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Geometry Common Core Curriculum Map for 2012-2013 School Year Geometry Mathematics, Quarter 1, Unit 1.1 Geometric Foundations, Constructions, and Relationships Established goals: What relevant goals; e.g. content standards; will this unit design address? Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Make geometric constructions [Formalize and explain processes] G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a 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. Prove geometric theorems [Focus on validity of underlying reasoning while using variety of ways of writing proofs] 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. G-CO.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. Construct viable arguments and critique the reasoning of others. • Use definitions, theorems, postulates, and properties to make formal arguments about lines, angles, and triangles in a variety of ways. • Justify conclusions using definitions, theorems, and postulates. • Identify flaws in arguments using logic and reasoning based on geometric concepts. Use appropriate tools strategically. • Make formal geometric constructions with a variety of tools including a compass, straightedge, string, reflexive devices, paper folding, dynamic software, etc. • Use technological tools to explore and deepen understanding of concepts. 1 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year Look for and make use of structure. • Identify relationships in diagrams to solve problems involving missing angles. • Recognize angle relationships formed by parallel lines cut by a transversal and use them to solve problems Clarifying the Standards Support Materials 2 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year What Learning experienced and instruction will enable students to achieve the desired results? How will the design: W = Help the students know Where the unit is go and What is expected? Help the teacher know where the students are coming from (prior knowledge, interests.? H = Hook all students and Hold their interest? E = Equip students, help them experience the key ideas and explore the issues? R= Provide opportunities to Rethink and Revise their understandings and work? E = Allow students to Evaluate their work and its implications? T= Be Tailored (personalized to the different needs, interests, and abilities of learners? O= Be organized to maximize initial and sustained engagement as well as effective learning? Big Ideas: Essential Questions: Content to be learned: What are the big ideas? What specific understanding s about them are desired? What misunderstand ings are predictable? What provocative questions will foster inquiry, understanding and transfer of learning? Students will be able to: What key knowledge and skill will students acquire as a result of this unit? Student will know? What should they eventually be able to do as a result of such knowledge and skill How does the use of precise definitions help you to understand more complex geometric ideas and theorems? 3 updated 7/12/12 Content to Be Learned Mathematical Practices to Be Integrated 1. Know precise definitions of geometric terms (e.g., angle, circle, Lessons Resources: What learning experiences design: and instruction will enable students to achieve the desired results? How will the 1. Word wall in classroom with vocab for the unit 1. Pre-assess vocabulary Assessments: Through what authentic performance task will students demonstrate the desired understanding? By what criteria will performance of understanding be judged? Through what other evidence; e.g. quizzes, tests, academic prompts, observations, homework, journals, will students show achievement of the desired results? How will students self asses and reflect upon their results? Pre assessment vocabulary Post assessment vocabulary Jeopardy unit review for angle relationships Geometry Common Core Curriculum Map for 2012-2013 School Year • What are the similarities and differences between a geometric sketch and a formal construction? • How do you use logic and reasoning when proving geometric concepts? • What are the basic terms of geometry and how are they used to describe figures? 4 updated 7/12/12 perpendicular line, parallel line, and line segment), based on the undefined notions of point, line, distance along a line, and distance around a circular arc 2. Make formal geometric constructions with a variety of tools and methods. Constructions include: Copying a segment/angle. Bisecting a segment/angle. 3. Constructing perpendicular lines. 4. Constructing a perpendicular bisector of a line segment. 5. Constructing a line parallel to a given line through a point not on the line. 6. Prove theorems about lines and angles. Theorems include, but are not limited to: Congruency of vertical angles. Relationship between angles formed by intersection of parallel lines and a transversal. 2. 