Laurie E. Bass and Art Johnson; Geometry Common Core ; Pearson Education, Inc.

Transcription

1 Course Title Damien High School Mathematics & Computer Science Department Curriculum Map Prerequisites Achieve a Qualifying Score on the Algebra 1 Challenge Exam or earn an A in Algebra 1 CSU/UC Approval Length of Course Yes Category c Year Brief Course Description This course covers the foundations of geometrical figures and their measurement. Beginning with the component part of geometrical figures points, lines, and planes and through the use of reasoning and proof, the course encompasses the study of triangles, quadrilaterals, other polygons, circles, and solids. Through the study of definitions, postulates, and theorems, in addition to other related mathematical topics, the properties of these figures are incorporated into an understanding and ability to construct and measure both plane figures and solids. Major topics in the course include deductive and inductive reasoning, triangle relationships and congruence, right triangle trigonometry, similarity, areas of plane figures, and surface areas and volumes of solids. Assigned Textbook(s) Supplemental Material(s) Laurie E. Bass and Art Johnson; Geometry Common Core ; Pearson Education, Inc. MathXL (Pearson) Common Assessments Utilized Common Final each semester ISOs Addressed Be academically prepared for a higher education Exhibit community and global awareness Elements/Themes of Catholic Identity Introduce Cavalieri, a Jesuat, and one of his famous works: Cavalieri s Principle (Theorem 11-5). Participate in Respect for Life activities.

2 Overview of Course / Skill Outcomes This section serves as a precursor to the Curriculum Map and, as such, should briefly describe the various units (major content chunks) that comprise the course as well as the skills / techniques necessary to be successful in the course. Major Content Outcomes I. Prove basic theorems, and prove triangle congruence and similarity. A Making a conjecture 1. Observe patterns leading to making conjectures. B. Prove geometric relationships using given information, definitions, properties, postulates, and theorems. 1. Proving angles are congruent. 2. Proving two lines are parallel. a. Using congruent alternate interior angles. b. Using congruent corresponding angles. c. Using supplementary same-side interior angles. 3. Proving two triangles are congruent. a. Using SSS, SAS, ASA, AAS b. Using HL for right triangles. 4. Proving two triangles are similar. a. Using AA similarity, SAS similarity, SSS similarity II. Measure in the coordinate plane. A. Segments will be measured with and without a coordinate grid. B. Use Midpoint and Distance Formulas. x 1 + x 2 y + y ( , 1 2) 2. d = (x 2 x 1 ) 2 + (y 2 y 2 1 ) C. Compare slopes of parallel and perpendicular lines. 1. Parallel lines have the same slope. 2. The product of the slopes of perpendicular lines is -1. III. Find measures of areas, surface areas, and volumes of plane and space figures. A. Areas of parallelograms, triangles, trapezoids, rhombuses, kites, and regular polygons. B. Areas of circles and sectors. C. Lateral areas of prisms, cylinders, pyramids, and cones. D. Surface areas of prisms and cylinders. 1. Lateral area + 2 Base areas. E. Surface areas of pyramids and cones. 1. Lateral area + 1 Base area. F. Volumes of prisms and cylinders. 1. Base area height G. Volumes of pyramids and cones Base area height 3 H. Surface areas and volumes of a sphere. 1. S.A. = 4 π r 2 2. V = 4 π r 3 3 Major Skill Outcomes By the end of the year, students should be able to: I. Prove basic theorems, and prove triangle congruence and similarity. II. Measure in the coordinate plane. III. Find measures of areas, surface areas, and volumes of plane and space figures. IV. Construct basic geometrical figures with a straightedge and compass. V. Apply geometric knowledge in solving selected real-world problems modeled with geometric figures, particularly in areas related to trigonometry and the Pythagorean Theorem. VI. Know how to use properties of geometric figures to determine equations and solve for unknown dimensions of the figures. Unit 1 Students should be able to describe each pattern and find the next terms in each sequence. Students should be able to name collinear and coplanar points. Students should be able to find the intersection of a line and a plane. Students should be able to identify opposite rays. Students should be able to find the length of segments using Algebra. Students should be able to find the length of segments based on distance from the midpoint. Students should be able to find the area of a square, rectangle, and circle. Students should be able to identify complementary angles, supplementary angles, and perpendicular bisectors of a given figure. Students should be able to correctly name lines, segments, rays, and angles using correct notation. Unit 2 Students should be able to identify the hypothesis and conclusion of a statement. Students should be able to use a statement to write a conditional. Students should be able to find a counterexample to show a statement is not true. Students should be able to write the converse of a conditional. Students should be able to write the inverse and contrapositive of a statement. Students should be able to explain why a statement is not a good definition. Students should be able to identify certain Geometric Properties Students should be able to find the measure of each angle in a given figure. Students should be able to rewrite a biconditionals as two conditionals and vice versa. Students should be able to use the Law of Detachment and the Law of Syllogism to draw conclusions from statements. Unit 3 Students should be able to find the measures of angles based on parallel lines and a transversal. Students should be able to find the values for a variable for which two lines must be parallel. Students should be able to write the equation of a line given a slope and a point the lines passes through. Students should be able to write the equation of a line given two points the line passes through. Students should be able to find the measures of interior/exterior angles in various polygons.

