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2016-2017 Mathematics Curriculum Map Geometry, Q1 Unifying Concept: Transformations, Congruence, Proof, and Constructions Quarter 1 Target Standards are emphasized every quarter and used in formal assessment to evaluate student mastery. Highly-Leveraged Standards1 G-CO.A.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. G-CO.A.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.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygons, describe the rotations and reflections that carry it onto itself. G-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.A.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-CO.B.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.B.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. G-CO.B.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.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. G-CO.C.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. G-CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Additional Standards Complementary Standards: (Standards to be assessed in classroom and/or future benchmarks) G-CO.D.12, 13 Standards for Mathematical Practice: (quarterly focus) Mathematically proficient students: SMP 3. Construct viable arguments and critique the reasoning of others. SMP 4. Model with mathematics. SMP 5. Use appropriate tools strategically. SMP 6. Attend to precision. Mathematical Practices Poster 6th – 12th Grade Literacy in History/Social Studies, Science, and Technical Subjects Focus Standards: (http://www.azed.gov/standardspractices/files/2015/04/accs-6-12-ela-contentliteracy-standards-final10_28_2013.pdf) 9-10.RST.4 9-10.WHST.1a-1e 9-10.WHST.2c 9-10.WHST.3 9-10.WHST.4 Supporting Standards2 - There are no Supporting Target Standards for this quarter. 1 Highly-Leveraged Standards are the most essential for students to learn because they have endurance, leverage and essentiality. This definition for highly-leveraged standards was adapted from the website of Millis Public Schools, K-12, in Massachusetts, USA. http://www.millis.k12.ma.us/services/curriculum_assessment/brochures TUSD Department of Curriculum and Instruction Curriculum 3.0 Revised 5/11/2016 3:05 PM Page 1 2016-2017 Mathematics Curriculum Map Geometry, Q1 Specifically for mathematics, the Highly-Leveraged Standards are the Major Content/Clusters as defined by the AZCCRS Grade Level Focus documents. They should encompass a range of at least 65%-75% for Major Content/Clusters and a range of 25%-35% for Supporting Cluster Instruction. See the Grade Level Focus documents at: https://cms.azed.gov/home/GetDocumentFile?id=57069f7baadebe0bccd0a8b5 2 Supporting Standards are related standards that support the highly-leveraged standards in and across grade levels. *Highly-Leveraged standards in bold and supporting standards are normal text. Adopted and Supplemental Texts Big Ideas Eureka Math/ Engage NY: Module 1 Holt McDougal Geometry: Chapter 1 – Sections 1-7 Chapter 2 – Section 7 Chapter 3 – Section 1-4 Chapter 4 – Sections 1-9 Chapter 5 – Sections 3 and 4 Chapter 6 – Sections 2 and 4 Chapter 9 – Sections 1-5 Instruction must be supplemented to meet the expectations of the standards. Additional Resources: https://www.khanacademy.org http://achievethecore.org https://www.illustrativemathematics.org/ www.insidemathematics.org https://learnzillion.com/resources/75114-math http://maccss.ncdpi.wikispaces.net/ (Choose your grade level on the left) http://www.pbslearningmedia.org/standards/1 http://nlvm.usu.edu/en/nav/vlibrary.html http://nrich.maths.org https://www.youcubed.org/week-of-inspirational-math/ http://illuminations.nctm.org/Lessons-Activities.aspx (choose grade level and connect to search lessons) http://www.yummymath.com/birds-eye-of-activities/ http://map.mathshell.org/tasks.php?collection=9&unit=HE06 http://www.shmoop.com/common-core-standards/math.html http://www.njcore.org/standards?processing=true# TUSD Department of Curriculum and Instruction Curriculum 3.0 Essential Concepts: The study of Geometry is based on three undefined terms: a point, a line, and a plane. These three notions are used to define all other terms in geometry. Some transformations preserve distance and angle, and some do not. Figures with symmetry can be transformed onto themselves. Rotations, reflections, and translations can be developed using mathematical definitions of general geometric terms. Using a variety of materials, we determine a sequence needed to carry a figure onto another. Rigid motions can be used to determine if two figures are congruent. When figures are congruent, all corresponding parts (e.g., angles, segments, and regions) are congruent. When certain corresponding parts of triangles are congruent, the triangles themselves are congruent. When lines intersect, certain relationships among the created angles are always true. The same is true of points on perpendicular bisectors of segments. In triangles, certain relationships among angles, sides, and other segments are always true. (Triangle Sum Theorem, Isosceles Triangle Theorem, Midsegment Theorem) In parallelograms, certain relationships among angles, sides, and other segments are always true. Essential Questions: What are the undefined terms in geometry and why are they undefined? Can you use these undefined terms to define angles, circles, parallel lines and line segments? What is the difference between a rigid transformation and a non-rigid transformation? What