3,4,5. Glencoe Geometry (orange) section 1.2, 1.6, 1.7 Web tools: videos with construction direction http://www.ma thopenref.com/ tocs/constructi onstoc.html http://www.onl inemathlearnin g.com/geometr y-help.html Use hand constructions using compass, straight edge 1. Web resource for proofs, pdf http://www.be aconlearningce nter.com/docu ments/1727_01 .pdf pdf http://hanlonm ath.com/pdfFil es/460Chapter Create test: Copy segment/angel Bisect segment /angle Construct perp lines Construct parallellines Construct perp bisector Geometry Common Core Curriculum Map for 2012-2013 School Year 7.Prove the triangle angle sum theorem. 3parallellines.p df regents site: http://regentsp rep.org/Regent s/math/geomet ry/GP8/Lparall el.htm 3Triangle sum paper proof Geometry, Quarter 1, Unit 1.2 Transformations Established goals: What relevant goals; e.g. content standards; will this unit design address? G-CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G-SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Model with mathematics. Use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Use appropriate tools strategically. Consider and use the available tools when they develop an understanding of transformations. These tools might include pencil and paper, concrete models, a ruler, a protractor, a 5 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Use technology tools to explore and deepen understanding of various transformation concepts. Attend to precision. Communicate precisely to others. Use clear definitions in discussion with others and your own reasoning. Examine claims and make explicit use of definitions. What Learning experienced and instruction will enable students to achieve the desired results? How will the design: W = Help the students know Where the unit is go and What is expected? Help the teacher know where the students are coming from (prior knowledge, interests.? H = Hook all students and Hold their interest? E = Equip students, help them experience the key ideas and explore the issues? R= Provide opportunities to Rethink and Revise their understandings and work? E = Allow students to Evaluate their work and its implications? T= Be Tailored (personalized to the different needs, interests, and abilities of learners? O= Be organized to maximize initial and sustained engagement as well as effective learning? Big Ideas: What are the big ideas? What specific understanding s about them are desired? What misunderstand ings are predictable? Essential Questions: Content to be learned: 1.Represent, construct, and draw transformations (reflections, translations, dilations, and rotations) in the plane using a What provocative questions will foster inquiry, understanding and transfer of learning? 6 updated 7/12/12 After a transformation has taken place on the coordinate plane, where does the image lie and what does it look like? What tools or methods would Students will be able to: What key knowledge and skill will students acquire as a result of this unit? Student will know? What should they eventually be able to do as a result of such knowledge and skill Lessons Resources: What learning experiences design: and instruction will enable students to achieve the desired results? How will the Assessments: Through what authentic performance task will students demonstrate the desired understanding? By what criteria will performance of understanding be judged? Through what other evidence; e.g. quizzes, tests, academic prompts, observations, homework, journals, will students show achievement of the desired results? How will students self asses and reflect upon their results? Glencoe (red)sections: 3.1,3.2,3.4, Web resource 3. http://regentsprep.or Pre assessment vocabulary Post assessment vocabulary Geometry Common Core Curriculum Map for 2012-2013 School Year you use to construct a figure under a reflection, translation, rotation, and dilation? Compare dilation to rigid motions, how are they similar? How are they different? Where would you find transformations in the world of art? variety of tools, such as transparencies and geometry software. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. State a sequence of transformations that will carry a given figure onto another. Verify experimentally the properties of dilations given by a center and a scale factor: The dilation of a line segment is longer or shorter depending on the scale factor. Dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. g/Regents/math/geo metry/GP4/indexGP4. htm Web resource 1,2,3. http://www.haesemat hematics.com.au/sam ples/igcse_20.pdf Geometry Mathematics, Quarter 2, Unit 2.1 Triangle Congruency Established goals: What relevant goals; e.g. content standards; will this unit design address? 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. 7 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year 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. 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.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 G-CO.