Oley Valley School District Oley Valley High School Curriculum: Geometry

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1 Oley Valley School District Oley Valley High School Curriculum: Geometry Name of Unit Priority Standards Supporting Standards Essential Questions - Dominant Focus (topics) Introduction to Geometry Define basic geometric terms Label basic geometric objects Use inductive reasoning to define number patterns. Measure, find the midpoint of, and compare segments using the Distance and Midpoint Formulas. Measure and compare angles using number operations. Be able to define and recognize special angle CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. CC.2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real world or mathematical CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. CC.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC.2.2.HS.D.10 Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and geometrically. Why do we need a specific method to label and name points, lines, planes, figures, and rays? How can we measure segments graphed on a number line or coordinate plane? How can we use number operations to find the measure of segments? How can we use number operations

2 pairs. Construct if-then statements Be able to test the truth value of if-then statements. Use the Laws of Syllogism and Detachment to make valid conclusions Use a biconditional to create good definitions. Solve algebraic equations and use algebraic properties to justify solution steps. Prove and apply theorems about angles. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. to find the measure of angles? How can we use the properties of special angle pairs to find angle measurements? How can we use ifthen statements to make valid conclusions? What makes a definition a good definition? Why are the Laws of Syllogism and Detachment methods of deductive reasoning? What is the difference between a postulate and a theorem? How can we use

3 Parallel & Perpendicular Lines Define special angle pairs created when a line intersects two or more lines. Discover properties of special angle pairs formed when a line intersects two parallel lines. Prove lines are parallel using angles Complete two-column proofs Determine the sum of all interior angles of an n-sided polygon Determine the sum of all exterior angles of an n-sided polygon Differentiate between convex and concave polygons. Define what makes a CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.C.2 Graph and analyze functions and use their properties to make connections between the different representations. CC.2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables. CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. CC.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. algebraic properties to solve problems and justify conclusions in geometry? What are the special angle pairs created when a line intersects two or more lines? How can you use the properties of special angle pairs formed when a line intersects two or more lines to find missing angle measures? How can you use and compare the measures of angles to prove lines are parallel or to prove lines are not parallel? How do you determine the sum of the interior angles of an n-sided polygon?

4 polygon a regular polygon Compare slopes to determine if two lines are parallel or perpendicular Write equations of parallel and perpendicular liens How do you determine the sum of the exterior angles of an n-sided polygon? If you know the sum of all the interior angles of an n-sided polygon can you determine the number of sides of the polygon? How can you use slopes of lines to determine if the lines are parallel, perpendicular, or neither. Properties of Triangles Recognize and apply properties of congruent figures Prove triangles congruent using SSS, CC.2.3.HS.A.5 Create justifications based on transformations to establish similarity of plane figures. CC.2.2.HS.D.10 Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and geometrically. CC.2.2.HS.D.8 Apply inverse What does it mean if two shapes are congruent? What short cuts can you use to prove

5 SAS, ASA, AAS using two-column proofs. Define and prove the properties of isosceles and equilateral triangles. Use congruent parts of triangles to prove other triangles congruent. Discover properties of midsegments of triangles. Discover properties of perpendicular and angle bisectors in a triangle. Discover Properties of Points of Concurrency Discover/prove the Triangle Inequality Theorem Define and apply properties of similar polygons. Prove triangles similar using AA, SSS, and SAS Similarity in right triangles Apply the Side Splitter CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures. CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.14 concepts to model and solve real world CC.2.3.HS.A.2 Apply rigid transformations to determine and explain congruence. operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. CC.2.2.HS.D.10 Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. triangles congruent? Once you prove triangles congruent what other conclusions can you make? What is the difference between isosceles and equilateral triangles? What other method can you use to prove right triangles congruent? How do you use coordinate geometry to prove relationships within triangles? How do you determine if three segments form a triangle? What makes polygons similar? How can you use

6 Theorem and Triangle- Angle Bisector Theorem to find missing measurements Apply the Pythagorean Theorem to find missing triangle segment measures. Apply the Converse of the Pythagorean Theorem to classify a triangle properties of similar polygons to find missing measurements? How do you proof two triangles are similar? Why does the altitude of a right triangle create three similar triangles? How can the Pythagorean Theorem be used to solve realworld problems? How do we use the Converse of the Pythagorean theorem to define a triangle as right, acute, or obtuse? Trigonometry Discover the properties of special right triangles. CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right CC.2.4.HS.B.2 Summarize, represent, and interpret data on How are the sides of and triangles related? How can you use the ratios of special

7 Use ratio of sides of special right triangles to introduce trigonometry. Apply sine, cosine, and tangent ratios to find missing angles and sides in right triangles. triangles. CC.2.3.HS.A.14 concepts to model and solve real world two categorical and quantitative variables. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. right triangles to find missing sides? How can you use trigonometric ratios to find a missing side length or angle measurement? How can trigonometric ratios be used to solve real world problems? How is using the Pythagorean Theorem and Trig Functions to find a missing side different/the same? Polygons & Quadrilaterals Classify polygons Find the sum of all interior angles of any polygon. Find the measure of one regular angle of a regular polygon. Classify Quadrilaterals Prove a quadrilateral is a parallelogram Complete Coordinate Proofs CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.14 concepts to model and solve real world CC.2.3.HS.A.11 Apply coordinate geometry to prove CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. What is the relationship between trig functions in similar triangles? How do you classify polygons? What are the properties of special quadrilaterals? How can you use the properties of special quadrilaterals to define the most accurate name of a quadrilateral? How can you use the Distance Formula and

8 simple geometric theorems algebraically. slope of lines to complete a coordinate proof? Area, Surface Area, and Volume Use formulas to calculate the area of polygons. Use formulas to calculate the surface area of threedimensional shapes. Use formulas to calculate the volume of three- dimensional figures. Application problems with area, surface area, and volume. CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.14 concepts to model and solve real world problems CC.2.3.HS.A.13 Analyze relationships between twodimensional and three dimensional objects. CC.2.3.HS.A.12 Explain volume formulas and use them to solve CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. G Use and/or develop procedures to determine or describe measures of perimeter, circumference, and/or area. G Describe how a change in one dimension of a twodimensional figure affects other measurements of that figure. G Use and/or develop procedures to determine or describe measures of surface area and/or volume. How do area, surface area, and volume of similar figures compare? When you change one dimension of a threedimensional figure how does that change the surface area or volume? How can use trigonometry to find the area of a figure? What makes finding the area of a regular polygon unique? Can you use different formulas to find the area of different polygons?

9 Properties of Circles Calculate arc length Calculate the area of a sector and a segment. Relate arcs and angles Explore properties of angles formed by lines inside and outside of circles. Use properties of tangent lines Solve problems with angles formed by secants and tangents. Discover the equation of a circle graphed in the coordinate plane. CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.8 theorems to verify properties of circles. CC.2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles. CC2.3.HS.A.10 Translate between the geometric description and the equation of a line. CC.2.3.HS.A.11 Apply coordinate CC.2.2.HS.D.10 Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and geometrically. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. How are arc length and area of a sector related? How can use the measure of an arc to find the measure of a central angle or inscribed angle? How can you use the measure of a central angle or inscribed angle to find the measure of length of an arc? How do you find the equation of a circle graphed in the coordinate plane? How can you use

10 geometry to prove simple geometric theorems algebraically. CC.2.3.HS.A.14 concepts to model and solve real world the properties of tangent lines and secant lines to find missing segment and angle measures? How are the definition of a circle, the definition of a locus, and the equation of a circle all related? How are the equation of a circle and the distance equation related>

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