Course Name Keystone Geometry (1.03 / 1.06 / 1.10) Grade Level Grade 10 Northern York County School District Curriculum Module Instructional Procedures Module 1: Geometric Properties and Reasoning Course Manual Contains Details Graphing Calculator Activities Computer Software Simulations/Activities Unit 1 Properties of Circles, Spheres, and Cylinders Time Frame 4-5 Weeks Key Concepts Essential Questions PA Common Core Content Standard Eligible Content Terminology G.1.1.1 Identify and/or use parts of circles and segments associated with circles, spheres, and cylinders. How do you find the measure of an arc of a circle? How can you tell if two chords in a circle are congruent? How do you find the measure of an inscribed angle? How do you find the measure of an angle formed by two chords that intersect inside a circle? What are some properties of chords, secants, and tangents to a circle? CC.2.3.HS.A.8 Apply geometric 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. CC.2.3.HS.A.13 Analyze relationships between twodimensional and three-dimensional objects. G.1.1.1.1 Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle. G.1.1.1.2 Identify, determine, and/or use the arcs, semicircles, sectors, and/or of a circle. G.1.1.1.3 Use chords, tangents, and secants to find missing arc measures or missing segment measures. G.1.1.1.4 Identify and/or use the properties of a sphere or cylinder. Circle Center Radius Diameter Chord Secant Tangent Central angle Minor arc Major arc Semicircle Congruent circles Congruent arcs Inscribed angle Intercepted arc Standard equation of a circle What do you need to know to write the standard equation of a circle? Geometry Northern York County School District Page 1 of 14
How do you find the length of an arc of a circle? How do you find the radian measure of an angle? How do you find the area of a sector? What are the components of a sphere? What are the components of a cylinder? Geometry Northern York County School District Page 2 of 14
Unit 2 Properties of Polygons and Polyhedra Timeframe 5-6 Weeks Key Concepts Essential Questions PA Common Core Content Standard Eligible Content Terminology G.1.2.1 Recognize and/or apply properties of, polygons, and polyhedra. How do you classify polygons? How can you find the measure of the third angle of a triangle if you know the measures of the other two? How do you find the possible lengths of the thirds side of a triangle if you know the lengths of the other two sides? How do you find the points of concurrency of a triangle? How do you find a missing angle measure in a convex polygon? How do you find missing angle and side measures in a parallelogram? How can you prove that a quadrilateral is a parallelogram? What are the properties of parallelograms? What are the properties of trapezoids and kites? How can you identify special quadrilaterals? When is a solid a polyhedron? CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.13 Analyze relationships between twodimensional and three-dimensional objects. G.1.2.1.1 Identify and/or use properties of tri. G.1.2.1.2 Identify and/or use properties of quadrilaterals. G.1.2.1.3 Identify and/or use properties of isosceles and equilateral tri. G.1.2.1.4 Identify and/or use properties of regular polygons. G.1.2.1.5 Identify and/or use properties of pyramids and prisms. Triangle Scalene Isosceles Equilateral Acute Right Obtuse Equiangular Interior Exterior Corollary Congruent figures Corresponding parts Right triangle Legs Hypotenuse Vertex angle Base Base Transformation Midsegment Coordinate proof Perpendicular bisector Equidistant Point of concurrency Circumcenter Incenter Orthocenter Centroid Median of a triangle Altitude of a triangle Geometry Northern York County School District Page 3 of 14
G.1.2.1 Recognize and/or apply properties of, polygons, and polyhedra. What are the components of a pyramid? What are the components of a prism? How do you write a coordinate proof? CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.13 Analyze relationships between twodimensional and three-dimensional objects. G.1.2.1.1 Identify and/or use properties of tri. G.1.2.1.2 Identify and/or use properties of quadrilaterals. G.1.2.1.3 Identify and/or use properties of isosceles and equilateral tri. G.1.2.1.4 Identify and/or use properties of regular polygons. G.1.2.1.5 Identify and/or use properties of pyramids and prisms. Indirect proof Diagonal Parallelogram Rhombus Rectangle Square Trapezoid Isosceles trapezoid Kite Midsegment of a trapezoid Polyhedron volume surface area Geometry Northern York County School District Page 4 of 14
Unit 3 Congruence, Similarity, and Proofs Timeframe 5-6 Weeks Key Concepts Essential Questions PA Common Core Content Standard Eligible Content Terminology G.1.3.1 Use properties of congruence, correspondence, and similarity in problemsolving settings involving 2- and 3- dimensional figures. What are congruent figures? How do you identify a rigid motion in the plane? How can you use side lengths to prove tri congruent? How can you use two sides and an angle to prove two tri congruent? If a side of one triangle is congruent to a side of another triangle, what information about the would allow you to prove the tri congruent? How can you use congruent tri to prove or sides congruent? How are the sides and of a triangle related if there are two or more congruent sides or? What transformations create an image congruent to the original figure? If two figures are similar how do you find the length of a missing side? How do you identify a similarity transformation in a plane? CC.2.3.HS.A.2 Apply rigid transformations to determine and explain congruence. CC.2.3.HS.A.5 Create justifications based on transformations to establish similarity of plane figures. CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures. G.1.3.1.1 Identify and/or use properties of congruent and similar polygons or solids. G.1.3.1.2 Identify and/or use proportional relationships in similar figures. Circumference Arc length Sector of a circle Center of a polygon Radius of a polygon Apothem Central angle of a regular polygon Probability Polyhedron Face Edge Vertex Platonic solids Cross section Volume Sphere Hemisphere Great circle Similar solids Image Preimage Isometry Vector Component form Dimensions Line of reflection Center of rotation Angle of rotation Glide reflection Composition of transformations Line symmetry Rotational symmetry Geometry Northern York County School District Page 5 of 14