3 IV. Construct basic geometrical figures with a straightedge and a compass. A. Construct congruent segments and congruent angles. B. Construct the perpendicular bisector and the angle bisector. C. Construct parallel and perpendicular lines. V. Apply geometric knowledge in solving selected real-world problems modeled with geometric figures, particularly in areas related to trigonometry and the Pythagorean Theorem. A. Use the Pythagorean Theorem - a 2 + b 2 = c 2 B. Special right triangles. 1. Ratio of the sides of a triangle. a. hypotenuse = 2 shorter leg b. longer leg = 3 shorter leg 2. Ratio of the sides of a triangle. a. hypotenuse = 2 leg 3. Solving real-world problems. a. Using SOHCAHTOA b. Using angle of elevation and angle of depression. C. Solve right triangles (find missing sides and angles) 1. Using Law of Sines 2. Using Law of Cosines VI. Know how to use properties of geometric figures to determine equations and solve for unknown dimensions of the figures. A. Properties of parallelograms, rhombuses, rectangles, and squares. C. Prove that a quadrilateral is a parallelogram. C. Conditions for rhombuses, rectangles, and squares. D. Properties of tangent lines. E. Relationships among chords, arcs, and central angles. F. Solve problems with angles formed by secants and tangents. G. Using the center and radius of a circle to find the equation of a circle. Students should be able to tell whether two lines are parallel or perpendicular based on their slopes. Unit 4 Students should be able to identify triangle congruency based on various given congruent angles and sides. Students should be able to identify which postulate can be used to prove triangle congruency based on the given information. Students should be able to find the values of sides and angles based on triangle congruency. Students should be able to identify triangle congruency in overlapping triangles. Unit 5 Students should be able to identify statements that contradict one another. Students should be able to list angles in order of size based on the length of opposite sides. Students should be able to list sides in order of size based on the length of opposite angles. Students should be able to find the measure of sides and angles using algebra. Students should be able to find the center of a circle that can be circumscribed about a triangle. Unit 6 Students should be able to classify triangles by their sides and angles. Students should be able to classify quadrilaterals in as many ways as possible. Students should be able to find the values of variables based on the properties of various quadrilaterals. Students should be able to find the measures of angles and sides in parallelograms. Students should be able to prove a quadrilateral is a parallelogram based on known properties. Students should be able to find the measure of sides and angles in a rhombus and rectangle. Students should be able to find the measure of sides and angles in a kite and isosceles trapezoid. Students should be able to place quadrilaterals in the coordinate plane and identify coordinates of each vertice. Unit 7 Students should be able to write ratios and solve various proportions. Students should be able to identify similar polygons and give the similarity ratio. Students should be able to find the values of variables in similar polygons. Students should be able to prove triangles are similar and write similarity statements. Students should be able to explain why triangles are similar by using algebra to solve. Students should be able to find the geometric mean in a pair of numbers. Students should be able to find the values of variables in right triangles. Students should be able to use proportions in triangles to solve for variables. Unit 8 Students should be able to find the lengths of sides of a right triangle using the Pythagorean Theorem. Students should be able to decides whether a set of numbers form a Pythagorean triple. Students should be able to determine whether a triangle is a right triangle based on given values. Students should be able to find the values of variables in and right triangles. Students should be able to write tangent ratios. Students should be able to find the values of variables based on the tangent ratio. Students should be able to write sine and cosine ratios. Students should be able to find the values of variables based on sine and cosine ratios. Students should be able to identify and find the angles of elevation and depression.

4 Unit 9 Students should be able to state whether a transformation image appears to be an isometry. Students should be able to find the image of a figure under a given translation. Students should be able to find the coordinates of reflection images in the coordinate plane. Students should be able to draw reflection images across a line of reflection. Students should be able to draw an image based on a given rotation. Students should be able to tell what type symmetry can be found in a given figure. Students should be able to draw the lines of symmetry in a given figure. Students should be able to describe the dilation image of a figure. Students should be able to find the image of points in the coordinate plane for a given scale factor. Students should be able to classify Isometries. Students should be able to find the glide reflection image of a given figure in the coordinate plane. Students should be able to identify whether a figure shows a tessellation of repeating figures. Students should be able to determine whether a figure will tessellate a plane. Students should be able to list the symmetries in each tessellation. Unit 10 Students should be able to find the area of a parallelogram. Students should be able to find the area of a triangle. Students should be able to find the area of a trapezoid. Students should be able to find the area of a kite. Students should be able to find the area of a rhombus. Students should be able to find the area of a regular polygon. Students should be able to find the measure of various angles in polygons based on given radii and apothem. Students should be able to find the perimeters and areas based on ratios of similar figures. Students should be able to find the areas of regular polygons using trigonometry. Students should be able to find the circumference of a circle and the measures of arcs in circles. Students should be able to find the area of an entire circle, a sector of a circle, and a shaded section of a circle. Students should be able to find geometric probability in various figures. Unit 11 Students should be able to find the number of vertices, edges, and faces in a polyhedron. Students should be able to use Euler s Formula to find the number of faces, edges, or vertices in a polyhedron. Students should be able to describe the cross section of a 3-D figure. Students should be able to find the surface area of a prism using nets. Students should be able to find the surface area of a cylinder. Students should be able to find the surface area and lateral area of a pyramid and cone. Students should be able to find the volume of a prism, cylinder, and composite space figure. Students should be able to find the volume of a square pyramid and cone. Students should be able to find the surface area of a sphere from a given diameter. Students should be able to find the volume of a sphere and surface area based on a given volume. Students should be able to identify similarity in 3-D figures and give the similarity ratio. Students should be able to use the similarity ratio to find volumes of similar figures.

