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Youngstown City Schools MATH: GEOMETRY Unit 1C: THEOREMS ABOUT LINES AND ANGLES (2 WEEKS) 2013-2014 SYNOPSIS: In this unit, students learn about theorems and proofs as they relate to angles and lines. Students learn the technology as well as the hands-on methods for solving problems that involve real-world applications of the concepts in the unit. Students apply their learning through taking a bridge or a section of city streets and addressing the concepts from the unit and explaining each. STANDARDS G.CO.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. 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.6 Learn to read mathematical text - - problems and explanations Communicate using correct mathematical terminology Represent and interpret data with and without technology MOTIVATION TEACHER NOTES 1. Teacher sets up connection of this unit to the previous one on use of Theorems, moving from congruent triangles to theorems about lines and angles. 2. Teacher gives some type of pre-assessment on vocabulary terms to be sure students have understanding of terms needed for the unit. 3. Preview expectations for end of Unit. 4. Have students set both personal and academic goals for this Unit or grading period. TEACHING-LEARNING Vocabulary Terms: bisector perpendicular equidistant supplementary angles congruent vertical angles parallel corresponding angles TEACHER NOTES transversal line segment complementary angles alternate interior angles right angles alternate exterior angles Teachers: if needed, take one day to focus on teaching students how to use the Geometer SketchPad or the TI-N-spire technology 6/30/2013 YCS Geometry: Unit 1C: Theorems About Lines and Angles 2013-2014 1 TEACHING-LEARNING TEACHER NOTES All Teaching Learning Activities refer to G. CO.9 1. Through self-discovery or teacher explanation inform students what happens when parallel lines are cut by a transversal; special Theorems are created because of these relationships (use Geometer Sketchpad or Geogebra). (MP.5, MP.7, MP.8, L.2) 2. Teacher models a transversal on the board, and then provides students with handouts containing parallel lines; students are to cut the lines with a transversal that is NOT perpendicular to the parallel lines and label the angles formed. Students then measure and record each angle measure to discover the relationship among the angles. Students may use a protractor or the TI-Nspire geometry graphing program. (MP.2, MP.4, MP.5, MP.6, MP.7, MP.8, .2) 3. From previous activities, the teacher helps students label the angles as either congruent (alternate interior, alternate exterior, vertical angles, corresponding angles) or supplementary angles (consecutive interior, consecutive exterior, or adjacent angles). (MP.4, MP.5, MP.6, MP.7, MP.8, L.2) 4. The teacher provides students with sample problems (pp. 127-137 from text or supplementals), stressing “real-world” applications of various shapes in the sample problems; students practice “angle” and “relationship” skills; students supply missing information, justify answers, and use algebraic expressions to represent missing information. Be sure to balance use of technology with hands-on solving. (MP.2, MP.4, MP.5, MP.6, MP.7, MP.8, L.2) 5. Teacher demonstrates proving theorems about parallel lines and transversals using a two-column proof method (e.g., p. 152 of text). Students take notes; teacher provides practice, giving students the “given” and “what’s to be proven,” and students work with the teacher to complete the proof. (MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.6) 6. Students work in 2’s to complete proofs for other problems-situations using 2-column method to be sure the students understand what they are doing. (MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.6) 7. Next the teacher models the narrative method for completing a proof, and finally moves to the flow diagram method. This is done using demonstration problems from Activity #6, so students see how the various methods work. Students work one problem, each, using the narrative and flow chart methods. (MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.6) 8. Teacher provides real-world problem for the idea of a line segment with a bisector. Tampa Bay Bridge: http://player.discoveryeducation.com/index.cfm?guidAssetId=6566F746-4FE4-4759-A15D5BA007CCE5C4&blnFromSearch=1&productcode=US. Teacher does demonstration of perpendicular bisector of a line segment to show students that when this happens the two parts of the line segment are equal. Students then use a protractor as a tool to prove each point on the two halves of the line segment are equidistant from the bisector. (MP.2, MP.3, MP.4, MP.5, MP.7, MP.8, L.2, L.6) 9. Teacher and students use SAS to prove that points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints: e.g., S → to show that the 2 pieces of the line segment cut by the perpendicular bisector are congruent; A → to show that the angles formed by the perpendicular bisector are congruent right angles; S → to show that the perpendicular bisector creates a side for each of the triangles formed and that they are congruent. (MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.6) 6/30/2013 YCS Geometry: Unit 1C: Theorems About Lines and Angles 2013-2014 2 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 AUTHENTIC ASSESSMENT 1. TEACHER NOTES Students evaluate goals they set at beginning of unit or on a weekly basis 2. Sunshine Skyway Bridge in St. Petersburg, FL with United Streaming Video 3. My City of Angles – where students use concepts in the unit to create angles, lines, etc. as per a given set of instructions below and on page 4 of Unit Plan. (G.CO.9, MP.1, MP.4, MP.5, MP.6, MP.8. L.1, L.2) AUTHENTIC ASSESSMENT: MY CITY OF ANGLES Choose a city: 1. 