COLLEGE-PREP GEOMETRY Course #242

Transcription

1 COLLEGE-PREP GEOMETRY Course #242 Course of Study Findlay City Schools 2013

2 TABLE OF CONTENTS 1. Findlay City Schools Mission Statement and Beliefs 2. College-Prep Geometry Course of Study 3. College-Prep Geometry Pacing Guide Course Description: In this course, the students will organize their observations about the properties and relationships of two and three dimensional figures into a logical thought system. The students will prove relationships using synthetic, coordinate and transformational methods. Compass and straightedge constructions will be explored. Students must combine good memorization skills with logical thinking in order to be successful in problem solving. Good study habits including study time spent outside class are necessary for success. This course is necessary to help prepare for the Ohio Graduation Test. COLLEGE-PREP GEOMETRY Course of Study Writing Team Ryan Headley Ellen Laube Karen Ouwenga Carrie Soellner Text: Geometry Common Core, 2012 edition; Pearson (publisher); ISBN:

3 Mission Statement The mission of the Findlay City Schools, a community partnership committed to educational excellence, is to instill in each student the knowledge, skills and virtues necessary to be lifelong learners who recognize their unique talents and purpose and use them in pursuit of their dreams and for service to a global society. This is accomplished through a passion for knowledge, discovery and vision shared by students, families, staff and community. Beliefs Our beliefs form the ethical foundation of the Findlay City Schools. We believe. every person has worth every individual can learn family is the most important influence on the development of personal values. attitude is a choice and always affects performance motivation and effort are necessary to achieve full potential honesty and integrity are essential for building trust. people are responsible for the choices they make. performance is directly related to expectations. educated citizens are essential for the survival of the democratic process. personal fulfillment requires the nurturing of mind, body and spirit. every individual has a moral and ethical obligation to contribute to the well-being of society. education is a responsibility shared by students, family, staff and community. the entire community benefits by investing its time, resources and effort in educational excellence. a consistent practice of shared morals and ethics is essential for our community to thrive.

4 Subject(s) Grade / Course Unit of Study Pacing FINDLAY CITY SCHOOLS Curriculum Design Grades 6 12 CP Geometry 10 th grade Chapter 1: Tools of Geometry 20 days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 1.1 3D objects can be represented with a 2D figure using special drawing techniques Prepares for G.CO Geometry is a mathematical system built on accepted facts, basic terms, and definitions G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 1.3 Number operations can be used to find and compare the lengths of segments and the measures of angles G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 1.4 Number operations can be used to find and compare the lengths of segments and the measures of angles G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 1.5 Special angle pairs can used to identify geometric relationships and to find angle measures Prepares for G.CO Geometric figures can be constructed using straightedge and compass G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on

5 the undefined notions of point, line, distance along a line, and distance around a circular arc. 1.7 Formulas can be used to find the midpoint and length of any segment in the coordinate plane Prepares for G.PE.4 Prepares for G.PE.7 G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 1.8 Perimeter and area are two different ways of measuring the size of geometric figures N.Q.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas Mathematical Practices: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unwrapped Skills (Students need to be able to do) Unwrapped Concepts (Students need to know) Bloom s Taxonomy Levels Know (G.CO.1) Definitions of Understand Angle Circle Perpendicular line Parallel line Line segment Find (G.GPE.6) Locations of points using segment addition Understand Make (G.CO.12) Formal geometric constructions Create

6 Use (N.Q.1) Appropriate units for solutions Understand Choose (N.Q.1) Appropriate units and scales: Apply Formulas Graphs Data Displays Interpret (N.Q.1) Appropriate units and scales: Analyze Formulas Graphs Data Displays Vocabulary 1.1 Net, Isometric drawing, orthographic drawing Resources Textbook with Supplementals 1.2 Point, Line, Plane, Collinear Points, Coplanar, Space, Segment, Ray, Opposites Rays, Postulate, Axiom, Intersection 1.3 Coordinate, Distance, Congruent Segments, Midpoint, Segment Bisector 1.4 Angle, Side of Angle, Vertex of an Angle, Measure of an Angle, Acute Angle, Right Angle, Obtuse Angle, Straight Angle, Congruent Angles 1.5 Adjacent Angles, Vertical Angles, Complementary Angles, Supplementary Angles, Linear Pair, Angle Bisector