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. Reason abstractly and quantitatively. Interpret a given situation abstractly, representing it symbolically and conversely. Manipulate symbolic representations to visualize situations in a diagram in order to solve quantitative problems with geometric figures. Know and flexibly use properties of rigid motion to make sense of congruent relationships. Construct viable arguments and critique the reasoning of others. Make conjectures and construct a logical argument to prove congruent triangles. Analyze a given situation and justify conclusions using triangle congruence criteria. Listen or read the arguments of others and decide whether they make sense; ask useful questions to clarify or improve the arguments. Look for and make use of structure. Recognize and explain the sequence of rigid motions to show triangle congruence. Predict the effects of certain rigid motions to transform a given figure. Recognize the significance of a perpendicular segment and angle bisector as well as their properties in relation to segments and angles. 8 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year Big Ideas: What are the big ideas? What specific understanding s about them are desired? What misunderstand ings are predictable? Essential Questions: Content to be learned: Lessons Resources: How do you identify transformations that are rigid motions? How do you use congruence criteria in proofs and to solve problems? What are the different methods that may be used to create a formal argument? Describe special segments and angles in triangles and explain how can they be used to solve problems? Given a proof, how can you use logical reasoning to critique, analyze, and improve the argument? Why are AAA and SSA invalid criteria for proving triangle 1. Apply and describe the effects of rigid motions (translation, reflection, rotation) on a given figure. Compare corresponding parts of triangles to determine congruency. Explain and utilize the criteria for triangle congruency. oASA, SAS, SSS oAAS, HL oCPCTC could be taught here. Prove theorems about lines and angles using the triangle congruency criteria, for example: o Points on a perpendicular Glencoe (red) Chapter 4: 4.3 – 4.5 What provocative questions will foster inquiry, understanding and transfer of learning? 9 updated 7/12/12 Students will be able to: What key knowledge and skill will students acquire as a result of this unit? Student will know? What should they eventually be able to do as a result of such knowledge and skill What learning experiences design: and instruction will enable students to achieve the desired results? How will the Anchor Task NCTM Illumintions: Pieces of Proofs Web resource: http://regentsprep.org /Regents/math/geomet ry/GP4/indexGP4.htm 3. NCTM journal activity, Letting the cat out of the hat. Assessments: Through what authentic performance task will students demonstrate the desired understanding? By what criteria will performance of understanding be judged? Through what other evidence; e.g. quizzes, tests, academic prompts, observations, homework, journals, will students show achievement of the desired results? How will students self asses and reflect upon their results? Pre assessment vocabulary Post assessment vocabulary Geometry Common Core Curriculum Map for 2012-2013 School Year congruence? bisector of a line segment are exactly those equidistant from the segment’s endpoints. o Points on an angle bisector are equidistant from sides of the angle. Prove theorems about triangles using the triangle congruency criteria. Examples include: o Base angles of isosceles triangles are congruent. o The segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. o The medians of a triangle meet at a point (centroid). Geometry Mathematics, Quarter 2, Unit 2.2 Triangle Similarity Established goals: What relevant goals; e.g. content standards; will this unit design address? Understand similarity in terms of similarity transformations G-SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G-SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Prove theorems involving similarity G-SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 10 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year Make sense of problems and persevere in solving them. Analyze givens, constraints, relationships, and goals. Translate verbal descriptions and problems into diagrams, including the important features and relationships. Construct viable arguments and critique the reasoning of others. Make conjectures and construct logical arguments to prove triangles are similar. Analyze a given situation and justify conclusions using triangle similarity criterion. Listen or read the arguments of others and decide whether they make sense; ask useful questions to clarify or improve the arguments. What Learning experienced and instruction will enable students to achieve the desired results? How will the design: W = Help the students know Where the unit is go and What is expected? Help the teacher know where the students are coming from (prior knowledge, interests.? H = Hook all students and Hold their interest? E = Equip students, help them experience the key ideas and explore the issues? R= Provide opportunities to Rethink and Revise their understandings and work? E = Allow students to Evaluate their work and its implications? T= Be Tailored (personalized to the different needs, interests, and abilities of learners? O= Be organized to maximize initial and sustained engagement as well as effective learning? 