Franklin County Community School Corporation Franklin County High School Brookville, Indiana s3

Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana Curriculum Map Course Title: Geometry Quarter: 3 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do you construct a basic construction? 2. How do you apply the basic constructions to more complex geometrical concepts? 3. How do you use coordinate geometry to find relationships within triangles? 4. How do you solve problems that involve measurements of triangles? 5. How do you write indirect proofs? 6. How can you find the sum of the measures of polygon angles? 7. How can you classify quadrilaterals? 8. How do you show two triangles are similar? 9. How do you identify corresponding parts of similar triangles?

Unit/Time Frame Standards Content Skills Assessment Resources

Constructions Units State Standards Basic Construction  To construct the basic Textbook Textbook 1. Congruent Process constructions assignments Prentice-Hall G.1.4  To apply the basic Geometry Segments and G.1.6 Application of the Basic constructions to construct Worksheet Foundation Segment G.1.7 Constructions more complex geometrical assignments Series 2011 Bisectors G.2.1 figures Edition G.2.3 Special Segments in  To use properties of Section Quizzes 2. Congruent G.2.6 Triangles midsegments to solve Power point Angles and G.2.7 problems Quizzes Presentations Angle Bisectors G.2.8 Indirect Proof  To use properties of G.2.9 Tests Intro to Geometry 3. Parallel and perpendicular bisectors and G.2.10 The Triangle Inequality angle bisectors Frank Schaffer Perpendicular G.2.12 Theorem  To identify properties of Oral Responses Publications Lines G.2.13 perpendicular bisectors G.2.14 Properties of Observations Teacher generated 4. Construction of  To identify properties of G.2.17 Parallelograms medians and altitudes of a worksheets Polygons G.4.2 triangle G4.3 Special Parallelograms Notebooks 5. Construction of  To identify the orthocenter of a G.5.4 triangle points of Ratios and Proportions  To use indirect reasoning to concurrency write proofs Proving Similar Triangles  To use inequalities involving Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana Curriculum Map Course Title: Geometry Quarter: 3 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do you construct a basic construction? 2. How do you apply the basic constructions to more complex geometrical concepts? 3. How do you use coordinate geometry to find relationships within triangles? 4. How do you solve problems that involve measurements of triangles? 5. How do you write indirect proofs? 6. How can you find the sum of the measures of polygon angles? 7. How can you classify quadrilaterals? 8. How do you show two triangles are similar? 9. How do you identify corresponding parts of similar triangles?

Proportions in Triangles angles and sides of triangles Common Core  To apply inequalities in two Chapter 5 (Part A) Standards triangles Relationships  To find the sum of the CC.G.CO.9 measures of the interior Within Triangles CC.G.CO.10 angles of a polygon CC.G.CO.11  To find the sum of the CC.G.CO.12 5-1 Midsegments of measures of the exterior CC.G.CO.13 angles of a polygon Triangles CC.G.MG.2  To use relationships among 5-2 Perpendicular and CC.G.MG.3 sides and angles of Angle Bisectors CC.G.C.1 parallelograms 5-3 Part 1 Bisectors in CC.G.C.3  To use relationships among CC.G.C.4 Triangles diagonals among diagonals of CC.G.SRT.1 parallelograms 5-3 Part 2 Bisectors in CC.G.SRT.2  To determine whether a Triangles CC.G.SRT.3 quadrilateral is a parallelogram CC.G.SRT.4 5-4 Part 1 Medians and  To define and classify special CC.G.SRT.5 Altitudes types of parallelograms CC.G.SRT.8  To use properties of diagonals 5-4 Part 2 Medians and CC.G.GPE.5 Altitudes of rhombuses and rectangles  To determine whether a Standards for parallelogram is a rhombus or Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana Curriculum Map Course Title: Geometry Quarter: 3 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do you construct a basic construction? 2. How do you apply the basic constructions to more complex geometrical concepts? 3. How do you use coordinate geometry to find relationships within triangles? 4. How do you solve problems that involve measurements of triangles? 5. How do you write indirect proofs? 6. How can you find the sum of the measures of polygon angles? 7. How can you classify quadrilaterals? 8. How do you show two triangles are similar? 9. How do you identify corresponding parts of similar triangles?