7 1.6 Straightedge, Compass, Construction, Perpendicular Lines, Perpendicular Bisector 1.8 Perimeter, Area Essential Questions 1. How can you represent a 3D figure with a 2D drawing? Understanding/Corresponding Big Ideas Students will make nets for solid figures. Isometric drawings and orthographic drawings will be used to show attributes of figures. 2. What are the building blocks of geometry? Students will define basic geometric figures. Undefined terms such as point, line, plane will be shown with visual representations. Postulates, which lead to proofs later in the text, will be presented. 3. How can you describe the attributes of a segment or angle? Segments will be measured with and without a coordinate grid. Students will use the midpoint and distance formulas. Protractors will be used to measure angles.

8 FINDLAY CITY SCHOOLS Curriculum Design Grades 6 12 Subject(s) CP Geometry Grade / Course 10 th Grade Unit of Study Chapter 2: Reasoning and Proof Pacing 7 days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 2.1 Patterns in some number sequences and some sequences of geometric figures can be used to discover relationships Prepares for G.CO.9 Prepares for G.CO.10 Prepares for G.CO Some mathematical relationships can be described using a variety of if-then statements Prepares for G.CO.9 Prepares for G.CO.10 Prepares for G.CO Algebraic properties of equality are used in geometry to solve problems and justify reasoning Prepares for G.CO.9 Prepares for G.CO.10 Prepares for G.CO Given information, definitions, properties, postulates, and previously proven theorems, can be used as a reason in a proof G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints.

9 Mathematical Practices: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unwrapped Skills Unwrapped Concepts Bloom s (Students need to be able to do) (Students need to know) Taxonomy Levels Prove (G.CO.9) Theorems about lines and angles Evaluate Vocabulary 2.1 Inductive Reasoning, Conjecture, Counterexample Resources Textbook with Supplementals 2.2 Conditional, Hypothesis, Conclusion, Negation, Converse, Inverse, Contrapositive, Equivalent Statements 2.5 Reflexive Property, Symmetric Property, Transitive Property, Proof, Two-Column Proof 2.6 Theorem, Paragraph Proof

10 Essential Questions 1. How can you make a conjecture and prove that it is true? Understanding/Corresponding Big Ideas Students will observe patterns leading to making conjectures. Students will solve equations giving their reasons for each step and connect this to simple proofs. Students will prove geometric relationships using given information, definitions, properties, postulates, and theorems.

11 FINDLAY CITY SCHOOLS Curriculum Design Grades 6 12 Subject(s) CP Geometry Grade / Course 10 th Grade Unit of Study Chapter 3: Parallel and Perpendicular Lines Pacing 16 Days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 3.1 Not all lines and not all planes intersect. When a line intersects two or more lines, the angles formed at the intersection points create special angle pairs. G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Prepares for G.CO The special angle pairs formed by parallel lines and a transversal are either congruent or supplementary. G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. 3.3 Certain angle pairs can be used to decide whether two lines are parallel. Extends G.CO The relationships of two lines to a third line can be used to decide whether two lines are parallel or perpendicular to each other. G.MG.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).* 3.5 The sum of the angle measures of a triangle is always the same. G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

12 3.6 Construct parallel and perpendicular lines using a compass and a straightedge. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 3.7 A line can be graphed and its equation written when certain facts about the line, such as its slope and a point on the line, are known. Prepares for G.GPE Comparing the slopes of two lines can show whether the lines are parallel or perpendicular. G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses 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). Mathematical Practices: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unwrapped Skills (Students need to be able to do) Know (G.CO.1) Unwrapped Concepts (Students need to know) Definitions of Angle Circle Bloom s Taxonomy Levels Understand