11 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year Big Ideas: What are the big ideas? What specific understanding s about them are desired? What misunderstand ings are predictable? Essential Questions: Content to be learned: Lessons Resources: What can you conclude about similar triangles, and how can you prove two triangles are similar? How can similar triangles be used to measure objects, and what are the benefits of using indirect measurement? What relationships exist within a triangle when a line is drawn parallel to one of the sides? How can you use triangle similarity to prove the Pythagorean Theorem? What are the similarities and differences between triangle similarity Establish the AA, SSS, and SAS triangle similarity criteria using similarity transformations. Determine if two triangles are similar by using the definition of similarity. Explain the meaning of similarity for triangles using similarity transformations. Prove theorems about triangles using triangle similarity; examples include: o A line parallel to one side of a triangle divides the other two proportionally. o The Pythagorean Theorem. Prove theorems and solve problems involving similarity using congruence and similarity criteria. 1,2. NCTM journal activity, Letting the cat out of the hat. 1,2,Glenco (red) Sections 7.2,7.3,7.5, What provocative questions will foster inquiry, understanding and transfer of learning? 12 updated 7/12/12 Students will be able to: What key knowledge and skill will students acquire as a result of this unit? Student will know? What should they eventually be able to do as a result of such knowledge and skill What learning experiences design: and instruction will enable students to achieve the desired results? How will the 4.Web resource: Kahn academy: Pythangorean them and similarity: http://www.khanacad emy.org/math/geome try/triangles/v/pytha gorean-theoremproof-using-similarity Web resource: parallel lines and Assessments: Through what authentic performance task will students demonstrate the desired understanding? By what criteria will performance of understanding be judged? Through what other evidence; e.g. quizzes, tests, academic prompts, observations, homework, journals, will students show achievement of the desired results? How will students self asses and reflect upon their results? Pre assessment vocabulary Post assessment vocabulary Geometry Common Core Curriculum Map for 2012-2013 School Year proportional sides http://www.mathwar ehouse.com/geometr y/similar/triangles/si de-splittertheorem.php Illustrated math.org Are they similar http://illustrativemat hematics.org/standar ds/hs Geometry, Quarter 2, Unit 2.3 Right Triangle Trigonometry Define trigonometric ratios and solve problems involving right triangles G-SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ Apply trigonometry to general triangles G-SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). G-SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. ***Teacher Note: (as per the PARCC Model Content Frameworks, mathematics grades 3–11, G-SRT.9, G-SRT.10, and G-SRT.11 may be an extension to right triangle trigonometry (p. 53).*** Big Ideas: What are the Essential Questions: What provocative questions will 13 updated 7/12/12 Content to be learned: Students will be able to: Lessons Resources: What learning Assessments: Through what authentic performance task will students Geometry Common Core Curriculum Map for 2012-2013 School Year big ideas? What specific understanding s about them are desired? What misunderstand ings are predictable? foster inquiry, understanding and transfer of learning? What key knowledge and skill will students acquire as a result of this unit? Student will know? What should they eventually be able to do as a result of such knowledge and skill experiences design: and instruction will enable students to achieve the desired results? How will the • How can you find a side (For each of the bullets below, the expressions and equations refer to, but are not limited to, linear, quadratic, and exponential relationships.) Define trigonometric ratios (sine, cosine, tangents) in right triangles by understanding that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Calculate the side lengths and the trigonometric ratios associated with special right triangles. Explain and use the relationship between the sine and cosine of complementary angles. Use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems.★ Angles of elevation and depression section 8-4 glenc 5.Law of sine 8.5 Law of cosine 8.6 On Core book: 6.1,6.2 1,2,3 4Glencoe (orange) Chapter 8 All sections 8.1-8.6 length or angle measure in a right triangle? How would you explain a solution to a real world scenario where you would apply your knowledge of right triangle relationships to find a height, distance, or an angle measure? What are the important ratios between sides of a right triangle, and how does changing the sides affect these ratios? 