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Youngstown City Schools MATH: GEOMETRY Unit 1B: PROVING THE CONGURENCE OF TRIANGLES (2 WEEKS) 2013-2014 SYNOPSIS: In this unit, students work with the Theorem of congruence as it relates to triangles. Students find real-life examples of congruence in their lives, and explain why they meet the definition of congruence in terms of angles and sides. STANDARDS G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. MATH PRACTICES: 1. 2. 3. 4. 5. 6. 7. 8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning LITERACY STANDARDS L.1 L.2 L.4 L.5 L.8 Learn to read mathematical text - - problems and explanations Communicate using correct mathematical terminology Listen to and critique peer explanations of reasoning Justify orally and in writing mathematical reasoning Read appropriate text, providing explanations for mathematics concepts, reasoning or procedures MOTIVATION TEACHER NOTES 1. Teacher shows students pictures that will be similar to those required in the Authentic Assessment (e.g., flukes on whales; butterfly wings; a person with fists on hips, etc.); asks students if they see two congruent triangles, demonstrates how to identify these on samples. 2. To introduce congruent triangles teachers can engage students in a discussion of congruent and not congruent triangles or use clip from Flatland or United Streaming on congruent triangles below (G.CO.6, 7, 8) http://player.discoveryeducation.com/index.cfm?guidAssetId=A38D422E-90C9-4758B1C5-3B6A8FE1835F&blnFromSearch=1&productcode=US 3. Preview expectations for end of Unit. 4. Have students set both personal and academic goals for this Unit or grade period. TEACHING-LEARNING 6/30/2013 TEACHER NOTES YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 1 TEACHING-LEARNING Vocabulary: Pythagorean theorem Corresponding sides SSA TEACHER NOTES Congruent Corresponding angles SAS ASA SSS postulate 1. Teacher demonstrates and defines congruent triangles by showing students two triangles from the prior unit (remind students that when they did the 3 transformations, the segments remained the same) modeling for them to see how they are congruent. Ask students what they notice about the triangles, and have them describe what does not change. Next, have students draw a trapezoid, and then do the transformations. Students then make a stencil of the original, and place atop the transformations. Teacher may use TI-Nspire activity for this, but must be connected to triangles from previous unit. They can also use sketchpad geogebra or other hands-on activity. http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5024&t=5056&id=13159 (G.CO.6) (L.2; L.5) (MP-2; MP-7) 2. Teacher leads discussion about the definition of two congruent triangles and their corresponding angles and sides through transformation (vocabulary). Students do examples with teacher including using the distance formula to show sides congruent and then individually (e.g., whiteboards, math notebook) with different triangles. Use real-life examples such as the spokes in a bicycle, landscaping, architecture, etc. for students to work. Have student write angles and sides that are congruent, using the appropriate notation (). The students write these to show congruence of several pairs of triangles. Make sure students understand that they have to identify corresponding sides and angles. They also need to solve algebraic problems dealing with corresponding sides and angles, including one and two variable equations. (G.CO.6, 7) (L.8) (MP-8) 3. Teacher explains that we need not measure all 6 parts to know if triangles are congruent; the “Theorem” was devised to determine congruence (ASA, SAS, and SSS). (G.CO.8) (L-2; L-5) (MP-4; MP-5) 4. Teacher give students additional examples with some that are not congruent and have students “prove” the congruence (or not). (L-1; L-5) (MP-1; MP-5) TRADITIONAL ASSESSMENT TEACHER NOTES 1. Paper-pencil test with M-C questions and 2-and 4-point questions TEACHER CLASSROOM ASSESSMENT TEACHER NOTES 1. Quizzes 2. In-class participation and practice problems for each concept 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 2 AUTHENTIC ASSESSMENT TEACHER NOTES 1. Students evaluate goals they set at beginning of unit or on a weekly basis. 