13 Perpendicular line Parallel line Line segment Prove (G.CO.9) Theorems about lines and angles Evaluate Prove (G.CO.11) Theorems about parallelograms Evaluate Make (G.CO.12) Formal geometric constructions Create Construct (G.CO.13) Construct Create Equilateral triangle Square Regular hexagon inscribed in a circle Apply (G.MG.3) Geometric methods to solve design problems Apply Vocabulary 3.1 Skew Lines, Parallel Lines, Parallel Planes, Transversal, Alternate Interior Angles, Same Side Interior Angles, Corresponding Angles, Alternate Interior Angles Resources Textbook with Resources 3.3 Flow Proof 3.5 Auxiliary Line, Exterior Angle of a Polygon, Remote Interior Angles 3.7 Slope, Slope Intercept Form, Point Slope Form Essential Questions 1. How to prove that two lines are parallel and perpendicular? Understanding/Corresponding Big Ideas Students will use postulates and theorems to explore lines in a plane. Students will use coordinate geometry to examine the slopes of parallel and perpendicular lines.

14 2. What is the sum of the measures of the angles of a triangle? 3. How do you write an equation of a line in the coordinate plane? Students will use the triangle angle sum theorem. Students will write equations using slope intercept form.

15 Subject(s) Grade / Course Unit of Study Pacing FINDLAY CITY SCHOOLS Curriculum Design Grades 6 12 CP Geometry 10 th Grade Chapter 4: Congruent Triangles 16 Days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 4.1 Comparing the two corresponding parts of two figures can show whether the figures are congruent Prepares for G.SRT Two triangles can be proven to be congruent without having to show that all corresponding parts are congruent. Triangles can be proven to be congruent by using SSS, SAS, ASA, AAS or HL. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 4.3 Two triangles can be proven to be congruent without having to show that all corresponding parts are congruent. Triangles can be proven to be congruent by using SSS, SAS, ASA, AAS or HL. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 4.4 If two triangles are congruent, then every pair of their corresponding parts is also congruent. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 4.5 The angles and the sides of isosceles and equilateral triangles have special relationships. G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

16 G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 4.6 Two triangles can be proven to be congruent without having to show that all corresponding parts are congruent. Triangles can be proven to be congruent by using SSS, SAS, ASA, AAS or HL. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 4.7 Congruent corresponding parts of one pair of congruent triangles can sometimes be used to prove another pair of triangles congruent. This often involves overlapping triangles. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unwrapped Skills (Students need to be able to do) Unwrapped Concepts (Students need to know) Bloom s Taxonomy Levels Use (G.SRT.5) Congruence and similarity criteria to solve Apply problems Make (G.CO.12) Formal geometric constructions Create Prove (G.CO.10) Theorems about triangles Evaluate Construct (G.CO.13) Construct Create

17 Equilateral triangle Square Regular hexagon inscribed in a circle 4.1 Congruent Polygons Vocabulary Resources Textbook with Supplementals 4.5 Legs of an Isosceles Triangle, Base of an Isosceles Triangle, Vertex Angle of an Isosceles Triangle, Base Angles of an Isosceles Triangle, Corollary 4.6 Hypotenuse, Legs of a Right Triangle Essential Questions Understanding/Corresponding Big Ideas 1. How do you identify corresponding parts of congruent triangles? Students will visualize the triangles placed on top of each other. Students will use tick marks and angle marks to label corresponding sides and corresponding angles. 2. How do you show that two triangles are congruent? Students will use SSS, SAS, ASA, AAS and HL. 3. How can you tell whether a triangle is isosceles or equilateral? Students will use the definitions and look at the number of congruent sides and angles.

