UNIT 1 GEOMETRY TEMPLATE CREATED BY REGION 1 ESA UNIT 1

Transcription

1 UNIT 1 GEOMETRY TEMPLATE CREATED BY REGION 1 ESA UNIT 1

2 Traditional Pathway: Geometry 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 s 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 six units are as follows. Critical Area 1: 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. Critical Area 2: 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. Critical Area 3: 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. Critical Area 4: 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. Template created by Region 1 ESA Page 2 of 16

3 Critical Area 5: 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. Critical Area 6: Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions. e: It is important to note that the units (or critical areas) are intended to convey coherent groupings of content. The clusters and standards within units are ordered as they are in the Common Core State s, and are not intended to convey an instructional order. Considerations regarding constraints, extensions, and connections are found in the instructional notes. The instructional notes are a critical attribute of the courses and should not be overlooked. For example, one will see that standards such as A.CED.1 and A.CED.2 are repeated in multiple courses, yet their emphases change from one course to the next. These changes are seen only in the instructional notes, making the notes an indispensable component of the pathways. (All instructional notes/suggestions will be found in italics throughout this document) - Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol Geometry Unit 1 Overview: 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. Template created by Region 1 ESA Page 3 of 16

4 Unit 1: Congruence, Proof, and Constructions- G.CO.1 Cluster: Experiment with transformations in the plane. Instructional es: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. I can indentify undefined notions used in geometry (point, line, plane, distance) and describe their characteristics. I can identify angles, circles, perpendicular lines, parallel lines, rays, and line segments. 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. I can define angles, circles, perpendicular lines, parallel lines, rays, and line segments precisely using the undefined terms and ifthen and if-and-only-if statements. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Undefined terms, point, line, plane, distance, angle, circle, perpendicular, parallel, line segment, arc, ray, vertex, equidistant, intersect, right angle Template created by Region 1 ESA Page 4 of 16

5 Unit 1: Congruence, Proof, and Constructions- G.CO.2 Cluster: Experiment with transformations in the plane. Instructional es: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. 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). I can draw transformations of reflections, rotations, translations, and combinations of these using graph paper, transparencies, and/or geometry software. I can determine the coordinates for the image (output) of a figure when a transformation rule is applied to the preimage (input). I can distinguish between transformations that are rigid (preserve distance and angle measure-reflections, rotations, translations, or combinations of these) and those that are not (dilations or rigid motions followed by dilations). #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Transformation, reflection, rotation, translation, dilation, image, pre-image, rigid motion, input, output, coordinates, distance, angle measure Template created by Region 1 ESA Page 5 of 16

6 Unit 1: Congruence, Proof, and Constructions- G.CO.3 Cluster: Experiment with transformations in the plane. Instructional es: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. I can describe and illustrate how a rectangle is mapped onto itself using transformations. I can describe and illustrate how a parallelogram is mapped onto itself using transformations. I can describe and illustrate how an isosceles trapezoid is mapped onto itself using transformations. I can calculate the number of lines of reflection symmetry and the degree of rotational symmetry of any regular polygon. #1 Make sense of problems and persevere in solving #2 Reason abstractly and quantitatively. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. #6 Attend to precision. #7 Look for and make use of structure #8 Look for and express regularity in repeated reasoning. In what ways can congruence be useful? Proving and applying congruence provides a basis for modeling situations geometrically. Assessments align to suggested learning targets. Drill and practice Multiple choice Short answer (written) Performance (verbal explanation) Product / Project Rectangle, parallelogram, trapezoid, isosceles trapezoid, regular polygon, rotational symmetry, reflection symmetry, mapped onto Template created by Region 1 ESA Page 6 of 16

7 Unit 1: Congruence, Proof, and Constructions- G.CO.4 Cluster: Experiment with transformations in the plane. Instructional es: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Geometry Math Tool Unit 1 I can construct the reflection definition by connecting any point on the pre-image to its corresponding point on the reflected image and describing the line segment s relationship to the line of reflection (i.e., the line of reflection is the perpendicular bisector of the segment. I can construct the translation definition by connecting any point on the pre-image to its corresponding point on the translated image and connecting a second point on the pre-image to its corresponding point on the translated image, and describing how the two segments are equal in length, point in the same direction, and are parallel. I can construct the rotation definition by connecting the center of rotation to any point on the pre-image and to its corresponding point on the rotated image, and describing the measure of the angle formed and the equal measures of the segments that formed the angle as part of the definition. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Rotation, reflection, translation, perpendicular bisector, line segment, pre-image, image, parallel lines, angle, center of rotation Template created by Region 1 ESA Page 7 of 16

8 Unit 1: Congruence, Proof, and Constructions- G.CO.5 Cluster: Experiment with transformations in the plane. Instructional es: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. 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. I can draw a specific transformation when given a geometric figure and a rotation, reflection, or translation. I can predict and verify the sequence of transformations (a composition) that will map a figure onto another. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Refection, rotation, translation, figure, map, transformation, composition Template created by Region 1 ESA Page 8 of 16

9 Unit 1: Congruence, Proof, and Constructions- G.CO.6 Cluster: Understand congruence in terms of rigid motions. Instructional es: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. 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. Geometry Math Tool Unit 1 I can define rigid motions as reflections, rotations, translations, and combinations of these, all of which preserve distance and angle measure. I can define congruent figures as figures that have the same shape and size and state that a composition of rigid motions will map one congruent figure onto another. I can predict the composition of transformations that will map a figure onto a congruent figure. I can determine if two figures are congruent by determining if rigid motions will turn one figure into the other. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Congruence, composition, rigid motions, map, reflections, rotation, translation, transformation, angle measure, distance Template created by Region 1 ESA Page 9 of 16

