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PRACTICAL GEOMETRY Course #240 Course of Study Findlay City Schools 2013 TABLE OF CONTENTS 1. Findlay City Schools’ Mission Statement and Beliefs 2. Practical Geometry Course of Study 3. Practical Geometry Pacing Guide Course Description: The student will describe two and three-dimensional figures; recognize properties of figures; draw, construct and make models of figures. Much emphasis will be placed upon measurement, including the use of different units of measures and working with various devices to measure segment lengths and angles. Students will use coordinate and transformational approaches to solving problems. Real life applications of geometric concepts will be explored. Calculators will be employed to solve problems. This course is necessary to help prepare for the OGT test and other standardized tests. PRACTICAL GEOMETRY Course of Study Writing Team Ryan Headley Ellen Laube Karen Ouwenga Carrie Soellner Text: Geometry Common Core, 2012 edition; Pearson (publisher); ISBN: 978-0-13-318582-9 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. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) Grade / Course Unit of Study Pacing 1.1 Practical Geometry 10th grade Chapter 1: Tools of Geometry 22 days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 3D objects can be represented with a 2D figure using special drawing techniques Prepares for G.CO.1 1.2 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.1 1.6 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 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) Know (G.CO.1) Find (G.GPE.6) Make (G.CO.12) “Unwrapped Concepts” (Students need to know) Definitions of Angle Circle Perpendicular line Parallel line Line segment Locations of points using segment addition Formal geometric constructions Bloom’s Taxonomy Levels Understand Understand Create Use (N.Q.1) Choose (N.Q.1) Appropriate units for solutions Appropriate units and scales: Formulas Graphs Data Displays Appropriate units and scales: Formulas Graphs Data Displays Interpret (N.Q.1) Vocabulary 1.1 Net, Isometric drawing, orthographic drawing 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 Understand Apply Analyze Resources Textbook with Supplementals 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? 2. What are the building blocks of geometry? 3. How can you describe the attributes of a segment or angle? Understanding/Corresponding Big Ideas Students will make nets for solid figures. Isometric drawings and orthographic drawings will be used to show attributes of figures. 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. 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. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) Grade / Course Unit of Study Pacing Practical Geometry 10th Grade Chapter 2: Reasoning and Proof 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.11 2.2 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.11 2.5 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.11 2.6 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. 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) Prove (G.CO.9) “Unwrapped Concepts” (Students need to know) Theorems about lines and angles Vocabulary 2.1 Inductive Reasoning, Conjecture, Counterexample Bloom’s Taxonomy Levels Evaluate 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 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. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) Grade / Course Unit of Study Pacing Practical Geometry 10th Grade Chapter 3: Parallel and Perpendicular Lines 18 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.9 3.2 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.9 3.4 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. 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.5 3.8 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 Perpendicular line Parallel line Line segment Bloom’s Taxonomy Levels Understand Prove (G.CO.9) Prove (G.CO.11) Make (G.CO.12) Construct (G.CO.13) Theorems about lines and angles Theorems about parallelograms Formal geometric constructions Construct Equilateral triangle Square Regular hexagon inscribed in a circle Geometric methods to solve design problems Apply (G.MG.3) Vocabulary 3.1 Skew Lines, Parallel Lines, Parallel Planes, Transversal, Alternate Interior Angles, Same Side Interior Angles, Corresponding Angles, Alternate Interior Angles Evaluate Evaluate Create Create Apply 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? 