2. Each student will be given a “Student Test” to score. The objective is to find if there are errors, and provide an explanation of what the errors were. On pages 5-12 of this Unit, you will find 4 different “student tests.” Your Geometry students are to review and grade one of the 4 tests to see where the “student who took that test” may have made errors. Answers are on pages 13-17. For each of the 4 Tests, the “student taking the test” made 12 errors. As the evaluator of this test, student will earn 2 points for each error found and one point for each error explained as to what was done incorrectly in the work on the “test.” (G.CO.7, G.CO.8,L.2; L.4; L.5) 3. Each student will: find real-world pictures that illustrate Unit concepts; students “draw” the triangles on the picture and then use a theorem to prove the congruence; students must justify orally and in writing why they chose a theorem and why it works for the picture chosen (page 18). (G.CO.6, G.CO.7, G.CO.8, L.5; L.8) . . 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 3 GEOMETRY UNIT 1B AUTHENTIC ASSESSMENT #1 CONGRUENT TRIANGLES (Standards: G.CO.7 and G.CO.8) DIRECTIONS FOR THE ASSESSMENT: Each student will be given a “Student Test” to score. The objective is to find if there are errors, and provide an explanation of what the errors were. On the pages 5-12, you will find 4 different “student tests.” Your Geometry students are to review and grade one of the 4 tests to see where the “student who took that test” may have made errors. Answers are on pages 13-17 For each of the 4 Tests, the “student taking the test” made 12 errors. As the evaluator of this test, student will earn 2 points for each error found and one point for each error explained as to what was done incorrectly in the work on the “test.” 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 4 AUTHENTIC ASSESSMENT GEOMETRY UNIT 1B STUDENT TEST #1: Analyze the 6 proofs for correctness, including (a) if the labels are correct and the definitions and/or theorems are incorrect; (b) correct theorems and/or definitions, but incorrect labels. Correct the errors and write a statement about the corrections and errors for each problem. Then choose one problem to orally defend. 1. Prove: ∆DAB ∆BCD A 1. <ADB <CBD <CDB <ABD 2. BD BD 3. ∆DAB ∆BCD B 1. Given 2. Reflexive 3. AAS D 2. Prove: ∆RVT C ∆RST 1. RV SR, ST 2. RT RT 3. ∆RVT ∆STR R VT 1. Given 2. Reflexive 3. SSS V S T 3. Prove: ∆DBW ∆TYW 1. DW WT, WB <DWB <TWY 2. ∆DBW ∆TYW D WY, 1. Given W 2. ASA B 6/30/2013 T Y YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 5 (Student Test #1 continued) 4. Prove: ∆BSE ∆USL (note: EL and UB intersect at S) 1. <ESB <EBS 2. ∆BSE <USL, BS <LSU ∆USL SU E U 1. Given S SSS 2. ASA B 5. Prove: ∆JOS L ∆HPE 1. JO PH, OS PE, 1. Given JO is perpendicular to PO and HP is perpendicular to PO 2. <O <P 2. All right angles are 3. ∆JOS ∆HPE 3. SSS J E O P S H 6. Prove: ∆PIK ∆NIK K 1. <PKI <NKI, 1. KI is perpendicular to PN 2. KI KI 2. 3. <KIP <NIK 3. 4. ∆KPI ∆KNI 4. Given Reflexive All right angles are congruent SAS P I N Scoring per problem: Two points for each correction, 2 points for each written explanation. Oral defense of one problem: 3 points 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 6 AUTHENTIC ASSESSMENT GEOMETRY UNIT 1B STUDENT TEST #2: Analyze the 6 proofs for correctness, including (a) if the labels are correct and the definitions and/or theorems are incorrect; (b) correct theorems and/or definitions, but incorrect labels. Correct the errors and write a statement about the corrections and errors for each problem. Then choose one problem to orally defend. 