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HCPSS Worthwhile Math Task Sailing with Congruency Common Core Standard G.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. Note: This activity focuses on perpendicular bisectors of segments. Students will need to use of theorems related to proving triangles congruent. MP1: MP3: MP5: MP8: Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Use appropriate tools strategically. Look for and express regularity in repeated reasoning. Common Core Geometry, Unit 2 The Task The world’s largest yacht and sailboat manufacturer have hired your team. The company wants your team to investigate ways to save time in the manufacturing process. A sailboat uses guy-wires that attach the deck of the boat and the boat’s mast. (See the diagram below – the arrows and numbers refer to the boats guy-wires) The guy-wires provide additional strength to the mast to withstand the forces the mast will encounter in severe winds. During the investigation, your team notices that workers measure each guy-wire individually before installing them on to the boat. The process of having two workers measure an individual guy-wire and install it takes 15 minutes per wire. Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. HCPSS Worthwhile Math Task http://upload.wikimedia.org/wikipedia/commons/1/18/Schema-greement-TETE.png In your analysis, your team recommends that measuring once and cutting two guy-wires could save 5 minutes, since the mast is the perpendicular bisector of a segment on the deck and the guy-wires are segment connecting a point on the perpendicular bisector of a segment to the endpoints of the segment. The company’s management agrees with your team, but before making the change in their manufacturing process, they want a proof that two guy-wires can be cut each time using a single measurement. As a group, create a proof to convince the company’s management that your suggestion is valid. Sample Diagram Mast Deck Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. HCPSS Worthwhile Math Task Facilitator Notes 1. Place students in groups of three. Have graph paper, patty paper, or dynamic geometry software available. (Look for evidence of MP1 and MP5.) 2. Present the task. Have students work through all parts of the task. Provide adequate time for students to complete the task. (Look for evidence of MP1 and MP8.) 3. Jigsaw students into different groups. Each member must show the convincing argument for the guy-wire having equal lengths. Have the other members of the new group rate the argument (scale 1-5) for its ability to convince them of the correctness. (Look for evidence of MP3.) 4. Have students return to their original group. Ask groups to tally their scores. Recognize the high scoring group. Allow groups with various scores to explain their method. Ask for alternate approaches. Allow students to defend their group’s argument. (Look for evidence of MP1 and MP3.) Follow-Up Questions 1. Where else do you see guy-wires? (Look for evidence of MP8.) 2. Would all pairs of guy-wires have to be congruent? (Look for evidence of MP8.) Solutions Possible Proof: Mast D C A B Deck Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. HCPSS Worthwhile Math Task Statements _____ _____ DC is the perpendicular bisector of AB Reasons Given _____ _____ Definition of Perpendicular Bisector _____ _____ Reflexive Property AC BC DC DC ACD BCD DCA DCB _____ _____ AD BD Definition of Perpendicular Bisector Side Angle Side Triangle Congruence (SAS) Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.