Chapter 5 (Part B) Mathematical rectangle Practice  To verify and use properties of Relationships trapezoids Within Triangles SMP1  To verify and use properties of SMP2 kites 5-5 Indirect Proof SMP3  To write ratios and solve 5-6 Part 1 Inequalities in SMP4 proportions One Triangle SMP5  To identify and apply similar 5-6 Part 2 Inequalities in SMP6 polygons SMP7 One Triangle  To use AA~ Postulate and the SMP8 SAS~ and SSS~ Theorems 5-7 Inequalities in Two  To use similarity to find Triangles indirect measurements  To find and use relationships Chapter 6 Polygons in similar right triangles  To use the Side-Splitter and Quadrilaterals Theorem and the Triangle- Angle-Bisector Theorem 6-1 The Polygon Angle- Sum Theorems 6-2 Part 1 Properties of Parallelograms Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana Curriculum Map Course Title: Geometry Quarter: 3 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do you construct a basic construction? 2. How do you apply the basic constructions to more complex geometrical concepts? 3. How do you use coordinate geometry to find relationships within triangles? 4. How do you solve problems that involve measurements of triangles? 5. How do you write indirect proofs? 6. How can you find the sum of the measures of polygon angles? 7. How can you classify quadrilaterals? 8. How do you show two triangles are similar? 9. How do you identify corresponding parts of similar triangles?

6-2 Part 2 Properties of Parallelograms 6-3 Proving That a Quadrilateral is a Parallelogram 6-4 Properties of Rhombuses, Rectangles, and Squares 6-5 Conditions for Rhombuses, Rectangles, and Squares 6-6 Part1 Trapezoids and Kites 6-6 Part 2 Trapezoids and Kites Chapter 7 Similarity

7-1 Ratios and Proportions 7-2 Similar Polygons Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana Curriculum Map Course Title: Geometry Quarter: 3 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do you construct a basic construction? 2. How do you apply the basic constructions to more complex geometrical concepts? 3. How do you use coordinate geometry to find relationships within triangles? 4. How do you solve problems that involve measurements of triangles? 5. How do you write indirect proofs? 6. How can you find the sum of the measures of polygon angles? 7. How can you classify quadrilaterals? 8. How do you show two triangles are similar? 9. How do you identify corresponding parts of similar triangles?

7-3 Proving Triangles Similar 7-4 Similar Right Triangles 7-5 Proportions in Triangles Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS State Standards

