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Grade Level/Course: Honors Geometry Content Area: Mathematics Grade Level/Course Overview: Honors Geometry is a rigorous proof-based course designed to develop spatial concepts and insight into geometric relationships. A precise mathematical language is developed with an emphasis on reading and real-life problem solving. Transformations on the coordinate plane provide opportunities for the formal study of congruence and similarity. The study of similarity leads to right triangle trigonometry and connects to quadratics and circles through Pythagorean relationships. The study of circles uses similarity and congruence to develop basic theorems related to circles and lines. The link between probability and data is explored through conditional probability. All units in this course will tie together geometric and previous advanced algebraic content knowledge such as systems of equations, factoring and solving quadratic equations. Advanced proofs are integrated throughout the course and will allow students to experience Geometry as a useful and logical subject. Strands/Domains 1. Geometry a. Congruence b. Similarity, Right Triangles, and Trigonometry c. Circles d. Expressing Geometric Properties with Equations e. Geometric Measurement and Dimension f. Modeling with Geometry 2. Statistics and Probability a. Conditional Probability b. Using Probability to Make Decisions Program Understandings (pk-12) 1. 2. 3. 4. 5. 6. 7. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Units of Study 1. Introduction to Geometry and Basic Constructions 2. Introduction to Proofs 3. Triangle Congruence 4. Triangles and Coordinate Proofs 5. Similarity 6. Trigonometry 7. Quadrilaterals and Coordinate Geometry 8. 2D vs 3D 9. Circles – Part 1 10. Circles – Part 2 11. Conics 12. Probability 13. Constructions Involving Circles Interdisciplinary Themes 1. 2. 3. 4. 5. Patterns Cause and Effect Scale, Proportion, and Quantity Systems and Systems Models Structure and Function Strand/Domain: Congruence Cluster: Experiment with transformations in the plane Understand (Conceptual): Students will understand that… • rotations / reflections and translations are based on the notions of point, line, distance along a line and distance around circular arc. Standard: G-CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Learning Targets Know (Factual) Students will know… Define: • • • • • • • • Do (Reasoning/Performance/Product) Students will… • angle circle perpendicular parallel line segment point line arc DOK • • • name an angle, line, line segment, ray, circle with the correct notation identify parallel and perpendicular lines from a diagram identify line / ray / line segment draw and label points, angles, lines, rays and segments correctly Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Experiment with transformations in the plane Understand (Conceptual): Students will understand that… • rotations / reflections and translations are based on the notions of point, line, distance along a line and distance around circular arc. Standard: G-CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Learning Targets Know (Factual) Students will know… • • • • • • • • • • rotations reflections triangle rectangle parallelogram trapezoid regular polygon symmetry dilation translation DOK Do (Reasoning/Performance/Product) Students will… • • • identify dilations, translations, rotations and reflections of triangles, rectangles, parallelograms, trapezoids or regular polygons plot a transformation given a set of points to be translated compare transformations that preserve size/length to those that do not Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Experiment with transformations in the plane Understand (Conceptual): Students will understand that… • rotations / reflections and translations are based on the notions of point, line, distance along a line and distance around circular arc. Standard: G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Learning Targets Know (Factual) Students will know… • • different ways to do transformations in a plane functions DOK Do (Reasoning/Performance/Product) Students will… • • draw a transformation in a plane describe rotations and reflections that map a polygon onto itself Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Experiment with transformations in the plane Understand (Conceptual): Students will understand that… • rotations / reflections and translations are based on the notions of point, line, distance along a line and distance around circular arc. Standard: G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Learning Targets Know (Factual) Students will know… • • • • • • • • rotations reflections translation angles circles perpendicular lines parallel lines line segment DOK Do (Reasoning/Performance/Product) Students will… • • create definitions of rotations / reflections / translations understand properties that are preserved in rotations, reflections and translations Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Experiment with transformations in the plane Understand (Conceptual): Students will understand that… • rotations / reflections and translations are based on the notions of point, line, distance along a line and distance around circular arc. Standard: G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Learning Targets Know (Factual) Students will know… • • • rotations translations reflections DOK Do (Reasoning/Performance/Product) Students will… • • • draw rotations / reflections / translations of a geometric figure using manipulatives recognize and draw compositions of transformations including mapping onto itself find the translation, rotation or reflection Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Understand congruence in Understand (Conceptual): Students will understand that… terms of rigid motions • rigid motions and their properties can be used to establish the triangle congruence criteria, which can then be used to prove other theorems. Standard: G-CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Learning Targets Know (Factual) Students will know… • • • rigid motion congruence transformation DOK Do (Reasoning/Performance/Product) Students will… • • • determine if two figures are congruent determine the effect of a given rigid motion transformation figures using geometric descriptions of rigid motion Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Understand congruence in Understand (Conceptual): Students will understand that… terms of rigid motions • rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Standard: G-CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Learning Targets Know (Factual) Students will know… • • • • congruence angles rigid motion corresponding angles DOK Do (Reasoning/Performance/Product) Students will… • verify two triangles are congruent • show that the triangles are congruent given triangles that have been transformed Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Understand congruence in Understand (Conceptual): Students will understand that… • terms of rigid motions rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Standard: G-CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Learning Targets Know (Factual) Students will know… • ASA • SAS • SSS • congruence • distance formula DOK Do (Reasoning/Performance/Product) Students will… • • use the definitions of congruence, based on rigid motion, to develop and explain the triangle congruence criteria complete proofs involving ASA, SAS, SSS Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Understand (Conceptual): Students will understand that… • in proving geometric theorems they need to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Cluster: Prove geometric theorems Standard: G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent: points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoint. Learning Targets Know (Factual) Students will know… • • • • • • • • • vertical angles alternate interior angles corresponding angles transversal parallel lines perpendicular perpendicular bisector equidistance segment DOK Do (Reasoning/Performance/Product) Students will… • • prove theorems: - vertical angles are congruent - transversal and parallel lines - alternate interior angles are congruent - corresponding angles are congruent - points on a perpendicular bisector of a line are equidistant from the endpoint apply proven theorems to a variety of problems Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Prove geometric theorems Understand (Conceptual): Students will understand that… • in proving geometric theorems they need to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Standard: G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°, base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Learning Targets Know (Factual) Students will know… • • • • isosceles triangle midpoint median triangle DOK Do (Reasoning/Performance/Product) Students will… • • prove theorems about triangles - in angles equal to 180 - base angles of isosceles triangles are congruent - segment joining midpoints of 2 sides of a triangle is parallel to the third side and ½ of length - medians of a triangle meet at a point apply proven theorems to a variety of problems Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Prove geometric theorems Understand (Conceptual): Students will understand that… • in proving geometric theorems they need to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Standard: G-CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Learning Targets Know (Factual) Students will know… • congruent • angles • parallelogram • bisector • rectangle • diagonals DOK Do (Reasoning/Performance/Product) Students will… • • prove and apply theorems about parallelograms: - opposite sides are congruent - opposite angles are congruent - diagonals of a parallelogram bisect each other - rectangles are parallelograms with congruent diagonals apply proven theorems to a variety of problems Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Make geometric constructions Understand (Conceptual): Students will understand that… • geometric constructions can be created using a variety of tools. Standard: G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.) Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Learning Targets Know (Factual) Students will know… • • • • • segment angle bisector perpendicular parallel DOK Do (Reasoning/Performance/Product) Students will… • Construct the following: - copy the segment - copy an angle - bisect a segment - bisect an angle - perpendicular lines including perpendicular bisector of a segment - parallel lines given a point not on a line Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Congruence Cluster: Make geometric constructions Understand (Conceptual): Students will understand that… • geometric constructions can be created using a variety of tools. Standard: G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Learning Targets Know (Factual) Students will know… • • • • • equilateral triangles square regular hexagon inscribed angles circle Do (Reasoning/Performance/Product) Students will… • Construct the following: - equilateral triangle inscribed in a circle - square inscribed in a circle - regular hexagon inscribed in a circle DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Understand (Conceptual): Students will understand that… • a dilation is an enlargement or reduction of a pre-image through a center point. Cluster: Understand similarity in terms of similarity transformations Standard: G-SRT.1. Verify experimentally the properties of dilations given by a center and a scale factor: a. b. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Learning Targets Know (Factual) Students will know… • • • • • • • dilation scale factor center of dilation enlargement reduction how to find scale factor between preimage and image the relationship between a pre-image, image, and center Do (Reasoning/Performance/Product) Students will… • determine the scale factor given a figure and its dilation • determine the dilation given a figure and a scale factor • find the center of dilation given a figure and its dilation • draw a dilation given a figure and a center of dilation DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Understand similarity in terms of similarity transformations Understand (Conceptual): Students will understand that… • similar figures have congruent corresponding sides and proportional sides. • triangles can be similar by various theorems. Standard: G-SRT.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Learning Targets Know (Factual) Students will know… • • definition of similar definition of proportions definition of corresponding parts of changes DOK Do (Reasoning/Performance/Product) Students will… • • • • • identify whether corresponding parts are similar by proportional sides and congruent angles identify the scale factor between two similar changes write a similarity statement identify/label the corresponding parts of the angles and sides using prime and now letters???? show that triangles are similar by SSS~ and SAS ~ Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Understand similarity in terms of similarity transformations Understand (Conceptual): Students will understand that… • two pairs of congruent angles are sufficient to prove two triangles are similar. (AA) Standard: G-SRT.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Learning Targets Know (Factual) Students will know… • Triangle Angle Sum Theorem: If 3 angles of one triangle are congruent to 3 angles of another triangle, then the triangles are dilations of one another, and therefore, similar DOK Do (Reasoning/Performance/Product) Students will… • • show that the triangles are similar given two pairs of congruent angles in two triangles derive the Third Angles Theorem Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Prove theorems involving similarity Understand (Conceptual): Students will understand that… • similarity is used to prove theorems about triangles. Standard: G-SRT.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Learning Targets Know (Factual) Students will know… • • properties of proportions recognize the 3 similar triangles when an altitude is drawn from the right angle of a right triangle DOK Do (Reasoning/Performance/Product) Students will… • • • show that the split sides are proportional given a line parallel to one side of a triangle that intersects the triangle find any other segment length given a right triangle with an altitude drawn from the right angle and 2 segment lengths find the geometric mean between two numbers Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Prove theorems involving similarity Understand (Conceptual): Students will understand that… • non-triangular geometric figures can be shown to be congruent or similar in the same way triangles are. Standard: G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Learning Targets Know (Factual) Students will know… • congruent figures are similar figures with a scale factor of 1 DOK Do (Reasoning/Performance/Product) Students will… • • • prove triangles are congruent or similar using similarity and congruency theorems (SSS, SAS, ASA, AAS, HL 𝐴𝐴~, 𝑆𝐴𝑆 ~, 𝑆𝑆𝑆~) prove other geometric figures are similar and/or congruent using the criteria found from triangles show all sides proportional and all angles congruent Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles Understand (Conceptual): Students will understand that… • similar right triangles are used to generate ratios between sides, leading to trigonometric functions. Standard: G-SRT.6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Learning Targets Know (Factual) Students will know… • • • • similar triangles right triangles ratio proportion DOK Do (Reasoning/Performance/Product) Students will… • • • use a corresponding angle to show the three side ratios are the same given the lengths of the sides of two similar right triangles define the trigonometric ratios (sine, cosine and tangent) discover the relationships in special right triangles Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles Understand (Conceptual): Students will understand that… • the sine and cosine of complementary angles are equivalent. Standard: G-SRT.7. Explain and use the relationship between the sine and cosine of complementary angles. Learning Targets Know (Factual) Students will know… • • • sine cosine complementary DOK Do (Reasoning/Performance/Product) Students will… • • express a sine ratio in terms of a cosine and vice-versa (co-functions) show that the sine of an angle is equal to the cosine of the angle’s complement Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Understand (Conceptual): Students will understand that… • six parts of right triangles are interdependent. • all missing parts of a right triangle can be found using trigonometric ratios and/or Pythagorean theorem. Cluster: Define trigonometric ratios and solve problems involving right triangles Standard: G-SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Learning Targets Know (Factual) Students will know… • • • • • • • • right triangles SOH CAH TOA Pythagorean theorem square roots inverse trigonometry opposite and adjacent legs hypotenuse angle of elevation and angle of depression DOK Do (Reasoning/Performance/Product) Students will… • • • • draw triangle from a word problem identify missing parts and choose appropriate trigonometry ratio or Pythagorean theorem to find missing sides solve equation to find missing part use the trig ratios and Pythagorean theorem to solve right triangles in applied problems Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Understand (Conceptual): Students will understand that… • the area of oblique (non-right) triangles can be found by A=1/2 ab sin C. Cluster: Apply trigonometry to general triangles Standard: G-SRT.9. (+) Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Learning Targets Know (Factual) Students will know… • • • definition of oblique triangles sine formula for the area of a triangle DOK Do (Reasoning/Performance/Product) Students will… • • apply formula A = ½ ab sin C to find area of oblique triangles derive A = ½ ab sin C from basic area formula (A = ½bh) using b A h C Level-2 Skill/Concept c B a Recall Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Apply trigonometry to general triangles Understand (Conceptual): Students will understand that… • the Law of Sines and Law of Cosines are used to find missing pieces of oblique (non-right) triangles. Standard: G-SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. Learning Targets Know (Factual) Students will know… • • • sine cosine when to use Law of Sines vs. Law of Cosines vs. SOH CAH TOA DOK Do (Reasoning/Performance/Product) Students will… • prove the Law of Sines and Law of Cosines using: b A h C Level-2 Skill/Concept c B a • Recall use Law of Sines and Law of Cosines to solve oblique triangles Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Similarity, Right Triangles, and Trigonometry Understand (Conceptual): Students will understand that… • the Law of Sines and Cosines can be used in applied problems to find missing sides and angles of any type of triangle. Cluster: Apply trigonometry to general triangles Standard: G-SRT.11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Learning Targets Know (Factual) Students will know… • • • Law of Sines Law of Cosines when to use the Law of Sines vs. the Law of Cosines DOK Do (Reasoning/Performance/Product) Students will… • use Law of Sines and Cosines to find unknown measures of right and oblique triangles in real-world problems Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Circles Cluster: Understand and apply theorems about circles Understand (Conceptual): Students will understand that… • circles are similar and therefore, useful ratios are created. Standard: G-C.1. Prove that all circles are similar. Learning Targets Know (Factual) Students will know… • • • • • • • • formula(s) for circumference radius diameter circle circumference similarity ratio proportions DOK Do (Reasoning/Performance/Product) Students will… • • find ratio of similarity using circumference/diameter and identify that the ratio is 𝜋 use similarity ratios to find missing information Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Circles Cluster: Understand and apply theorems about circles Understand (Conceptual): Students will understand that… • segments drawn in circles create relationships between arcs and angles. Standard: G-C.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Learning Targets Know (Factual) Students will know… • • • • • • • • chords tangent arc measure inscribed angle central angle diameter secant arc length DOK Do (Reasoning/Performance/Product) Students will… • • • use relationships between diameter, radii, chords, tangents, and secants to find angles and arcs find measure of inscribed, central, circumscribed, etc., angles and their intercepted arcs use relationships to find segment lengths Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Circles Cluster: Understand and apply theorems about circles Understand (Conceptual): Students will understand that… • polygons can be constructed in and around circles. Standard: G-C.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Learning Targets Know (Factual) Students will know… • incenter • • • • • • • angle bisector perpendicular bisector circumcenter angles of a quadrilateral add to 360˚ inscribed angles = ½ intercepted arc use a compass and straightedge how to write a proof DOK Do (Reasoning/Performance/Product) Students will… • • • use incenter to construct inscribed circle of a triangle use circumcenter to construct circumscribed circle of a triangle show opposite angles of inscribed quadrilateral are supplementary Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Circles Cluster: Understand and apply theorems about circles Understand (Conceptual): Students will understand that… • figures can be constructed in and around circles. Standard: G-C.4. (+) Construct a tangent line from a point outside a given circle to the circle. Learning Targets Know (Factual) Students will know… • • • • • • construct midpoint draw a circle with a given radius radius midpoint use a compass tangent DOK Do (Reasoning/Performance/Product) Students will… • construct tangent line from a point outside a given circle to the circle Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Circles Cluster: Find arc lengths and areas of sectors of circles Understand (Conceptual): Students will understand that… • circles are similar, and therefore, useful ratios are created. Standard: G-C.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Learning Targets Know (Factual) Students will know… • • • • circumference of a circle area of a circle definition of an arc/arc length definition of a sector DOK Do (Reasoning/Performance/Product) Students will… • • • • • • • find the circumference of a circle find the arc length of a sector measure several radii and arc lengths and compare their proportionality recognize that proportionality ratio is the angle measure in radians compare full circle to part of circle derive the formula for the area of a sector apply the area of a sector to a wide variety of problems (find area, find missing information) Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Expressing Geometric Properties with Equations Understand (Conceptual): Students will understand that… • the equation of a circle can be derived from Pythagorean Theorem and that they can change from standard form to vertex form by completing the square. Cluster: Translate between the geometric description and the equation for a conic section Standard: G-GPE.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Learning Targets Know (Factual) Students will know… • • • • • distance formula Pythagorean Theorem properties of radicals completing the square factoring DOK Do (Reasoning/Performance/Product) Students will… • • • derive the equation of a circle given center and radius using Pythagorean Theorem (distance formula) complete the square to find the center and radius of a circle manipulate the equations of circles from vertex to standard form Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Expressing Geometric Properties with Equations Cluster: Translate between the geometric description and the equation for a conic section Understand (Conceptual): Students will understand that… • given a focus and directrix they can derive the equation of a parabola. Standard: G-GPE.2. Derive the equation of a parabola given a focus and directrix. Learning Targets Know (Factual) Students will know… • • • • distance formula standard form vertex form FOIL DOK Do (Reasoning/Performance/Product) Students will… • • derive the equation of a parabola given a focus and directrix find the equation of a parabola given a focus and directrix Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Expressing Geometric Properties with Equations Understand (Conceptual): Students will understand that… • algebra can be applied to geometric proofs. Cluster: Use coordinates to prove simple geometric theorems algebraically Standard: G-GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Learning Targets Know (Factual) Students will know… • • • • • • • • • slope distance formula midpoint formula coordinate plane theorems on quadrilaterals theorems on triangles definitions of rectangle, square, kite, rhombus, trapezoid, parallelogram, circle, triangle how to classify quadrilaterals how to classify/name triangles DOK Do (Reasoning/Performance/Product) Students will… • • • prove a figure is a specific type of quadrilateral using distance and slope. prove a triangle is either isosceles, equilateral or scalene prove a point lies on a circle Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Expressing Geometric Properties with Equations Understand (Conceptual): Students will understand that… • the slope criteria for parallel and perpendicular lines can be used to solve geometric problems. Cluster: Use coordinates to prove simple geometric theorems algebraically Standard: G-GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Learning Targets Know (Factual) Students will know… • • • • • • • • slope parallel lines perpendicular lines slope-intercept form point-slope form perpendicular bisector altitude midpoint DOK Do (Reasoning/Performance/Product) Students will… • • • • • • find equations of a line parallel to a line through a given point find equation of a line perpendicular to a line through a given point find a perpendicular bisector to a line or side of triangle find an altitude of a triangle find a median of a triangle find the median of a trapezoid Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically Understand (Conceptual): Students will understand that… • line segments can be partitioned proportionally. Standard: G-GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Learning Targets Know (Factual) Students will know… • ratio • distance formula • midpoint formula • proportion • endpoint • line segment • triangle proportionality theorem DOK Do (Reasoning/Performance/Product) Students will… • • determine the coordinate of a point on a given line segment in given ratio: - number line - coordinate plane find lengths of segments with proportional relationships: - triangles with altitudes - triangles with line parallel to a side - 3 parallel lines cut by a transversal Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically Understand (Conceptual): Students will understand that… • the area or perimeter of a figure can be found by applying geometric concepts to points on a coordinate plane. Standard: G-GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Learning Targets Know (Factual) Students will know… • • • • • distance formula area formula perimeter simplify radicals identify polygons Do (Reasoning/Performance/Product) Students will… • use the distance formula to find the lengths of sides • find perimeter of polygons drawn in the coordinate plane • find areas of triangles and rectangles drawn on the coordinate plane DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Geometric Measurement and Dimension Cluster: Explain volume formulas and use them to solve problems Understand (Conceptual): Students will understand that… • perimeter, area and volume of two dimensional and three dimensional shapes can be derived. Standard: G-GMD.