2. 3. 4. Find a map of your city on the internet, Zoom in until the streets are visible, Look for a section of town that has some parallel streets intersected by transversals, at least a total of 8 streets. Chose an area where not all transversals are perpendicular to the parallel streets, at least one needs to be slanted at an angle different than 900 and at least one needs to be perpendicular. 5. Print this area of town. 6. On a piece of graph paper with a straight-edge, draw this section of city. 7. Label intersection points with capital letters and number angles. On your City Map Section - - List and identify the following: 4 pairs of vertical angles 4 pairs of corresponding angles 4 pairs of alternate interior angles 4 pairs of supplementary angles 4 pairs of complementary angles 4 pairs of alternate exterior angles Use a protractor to find the angle measurements of all angles and show them on your drawing. Create and answer a question to solve for a missing value (x) using alternate interior angle or same side interior angle relationship between the angles in the city. 6/30/2013 YCS Geometry: Unit 1C: Theorems About Lines and Angles 2013-2014 3 ELEMENTS OF THE PROJECT Printed area of city that has parallel streets intersected by transversals ( at least 8 streets) that are both slanted and perpendicular CITY SECTION RUBRIC 2 0 1 3 4 Did not attempt Used section with12 streets; had streets intersected by either perpendicular or slanted transversal Did not use straightedge, drawing; does not resemble section of town Labeled only a few of the angles and points of intersection Correctly listed 1 pair of vertical angles Used 3-5 streets; had streets intersected by either perpendicular or slanted transversal Used section with 57 streets; a perpendicular or slanted transversal NA NA At least 8 streets, at least one perpendicular, at least on intersecting at an angle different than 900 Used straight-edge, drawn in likeness to the selection of town Labeled half of the angles and points of intersection Correctly listed 2 pairs of vertical angles Labeled most of the angles and points of intersection Correctly listed 3 pairs of vertical angles All points and angles are labeled correctly Correctly listed 1 pair of corresponding angles Correctly listed 1 pair of alternate interior angles Correctly listed 2 pairs of corresponding angles Correctly listed 3 pairs of corresponding angles Correctly listed 3 pairs of alternate interior angles Correctly listed 4 pairs of corresponding angles Correctly listed 4 pairs of alternate interior angles Correctly listed 1 pair of supplementary angles Correctly listed 1 pairs of complementary angles Correctly listed 2 pairs of supplementary angles Correctly listed 3 pairs of supplementary angles Correctly listed 3 pairs of complementary angles Correctly listed 4 pairs of supplementary angles Correctly listed 4 pairs of complementary angles Listed 4 pairs of alternate exterior angles Did not Correctly listed 1 attempt or pair of alternate all were exterior angles incorrect Correctly listed 2 pairs of alternate exterior angles Correctly listed 3 pairs of alternate exterior angles Correctly listed 4 pairs of alternate exterior angles Find measures of all angles on drawing Did not attempt or all were incorrect Did not attempt Found measures of 1-2 angles on the drawing Found measures of 35 angles on the drawing Found measures of 6-7 angles on the drawing Found measures of 8 angles on the drawing Created problem, did not find the solution Created problem, but solution was incorrect NA Created problem and found correct solution Prepared drawing of the section of city with straightedge Did not attempt Labeled intersection points with letters and angles with numbers Listed 4 pairs of vertical angles Did not attempt Listed 4 pairs of corresponding angles Listed 4 pairs of alternate interior angles Listed 4 pairs of supplementary angles Listed 4 pairs of complementary angles Created problem and found solution 6/30/2013 Did not attempt or all were incorrect Did not attempt or all were incorrect Did not attempt or all were incorrect Did not attempt or all were incorrect Did not attempt or all were incorrect Correctly listed 2 pairs of alternate interior angles Correctly listed 2 pairs of complementary angles Correctly listed 4 pairs of vertical angles YCS Geometry: Unit 1C: Theorems About Lines and Angles 2013-2014 4