G.1: Students find lengths and midpoints of line segments. They describe and use parallel and perpendicular lines. They find slopes and equations of lines. G.1.1: Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems. G.1.2: Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass, explaining and justifying the process used. G.1.3: Understand and use the relationships between special pairs of angles formed by parallel lines and transversals. G.1.4: Use coordinate geometry to find slopes, parallel lines, perpendicular lines, and equations of lines. G.2: Students identify and describe polygons and measure interior and exterior angles. They use congruence, similarity, symmetry, tessellations, and transformations. They find measures of sides, perimeters, and areas. G.2.2: Find measures of interior and exterior angles of polygons, justifying the method used. G.2.3: Use properties of congruent and similar polygons to solve problems. G.2.4: Apply transformations (slides, flips, turns, expansions, and contractions) to polygons in order to determine congruence, similarity, symmetry, and tessellations. Know that images formed by slides, flips and turns are congruent to the original shape. G.2.5: Find and use measures of sides, perimeters, and areas of polygons, and relate these measures to each other using formulas. G.2.6: Use coordinate geometry to prove properties of polygons such as regularity, congruence, and similarity. G.3: Students identify and describe simple quadrilaterals. They use congruence and similarity. They find measures of sides, perimeters, and areas. G.3.1: Describe, classify, and understand relationships among the quadrilaterals square, rectangle, rhombus, parallelogram, trapezoid, and kite. G.3.2: Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas. G.3.3: Find and use measures of sides, perimeters, and areas of quadrilaterals, and relate these measures to each other using formulas. Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS G.3.4: Use coordinate geometry to prove properties of quadrilaterals such as regularity, congruence, and similarity. G.4: Students identify and describe types of triangles. They identify and draw altitudes, medians, and angle bisectors. They use congruence and similarity. They find measures of sides, perimeters, and areas. They apply inequality theorems. G.4.1: Identify and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular. G.4.2: Define, identify, and construct altitudes, medians, angle bisectors, and perpendicular bisectors. G.4.3: Construct triangles congruent to given triangles. G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and areas. G.4.5: Prove and apply theorems involving segments divided proportionally. G.4.6: Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. G.4.7: Find and use measures of sides, perimeters, and areas of triangles, and relate these measures to each other using formulas. G.4.8: Prove, understand, and apply the inequality theorems: triangle inequality, inequality in one triangle, and hinge theorem. G.4.9: Use coordinate geometry to prove properties of triangles such as regularity, congruence, and similarity. G.5: Students prove the Pythagorean Theorem and use it to solve problems. They define and apply the trigonometric relations sine, cosine, and tangent. G.5.1: Prove and use the Pythagorean Theorem. G.5.2: State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. G.5.4: Define and use the trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) in terms of angles of right triangles. G.5.5: Know and use the relationship sin²x + cos²x = 1. G.5.6: Solve word problems involving right triangles. G.6: Students define ideas related to circles: e.g., radius, tangent. They find measures of angles, lengths, and areas. They prove theorems about circles. They find equations of circles. G.6.2: Define and identify relationships among: radius, diameter, arc, measure of an arc, chord, secant, and tangent. Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS G.6.3: Prove theorems related to circles. G.6.5: Define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents). G.6.6: Define and identify congruent and concentric circles. G.6.7: Define, find, and use measures of circumference, arc length, and areas of circles and sectors. Use these measures to solve problems. G.6.8: Find the equation of a circle in the coordinate plane in terms of its center and radius. G.7: Students describe and make polyhedra and other solids. They describe relationships and symmetries, and use congruence and similarity. G.7.2: Describe the polyhedron that can be made from a given net (or pattern). Describe the net for a given polyhedron. G.7.4: Describe symmetries of geometric solids. G.7.5: Describe sets of points on spheres: chords, tangents, and great circles. G.7.6: Identify and know properties of congruent and similar solids. G.7.7: Find and use measures of sides, volumes of solids, and surface areas of solids, and relate these measures to each other using formulas. G.8: Mathematical Reasoning and Problem Solving G.8.6: Identify and give examples of undefined terms, axioms, and theorems, and inductive and deductive proof. G.8.8: Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, and two-column and indirect proofs.

Common Core Standards

Congruence G-CO Experiment with transformations in the plane 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Make geometric constructions 12. Make formal geometric constructions with a variety of tools and Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Similarity, Right Triangles, and Trigonometry G-SRT Understand similarity in terms of similarity transformations 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Define trigonometric ratios and solve problems involving right triangles 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply trigonometry to general triangles 9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. 11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Circles G-C Understand and apply theorems about circles 1. Prove that all circles are similar. 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 4. (+) Construct a tangent line from a point outside a given circle to the circle. Find arc lengths and areas of sectors of circles 5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Expressing Geometric Properties with Equations G-GPE Translate between the geometric description and the equation for a conic section 1. Derive the equation of a circle of given center and radius using the Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2. Derive the equation of a parabola given a focus and directrix. 3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Use coordinates to prove simple geometric theorems algebraically 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Geometric Measurement and Dimension G-GMD Explain volume formulas and use them to solve problems 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Visualize relationships between two-dimensional and three dimensional objects 4. Identify the shapes of two-dimensional cross-sections of three dimensional objects, and identify three-dimensional objects generated by Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS rotations of two-dimensional objects.

Modeling with Geometry G-MG Apply geometric concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). 3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Standards for Mathematical Practice SMP1. Make sense of problems and persevere in solving them. SMP2. Reason abstractly and quantitatively. SMP3. Construct viable arguments and critique the reasoning of others. SMP4. Model with mathematics. SMP5. Use appropriate tools strategically. SMP6. Attend to precision Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS