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Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 8-1. Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of two-dimensional figures and threedimensional objects. Lesson 0.1 Topic Symmetry Activities/Investigation Symmetry show and tell Materials Work sheet Pretty pictures Rulers 0.2 Line designs Astrid, 8 point slot 2&3 Days Compass Straightedge 0.3 Construction tools Work sheet Curves by lines Ruler Dot paper 2&3 Days .5 Day Ruler 1.5 Days Radius Regular hexagon Recognize and explain Point Line Plane Collinear Coplanar Line segment Endpoints Congruent Midpoint Bisects Daisy Designs 0.99 Test Geometry and Spatial Sense Standard 10-2. Recognize and explain the necessity for certain terms to remain undefined, such as point, line and plane. 10- 1. Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles of polygons; b. segments related to triangles (median, altitude, midsegment); c. points of concurrency related to triangles (centroid, incenter, orthocenter, circumcenter); d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). Patterns, Functions and Algebra Standard 8-13. Compute and interpret slope, midpoint and distance given a set of ordered pairs. 1.1 1.1A Basic definitions Definition List Midpoint Formula Technology Pacing 1 Day 1 Day 1 Vocabulary Symmetry (reflection, bilateral, rotational) Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles of polygons; b. segments related to triangles (median, altitude, midsegment); c. points of concurrency related to triangles (centroid, incenter, orthocenter, circumcenter); d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). Lesson 1.2 Topic Activities/Investigation Poolroom math Materials Technology Protractor Pacing 2 Days Protractor Defining angles 1.3 Writing definitions 1 Day Straightedge 1.4 1.5 Days Polygons Quiz .5 Day Geometry and Spatial Sense Standard 8-4. Represent and analyze shapes using coordinate geometry; e.g., given three vertices and the type of quadrilateral, find the coordinates of the fourth vertex. 10-1. Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles of polygons; b. segments related to triangles (median, altitude, midsegment); c. points of concurrency related to triangles (centroid, incenter, orthocenter, circumcenter); d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). Sketchpad 1.5 Triangles Special quadrilaterals Triangles Special quadrilaterals 2 Rulers Protractors 2 Days Vocabulary Angle Vertex Sides Degrees Counterexample Skew Right Acute Obtuse Vertical angles Linear pair Complementary Supplementary Polygon Diagonal Convex Concave Equilateral Equiangular Regular Scalene Equilateral Isosceles Base (angles) Trapezoid Kite Parallelogram Rhombus Rectangle Square Discovering Geometry Indicators/Benchmarks Measurement Standard 10-5. Determine the measures of central and inscribed angles and their associated major and minor arcs. Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: d. Circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). Geometry and Spatial Sense Standard 8-6. Draw nets for a variety of prisms, pyramids, cylinders and cones. 11-5. Identify, sketch and classify the cross sections of three-dimensional objects. Lesson Topic Activities/Investigation Materials Technology Pacing 1.6 Circles Defining circle terms Construction tools 1 Day 1.8 Space geometry Space Geometry Dot paper 1 Day 1.98 1.99 Chapter 1 Review Chapter 1 Test 2 Days 1 Day 3 Vocabulary Circle Radius Diameter Concentric Arc Semicircle Minor arc Space Isometric drawing Pyramid Cylinder Prism Cone Sphere Hemisphere Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: c. prove theorems involving properties of lines, angles, triangles and quadrilaterals. Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: c. prove theorems involving properties of lines, angles, triangles and quadrilaterals. Patterns, Functions and Algebra Standard 10-1. Define function formally and with f(x) notation. 8-2. Generalize patterns and sequences by describing how to find the nth term. Patterns, Functions and Algebra Standard 10-4. Use algebraic representations and functions to describe and generalize geometric properties and relationships. Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: c. prove theorems involving properties of lines, angles, triangles and quadrilaterals. Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: c. prove theorems involving properties of lines, angles, triangles and quadrilaterals. 8-2. Recognize the angles formed and the relationship between the angles when two lines intersect and when parallel lines are cut by a transversal. Lesson 2.1 Topic Inductive reasoning 2.2 Deductive reasoning 2.3 Number patterns Finding the rule 2 Days 2.4 Math models Party handshakes 1 2.4A 2.5 2.6 Activities/Investigation Shape Shifters Materials Technology Pacing 1 Day 1 Day Networks Angle relationships Linear pair conjecture Vertical angle conjecture Parallel lines Transversals Which angles are congruent? Is the converse true? 4 Konigsburg Worksheet Protractors Patty paper Protractor Sketchpad Vocabulary Inductive reasoning Conjecture Deductive Function rule Linear function Mathematical models Triangular number Rectangular number 1 Day 1 Day 2 Days Paragraph proof Converse Transversal Corresponding angles Alternate interior angles Alternate exterior angles Discovering Geometry Indicators/Benchmarks Patterns, Functions and Algebra Standard 8-13. Compute and interpret slope, midpoint and distance given a set of ordered pairs. Lesson Topic Activities/Investigation 2.4A 2.98 2.99 Materials Technology Pacing 1 Day Chapter 2 Review Chapter 2 Test 2 Days 1 Day 5 Vocabulary Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: d. test a conjecture using basic constructions made with a compass and straightedge or technology. 10-4. Construct right triangles, equilateral triangles, parallelograms, trapezoids, rectangles, rhombuses, squares and kites, using compass and straightedge or dynamic geometry software. 10-5. Construct congruent figures and similar figures using tools, such as compass, straightedge, and protractor or dynamic geometry software. Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: b. segments related to triangles (median, altitude, midsegment). 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: d. test a conjecture using basic constructions made with a compass and straightedge or technology. 10-4. Construct right triangles, equilateral triangles, parallelograms, trapezoids, rectangles, rhombuses, squares and kites, using compass and straightedge or dynamic geometry software. 10-5. Construct congruent figures and similar figures using tools, such as compass, straightedge, and protractor or dynamic geometry software. Patterns, Functions and Algebra Standard 10-9. Recognize and explain that the slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals. Lesson Topic Activities/Investigation Materials Technology Pacing Vocabulary 3.1 Constructing a duplicate Copying a segment Copying an angle Construction tools 1 Day Sketch Draw Construct 3.2 Constructing perpendicular bisectors Finding the right bisector, right down the middle Patty paper Construction tools 1 Day Perpendicular bisector Coincide Median Midsegment 3.3 Constructing Perpendiculars to a line Constructing angle Bisectors 3.1 – 3.4 Finding the right line Patty paper Perpendiculars Angle bisecting by folding Angle bisecting by compass Constructing parallel lines by folding Algebra skills 3 3.4 Review Quiz 3.5 Constructing parallel lines 2.6A Altitude 1 Day 1 1 Day 1 Day Slope 1 Day 3.5A Slopes of parallel and Perpendicular lines 1 Day 6 Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: d. test a conjecture using basic constructions made with a compass and straightedge or technology. 10-4. Construct right triangles, equilateral triangles, parallelograms, trapezoids, rectangles, rhombuses, squares and kites, using compass and straightedge or dynamic geometry software. 10-5. Construct congruent figures and similar figures using tools, such as compass, straightedge, and protractor or dynamic geometry software. Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: b. segments related to triangles (median, altitude, midsegment); c. points of concurrency related to triangles (centroid, incenter, orthocenter, circumcenter). Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: c. points of concurrency related to triangles (centroid, incenter, orthocenter, circumcenter). Lesson Topic Activities/Investigation Materials 3.6 Construction problems Perspective drawing Construction tools 3.7 Constructing points of concurrency Concurrence Incenter Circumcenter Patty paper Construction tools 3.8 The centroid Are medians concurrent? Construction tools Patty paper Balancing act 3.98 3.99 Chapter 3 Review Chapter 3 Test Technology Pacing 1 Day Sketchpad 2 Days Concurrent Incenter Circumcenter Orthocenter Circumscribed Inscribed 1 Days Center of gravity 2 Days 1 Day 7 Vocabulary Perspective Vanishing point Horizon line One (two) point perspective Discovering Geometry Indicators/Benchmarks Measurement Standard 8-8. Find the sum of the interior and exterior angles of regular convex polygons with and without measuring the angles with a protractor. Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles of polygons. Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles of polygons. Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: b. prove theorems involving triangle similarity and congruence. Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: b. prove theorems involving triangle similarity and congruence. Lesson Topic Activities/Investigation Materials Technology Sketchpad Demo Pacing 1 Day Vocabulary Auxiliary line 4.1 Triangle sum Investigation: Triangle sum Straightedge 4.2 Special triangles Investigation: Angles (isosceles triangle) Ruler Protractor 1 Day 4.3 Quiz Triangle inequalities Investigation: Shortest path worksheet Construction tools 1 Day Exterior angle Adjacent interior angles Remote interior angles 4.4 Congruence theorems Investigation: Side, Side, Side Side, Angle, Side Construction tools 1 Day Included angle Included side 4.5 More congruence theorems Investigation ASA, SAA 4.6 4.7 4.8 4.98 4.99 CPCTC Flowchart proofs Isosceles triangle proofs Construction tools Legs 1 Day 1 Day 1 Day Proof Puzzles Investigation: Symmetry line in an isosceles triangle Review Assessment 1 Day Construction tools 2 Days 1 Day 8 Flowchart Flowchart proof Biconditional Discovering Geometry Indicators/Benchmarks Measurement Standard 8-8. Find the sum of the interior and exterior angles of regular convex polygons with and without measuring the angles with a protractor. Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles of polygons. Lesson Topic Activities/Investigation Materials Pacing 1 Day 5.1 Polygon sum Investigation: Polygon sum Ruler Protractor Calculators 5.2 Exterior angles Investigation: Exterior angle sum Ruler Protractor Sketchpad demo Calculators 5.3 Kites/trapezoid Investigation: Kites/trapezoid Ruler Protractor 2 Days 5.4 Mid segments Investigation: Trapezoid 5.5 Parallelogram Parallelogram 1 Day Systems of Linear equations Substitution/Elimination worksheet 1 Day More Investigation: Rhombus Rectangle 5.5A 5.6 Parallelogram Ruler Protractor Ruler Protractor Graph paper 1 Day 1 Day 5.7 5.98 5.99 Vocabulary N-gon 1 Day 1 Day Quiz Patterns, Functions and Algebra Standard 10-6. Solve equations and inequalities having rational expressions as coefficients and solutions. 10-11. Solve real-world problems that can be modeled, using systems of linear equations and inequalities. 8-10. Solve 2 by 2 systems of linear equations graphically and by simple substitution. 10.3 Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or twocolumn proof, including: c. prove theorems involving properties of lines, angles, triangles and quadrilaterals. Technology Trapezoid Bases Base angles Vertex angles Non vertex angles Vector Resultant vector Vector sum Rectangle Square Dart Quadrilateral proofs 1 Day Chapter 5 Review Chapter 5 Test 2 Days 1 Day 9 Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). 10-10. Solve problems involving chords, radii and arcs within the same circle. Measurement Standard 10-5. Determine the measures of central and inscribed angles and their associated major and minor arcs. Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). Number, Number Sense and Operations Standard 10-1. Connect physical, verbal and symbolic representations of irrational numbers; e.g., construct √2 as a hypotenuse or on a number line. Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). Lesson Topic Activities/Investigation Materials 6.1 Chords Investigation: Angles Central angles Center Perpendicular Bisector Construction tools 6.2 Tangents Investigation: Tangents Segments Construction tools 6.3 Arcs Angles Investigation: Inscribed angle Same arc Semicircle Quadrilaterals Parallel lines Construction tools 6.4 Circle proofs 6.5 Circumference 6.6 Technology Pacing Vocabulary 2 Days Sketchpad demo 1 Day Tangent circles Internally tangent Externally tangent Tangent segments 2 Days 1 Day Investigation: Pi Pi problem solving Sketchpad demo Calculators 10 1 Day 1 Day Circumference Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). 