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Course Title: Geometry Grade Level(s): 8th-12th Materials: Objectives: All students will: Review and master Algebra concepts previously learned. Construct, formulate, and evaluate area, perimeter, and volume of two-dimensional and three-dimensional. Explore angle relationships including parallel lines, straight line, angle measurements, and polygon properties. Deduce and justify proofs using logic, conjectures, and indirect proofs to establish relationships and solve application problems. Demonstrate congruence between polygons specifically triangles. Use spatial visualization to construct models, draw diagrams, and solve problems. Relate trigonometry to right triangles. Demonstrate similarity between polygons and polyhedra; specifically triangles. Explore the basic properties, theorems, and measurements of circles Essential Questions: Algebra 1. How calculating slope and identifying the y-intercept work together to write and graph an equation? 2. How do we use patterns, tables, systematic lists of data, and diagrams to problem solve? 4. How do you find the length and midpoint of a line segment? 5. How do we solve and graph solutions to single-variable and linear inequalities? 6. How do we simplify radicals? 7. What are the essential components of solving a rational single variable, or Algebraic, fraction to eliminate fractions in an equation prior to solving? 8. How do we use the law of exponents to solve problems? 9. How do you use the substitution and elimination methods of systems of equations to find the point of intersection? 10. How can we determine the slope of perpendicular, parallel, horizontal, and vertical linear equations? 11. How do we solve quadratic equations using to factoring and the Quadratic Formula? 12. How do we solve factored quadratic equations using to Zero Product Property? 13. How do the solutions of quadratic equations relate graphically? 13. How can we graph non-linear relationships? 14. How to complete arithmetic operations of polynomial expressions and simplification of polynomial expressions. Area, Perimeter, Volume 1. What ways can we demonstrate how to use visual representation to calculate area and perimeter. 2. What are effective methods for finding perimeter and area of polygons? 3. How do we use patterns, tables, systematic lists of data, and diagrams to problem solve? 4. How would we develop and solve the formulas for Pythagorean Theorem and area and volume of two-dimensional and three-dimensional figures including polygons. Angle Relationships 1. How can we identify angle vocabulary and angle relationships to solve problems? 2. How would you recognize angle relationships formed by parallel lines and transversals when you use a diagram? 3. What relationships exist in the sum of the angles of any triangle is 180o and exterior angles of triangles? 4. How do you find the length and midpoint of a line segment? 5. How can angle relationships be used to solve multi-step problems? Proofs and Logic 1. What are the essential components of constructing logical arguments to justify your solutions to logic games and simple number theory principles? 2. How can you use observations and patterns to write a conjecture? 3. What information can be extracted from word problems and then apply to draw diagrams? 4. How do we solve previous conjectures using a proof? 5. What are representations of various proof formats? 6. How would you use an indirect proof or proof by contradiction to justify conclusions? 7. What are the essential components of constructing logical arguments to construct a proof? Congruence 1. How do we transform polygons on a flat surface while preserving their size and shape? 2. What are the most effective ways to discover and use the conditions user which pairs of triangles must be congruent? 3. How would you prove that parts of polygons, specifically triangles and quadrilaterals, are congruent to corresponding parts of other polygons, specifically triangles and quadrilaterals? 4. How do we determine the limits to the length of the third side of a triangle? 5. How can we determine the slope of perpendicular, parallel, horizontal, and vertical linear equations? 6. How do we solve quadratic equations using to factoring and the Quadratic Formula? Trigonometry: Triangle Ratios 1. How do we find the lengths of sides in right triangles when you only know two of the six measurements? 2. What strategies can we use to solve trigonometric application problems? 3. How do we find the measures of angles in right triangles when you only know two of the six measurements? 4. What are the essential components of solving a rational single variable fraction to eliminate fractions in an equation prior to solving? 5. How do we simplify radicals? 