How can you show two tri are similar using the similarity theorems? How do you dilate a figure in the coordinate plane? How do you find the point that partitions a directed line segment in a given ratio? How do you translate a figure using a vector? How do you use matrix operations to translate a figure? How do you reflect a figure on the coordinate plane? How do you rotate a figure about the origin? What is a glide reflection? What regular polygons can tessellate a plane? When does a figure have line symmetry? How do you use matrices to draw a dilation? If two solids are similar what is the ratio of their surface areas and what is the ratio of their volumes? Geometry Northern York County School District Page 6 of 14
G.1.3.2 Write formal proofs and/or use logic statements to construct or validate arguments. How do you use inductive reasoning in mathematics? How do you rewrite a biconditional statement? How do you construct a logical argument? How do you represent a logical statement symbolically? How do you write a two-column proof in geometry? How do you write an indirect proof/proof by contradiction? CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures. CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles. 2.2.HS.C.9 Prove the Pythagorean identity and use it to calculate trigonometric ratios. G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction). Conjecture Inductive reasoning Counter example Conditional statement Converse Inverse Contrapositive If-then form Hypothesis Conclusion Negation Equivalent statements Biconditional statement Deductive reasoning Proof Two-column proof Theorem Geometry Northern York County School District Page 7 of 14
Unit 4 Coordinate Geometry and Right Tri Timeframe 4-5 Weeks Key Concepts Essential Questions PA Common Core Content Standard Eligible Content Terminology G.2.1.2 Solve problems involving right tri. If you know the lengths of two sides of a right triangle how do you find the length of the third side? How do you find the length of the altitude to the hypotenuse of a right triangle? How do you find the lengths of the sides of 45-45-90 and 30-60-90 tri? How can you find a leg of a right triangle when you know the other leg and one acute angle? CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right tri. 2.2.HS.C.9 Prove the Pythagorean identity and use it to calculate trigonometric ratios. G.2.1.1.1 Use the Pythagorean theorem to write and/or solve problems involving right tri. G.2.1.1.2 Use trigonometric ratios to write and/or solve problems involving right tri. Conjecture Inductive reasoning Counter example Conditional statement Converse Inverse Contrapositive If-then form Hypothesis Conclusion Negation Equivalent statements Biconditional statement Deductive reasoning Proof Two-column proof Theorem G.2.1.2 Solve problems using analytic geometry. How do you find the distance and the midpoint between two points in the coordinate plane? How do you determine if two lines are parallel/perpendicular using their slopes? CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically. G.2.1.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction). G.2.1.2.2 Calculate the distance and/or midpoint between two points on a number line or on a coordinate plane. G.2.1.2.3 Relate slope to perpendicularity and/or parallelism (limit to linear algebraic equations). Undefined terms Point Line Plane Defined terms Line segment End points Ray Opposite rays Postulate Axiom Congruent segment Midpoint Segment bisector Acute Geometry Northern York County School District Page 8 of 14
Right Obtuse Straight angle Congruent Angle bisector Linear pair Vertical Polygon Convex Concave N-gon Equilateral Equiangular Regular Slope Parallel Perpendicular Distance G.2.1.3 Compute and/or use the slope of a line. How do you find the slope of a line given two points on the line? How do you use slope to determine whether lines are parallel, perpendicular or neither? CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically. G.2.1.3.1 Apply the concept of the slope of a line to solve problems. Parallel lines Skew lines Parallel planes Transversal Corresponding Alternate interior Alternate exterior Consecutive interior Slope Slope-intercept form Standard form of a line Distance from a point to a line Geometry Northern York County School District Page 9 of 14
G.2.1.4 Solve and/or graph systems of equations and inequalities using coordinate geometry. How do you write an equation of a line? How do you use systems of equations to solve real-world problems? 2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. 2.2.HS.D.10 Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. G.2.1.4.1 Solve or graph systems of equations or systems of inequalities within a problem situation using coordinate geometry. Parallel lines Skew lines Parallel planes Transversal Corresponding Alternate interior Alternate exterior Consecutive interior Slope Slope-intercept form Standard form of a line Distance from a point to a line Geometry Northern York County School District Page 10 of 14
