2016 Mathematics Standards of Learning Geometry Overview of

2016 Mathematics Standards of Learning Geometry Overview of Changes from 2009 to 2016 Related documents available on VDOE Mathematics 2016 webpage 1

Purpose • Overview of the 2016 Mathematics Standards of Learning and the Curriculum Framework • Highlight information included in the Essential Knowledge and Skills and the Understanding the Standard sections of the Curriculum Framework 2

Agenda • Overview and Implementation Timeline • Resources Currently Available – Crosswalk (Summary of Revisions) – Standards and Curriculum Frameworks • Comparison of 2009 to 2016 Standards – – Reasoning, Lines, and Transformations Triangles Polygons and Circles Three-Dimensional Figures 3

Implementation Timeline 2016 -2017 School Year – Curriculum Development VDOE staff provides a summary of the revisions to assist school divisions in incorporating the new standards into local written curricula for inclusion in the taught curricula during the 2017 -2018 school year. 2017 -2018 School Year – Crossover Year 2009 Mathematics Standards of Learning and 2016 Mathematics Standards of Learning are included in the written and taught curricula. Spring 2018 Standards of Learning assessments measure the 2009 Mathematics Standards of Learning and include field test items measuring the 2016 Mathematics Standards of Learning. 2018 -2019 School Year – Full-Implementation Year Written and taught curricula reflect the 2016 Mathematics Standards of Learning assessments measure the 2016 Mathematics Standards of Learning. 4

2016 SOL Revisions • Improve the vertical progression of mathematics content • Ensure developmental appropriateness of student expectations • Increase support for teachers in mathematics content • Clarify expectations for teaching and learning • Improve precision and consistency in mathematical language and format • Ensure proficiency of elementary students in computational skills 5

Support for Teachers • Significant additions to the Understanding the Standard column including – – Definitions Explanations Examples Instructional connections • Improvements in precision, clarity, and consistency in language K-12 • Indicators of SOL sub-bullet added to each bullet within the Essential Knowledge and Skills 6

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Overview of Changes Reporting Category # of Standards (2009) # of Standards (2016) Reasoning, Lines, and Transformations 4 4 Triangles 4 4 Polygons and Circles 4 4 Three-Dimensional Figures 2 2 14 14 Total 8

Mathematics Process Goals for Students “The content of the mathematics standards is intended to support the five process goals for students” - 2009 and 2016 Mathematics Standards of Learning Communication Connections Problem Solving Representations Mathematical Understanding Reasoning 9

Standards of Learning Curriculum Frameworks Introduction includes: • Mathematical Process Goals for Students • Instructional Technology • Computational Fluency New • Algebra Readiness • Equity 10

2009 2016 11

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REASONING, LINES, AND TRANSFORMATIONS 13

2009 SOL 2016 SOL G. 1 The student will construct and judge the validity G. 1 The student will use deductive reasoning to of a logical argument consisting of a set of construct and judge the validity of a logical premises and a conclusion. This will include argument consisting of a set of premises and a a) identifying the converse, inverse, and conclusion. This will include contrapositive of a conditional statement; a) identifying the converse, inverse, and b) translating a short verbal argument into contrapositive of a conditional statement; symbolic form; b) translating a short verbal argument into c) using Venn diagrams to represent set symbolic form; and relationships; [Moved to DM. 12] and c) determining the validity of a logical argument. d) using deductive reasoning. Revisions: • 2009 SOL G. 1 c – Use of Venn diagrams to represent set relationships and interpreting Venn diagrams was moved to Discrete Mathematics (SOL DM. 12) • Determine the validity of a logical argument was moved from standard stem of 2009 SOL G. 1 to 2016 SOL G. 1 c • Using deductive reasoning was moved from the 2009 SOL G. 1 d to the stem of the 2016 SOL G. 1 • SOL G. 1 c EKS - Determine that an argument is false using a counterexample is now included 14

2009 SOL 2016 SOL G. 2 The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. us G. 2 The student will use the relationships between angles formed by two lines intersected by a transversal to a) prove two or more lines are parallel; and b) solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. Revisions: • 2009 SOL G. 2 c is now SOL G. 2 b in the 2016 standards • 2016 SOL G. 2 a combines SOL G. 2 a and SOL G. 2 b from the 2009 standards • 2016 SOL G. 2 b EKS - Includes same-side exterior angles as a required relationship to be included between pairs of angles • 2016 SOL G. 2 b - Real world problems are now referred to as practical problems in 2016 15