6. How do we apply the ratios of Special Right Triangles? Spatial Visualization 3. How do we explore and discuss visualization activities? 4. How do we use cube stacks to draw Isometric representations? 5. How would you recognize relationships among properties, constructions and 2 dimensional drawings of prisms and pyramids? 6. How do I calculate surface areas and volumes of polyhedra? Similarity 1. How do we use the properties of similarity to determine whether two figures are similar? 2. What strategies can we use to solve application problems? 2 3 3. What are the essential components of r : r : r ? 4. How do we prove triangle and polygonal similarity? 5. What are the basic theorems of similarity? 6. How do the basic theorems of similarity apply to polygons. Circles 1. What is the essential vocabulary in terms of circles and coordinate geometry? 2. How is the vocabulary used to solve and justify complex problems? 3. How do geometric theorems justify properties of circles? 4. How is the tangent/diameter theorem used to prove and solve problems? 5. How is the diameter/chord theorem used to prove and solve problems? 6. How can we use the concept of similarity to determine arc lengths and areas of sectors of circles? 7. Determine the difference between the description, length, and measurement of arcs including inscribed angles, intercepted arcs, and central angles. 8. What are the essential components used to apply coordinate geometry to formulate simple geometric theorems algebraically? Standards Content (What students will know) Performance (Student will do) CC.2.3.HS.A.1 Use geometric figures and their properties to represent transformations in the plane. Use mirrors and other techniques to transform polygons. G.1.3.1.1, G.1.3.1.2 Explore and discuss visualization activities. Represent transformation word problems through spatial visualization and diagrams. Learn to draw transformation representations of polygons CC.2.3.HS.A.2 G.1.3.1.1, G.1.3.1.2 CC.2.3.HS.A.3 G.1.2.1.1, G.1.2.1.2, G.1.2.1.3, G.1.2.1.4, G.1.2.1.5, G.1. 3.2.1, G.2.2.1.1, G.2.2.1.2, G.2.2.2.1, G.2.2.2.2, G.2.2.2.3, G.2.2.2.4, G.2.2.2.5 Transformation of polygons Apply rigid transformations to determine and explain congruence. How to transform polygons and determine where congruence exists. Verify and apply geometric theorems as they relate to geometric figures. Prove triangle congruence. Construct models to assess the relationships between congruent triangles. Apply geometric theorems as they relate to geometric figures to solve problems. 2 3 Develop and use the components of r : r : r . Activities/Assessments CC.2.3.HS.A.4 CC.2.3.HS.A.5 G.1.3.1.1, G.1.3.1.2 CC.2.3.HS.A.6 G.1.3.1.1, G.1.3.1.2, G.1.3.2.1 Apply the concept of congruence to create geometric constructions. Construct proofs and logical argumentation to prove triangle congruence properties/theorems. Develop and prove triangle congruence properties/theorems. Create justifications based on transformations to establish similarity of plane figures. Explore properties of figures with the same shape: similarity. Understand the properties of figures with the same shape: similarity. Discover the basic theorems of similarity and apply them to triangles. Discover the basic theorems of similarity and apply them to triangles. Verify and apply theorems involving similarity as they relate to plane figures. Explore properties of plane figures with the same shape: similarity. Understand the properties of plane figures with the same shape: similarity. Discover the basic theorems of similarity and apply them to plane figures. Discover the basic theorems of similarity and apply them to plane figures. 2 3 Develop and use the components of r : r : r . CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles. Compute the lengths of sides in right triangles when given exactly two measurements. G.2.1.1.1, G.2.1.1.2 Find the lengths of sides in right triangles when you only know two of the six measurements. Solve interesting application problems. How to solve interesting application problems. Compute the measurements of angles in right triangles when given exactly two measurements. Find the measures of angles in right triangles when you only know two of the six measurements. Solve algebraic fractions. Review the simplification process for radicals. How to eliminate fractions in an algebraic fraction prior to solving equations. CC.2.3.HS.A.8 G.1.1.1.1, G.1.1.1.2, G.1.1.1.3, G.1.1.1.4, G.1.3.2.1, G.2.2.3.1 How to simplify radicals. Apply geometric theorems to verify properties of circles. Compose and explain geometric theorems to verify properties of circles. Geometric theorems to justify properties of circles. Develop and use the tangent/diameter theorem and diameter/chord