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Integrated 2 Unit 3: Triangle Properties and Proof Lesson Plan 5 weeks 2011-12 Use deductive reasoning to prove that a valid geometric statement is true. Valid proofs may be presented in paragraph, two-column, or flow-chart formats. Proof by contradiction is a form of deductive reasoning. Embed M2.3.B Identify errors or gaps in a mathematical argument and develop counterexamples to refute invalid statements about geometric relationships. M2.3.D Distinguish between definitions and undefined geometric terms and explain the role of definitions, undefined terms, postulates (axioms), and theorems. M2.3.E Know, explain, and apply basic postulates and theorems about triangles and the special lines, line segments, and rays associated with a triangle. M2.3.F Determine and prove triangle congruence (and similarity) and other properties of triangles. M2.3.G Know, prove, and apply the Pythagorean Theorem and its converse. M2.6.D Generalize a solution strategy for a single problem to a class of related problems, and apply a strategy for a class of related problems to solve specific problems. M2.6.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics. M2.6.G Synthesize information to draw conclusions and evaluate the arguments and conclusions of others. Sect and Std M2.3.A 3.1 T4 L1 M2.3.F Key ideas and Vocabulary parallel lines transversal vertical angles corresponding angles alternate interior congruent sum of angles of triangle Supplies for this unit: Patty Paper can be purchased at Cash and Carry or on line at www.eaieducation.com Resources T: Teacher Page S: 1 copy per student G: 1 copy per group Classwork Assessments C: 1 class set EX: Extensions I: Internet Resource Create a graphic organizer or reference page for student to use during the unit. What do you remember about Parallel Lines? G: 3.1 Memory (from Integrated 1 Unit 6) This is needed for the Triangle Sum Proof. There are 2 choices. 1. Game: Memory Angle Vocabulary Cards must be cut in advance. Have students lay all the cards facedown in an array. Take turns. Turn over two cards. If the picture matches with the term, they have a match. If the cards do not match, turn them back over. Summary – post a list of vocabulary. Have students write definitions for words they did not remember. 2. Put it All Together. Review assignment from Unit 6. Angle Vocabulary © Evergreen Public Schools 2010 9/27/11 doc pdf S: 3.1 Put it All Together doc pdf 3.1 Parallel Lines TICKET More Practice 3.1 Practice U6 L 1 doc pdf Note: U3 Foundations Assessment 3.1 Practice U6 L 2 should be done in doc pdf advance so you know what review Need Keys your students need. Page 1 of 8 Integrated 2 3.2 Unit 3: Triangle Properties and Proof Angle Sum of Triangle T4 L1 Why is the sum of the angles of a triangle 180? Conjecture: Students tear the angles of a triangle and put them together to show the angle sum is 180. Proof: As a class do a proof. Focus on reason and not form. Note: Students need these notes from section 6. Add problems solved algebraically. M2.3.A M2.3.E M2.6.E Lesson Plan T: 3.2 Teacher Notes C: 3.2 Triangle Sum Conjecture TASK CARD doc pdf S: 3.2 Triangle Sum Conjecture Cut Triangles (2 on a page) doc pdf 5 weeks 3.2 Triangle Sum TICKET 2011-12 3.2 Practice Angles in Triangles KUTA TPR Target Check 1 doc pdf 3.2 KEY 3.3 Exterior Angle TICKET 3.3 Practice 1 doc pdf 3.3 KEY 1 S: 3.2 Triangle Sum Conjecture STUDENT NOTES doc pdf 3.3 T3 L3 M2.3.E M2.6.E angles complementary supplementary vertical interior remote interior exterior perpendicular What is an exterior angle of a triangle? Triangle Investigation 1 T: 3.3Teacher Notes Triangle Exterior Angle Conjecture The lesson starts with students reviewing vertical, complementary and supplementary angles. Then use inductive reasoning to find the conjecture “The exterior angle of a triangle is equal to the sum of the two remote interior angles”. Note: It remains a conjecture until we prove it. Add problems solved algebraically. S: 3.3 Triangle Investigation doc pdf 3.3 Practice 2 doc pdf 3.3 KEY 2 3.3 Exterior Angle Theorem KUTA 3.3 KEY Exterior angle © Evergreen Public Schools 2010 9/27/11 Page 2 of 8 Unit 3: Triangle Properties and Proof Integrated 2 3.4 T3 L3 M2.3.E M2.6.E altitude congruent construct endpoint isosceles triangle perpendicular bisector equilateral triangle equiangular A. What is special about the angles of an isosceles triangle? Isosceles Triangle Conjecture 1) copy a segment 2) construct an isosceles triangle 3) make conjecture about angles of isosceles triangle. 