2009 SOL 2016 SOL G. 3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include G. 3 The student will solve problems involving symmetry and transformation. This will include a) investigating and using formulas for determining distance, midpoint, and slope; b) applying slope to verify and determine whether lines are parallel or perpendicular; c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. a) investigating and using formulas for finding distance, midpoint, and slope; b) applying slope to verify and determine whether lines are parallel or perpendicular; c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Revisions: • “Use of pictorial representations, including computer software…. ” wording that was included in the stem of the 2009 G. 3 standard is now included in the Understanding the Standard section of the 2016 G. 3 standard • SOL G. 3 a EKS - Includes the use of the midpoint formula to determine the endpoint of a segment in addition to the midpoint • SOL G. 3 d EKS – Provides clarification that transformations may be a combination of transformations, and a list of possible transformations is included; reflections limited to reflections over any horizontal line, vertical line, the line y = x, or the line y = -x; rotations limited to 90°, 180°, 270°, or 360° on a coordinate grid where the center of rotation at the origin; dilations limited to those from a fixed point on the coordinate grid 16

2009 SOL 2016 SOL G. 4 The student will construct and justify the constructions of a) a line segment congruent to a given line segment; b) the perpendicular bisector of a line segment; c) a perpendicular to a given line from a point not on the line; d) a perpendicular to a given line at a given point e) the bisector of a given angle; on the line; f) an angle congruent to a given angle; e) the bisector of a given angle; g) a line parallel to a given line through a point not f) an angle congruent to a given angle; and on the line; and g) a line parallel to a given line through a point h) an equilateral triangle, a square, and a regular not on the given line. hexagon inscribed in a circle. G. 4 The student will construct and justify the constructions of Revisions: • 2009 SOL G. 4 EKS addressing constructing the inscribed and circumscribed circles of a triangle and constructing a tangent line from a point outside a given circle to the circle are NOT included in 2016 SOL G. 4 • 2009 SOL G. 4 EKS of constructing an equilateral triangle, a square, and a regular hexagon inscribed in a circle are now included in SOL G. 4 h 17

TRIANGLES 18

2009 SOL 2016 SOL G. 5 The student, given information concerning the G. 5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will solve problems, including practical a) order the sides by length, given the angle problems. This will include measures; a) ordering the sides by length, given angle b) order the angles by degree measure, given the measures; side lengths; b) ordering the angles by degree measure, given c) determine whether a triangle exists; and side lengths; d) determine the range in which the length of the third side must lie. c) determining whether a triangle exists; and These concepts will be considered in the context d) determining the range in which the length of of real-world situations. the third side must lie. Revisions: • SOL G. 5 - Real world problems are now referred to as practical problems in 2016 19

2009 SOL 2016 SOL G. 6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. G. 6 The student, given information in the form of a figure or statement, will prove two triangles are congruent. Revisions: • Wording from 2009 SOL G. 6 standard regarding using algebraic and coordinate methods as well as deductive proofs is now included in the Understanding the Standard section of the 2016 SOL G. 6. • In 2016 SOL G. 6, G. 7, and G. 9 proofs are referred to as “direct proofs” and the following statement was added to the Understanding the Standard section: “Deductive or inductive reasoning is used in mathematical proofs. In this course, deductive reasoning and logic are used in direct proofs. Direct proofs are presented in different formats (typically two-column or paragraph) and employ definitions, postulates, theorems, and algebraic justifications including coordinate methods” 20

2009 SOL G. 7 The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. 2016 SOL G. 7 The student, given information in the form of a figure or statement, will prove two triangles are similar. Revisions: • Wording from 2009 SOL G. 7 regarding using algebraic and coordinate methods as well as deductive proofs is now included in the Understanding the Standard section of 2016 SOL G. 7. • In 2016 SOL G. 6, G. 7, and G. 9 proofs are referred to as “direct proofs” and the following statement was added to the Understanding the Standard section: “Deductive or inductive reasoning is used in mathematical proofs. In this course, deductive reasoning and logic are used in direct proofs. Direct proofs are presented in different formats (typically two-column or paragraph) and employ definitions, postulates, theorems, and algebraic justifications including coordinate methods” 21

2009 SOL G. 8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. 2016 SOL G. 8 The student will solve problems, including practical problems, involving right triangles. This will include applying a) the Pythagorean Theorem and its converse; b) properties of special right triangles; and c) trigonometric ratios. Revisions: • SOL G. 8 b EKS – Clarified that solving problems for missing lengths in right triangles may include situations where rationalizing denominators may be necessary • SOL G. 8 c EKS – Clarified that solving problems involving right triangles may include determining missing side lengths or angle measurements • SOL G. 8 - Real world problems are now referred to as practical problems in 2016 22