1. Prove: ∆DBA ∆BDC 1. AD <CB <CBD <ADB 2. BD BD 3. ∆DAB ∆BCD 2. Prove: ∆RVT A B D C 1. Given 2. Reflexive 3. SSS ∆RST 1. RV SR, ST 2. RT RT 3. ∆RVT ∆RST R VT 1. Given 2. Transitive 3. SAS V S T 3. Prove: ∆DBW ∆TYW 1. DW WT, WB <DWB <TWY 2. ∆DBW ∆WTY D WY, 1. Given T W 2. ASA Y B (Student Test #2 continued) 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 7 4. Prove: ∆BSE 1. <ESB <EBS 2. ∆BSE ∆USL (note: EL and UB intersect at S) <USL, BS <LUS ∆USL US E U 1. Given S SSS 2. SSA B 5. Prove: ∆JOS L ∆HPE 1. JO HP, OS SE, SE JO is perpendicular to OP HP is perpendicular to OP 2. <O <P 3. OS PE 4. ∆JOS ∆HEP PE 1. Given J E 2. All right angles are 3. Reflexive 4. SAS O P S H 6. Prove: ∆PIK ∆NIK K 1. <PKI <NKI, 1. KI is perpendicular to PN 2. KI IK 2. 3. <KIP <KIN 3. 4. ∆KPI ∆KNI 4. Given Transitive All right angles are congruent ASA P I N Scoring per problem: Two points for each correction, 2 points for each written explanation. Oral defense of one problem: 3 points 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 8 AUTHENTIC ASSESSMENT GEOMETRY UNIT 1B STUDENT TEST #3: Analyze the 6 proofs for correctness, including (a) if the labels are correct and the definitions and/or theorems are incorrect; (b) correct theorems and/or definitions, but incorrect labels. Correct the errors and write a statement about the corrections and errors for each problem. Then choose one problem to orally defend. E 1. Prove: ∆ABE ∆DCE 1. <A <D <ABE <DCE AB BC, BC DC 2. AB DC 3. ∆ABE ∆ECD 1. Given 2. Reflexive 3. ASA A 2. Prove: ∆GAP ∆PSG P 1. SG AP, SP AG 2. GP GP 3. ∆GAP ∆PSG B C D S 1. Given 2. Reflexive 3. SAS G A G 3. Prove: ∆DGO ∆ CTA 1. <D <C, <O <A DO CA 2. ∆DGO ∆ ATC A 6/30/2013 T 1. Given 2. SAS D O C YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 9 (Student Test #3 continued) 4. Prove: ∆SRU ∆ STU 1. <RSU <UST RS TS 2. SU SU 3. ∆SRU ∆ STU R 1. Given 2. Reflexive 3. AAA S U T J 5. Prove: ∆JEF ∆ JIF 1. <EJF <FJI, EJ 2. JF FJ 3. ∆JEF ∆ JIF IJ 1. Given 2. Reflexive 3. SAS E F X 6. Prove: ∆RXS ∆ UTW 1. XS is perpendicular to RT WT is perpendicular to XU <R <U RS UW 2. <RSX <TWU 3. ∆RXS ∆ TWU I W U 1. Given R S 2. All right angles are congruent 3. ASA T Scoring per problem: Two points for each correction, 2 points for each written explanation. Oral defense of one problem: 3 points 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 10 AUTHENTIC ASSESSMENT GEOMETRY UNIT 1B STUDENT TEST #4: Analyze the 6 proofs for correctness, including (a) if the labels are correct and the definitions and/or theorems are incorrect; (b) correct theorems and/or definitions, but incorrect labels. Correct the errors and write a statement about the corrections and errors for each problem. Then choose one problem to orally defend. E 1. Prove: ∆ABE ∆DCE 1. <A <D <ABE <DCE AB BC, BC DC 2. AB CD 3. ∆ABE ∆DCE 1. Given 2. Transitive 3. AAS A 2. Prove: ∆GAP ∆PSG P 1. SG AP, SP AG 2. GP GP 3. ∆GAP ∆PSG B C D S 1. Given 2. Reflexive 3. SAS G A G 3. Prove: ∆DGO ∆ CTA 1. <D <C, <O <A DO AC 2. ∆DGO ∆ ATC A 6/30/2013 T 1. Given 2. ASA D O C YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 11 (Student Test #4 continued) 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 12 4. Prove: ∆SRU ∆ STU 1. <RSU <TSU RS ST 2. SU SU 3. ∆SRU ∆ STU R 1. Given 2. Transitive 3. SAS S U T J 5. Prove: ∆JEF ∆ JIF 1. <EJF <IJF, EJ 2. JF FJ 3. ∆JEF ∆ JFI IJ 1. Given 2. Reflexive 3. SAS E F X 6. Prove: ∆RXS ∆ UTW 1. XS is perpendicular to RT WT is perpendicular to XU <R <U RS WU 2. <RSX <UWT 3. ∆RXS ∆ UTW I W U 1. Given R S 2. All right angles are congruent 3. SAS T Scoring per problem: Two points for each correction, 2 points for each written explanation. Oral defense of one problem: 3 points 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 13 ANSWERS TO GEOMETRY UNIT 1B AUTHENTIC ASSESSMENT #1 Student Test #1 1. Step #2: DB BD (letters are to be in the reverse order) Step #3: ASA Explanations: The triangles are set up in the prove statement such that D→B, A→C, and B→D so DB→BD. ASA because side is included between the two angles. 