
Triangle Congruence
Creating Congruent Triangles (Part 1)
Creating Congruent Triangles (Part 2)
Triangle Congruence Constructions
Additional Websites:
Geodesic Domes and Congruence http://www.teachersdomain.org/resource/phy03.sci.phys.mfw.bbgeodesic/
Congruent Triangles
http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=8516&MICROSITE=TI%20Math
Congruent Triangles
http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=8817&MICROSITE=ACTIVITYEXCHANGE
Sailing Away
http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=11064&MICROSITE=TI%20Math
Similar or Congruent?
http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=11063&MICROSITE=TI%20Math
Side-Side-Angle
http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5024&t=5056&id=13157
Corresponding Parts of Congruent http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=13159&MICROSITE=TI%20Math Lesson 10:
Rulers, protractors, and compasses
Patty paper
Bisectors – Perpendiculars
Constructions
Challenge Problem
Simple Constructions
http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=4005&MICROSITE=ACTIVITYEXCHANGE
Creating Perpendicular Bisectors http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=13170&MICROSITE=ACTIVITYEXCHANGE
Traditional Compass Construction: Angle Bisector Using Cabri Jr.
http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=7316&MICROSITE=ACTIVITYEXCHANGE
Traditional Copy an Angle: Compass Construction Using Cabri Jr. - Students use Cabri Jr. http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=7314&MICROSITE=ACTIVITYEXCHANGE Lesson 11:
Rulers, protractors, and compasses
Patty paper Triangle Segments
Triangle Construction
More Constructions
Math Open Reference (Constructions) provides animations of a wide variety of construction techniques as well as proofs of the validity of the construction technique.
http://www.mathopenref.com/tocs/constructionstoc.html Lesson 12:
Presentation display materials such as construction paper, glue, and markers
Logo Construction Directions and Rubric
Introductory Constructions Designs
Bisect an Angle – Directions and Justification Perpendicular Bisector – Directions and Justification
Parallel using Corresponding Angles – Directions and Justification
Perpendicular from a Point to Line – Directions and Justification
Perpendicular to a Point on a Line – Directions and Justification
Inscribed Triangle Inscribed Triangle – Alternate Version
Inscribed Square
Inscribed Hexagon Inscribed Hexagon - Alternate Version
GeoGebra http://www.geogebra.org/cms/ |
Assessment Plan:
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Each lesson identifies opportunities for formative assessment. Students are asked to apply and demonstrate their current level of understanding through a variety of performance tasks. Students create and notate illustrations demonstrating their mastery of vocabulary terms in a Draw Along Activity and as they create their glossaries. In Geometric Transformation Workout, students are asked to identify the translations and describe in words the transformation that maps a figure to other figures. They utilize translations, reflections, and rotations on a golf ball in the coordinate plane as they attempt to find the most efficient method (minimum number of transformations) to get the golf ball into the “hole” in Transformation Golf. Students are asked to apply properties of translations (Translations in the Plane (Part 2)), properties of reflections (Reflections in the Plane with GeoGebra), and properties of rotations (Rotations in the Plane with GeoGebra). In Assessment of Learning, students are asked to apply properties of rotations for 2-dimensional figures in the plane and on the coordinate plane. Algebraic Transformations is a culminating task that offers students an opportunity to demonstrate their understanding of various transformations from an algebraic perspective. Students are asked to create illustrations using appropriate notation of two triangles that are congruent by SSS, SAS, and ASA as they demonstrate an understanding of triangle congruence conjectures in a Draw Along Activity. Logo Construction provides students an opportunity to demonstrate their ability to combine and apply their constructions skills. Students are asked to create original constructions to inscribe equilateral triangles, squares and regular hexagons. Students are asked to justify constructions of a two of the following: parallel lines, a perpendicular from a point to a line, or a perpendicular to a point on the line. |
Major Products:
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Logo Constructions (Individual)
Your school club has decided to create a new logo. The possible formats have been narrowed to three options: an equilateral triangle inscribed in a circle; a square inscribed in a circle; or a hexagon inscribed in a circle. Construct each of the three possibilities. Choose your favorite of the three possibilities and include your design for the new logo. (Create this logo design on a second copy of the construction. This second copy can be created using GeoGebra construction tools or formal geometric construction techniques.)