5 Unit 12 Students should be able to find the values of variables based on tangent lines and the center of a circle. Students should be able to determine whether a line on a circle is a tangent line. Students should be able to find the values of variables based on given chords and arcs of circles. Students should be able to identify inscribed angles and their intercepted arcs. Students should be able to find the values of variables of inscribed angles within circles. Students should be able to find the values of variables based on given angle measures and segment lengths in circles. Students should be able to write the standard equation of a circle with a given center. Students should be able to find the center and radius of a circle, then graph the circle in the coordinate plane. Students should be able to draw and describe a locus in a plane. Students should be able to draw and describe a locus in the coordinate plane.

6 Unit 1 What are the basic tools of Geometry? Students will be able to represent a 3-D object with a 2-D figure using special drawing techniques such as nets, isometric drawings, and orthographic drawings. How are isometric, orthographic, and foundation drawings drawn and interpreted? How are nets drawn and interpreted? How can a 3-D figure be represented with a 2-D drawing? Definitions of isometric and orthographic drawings Definition of nets Drawing isometric, orthographic, and nets Students will be able to show visual representations of points, lines, and planes. What are points, lines, and planes? What are postulates and how are they understood? Definitions of points, lines, and planes Using postulates CCSS: G-CO.A.1 Students will be able to show visual representations of segments and rays. How are segments and rays identified? How is a segment different from a line? Definitions of segments and rays CCSS: G-CO.A.1 Students will be able to measure segments. How are the lengths of segments determined? Finding the length of segments CCSS: G-CO.A.1, G-GPE.B.6 Students will be able to find and compare lengths of segments. What are congruent segments? What is a midpoint of a segment? Definitions of midpoints and segment bisectors What is a segment bisector? Students will be able to use protractors to measure angles. Students will be able to find and compare the measures of angles. How are the measures of angles determined? How are angles classified? Definition of angles Measuring angles Classifying angles according to their measures. CCSS: G-CO.A.1 Students will be able to identify special angle pairs and use their relationship to find angle measures. How are special angle pairs identified? What is an angle bisector? Definitions of vertical, complementary, and supplementary angles Definition of angle bisectors Students will be able to make basic constructions using a straightedge and a compass. How is using a compass and a straightedge more accurate than sketching and drawing? How is a compass and a straightedge used to construct a figure without measuring? Definition of constructions Definitions of perpendicular lines and perpendicular bisectors CCSS: G-CO.A.1, G-CO.D.12 Students will be able to use the midpoint and distance formulas. What is the coordinate plane? How is the coordinate plane used to find the distance between two points? Definition of the coordinate plane and its uses (distance/midpoint) How is the coordinate plane used to find the midpoint of a segment?

7 Students will be able to find perimeter, circumference, and area of basic shapes. How are perimeters and areas different in measuring the size of geometric figures? Formulas for perimeter/area of squares and rectangles CCSS: N-Q.A.1 How are the perimeters of rectangles and squares calculated? Formulas for circumference/area of circles How is the circumference of a circle calculated? How are the areas of squares, rectangles, and circles calculated? Unit 2 How are the concepts of reasoning and proofs used in Geometry? Students will be able to use inductive reasoning to make conjectures. Students will be able to use patterns in number sequences and sequences of geometric figures to discover relationships. Students will be able to use a counterexample to prove a conjecture is false. How is inductive reasoning used to make conjectures? What are some ways to find and use patterns? What are counterexamples and how are they found? Definitions of inductive reasoning and conjecture Identifying patterns and counterexamples Students will be able to recognize conditional statements and their parts. Students will be able to write converses, inverses, and contrapositives of conditionals. How are conditional statements recognized? How are the converses of conditional statements written? How is the hypothesis and conclusion in a conditional statement identified? How are Venn Diagrams used? How is the truth value of a conditional statement found? Definition of a conditional statement (if-then statement) Identifying the hypothesis and conclusion in a conditional statement Identifying counterexamples to conditional statements Definition/identification of Venn Diagram Identifying truth value of conditional statements How is the negation of a statement written? How are the inverse and contrapositive of a conditional statement written? Definitions of converses, inverses, and contrapositives Application of contrapositive to a statement Does each conditional have a converse, an inverse, and a contrapositive?