4) identify altitude and perpendicular bisector in an isosceles triangle 5) apply conjecture 6) construct an equilateral triangle. B. Continue. . . And add special case of Equilateral Triangle. Lesson Plan Triangle Investigation 2 T: 3.4ATeacher Notes C: 3.4A Task Card doc pdf 5 weeks 2011-12 3.4A Isosceles 3.4 Isosceles and Triangle TICKET Equilateral Triangles KUTA TPR Target Check 2 3.4 KEY doc pdf I: Constructions: 3.4A Copy a segment pdf 3.4A Construct an Isosceles Triangle How to construct with a bullseye compass doc pdf 3.4A Construct Isosceles Triangle with standard compass pdf How to construct with bullseye compass 3.4B Construct an Equilateral Triangle doc pdf 3.4B Isosceles Triangle TICKET. How to construct with a standard compass 3.4B Construct Equilateral Triangle pdf 3.5 T3 L3 T4 L1 M2.3.F M2.6.E C: 3.5 Triangle Properties TEAM PRACTICE Team Practice Problem Solving: Triangle Properties doc doc pdf Use problems from KUTA software worksheets 3.5 KEY M2.3.A M2.3.E © Evergreen Public Schools 2010 pdf Concept Check T3 L3 9/27/11 Page 3 of 8 Unit 3: Triangle Properties and Proof Integrated 2 3.6 T4 L3 theorem flowchart M2.3.A M2.3.D M2.3.F M2.6.G 3.7 T4 L3 Lesson Plan T: 3.6 ppt pps What is a theorem? Introduce Flowchart proofs. They write a flowchart to make a peanut butter and jelly sandwich. You might want PB &J and knife for demonstration from one student’s flowchart. Do exactly what the flowchart says. T: 3.6 Triangle Sum TEACHER NOTES S: 3.6 Triangle Sum STUDENT NOTES doc exterior triangle theorem C: 3.7 Ext Angle Proof Parts Prove Exterior Angle Conjecture M2.3.A M2.3.D M2.3.F M2.6.G doc 3.8 T4 L3 M2.3.F M2.6.E congruent figures congruent triangles corresponding sides corresponding angles A. How do we know if two figures are congruent? Congruent Figures – students are asked to identify congruent figures. . Do not define congruent figures. Have patty paper available. TPF Target Check 1 doc pdf Gradebook Quiz T3 L3 3.7 doc pdf Concept Check T3 L3 3.7 continued doc 3.7 KEY pdf T: 3.8A ppt T: 3.8A Teacher Notes This should be review, but model a few before you send them home. 3.8A Information in Geometric Diagrams KUTA S: 3.8A Congruent Figures STUDENT doc pdf Where do we see congruent triangles? 3.8A KEY T: 3.8B Teacher Notes B. What makes a triangle congruent? Congruent Triangles – use protractors and rulers to measure sides and find congruent corresponding parts. © Evergreen Public Schools 2010 3.7 Exterior Angle Theorem KUTA (same as section 3.3) pdf Building the “Why?” -- see for photos 3.8 3.6 Practice Flowchart pdf S: 3.7 Ext Angle Thm STUDENT NOTES doc 3.6 Triangel Sum TICKET 2011-12 pdf T: 3.7 Teacher Notes How can we prove the Exterior Angle Conjecture? 5 weeks 9/27/11 3.8 Congruent Triangles TICKET 3.8B Congruence and Triangles KUTA S: 3.8B Congruent Triangles doc pdf 3.8BKEY Page 4 of 8 Integrated 2 3.9 T4 L3 M2.3.E M2.3.F M2.6.E Unit 3: Triangle Properties and Proof AA Similarity SSS SAS ASA AAS Lesson Plan Triangle Investigation 3 G: 3.9A Triangle Investigation TASK CARD A. How can we guarantee two triangles are congruent? AA, SSS, SAS Explorations doc 5 weeks 3.9A Congruent Triangles TICKET 3.9B Congruent Triangles TICKET 3.9B ASA and AAS Congruence KUTA Gradebook Quiz T3 L3 3.9B ASA and AAS Congruence KEY doc G: 3.9C Congruent Triangle TEAM PRACTICE C. Congruent Triangle TEAM PRACTICE doc KEY Look at edits in MTV groupshare pdf 3.9A SSS, and SAS Congruence KUTA 3.9A KEY pdf B. ASA, AAS, SSA Explorations 2011-12 pdf TPF Target Check 2 doc pdf T: 3.9C KEY 3.10 A. Extension: Level 4 OPTIONAL T4 L3 How can we prove SSS, etc? SSS and SAS are postulates Prove SAA and ASA using the postulates. This is best to do with direct instruction. Our book