10-10. Solve problems involving chords, radii and arcs within the same circle. Lesson Topic 6.7 Arc length 6.98 6.99 Chapter 6 Review Chapter 6 Test Activities/Investigation Investigation: Arc length Materials Construction tools Technology Pacing Calculators 1 Day 1 Day 1 Day 11 Vocabulary Length of an arc Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 10-6. Identify the reflection and rotation symmetries of two- and three dimensional figures. 8-5. Draw the results of translations, reflections, rotations and dilations of objects in the coordinate plane, and determine properties that remain fixed; e.g., lengths of sides remain the same under translations. 11-2. Represent translations using vectors. 10-7. Perform reflections and rotations using compass and straightedge constructions and dynamic geometry software. Geometry and Spatial Sense Standard 10-8. Derive coordinate rules for translations, reflections and rotations of geometric figures in the coordinate plane. 8-5. Draw the results of translations, reflections, rotations and dilations of objects in the coordinate plane, and determine properties that remain fixed; e.g., lengths of sides remain the same under translations. Geometry and Spatial Sense Standard 10-9. Show and describe the results of combinations of translations, reflections and rotations (compositions); e.g., perform compositions and specify the result of a composition as the outcome of a single motion, when applicable. 8-5. Draw the results of translations, reflections, rotations and dilations of objects in the coordinate plane, and determine properties that remain fixed; e.g., lengths of sides remain the same under translations. Lesson Topic 7.1 Transformations Investigation: Reflection 7.2 Isometries Investigation: Coordinate plane Minimal path Protractor 7.3 Compositions Investigation: Parallel lines Intersecting lines Protractor 7.4 Tessellations Investigation: Semi-regular 7.98 7.99 Activities/Investigation Chapter 7 Review Chapter 7 Test 12 Materials Technology Sketchpad Demo Sketchpad demo Sketchpad demo Pacing 1 Day Vocabulary Image Isometry Nonrigid transformation Translation Distance Direction Translation vector Rotation reflection Line of reflection Reflectional symmetry Line of symmetry Rotational symmetry Point symmetry 3 Days Ordered pair rules 2 Days Composition 1 Day Tessellation Monohedral tiling Regular tessellation Vertex arrangement Semi-regular tessellation Archimedean tilings Discovering Geometry Indicators/Benchmarks Patterns, Functions and Algebra Standard 10-3. Solve equations and formulas for a specified variable; e.g., express the base of a triangle in terms of the area and height. Measurement Standard 8-9. Demonstrate understanding of the concepts of perimeter, circumference and area by using established formulas for triangles, quadrilaterals, and circles to determine the surface area and volume of prisms, pyramids, cylinders, spheres and cones. (Note: Only volume should be calculated for spheres and cones.) 10-1. Explain how a small error in measurement may lead to a large error in calculated results. 10-2. Calculate relative error. 10-3. Explain the difference between absolute error and relative error in measurement. 10-4. Give examples of how the same absolute error can be problematic in one situation but not in another; e.g., compare “accurate to the nearest foot’ when measuring the height of a person versus when measuring the height of a mountain. Data Analysis & Probability Standard 10.7. Model problems dealing with uncertainty with area models (geometric probability). Patterns, Functions and Algebra Standard 10.3. Solve equations and formulas for a specified variable; e.g., express the base of a triangle in terms of the area and height. Lesson Topic 8.1 Area of rectangles Parallelogram Area of triangles Trapezoids Kites 8.2 8.3 Area problems Activities/Investigation Materials Scissors 1 Day Investigation: Triangles Trapezoids Kites Investigation: Area formulas Scissors 1 Day Investigation: Regular polygon Calculators Investigation: Circle area Vocabulary Area Base Height Altitude Commutative Associative Calculators 1 Day Apothem 8.5 Circle area Pacing Investigation: Parallelogram 8.4 Areas of regular polygon Technology Calculators Scissors 1 Day 8.6 Sectors 1 Day 1 Day Sector of a circle Segment of a circle Annulus Where the Chips Fall 8.6A Geometric probability 8.7 Surface area 1 Day Investigation: Regular pyramid Cone 13 2 Days Surface area Bases Lateral faces Slant height Discovering Geometry Indicators/Benchmarks Measurement Standard 8-4. Derive formulas for surface area and volume and justify them using geometric models and common materials. For example, find: a. the surface area of a cylinder as a function of its height and radius. 