theorem. CC.2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles. G.1.1.1.1, G.1.1.1.2, G.1.1.1.3, G.1.1.1.4, G.2.2.2.1, G.2.2.2.2, G.2.2.2.3, G.2.2.2.4, G.2.2.2.5, G.2.2.3.1 CC.2.3.HS.A.11 How to use the concept of similarity to determine arc lengths and areas of sectors of circles. G.2.1.2.1, G.2.1.2.2, G.2.1.2.3 How to apply coordinate geometry to formulate simple geometric theorems algebraically. Justify the concept of similarity to determine arc lengths and areas of sectors of circles. Difference between the description, length, and measurement of arcs including inscribed angles, intercepted arcs, and central angles. Apply coordinate geometry to prove simple geometric theorems algebraically. Support and defend the application of coordinate geometry to prove simple geometric theorems algebraically. Develop and use the tangent/diameter theorem and diameter/chord theorem. CC.2.3.HS.A.12 Explain volume formulas and use them to solve problems. Formulate volume formulas of polygons and use them to solve application problems. G.2.3.1.1, G.2.3.1.2, G.2.3.1.3 Efficient ways to calculate lengths, areas, and volumes between similar two and three dimensional figures. Construct three-dimensional models and represent them in two-dimensional space. How to explore and discuss visualization activities. Construct and draw Isometric representations of cube stacks. Explore and discuss visualization activities. Learn how to draw Isometric representations of cube stacks Understand properties, build and draw selected polyhedra, specifically prisms and pyramids, on 2-dimensional surfaces. How to calculate surface areas and volumes of polyhedra. Build polyhedra models, including prisms and pyramids, to formulate surface area and volume formulas. 2 3 Develop and use the components of r : r : r . CC.2.3.HS.A.13 Analyze relationships between two‐ dimensional and three‐dimensional objects. G.1.1.1.1, G.1.1.1.2, G.1.1.1.3, G.1.1.1.4, G.1.2.1.1, G.1.2.1.2, G.1.2.1.3, G.1.2.1.4, G.1.2.1.5, G.2.3.2.1 Efficient ways to calculate lengths, areas, and volumes between similar two and three dimensional figures. How to explore and discuss visualization activities. Learn how to draw Isometric representations of cube stacks Compare and assess relationships between two-dimensional and three-dimensional objects. Build models to analyze relationships between two-dimensional and three-dimensional objects. Construct and draw Isometric representations of cube stacks. Explore and discuss visualization activities. Understand properties, build and draw selected polyhedra, specifically prisms and pyramids, on 2-dimensional surfaces. CC.2.3.HS.A.14 G.2.2.4.1, G.2.3.1.1, G.2.3.1.2, G.2.3.1.3 How to calculate surface areas and volumes of polyhedra. How to apply geometric concepts to model and Construct models using geometric concepts to solve real word problems. solve real word problems. Create diagrams using geometric concepts to solve real work problems. Title: Algebra Concepts Standards Content (What students will know) Performance (Student will do) CC.2.2.HS.D.1 How to interpret the structure of expressions to represent a quantity in terms of its context. Interpret and justify the structure of expressions to represent a quantity in terms of its context. How to write expressions in equivalent forms to solve problems. Write expressions in equivalent forms to solve problems. Knowledgeable of arithmetic operations and apply to polynomials. Demonstrate knowledge of arithmetic operations and apply to polynomials. A1.1.1.5.1, A1.1.1.5.2, A1.1.1.5.3, A2.1.2.2.1, A2.1.2.2.2 CC.2.2.HS.D.2 A1.1.1.5.1, A1.1.1.5.2, A1.1.1.5.3, A2.1.2.1.1, A2.1.2.1.2, A2.1.2.1.3, A2.1.2.1.4, A2.1.2.2.1, A2.1.2.2.2 CC.2.2.HS.D.3 A1.1.1.5.1, A1.1.1.5.2, A1.1.1.5.3, A2.1.2.2.1, A2.1.2.2.2 Activities/Assessments CC.2.2.HS.D.4 The relationship between zeroes and factors of polynomials. A2.1.2.2.1, A2.1.2.2.2 How to make generalizations about functions and their graphs Demonstrate an understanding of the relationship between zeroes and factors of polynomials to make generalizations about functions and their graphs. Use the concept of factoring and the Zero Product Property to establish a relationship between zeroes and factors of polynomials. CC.2.2.HS.D.5 A1.1.1.5.1, A1.1.1.5.2, A1.1.1.5.3, A2.1.2.2.1, A2.1.2.2.2, A2.1.3.1.1, A2.1.3.1.2, A2.1.3.1.3, A2.1.3.1.4 CC.2.2.HS.D.6 A1.1.1.5.1, A1.1.1.5.2, A1.1.1.5.3, A2.1.3.1.1, A2.1.3.1.2, A2.1.3.1.3, A2.1.3.1.4 How to use polynomial identities to solve problems. How to extend their knowledge of rational functions to rewrite in equivalent forms. Construct and assess quadratic equations graphically to establish a relationship between zeroes and factors of polynomials. Use polynomial identities to solve problems. Demonstrate an extended knowledge of rational function to rewrite in equivalent forms. CC.2.2.HS.D.7 