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2013 – 2014 Geometry=>MGS-41 MATHEMATICS Curriculum Map ITHS Common Core State Standards Pierre-Max Foucault-Room 340 Mathematics Department ITHS CCSS Geometry Curriculum Map Common Core State Standards Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. (MP.1) Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process which sometimes requires perseverance, flexibility, and a bit of ingenuity. 2. Reason abstractly and quantitatively. (MP.2) The concrete and the abstract can complement each other in the development of mathematical understanding: representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete context can help make sense of abstract symbols. 3. Construct viable arguments and critique the reasoning of others. (MP.3) A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and supporting evidence. 4. Model with mathematics. (MP.4) Many everyday problems can be solved by modeling the situation with mathematics. 5. Use appropriate tools strategically. (MP.5) Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen mathematical understanding. 6. Attend to precision. (MP.6) Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical explanations. 7. Look for and make use of structure. (MP.7) Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea. 8. Look for and express regularity in repeated reasoning. (MP.8) Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results more quickly and efficiently. Mathematics Department ITHS CCSS Geometry Curriculum Map Geometry: Common Core State Standards The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school CCSS. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas, organized into five units are as follows. Congruence, Proof, and Constructions: In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. Similarity, Proof, and Trigonometry: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles, building on students’ work with quadratic equations done in the first course. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles. Extending to Three Dimensions: Students’experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of crosssections and the result of rotating a two-dimensional object about a line. Connecting Algebra and Geometry Through Coordinates: Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, which relates back to work done in the first course. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. Circles With and Without Coordinates: In this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a radius, inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths and angle measures.They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to determine intersections between lines and circles or parabolas and between two circles. Geometry: Common Core State Standards At A Glance Mathematics Department ITHS CCSS Geometry Curriculum Map September Unit 1- Geometry Fundamentals G-GPE.2.6 G-GPE.2.7 G-CO.4.12 G-CO.4.13 October Unit 2- Geometric Reasoning D.6.2 D.6.3 D.6.4 G-CO.3.9 November Unit 3- Parallel and Perpendicular Lines G-CO.1.1 G-CO.3.9 G-GPE.2.5 December/January Unit 4- Triangle Fundamentals G-CO.3.10 Mathematics Department ITHS CCSS Geometry Curriculum Map Unit 5- Triangle Congruence G-CO.2.7 G-CO.2.8 G-CO.3.10 Unit 6- Triangle Properties and Attributes G-CO.3.10 G-SRT.2.4 G-SRT.2.5 The following English Language Arts CCSS should be taught throughout the course: RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements or performing tasks, attending to special cases or exceptions defined in the text. RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in context and topics. RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form and translate information expressed visually or mathematically into words. L.1.1: Initiate and participate effectively in a range of collaborative discussions with diverse partners. L.1.2: Integrate multiple sources of information presented in diverse media or formats evaluating the credibility and accuracy of each source. L.1.3: Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence. SL.2.4: Present information, findings and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning. Mathematics Department ITHS CCSS Geometry Curriculum Map WHST.1.1: Write arguments focused on discipline-specific content. WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research. Course: Geometry Unit 1- Geometry Fundamentals Essential Question(s): How can algebra be useful when expressing geometric properties? Standard Learning Goals Remarks I can: The students will: G-GPE.2.6 calculate the point(s) on a directed This includes the midpoint formula. Find the point on a directed line Find a missing midpoint given an line segment whose endpoints are segment between two given points (x1, y1) and (x2, y2) that partitions endpoint and the midpoint of the that partitions the segment in a given segment. the line segment into a given ratio. ratio. In addition to using the formula, MP #7 students may find the midpoint graphically using slope. G-GPE.2.7 This standard provides practice use the distance formula to Use coordinates to compute compute segment length given two with the distance formula and its perimeters of polygons and areas of connection with the Pythagorean coordinates triangles and rectangles, e.g., using Theorem. the distance formula. MP #1 Mathematics Department ITHS CCSS Resources Partitioning Segments: http://learni.st/users/S3357 2/boards/3128-partitioningsegments-into-a-particularratio-common-corestandard-9-12-g-gpe-6 TI-Nspire Lessons: http://ccssmath.org/?page_i d=2315 Describing Constructions Worksheet: http://www.shmoop.com/co mmon-corestandards/handouts/g-coworksheet_12.pdf Geometry Curriculum Map G-CO.4.