2. Step #1: RV RS Step #3: ∆RVT ∆RST Explanations: The triangles are set up in the prove statement such that R→R, V→S, and T→T so RV→RS (R goes with R not S). ∆RVT → ∆RST because V→S, not T and T→T, not R 3. Step #1: DW TW Step #2: SAS Explanations: The triangles are set up in the prove statement such that D→T, W→W, and B→Y so DW→TW. SAS because the angle is included between the two sides and there are two pairs of sides and one pair of angles 4. Step #1: <ESB <LSU, BS US Step #3: SAS Explanations: The triangles are set up in the prove statement such that E→L, S→S, and B→U so <ESB→<LSU and BS . 5. Step #1: JO HP Step #3: ∆JOS ∆HPE Explanations: The triangles are set up in the prove statement such that J→H, O→P, and S→E so JO→HP. SAS because there is one pair of angles and two pairs of sides congruent and the angle is included between the two sides. 6. 6/30/2013 Step #3: <KIP <KIN Step #4: ASA Explanations: The triangles are set up in the prove statement such that P→N, I→I, and K→K so <KIP→<KIN. ASA because the side is included between the two angles and there are two pairs of angles and one pair of sides. YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 14 Student Test #2: 1. Step #2: BD DB (letters are to be reversed) Step #3: SAS Explanations: The triangles are set up in the prove statement such that D→B, B→D, and A→C so BD→DB. SAS because the angle is included between the two sides and there are two pairs of sides and one pair of angles congruent. 2. Step #1: RV RS Step #2: Reflexive Explanations: The triangles are set up in the prove statement such that R→R, V→S, and T→T so RV→RS (R goes with R not S. V goes with S). Transitive states that a first quantity is congruent to a second quantity and a second quantity is congruent to a third quantity, then the first quantity is congruent to the third quantity. We do not have that here. Reflexive states a quantity is congruent to itself, which is what this step states. 3. Step #1: DW TW Step #2: SAS Explanations: The triangles are set up in the prove statement such that D→T, W→W, and B→Y so DW→TW. SAS because the angle is included between the two sides and there are two pairs of sides and one pair of angles 4. Step #1: <ESB <LSU Step #3: ASA Explanations: The triangles are set up in the prove statement such that E→L, S→S, and B→U so <ESB→<LSU. ASA because there are two pairs of angles and one pair of sides congruent and the side is included between the angles. Also SSA is not a method of proving triangles congruent. 5. Step #3: Transitive Step #4: ∆JOS ∆HPE Explanations: Transitive states that a first quantity is congruent to a second quantity and a second quantity is congruent to a third quantity, then the first quantity is congruent to the third quantity, which is what we have here. Reflexive states a quantity is congruent to itself, which we do not have that here. The triangles are set up in the prove statement such that J→H, O→P, and S→E so ∆JOS→ ∆HPE. 6. Step #2: KI <KI Step #2: Reflexive Explanations: The triangles are set up in the prove statement such that P→N, I→I, and K→K so KI→KI. Transitive states that a first quantity is congruent to a second quantity and a second quantity is congruent to a third quantity, then the first quantity is congruent to the third quantity, which is not what we have here. Reflexive states a quantity is congruent to itself, which we do have here. 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 15 Student Test #3: 1. Step #2: Transitive Step #3: ∆ABE ∆DCE Explanations: Transitive states that a first quantity is congruent to a second quantity and a second quantity is congruent to a third quantity, then the first quantity is congruent to the third quantity. We have that here. Reflexive states a quantity is congruent to itself, which is not what this step states. The triangles are set up in the prove statement such that A→D, B→C, and E→E so ∆ABE→ ∆DCE. 2. Step #2: GP PG Step #3: SSS Explanations: The triangles are set up in the prove statement such that G→P, A→S, and P→G so GP→PG (letters are to be reversed). There are three pairs of sides congruent not two pairs of sides and a pair of angles. 