8 Students will be able to write biconditionals and recognize good definitions. How are biconditionals written? How are biconditionals written as two conditionals? How are good definitions recognized? How are biconditionals written from definitions? Definition of biconditionals Writing biconditionals into two conditionals that are converses Recognizing good definitions (necessary requirements of good definitions) Students will be able to use deductive reasoning. What is the Law of Detachment and how is it used? Definition of the Law of Detachment Students will be able to use the Law of Detachment and the Law of Syllogism What is the Law of Syllogism and how is it used? If only one of the laws is to be used, how can it be determined which law to use? Definition of the Law of Syllogism Definition of deductive reasoning What is deductive reasoning and how is it used? Students will be able to apply logical reasoning in algebraic and geometric situations How is reasoning in algebra connected to reasoning in geometry? What tools are used to justify steps in solving equations? Properties of Equality in Algebra The Distributive Property Properties of Congruence How is logical reasoning used to build a proof? Connecting algebra to geometry What is the main difference between a Property of Equality and a Property of Congruence? What different types of reasons can be used to build a proof? Students will be able to prove and apply theorems about angles. How are theorems about angles used and applied in Geometry? What is the Vertical Angles theorem? Definition of theorems Vertical Angles Theorem CCSS: G-CO.C.9 How are postulates difference from theorems?

9 Unit 3 How are parallel and perpendicular lines identified and used in Geometry? Students will be able to identify relationships between figures in space. Students will be able to identify angles formed by two lines and a transversal. What is a transversal? How are angles formed by two lines and transversal identified? Definition of a transversal Identifying angle pairs Definitions of alternate interior angles, same-side-interior angles, corresponding angles CCSS: G-CO.A.1 Students will be able to prove theorems about parallel lines Students will to use the properties of parallel lines to find angle measures. What are some properties of parallel lines and how are they used? Same-Side Interior Angles Postulate Alternate Interior/Exterior Angles Theorem Corresponding Angles Theorem CCSS: G.CO.C.9 Students will be able to determine whether two lines are parallel. Students will be able to use the three different forms of proofs. How can certain angle pairs be used to prove two lines are parallel? What are the three forms of proofs? Converse of the Corresponding Angles Theorem Converse of The Alternate Interior Angles Theorem Converse of the Same-Side Interior Angles Postulate CCSS: G-CO.C.9 Paragraph, Two-Column, and Flow Proofs Students will be able to relate parallel and perpendicular lines. What are perpendicular lines? How are parallel and perpendicular lines related? Multiple Parallel/Perpendicular Lines Theorem CCSS:G-MG.A.3 Students will be able to use parallel lines to prove theorems about triangles Students will be able to find measures of angles of triangles. How are the measures of the angles in a triangle found? How are the exterior angles of triangles used? Triangle Angle-Sum Theorem Triangle Exterior Angle Theorem CCSS: G-CO.C.10 Students will be able to construct parallel and perpendicular lines How are parallel lines and perpendicular lines constructed using a straightedge and a compass? Perpendicular Postulate CCSS: G-CO.D.12, G-CO.D.13 Students will be able to graph and write linear equations. What is slope and how is it used to graph lines? Given their equations, how are lines graphed in the coordinate plane? What are slope-intercept form, standard form, and point-slope form of linear equations? Definition of Slope Slope-intercept Form Standard Form of a Linear Equation Point-Slope Form What are the equations for horizontal and vertical lines?

10 Students will be able to determine whether two lines are parallel or perpendicular lines by comparing their slopes. How are slope and parallel/perpendicular lines related? How are the equations of parallel/perpendicular lines written? Slopes of Parallel Lines Slopes of Perpendicular Lines CCSS: G-GPE.B.5 How can it be determined if two lines intersect? Unit 4 What are the different ways to identify triangle congruency? Students will be able to recognize congruent figures and their corresponding parts. What are congruent figures and how are they recognized? How are the corresponding parts of congruent figures identified and named? Definition of Congruent Polygons Triangle 3rd Angle Congruency Theorem Students will be able to prove triangle congruency by SSS and SAS. How are two triangles proved congruent by SSS? How are two triangles proved congruent by SAS? Side-Side-Side (SSS) Postulate Side-Angle-Side (SAS) Postulate CCSS: G-SRT.B.5 What is the minimum number of conditions necessary to prove that two triangles are congruent? Students will be able to prove triangle congruency by ASA and AAS. How are two triangles proved congruent by ASA? How are two triangles proved congruent by AAS? Angle-Side-Angle (ASA) Postulate Angle-Angle-Side (AAS) Theorem CCSS: G-SRT.B.5 Students will be able to use triangle congruence and corresponding parts of congruent triangles (CPCTC) to prove that parts of two triangles are congruent. What is CPCTC and how is it applied to congruent triangles? How is triangle congruence and CPCTC used to prove other parts of two triangles congruent? Corresponding Parts of Congruent Triangles are Congruent (CPCTC) CCSS:G-CO.D.12, G-SRT.B.5 What postulates or theorems can be used to prove two triangles are congruent? Students will be able to use and apply properties of isosceles and equilateral triangles. What are the properties of isosceles triangles and how are they applied? Definitions of isosceles, equilateral, and equiangular triangles CCSS: G-CO.C.10, G-CO.D.13, G-SRT.B.5 What are the properties of equilateral triangles? Definition of a corollary Isosceles Triangle Theorem and its Corollary

