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Math Standards Document Geometry Final Draft Hawaii School Districts Prepared by Marzano and Associates 1 Geometry Summary Chapter Strand(s) Topic(s) Basics of Geometry and Geometry Spatial Sense Standard(s) 5 Reasoning and Proof Geometry and Spatial Sense 5 Perpendicular and Parallel Lines Congruent Triangles Geometry and Spatial Sense 5 Geometry and Spatial Sense 5 Geometric Shapes and Their Properties and Relationships Quadrilaterals Geometry and Spatial Sense 5 Geometry and Spatial Sense 5 Transformations Numbers and Operation 1,3,8 Similarity Geometry and Spatial Sense 5 Right Triangles and Trigonometry Circles Measurement, Geometry and Spatial Sense Geometry and Spatial Sense 4 Areas of Polygons and Circles Surface Area and Volume Measurement 4 Measurement 4, 7 5 Number of Elements Level 4.0- 1 Level 3.0- 1 Level 2.0- 1 Level 4.0- 1 Level 3.0- 3 Level 2.0- 3 Level 4.0- 1 Level 3.0- 3 Level 2.0- 3 Level 4.0- 1 Level 3.0- 3 Level 2.0- 2 Level 4.0- 2 Level 3.0- 4 Level 2.0- 4 Level 4.0- 1 Level 3.0- 2 Level 2.0- 1 Level 4.0- 4 Level 3.0- 2 Level 2.0- 3 Level 4.0- 1 Level 3.0- 2 Level 2.0- 2 Level 4.0- 2 Level 3.0- 4 Level 2.0- 4 Level 4.0- 1 Level 3.0- 2 Level 2.0- 2 Level 4.0- 1 Level 3.0- 2 Level 2.0- 3 Level 4.0-2 Level 3.0- 4 Level 2.0- 5 32 Total of Levels 3.0 2 Chapter Topic: Basics of Geometry Strand: Geometry and Spatial Sense Standard 5: PROPERTIES AND RELATIONSHIPS: Analyze properties of objects and relationships among the properties. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: using logic skills, deduces what kind of figure to draw Level 3.5 Level 3.0 While engaged in tasks regarding geometric shapes and their properties and relationships the student will: (MA.G.5.1) use inductive and deductive reasoning to create and defend geometric conjectures (e.g., write a logically sound proof (two-column, paragraph form) to defend conjectures) The student exhibits no major errors or omissions. Level 2.5 Level 2.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o point o line o plane o ray o line segment o inductive reasoning o deductive reasoning performs basic processes such as: o coming up with rational explanations to defend a geometric conjecture However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 3 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Given certain conditions, ask students to use logical deduction to draw a geometric shape. Level 3.0 Ask students to use inductive and deductive reasoning to create and defend geometric conjectures. Level 2.0 Ask students to define “point,” “line,” “plane,” “ray,” “line segment.” Ask students to come up with rational explanations to defend a geometric conjecture. 4 Chapter Topic: Reasoning and Proof Strand: Geometry and Spatial Sense Standard 5: PROPERTIES AND RELATIONSHIPS: Analyze properties of objects and relationships among the properties. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: proves geometric conjectures Level 3.5 Level 3.0 While engaged in tasks regarding geometric shapes and their properties and relationships the student will: (MA.G.5.1) use inductive and deductive reasoning to create and defend geometric conjectures (e.g., write a logically sound proof (two-column, paragraph form) to defend conjectures) (MA.G.5.3) explain properties and characteristics of angle bisectors, perpendicular bisectors, and parallel lines (e.g., use a straight edge and compass to construct angle bisectors, perpendicular bisectors and parallel lines) (First Quarter) (MA.G.5.4) use the relationship between pairs of angles (complimentary, supplementary, vertical, exterior, interior) to determine unknown angle measures or definitions of properties (e.g., use complimentary, supplementary, vertical, exterior, and interior angles that are formed when two parallel lines are cut by a transversal to find the measure of an unknown angle) (First Quarter) The student exhibits no major errors or omissions. Level 2.5 Level 2.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o bisect o complementary angles o supplementary angles o congruent angles performs basic processes such: o coming up with rational explanations to defend a geometric conjecture o defining perpendicular and bisector; explaining properties and characteristics of angle bisectors and parallel lines o defining and recognizing or recalling examples of complimentary, supplementary, vertical, interior and exterior angles However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 5 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to prove a geometric conjecture (in a statement – reason or paragraph form). Ask students to explain/justify why their constructions are bisected or parallel by naming the appropriate angle relationship. Level 3.0 Ask students to use inductive and deductive reasoning to create and defend geometric conjectures. Ask students to explain properties and characteristics of angle bisectors, perpendicular bisectors, and parallel lines. Ask students to use the relationship between pairs of angles (complimentary, supplementary, vertical, exterior, interior) to determine unknown angle measures or definitions of properties. Level 2.0 Ask students to define “bisect,” “complementary angles,” “supplementary angles,” “congruent angles.” Ask students to come up with rational explanations to defend a geometric conjecture. Ask students to define perpendicular and bisector; be able to explain the properties and characteristics of angle bisectors and parallel line. Ask students to define and recognize or recall examples of complimentary, supplementary, vertical, interior and exterior angles. 