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Learning Targets Know (Factual) Students will know… • volume formulas • area formulas DOK Do (Reasoning/Performance/Product) Students will… • • use Cavalieri’s principles with cross sections of cylinders, pyramid and cones to compare the volumes use a combination of concrete models and formal reasoning to formulate conceptual understanding of the volume formulas Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Geometric Measurement and Dimension Cluster: Explain volume formulas and use them to solve problems Understand (Conceptual): Students will understand that… • volume formulas are useful for solving real-world problems. Standard: G-GMD.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Learning Targets Know (Factual) Students will know… • • volume formulas for: - cylinders - pyramids - spheres - cones area formulas for: rectangles circles triangles DOK Do (Reasoning/Performance/Product) Students will… • • • calculate the volume of cylinders, pyramids, spheres and cones use the volume formulas to solve problems in a real-world context solve for a missing variable in a formula given the volume Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Geometric Measurement and Dimension Cluster: Visual relationships between two-dimensional and three-dimensional objects Understand (Conceptual): Students will understand that… • there is a relationship between two and three dimensional shapes. Standard: G-GMD.4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Learning Targets Know (Factual) Students will know… • • • • names of the 2-D shapes names of the 3-D shapes definition of crosssection definition of rotation DOK Do (Reasoning/Performance/Product) Students will… • • • draw/visualize cross-sections created when 2-D shapes intersect 3-D shapes determine the different cross- sections created when cutting the 3-D shape at various angles identify the 3-D objects generated by rotations of 2-D objects Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Modeling with Geometry Cluster: Apply geometric concepts in modeling situations Understand (Conceptual): Students will understand that… • real life objects can be modeled using two dimensional and three dimensional geometric shapes. Standard: G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Learning Targets Know (Factual) Students will know… • • two dimensional shapes properties three dimensional shapes properties DOK Do (Reasoning/Performance/Product) Students will… • • recognize two dimensional and three dimensional shapes in real life situations create three dimensional objects and discuss their properties Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Modeling with Geometry Cluster: Apply geometric concepts in modeling situations Understand (Conceptual): Students will understand that… • density utilizes concepts of the area and volume of 2-D and 3-D figures. Standard: G-MG.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTU’s per cubic foot). Learning Targets Know (Factual) Students will know… • • • • 2-D and 3-D shapes and their area and volume formulas proportions density unit analysis DOK Do (Reasoning/Performance/Product) Students will… • • calculate area and volume of 2-D and 3-D shapes and apply it to density problems calculate surface area and lateral area Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Modeling with Geometry Cluster: Apply geometric concepts in modeling situations Understand (Conceptual): Students will understand that… • real life objects can be modeled using 2-D and 3-D geometric shapes. Standard: G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Learning Targets Know (Factual) Students will know… • • 2-D and 3-D shapes and their properties formulas for area, surface area and volume of 2-D and 3-D DOK Do (Reasoning/Performance/Product) Students will… • • shapes • • calculate area, surface area and volume of 2-D and 3-D shapes in real-world context find the dimension of 2-D and 3-D shapes that satisfy certain physical constraints given a real-world example, find errors and re-calculate area, surface area and volume (find errors in an estimate to build) create a 3D project that involves surface area and volume calculations and then build it Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence Understand (Conceptual): Students will understand that… and conditional probability and use them to interpret data • independence and conditional probability can be used to interpret data. Standard: S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,””not”). Learning Targets Know (Factual) Students will know… Definitions • union (“or”) of an event • intersection (“and”) of two events • complement (“not”) of an event • sample space • subset appropriate symbols of union, intersection, and complement Do (Reasoning/Performance/Product) Students will… • identify sample space and events within a sample space • identify subsets from within the sample space DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence Understand (Conceptual): Students will understand that… and conditional probability and use them to interpret data • independence and conditional probability can be used to interpret data. Standard: S-CP.2. Understand that two events A and B are independent of the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Learning Targets Know (Factual) Students will know… Definitions • independent events • conditional probability Do (Reasoning/Performance/Product) Students will… • determine if two events are independent DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence Understand (Conceptual): Students will understand that… and conditional probability and use them to interpret data • independence and conditional probability can be used to interpret data. Standard: S-CP.3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Learning Targets Know (Factual) Students will know… • multiplication principle Do (Reasoning/Performance/Product) Students will… • use the multiplication principle to calculate conditional probabilities DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence Understand (Conceptual): Students will understand that… and conditional probability and use them to interpret data • independence and conditional probability can be used to interpret data. Standard: S-CP.4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Learning Targets Know (Factual) Students will know… • sample space • two-way table • conditional probability • independent events Do (Reasoning/Performance/Product) Students will… • construct and interpret two-way frequency tables for two categorical variables • calculate probabilities from the two-way tables • use probabilities from the table to evaluate independence DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence Understand (Conceptual): Students will understand that… and conditional probability and use them to interpret data • independence and conditional probability can be used to interpret data. Standard: S-CP.5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Learning Targets Know (Factual) Students will know… • independent events • conditional probability Do (Reasoning/Performance/Product) Students will… • recognize and explain the concepts of conditional probability and independence in a real-life setting DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Use the rules of Understand (Conceptual): Students will understand that… • probability to compute probabilities of compound events in a uniform probability model independence and conditional probability can be used to interpret data. Standard: S-CP.6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Learning Targets Know (Factual) Students will know… • conditional probability formula Do (Reasoning/Performance/Product) Students will… • calculate and interpret conditional probability of A given B DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Use the rules of Understand (Conceptual): Students will understand that… probability to compute probabilities of compound events in a uniform probability model • different probability formulas can use be used to calculate and interpret real world phenomena. Standard: S-CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Learning Targets Know (Factual) Students will know… Definitions • disjoint events • mutually exclusive events addition rule of probability Do (Reasoning/Performance/Product) Students will… • calculate and interpret probabilities using the addition rule DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Use the rules of Understand (Conceptual): Students will understand that… probability to compute probabilities of compound events in a uniform