transformations are isometries? How is symmetry defined? Revised 5/11/2016 3:05 PM Page 2 2016-2017 Mathematics Curriculum Map Geometry, Q1 https://hcpss.instructure.com/courses/162 https://www.desmos.com/ http://www.geogebra.org/ http://ccssmath.org/?page_id=1306 http://www.cpalms.org/Public/ToolkitGradeLevelGroup/Toolkit?id=14 Can a figure have more than one type of symmetry? How do you draw the image of a figure under a reflection, rotation, and translation? What mathematical definitions do we use to rotate, reflect, and/or translate so that we can transform figures? What sequence of transformations will carry a figure onto another? How can congruence be represented through the transformations of figures? What is required to show that two triangles are congruent? What can we conclude about two triangles that are congruent? How do triangle congruence criteria follow from the rigid motion definition of congruence? What relationships between angles formed by intersections of lines are always true? What is true about the endpoints of a segment and its perpendicular bisector? What relationships among angles, sides, and other segments in a triangle are always true? What relationships among angles, sides, and other segments in parallelograms are always true? Vocabulary AAS (angle-angle-side congruence) acute triangle alternate exterior angles angle angle of rotation arc ASA (angle-side-angle congruence) base base angles bisect bisector center of rotation centroid circle circumcenter circumference Additional Resource: http://www.graniteschools.org/mathvocabulary/ dilation midpoint distance midsegment end point non-rigid transformation equiangular obtuse triangle equiangular triangle orthocenter equidistant output equilateral parallel lines equilateral triangle parallelogram expansion perpendicular exterior angle perpendicular lines plane horizontal stretch point hypotenuse point of symmetry image pre-image incenter quadrilateral initial point radius input TUSD Department of Curriculum and Instruction Curriculum 3.0 Revised 5/11/2016 3:05 PM rigid motion rigid transformation rotation rotational symmetry rotations same-side interior angles SAS (side-angle-side congruence) scalene triangle sequence of transformations sides of an angle square SSS (side-side-side congruence) supplementary supplementary angles symmetry terminal point Page 3 2016-2017 Mathematics Curriculum Map Geometry, Q1 collinear/noncollinear complementary angles composition of transformations consecutive angles contraction corresponding angles corresponding parts CPCTC (corresponding parts of congruent triangles are congruent) CPFTC (corresponding parts of congruent figures are congruent) diagonal diameter interior angle isometry isosceles triangle leg line line of reflection line of symmetry line segment linear pair locus mapping ray rectangle reflection reflectional symmetry reflections regular polygon remote angle rhombus right angle right triangle transformation translation transversal line trapezoid undefined term vector vertex angle vertex of an angle vertical angles vertical stretch Interdisciplinary Connections Multicultural Math Connections: https://www.deltacollege.edu/dept/basicmath/Multicultural_Math.htm http://www.edchange.org/multicultural/sites/math.html http://users.wfu.edu/mccoy/mgames.pdf http://www.ericdigests.org/1996-1/more.htm A Course in Multicultural Mathematics Integrating Mathematics of Worldwide Cultures into K-12 Teaching Mathematics through Multicultural Literature http://www.nea.org/tools/lessons/47756.htm https://www.teachervision.com/ Math Modeling Projects https://docs.google.com/spreadsheets/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkT QEsfqouLWNNC6Z4/pub?output=html Literacy Connections: http://mathsolutions.com/wpcontent/uploads/1995_Writing_in_Elem_Grades.pdf http://www.edutopia.org/blog/four-tips-writing-math-classroom-heatherwolpert-gawron http://files.eric.ed.gov/fulltext/ED544239.pdf http://www-tc.pbs.org/teacherline/courses/rdla230/docs/session_1_burns.pdf Writing Prompts for Math http://writingfix.com/wac/numberfix.htm http://www.nea.org/tools/lessons/47756.htm https://www.teachervision.com/ http://msms.ehe.osu.edu/2010/05/20/teaching-with-trade-books-math/ http://letsreadmath.com/math-and-childrens-literature/ Assessment Guides and Resources http://www.azed.gov/assessment/files/2015/12/math-pld-geometry.pdf http://achievethecore.org/category/1020/mathematics-assessments http://www.azed.gov/assessment/azmeritsupportmaterials/ http://www.ccsstoolbox.org/ http://www.insidemathematics.org/tools-for-educators http://map.mathshell.org/materials/index.php http://www.orglib.com/home.aspx http://schools.nyc.gov/Academics/CommonCoreLibrary/TasksUnitsStudentWor k/default.htm?s=Z3JhZGVzPTgmc3ViamVjdD0y TUSD Department of Curriculum and Instruction Curriculum 3.0 Instructional Resources DOK Levels http://www.azed.gov/assessment/files/2014/11/dok-levels.pdf DOK Stems http://www.azed.gov/assessment/files/2014/11/dok-question-stems.pdf Hess’s Matrix http://static.pdesas.org/content/documents/M2-Activity_2_Handout.pdf Bloom’s Taxonomy http://www.bloomstaxonomy.org/Blooms%20Taxonomy%20questions.pdf Revised 5/11/2016 3:05 PM Page 4 2016-2017 Mathematics Curriculum Map Geometry, Q1 https://www.illustrativemathematics.org/8 https://hcpss.instructure.com/courses/99 http://www.cpalms.org/Public/ToolkitGradeLevelGroup/Toolkit?id=14 TUSD Department of Curriculum and Instruction Curriculum 3.0 Revised 5/11/2016 3:05 PM Page 5