11 Converse of Isosceles Triangle Theorem and its Corollary Isosceles Triangle Bisector Theorem Students will be able to prove right triangles congruent using the Hypotenuse-Leg Theorem. What is the HL Theorem? What conditions must the triangles meet to be able to use the HL Theorem? Hypotenuse-Leg (HL) Theorem CCSS: G-SRT.B.5 How is the HL Theorem used to prove triangles congruent? Students will be able to identify congruent overlapping triangles. Students will be able to prove two triangles congruent using other congruent triangles. How are congruent overlapping triangles identified? How are common parts within overlapping triangles identified and used? How can two triangles be proven congruent using another pair of triangles? Identifying overlapping congruent triangles CCSS: G-SRT.B.5 What are ways to separate overlapping triangles? Unit 5 What are the various relationships within a Triangle? Students will be able to use properties of midsegments to solve problems. What is the midsegment of a triangle? How are the properties of midsegments used to solve problems in geometry? In what ways is the midsegment of a triangle related to the third side? Definition of a Midsegment Triangle Midsegment Theorem CCSS: G-CO.C.10, G-CO.D.12, G-SRT.B.5 Students will be able to use properties of perpendicular bisectors and angle bisectors. What are the properties of perpendicular bisectors and how are they applied? What are the properties of angle bisectors and how are they applied? Perpendicular Bisector Theorem and its converse Angle Bisector Theorem and its converse CCSS: G-CO.C.9, G-CO.D.12, G-SRT.B.5 Students will be able to identify properties of perpendicular bisectors and angle bisectors. How are the properties of perpendicular bisectors identified? Definitions of concurrent lines and point of concurrency CCSS: G-C.A.3

12 How are the properties of angle bisectors identified? What are the properties of the circumcenter of a triangle? Concurrency of Perpendicular Bisectors Theorem Definitions of circumcenter and incenter Concurrency of Angle Bisectors Theorem What are the properties of the incenter of a triangle? Students will be able to identify properties of medians and altitudes of a triangle. How are the properties of the medians of a triangle identified? Definitions of a median of a triangle and altitude of a triangle CCSS: G-CO.C.10, G-SRT.B.5 How are the properties of altitudes of triangles identified? Where do the medians of a triangle intersect? Where do the altitudes of a triangle intersect? Definitions of the centroid and orthocenter Concurrency of Medians Theorem Concurrency of Altitudes Theorem Students will be able to use indirect reasoning to write Indirect proofs. How is indirect reasoning used in proofs? What are the steps in writing an Indirect proofs? Definitions of indirect reasoning and indirect proofs CCSS: G-CO.C.10 How does an indirect proof lead to the desired conclusion? Students will be able to use inequalities involving angles and sides of triangles. How are inequalities used when involving angles of triangles? How are inequalities used when involving sides of triangles? What is the Triangle Inequality Theorem and how is it applied to triangles? Comparison Property of Inequality Corollary to the Triangle Exterior Angle Theorem Unequal Triangle Sides Theorem Unequal Triangle Angles Theorem CCSS: G-CO.C.10 What is the relationship between the lengths of the sides and the measures of the angles of a triangle? Triangle Inequality Theorem Students will be able to apply inequalities in two triangles. In triangle that have two pairs of congruent sides, what is the relationship between the included angles and the the third pairs of sides? The Hinge Theorem (SAS Inequality Theorem) Converse of the Hinge Theorem (SSS Inequality) CCSS: G-CO.C.10 Unit 6 What are the various Quadrilaterals and their properties? Students will be able to find the sum of the measures of the interior angles of a polygon. Students will be able to find the sum of the measures of the exterior angles of a polygon What is a polygon and how are they classified? How are the sums of the measures of the interior/exterior angles of a polygon found? Definition of a polygon Polygon Angles-Sum Theorem and its corollary Polygon Exterior Angle-Sum Theorem CCSS: G-SRT.B.5