18 FINDLAY CITY SCHOOLS Curriculum Design Grades 6 12 Subject(s) CP Geometry Grade / Course 10 th Grade Unit of Study Chapter 5: Relationships within Triangles Pacing 9 Days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 5.1 The midsegment of a triangle can be used to uncover relationships within a triangle G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 5.2 Triangles play a key role in relationships involving perpendicular bisectors and angle bisectors. Geometric figures such as angle bisectors and perpendicular bisectors can be used to cut the measure of an angle or segment in half. G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 5.3 There are special parts of a triangle that are always concurrent. A triangle s three perpendicular bisectors are always concurrent, as are a triangle s three angle bisectors, it s three medians, and it s three altitudes. Angle bisectors and segment bisectors can be used in triangles to determine various angle and segment measures. G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

19 5.4 There are special parts of a triangle that are always concurrent. A triangle s three perpendicular bisectors are always concurrent, as are a triangle s three angle bisectors, it s three medians, and it s three altitudes. The length of medians and altitudes in a triangle can be determined given the measures of other triangle segments. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 5.6 The measures of the angles of a triangle are related to the lengths of the opposites sides. Extends G.CO.10 Mathematical Practices: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unwrapped Skills (Students need to be able to do) Unwrapped Concepts (Students need to know) Bloom s Taxonomy Levels Prove (G.CO.10) Theorems about triangles Evaluate Make (G.CO.12) Formal geometric constructions Create Use (G.SRT.5) Congruence and similarity criteria to solve Apply problems Prove (G.SRT.5) Relationships in geometric figures Evaluating Prove (G.CO.9) Theorems about lines and angles Evaluate Construct (G.C.3) Inscribed and circumscribed circles of a triangle Create

20 Prove (G.C.3) Properties of a cyclic quadrilateral Evaluate Vocabulary Resources 5.1 Textbook with Supplementals Midsegment of a Triangle 5.2 Equidistant, Distance from a point to a line 5.3 Concurrent, Point of Concurrency, Circumcenter, Circumscribe, Incenter of a Triangle, Inscribed in 5.4 Median of a Triangle, Centroid of a Triangle, Altitude of a Triangle, Orthocenter of a Triangle Essential Questions 1. How do you use coordinate geometry to find relationships within triangles? 2. How do you solve problems that involve measurements of triangles? Understanding/Corresponding Big Ideas Students will use the midpoint formula to find midsegments of triangles. Students will use the distance formula to examine relationships in triangles. Students will examine inequalities in one triangle. Students will examine inequalities in two triangles.

21 FINDLAY CITY SCHOOLS Curriculum Design Grades 6 12 Subject(s) CP Geometry Grade / Course 10 th Grade Unit of Study Chapter 6: Polygons and Quadrilaterals Pacing 22 days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 6.1 The sum of the angle measures of a polygon depends on the number of sides the polygon has. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 6.2 Parallelograms have special properties regarding their sides, angles, and diagonals. G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 6.3 If a quadrilateral s sides, angles, and diagonals have certain properties, it can be shown that the quadrilateral is a parallelogram. G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 6.4 The special parallelograms, rhombus, rectangle and square, have basic properties of their sides, angles, and diagonals that help to identify them. G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in

22 geometric figures. 6.5 The special parallelograms, rhombus, rectangle and square, have basic properties of their sides, angles, and diagonals that help to identify them. G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 6.6 The angles, sides, and diagonals of a trapezoid have certain properties. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 6.7 The formulas for slope, distance and midpoint can be used to classify and to prove geometric relationships for figures in the coordinate plane. Using variables to name the coordinates of a figure allow relationships to be shown to be true for a general case. G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 6.8 Variables can be used to name the coordinates of a figure in the coordinate plane. This allows relationships to be shown to be true for general case. The formulas for slope, distance and midpoint can be used to classify and to prove geometric relationships for figures in the coordinate plane. Using variables to name the coordinates of a figure allow relationships to be shown to be true for a general case. Prepares for G.GPE Variables can be used to name the coordinates of a figure in the coordinate plane. This allows relationships to be shown to be true for general case. The formulas for slope, distance and midpoint can be used to classify and to prove geometric relationships for figures in the coordinate plane. Using variables to name the coordinates of a figure allow relationships to be shown to be true for a general case. G.GPE.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).