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? 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. Students will use the triangle angle sum theorem. Students will write equations using slope intercept form. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) Grade / Course Unit of Study Pacing Practical Geometry 10th 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.5 4.2 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. 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) Use (G.SRT.5) Make (G.CO.12) Prove (G.CO.10) Construct (G.CO.13) “Unwrapped Concepts” (Students need to know) Congruence and similarity criteria to solve problems Formal geometric constructions Theorems about triangles Construct Equilateral triangle Bloom’s Taxonomy Levels Apply Create Evaluate Create Square Regular hexagon inscribed in a circle Vocabulary 4.1 Congruent Polygons 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 1. How do you identify corresponding parts of congruent triangles? 2. How do you show that two triangles are congruent? 3. How can you tell whether a triangle is isosceles or equilateral? Understanding/Corresponding Big Ideas 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. Students will use SSS, SAS, ASA, AAS and HL. Students will use the definitions and look at the number of congruent sides and angles. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) PRACTICAL Geometry Grade / Course 10th Grade Unit of Study Chapter 5: Relationships within Triangles Pacing 10 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. 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) Prove (G.CO.10) Make (G.CO.12) Use (G.SRT.5) Prove (G.SRT.5) Prove (G.CO.9) Construct (G.C.3) Prove (G.C.3) “Unwrapped Concepts” (Students need to know) Theorems about triangles Formal geometric constructions Congruence and similarity criteria to solve problems Relationships in geometric figures Theorems about lines and angles Inscribed and circumscribed circles of a triangle Properties of a cyclic quadrilateral Bloom’s Taxonomy Levels Evaluate Create Apply Evaluating Evaluate Create Evaluate Vocabulary 5.1 Midsegment of a Triangle Resources Textbook with Supplementals 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. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) PRACTICAL Geometry Grade / Course 10th Grade Unit of Study Chapter 6: Polygons and Quadrilaterals Pacing 19 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 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.★ 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) Prove (G.SRT.5) Prove (G.CO.11) Use (G.GPE.7) “Unwrapped Concepts” Bloom’s (Students need to know) Taxonomy Levels Relationships in geometric figures Evaluating Theorems about parallelograms Evaluate Coordinate geometry to compute perimeters and Apply areas of polygons 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 Trapezoid, Base, Leg, Base Angle, Isosceles Trapezoid, Midsegment of a Trapezoid, Kite Essential Questions 1. How can you find the sum of the measures of polygon angles? 2. How can you classify quadrilaterals? 3. How can you use coordinate geometry to prove general relationships? Understanding/Corresponding Big Ideas The formula for angle measures of a polygon will be derived using diagonals. Students will use the properties of parallel and perpendicular lines and diagonals to classify quadrilaterals. Students will use coordinate geometry to classify special parallelograms. Students will examine slope and segment length in the coordinate plane. Students will use the distance formula in the coordinate plane. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) Grade / Course Unit of Study Pacing Practical Geometry 10th Grade Chapter 7 – Similarity 17 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.5 7.2 – 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. 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) Use (G.SRT.5) Prove (G.SRT.5) Prove (G.GPE.5) Use (G.GPE.5) Prove (G.SRT.4) “Unwrapped Concepts” (Students need to know) Congruence and similarity criteria to solve problems Relationships in geometric figures Relationship of the slopes of parallel and perpendicular lines Slopes to solve geometric problems Theorems about triangles Bloom’s Taxonomy Levels Apply Evaluating Evaluate Apply Evaluate Vocabulary 7.1 Ratio, extended ratio, proportion, extremes, means, cross products properties Resources Textbook with Supplementals 7.2 Similar figures, similar polygons, extended proportions, scale factor, scale drawing, scale 7.3 Indirect measurement 7.4 Geometric mean Essential Questions How do you use proportions to find side lengths in similar polygons? 