8-5. Determine surface area for pyramids by analyzing their parts. 8-9. Demonstrate understanding of the concepts of perimeter, circumference and area by using established formulas for triangles, quadrilaterals, and circles to determine the surface area and volume of prisms, pyramids, cylinders, spheres and cones. (Note: Only volume should be calculated for spheres and cones.) 11-3. Derive a formula for the surface area of a cone as a function of its slant height and the circumference of its base. Lesson 8.7 Topic Activities/Investigation Materials Technology Pacing Chapter 8 Review 1 Day Chapter 8 Test 1 Day 14 Vocabulary Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometrid properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: a. prove the Pythagorean Theorem. Number, Number Sense and Operations Standard 8-7. Find the square root of perfect squares, and approximate the square root of non-perfect squares as consecutive integers between which the root lies, e.g., √130 is between 11 and 12. Patterns, Functions and Algebra Standard 10-8. Graph the quadratic relationship that defines circles. 8-13. Compute and interpret slope, midpoint and distance given a set of ordered pairs. Lesson Topic Activities/Investigation Materials 9.1 Pythagorean Theorem Investigation: Pythagorean Theorem Scissors 9.2 Converse Pythagorean Theorem Special right triangles Investigation: Converse Ruler Protractor 9.3 9.4 Story problems 9.5 Distance 9.6 Circles Pythagorean Theorem 9.98 9.99 Investigation: Isosceles 30-60-90 Isosceles Right Triangle Technology Sketchpad demo Pacing 1 Day Calculators 1 Day Investigation Formula Equation of circle 1 Day 1 Day 15 Theorem Pythagorean triples 1 Day 2 Days Quiz (9.1-9.3) Chapter 9 Review Chapter 9 Test Vocabulary Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 10-1. Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles of polygons; b. segments related to triangles (median, altitude, midsegment); c. points of concurrency related to triangles (centroid, incenter, orthocenter, and circumcenter); d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). Lesson 10.1 Topic Solids Activities/Investigation Definition List Measurement Standard 8-4. Derive formulas for surface area and volume and justify them using geometric models and common materials. For example, find: b. that the volume of a pyramid (or cone) is one-third of the volume of a prism (or cylinder) with the same base area and height. 8-6. Solve and determine the reasonableness of the results for problems involving rates and derived measurements, such as velocity and density, using formulas, models and graphs. 8-10. Use conventional formulas to find the surface area and volume of prisms, pyramids and cylinders and the volume of spheres and cones to a specified level of precision. 10.2 Prism Cylinder Volume Investigation: Volume 10.3 Pyramid Cone Volume Investigation: Volume Volume problems Quiz (10.1-10.3) 10.4 10.3A Net Project Platonic Solids Materials Technology Pacing 1 Day Vocabulary Polyhedron Faces Edge Vertex Tetrahedron Regular polyhedron Prism Bases Lateral faces Lateral edges Right Oblique Altitude Height Pyramid Sphere Radius Center Hemisphere Great Circle Cylinder Axis cone 1 Day 1 Day 1 Day Octahedron Dodecahedron Icosohedron 1 Day 10.5 Displacement Density Quiz (10.3-10.5) Displacement Density 1 Day 16 Discovering Geometry Indicators/Benchmarks Measurement Standard 8-10. Use conventional formulas to find the surface area and volume of prisms, pyramids and cylinders and the volume of spheres and cones to a specified level of precision. Lesson Topic Activities/Investigation 10.6 Sphere volume Investigation: Volume formula 10.7 Sphere Surface area Investigation: Surface area Chapter 10 Review Chapter 10 Test Quiz (10.6-10.7) Materials Technology Pacing 1 Day Calculators 1 Day Number, Number Sense and Operations 10-2. Explain the meaning of the nth root. 10-4. Approximate the nth root of a given number greater than zero between consecutive integers when n is an integer; e.g., the 4th root of 50 is between 2 and 3. 