probability model • different probability formulas can use be used to calculate and interpret real world phenomena. Standard: S-CP.8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model. Learning Targets Know (Factual) Students will know… • multiplication rule of probability • conditional probability • independent events Do (Reasoning/Performance/Product) Students will… • calculate and interpret a probability using the multiplication rule DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Use the rules of Understand (Conceptual): Students will understand that… probability to compute probabilities of compound events in a uniform probability model • different probability formulas can use be used to calculate and interpret real world phenomena. Standard: S-CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Learning Targets Know (Factual) Students will know… Definitions • factorials • combination • permutation Formulas to calculate probabilities of a • combination • permutation Do (Reasoning/Performance/Product) Students will… • determine the difference between a permutation and a combination • calculate probabilities using the appropriate permutation or combination formula DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Using Probability to Make Decisions Cluster: Use probability to evaluate outcomes of decisions Understand (Conceptual): Students will understand that… • probabilities exhibit relationships that can be extended, described, and generalized to make decisions. Standard: S-MD.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Learning Targets Know (Factual) Students will know… • definition of random • how to use a random number generator Do (Reasoning/Performance/Product) Students will… • understand factors that make decisions fair and random o toss a die o flip a coin o use a spinner DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Strand/Domain: Using Probability to Make Decisions Cluster: Use probability to evaluate outcomes of decisions Understand (Conceptual): Students will understand that… • probabilities exhibit relationships that can be extended, described, and generalized to make decisions. Standard: S-MD.7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Learning Targets Know (Factual) Students will know… • multiplication rule • addition rule • permutations • combinations Do (Reasoning/Performance/Product) Students will… • use multiplication rule to find the intersection of independent events 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵) • use addition rule to find probabilities (with Venn diagrams for example) of AND and OR events • analyze decisions and strategies using probability concepts DOK Recall Level-2 Skill/Concept Level-3 StrategicThinking Level-4 Extended Thinking Honors Geometry vs. Regular Geometry Honors Geometry will do more rigorous proofs and use more complex Algebra throughout the course. Additional Topics in Honors Geometry: • • • • • • • • • • • • • • • Derive midpoint and angle bisector theorem. (Unit 1) Biconditionals, inverses and contrapositives (Unit 2) Indirect proofs (Unit 3) Prove two triangles are congruent and use corresponding parts to prove that a second set of triangles are congruent. (Unit 3) Ambiguous case (Unit 3) Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (Unit 7) Tangent equals sine over cosine (Unit 6) Interior and exterior angle sum theorem (Unit 8) Apply areas and volumes of 2D and 3D shapes to density. (Unit 8) Demonstrate the derivation of and find the equation of a parabola given the focus and directrix. (Unit 11) Find lengths of segments involving secants and tangents. (Unit 10) Construct an equilateral triangle, a square and a regular hexagon inscribed in a circle. (Unit 13) Construct a tangent line given a point not on the circle to the circle. (Unit 13) Construct the inscribed and circumscribed circle of a triangle. (Unit 13) Prove properties of angles for a quadrilateral inscribed in a circle. (opposite angles are supplementary, inscribed rhombi must be squares, inscribed parallelograms must be rectangles) (Unit 13) COURSE: Honors Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Introduction to Geometry and Basic Constructions 3 weeks ESSENTIAL QUESTION(S): How are points, lines, rays and segments related ? How does each pre-image relate to its image? How do transformations relate to congruence? In what ways is it possible to construct different geometric figures? In what ways can congruence be useful? REFERENCE/ STANDARD # G-CO.1 G-CO.2 G-CO.3 G-CO.4 G-CO.5 G-CO.12 WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? STANDARDS: A listing of all standards included in the unit Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.) Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Prove theorems about lines and angles. Theorems include the angle bisector theorem and the midpoint theorem. MA.HG.CO.03 UNIT DESCRIPTION: UNIT VOCABULARY Students will understand the defined and undefined terms of Geometry, analyze and apply transformations to geometric figures and understand properties that are preserved by these transformations. Students will construct geometric figures using a variety of tools and resources. Point, line, plane, ray, collinear, coplanar, intersection, opposite rays, segments, angle, vertex, circle, perpendicular, parallel, distance, circumference, rectangle, parallelogram, trapezoid, regular polygon, transformation, rotation, reflections, translation, dilation, vector, symmetry, congruence, construct, bisect HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT MAJOR SUPPORTING STANDARD STANDARD (M) (S) UNIT SCORING GUIDE A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # G-CO.1 TEACHER INSTRUCTION STUDENT LEARNING Use constructions to develop formal definitions of midpoint, angle bisector, segment bisector, perpendicular bisector and congruence Skills Checks – p. 1-2 McDougal Littell Textbook – p. 46 Written Exercises Derive Midpoint and Angle Bisector Theorems - McDougal Littell Textbook (pages 43/44) G-CO.2 Glencoe Geometry Textbook Rotation Demonstration Activity Rotation Demonstration (Geogebra) Reflection Demonstration Activity Reflection Demonstration (y=x) (Geogebra) Reflection Demonstration Practice Worksheets 9.1-9.6, Glencoe Book HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS (hor/ver) (Geogebra) Translation Demonstration Translation Demonstration (vector) (Geogebra) G-CO.3 Horizontal stretch of a parabola illustrations Discover and demonstrate rotational symmetry to map a figure onto itself. Discover how a reflection or series of reflections can map a figure onto itself. Rotational symmetry activity for students Rotational symmetry demonstration for students Rotation/Reflection Lesson Glencoe Book 9.1 (reflections) and 9.3 (rotations) G-CO.4 New York Curriculum Lesson 15 (p. 111) Rotation/Reflection relationship New York Curriculum Lesson 16 (p. 117) Discovery of Definition of Translation New York Curriculum Lesson 18 (p. 131) Reflection discovery activity for students Video introducing transformations G-CO.5 G-CO.12 Discovering parallel lines using reflection Compositions of Transformations Use technology (Geogebra, SMART Notebook, Core Math Tools, etc.) to contstruct Construction Tutorials Compositions of Transformations WS Use compass and straight edge to construct Construction Instruction Packet Construction Tutorials ADDITIONAL UNIT RESOURCES Video: Review of Points, Lines, and Planes – go to LearnZillion.com and search for LZ4568 Video: Importance of Precise Geometric Terms – go to LearnZillion.com and search for LZ4571 Transformations Review Packet Website with additional resources (under construction) COURSE: Honors Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Introduction to Proofs 2 weeks ESSENTIAL QUESTION(S): What is the congruence relationship between the angle pairs formed from intersecting lines? REFERENCE/ STANDARD # G-CO.9 WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? STANDARDS: A listing of all standards included in the unit Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent: points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoint. Prove theorems about triangles using an indirect proof. MAJOR SUPPORTING STANDARD STANDARD (M) (S) MA.HG.CO.02 UNIT DESCRIPTION: UNIT VOCABULARY Students will be able to identify and prove angle relationships that occur with parallel lines that are cut by a transversal, intersecting lines and perpendicular lines. Parallel lines, intersecting lines, perpendicular lines, vertical angles, transversal, alternate interior angles, corresponding angles, perpendicular bisector HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # TEACHER INSTRUCTION G-CO.9 Glencoe book 171-184 STUDENT LEARNING HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS Introduction to angle theorems Angle theorems with illustrations Parallel lines and transversals worksheet Re-teach site with practice problems New York Common Core Curriculum Lesson 9 (unknown angle proofs) p. 66 Khan Academy video proving vertical angles congruent Foldable for parallel lines cut by a transversal Dummies.com perpendicular bisector proof Practice with parallel lines Parallel lines resource Parallel lines and angle relationship Parallel lines task Perpendicular bisector practice Glencoe book pg 327 Algebraic Proofs Segment addition and Angle addition proofs Parallel line proofs Perpendicular bisector practice ADDITIONAL UNIT RESOURCES Geometry Teacher – Unit 2 -http://www.geometry-teachers.com/ COURSE: Honors Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Triangle Congruence 3 weeks ESSENTIAL QUESTION(S): What processes are valid to prove two triangles are congruent? What can you conclude about two triangles that are congruent? REFERENCE/ STANDARD # G-CO.6 G-CO.7 G-CO.8 MA.HG.CO.02 MA.G.CO.03 WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? STANDARDS: A listing of all standards included in the unit Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove theorems about triangles using an indirect proof. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. MA.HG.SRT.01 Understand the Ambiguous Case (SSA) and how it can lead to two unique triangles. UNIT DESCRIPTION: UNIT VOCABULARY Students will be able to determine if and prove that two triangles are congruent. Rigid motion, corresponding parts, ASA, SAS,SSS,ASA,SAS, SSS, HL, CPCTC MAJOR SUPPORTING STANDARD STANDARD (M) (S) HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # TEACHER INSTRUCTION STUDENT LEARNING HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS G-CO.6 G-CO.7 Page 232-307 Glencoe Geometry Textbook Section 4.3 – Identify congruent figures and name by corresponding parts Worksheet – Triangle Angle Sum Video Showing Triangle Angle Sum Theorem – cutting angles to form a line Worksheet – Correspodning parts Show Triangles are Congruent – Also links to SSS, SAS, AAS, ASA, HL Best Strategies by Benson – p. 27 #46 Triangle Congruence Skills Checks – Page 2 Multiple Choice Questions Activity to Discover Triangle Congruences – SSS, SAS, ASA, AAS, and HL, and why AAA and SSA don’t work Activity to Discover SSS and SAS – Easy to apply to ASA Triangle Congruence Skills Checks – Page 1 G-CO.8 Why AAA Doesn't Work Why SSA Doesn't Work Glencoe Geometry Textbook – Section 4-4 p 268 #23, 30, 31, 33 Activity to Discover Triangle Congruences – SSS, SAS, ASA, AAS, and HL, and why AAA and SSA don’t work Best Strategies by Benson-Proofs on p. Best Strategies by Benson – p. 9 #12 5 #4-5, p. 6 #6, p. 8 #10 Why AAA and SSA Don't Work Video – Why SSA sometimes works Worksheet – Using SSS et al to determine congruency Worksheet – Using SSS et al in proofs ADDITIONAL UNIT RESOURCES Jeopardy – Triangle Angle Sum, Congruence, CPCTC Worksheet – Using CPCTC in proofs Worksheet – Using CPCTC in proofs Project – Proof Puzzles COURSE: Honors Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Triangle Theorem Proofs 2 weeks ESSENTIAL QUESTION(S): How do you use prior knowledge to prove a new idea? How do algebraic concepts relate to the segments and angles within a triangle? How can the coordinate plane be used to prove properties of triangles? WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? REFERENCE/ STANDARDS: MAJOR SUPPORTING STANDARD A listing of all standards included in the unit STANDARD STANDARD # (M) (S) G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°, base angles of G-GPE.4 G-GPE.5 isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). UNIT DESCRIPTION: UNIT VOCABULARY The student will define midsegment, median, centroid, Altitude, Angle bisector, Centroid, Equilateral, Isosceles, Median, perpendicular and angle bisectors, and altitude of triangles. Midsegment, Parallel, Perpendicular, Perpendicular bisector, Scalene The student will apply and prove properties of these parts of triangles. Students will use the coordinate plane to complete proofs. HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning HOW WILL WE RESPOND WHEN HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? STANDARD # G-CO.10 TEACHER INSTRUCTION Prove/discover that base angles of isosceles triangles are congruent. -Proving base angles of isosceles triangles congruent STUDENT LEARNING STUDENTS HAVE NOT LEARNED? INTERVENTIONS EXTENSIONS -Calibrating Consoles (ProblemBased Tasks: Math II, Pg 181) -Isosceles Triangle Proof (Best Strategies by Benson, #12) -Angle Bisector Application (Best Strategies by Benson, #17) -Jigsaw Vocabulary Activity -Isosceles Triangle Discovery and Application In Glencoe text on page 283-291 section 4-6 G-CO.10 Define and apply midsegment, median, centroid, perpendicular and angle bisectors, and altitudes of triangles. -Finding Medians (Best Strategies by Benson, #149) -Finding Centroid and -Centroid Application (Best Strategies by Benson, #153) -Medians and Altitude Notes & Problems -Finding lengths of medians in a right triangle (Best Strategies by Benson, #155) Orthocenter -Median of a Triangle Notes -Concurrent Medians Construction In Glencoe text on page 322-291 sections 5-1 and 5-2 -Orthocenters and Altitudes (Best Strategies by Benson, #156) -Finding Bisectors, Medians and Altitudes (Geometry Stations, Pg 5054) -9 Point Circle Project G-GPE.5 Review slope-intercept form, point-slope form, perpendicular bisector, altitude, and midpoint. In Glencoe text on page 196-204 sections 3-4 Introduce slopes of parallel and perpendicular lines. In Glencoe text on page 186-195 sections 3-3 Find equation of a line parallel/perpendicular to a line through a given point. Find a perpendicular bisector to a line or side of a triangle. Find the equation for the altitude/median of a triangle given vertices. -Review distance with triangles (Best Strategies by Benson, #69) -Review distance in coordinate plane (Best Strategies by Benson, #70) -Equation of Perpendicular Bisector (Best Strategies by Benson, #132) -Centroid Problem (Best Strategies by Benson, #133) -Review slope (Best Strategies by Benson, #134) -Altitude Equation with Median (Best Strategies by Benson, #135) -Equations of Medians and Altitudes -Equations of Medians and Altitudes -Equations of Medians and Altitudes -Equations of Altitudes and Perpendicular Bisectors -Bisectors, Medians and Altitudes -Distance and Perpendicular lines (Geometry Stations, Pg 205-218) -Distance and Parallel Lines (Geometry Stations, Pg 192-204) Altitudes and Medians (See Teacher Resource Files) G-CO.10 G-CO.10 Applications using midsegment, median, centroid, perpendicular and angle bisectors, and altitude. Coordinate proofs involving midsegment, median, centroid, perpendicular and angle bisectors, and altitude. -Sailing Centroid (Problem-Based Tasks: Math II, Pg 189) -Median Application Problems -Median Application Problems -Coordinate Proofs -Constellation Coordinate Proof -Coordinate Proofs G-GPE.4 Coordinate proofs to determine if a triangle is isosceles, equilateral, scalene or right using distance. -Coordinate Proofs -Prove a triangle is isosceles ADDITIONAL UNIT RESOURCES -Right Triangle Proofs (#2 and #6) -Classify Triangle by Sides -Coordinate Proofs COURSE: Honors Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Similarity 3 weeks ESSENTIAL QUESTION(S): How do you prove triangles or polygons similar? What are the differences between similar and congruent figures? How might the features of one figure be useful whn solving problems about similar figure? WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? REFERENCE/ STANDARDS: MAJOR SUPPORTING STANDARD A listing of all standards included in the unit STANDARD STANDARD # (M) (S) Verify experimentally the properties of dilations given by a center and a scale factor: G-SRT.1 c. G-SRT.2 G-SRT.3 G-SRT.5 G-SRT.4 A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. UNIT DESCRIPTION: UNIT VOCABULARY Students will identify and apply similarity properties. Dilation, similarity, scale factor, corresponding parts, proportion, ratio, • • • • geometric mean, altitude Prove polygons are similar/congruent Write similarity statements Identify scale factors Prove triangles are similar/congruent by SSS, SAS, ASA, AAS, HL, SSS~, SAS~, AA~ • Use properties of similar triangles to solve application and algebraic problems HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # G-SRT.1 TEACHER INSTRUCTION Website for Illustration Real-world examples of centers in dilations Website page 9 of PDF STUDENT LEARNING Task Problems - CCSS Problem Based Tasks for Mathematics II – Prettying Up the Pentagon pg. 209 & The Bigger Picture pg. 213 Task Problems - CCSS Problem Based Tasks for Mathematics II – Video Game Transformations pg. 218 G-SRT.2 Book - Geometry Station Activities Book pg. 112 G-SRT.3 G-SRT.5 Worksheet pages 27-29 Worksheet over Similarity Answers to worksheet CCSS Problem Based Tasks for Mathematics II – True Tusses pg. 223 Task Problems - CCSS Problem Based Tasks for Mathematics II – Too Tall? Pg. 238 HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS G-SRT.4 Worksheet pages 34-36 Task Problems - CCSS Problem Based Tasks for Mathematics II – Down, Down, Down pg. 226 Suddenly Sinking pg. 230 Geometry Station Activities Book pg. 128 Identifying Similar Triangles Activity – Grou p Work PowerPoint Website Here Task Problems - CCSS Problem Based Tasks for Mathematics II – Towering Heights pg. 234 ADDITIONAL UNIT RESOURCES www.learnzillion.com and search Similarity http://ccssmath.org/?s=geometry Common Core website COURSE: Honors Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: 3 weeks Right Triangles and Trigonometry ESSENTIAL QUESTION(S): How does the measure of one acute angle relate to the ratio of two side measures in any right triangle? How do trigonometric ratios relate to similar right triangles? How are missing side lengths and angle measures found in a right or oblique triangle? What strategies can be used to find missing parts of triangles and how can they be used to apply to real world problems? Can trigonometry be used to find the area of a triangle? REFERENCE/ STANDARD # G-SRT.6 G-SRT.7 G-SRT.8 G-SRT.10 G-SRT.11 G-SRT.9 MA.HG.SRT.02 WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? STANDARDS: A listing of all standards included in the unit Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Explain and use the relationship between the sine and cosine of complementary angles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Prove the Laws of Sines and Cosines and use them to solve problems. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. MAJOR SUPPORTING STANDARD STANDARD (M) (S) UNIT DESCRIPTION: UNIT VOCABULARY Students will find missing sides and angles of a triangle using trigonometry. Pythagorean Theorem, Pythagorean Triple, Trigonometry, Trigonometry Ratio, Sine, Cosine, Tangent, Inverse Sine, Inverse Cosine, Inverse Tangent, Complementary, Co-Functions, Angle of Elevation, Angle of Depression, Oblique Triangle, Law of Sine, Law of Cosine • • • • • SOH CAH TOA Law of Sines, Law of Cosines Pythagorean Theorem Special right triangle relationships Application problems Students will find the area of an oblique triangle. HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # G-SRT.6 TEACHER INSTRUCTION STUDENT LEARNING Instructional stragies, links to websites, resources, etc. Anything that will help teacher provide instruction related to the standard(s) Tasks, activities, links to practice, etc. Understand that similar triangles share angle measures and side ratios Worksheet – Special Triangles (Answer Key) 45-45-90 Triangle Worksheet – Special Right Triangles (Answer Key) 30-60-90 Triangle SOH-CAH-TOA Find sine value using side ratios Find cosine value using side ratios Find tangent value using side ratios G-SRT.7 Sine and Cosine of Complementary Angles Exit Slip – Special Right Triangles (Answer Key) Special Right Triangles Problems Geometry Station Activity for Common Core pgs. 139-150 Complimentary Angles Activity HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS G-SRT.8 Inverse Function Notes Inverse Function WS G-SRT.10 Angle of Elevation and Depression Prove Law of Sines and Law of Cosines Angle of Elevation and Depression WS Glecoe Secondary Math Aligned to the CC Pgs. 12-16 #1-6, 8-20, 22-27, 31-42, 47-50 Glecoe Secondary Math Aligned to the CC Pg. 15 #45 and #46 G-SRT.11 Glecoe Secondary Math Aligned to the CC Pgs. 8-12 G-SRT.9 Derive A=1/2ab sin C from basic area formula using A=1/2bh ADDITIONAL UNIT RESOURCES Law of Sines Problems Law of Cosines Problems Glecoe Secondary Math Aligned to the CC Pgs. 12-16 #7, 21, 28, 29, 30, 43-44, 51-53 Apply formula to find area of oblique triangles COURSE: Honors Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Quadrilaterals and Coordinate Geometry 3 weeks ESSENTIAL QUESTION(S): How can you use your prior knowledge to derive and apply properties of special quadrilaterals? How can the coordinate plane used to measure, model, and calculate area and perimeter of polygons? WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? REFERENCE/ STANDARDS: MAJOR SUPPORTING STANDARD A listing of all standards included in the unit STANDARD STANDARD # (M) (S) Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, G-CO.11 G-GPE.4 G-GPE.5 G-GPE.6 G-GPE.7 the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. (Honors Geometry only) UNIT DESCRIPTION: UNIT VOCABULARY The student will be able to derive and use the properties of special quadrilaterals using geometric and algebraic concepts on a coordinate plane. The student will be able to calculate the area and perimeter of polygons in the coordinate plane. Quadrilateral, parallelogram, rectangle, rhombus, square, kite, trapezoid, isosceles trapezoid, distance, midpoint, slope, parallel, perpendicular, ratio, diagonal, coordinate plane, triangle, perimeter, area, polygon HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # G-CO.11 TEACHER INSTRUCTION Instructional stragies, links to websites, resources, etc. Anything that will help teacher provide instruction related to the standard(s) Review Lesson- Slope lesson, practice, and teacher resource Review Lesson- mispoint lesson, practice, and teacher resource Review lesson- Distance formula lesson, practice, and teacher resource Have the students discover the properties of quadrilaterals using a discovery activity like the NCSM Great tasks or the discovery examples below Lesson- Discover properties about special quadriaterals using variable coordinates on the coordinate plane. STUDENT LEARNING Tasks, activities, links to practice, etc. NCSM Great Tasks p.145-148- discovery activity to figure out all of the properties of the special quadrilaterals Task- Have students draw a venn diagram showing the relationship between all special quadrilaterals Example problemsBest Strategies by Benson #’s 29, 30, 31, 32, 33 HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS G-GPE.4 G-GPE.5 G-GPE.6 G-GPE.7 Lesson- Coordinate geometry lesson, practice, and teacher resource Lesson- Median of a trapezoid applet Lesson- Area and perimeter of rectangle and triangle on coordinate plane ADDITIONAL UNIT RESOURCES Geometry Station Activities p.219-229Practice- Coordinate proofs for triangles and special quads. Practice-Area and perimeter of rect and triangle on coordinate plane COURSE: Honors Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Two-Dimension vs. Three-Dimension 3 weeks ESSENTIAL QUESTION(S): How can two-dimensional figures be used to understand three-dimensional objects? Where did area and volume formulas come from? How can geometric figures be used in real-life area and volume situations? REFERENCE/ STANDARD # G-GMD.1 G-GMD.3 G-GMD.4 G-MG.1 G-MG.2 G-MG.3 WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? STANDARDS: A listing of all standards included in the unit Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTU’s per cubic foot). Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Prove theorems about convex polygons and the measures of their interior and exterior angle sums. MAJOR SUPPORTING STANDARD STANDARD (M) (S) MA-HG-CO.04 UNIT DESCRIPTION: UNIT VOCABULARY This unit investigates area and volume paying particular attention to modeling situations. Two dimensions, three dimensions prisms, pyramids, cylinders, cones, spheres, similar solids HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? STANDARD # G-GMD.1 TEACHER INSTRUCTION Instructional stragies, links to websites, resources, etc. Anything that will help teacher provide instruction related to the standard(s) Tasks, activities, links to practice, etc. Relate diameter and circumference Glecoe 1.6 Circle Poster Circumference Informally prove the area of a circle Glencoe 11.3 Area of Circles Calculate volume of prisms and cylinders using the cavalieri principle Glencoe 12.4 G-GMD.3 STUDENT LEARNING Relate the volume of prisms/cylinders to pyramids/cones Glencoe 12.4-12.5 Solve real-world problems involving cones Glencoe 12.3 and 12.5 Solve real-world problems involving pyramids Cavalieri's Principle Worksheet Area of Prisms, Pyramids, Cylinders, and Cones Surface Area and Volume - All Surface Area and Volume – Prisms and Cylinders Surface Area and Volume – Spheres Online Activity - Volume INTERVENTIONS EXTENSIONS Glencoe 12.3 and 12.5 Solve real-world problems involving cylinders Glencoe 12.2 and 12.4 G-GMD.4 Solve real-world problems involving spheres Glencoe 12.6 Visualize cross-sections of prisms Glencoe 12.2 and 12.4 Visualize cross-sections of pyramids Glencoe 12.3 and 12.5 Visualize cross-sections of cylinders Glencoe 12.2 and 12.4 Visualize cross-sections of cones Glencoe 12.3 and 12.5 G-MG.1 Predict 3D results of rotating simple figures Volume of prisms, cylinders, pyramids, spheres, cones NOTES.notebook of Cones, Cylinders, and Spheres G-MG.2 G-MG.3 2D vs. 3D - Volume.ksig Prism and Cylinders LA SA Pyramids and Cones LA SA A Day at the Beach ..\..\Geometry\Chapter 12\Extra Practice Word Problems Prisms, Cylinders and Spheres.docx ADDITIONAL UNIT RESOURCES Performance Task – A Day at the Beach COURSE: Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Circles –Part 1 2 week ESSENTIAL QUESTION(S): 1. Why are all circles similar? 2. How can the arc length and area of sector formulas be derived using similarity? 3. What are radians and how were they derived? REFERENCE/ STANDARD # G-C.1 G-C.5 STANDARDS: WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? A listing of all standards included in the unit Prove that all circles are similar. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. UNIT DESCRIPTION: Students will discover that all circles are similar by using the length of the arc and other measurements. The proportionality of the length of an arc intercepted by an angle to the radius will be discovered. Students will also derive the formula for the area of a sector. Students will learn the origin of radians and its role as the constant of proportionality. MAJOR SUPPORTING STANDARD STANDARD (M) (S) UNIT VOCABULARY Tangents, secants, arc, chords, ratio, diameter, radius HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? STANDARD # G-C.1 G-C.5 G-C.5 G-C.1 G-C.5 G-C.5 G-C.5 G-C.1 G-C.1 G-C.5 TEACHER INSTRUCTION Lesson for proving circles similar using similar triangles Website that illustrates area of sectors and introduces radians & area of a sector Website that helps explain radians in plain terms Explanation and practice of proving that all circles are similar using the concept of transformations and dilations Glencoe Geometry Baseball book section 113 Sectors and segments of circles website Activity with sectors and pizza Balloon activity showing circles are similar Website for proving circles similar using translations Applet demonstrating arc length STUDENT LEARNING Worksheet practice for similar circles Activity investigating radians Problem Based Tasks for Math II (orange book) Similar Circles Pg. 265 Following in Archimedes’ Footsteps Problem Based Tasks for Matt II (orange book) Defining Radians pg. 290 Around the MerryGo-Round Practice for sectors and segments Activity investigating arc length and area of a sector INTERVENTIONS EXTENSIONS ADDITIONAL UNIT RESOURCES www.learnzillion.com Online practice that covers the entire Unit. Multiple Choice practice over Circles (PDF available) COURSE: Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Circles – Part 2 2 week ESSENTIAL QUESTION(S): What are the relationships between parts of a circle? Can those relationships be used to find unknown parts of a circle? REFERENCE/ STANDARD # G-C.2 MA-HG-C.01 MA-HG-C.03 WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? STANDARDS: A listing of all standards included in the unit Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Derive and identify the relationships between segments created by chords, secants, and tangents in circles. Construct the inscribed and circumscribed circles of triangle, and prove properties of angles for a quadrilateral inscribed in a circle. UNIT DESCRIPTION: The goal of this unit is to establish the numerical relationship between arcs and angles of a circle and to provide ways of calculating segments related to circles. MAJOR SUPPORTING STANDARD STANDARD (M) (S) UNIT VOCABULARY Arc, central angle, chord, circumscribed angle, inscribed angle, major arc, minor arc, point of tangency, radii, secant, semicircle, tangent HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # TEACHER INSTRUCTION STUDENT LEARNING HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS G-C.2 G-C.2 G-C.2 G-C.2 G-C.2 G-C.2 G-C.2 G-C.2 G-C.2 G-C.2 Website that illustrates inscribed angles in a semicircle are right angles. Presentation that shows central and inscribed angles Website that explains that a tangent is perpendicular to the radius to the circle of the radius of the circle at the point where the tangent intersects the circle Glencoe Geometry Baseball Book resources section 10-2 through 107 Insider Teacher Exchange Files for Unit Website for chords and circles Website for tangents and circles Website for special segments in circles Website for Constructing Tangents Geometry Station Activities for Common Core State Standards Pages 151165 Problem Based Tasks for Math II (orange book) Chord Central Angles Conjecture Masking the Problem pg. 268 Practice for chords and circles Practice for tangents and circles Practice for special segments in circles Problem Based Tasks for Math II (orange book) Properties of Tangents of a Circle The Circus is in Town! Is it Safe? pg. 271 G-C.2 G-C.2 Website for special segments in circles (Sketchpad) Website for special segments are two intersecting lines and a circle ADDITIONAL UNIT RESOURCES Explanation of standards in friendly language with example problems Benson Workshop Problems on District Teacher Files Clock Problem (pg. 9-10) Circle-Angle #45 (pg. 27) Circles #80-87 (pgs. 46-51) Radius #90 (pg. 52) Circumference #91 (pg. 53) Concentric Circles and Circumference #93 (pg. 54) Tangent line and circles on coordinate plane #137 (pg. 79) COURSE: Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Probability 3 weeks ESSENTIAL QUESTION(S): What is a sample space and how do you represent it? When do you use permutations and combinations with probability? What does it mean to be independent, dependent, and mutually exclusive? REFERENCE/ STANDARD # S-CP.1 S-CP.2 S-CP.3 S-CP.4 S-CP.5 S-CP.6 S-CP.7 S-CP.8 S-CP.9 S-MD.6 S-MD.7 WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? STANDARDS: A listing of all standards included in the unit Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,””not”). Understand that two events A and B are independent of the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model. Use permutations and combinations to compute probabilities of compound events and solve problems. Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). MAJOR SUPPORTING STANDARD STANDARD (M) (S) X X X X X X X X X X X UNIT DESCRIPTION: UNIT VOCABULARY In this unit students will use conditional probability, represent sample space, use permutations and combinations, and find probabilities of compound events. Sample space, complement, union, intersection, tree diagram, permutation, combination, independent events, dependent events, conditional probability, mutually exclusive, classical probability, empirical probability, frequency table HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # TEACHER INSTRUCTION STUDENT LEARNING S-CP.1 Introduce sample space, outcome, classical/empirical with deck of cards, coins, dice, skittles, spinners Empirical probability: Activites with concrete maniuplatives Spinner Activity Probability and Data Analysis Activities Lesson over sample space Problem Based Tasks for Math II (orange book) Describing Events pg. 325 S-CP.2 Explanation of standard Lesson about Independence Problem Based Tasks for Math II (orange book) Understanding Independent Events pg. 330 Worksheet Titanic Problem HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS S-CP.3, 5, 6 Conditional Probability Lesson Problem Based Tasks for Math II (orange book) Introducing Conditional Probability pg. 334 S-CP.4, 5, 6 Addition Rule lesson Problem Based Tasks for Math II (orange book) Using Two-Way Frequency Tables pg. 337 Two-way table lesson Two-way table worksheet S-CP. 6 Using probability to make fair decisions Resource for teachers Conditional probability demonstrated Problem Based Tasks for Math II (orange book) Making Decisions pg. 358 Worksheets for fair decisions Interactive Activities for students S-CP.7 Lesson Decision Trees Help with Addition Rule S-CP.7 Additional lesson on decision trees Lesson on Addition Rule Problem Based Tasks for Math II (orange book) Analyzing Decisions pg. 362 Problem Based Tasks for Math II (orange book) The Addition Rule pg. 341 S-CP.8 Video explaining Multiplication Rule Problem Based Tasks for Math II (orange book) The Multiplication Rule pg. 345 Explanation of Multiplication Rule S-CP.9 Lesson over Permutations and Combinations Problem Based Tasks for Math II (orange book) Combinations and Permutations pg. 350 Permutations and Combinations Student Resource ADDITIONAL UNIT RESOURCES Glencoe Geometry textbook sections: 0-3, 13-1, 13-2, 13-4, 13-5, 13-6 Math is Fun website – explains concepts pretty basic COURSE: Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Constructions Involving Circles 2 weeks ESSENTIAL QUESTION(S): How are constructions used to inscribe polygons in a circle or circumscribe polygons about a circle? What properties can be proved about angles of quadrilaterals inscribed in a circle? REFERENCE/ STANDARD # G-CO.13 G-C.3 G-C.4 MA-HG-C.02 WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? STANDARDS: A listing of all standards included in the unit Construct an equilateral triangle, a square, and a regular hexagon. Construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle. Construct a tangent line from a point outside a given circle to the circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. MAJOR SUPPORTING STANDARD STANDARD (M) (S) UNIT DESCRIPTION: UNIT VOCABULARY Students will construct equilateral triangles, squares, and regular Circumscribe, Inscribe, Equilateral, Regular, Tangent line hexagons inscribed in a circle. Students will also construct inscribed and circumscribed circles of triangles. Properties of angles for quadrilaterals inscribed in a circle will also be discovered. Geogebra will be used throughout the unit. HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # TEACHER INSTRUCTION STUDENT LEARNING HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS G-CO.13 Interactive lesson on inscribed equilateral triangle construction. McDougal Littell Pg 395 #9 Video on inscribed square construction. McDougal Littell Pg 395 #10 Video on inscribed regular hexagon construction. G-C.3 Interactive lesson on inscribing a circle in a triangle. Interactive lesson on circumscribing a circle on a triangle. Video showing opposite angles of a inscribed quadrilateral are supplementary. G-C.4 Interactive lesson on constructing a tangent line from a point not on the circle. ADDITIONAL UNIT RESOURCES McDougal Littell Pg 395 #6-8 McDougal Littell Pg 395 #3-5 McDougal Littell Pg 342 #13 McDougal Littell Pg 331 #6, 7, 9, 11 McDougal Littell Pg 395 #2