2. How do you show two triangles are similar? 1. 3. How do you identify corresponding parts of similar triangles? Understanding/Corresponding Big Ideas Student will form proportions based on known lengths of corresponding sides. 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. A key to understanding corresponding parts of similar triangle is to show the triangle in like orientations. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) Grade / Course Unit of Study Pacing Practical Geometry 10th Grade Chapter 8 – Right Triangles and Trigonometry 14 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.★ 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) Prove (G.SRT.4) Use (G.SRT.8) Understand (G.SRT.6) Explain (G.SRT.7) Use (G.SRT.7) Use (G.MG.1) “Unwrapped Concepts” (Students need to know) Theorems about triangles Trig ratios and Pythagorean theorem to solve application problems That right triangle similarity leads to trig ratios Relationship between the sine and cosine of complementary angles Relationship between the sine and cosine of complementary angles Geometric shapes, measures and properties to describe objects Vocabulary 8.1 Pythagorean triple 8.3 Trigonometric ratios, sine, cosine, tangent 8.4 Angle of elevation, angle of depression Bloom’s Taxonomy Levels Evaluate Apply Understand Understand Apply Apply Resources Textbook with Supplementals Essential Questions 1. How do you find a side length or angle measure in a right triangle? 2. How do trig ratios relate to similar right triangles? Understanding/Corresponding Big Ideas Students will use the Pythagorean Theorem. Students will use concepts of 30-60-90 and 45-45-90 triangles. Students will use trig ratios to form proportions. Student will examine the sine ratio. Student will examine the cosine ratio. Student will examine the tangent ratio. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) Grade / Course Unit of Study Pacing Practical Geometry 10th Grade Chapter 9– Transformations 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. 9.2. – 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. 9.3 – 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 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) Represent (G.CO.2) Describe (G.CO.2) Compare (G.CO.2) Develop (G.CO.4) Bloom’s Taxonomy Levels Understand Understand Analyze Creating Draw (G.CO.5) Identify (G.CO.5) Use (G.CO.6) “Unwrapped Concepts” (Students need to know) Transformations in the plane Transformations as functions Congruent and non-congruent transformations Definitions of rotations, reflections and translations in terms of Angles Circles Perpendicular lines Parallel lines Line segments A transformed geometric figure A composition of transformations Geometric descriptions to transform figures Predict (G.CO.6) The effect of a given rigid motion Analyze Create Understand Apply 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. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) Grade / Course Unit of Study Pacing Practical Geometry 10th 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).* 10.6 – 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. 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. 10.7 – 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) Know (G.CO.1) “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 Square Regular hexagon inscribed in a circle Definitions of Angle Circle Perpendicular line Parallel line Bloom’s Taxonomy Levels Apply Apply Create Understand Line segment Circles are similar Relationships among inscribed angles, radii and chords Relationships among inscribed angles, radii and chords Arc length Formula for the area of a sector Prove (G.C.1) Identify (G.C.2) Describe (G.C.2) Derive (G.C.5) Derive (G.C.5) Vocabulary 10.1 Base of a parallelogram, altitude of a parallelogram, height of a parallelogram, base of a triangle, height of a triangle Evaluate Understand Understand Create Create 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 10.7 Sector of a circle, segment of a circle Essential Questions 1. How do you find the area of a polygon or find the circumference and area of a circle? Understanding/Corresponding Big Ideas Students will use formulas to find areas of parallelograms, triangles, trapezoids, rhombuses, and kites. Students will explore area concepts related to regular 2. How do perimeters and areas of similar polygons compare? polygons. Students will use trigonometry to find areas. Students will find circumferences and areas of circles. Students will examine ratios among similar figures. Given a figure and its area, students will be able to find the area of a figure similar to the original figure. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) Grade / Course Unit of Study Pacing Practical Geometry 10th Grade Chapter 11 Surface Area and Volume 10 days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 11.1 – A three-dimensional figure can be analyzed by describing the relationships among its vertices, edges, and faces. G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects. 11.2 – The surface area of a three-dimensional figure is equal to the sum of the areas of each surface of the figure. 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).* 11.3 - The surface area of a three-dimensional figure is equal to the sum of the areas of each surface of the figure. 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).* 11.4 – The volume of a prism and a cylinder can be found when its height and the area of its base are known. G.GMD.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. G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve 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).* 11.5 – The volume of a pyramid is related to the volume of a prism with the same base and height. G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve 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).* 11.6 – The surface area and the volume of a sphere can be found when its radius is known. G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve 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).* 11.7 – Ratios can be used to compare the areas and volumes of similar solids. 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).* G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* 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) Identify (G.GMD.4) Identify (G.GMD.4) Use (G.MG.1) Give (G.GMD.1) “Unwrapped Concepts” (Students need to know) The shapes of cross sections of 3D figures 3D objects generated by rotations of 2D objects Geometric shapes, measures and properties to describe objects An informal argument for the formulas of Circumference Area of a circle Volume of a cylinder Volume of a pyramid Bloom’s Taxonomy Levels Understand Understand Apply Evaluate Use (G.GMD.3) Use (G.MG.1) Apply (G.MG.2) Volume of a cone Volume formulas to solve problems for Cylinders Pyramids Cones spheres Geometric shapes, measures and properties to describe objects Concepts of density based on area and volume in modeling situations Vocabulary 11.1 Polyhedron, face, edge, vertex, cross section 11.2 Prism(base, lateral face, altitude, height, lateral area, surface area), right prism, oblique prism, cylinder (base, altitude, height, lateral area, surface area), right cylinder, oblique cylinder 11.3 Pyramid (base, lateral face, vertex, altitude, height, slant height, lateral area, surface area), regular pyramid, cone (base, altitude, vertex, height, slant height, lateral area, surface area), right cone 11.4 Volume, composite space figure 11.5 Volumes of Pyramids and Cones Apply Apply Apply Resources Textbook with Supplementals 11.6 Sphere, center of a sphere, radius of a sphere, diameter of a sphere, circumference of a sphere, great circle, hemisphere 11.7 Similar solids Essential Questions 1. How can you determine the intersection of a solid and a plane? 2. How do you find the surface area and volume of a solid? Understanding/Corresponding Big Ideas Students will examine cross sections. 3. How do the surface areas and volumes of similar solids compare? Students will use formulas to find surface areas and volumes of prisms and cylinders. Students will use formulas to find surface areas and volumes of pyramids and cones. Students will use formulas to find surface areas and volumes of spheres. Students will examine ratios among similar solids. Given a figure and its surface area, students will be able to find the surface area of a solid similar to the original solid. Given a figure and its volume, students will be able to find the volume of a solid similar to the original solid. FINDLAY CITY SCHOOLS Curriculum Design – Grades 6 – 12 Subject(s) Grade / Course Unit of Study Pacing Practical Geometry 10th Grade Chapter 12 Circles 14 days ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS 12.1 – A radius of a circle and the tangent that intersects the endpoint of the radius on the circle have a special relationship. Prepares for G.C.2 12.2 – Information about congruent parts of a circle (or congruent circles) can be used to find information about other parts of the circle (or circles). 