10.98 10.99 17 1 Day 1 Day Vocabulary nth root Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 8-3. Use appropriate levels of precision when calculating with measurements. Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: b. prove theorems involving triangle similarity Measurement Standard 8-7. Apply proportional reasoning to solve problems involving indirect measurements or rates. Measurement Standard 9-3. Use the ratio of lengths in similar twodimensional figures or three-dimensional objects to calculate the ratio of their areas or volume respectively. Lesson Topic 11.1 Similar polygons Investigation: Similarity Dilations Rulers Protractors Calculators 1 Day 11.2 Similar triangles Investigation: Angle, angle Side, side, side Side, angle, side Rulers Protractors 1 Day 11.3 Indirect measurement Investigation: Mirror, mirror Mirrors 1 Day 11.4 Corresponding parts of similar triangles Investigation: Corresponding parts Opposite side ratios Rulers Calculators 11.5 Proportions with area/volume Investigation: Area Volume 11.6 Proportional segments between parallel lines Investigation: Parallels Proportionality Extended parallel 11.98 11.99 Activities/Investigation Chapter 11 Review Chapter 11 Test Materials Technology 1 Day 1 Day 1 Day 1 Day 18 Vocabulary Similar figures Similar polygons Scale factor Dilation Indirect measurement 1 Day Sketchpad demo Rulers Protractors Pacing Similar solids Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 9-1. Define the basic trigonometric ratios in right triangles: sine, cosine and tangent. Measurement Standard 9-4. Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures. Geometry and Spatial Sense Standard 9-2. Apply proportions and right triangle trigonometric ratios to solve problems involving missing lengths and angle measures in similar figures. Measurement Standard 9-4. Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures. Geometry and Spatial Sense Standard 11-4. Use trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of cosines. Measurement Standard 9-4. Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures. Lesson 12.1 Topic Trigonometry ratios Activities/Investigation Investigation: Trigonometry tables 12.2 Problem solving 12.3 Law of sines Quiz (12.1-12.2) Investigation: Area of triangle Law of sines 12.4 Law of cosine Investigation: Pythagorean identity 12.5 Problem solving with trigonometry 12.98 12.99 Chapter 12 Review Chapter 12 Test Materials Technology Pacing Calculators Protractors Rulers 1 Day Calculators 1 Day Calculators Sketchpad demo Angle of elevation Angle of depression Law of sines 1 Day Law of cosines 1 Day 19 Opposite side Adjacent side Tangent Sine Cosine Trigonometry Inverse tangent 1 Day Calculators Quiz (12.3-12.5) Vocabulary 1 Day 1 Day Discovering Geometry Indicators/Benchmarks Geometry and Spatial Sense Standard 10-2. Recognize and explain the necessity for certain terms to remain undefined, such as point, line and plane. Geometry and Spatial Sense Standard 10-3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: a. prove the Pythagorean Theorem; b. prove theorems involving triangle similarity and congruence; c. prove theorems involving properties of lines, angles, triangles and quadrilaterals; d. test a conjecture using basic constructions made with a compass and straightedge or technology. Geometry and Spatial Sense Standard 9-3. Analyze two-dimensional figures in a coordinate plane; e.g., use slope and distance formulas to show that a quadrilateral is a parallelogram. Lesson 13.1 13.3 13.4 Topic Premises of geometry Triangle proofs Quadrilateral proofs 13.5 Indirect proof 13.6 13.7 Circle proofs Similarity proofs Activities/Investigation Postulates Investigation: Parallelogram Conjectures Investigation: Tangent Conjecture Investigation: Side, side, side Similarity Conjecture Materials Technology Pacing 1 Day 1 Day 1 Chapter 13, Properties of similarity 1 Day 1 Day 1 Day 1 Day Review Chapter 13, Test 20 Lemma Indirect proof 1 Day 13.7A 13.98 13.99 Vocabulary Deductive system Premises Postulates Theorems Properties of arithmetic Properties of equality Definition of congruence Postulte of geometry Discovering Geometry Indicators/Benchmarks Lesson Topic Pacing Activities/Investigation Materials Technology Vocabulary