Course Name Keystone Geometry (1.03 / 1.06 / 1.10) Grade Level Grade 10 Northern York County School District Curriculum Module Instructional Procedures Module 2: Coordinate Geometry and Measurement Course Manual Contains Details Graphing Calculator Activities Computer Software Simulations/Activities Unit 5 Time Frame Measurements of Two-Dimensional Shapes and Figures 6-7 Weeks Key Concepts Essential Questions PA Common Core Content Standard Eligible Content Terminology G.2.2.1 Use and/or compare measurements of. What is the relationship between vertical, between two that are supplementary to the same angle, and between two that are complementary to the same? What angle pairs are formed by transversals? How are corresponding, consecutive interior, and alternate interior related for two parallel lines and a transversal? CC 2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. G.2.2.1.1 Use properties of formed by intersecting lines to find the measures of missing. G.2.2.1.2 Use properties of formed when two parallel lines are cut by a transversal to find the measures of missing. Parallel lines Skew lines Parallel planes Transversal Corresponding Alternate interior Alternate exterior Consecutive interior Slope Slope-intercept form Standard form of a line Distance from a point to a line Geometry Northern York County School District Page 11 of 14
G.2.2.2 Use and/or develop procedures to determine or describe measures of perimeter, circumference, and/or area. (May require conversions within the same system.) How do you find the area and perimeter of a regular polygon? How do you find the area and perimeter of irregular figures? How can you use perimeter and area to find missing lengths in a figure? How do you determine the area of a sector of a circle? How do you use similarity to determine the area and arc lengths of sectors of two circles? CC 2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC 2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles. CC 2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context. G.2.2.2.1 Estimate area, perimeter, or circumference of an irregular figure. G.2.2.2.2 Find the measurement of a missing length, given the perimeter, circumference, or area. G.2.2.2.3 Find the side lengths of a polygon with a given perimeter to maximize the area of the polygon. G.2.2.2.4 Develop and/or use strategies to estimate the area of a compound/composite figure. G.2.2.2.5 Find the area of a sector of a circle. Dilations Scale factor of a dilation Similar polygons Scale factor of two similar polygons Center of dilation Reduction Enlargement Sector of a circle Circumference Arc length How do you find the side lengths of a polygon with a given perimeter to maximize the area of the polygon? G.2.2.3 Describe how a change in one dimension of a 2 dimensional figure affects other measurements of that figure. How does a change in one dimension of a two-dimensional figure affect other measurements of that figure? CC 2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles. CC 2.3.HS.A.8 Apply geometric theorems to verify properties of circles. G.2.2.3.1 Describe how a change in the linear dimension of a figure affects its perimeter, circumference, and area (e.g., How does changing the length of the radius of a circle affect the circumference of the circle?). Dilations Scale factor of a dilation Similar polygons Scale factor of two similar polygons Center of dilation Reduction Enlargement G.2.2.4 Apply probability to practical situations. How do you find the probability that a point randomly selected in a region is in a particular part of that region? CC 2.3.HS.A.14 Apply geometric concepts to model and solve real world problems. G.2.2.4.1 Use area models to find probabilities. Probability Geometric probability Geometry Northern York County School District Page 12 of 14
Unit 6 Measurements of Three-Dimensional Shapes and Figures Timeframe 6-7 Weeks Key Concepts Essential Questions PA Common Core Content Standard Eligible Content Terminology G.2.3.1 Use and/or develop procedures to determine or describe measures of surface area and/or volume. (May require conversions within the same system.) How do you find the volume of a right prism or a right cylinder? How do you find the volume/surface area of a pyramid or cone? How do you find the volume/surface area of a sphere? How do you apply geometric properties of 3-dimensional figures in real-world problems? CC 2.3.HS.A.14 Apply geometric concepts to model and solve real world problems. CC 2.3.HS.A.12 Explain volume formulas and use them to solve problems. G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet. G.2.3.1.2 Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet. G.2.3.1.3 Find the measurement of a missing length, given the surface area or volume. Circumference Arc length Sector of a circle Center of a polygon Radius of a polygon Apothem Central angle of a regular polygon Probability Geometric probability Polyhedron Face Edge Vertex Platonic solids Cross section Volume Sphere Great circle Hemisphere Similar solids G.2.3.2 Describe how a change in one dimension of a 3- dimensional figure affects other measurements of that figure. How does a change in one dimension of a three-dimensional figure affect other measurements of that figure? CC 2.3.HS.A.13 Analyze relationships between twodimensional and three-dimensional objects. G.2.3.2.1 Describe how a change in the linear dimension of a figure affects its surface area or volume (e.g., How does changing the length of the edge of a cube affect the volume of the cube?). Circumference Arc length Sector of a circle Center of a polygon Radius of a polygon Apothem Central angle of a regular polygon Probability Geometric probability Polyhedron Geometry Northern York County School District Page 13 of 14
Face Edge Vertex Platonic solids Cross section Volume Sphere Great circle Hemisphere Similar solids The Standard (1.03) level and the Academic (1.06) level Geometry courses will be differentiated through assessment expectations, where the Academic level requires a greater volume of questions and a faster pace. The Honors (1.10) level Geometry will be differentiated to include additional topics that are high-lighted in yellow as well as additional challenges through assessment. Geometry Northern York County School District Page 14 of 14