6 Topic: Perpendicular and Parallel Lines Strand: Geometry and Spatial Sense Standard 5: PROPERTIES AND RELATIONSHIPS: Analyze properties of objects and relationships among the properties. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: explains/justifies why their constructions are bisected or parallel by naming the appropriate angle relationship Level 3.5 Level 3.0 While engaged in tasks regarding geometric shapes and their properties and relationships the student will: (MA.G.5.1) use inductive and deductive reasoning to create and defend geometric conjectures (e.g., write a logically sound proof (two-column, paragraph form) to defend conjectures) (MA.G.5.3) explain properties and characteristics of angle bisectors, perpendicular bisectors, and parallel lines (e.g., use a straight edge and compass to construct angle bisectors, perpendicular bisectors and parallel lines) (MA.G.5.4) use the relationship between pairs of angles (complimentary, supplementary, vertical, exterior, interior) to determine unknown angle measures or definitions of properties (e.g., use complimentary, supplementary, vertical, exterior, and interior angles that are formed when two parallel lines are cut by a transversal to find the measure of an unknown angle) The student exhibits no major errors or omissions. Level 2.5 Level 2.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o bisect o complementary angles o supplementary angles o congruent angles performs basic process such as: o coming up with rational explanations to defend a geometric conjecture o defining perpendicular and bisector; explaining properties and characteristics of angle bisectors and parallel lines o defining and recognizing or recalling examples of complimentary, supplementary, vertical, interior and exterior angles However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 7 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to explain/justify why their constructions are bisected or parallel by naming the appropriate angle relationship. Level 3.0 Ask students to use inductive and deductive reasoning to create and defend geometric conjectures. Ask students to explain properties and characteristics of angle bisectors, perpendicular bisectors, and parallel lines. Ask students to use the relationship between pairs of angles (complimentary, supplementary, vertical, exterior, interior) to determine unknown angle measures or definitions of properties. Level 2.0 Ask students to define “bisect,” “complementary angles,” “supplementary angles,” “congruent angles.” Ask students to come up with rational explanations to defend a geometric conjecture. Ask students to define perpendicular and bisector; be able to explain the properties and characteristics of angle bisectors and parallel line. Ask students to define and identify complimentary, supplementary, vertical, interior and exterior angles. 8 Chapter Topic: Congruent Triangles Strand: Geometry and Spatial Sense Standard 5: PROPERTIES AND RELATIONSHIPS: Analyze properties of objects and relationships among the properties. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: uses the knowledge of interior angles of a polygon to determine unknown angle measures Level 3.5 Level 3.0 While engaged in tasks regarding geometric shapes and their properties and relationships the student will: (MA.G.5.1) use inductive and deductive reasoning to create and defend geometric conjectures (e.g., write a logically sound proof (two-column, paragraph form) to defend conjectures) (MA.G.5.2) use the concept of corresponding parts to prove that triangles and other polygons are congruent (e.g., identify enough corresponding parts to determine if two shapes are congruent) (MA.G.5.4) use the relationship between pairs of angles (complimentary, supplementary, vertical, exterior, interior) to determine unknown angle measures or definitions of properties (e.g., use complimentary, supplementary, vertical, exterior, and interior angles that are formed when two parallel lines are cut by a transversal to find the measure of an unknown angle) The student exhibits no major errors or omissions. Level 2.5 Level 2.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o corresponding parts o congruent shapes performs basic processes such as: o coming up with rational explanations to defend a geometric conjecture o recognizing or recalling examples of corresponding parts of polygons; proving triangles are congruent; recognizing or recalling accurate statements about the difference between similarity and congruence However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 9 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to use knowledge of interior angles of a polygon to determine unknown angle measures. Level 3.0 Ask students to use inductive and deductive reasoning to create and defend geometric conjectures. Ask students to use the concept of corresponding parts to prove that triangles and other polygons are congruent. Level 2.0 Ask students to define “corresponding parts,” “congruent shapes.” Ask students to come up with rational explanations to defend a geometric conjecture. Ask students to recognize or recall examples of corresponding parts of polygons; prove triangles are congruent; recognize or recall accurate statements about the difference between similarity and congruence. 