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc. This includes: copying a segment; copying an angle; bisecting a identify the tools used in formal constructions. use tools and methods to precisely copy a segment use tools and methods to precisely copy an angle use tools and methods to precisely bisect a segment Students should be able to informally perform the constructions listed using string, reflective devices, paper folding, and/or dynamic geometric software. Some students may believe that a construction is the same as a sketch or drawing. Emphasize the need for First to Finish Task (toward the bottom of this webpage) https://commoncoregeomet ry.wikispaces.hcpss.org/Uni t+1 Midpoint resources: http://www.shmoop.com/co mmon-core-standards/ccsshs-g-gpe-6.html Course: Geometry Unit 1- Geometry Fundamentals Essential Question(s): How can algebra be useful when expressing geometric properties? Standard Learning Goals I can: Remarks The students will: G-GPE.2.6 calculate the point(s) on a directed This includes the midpoint formula. Find the point on a directed line Find a missing midpoint given an line segment whose endpoints are segment between two given points endpoint and the midpoint of the (x1, y1) and (x2, y2) that partitions that partitions the segment in a given segment. the line segment into a given ratio. ratio. In addition to using the formula, MP #7 students may find the midpoint graphically using slope. Mathematics Department ITHS CCSS Resources Partitioning Segments: http://learni.st/users/S3357 2/boards/3128-partitioningsegments-into-a-particularratio-common-corestandard-9-12-g-gpe-6 TI-Nspire Lessons: Geometry Curriculum Map G-GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. MP #1 use the distance formula to compute segment length given two coordinates This standard provides practice with the distance formula and its connection with the Pythagorean Theorem. G-CO.4.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc. This includes: copying a segment; copying an angle; bisecting a identify the tools used in formal constructions. use tools and methods to precisely copy a segment use tools and methods to precisely copy an angle use tools and methods to precisely bisect a segment Students should be able to informally perform the constructions listed using string, reflective devices, paper folding, and/or dynamic geometric software. Some students may believe that a construction is the same as a sketch or drawing. Emphasize the need for Mathematics Department ITHS CCSS http://ccssmath.org/?page_i d=2315 Describing Constructions Worksheet: http://www.shmoop.com/co mmon-corestandards/handouts/g-coworksheet_12.pdf First to Finish Task (toward the bottom of this webpage) https://commoncoregeomet ry.wikispaces.hcpss.org/Uni t+1 Midpoint resources: http://www.shmoop.com/co mmon-core-standards/ccsshs-g-gpe-6.html Geometry Curriculum Map segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.) MP #5, #6 G-CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. MP #5, #6 Mathematics Department ITHS use tools and methods to precisely bisect an angle construct perpendicular lines and bisectors construct a line parallel to a given line through a point not on the line. precision and accuracy when doing constructions. Stress the idea that a compass and straightedge are identical to a protractor and ruler. Explain the difference between measurement and construction. As the Crow Flies Task: http://www.nctm.org/upload edFiles/Journals_and_Book s/Books/FHSM/RSMTask/Crow.pdf define inscribed polygons. construct an equilateral triangle, a square, a hexagon inscribed in a circle. explain the steps to constructing and equilateral triangle, a square, and a regular hexagon inscribed in a circle. CCSS Geometry Curriculum Map Unit 2-Geometric Reasoning Essential Question(s): How can inductive and deductive reasoning help to construct and defend logical arguments? Standard Learning Goals I can: Remarks The students will: D.6.2 Emphasize all forms of a conditional identify and write a conditional Find the converse, inverse, and statement. statement and its converse, inverse, contrapositive of a statement. and contrapositive. MP #2, #3 D.6.3 Determine whether two propositions are logically equivalent. MP #2, #3 D.6.4 Use methods of direct and indirect proof and determine whether a