13 Students will be able to use relationships among sides and angles of parallelograms. Students will be able to use relationships among diagonals of parallelograms. What are the relationships among the sides of parallelograms and how are they used? What are the relationships among the angles of parallelograms and how are they used? Definition of a parallelogram Opposite Sides/Angles of a Parallelogram Theorem Consecutive Angles of a Parallelogram Theorem CCSS: G.CO.C.11, G-SRT.B.5 How are the relationships involving diagonals of parallelograms or transversals used? Diagonals of a Parallelogram Theorem Parallel Lines and Transversals Theorem Students will be able to determine whether a quadrilateral is a parallelogram. How is it determined whether a quadrilateral is a parallelogram? How are the values for parallelograms found? Quadrilateral Opposite Sides/Angles Theorem Quadrilateral Consecutive Angles and Bisecting Diagonals Theorem CCSS: G.CO.C.11, G-SRT.B.5 What biconditional can be written concerning a parallelogram and its opposite angles? Quadrilateral Opposite Sides Congruent and Parallel Theorem Students will be able to define and classify special types of parallelograms. Students will be able to use properties of diagonals of rhombuses and rectangles. What are the properties of diagonals of rhombuses and rectangles and how are they used? Definitions of rhombus, rectangle, square Rhombus Diagonal Bisector and Perpendicular Diagonals Theorems Rectangle Diagonals Theorem CCSS: G.CO.C.11, G-SRT.B.5 Students will be able to determine whether a parallelogram is a rhombus or rectangle. How is it determined whether a parallelogram is a rhombus or a rectangle? Parallelogram Diagonal Bisector and Perpendicular Diagonals Theorems CCSS: G.CO.C.11, G-SRT.B.5 Parallelogram Diagonal Congruency Theorem Students will be able to verify and use properties of trapezoids and kites. What are the properties of trapezoids and kites? What is another way to describe the relationship between the midsegment of a trapezoid and the bases? Definitions trapezoid, isosceles trapezoid, midsegment of a trapezoid, kite Isosceles Trapezoid Base Angles and Diagonals Theorem CCSS: G-SRT.B.5 Are either kites or trapezoids a subset of parallelograms? Trapezoid Midsegment Theorem Kite Perpendicular Diagonals Theorem Students will be able to classify polygons in the coordinate plane. How are the formulas for slope, distance, and midpoint used to classify figures in the coordinate plane? Formulas of slope, distance, midpoint Naming coordinates of special figures CCSS: G-GPE.B.7 How are the formulas for slope, distance, and midpoint used to prove geometric relationships for figures in the coordinate plane? Characteristics of scalene, isosceles, and equilateral triangles Students will be able to name coordinates of special figures by using their properties. How can the properties of special figures be used to name their coordinates in the coordinate plane? Definition of Coordinate Proof

14 What is needed to plan a coordinate geometry proof? Coordinate proofs What is a coordinate proof and how is it used? Students will be able to prove theorems using figures in the coordinate plane. How can coordinate geometry be used to prove theorems? Writing a Coordinate Proof CCSS: G-GPE.B.4 What three formulas are important for completing coordinate proofs? Unit 7 What is Similarity and how it is used in Geometry? Students will be able to write ratios and solve proportions. How are ratios written? How are proportions solved? Definition of a ratio Definition of a proportion/extended proportion What are the properties of proportions? Properties of Proportions (Cross-Product Property) What are scale and scale drawings? Definition of scale/scale drawings Students will be able to identify and apply similar polygons. What are similar polygons? How are similar polygons identified and how are they used in geometry? Definition of similar/similarity ratio Definition of golden rectangle/golden ratio Using/identifying similar polygons CCSS: G-SRT.B.5 Students will be able to use theorems to prove triangles similar. Students will be able to use similarity to find indirect measurements. What are AA, SAS, and SSS similarity statements and how are they applied? How can indirect measurement be used to measure objects that are otherwise difficult to measure? Angle-Angle Similarity (AA~) Postulate Side-Angle-Side Similarity (SAS~) Theorem Side-Side-Side Similarity (SSS~) Theorem CCSS: G-SRT.B.5, G-GPE.B.5 Definition of an indirect measurement Students will be able to find and use relationships in similar right triangles. What are the relationships in similar right triangles and how are they used? Right Triangle Altitude Hypotenuse Theorem Definition of Geometric Mean CCSS: G-SRT.B.5, G-GPE.B.5

15 . What is the geometric mean? First and Second Corollary to Right Triangle Altitude Hypotenuse Theorem Students will be able to use the Side-Splitter Theorem and the Triangle-Angle-Bisector Theorem. What is the Side-Splitter Theorem and how is it applied? What is the Triangle-Angle-Bisector Theorem and how is it applied? Side-Splitter Theorem Corollary to Side-Splitter Theorem Triangle-Angle-Bisector Theorem CCSS: G-SRT.B.4 Unit 8 What is the relationship between Right Triangles and Trigonometry? Students will be able to use the Pythagorean Theorem and its converse. What is the Pythagorean Theorem and how is it applied in right triangles? What is the Converse of the Pythagorean Theorem and how is it used in right triangles? Pythagorean Theorem Definition of Pythagorean triple Converse of Pythagorean Theorem CCSS: G-SRT.B.4, G-SRT.C.8 What other theorems can be used to identify obtuse and acute triangles? Obtuse Triangle Theorem Acute Triangle Theorem Students will be able to use the properties of special right triangles. What are the properties of triangles and how can they be used to find the length of the hypotenuse and legs of a triangle? Triangle Theorem Triangle Theorem CCSS: G-SRT.C.8 What are the properties of triangles and how can they be used to find the length of the hypotenuse and legs of a triangle? Students will be able to use the sine, cosine, and tangent ratios to determine side lengths and angle measurements in right triangles. What are sine, cosine, and tangent ratios and how are they written? How are sine, cosine, and tangent ratios used to determine side lengths in triangles? Definitions of sine, cosine, tangent Writing sine, cosine, tangent ratios CCSS: G-SRT.C.6, G-SRT.C.7, G-SRT.C.8, G-MG.A.1 Students will be able to angles of elevation and depression. What is an angle of elevation? What is an angle of depression? Definition of angle of elevation Definition of angle of depression CCSS: G-SRT.C.8