10 Chapter Topic: Geometric Shapes and Their Properties and Relationships Strand: Geometry and Spatial Sense Standard 5: PROPERTIES AND RELATIONSHIPS: Analyze properties of objects and relationships among the properties. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: explains/justifies why their constructions are bisected or parallel by naming the appropriate angle relationship uses the knowledge of interior angles of a polygon to determine unknown angle measures applies the properties of angle, perpendicular bisectors, parallel lines and pairs of angles in a given situation Level 3.5 Level 3.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. While engaged in tasks regarding geometric shapes and their properties and relationships the student will: (MA.G.5.1) use inductive and deductive reasoning to create and defend geometric conjectures (e.g., write a logically sound proof (two-column, paragraph form) to defend conjectures) (Throughout the year) (MA.G.5.2) use the concept of corresponding parts to prove that triangles and other polygons are congruent (e.g., identify enough corresponding parts to determine if two shapes are congruent) (Second Quarter) (MA.G.5.3) explain properties and characteristics of angle bisectors, perpendicular bisectors, and parallel lines (e.g., use a straight edge and compass to construct angle bisectors, perpendicular bisectors and parallel lines) (First Quarter) (MA.G.5.4) use the relationship between pairs of angles (complimentary, supplementary, vertical, exterior, interior) to determine unknown angle measures or definitions of properties (e.g., use complimentary, supplementary, vertical, exterior, and interior angles that are formed when two parallel lines are cut by a transversal to find the measure of an unknown angle) (First Quarter) The student exhibits no major errors or omissions. Level 2.5 No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. 11 Level 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o corresponding parts o congruent shapes o bisect o complementary angles o supplementary angles o congruent angles o chords o tangents o circumference o arcs o radius o diameter o inscribe o circumscribe performs basic processes such as: o coming up with rational explanations to defend a geometric conjecture (all 4 quarters) o recognizing or recalling examples of corresponding parts of polygons; proving triangles are congruent; recognizing or recalling accurate statements about the difference between similarity and congruence (second quarter) o defining perpendicular and bisector; explaining properties and characteristics of angle bisectors and parallel lines (first quarter) o defining and recognizing or recalling examples of complimentary, supplementary, vertical, interior and exterior angles (first quarter) However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 12 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to use knowledge of interior angles of a polygon to determine unknown angle measures. Ask students to explain/justify why their constructions are bisected or parallel by naming the appropriate angle relationship. Ask students to apply the properties of angle bisectors, perpendicular bisectors, parallel lines and pairs of angles in a given situation. Level 3.0 Ask students to use inductive and deductive reasoning to create and defend geometric conjectures. Ask students to use the concept of corresponding parts to prove that triangles and other polygons are congruent. Level 2.0 Ask students to define “corresponding parts,” “congruent shapes.” Ask students to come up with rational explanations to defend a geometric conjecture. Ask students to recognize or recall examples of corresponding parts of polygons; prove triangles are congruent; recognize or recall accurate statements about the difference between similarity and congruence. Ask students define perpendicular and bisector; explain properties and characteristics of angle bisectors and parallel lines. Ask students to define and recognize or recall examples of complimentary, supplementary, vertical, interior and exterior angles. 13 Chapter Topic: Quadrilaterals Strand: Geometry and Spatial Sense Standard 5: PROPERTIES AND RELATIONSHIPS: Analyze properties of objects and relationships among the properties. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: explains/justifies why a quadrilateral is classified as a special quadrilateral by it’s properties Level 3.5 Level 3.0 While engaged in tasks regarding geometric shapes and their properties and relationships the student will: (MA.G.5.1) use inductive and deductive reasoning to create and defend geometric conjectures (e.g., write a logically sound proof (two-column, paragraph form) to defend conjectures). (MA.G.5.2) use the concept of corresponding parts to prove that triangles and other polygons are congruent (e.g., identify enough corresponding parts to determine if two shapes are congruent). The student exhibits no major errors or omissions. Level 2.5 Level 2.