Students also create clear, thorough and accurate justifications for two GeoGebra constructions, marking their diagrams to support their proof. Algebraic Transformations.(individual)
Students demonstrate in this culminating task their understanding of various transformations from an algebraic perspective. Unit Assessment Tasks
Additional assessment prompts or tasks have been compiled. A selection of these prompts/tasks can be compiled and used as a culminating assessment. These prompts/tasks may also be used throughout the unit to assess student learning. |
Unit Reflection:
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The formative assessment process requires teachers to reflect on the performance of each student during the unit of study by asking these questions: Have all students mastered the Next Generation standards targeted for this unit of study? Is it necessary to re-teach a concept to some members of the class while others benefit from an exercise that enriches or extends their learning during the unit? Teachers should also cause students to reflect upon their learning during the unit of study by having them reflect on questions such as: What have I learned? Are there concepts or skills I believe I need to continue to work with? We often neglect reflection, this very important stage in the learning process. By taking time to reflect upon where our students are in their learning, we can design the next unit of study to better meet their identified needs. |
| Tagged Next Generation Content Standards and Objectives |
| NxG ID |
NxG Objectives |
| M.1HS.CPC.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. (CCSS.Math.Content.HSG-CO.A.1) |
| M.1HS.CPC.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). (CCSS.Math.Content.HSG-CO.A.2) |
| M.1HS.CPC.3 |
given a rectangle, parallelogram, trapezoid or regular polygon, describe the rotations and reflections that carry it onto itself. (CCSS.Math.Content.HSG-CO.A.3) |
| M.1HS.CPC.4 |
develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments. (CCSS.Math.Content.HSG-CO.A.4) |
| M.1HS.CPC.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. (CCSS.Math.Content.HSG-CO.A.5) |
| M.1HS.CPC.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. (CCSS.Math.Content.HSG-CO.B.6) |
| M.1HS.CPC.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. (CCSS.Math.Content.HSG-CO.B.7) |
| M.1HS.CPC.8 |
explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (CCSS.Math.Content.HSG-CO.B.8) |
| M.1HS.CPC.9 |
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. (CCSS.Math.Content.HSG-CO.D.12) |
| M.1HS.CPC.10 |
construct an equilateral triangle, a square and a regular hexagon inscribed in a circle. (CCSS.Math.Content.HSG-CO.D.13) |
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| Files Uploaded |
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| Date Created: |
May 30, 2012 |
| Date Modified: |
August 06, 2012 |
Unit Plan Outline
(Lesson Plans) |
Lesson 1: Geometry Vocabulary
Lesson 2: Geometry Glossary
Lesson 3: An Introduction to Transformations
Lesson 4: Transformations Viewed Algebraically
Lesson 5: Properties Preserved by Translations
Lesson 6: Properties Preserved by Reflections
Lesson 7: Properties Preserved by Rotations
Lesson 8: Rigid versus Non-Rigid Motion
Lesson 9: Triangle Congruence Shortcuts
Lesson 10: Geometric Constructions (Part I)
Lesson 11: Geometric Constructions (Part 2)
Lesson 12: Geometric Constructions in a Circle
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Career Connections:
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To be well-equipped for the career options of this new century, understanding of transformational geometry will prove essential. What professions/jobs need this tool? Construction-based careers such as drafters, architects, and carpenters use transformational geometry when reflecting, translating, rotating, and varying the sizes of geometric shapes in the design of various structures. Bricklayers utilize transformational geometry to create symmetrical designs for walkways and patios. Landscape designers apply transformational geometry to diagram layout options. Robotic engineers use transformational geometry in programming robots to perform various tasks. Fabric, jewelry, and other designers use transformational geometry to create artistic patterns on a variety of products. Journalists use transformational geometry to display text, pictures, etc. in order to encourage the reader to focus on particular ideas. |
| Key Word Search Fields |
congruence, proof, construction
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