POLYGONS AND CIRCLES 23

2009 SOL 2016 SOL G. 9 The student will verify characteristics of G. 9 The student will verify and use properties of quadrilaterals to solve problems, including to solve real-world problems. practical problems. Revisions: • SOL G. 9 - Students will verify and use properties quadrilaterals; the term “properties” vs. characteristics is used in 2016 • • SOL G. 9 - Real world problems are now referred to as practical problems in 2016 In 2016 SOL G. 6, G. 7, and G. 9 proofs are referred to as “direct proofs” and the following statement was added to the Understanding the Standard section: “Deductive or inductive reasoning is used in mathematical proofs. In this course, deductive reasoning and logic are used in direct proofs; direct proofs are presented in different formats (typically two-column or paragraph) and employ definitions, postulates, theorems, and algebraic justifications including coordinate methods” SOL G. 9 EKS - To prove properties of angles for a quadrilateral inscribed in a circle moved to the Understanding the Standard section of 2016 G. 11 • 24

2009 SOL G. 10 The student will solve real-world problems involving angles of polygons. 2016 SOL G. 10 The student will solve problems, including practical problems, involving angles of convex polygons. This will include determining the a) sum of the interior and/or exterior angles; b) measure of an interior and/or exterior angle; and c) number of sides of a regular polygon. Revisions: • G. 10 b EKS - Includes determining angle measures of a regular polygon in a tessellation • SOL G. 10 clarifies that solving problems involves angles of convex polygons • SOL G. 10 - Real world problems are now referred to as practical problems in 2016 25

2009 SOL 2016 SOL G. 11 The student will use angles, arcs, chords, G. 11 The student will solve problems, including tangents, and secants to practical problems, by applying properties of a) investigate, verify, and apply properties of circles. This will include determining circles; a) angle measures formed by intersecting chords, b) solve real-world problems involving properties secants, and/or tangents; of circles; and b) lengths of segments formed by intersecting c) find arc lengths and areas of sectors in circles. chords, secants, and/or tangents; c) arc length; and d) area of a sector. Revisions: • 2009 SOL G. 11 a (investigate, verify, and apply properties of circles) was moved to the standard stem of 2016 SOL G. 11 • 2009 SOL G. 11 b (solve real-world problems involving properties of circles) was moved to the standard stem of 2016 SOL G. 11 • 2009 SOL G. 11 c (area of a sector ) was moved to 2016 SOL G. 11 d • SOL G. 11 - Real world problems are now referred to as practical problems in 2016 26

2009 SOL G. 12 The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle. 2016 SOL G. 12 The student will solve problems involving equations of circles. Revisions: • SOL G. 12 EKS - Clarifies expectations for solving problems involving equations of circles 27

THREE-DIMENSIONAL FIGURES 28

2009 SOL G. 13 The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems. 2016 SOL G. 13 The student will use surface area and volume of three-dimensional objects to solve practical problems. Revisions: • SOL G. 13 EKS - Includes determining the surface area and volume of hemispheres • SOL G. 13 EKS – Includes solving problems involving the lateral area of cylinders, prisms, and regular pyramids • The Understanding the Standard portion of the 2016 SOL G. 13 clarifies the types of cylinders, cones, prisms, and pyramids used in solving practical problems involving volume and surface area 29

2009 SOL 2016 SOL G. 14 The student will use similar geometric objects in G. 14 The student will apply the concepts of similarity two- or three-dimensions to to two- or three-dimensional geometric figures. a) compare ratios between side lengths, This will include perimeters, areas, and volumes; a) comparing ratios between lengths, b) determine how changes in one or more perimeters, areas, and volumes of similar dimensions of an object affect area and/or figures; volume of the object; b) determining how changes in one or more c) determine how changes in area and/or volume dimensions of a figure affect area and/or of an object affect one or more dimensions of volume of the figure; the object; and c) determining how changes in area and/or d) solve real-world problems about similar volume of a figure affect one or more geometric objects. dimensions of the figure; and d) solving problems, including practical problems, about similar geometric figures. Revisions: • Geometric “objects” from 2009 SOL G. 14 are now referred to as geometric “figures” in the 2016 G. 14 standard 30

Questions? Please contact the VDOE Mathematics Team Mathematics@doe. virginia. gov 31