A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3, A1.1.2.2.1, A1.1.2.2.2, A1.1.3.1.1, A1.1.3.1.2, A1.1.3.1.3, A1.1.3.2.1, A1.1.3.2.2, A2.1.3.1.1, A2.1.3.1.2, A2.1.3.1.3, A2.1.3.1.4, A2.1.3.2.1, A2.1.3.2.2, A2.2.2.1.1, A2.2.2.1.2, A2.2.2.1.3, A2.2.2.1.4 CC.2.2.HS.D.8 A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3, A2.1.3.1.1, A2.1.3.1.2, A2.1.3.1.3, A2.1.3.1.4, A2.1.3.2.1, A2.1.3.2.2 How to create and graph equations or inequalities to describe numbers or relationships. Create and graphically represent equations or inequalities to describe numbers or relationships. When and how to apply inverse operations to solve equations or formulas for a given variable. Apply inverse operations including reciprocal to solve equations or formulas for a given variable. CC.2.2.HS.D.9 A1.1.1.4.1, A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3, A1.1.2.2.1, A1.1.2.2.2, A1.1.3.1.1, A1.1.3.1.2, A1.1.3.1.3, A2.1.3.1.1, A2.1.3.1.2, A2.1.3.1.3, A2.1.3.1.4, A2.1.3.2.1, A2.1.3.2.2 How to use reasoning to solve equations and justify the solution method. Demonstrate the use of reasoning to solve equations and justify the solution method. CC.2.2.HS.D.10 A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3, A1.1.2.2.1, A1.1.2.2.2, A1.1.3.1.1, A1.1.3.1.2, A1.1.3.1.3, A1.1.3.2.1, A1.1.3.2.2, A2.1.3.1.1, A2.1.3.1.2, A2.1.3.1.3, A2.1.3.1.4 How to explore relationships between points on a straight line, leading to an understanding of and ability to use slope and the y-intercept to graph linear equations. Explore relationships between points on a straight line, leading to an understanding of and ability to use slope and the y-intercept to graph linear equations. Develop the problem solving skills of looking for patterns, making tables, and systematic lists of data, and drawing diagrams to create equations/inequalities. Find the length and midpoint of a line segment. How to graph solutions to linear inequalities, equations, and system of equations. How to distinguish between linear, quadratic, and non-linear functions algebraically and graphically. How to solve and graph single variable inequalities. Use problem solving skills of looking for patterns, making tables, and systematic lists of data, and drawing diagrams to create equations/inequalities. Graph solutions to linear inequalities, equations, and system of equations. Distinguish between linear, quadratic, and non-linear functions algebraically and graphically. How to solve factored quadratic equations using Demonstrate how to solve and graph single variable inequalities. to Zero Product Property. How to extend graphing to non-linear relationships; parabolas. Solve factored quadratic equations using to Zero Product Property. Extend graphing to non-linear relationships; parabolas. Title: Number and Quantity Standards Content (What students will know) Performance (Student will do) CC.2.1.HS.F.1 How to apply and extend the properties of exponents to solve problems with rational exponents. Apply and extend the properties of exponents to solve problems with rational exponents. How to apply properties of rational and irrational numbers to solve real world or mathematical problems. Apply properties of rational and irrational numbers to solve real world or mathematical problems. How to apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. A1.1.1.1.1, A1.1.1.1.2, A1.1.1.3.1, A2.1.2.1.1, A2.1.2.1.2, A2.1.2.1.3, A2.1.2.1.4 CC.2.1.HS.F.2 A1.1.1.1.1, A1.1.1.1.2, A1.1.1.3.1, A1.1.1.2.1 CC.2.1.HS.F.3 A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3, A1.2.1.2.1, A1.2.1.2.2, A2.2.2.1.1, A2.2.2.1.2, A2.2.3.1.1, A2.2.3.1.2 Activities/Assessments CC.2.1.HS.F.4 A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3, A1.2.1.2.1, A1.2.1.2.2, A2.2.2.1.1, A2.2.2.1.2 CC.2.1.HS.F.5 A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3, A1.1.2.2.1, A1.1.2.2.2, A1.1.3.1.1, A1.1.3.1.2, A1.1.3.1.3, A1.1.3.2.1, A1.1.3.2.2, A2.2.3.1.1, A2.2.3.1.2 CC.2.1.HS.F.6 A2.1.1.1.1, A2.1.1.1.2, A2.1.1.2.1, A2.1.1.2.2 How to use units as a way to understand Use units as a way to understand problems problems and to guide the solution of multi‐step and to guide the solution of multi‐step problems. problems. How to choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Demonstrate a how to choose a level of accuracy appropriate to limitations on measurement when reporting quantities. How to extend the knowledge of arithmetic operations and apply to complex numbers. Demonstrate how to extend the knowledge of arithmetic operations and apply to complex numbers. CC.2.1.HS.F.7 A2.2.1.1.1, A2.2.1.1.2, A2.2.1.1.3, A2.2.1.1.4 How to apply concepts of complex numbers in polynomial identities and quadratic equations to solve problems. Apply concepts of complex numbers in polynomial identities and quadratic equations to solve problems.