short proof is logically valid. MP #2, #3 G-CO.3.9 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.) MP #2, #3 Mathematics Department ITHS determine whether two statements are logically equivalent. identify a bi-conditional statement. Conditional and Contrapositive/Inverse and Converse are logically equivalent. If conditional and converse are both true, then the statement can be written as a biconditional statement. use methods of proof to validate a postulate, theorem, corollary, definition, or property. identify and use the properties of congruence and equality (reflexive, symmetric, transitive) in my proofs. order statements based on the Law of Syllogism when constructing my proof. interpret geometric diagrams and identifying what can and cannot be assumed. use theorems, postulates, or definitions to prove theorems about lines and angles Common misconceptions: students believe that exceptions to a proven generalization exist one counterexample is not sufficient to prove a conjecture false the converse of a statement is true a conjecture is true because it worked in all examples that were explored Theorems/postulates students may use in their proofs include, but are not limited to: a. Vertical angles are congruent. b. Ruler Postulate c. Segment Addition Postulate d. Protractor Postulate e. Angle Addition Postulate f. Definitions from Unit 1 CCSS Geometry Curriculum Map Resources http://www.youtub e.com/watch?v=B Xd9SWVZEzo&sa fety_mode=true&p ersist_safety_mod e=1&safe=active Unit 3- Parallel and Perpendicular Lines Essential Question(s): In what ways can parallel and perpendicular lines be used to solve real world problems? Standard Learning Goals I can: Remarks The students will: G-CO.1.1 identify the undefined notions used in Know precise definitions of angle, geometry (point, line, plane, distance) and circle, perpendicular line, parallel describe their characteristics line, and line segment, based on the identify angles, perpendicular lines, parallel undefined notions of point, line, lines, rays, and line segments. distance along a line, and distance define angles, perpendicular lines, parallel around a circular arc. lines, rays, and line segments precisely using MP #6 the undefined terms and “if-then” and “if-andonly-if” statements. G-CO.3.9 identify and use the properties of congruence Theorems include: Angle Prove theorems about lines and pairs formed by parallel and equality (reflexive, symmetric, transitive) angles. (Theorems include: vertical lines cut by a transversal in my proofs. angles are congruent; when a (alternate interior angles, order statements based on the Law of transversal crosses parallel lines, corresponding angles, Syllogism when constructing my proof. alternate interior angles are alternate exterior angles interpret geometric diagrams and identifying congruent and corresponding angles are congruent; same-side what can and cannot be assumed. are congruent; points on a interior angles are use theorems, postulates, or definitions to perpendicular bisector of a line supplementary) prove theorems about lines and angles. segment are exactly those Vertical angles are equidistant from the segment’s congruent; linear pairs are endpoints.) supplementary. MP #2, #3 Mathematics Department ITHS CCSS Resources http://www.purplemath.c om/modules/slope2.htm http://www.mathsisfun.c om/geometry/parallellines.html http://mathforum.org/mat htools/cell/g,10.5,ALL,AL L/ http://interactmath.com/ Exercises.aspx?chapterI d=3§ionId=2&objecti veId=1 http://www.shodor.org/int eractivate/activities/Slop eSlider/ http://www.shodor.org/int eractivate/activities/Slop eSlider/ Geometry Curriculum Map Unit 3- Parallel and Perpendicular Lines (cont) Essential Question(s): In what ways can parallel and perpendicular lines be used to solve real world problems? Standard Learning Goals I can: Remarks The students will: G-GPE.2.5 May need to review state that parallel lines have the same slope. Prove the slope criteria for parallel graphing coordinates and determine if lines are parallel using their slopes. and perpendicular lines and use finding slope before using it state that parallel lines have the opposite them to solve geometric problems to determine if lines are reciprocal slopes. (e.g., find the equation of a line parallel or perpendicular. determine if lines are perpendicular using their parallel or perpendicular to a given slopes. line that passes through a given point). MP #3, #8 Mathematics Department ITHS CCSS Resources Geometry Curriculum Map Unit 4- Triangle Fundamentals Standard The students will: G-CO.3.10 Prove theorems about triangles. (Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.) MP #2, #3 Mathematics Department ITHS Essential Question(s): What are the properties of a triangle? Learning Goals I can: interpret geometric diagrams (what can and cannot be assumed). apply theorems, postulates, or definitions to prove theorems about triangles, including: a) Measures of interior angles of a triangle sum to 180; b) Base angles of isosceles triangles are