Distance, Midpoint, and Pythagorean Theorem

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1 Geometry, Quarter 1, Unit 1.1 Distance, Midpoint, and Pythagorean Theorem Overview Number of instructional days: 8 (1 day = 45 minutes) Content to be learned Find distance and midpoint. (2 days) Identify and model points, lines, and planes. (1 day) Solve problems on the coordinate plane using distance and midpoint formulas. (1 day) Apply the distance and midpoint formula to solve word problems. (1 day) Solve problems using the Pythagorean Theorem and the converse of the Pythagorean Theorem. (1 day) Derive the distance formula using the Pythagorean Theorem. (1 day) Mathematical practices to be integrated Make sense of problems and persevere in solving them. Read problems involving Pythagorean Theorem; plan a strategy to solve and check the problem; modify and revise, if necessary. Think about simpler problems to help solve more complex problems with distance and midpoint. Construct viable arguments and critique the reasoning of others. Think logically; determine if there are errors, and explain reasoning. Communicate conclusions. Essential questions What are the differences and similarities between midpoint and distance? When is the distance or midpoint formula used? How is the Pythagorean Theorem related to the distance formula? What is the difference between how the Pythagorean Theorem and its converse are used? Where can the Pythagorean Theorem be applied in the real world? Cumberland, Lincoln, and Woonsocket Public Schools C-1

2 Geometry, Quarter 1, Unit 1.1 Final, July 2011 Distance, Midpoint, and Pythagorean Theorem (8 days) Written Curriculum Grade-Span Expectations M(G&M) 10 2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g., Pythagorean Theorem, Triangle Inequality Theorem). (State) M(G&M) 10 9 Solves problems on and off the coordinate plane involving distance and midpoint. (State) Clarifying the Standards Prior Learning In grade 4, students plotted points in the first quadrant and found horizontal and vertical distances. In fifth grade, students plotted points in all four quadrants and reflected, translated, and rotated polygons on the coordinate plane. In grade 7, students calculated perfect squares and related square roots. In grade 8, students were introduced to and appled the Pythagorean Theorem. Current Learning Students make and defend conjectures to solve problems using the Pythagorean Theorem and the converse of the Pythagorean Theorem. They find distance and midpoint on the coordinate plane. Students derive the distance formula using the Pythagorean Theorem. In unit 4.3, students use the distance formula to find the equation of a circle. Future Learning In precalculus and calculus, students will use the Pythagorean Theorem when studying trigonometry and trigonometric identities. Additional Research Findings According to Principles and Standards for School Mathematics, students in grades 9 12 are to specify locations and describe spatial relationships using coordinate geometry and other representational systems (p. 308). According to Beyond Numeracy, students at this level should be introduced to the coordinate system and its uses in analytic geometry (pp ). According to Principles and Standards for School Mathematics, in grades 9 12 students have to write an algebraic justification of the Pythagorean Theorem, and students use the Pythagorean Theorem as one of the many multiple approaches to solving geometric problems (p. 301). According to Benchmarks for Science Literacy, in grades 6 8 students should use the Pythagorean Theorem as a model for problem solving (p. 269). Cumberland, Lincoln, and Woonsocket Public Schools C-2

3 Geometry, Quarter 1, Unit 1.2 Applications of Slope Overview Number of instructional days: 8 (1 day = 45 minutes) Content to be learned Solve problems algebraically and graphically involving slope, parallel lines, and perpendicular lines. (3 days) Determine the rate of change and equation of a line. (2 days) Solve real-world problems involving lines on the coordinate plane. (2 days) Essential questions What does slope tell you about a line(s), including parallel and perpendicular lines? What do the graphs of linear equations have in common? Mathematical Practices to be Integrated Model with mathematics. Relate applications of slope to everyday life. Use appropriate tools strategically. Use technology to visualize results. Use tools and technology to explore and deepen understanding of slope and be able to detect errors. What are the differences and similarities between equations of parallel and perpendicular lines? Cumberland, Lincoln, and Woonsocket Public Schools C-3