16 How are angles of elevation and depression used to solve problems in Geometry? Students will be able to apply the Law of Sines. Given two angles and a side, how are the remaining parts of the triangle found? Law of Sines CCSS: G-SRT.D.10, G-SRT.D.11 Students will be able to apply the Law of Cosines. Given three sides or two sides and an angle, how are the remaining parts of the triangle found? Law of Cosines CCSS: G-SRT.D.10, G-SRT.D.11 Students will be able to describe a vector using an ordered pair notation. Students will be able to find the magnitude and direction of a vector. How is a vector described using an ordered pair notation? How is the magnitude and direction of a vector found? Definition of a vector Definition of the magnitude of a vector CCSS: N-VM.B.4a, N-VM.B.4b, N-VM.B.4c, N-VM.B.5a, N-VM.B.5a Students will be able to describe a vector. What is a vector? Students will be able to sketch a vector. How is a vector sketched? Unit 9 What are the various Transformations and their uses?

17 Students will be able to identify rigid motions. Students will be able to find translation images of figures. What is an isometry? How is the transformation of a geometric figure identified? Definitions of transformation/pre-image/image Definitions of isometry, translation, composition CCSS: G-CO.A.2, G-CO.A.4, G-CO.A.5, G-CO.B.6 What is a translation image and how is it found? Students will be able to find reflection images of figures. What is a reflection image and how is it found? Definition of reflection Finding reflections CCSS: G-CO.A.2, G-CO.A.4, G-CO.A.5, G-CO.B.6 Students will be able to draw and identify rotation images of figures. What are rotation images and how are they drawn and identified? Definition of rotation Drawing rotation images CCSS: G-CO.A.2, G-CO.A.4, G-CO.A.5, G-CO.B.6 Definition of the center of an image Students will be able to identify reflectional (line), rotational, and point symmetry using reflections and rotations. What is symmetry? What are the various types of symmetry in figures and how are they identified in a figure? Definition of symmetry Definitions of reflectional symmetry/line symmetry/rotational symmetry/point symmetry CCSS: G-CO.A.3 Identifying symmetry Students will be able to find compositions of isometries. Students will be able to classify isometries. What is a composition of reflections? How is composition of reflections used? What are glide reflections? How are glide reflections identified? Parallel Lines Composition of Reflections Theorem Intersection Lines Composition of Reflections Theorem Fundamental Theorem of Isometries Definition of glide reflection CCSS: G-CO.A.2, G-SRT.A.1a, G-SRT.A.1b, G-SRT.A.2 Isometry Classification Theorem Students will be able to identify a dilation as a similarity transformation. What is a dilation image and how are they located? What is the difference between a reduction and an enlargement? Definition of dilation Definitions of enlargement/reduction Translation or Rotation Theorem CCSS: G-CO.A.2, G-CO.A.5, G-CO.B.6 Unit 10 How are the Areas of Various Polygons calculated? Students will be able to find areas of parallelograms and triangles. What formula is used to find the area of a rectangle? Area of Rectangle, Parallelogram, Triangle Theorems CCSS: G-GPE.B.7, G-MG.A.1 What formula is used to find the area of a parallelogram? Definition of base/altitude/height of a parallelogram Definition of base/height of a triangle

18 What is the formula to find the area of a triangle? Students will be able to find areas of trapezoids, rhombuses, and kites. What formula is used to find the area of a trapezoid? What formula is used to find the area of a rhombus? What formula is used to find the area of a kite? Definition of height of a trapezoid Area of a Trapezoid Theorem Area of a Rhombus or a Kite Theorem CCSS: G-MG.A.1 Students will be able to find areas of regular polygons. What formula is used to find the area of a regular polygon? Definition of radius/apothem of a regular polygon Area of a Regular Polygon Theorem CCSS: G-CO.D.13, G-MG.A.1 Students will be able to find perimeters and areas of similar polygons. What formula is used to find the perimeter of similar figures? Perimeters and Areas of Similar Figures Theorem What formula is used to find the area of similar figures? Students will be able to find areas of regular polygons and triangles using trigonometry. How can trigonometry be used to find the area of a regular polygon? Relationship between trigonometry and area of regular polygons CCSS: G-SRT.D.9 How can trigonometry be used to find the area of a triangle? Area of a Triangle Given SAS Theorem Students will be able to find the measures of central angles and arcs. Students will be able to find the circumference and arc length. What is an arc? What is the central angle of a circle? How are the measures of central angles and arcs found? Definitions of circle/circle center/central angle Definitions of radius/diameter Definitions of semicircle/minor arc/major arc / adjacent arcs CCSS: G-C0.A.1, G-C.A.1, G-C.A.2, G-C.B.5 How are circumference and arc length found in circles? Arc Addition Postulate Definition of circumference/concentric circles Circumference of a Circle Theorem Definition of arc length Arc Length Theorem Students will be able to find the areas of circles, sectors, and segments of circles. What is the sector of a circle? What equation is used to find the area of a circle? Area of a Circle Theorem Definition of a sector of a circle CCSS: G-C.B.5 What equation is used to find the area of a circle sector? What equation is used to find the area of a circle segment? Area of a Sector of a Circle Theorem Definition of a segment of a circle