10 Unit 1: Congruence, Proof, and Constructions G.CO.7 Cluster: Understand congruence in terms of rigid motions. Instructional es: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. 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. I can identify corresponding sides and corresponding angles of congruent triangles. I can explain that in a pair of congruent triangles, corresponding sides are congruent (distance is preserved) and corresponding angles are congruent (angle measure is preserved). I can 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. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Rigid motions, reflection, rotation, translation, distance, angle measure, congruent, composition, map, figure, corresponding sides, corresponding angles, triangle Template created by Region 1 ESA Page 10 of 16

11 Unit 1: Congruence, Proof, and Constructions G.CO.8 Cluster: Understand congruence in terms of rigid motions. Instructional es: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. I can define rigid motions as reflections, rotations, translations, and combinations of these, all of which preserve distance and angle measure. I can list the sufficient conditions to prove triangles are congruent. 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. I can map a triangle with one of the sufficient conditions (e.g., SSS) onto the original triangle and show that corresponding sides and corresponding angles are congruent. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Rigid motion, reflection, rotation, translation, congruent, composition, map, SSS, SAS, ASA Template created by Region 1 ESA Page 11 of 16

12 Unit 1: Congruence, Proof, and Constructions G.CO.9 Cluster: Prove geometric theorems. Instructional es: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in twocolumn format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. 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. Geometry Math Tool Unit 1 I can identify and use the properties of congruence and equality (reflexive, symmetric, transitive) in my proofs. I can order statements based on the Law of Syllogism when constructing my proof. I can correctly interpret geometric diagrams by identifying what can and cannot be assumed. I can use the theorems, postulates, or definitions to prove theorems about lines and angles, including: a) vertical angles are congruent, b) when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent, and same-side interior angles are supplementary, c) points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Theorem, linear pair, vertical angles, alternate interior angles, alternate exterior, same-side interior angles, corresponding angles, perpendicular bisector, supplementary angles, complimentary angles, equidistant, congruent, adjacent, consecutive/non-consecutive, reflection, Law of Syllogism Template created by Region 1 ESA Page 12 of 16

13 Unit 1: Congruence, Proof, and Constructions G.CO.9 Cluster: Prove geometric theorems. Instructional es: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in twocolumn format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of G.CO.10 may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for G.C.3 in Unit 5. 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. Geometry Math Tool Unit 1 Suggested Learning Target I can order statements based on the Law of Syllogism when constructing my proof. O can correctly interpret geometric diagrams (what can and cannot be assumed). I can use theorems, postulates, or definitions to prove theorems about triangles, including: a) measures of interior angles of a triangle sum to 180⁰, b) base angles of isosceles triangles are congruent, c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, d) the medians of a triangle meet at a point. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Midpoint, mid-segment isosceles triangle, median, centroid, coordinate proof, adjacent, consecutive/non-consecutive, Law of Syllogism Template created by Region 1 ESA Page 13 of 16

14 Unit 1: Congruence, Proof, and Constructions G.CO.11 Cluster: Prove geometric theorems. Instructional es: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in twocolumn format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning G.CO.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. Geometry Math Tool Unit 1 I can use theorems, postulates, or definitions to prove theorems about parallelograms, including: Prove opposite sides of a parallelogram are congruent Prove opposite angles of a parallelogram are congruent Prove the diagonals of a parallelogram bisect each other Prove that rectangles are parallelograms with congruent diagonals. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Quadrilateral, parallelogram, rectangle, diagonals, distance formula, midpoint formula, slope, bisector, congruence properties Template created by Region 1 ESA Page 14 of 16

15 Unit 1: Congruence, Proof, and Constructions G.CO.12 Cluster: Make geometric constructions. Instructional es: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with 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. and Type I can identify the tools used in formal constructions. I can use tools and methods to precisely copy a segment, copy an angle, bisect a segment, bisect and angle, construct perpendicular lines and bisectors, and construct a line parallel to a give line through a point not on the line. I can informally perform the constructions listed above using string, reflective devices, paper folding, and/or dynamic geometric software. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Segment, angle, perpendicular lines, perpendicular bisector, parallel lines, bisect, formal construction, informal construction, compass, straightedge Template created by Region 1 ESA Page 15 of 16

16 Unit 1: Congruence, Proof, and Constructions G.CO.13 Cluster: Make geometric constructions. Instructional es: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with and Type I can define inscribed polygons (the vertices of the figure must be points on the circle. I can construct an equilateral triangle inscribed in a circle. G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. I can construct a square inscribed in a circle. I can construct a hexagon inscribed in a circle. I can explain the steps to construction an equilateral triangle, a square, and a regular hexagon inscribed in a circle. #1 Make sense of problems and persevere in solving In what ways can congruence be useful? Assessments align to suggested learning targets. Proving and applying congruence provides a #2 Reason abstractly and quantitatively. basis for modeling situations geometrically. #3 Construct viable arguments and critique the reasoning #4 Model with mathematics #5 Use appropriate tools strategically. Drill and practice Multiple choice Short answer (written) #6 Attend to precision. Performance (verbal explanation) #7 Look for and make use of structure Product / Project #8 Look for and express regularity in repeated reasoning. Construction, equilateral triangle, square, regular hexagon inscribe, circle Template created by Region 1 ESA Page 16 of 16