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. 12.3 – Angles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept. This includes (1) arcs formed by chords that inscribe angles, (2) angles and arcs formed by lines intersecting either within a circle of outside a circle, and (3) intersecting chords, intersecting secants, or a secant that intersects a tangent. 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.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 12.4 - Angles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept. This includes (1) arcs formed by chords that inscribe angles, (2) angles and arcs formed by lines intersecting either within a circle of outside a circle, and (3) intersecting chords, intersecting secants, or a secant that intersects a tangent. Extends G.C.2 12.5 - The information in the equation of a circle allows the circle to be graphed. The equation of a circle can be written if its center and radius are known. G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 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) Identify (G.C.2) Describe (G.C.2) Construct (G.C.3) Prove (G.C.3) Derive (G.GPE.1) Vocabulary 12.1 Tangent to a circle, point of tangency 12.2 Chord “Unwrapped Concepts” (Students need to know) Relationships among inscribed angles, radii and chords Relationships among inscribed angles, radii and chords Inscribed and circumscribed circles of a triangle Properties of a cyclic quadrilateral The equation of a circle Bloom’s Taxonomy Levels Understand Understand Create Evaluate Create Resources Textbook with Supplementals 12.3 Inscribed angle, intersected arc 12.4 Secant 12.5 Standard form of an equation of a circle Essential Questions 1. How can you prove relationships between angles and arcs in a circle? 2. When lines intersect a circle, or within a circle, how do you find the measures of resulting angles, arcs, and segments? 3. How do you find the equation of a circle in the coordinate plane? Understanding/Corresponding Big Ideas Students will examine angles formed by lines that intersect inside and outside a circle. Students will relate arcs and angles. Students will use properties of tangent lines. Students will use the relationships among chords, arcs, and central angles. Students will solve problems with angles formed by secants and tangents. The center and radius of a circle in a coordinate plane can be used to find the equation of a circle. Chapter 1 Section CP Geometry Review how to solve equations Pacing 1 Students take SLO test and do more algebra review Also plan on getting books this day 1 Students take SLO test and do more algebra review 1 1 Also plan on getting books this day 1 Teach 1.2 and 1.3 together. Spend one day on each, then a 3rd day combining them See above 1½ Teach 1.2 and 1.3 together. Spend one day on each, then a 3rd day combining them See above 1½ 1 1.4 Not a full period…begin 1.4 After quiz, do measuring and classifying activity 1.5 On day 3, pull 1.4 and 1.5 together 3 On day 3, pull 1.4 and 1.5 together 3 Only half period quiz Then practice with compass and concept byte Day 1: #1 and #3 Day 2: #2 and #4 1 Only half period quiz Then practice with compass and concept byte 1 2 Day 1: #1 and #3 Day 2: #2 and #4 2 2 Also do page 57 Review 2 First Day of School Second Day of School 1.1 1.2 1.3 QUIZ QUIZ 1.6 Also do page 57 Review 1½ 1½ Practical Geometry Review how to solve equations Not a full period…begin 1.4 After quiz, do measuring and classifying activity Pacing 1 1½ 1 1½ 1.7 Concept 1.8 Performance Task Review Test Total Days No performance task on this test Concept 1 2 1 1 1 2 1 20 No performance task on this test 1 22 Chapter 2 Section 2.1 2.2 2.3 2.4 CP Geometry Teach 2.1 and 2.2 together…only do #1 and #2 See above ½ Practical Geometry Teach 2.1 and 2.2 together…only do #1 and #2 See above SKIP 0 SKIP 0 SKIP 0 SKIP 0 2.5 2.6 Performance Task Review Test Total Days SKIP!!! Pacing ½ Pacing ½ ½ 2 2 2 2 0 SKIP!!! 0 1 1 1 1 7 7 Chapter 3 Section 3.1 3.2 CP Geometry Combine 3.1 and 3.2 Day 1: naming and looking at relationships Day 2: Proofs Day 3: Start angle project Day 4: Workday on angle project Only ½ day quiz, then start 3.4 Day 1: Teach 3.5 Day 2: Teach 5.6 Day 3: Practice problems finding missing angles NO point slope form 3 1 Do #2 and #3 1 2 ½ 1½ Day 1: Teach 3.5 Day 2: Teach 5.6 Day 3: Practice problems finding missing angles 3 1 NO point