3. Step #2: ∆DGO ∆ CTA Step #2: ASA Explanations: The triangles are set up in the prove statement such that D→C, G→T, and O→A so ∆DGO→ ∆ CTA. ASA because the side is included between the two angles and there are two pairs of angles and one pair of sides. 4. Step #1: <RSU <TSU Step #3: SAS Explanations: The triangles are set up in the prove statement such that R→T, S→S, and U→U so <RSU→<TSU. SAS because there are two pairs of sides and one pair of angles congruent and the angle is included between the two sides. Also AAA is not a method of proving triangles congruent. 5. Step #1: <EJF < IJF Step #2: JF JF Explanations: The triangles are set up in the prove statement such that J→J, E→I, and F→F so <EJF < IJF in step #1 and also JF→ JF in step #2. SAS because there are two pairs of sides and one pair of included angles congruent, not two pairs of angles and a pair of sides. 6. Step #2: <RSX <UWT Step #3: ∆RXS ∆ UTW Explanations: The triangles are set up in the prove statement such that R→U, X→T, and S→W so <RSX→ <UWT, also when naming the triangles this correspondence is preserved so ∆RXS ∆ UTW. 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 16 Student Test #4: 1. Step #2: AB DC Step #3: ASA Explanations: The triangles are set up in the prove statement such that A→D, B→C, and E→E so AB→ DC. ASA because there are two pairs of angles and the pair of included sides congruent and AAS is not a way we learned to prove triangles congruent. 2. Step #2: GP PG Step #3: SSS Explanations: The triangles are set up in the prove statement such that G→P, A→S, and P→G so GP→PG (letters are to be reversed). There are three pairs of sides congruent not two pairs of sides and a pair of angles. 3. Step #1: DO CA Step #2: ∆DGO ∆ CTA Explanations: The triangles are set up in the prove statement such that D→C, G→T, and O→A, so DO→ CA and also ∆DGO ∆ CTA. 4. Step #1: RS TS Step #2: Reflexive Explanations: The triangles are set up in the prove statement such that R→T, S→S, and U→U so RS→TS. Transitive states that a first quantity is congruent to a second quantity and a second quantity is congruent to a third quantity, then the first quantity is congruent to the third quantity, which is not what we have here. Reflexive states a quantity is congruent to itself, which we do have here. 5. Step #2: JF JF Step #3: ∆JEF ∆ JIF Explanations: The triangles are set up in the prove statement such that J→J, E→I, and F→F so JF→ JF and also with ∆JEF ∆ JIF. 6. Step #1: RS UW Step #3: ASA Explanations: The triangles are set up in the prove statement such that R→U, X→T, and S→W so RS→UW. ASA because there are two pairs of angles and one pair of included sides congruent, not two pairs of sides and one pair of included angles congruent. 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 17 6/30/2013 YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 18 Geometry Unit 1B Authentic Assessment #2 Standards: G.CO.6, G.CO.7, G.CO.8 Find three real-world pictures that illustrate two congruent triangles (e.g., flukes of whale’s tail; wings of butterfly, etc.); draw the triangles on the pictures and then use a theorem to prove the congruence. Justify orally and in writing why the particular theorem was chosen and why it works for the picture. ELEMENTS OF PROJECT Submit pictures 0 No pictures submitted Draw congruent Triangles not triangles on congruent picture Written proof Use incorrect theorems on all three Oral Proof Use incorrect theorems on all three 6/30/2013 Rubric 1 2 3 1 picture submitted Congruent on one drawing 2 pictures submitted Congruent on two drawings 3 pictures submitted Congruent on three drawings Correct theorems on one proof Correct theorems on one proof Correct theorems on two proofs Correct theorems on two proofs Correct theorems on three proofs Correct theorems on three proofs YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14 19