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o quadrilateral o parallelogram o rectangle o square o trapezoid o isosceles trapezoid o kite o rhombus performs basic processes such as: o coming up with rational explanations to defend a geometric conjecture However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 14 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to explain/justify why their constructions are bisected or parallel by naming the appropriate angle relationship. Ask students to use knowledge of interior angles of a polygon to determine unknown angle measures. Level 3.0 Ask students to use inductive and deductive reasoning to create and defend geometric conjectures. Ask students to use the concept of corresponding parts to prove that triangles and other polygons are congruent. Level 2.0 Ask students to define “quadrilateral,” “parallelogram,” “rectangle,” “square,” “rhombus,” “kite,” “trapezoid,” “isosceles trapezoid.” Ask students to come up with rational explanations to defend a geometric conjecture. 15 Chapter Topic: Transformations Strand: Numbers and Operations Standard 1: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Standard 3: COMPUTATION STRATEGIES: Use computational tools and strategies fluently and, when appropriate, use estimation. Strand: Geometry and Spatial Sense Standard 8: REPRESENTATIONAL SYSTEMS: Select and use different representational systems, including coordinate geometry. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: justifies why a situation can be represented by vectors uses vector addition, subtraction, and scalar multiplication to solve problems, then justify why their answer(s) is/are correct applies knowledge of the formulas to solve real-world problems creates an original Frieze pattern that includes more than one transformation (TRHVG) Level 3.5 Level 3.0 While involved in tasks related to vectors the student will: (MA.G.1.1) recognize situations that can be represented by vectors (e.g., decide if the information in a problem can be represented with vectors, and if it can, show how to use the vectors) (MA.G.3.1) use vector addition, subtraction, and scalar multiplication to solve problems (e.g., represent the information in a problem with vectors, and perform the appropriate operation(s) numerically and geometrically to solve the problem) The student exhibits no major errors or omissions. Level 2.5 Level 2.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o direction o magnitude o scalar multiplication o vector o midpoint o rotate o translate o reflect performs basic processes such as: o defining vector (direction and magnitude) o using vector addition and subtraction to solve problems o given coordinates, recognizing or recalling a similar situation and applying prior knowledge of the appropriate formula However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. 16 Level 0.0 Even with help, no understanding or skill demonstrated. 17 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to justify why a situation can be represented by vectors. Ask students to use vector addition, subtraction, and scalar multiplication to solve problems, then justify why their answer(s) is/are correct. Ask students to apply knowledge of the formulas to solve real-world problems. Ask students to create an original Frieze pattern that includes more than one transformation. Level 3.0 Ask students to determine if a situation can be represented by vectors. Ask students to use vector addition, subtraction, and/or scalar multiplication to solve word problems. Ask students to use coordinate geometry to produce formulas and prove theorems for the midpoint of a line segment, the distance formula, and forms of equations of lines and circles. Ask the students to describe the concept of rigid motion on figures in the coordinate plane, including rotation, translation, and reflection. Level 2.0 Ask student s to define “direction” and “magnitude” as they relate to vectors. Ask students to define vector (must state magnitude and direction in the answer). Ask students to define “vector” and “scalar multiplication.” Ask students to use vector addition and subtraction to solve problems where the vectors are given. Ask students to define “midpoint,” “rotate,” “translate,” and “reflect” as they relate to geometry. Ask students to recognize or recall a similar situation and apply prior knowledge of the appropriate formula, when given coordinates. 18 Chapter Topic: Similarity Strand: Geometry and Spatial Sense Standard 5: PROPERTIES AND RELATIONSHIPS: Analyze properties of objects and relationships among the properties. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: uses the idea of similar triangles to solve complex problems Level 3.5 Level 3.0 While engaged in tasks regarding geometric shapes and their properties and relationships the student will: (MA.G.5.1) use inductive and deductive reasoning to create and defend geometric conjectures (e.g., write a logically sound proof (two-column, paragraph form) to defend conjectures) (MA.G.5.2) use the concept of corresponding parts to prove that triangles and other polygons are congruent or similar (e.g., identify enough corresponding parts to determine if two shapes are congruent) (Second Quarter) The student exhibits no major errors or omissions. Level 2.5 Level 2.