slope form 2 3.8 Performance Task Only ½ day quiz, then start 3.4 1 3.6 3.7 ½ Pacing 2 2 1½ 3.4 3.5 2 Practical Geometry Combine 3.1 and 3.2 Day 1: naming and looking at relationships Day 2: Proofs Day 3: Start angle project Day 4: Workday on angle project **easier angle project than CP** 1 3.3 QUIZ Pacing 2 1 2 Do #2 and #3 1 1 Review Test Total Days Include a performance task question 1 16 2 Include a performance task question 1 18 Chapter 4 Section CP Geometry 4.1 Concept Byte 4.2 4.3 Page 225 1 Combine 4.2 and 4.3 Day 1: Identify with pics Day 2: Proving See above 1 Combine 4.2 and 4.3 Day 1: Identify with pics Day 2: Proving See above 1 Total Days 1 2 Send home graded HW problem after day 2 2 Day 1: all overlapping Day 1: all proofs Day 2: Find angles and performance tasks 1 2 Send home graded HW problem after day 2 1 Performance See below Task Do performance task #1 Review with review Test Pacing 1 1 4.6 4.7 Practical Geometry Page 225 4.4 4.5 Pacing 1 2 1 2 Day 1: all overlapping 2 0 See below 0 2 Do performance task #1 with review 2 2 Day 1: all proofs Day 2: Find angles and performance tasks 2 16 16 Chapter 5 Section CP Geometry 5.1 1 5.2 5.7 Test Total Days 2 2 Day 3: Review 5.3-5.4 3 SKIP 0 SKIP 0 TEACH THIS AFTER SECTION 3.5 0 TEACH THIS AFTER SECTION 3.5 0 SKIP 0 SKIP 0 1 Only #1 Use more from resources 1 Only #1 Performance Use more from resources Task Review Day 1: Teach 5.2 Day 2: Review 5.1-5.2 Pacing 1 2 5.4 5.6 Practical Geometry 2 5.3 5.5 Pacing 1 1 1 1 1 9 10 Chapter 6 Section 6.1 6.2 CP Geometry Also talk about the Quadrilateral Family Tree Day 1: #1 and #2 Day 2: #3 and #4 6.3 Teach 6.4 and 6.5 together REVIEW Day 1: #1 and #2 Day 2: #3 and #4 2 2 SKIP PROOFS 1 ½ 1 Teach 6.4 and 6.5 together 1 1 Day 1: Trapezoids Day 2: Kites 2 Day 1: Trapezoids Day 2: Kites 2 Review 6.1 – 6.6 1 Review 6.1 – 6.6 2 QUIZ 6.7 6.8 Pacing 1 2 ½ 6.5 6.6 Practical Geometry Also talk about the Quadrilateral Family Tree 1 QUIZ 6.4 Pacing 1 Day 1: Instruction Day 2: Group work 1 1 2 3 2 SKIP 0 6.9 Performance Task Day 1: Instruction Day 2: Group Work 2 SKIP 0 All 3 1 Do only #1 and #3 1 2 Review Test Total Days Day 1: 6.1 – 6.6 Day 2: 6.7 – 6.9 2 22 2 Only 6.1 – 6.7 1 19 Chapter 7 Section PROJECT CP Geometry Start the chapter with the Similarity Cartoon/Comic project (see Ellen, Karen or Carrie for help) 7.1 7.2 7.3 QUIZ 7.4 7.5 Performance Task Review Test Total Days Pacing 1 Practical Geometry Start the chapter with the Similarity Cartoon/Comic project (see Ellen, Karen or Carrie for help) Pacing 1 1 1 1 1 Days 1-2: Inside Day 3: Go outside! 3 Days 1-3: Inside Day 4: Go outside! 4 Only half the period Review how to simplify radicals the other half 1 Day 1: Quiz Day 2: Review how to simplify radicals Skip #4 2 2 2 2 2 1 1 1 2 1 1 14 17 Chapter 8 Section CP Geometry Pacing Practical Geometry Pacing 8.1 1 1 8.2 2 3 Quiz ½ ½ 8.3 8.4 ½ Trig ratios 1 day solving angles and sides ½ day solving angles and sides ½ day application 1 day of application 1 day outside (p 515) 8.5 8.6 8.5 – 8.6 2 2.5 1 Teach only one formula Pull it all together 2 1 ½ Trig ratios 2 days of solving for sides and angles 2 days of application 1 day outside (p 515) 2.5 3 SKIP SKIP SKIP Performance Task 1 1 Review 2 2 Test 1 1 Total Days 16 14 Chapter 9 Section 9.1 9.2 CP Geometry Combine 9.1 and 9.2 Pacing 1 See above Practical Geometry Combine 9.1 and 9.2 Pacing 1 See above 9.3 1 1 9.4 1 1 9.5 9.6 9.7 Performance Task Review SKIP Talk about compositions SKIP 1 Talk about compositions SKIP SKIP SKIP SKIP 1 ½ Test ½ Total Days 5 Use a project instead of a test 1 5 Chapter 10 Section 10.1 10.2 CP Geometry Combine 10.1 and 10.2 1 See above 10.3 10.4 Pacing Practical Geometry Combine 10.1 and 10.2 Pacing 1 See above 2 SKIP 2 SKIP SKIP 10.5 1 Quiz 1 1 10.6 2 2 10.7 2 2 10.8 Performance Task Review SKIP SKIP Only problems 1 and 2 Only problems 1 and 2 1 1 1 2 Test 1 1 Total Days 12 12 Chapter 11 Section 11.1 CP Geometry Include rotations of 2D objects 11.2 Combine 11.2 and 11.3 *give formulas 11.3 11.4 11.5 Pacing 1 Practical Geometry Include rotations of 2D objects 1 Combine 11.2 and 11.3 *give formulas See above See above Combine 11.4 and 11.5 Combine 11.4 and 11.5 1 See above Pacing 1 2 1 See above 11.6 1 1 11.7 2 2 Performance Task 1 1 Review 1 1 Test 1 1 Total Days 9 10 Chapter 12 Section CP Geometry Pacing Practical Geometry Pacing 12.1 1 1 12.2 2 2 12.3 2 2 Quiz 1 1 1 day – angles 1 day – lengths 2 1 Pull it all together 1 2 1 1 12.4 12.1 – 12.4 12.5 12.6 SKIP SKIP Performance Task 1 1 Review 1 1 Test 1 1 Total Days 13 14