14 updated 7/12/12 Illustrativemathemat htics.org Setting up sprinklers http://illustrativemath ematics.org/illustration s/607 demonstrate the desired understanding? By what criteria will performance of understanding be judged? Through what other evidence; e.g. quizzes, tests, academic prompts, observations, homework, journals, will students show achievement of the desired results? How will students self asses and reflect upon their results? Geometry Common Core Curriculum Map for 2012-2013 School Year + Apply trigonometry to general triangles: Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Prove the Laws of Sines and Cosines and use them to solve problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Geometry, Quarter 3, Unit 3.1 Polygons: Theorems, Proofs, and Applications On and Off the Coordinate Plane Established goals: What relevant goals; e.g. content standards; will this unit design address? Use coordinates to prove simple geometric theorems algebraically [Include distance formula; relate to Pythagorean theorem] G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Experiment with transformations in the plane 15 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Construct viable arguments and critique the reasoning of others. Make sense of problems and persevere in solving them. Analyze givens, constraints, relationships, and goals when proving theorems or solving problems with polygons. Make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. Monitor and evaluate self-progress and change course if necessary. Find correspondences between equations, verbal descriptions, and graphs or draw diagrams of important features and relationships and search for regularity or trends when solving problems involving polygons. Check answers to problems using a different method, and continually ask, “Does this make sense?” Understand the approaches of others to solving complex problems and identify correspondences between different approaches. Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations involving polygons. Create a coherent representation of the problem at hand by drawing a diagram from the verbal situation; consider the units involved when calculating the distance; attend to the meaning of quantities, not just how to compute them; and know and flexibly use different properties of objects and operations. 16 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year design: W = Help the students know Where the unit is go and What is expected? Help the teacher know where the students are coming from (prior knowledge, interests.? H = Hook all students and Hold their interest? E = Equip students, help them experience the key ideas and explore the issues? R= Provide opportunities to Rethik and Revise their understandings and work? E = Allow students to Evaluate their work and its implictions? T= Be Tailored (personalized to the different needs, interests, and abilities of learners? O= Be organized to maximize initial and usustained engagement as well as effective learning? Big Ideas: What are the big ideas? What specific understanding s about them are desired? What misunderstand ings are predictable? Essential Questions: What provocative questions will foster inquiry, understanding and transfer of learning? How do properties of polygons help establish methods to find unknown areas? How can you use the coordinate plane to prove general relationships in polygons? How can the Pythagorean Theorem help the understanding of the distance formula? How can you algebraically 17 updated 7/12/12 Content to be learned: Students will be able to: What key knowledge and skill will students acquire as a result of this unit? Student will know? What should they eventually be able to do as a result of such knowledge and skill Lessons Resources: What learning experiences design: and instruction will enable students to achieve the desired results? How will the Content to be learned 1.On Core 4.2 thru 4.5 Mathematical practices to be Has all proofs integrated 2. On Core 8.6 Prove theorems about Glencoe (red) 6.3-6.6 parallelograms. 3. on core 9.2 Theorems include the following: o Opposite sides and angles are congruent. o The diagonals of a parallelogram bisect each other. 4. on core 4.1 Glencoe 13.5 & 13..7 1,2.Glenco (orange) Chapter 6 Exploring Assessments: Through what authentic performance task will students demonstrate the desired understanding? By what criteria will performance of understanding be judged? Through what other evidence; e.g. quizzes, tests, academic prompts, observations, homework, journals, will students show achievement of the desired results? How will students self asses and reflect upon their results? WHS Common Task Prove parallelogram A midpoint miracle http://www.illustrativemath ematics.org/illustrations/60 5 Geometry Common Core Curriculum Map for 2012-2013 School Year prove that a polygon is a specific quadrilateral? What information is sufficient to justify that claim? How can you classify quadrilaterals? What properties do they share, and how may they differ? (not on the coordinate grid) 18 updated 7/12/12 o Rectangles are parallelograms with congruent diagonals. Use coordinates to prove simple geometric theorems algebraically. Theorems include the following: o Find the coordinates of midpoint and points along directed line segments given a ratio. o Use the slope criteria for parallel and perpendicular lines to solve and prove polygon-related problems and properties. o Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle/parallelogram/rhombus/ square/trapezoid. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles by employing the Pythagorean Theorem or distance formula (from a modeling perspective). Describe the rotations and reflections that carry polygons such as rectangles, parallelograms, trapezoids, or regular polygons quadrilaterals Properties of polygons Chapter 1. chapter 1 1.5 chapter 12 12.6 midpoint 2 interactive site http://www.mathope nref.com/coordparall elogram.html 2. regents site http://regentsp rep.org/Regent s/math/geomet ry/GCG4/Coord inatepRACTICE. htm http://www.ne xuslearning.net /books/MLGeometry/Chap ter6/ML%20Ge ometry%2063%20Proving% 20Quadrilatera ls%20are%20P arallelograms.p df Geometry Common Core Curriculum Map for 2012-2013 School Year onto themselves. 1.2., http://www.ve nicehigh.net/o urpages/auto/ 2007/9/11/11 89558895358/ 6_3%20Geo%2 0A.pdf 2 Tutorial: task similar to ommon task http://www.so phia.org/coordi nate-geometryofparallelograms --2/coordinategeometry-ofparallelograms --6tutorial?pathw ay=geometry Geometry, Quarter 3, Unit 3.2 2-D and 3-D Measurements and Modeling Apply geometric concepts in modeling situations G-MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★ Explain volume formulas and use them to solve problems G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, 19 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★ Model with mathematics. Apply the area and volume formulas to everyday situations. Model situations to apply concepts of population density based on area (persons per square mile). Model situations to apply concepts of density based on volume (BTUs per cubic foot). Understand that when developing mathematical models, there is often a trade-off between a model that is more precise and one that is easier to work with (e.g., choosing a trapezoid versus a more complex polygon when calculating area). Attend to precision. Communicate convincing arguments and accurate responses in multiple formats, including technological, written, and oral forms. Express numerical answers with a specific degree of precision appropriate for the problem context. Specify units of measure when calculating circumference, area, and volume. 20 updated 7/12/12 Geometry Common Core Curriculum Map for 2012-2013 School Year Big Ideas: What are the big ideas? What specific understanding s about them are desired? What misunderstand ings are predictable? Essential Questions: Content to be learned: How can you use geometric shapes, their measures, and their properties to describe and model objects in the world around you? How can you use an informal limit argument to determine the circumference of a circle? How can you use an informal dissection argument to determine the area of a circle? How have you utilized the geometry concepts learned in this unit for solving and modeling real-world problems? How can you use Cavalieri’s Principle to give an informal argument for the volume of Describe objects using geometric shapes, their measures, and their properties (e.g., tree trunk as a cylinder). Make informal arguments about formulas for (1) circumference and area of a circle and (2) volume of cylinder, pyramid, and cone. (Use dissection arguments, Cavalieri’s Principle, and informal limit arguments.) Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply concepts of density, What provocative questions will foster inquiry, understanding and transfer of learning? 21 updated 7/12/12 Students will be able to: What key knowledge and skill will students acquire as a result of this unit? Student will know? What should they eventually be able to do as a result of such knowledge and skill Lessons Resources: What learning experiences design: and instruction will enable students to achieve the desired results? How will the Assessments: Through what authentic performance task will students demonstrate the desired understanding? By what criteria will performance of understanding be judged? Through what other evidence; e.g. quizzes, tests, academic prompts, observations, homework, journals, will students show achievement of the desired results? How will students self asses and reflect upon their results? Geometry Common Core Curriculum Map for 2012-2013 School Year different solids? What are some examples of how you can use geometry to design a structure, given specific constraints? Given a map and the population of Rhode Island, how you would estimate the number of persons per square mile? 22 updated 7/12/12 based on area and volume in modeling situations. Apply geometric methods to solve design problems such as (1) designing an object or structure to satisfy physical constraints or minimize cost or (2) working with typographic grid systems based on ratios. Explain volume formulas and give an informal argument using Cavalieri’s Principle for formulas for volumes of spheres and other solid figures.