19 Students will be able to use segment and area models to find the probabilities of events. What is geometric probability? How are segment and area models used to find the probabilities of events in Geometry? Definition of geometric probability Geometric Probability Equation Unit 11 How are Surface Area and Volume of various 3-D Figures calculated? Students will be able to recognize polyhedrons and their parts. Students will be able to visualize cross sections of space figures. What is a polyhedron? How are polyhedral and their parts recognized? What are cross sections of space figures? Definition of polyhedron Definitions of face/edge/vertex Euler s Formula CCSS: G-GMD.B.4 How are cross sections of space figures visualized? Definition of cross section Drawing cross sections Students will be able to find the surface areas of prisms and cylinders. What is a prism and what formula is used to find the surface are of a prism? What is a cylinder and what formula is used to find the surface area of a cylinder? Definitions of prisms, cylinders Definitions of bases/lateral faces/altitude/height Definitions of right/oblique prisms CCSS: G-MG.A.1 Lateral and Surface Areas of Prisms Theorem Definitions of right/oblique cylinders Lateral and Surface Areas of Prisms Theorem Students will be able to find the surface areas of pyramids and cones. What is a pyramid and what formula is used to find the surface area of a pyramid? What is a cone and what formula is used to find the surface area of a cone? Definition of pyramids, cones Definitions of regular pyramid/slant height Lateral and Surface Areas of a Regular Pyramid Theorem CCSS: G-MG.A.1 Lateral and Surface Areas of a Cone Theorem

20 Students will be able to find the volumes of prisms and cylinders. What is volume? What formula is used to find the volume of a prism? What formula is used to find the volume of a cylinder? Definition of volume Cavalieri s Principle Theorem Volume of a Prism, Cylinder Theorems Definition of composite space figure CCSS: G-GMD.A.1, G-GMD.A.2, G-GMD. A.3, G-MG.A.1 Students will be able to find the volumes of pyramids and cones. What formula is used to find the volume of a pyramid? What formula is used to find the volume of a cone? Volume of a Pyramid Theorem Volume of a Cone Theorem CCSS: G-GMD. A.3, G-MG.A.1 Students will be able to find the surface areas and volumes of spheres. What is a sphere? What formula is used to find the surface area of a sphere? What formula is used to find the volume of a sphere? Definition of a sphere Definitions of great spheres/hemispheres Surface Area of a Sphere Theorem Volume of a Sphere Theorem CCSS: G-GMD. A.3, G-MG.A.1 Students will be able to compare and find the areas and volumes of similar solids. What are similar solids? What formulas are used to find the relationships between the ratios of the areas and volumes of similar solids? Definition of similar solids Definition of similarity ratio Areas and Volumes of Similar Solids Theorem CCSS: G-MG.A.1, G-MG.A.2 Unit 12 What are the properties of Circles and how are they applied? Students will be able to use properties of a tangent to a circle. What is the tangent to a circle? How is the relationship between a radius and a tangent used in Geometry? How is the relationship between two tangents from one point used in Geometry? Definition of tangent to a circle Definition of point of tangency Perpendicular Relationship of tangent to radius Theorem Perpendicular to Radius Endpoint Theorem Definitions of inscribed/circumscribed about Dual Tangent Segment Congruency Theorem CCSS: G-C.A.2

21 Students will be able to use congruent chords, arcs and central angles. Students will be able to use perpendicular bisectors to chords. Students will be able to find the measure of inscribed angles. Students will be able to find the measure of an angle formed by a tangent and a chord. What is a chord? How are chords used in conjunction with arcs and central angles in circles? What are the various properties of lines through the center of a circle? What is an inscribed angle? How is the measure of a inscribed angle found? What technique is used to find the measure of an angle formed by a tangent and a chord? Definition of a chord Congruent Central Angles/Chords/Arcs Relationships Theorem Equidistant Chords Theorem Perpendicular Diameter to Chord Bisector Theorem Diameter Bisecting Chord Theorem Perpendicular Bisector of a Chord Circle Center Theorem Definition of Inscribed Angle Inscribed Angle Theorem Three Corollaries to the Inscribed Angle Theorem Tangent/Chord Angle Measure Theorem CCSS: G-C.A.2 CCSS: G-C.A.2, G-C.A.3, G-C.A.4 Students will be able to find measures of angles formed by chords, secants, and tangents. Students will be able to find the lengths of segments associated with circles. What is a secant? What technique is used to find the measures of angles formed by chords, secants, and tangents? How are the lengths of segments associated with circles found? Definition of a secant Angle Measure of Intersecting Lines Inside/Outside of a Circle Theorem Product of Segment Length for a Given Point and Circle Theorem CCSS: G-C.A.2 Students will be able to write the equation of a circle. Students will be able to find the center and radius of a circle. How is the equation of a circle written? What technique is used to find the center and radius of a circle? Standard Equation of a Circle Theorem Naming a circle s radius/center CCSS: G-GPE.A.1 Students will be able to draw and describe a locus. What is a locus? Definition of a locus CCSS: G-GMD.B.4 How is a locus drawn? Drawing a locus How is a locus described? Describing a locus