uses AA~ Postulate and the fact that congruent triangles have a scale factor of 1. But only proves AAS. M2.3.A M2.3.E M2.3.F M2.6.E M2.6.G 3.10 continued prove congruent triangles I: Done with proof by contradition 3.10 SSS, SAS, ASA KEY B. How can we prove two triangles are congruent? Proof Manipulatives T: 3.10B Teacher Notes In team, practice applying SSS, SAS, ASA and AAS. C: 3.10B Proof Manipulatives C. Continue… S: 3.10B Student Notes © Evergreen Public Schools 2010 9/27/11 3.10 SSS, SAS, ASA, and AAS Congruence KUTA doc Concept Check T4 L3 pdf Page 5 of 8 Unit 3: Triangle Properties and Proof Integrated 2 Lesson Plan doc doc 3.11 T4L3 M2.3.F M2.6.D parts of congruent triangles Corresponding parts of congruent triangles are congruent. (CPCTC) Launch: Read letter to Dr. Math. There are 13 proofs. DO NOT ASSIGN THEM ALL!!! This s a differentiation opportunity. 2011-12 pdf S: 3.10D Prove Congruent Triangle TEAM PRACTICE D. TEAM PRACTICE 5 weeks 3.10 Ticket pdf T: 3.11 CPCTC Teacher Model 3.11 Practice C: 3.11 Letter to Dr. Math 3.11 KEY doc doc pdf pdf C: 3.11 CPCTC Proofs doc pdf 3.11 KEY 3.12 T3L3 M2.3.E M2.6.E T3L4 M2.3.F construct perpendicular through point angle bisector perpendicular bisector median A. How can I draw accurately? S: 3.12A angle bisector task Perform basic constructions with compass and straight edge. These skills are needed to find the segments in triangles (next lesson). S: 3.12A perpendicular task doc doc pdf 3.12A Practice Constructions pdf S: 3.12A perp thru point (altitude) task doc pdf S: 3.12A median task doc pdf C: How to sheets from Math Open Reference 3.12A Construct angle bisect 3.12A Construct perp bisect 3.12A Construct perpendicular thru point 3.12A Construct medians I: www.mathopenref © Evergreen Public Schools 2010 9/27/11 Page 6 of 8 Integrated 2 Unit 3: Triangle Properties and Proof altitude angle bisector median perpendicular bisector B. Which triangle segments make 2 new congruent triangles? Students draw the 4 segments in scalene, obtuse, isosceles, equilateral and right triangles. They must prove the 2 new triangles are congruent. Lesson Plan T: 3.12B Teacher Notes S: 3.12B Altitude Invest doc pdf S: 3.12B Angle Bisector Invest doc pdf 5 weeks Gradebook Quiz T4 L3 doc 2011-12 3.12B Angle Bisectors of Triangles KUTA pdf 3.12B KEY Angle Bisect 3.12B Medians KUTA S: 3.12B Median Invest doc pdf 3.12B KEY Median S: 3.12B Perpendicular Bisector Invest Section C is on the next page doc pdf I: www.mathopenref.com I: www.mathopenref.com I Need keys! 3.12 continued orthocenter incenter centroid circumcenter C. Extension (Level 4): What is special about these segments? S: 3.12C Altitude Invest L4 Make conjectures about the intersections of all 3 segments (from yesterday). S: 3.12C Median Invest L4 doc pdf S: 3.12C Angle Bisector Invest L4 doc doc 3.12C Practice Centers of Triangles JMAP doc pdf pdf pdf S: 3.12C Perpendicular Bisector Invest L4 doc pdf C: Instructions from Math Open Reference 3.12C Construct Orthocenter 3.12C Construct Incenter 3.12C Construct Circumcenter 3.12C Construct Centroid Need keys! They can be done by hand and I will scan them. © Evergreen Public Schools 2010 9/27/11 Page 7 of 8 Integrated 2 Unit 3: Triangle Properties and Proof 3.13 T4L4 M2.3.F M2.6.G Why are the base angles of an isosceles triangle congruent? There are 3 proofs Level 3 with median or angle bisector Level 4 with altitude Hint cards are available for all 3 proofs. You can have half to one-third of the pairs doing one of the proof methods. Have pairs show their method to another pair. Lesson Plan T: 3.13 Isosceles Triangle Thm Proof TEACHER NOTES Can I Do This? 15 Targeted Review 16 Assessment 2011-12 3.13 Angles in Triangles KUTA 3.13 KEY T: 3.13 clue card organizer S: 3.13 Isosceles Triangle Thm Proof STUDENT NOTES -each student gets one of the 2 (or 3) proof pages. doc 14 5 weeks pdf Unit 4 Foundations Int2 U3 T3 2010 Assmt Form A doc pdf Int2 U3 T3 2010 Assmt Form B doc pdf Int2 U3 T4 2010 Assmt Form A doc pdf Int2 U3 T4 2010 Assmt Form B doc pdf Consider letting students use a list of theorems and properties © Evergreen Public Schools 2010 9/27/11 Page 8 of 8