23 Mathematical Practices: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unwrapped Skills (Students need to be able to do) Unwrapped Concepts (Students need to know) Bloom s Taxonomy Levels Prove (G.SRT.5) Relationships in geometric figures Evaluating Prove (G.CO.11) Theorems about parallelograms Evaluate Use (G.GPE.7) Coordinate geometry to compute perimeters and Apply areas of polygons Use (G.GPE.4) Coordinates to prove simple geometric theorems algebraically Apply Vocabulary Resources 6.1 Textbook with Supplementals Equilateral Polygon, Equiangular Polygon, Regular Polygon 6.2 Parallelogram, Opposite Sides, Opposite Angles, Consecutive Angles 6.4 Rhombus, Rectangle, Square 6.6

24 Trapezoid, Base, Leg, Base Angle, Isosceles Trapezoid, Midsegment of a Trapezoid, Kite 6.8 Coordinate Proof Essential Questions Understanding/Corresponding Big Ideas 1. How can you find the sum of the measures of polygon angles? The formula for angle measures of a polygon will be derived using diagonals. 2. How can you classify quadrilaterals? Students will use the properties of parallel and perpendicular lines and diagonals to classify quadrilaterals. Students will use coordinate geometry to classify 3. How can you use coordinate geometry to prove general relationships? special parallelograms. Students will examine slope and segment length in the coordinate plane. Students will use the distance formula in the coordinate plane.

25 FINDLAY CITY SCHOOLS Curriculum Design Grades 6 12 Subject(s) CP Geometry Grade / Course 10 th Grade Unit of Study Chapter 7 Similarity Pacing 14 days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 7.1 An equation can be written stating that two ratios are equal, and if the equation contains a variable, it can be solved to find the value of the variable. Prepares for G.SRT Ratios and proportions can be used to decide whether two polygons are similar and to find unknown side lengths of similar figures. 7.2 Ratios and proportions can be used to prove whether two polygons are similar and to find unknown side lengths. Triangles can be shown to be similar based on the relationship of two or three pairs of corresponding parts. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 7.3 Ratios and proportions can be used to prove whether two polygons are similar and to find unknown side lengths. Triangles can be shown to be similar based on the relationship of two or three pairs of corresponding parts. 7.3 Two triangles can be shown to be similar. Drawing in the altititude to the hypotenuse of a right triangle forms three pairs of similar right triangles. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses 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). 7.4 Drawing in the altitude to the hypotenuse of a right triangle forms the three pairs of similar right triangles. 7.4 It can be proven that the three pairs of right triangles formed by drawing in the altitude to the hypotenuse are similar. 7.4 Two triangles can be shown to be similar. Drawing in the altitude to the hypotenuse of a right triangle forms three pairs of similar right triangles.

26 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses 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). 7.5 When two or more parallel lines intersect other lines, proportional segments are formed. G.SRT.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. Mathematical Practices: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unwrapped Skills (Students need to be able to do) Unwrapped Concepts (Students need to know) Bloom s Taxonomy Levels Use (G.SRT.5) Congruence and similarity criteria to solve Apply problems Prove (G.SRT.5) Relationships in geometric figures Evaluating Prove (G.GPE.5) Relationship of the slopes of parallel and Evaluate perpendicular lines Use (G.GPE.5) Slopes to solve geometric problems Apply Prove (G.SRT.4) Theorems about triangles Evaluate Vocabulary Resources 7.1 Textbook with Supplementals

27 Ratio, extended ratio, proportion, extremes, means, cross products properties 7.2 Similar figures, similar polygons, extended proportions, scale factor, scale drawing, scale 7.3 Indirect measurement 7.4 Geometric mean Essential Questions Understanding/Corresponding Big Ideas 1. How do you use proportions to find side lengths in similar polygons? Student will form proportions based on known lengths of corresponding sides. 2. How do you show two triangles are similar? Students will use the Angle-Angle Similarity Postulate. Students will use the Side-Angle-Side Similarity Theorem. Students will use the Side-Side-Side Similarity Theorem. 3. How do you identify corresponding parts of similar triangles? A key to understanding corresponding parts of similar triangle is to show the triangle in like orientations.