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o congruent o similar performs basic processes such as: o coming up with rational explanations to defend a geometric conjecture o recognizing or recalling examples of corresponding parts of polygons; proving triangles are congruent; recognizing or recalling accurate statements about the difference between similarity and congruence However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 19 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to use all concepts for polygons to solve nontraditional problems and defend their solutions. Level 3.0 Ask students to use inductive and deductive reasoning to create and defend geometric conjectures. Ask students to use the concept of corresponding parts to prove that triangles and other polygons are similar. Level 2.0 Ask students to define “congruent,” “similar.” Ask students to come up with rational explanations to defend a geometric conjecture. Ask students to recognize or recall examples of corresponding parts of polygons; prove triangles are congruent; recognize or recall accurate statements about the difference between similarity and congruence. 20 Chapter Topic: Right Triangles and Trigonometry Strand: Measurement Standard 4: FLUENCY WITH MEASUREMENT: Understand attributes, units, and systems of units in measurement and develop and use techniques, tools, and formulas for measuring. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: uses right triangle trigonometric ratios to solve for 2 unknown lengths of sides (given an angle measure and the length of one side) solves problems using the formulas for area and volume of two- and threedimensional figures and solids that are made of mixed “shapes” Level 3.5 Level 3.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. While engaged in tasks involving measurement formulas the student will: (MA.G.4.1) use right angle trigonometric ratios to solve for an unknown length of a side or measure or angle (e.g., use sine, cosine, and tangent to find the length of a side of a right triangle) (Fourth Quarter) (MA.G.4.2) solve problems using the formulas for perimeter, circumference, area, and volume of two- and three- dimensional figures and solids (e.g., select and apply the appropriate formula to solve surface area and volume problems. i.e., calculate the volume of a cone shaped vase to determine the amount of water needed to fill it) (Fourth Quarter) (MA.G.4.3) determine the effect of dimension changes to perimeter, area, and volume for common geometric figures and solids (e.g., alter the measure of one of the attributes of a 3-D shape, then determine how that changed affected the original shape’s surface area and volume. i.e., finds how the volume changes if the length of the cylinder is doubled.) (Fourth Quarter) (MA.G.5.5) apply the concepts of special right triangles to real-world situations (e.g., recognize if a right triangle is “special” (45-45-90 or 30-6090) and apply the appropriate concepts to find the length of an unknown side) The student exhibits no major errors or omissions. Level 2.5 No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. 21 Level 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o perimeter o area o circumference o volume o 2 and 3 dimensional o base o height o altitude performs basic processes such as: o recognizing or recalling examples of right triangle trig ratio formulas; placing given information in correct place on right triangle pictures o recognizing or recalling examples of formulas for perimeter, circumference, area, and volume o plugging information correctly into formulas o comparing original dimensions calculations to changed dimension calculations However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 22 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to use right triangle trigonometric ratios to solve for 2 unknown lengths of sides (given an angle measure and the length of one side). Ask students to solve problems using the formulas for area and volume of twoand three-dimensional figures and solids that are made of mixed “shapes.” Level 3.0 Ask students to use right triangle trigonometric ratios to solve for an unknown length of a side or measure of an angle. Ask students to solve problems using the formulas for perimeter, circumference, area, and volume of two- and three-dimensional figures and solids. Ask student to determine the effect of dimension changes to perimeter, area, and volume for common geometric figures and solids. Level 2.0 Ask students to define “perimeter,” “area,” “circumference,” “volume,” “two- and three-dimensional,” “base,” “height,” “altitude.” Ask students to place given information in the correct place on right triangle pictures and state the right triangle trigonometric formulas. Ask students to write the formulas for perimeter, circumference, area and volume. Ask students to plug information correctly into formulas. Ask students to compare original dimension calculations to changed dimension calculations. 23 Chapter Topic: Circles Strand: Geometry and Spatial Sense Standard 5: PROPERTIES AND RELATIONSHIPS: Analyze properties of objects and relationships among the properties. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: connects two or more circle concepts to solve a complex problem Level 3.5 Level 3.0 While engaged in tasks regarding geometric shapes and their properties and relationships the student will: (MA.G.5.1) use inductive and deductive reasoning to create and defend geometric conjectures (e.g., write logically sound proof (two-column, paragraph form) to defend conjectures) (MA.G.5.6) use the relationships among properties of circles (chords, secants, tangents, arcs, circumference, radius, diameter, inscribed polygons) to solve problems (e.g., use properties of circles to find the length of the chords of an inscribed regular hexagon) The student exhibits no major errors or omissions. Level 2.5 Level 2.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o chords o tangents o circumference o arcs o radius o diameter o inscribe o circumscribe performs basic processes such as: o coming up with rational explanations to defend a geometric conjecture o recognizing or recalling examples of chords, secants, tangents, and arcs of circles; solving problems involving one of the above and either circumference, radius, or diameter However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 24 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to use all concepts for circles to solve nontraditional problems and defend their solutions. Level 3.0 Ask students to use inductive and deductive reasoning to create and defend geometric conjectures. Ask students to use the relationships among properties of circles (chords, secants, tangents, arcs, circumference, radius, diameter, inscribed polygons) to solve problems. Level 2.0 Ask students to define “chords,” “tangents,” “circumference,” “arcs,” “radius,” “diameter,” “inscribe,” “circumscribe.” Ask students to come up with rational explanations to defend a geometric conjecture. Ask students to recognize or recall examples of chords, secants, tangents, and arcs of circles; solve problems involving one of the above and either circumference, radius, or diameter. 25 Chapter Topic: Areas of Polygons and Circles Strand: Measurement Standard 4: FLUENCY WITH MEASUREMENT: Understand attributes, units, and systems of units in measurement and develop and use techniques, tools, and formulas for measuring. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: solves problems using the formulas for area and volume of two- and threedimensional figures and solids that are made of mixed “shapes” Level 3.5 Level 3.0 While engaged in tasks involving measurement formulas the student will: (MA.G.4.2) solve problems using the formulas for perimeter, circumference, area, of two- and three- dimensional figures and solids (e.g., select and apply the appropriate formula to solve surface area problems) (MA.G.4.3) determine the effect of dimension changes to perimeter and area for common geometric figures and solids (e.g., alter the measure of one of the attributes of a 3-D shape, then determine how that change affected the original shape’s surface area) The student exhibits no major errors or omissions. Level 2.5 Level 2.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o perimeter o area o circumference o 2 and 3 dimensional o base o height o altitude performs basic processes such as: o recognizing or recalling examples of right triangle trig ratio formulas; placing given information in correct place on right triangle pictures o recognizing or recalling examples of formulas for perimeter, circumference and area o plugging information correctly into formulas o comparing original dimension calculations to changed dimension calculations However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 26 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to use right triangle trigonometric ratios to solve for 2 unknown lengths of sides (given an angle measure and the length of one side). Ask students to solve problems using the formulas for area of two- and threedimensional figures and solids that are made of mixed “shapes.” Level 3.0 Ask students to use right triangle trigonometric ratios to solve for an unknown length of a side or measure of an angle. Ask students to solve problems using the formulas for perimeter, circumference and area of two- and three-dimensional figures and solids. Ask students to determine the effect of dimension changes to perimeter and area for common geometric figures and solids. Level 2.0 Ask students to define “perimeter,” “area,” “circumference,” “two- and threedimensional,” “base,” “height,” “altitude.” Ask students to place given information in the correct place on right triangle pictures and state the right triangle trigonometric formulas. Ask students to write the formulas for perimeter, circumference and area. Ask students to plug the information correctly into formulas. Ask students to plug information into the formulas correctly; know how to compare original dimension calculations to changed dimension calculations. 27 Chapter Topic: Surface Area and Volume Strand: Measurement Standard 4: FLUENCY WITH MEASUREMENT: Understand attributes, units, and systems of units in measurement and develop and use techniques, tools, and formulas for measuring. Strand: Geometry and Spatial Sense Standard 7: VISUAL AND SPATIAL SENSE: Use visualization and spatial reasoning to solve problems both within and outside of mathematics. Geometry Level 4.0 In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught such as: solves problems using the formulas for area and volume of two- and threedimensional figures and solids that are made of mixed “shapes” given a three-dimensional object, designates what other two-dimensional shapes can be made from cross sections of the object Level 3.5 Level 3.0 In addition to Level 3.0 performance, in-depth inferences and applications with partial success. While engaged in tasks involving measurement formulas the student will: (MA.G.4.2) solve problems using the formulas for perimeter, circumference, area, and volume of two- and three- dimensional figures and solids (e.g., select and apply the appropriate formula to solve surface area and volume problems. i.e., calculate the volume of a cone shaped vase to determine the amount of water needed to fill it) (MA.G.4.3) determine the effect of dimension changes to perimeter, area, and volume for common geometric figures and solids (e.g., alter the measure of one of the attributes of a 3-D shape, then determine how that change affected the original shape’s surface area and volume. i.e., finds how the volume changes if the length of the cylinder is doubled) (MA.G.7.2) use concrete objects, pictorial representations, computer software, or graphing calculators to solve geometric problems (e.g., select an appropriate representation/strategy to solve a geometric problem and show or explain how the representation/strategy aided in solving the problem) (MA.G. 7.1) given a three-dimensional object, designate where to make the cross section to get a designated two-dimensional shape (e.g., sketch (or uses computer software) to show two-dimensional drawings of different views, truncations, and cross-sections of a three-dimensional solid) The student exhibits no major errors or omissions. Level 2.5 No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes. 28 Level 2.0 There are no major errors or omissions regarding the simpler details and processes as the student: recognizes or recalls specific terminology such as: o perimeter o area o circumference o volume o 2 and 3 dimensional o base o height o altitude o cross section o truncation performs basic processes such as: o recognizing or recalling examples of triangle trig ratio formulas; placing given information in correct place on right triangle pictures o recognizing or recalling examples of formulas for perimeter, circumference, area, and volume o plugging information correctly into formulas o comparing original dimension calculations to changed dimension calculations o given a three-dimensional object, recognizing or recalling what two-dimensional shape is visible from a designated cross-section However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Level 1.5 Level 1.0 Level 0.0 Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures. With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes. Even with help, no understanding or skill demonstrated. 29 Sample Tasks for Levels 4.0, 3.0, & 2.0 Level 4.0 Ask students to use right triangle trigonometric ratios to solve for 2 unknown lengths of sides (given an angle measure and the length of one side). Ask students to solve problems using the formulas for area and volume of twoand three-dimensional figures and solids that are made of mixed “shapes.” Ask students to state what other two-dimensional shapes can be made from cross sections of a three-dimensional object. Level 3.0 Ask students to use right triangle trigonometric ratios to solve for an unknown length of a side or measure of an angle. Ask students to solve problems using the formulas for perimeter, circumference, area, and volume of two- and three-dimensional figures and solids. Ask students to determine the effect of dimension changes to perimeter, area, and volume for common geometric figures and solids. Ask students to use concrete objects, pictorial representations, computer software, or graphing calculators to solve geometric problems. Ask students to designate where to make a cross section to get a designated twodimensional shape when given a three-dimensional object. Level 2.0 Ask students to define “perimeter,” “area,” “circumference,” “volume,” “two- and three-dimensional,” “base,” “height,” “altitude.” Ask students to place given information in the correct place on right triangle pictures and state the right triangle trigonometric formulas. Ask students to write the formulas for perimeter, circumference, area and volume. Ask students to plug the information correctly into formulas. Ask students to compare original dimension calculations to changed dimension calculations. Ask students to identify appropriate real-life objects/models that represent the situation; draw a picture that represents the situation. Ask students to define “cross section,” and “truncation” as they relate to threedimensional objects. Ask students to recognize or recall what two-dimensional shape is visible from a designated cross-section when given a three-dimensional object. 30