congruent; c) An exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. CCSS Remarks Classifying triangles should be reviewed in this unit. Resources http://teachers.henrico.k 12.va.us/math/IGO/03Tr ianglesPolygons/3_1.ht ml http://www.mathsisfun.c om/triangle.html Geometry Curriculum Map Unit 5- Triangle Congruence Standard The students will: G-CO.2.7 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. MP #3 G-CO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. MP #2, #3 Mathematics Department ITHS Essential Question(s): In what ways can congruence be useful in the real world? Learning Goals I can: Remarks Some students may believe: identify corresponding sides and Combinations such as SSA or AAA corresponding angles of congruent are also a congruence criterion for triangles. explain that in a pair of congruent triangles, triangles. That all transformations, including corresponding sides are congruent dilations, are rigid motions (distance is preserved) and corresponding That any two figures that have the angles are congruent (angle measure is same area represent a rigid preserved). transformation. demonstrate that when distance is preserved (corresponding sides are congruent) and angle measure is preserved (corresponding angles are congruent) the triangles must also be congruent. Include AAS and HL as a criterion list the sufficient conditions to prove for congruence. triangles are congruent. Once you prove your triangles are map a triangle with one of the sufficient congruent, the "left-over" pieces conditions (e.g., SSS) onto the original that were not used in your method triangle and show that corresponding sides of proof are also congruent. and corresponding angles are congruent. Remember, congruent triangles have 6 sets of congruent pieces. We now have a "follow-up" theorem to be used AFTER the triangles are known to be congruent: CPCTC. CCSS Resources http://illuminations. nctm.org/ActivityDe tail.aspx?id=4 http://nlvm.usu.edu/ ennav/frames_asid _165_g_1_t_3.html ?open=instructions &from=topic_t_3.ht ml http://ccssmath.org/ ?page_id=2261 Geometry Curriculum Map Unit 5- Triangle Congruence (cont) Standard The students will: G-CO.3.10 Prove theorems about triangles. (Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.) MP #2, #3 Mathematics Department ITHS Essential Question(s): In what ways can congruence be useful in the real world? Learning Goals I can: Remarks Include proofs of triangle interpret geometric diagrams (what can and congruence. cannot be assumed). apply theorems, postulates, or definitions to prove theorems about triangles. CCSS Resources Geometry Curriculum Map Unit 6- Triangle Properties and Attributes Essential Question(s): What conclusions can be drawn from triangles and their congruency? Standard Learning Goals I can: Remarks The students will: G-CO.3.10 Include perpendicular interpret geometric diagrams (what can and Prove theorems about triangles. bisectors, altitudes, cannot be assumed). (Theorems include: measures of medians, angle bisectors apply theorems, postulates, or definitions to interior angles of a triangle sum to and their points of prove theorems about triangles, including: 180°; base angles of isosceles concurrency. a) The segment joining midpoints of two triangles are congruent; the segment sides of a triangle is parallel to the third joining midpoints of two sides of a side and half the length; triangle is parallel to the third side and b) The medians of a triangle meet at a point. half the length; the medians of a triangle meet at a point.) MP #2, #3 G-SRT.2.4 Some students may apply theorems, postulates, or definitions to Prove theorems about triangles. confuse the alternate prove theorems about triangles, including: (Theorems include: a line parallel to interior angle theorem and a) A line parallel to one side of a triangle one side of a triangle divides the other its converse as well as the divides the triangle proportionally; two proportionally, and conversely; the Pythagorean theorem and b) If a line divides two sides of a triangle Pythagorean Theorem proved using proportionally, then it is parallel to the third its converse. triangle similarity.) side; MP #3 c) The Pythagorean Theorem proved using triangle similarity. G-SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MP #1 Mathematics Department ITHS use triangle congruence and triangle similarity to solve problems (e.g., indirect measure, missing sides/angle measures, side splitting). use triangle congruence and triangle similarity to prove relationships in geometric figures. CCSS Resources http://www.showme. com/sh/?h=Epf0Rk0 https://ccgps.org/GSRT_G6QQ.html http://staff.argyll.eps b.ca/jreed/math9/str and3/triangle_congr uent.htm https://docs.google.c om/viewer?a=v&pid =sites&srcid=ZGVm YXVsdGRvbWFpbn xnZW9tZXRyeXdpd Ghtb3N0ZWxsZXJ8 Z3g6MmQ5MTRmN GNkMzk3Y2E3OQ Some students may assume that segments, angles, and triangles are congruent/ similar without given that information. Geometry Curriculum Map Mathematics Department ITHS CCSS Geometry Curric