28 FINDLAY CITY SCHOOLS Curriculum Design Grades 6 12 Subject(s) CP Geometry Grade / Course 10 th Grade Unit of Study Chapter 8 Right Triangles and Trigonometry Pacing 16 ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 8.1 If the lengths of any two sides of a right triangle are known, the length of the third side can be found by using the Pythagorean Theorem. G.SRT.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. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 8.2 Certain right triangles have properties that allow their side lengths to be determined without using the Pythagorean Theorem. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 8.3 If certain combinations of side lengths and angle measure of a right triangle are known, ratios can be used to find other side lengths and angle measures. 8.3 Ratios can be used to find side lengths and angle measures of a right triangle when certain combinations of side lengths and angle measures are known. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.MG.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).* 8.4 The angles of elevation and depression are the acute angles of right triangles formed by a horizontal distance and a vertical height. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 8.5 If you know the measures of two angles and any side, or two sides and a nonincluded angle, you can use the Law of Sines to find the other measures of the triangle.

29 G.SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.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). 8.6 If you know the measures of two sides and the included angles, or the measures of all three sides you can use the Law of Cosines to find the other measures of the triangle. G.SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.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). Mathematical Practices: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unwrapped Skills (Students need to be able to do) Unwrapped Concepts (Students need to know) Bloom s Taxonomy Levels Prove (G.SRT.4) Theorems about triangles Evaluate Use (G.SRT.8) Trig ratios and Pythagorean theorem to solve Apply application problems Understand (G.SRT.6) That right triangle similarity leads to trig ratios Understand Explain (G.SRT.7) Relationship between the sine and cosine of Understand complementary angles Use (G.SRT.7) Relationship between the sine and cosine of Apply complementary angles Use (G.MG.1) Geometric shapes, measures and properties to Apply

30 describe objects Prove (G.SRT.10) Law of Sines and Cosines Evaluate Use (G.SRT.10) Law of Sines and Cosines to solve problems. Apply Understand (G.SRT.11) Law of Sines and Cosines to find unknown Understand measurements Apply (G.SRT.11) Law of Sines and Cosines to find unknown measurements Apply 8.1 Pythagorean triple Vocabulary Resources Textbook with Supplementals 8.3 Trigonometric ratios, sine, cosine, tangent 8.4 Angle of elevation, angle of depression 8.5 Law of Sines 8.6 Law of Cosines Essential Questions Understanding/Corresponding Big Ideas 1. How do you find a side length or angle measure in a Students will use the Pythagorean Theorem. right triangle? Students will use concepts of and triangles. Students will use trig ratios to form proportions. 2. How do trig ratios relate to similar right triangles? Student will examine the sine ratio. Student will examine the cosine ratio. Student will examine the tangent ratio.

31 FINDLAY CITY SCHOOLS Curriculum Design Grades 6 12 Subject(s) CP Geometry Grade / Course 10 th Grade Unit of Study Chapter 9 Transformations Pacing 5 ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 9.1 The distance between any two points and the angles in a geometric figure stay the same when (1) its location and orientation changes, (2) it is flipped across a line, or (3) it is turned about a point. G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent The distance between any two points and the angles in a geometric figure stay the same when (1) its location and orientation changes, (2) it is flipped across a line, or (3) it is turned about a point. G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid

32 motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 9.3 The distance between any two points and the angles in a geometric figure stay the same when (1) its location and orientation changes, (2) it is flipped across a line, or (3) it is turned about a point. G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 9.4 One of two congruent figures in a plane can be mapped onto the other by a single reflection, translation, oration, or glide reflection. G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 9.6 A scale factor can be used to make a larger or smaller copy of a figure that is similar to the original figure. 9.6 You can use coordinate geometry to prove triangle congruence and verify properties of similarity. G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare

33 transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Mathematical Practices: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unwrapped Skills (Students need to be able to do) Unwrapped Concepts (Students need to know) Bloom s Taxonomy Levels Represent (G.CO.2) Transformations in the plane Understand Describe (G.CO.2) Transformations as functions Understand Compare (G.CO.2) Congruent and non-congruent transformations Analyze Develop (G.CO.4) Definitions of rotations, reflections and translations in terms of Creating Angles Circles Perpendicular lines Parallel lines Line segments Draw (G.CO.5) A transformed geometric figure Create Identify (G.CO.5) A composition of transformations Understand Use (G.CO.6) Geometric descriptions to transform figures Apply Predict (G.CO.6) The effect of a given rigid motion Analyze

34 Vocabulary 9.1 Transformation, pre-image, image, rigid motion, translation, composition of transformations Resources Textbook with Supplementals 9.2 Reflection, Line of Reflection 9.3 Rotation, center of rotation, angle of rotation 9.4 Glide reflection, isometry 9.6 Dilation, center of dilation, scale factor of dilation, enlargement, reduction Essential Questions 1. How can you change a figure s position without changing it size and shape? How can you change a figure s size without changing its shape? 2. How can you represent a transformation in the coordinate plane? Understanding/Corresponding Big Ideas Students will explore translations, reflections, and rotations. Students will explore dilations. Transformations will be conducted both on and off a coordinate plane. Students will determine the new coordinates of a polygon after any given transformation.

35 Subject(s) Grade / Course Unit of Study Pacing FINDLAY CITY SCHOOLS Curriculum Design Grades 6 12 CP Geometry 10 th Grade Chapter 10-Area 12 days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 10.1 The area of a parallelogram or a triangle can be found when the length of its base and its height are known. G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G.MG.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).* 10.2 The area of a trapezoid can be found when the height and the lengths of its bases are known. The area of a rhombus or a kite can be found when the lengths of its diagonals are known. G.MG.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).* 10.3 The area of a regular polygon is a function of the distance from the center to a side and the perimeter. G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.MG.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).* 10.5 Trigonometry can be used to find the area of a regular polygon when the length of a side, radius, or apothem is known or to find the area of a triangle when the length of two sides and the included angle is known. G.SRT.9 Derive the formula A =.5absin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side The length of part of a circle s circumference can be found by relating it to an angle in the circle. G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

36 G.C.1 Prove that all circles are similar. G.C.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. G.C.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 The area of parts of a circle formed by radii and arcs can be found when the circle s radius is known. G.C.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. Mathematical Practices: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Unwrapped Skills (Students need to be able to do) Use (G.GPE.7) Use (G.MG.1) Construct (G.CO.13) Unwrapped Concepts (Students need to know) Coordinate geometry to compute perimeters and areas of polygons Geometric shapes, measures and properties to describe objects Construct Equilateral triangle Bloom s Taxonomy Levels Apply Apply Create

37 Square Regular hexagon inscribed in a circle Derive (G.SRT.9) The formula A =.5absin (C) to find the area of a Create triangle Know (G.CO.1) Definitions of Understand Angle Circle Perpendicular line Parallel line Line segment Prove (G.C.1) Circles are similar Evaluate Identify (G.C.2) Relationships among inscribed angles, radii and Understand chords Describe (G.C.2) Relationships among inscribed angles, radii and Understand chords Derive (G.C.5) Arc length Create Derive (G.C.5) Formula for the area of a sector Create Vocabulary 10.1 Base of a parallelogram, altitude of a parallelogram, height of a parallelogram, base of a triangle, height of a triangle Resources Textbook with Supplementals 10.2 Height of a trapezoid 10.3 Radius of a regular polygon, apothem 10.6 Circle, center, diameter, radius, congruent circles, central angle, semicircle, minor arc, major arc, adjacent arcs, circumference, pi, concentric circles, arc length