4 Geometry, Quarter 1, Unit 1.2 Final, July 2011 Applications of Slope (8 days) Written Curriculum Grade-Span Expectations M(G&M) 10 9 Solves problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope. (State) Clarifying the Standards Prior Learning In grade 4, students were introduced to constant rate of change. Grade 6 students continued learning about slope through the use of graphs. In grade 7, students solved problems involving constant rates of change, meaning of slope, translating slope from tables to equations and vice versa (F&A 7-2). Grade 8 students again discussed the meaning of slope and solved problems involving slope and rate of change through tables, graphs, and equations (F&A 8-2). In grade 9, students found the equation of lines, using two points, point and slope, etc. (F&A 10-4). Current Learning Students solve problems on the coordinate plane involving parallel and perpendicular lines and slope (developmental); they also prove lines to be parallel and perpendicular by comparing slopes (developmental). Throughout the unit, the concepts of slope and writing equations of lines are reinforced. Also on the developmental level, students use coordinate geometry to examine the slopes of parallel and perpendicular lines. In unit 1.4, students solve problems involving parallel and perpendicular lines (off the coordinate plane). Future Learning In grades 11 and 12, students will use rate of change for polynomial, rational, and exponential functions. Additional Research Findings According to Benchmarks for Science Literacy, by the end of eighth grade students should know that lines can be parallel, perpendicular, or oblique (p. 224). Also, as stated in the Principles and Standards for School Mathematics text, by grade 8 students should understand the relationship between symbolic expressions and graphs of lines using slope and intercepts (p. 222). A Research Companion to Principles and Standards for School Mathematics adds that, by eighth grade students learn that parallel lines should not intersect and are equidistant (p. 164). Cumberland, Lincoln, and Woonsocket Public Schools C-4

5 Geometry, Quarter 1, Unit 1.3 Angles and Segments Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Define angles, segments, and rays. (2 days) Make and defend conjectures about angle and segment addition postulates. (2 days) Make and defend conjectures about angle pairs (adjacent, linear, vertical, supplementary, and complementary) and angle bisectors. (5 days) Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Use proofs to make logical arguments and conclusions regarding angles and segments. Make conjectures regarding angles and segments and prove them, including using counter-examples. Attend to precision. Use precise mathematical vocabulary and symbols to communicate efficiently and effectively. Essential questions What are the differences and similarities between angle pairs? How do you know if an angle or segment has been bisected? How can vertical, linear, adjacent, supplementary, and complementary angle pairs be used to solve problems? Cumberland, Lincoln, and Woonsocket Public Schools C-5

6 Geometry, Quarter 1, Unit 1.3 Final, July 2011 Angles and Segments (10 days) Written Curriculum Grade Span Expectations M(G&M) 10 2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g., Pythagorean Theorem, Triangle Inequality Theorem). (State) Clarifying the Standards Prior Learning Students in grade 4 classified angles relative to 90 degrees. In grade 7, vertical, adjacent, linear, supplementary, and complementary angle properties were introduced to and reinforced. Current Learning At this grade level, students master and solve problems involving angle pairs (vertical, adjacent, linear, supplementary, and complementary). Future Learning In future units, students will create formal proofs involving angle properties. Additional Research Findings See NCTM Principles and Standards for School Mathematics, p According to A Research Companion to Principles and Standards for School Mathematics, Angles are the turning points in the study of geometry and special relationships (p. 162). Cumberland, Lincoln, and Woonsocket Public Schools C-6

7 Geometry, Quarter 1, Unit 1.4 Line and Angle Relationships Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Make and defend conjectures about parallel and perpendicular lines off the coordinate plane. (5 days) Use angles formed by parallel lines and transversals to solve real-world problems. (4 days) Essential questions What is different about the angle pairs that form when a transversal intersects parallel lines versus the angle pairs formed when a transversal intersects nonparallel lines? How can special angle pairs help you determine if two lines are parallel? Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Think logically, determine if there are errors, and explain their reasoning. Model with mathematics. Relate what they have learned regarding line and angle relationships to everyday life. Look for and make use of structure. Apply prior learning about angles and lines to new situations. How do you prove that two lines cut by a transversal are parallel or perpendicular? Cumberland, Lincoln, and Woonsocket Public Schools C-7

8 Geometry, Quarter 1, Unit 1.4 Final, July 2011 Line and Angle Relationships (10 days) Written Curriculum Grade-Span Expectations M(G&M) 10 9 Solves problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope. (State) M(G&M) 10 2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g., Pythagorean Theorem, Triangle Inequality Theorem). (State) Clarifying the Standards Prior Learning In grades 4 6, students were introduced to the meaning of parallel and perpendicular lines when identifying and describing shapes. In grade 7, students learned about angles formed by two parallel and nonparallel lines cut by a transversal. Current Learning Through solving real-world problems, students develop, reinforce, and master relationships of angles formed by parallel and nonparallel lines cut by transversals off the coordinate plane. This year, students solve problems involving parallel and perpendicular lines on the coordinate plane. Later this year, the angle relationships between parallel and perpendicular lines cut by a transversal are applied to show that polygons are parallelograms, rectangles, squares, etc. Future Learning In grade 12, students will create formal proofs of propositions involving angles and lines. Additional Research Findings According to Benchmarks for Science Literacy, by the end of eighth grade students should know that lines can be parallel, perpendicular, or oblique (p. 224). Also, as shown the Principles and Standards for School Mathematics text, in grades 9-12 students should be able to apply the angle relationships formed by two parallel lines and a transversal to solve problems (p. 310). Also, according to A Research Companion to Principles and Standards for School Mathematics, by eighth grade students should know that parallel lines should not intersect and that they are equidistant (p. 164). Cumberland, Lincoln, and Woonsocket Public Schools C-8

9 Geometry, Quarter 1, Unit 1.5 Polygons Overview Number of instructional days: 4 (1 day = 45 minutes) Content to be learned Use properties of polygons to classify and solve problems involving interior and exterior angles. (1 day) Use the exterior angle theorem to solve problems. (1 day) Classify polygons as convex or concave. (1 day) Derive the formula for the sum of the interior angles of a polygon. (1 day) Mathematical Practices to be Integrated Construct viable arguments and critique the reasoning of others. Use prior material and properties of polygons to solve more complex problems involving deductive and inductive reasoning. Look for and express regularity in repeated reasoning. Look for general methods, patterns, and repeated calculations when working with polygons. Look for patterns of polygons to find generalizations. Essential questions What are the differences and similarities among polygons? What is the relationship between the interior and exterior angles of a polygon? How would you derive the formula for the sum of the interior angles of a polygon? Cumberland, Lincoln, and Woonsocket Public Schools C-9

10 Geometry, Quarter 1, Unit 1.5 Final, July 2011 Polygons (4 days) Written Curriculum Grade-Span Expectations M(G&M) 10 2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g., Pythagorean Theorem, Triangle Inequality Theorem). (State) Clarifying the Standards Prior Learning In grade 3, students developed conceptual understanding of triangles and quadrilaterals by copying, comparing, and drawing. In sixth grade, students classified triangles and quadrilaterals. Current Learning At this grade level, students solve problems involving convex polygons. Students also create formal proofs involving polygons. Future Learning In grade 12, students will solve problems involving area of polygons. Additional Research Findings See Principles and Standards for School Mathematics, page 233. A Research Companion to Principles and Standards for School Mathematics states that, [according to the theory of Pierre and Dina van Hiele, Student Progress Though Levels of Thought in Geometry (levels 0-4) ], at level 3 students should be at the abstract level. Students can form abstract definitions, distinguish between necessary and sufficient sets of conditions for a concept, and understand and sometimes even provide logical arguments in the geometric domain (p. 152). Cumberland, Lincoln, and Woonsocket Public Schools C-10

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