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2015 Algebraic Geometry Grade Level/Course: Algebraic Geometry Content Area: Mathematics Grade Level/Course Overview: The fundamental purpose of this course is to formalize and extend students’ geometric experiences from the earlier grades. Students will explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. They will prove basic theorems and solve problems about triangles, quadrilaterals, and other polygons; establish triangle congruence criteria based on analyses of rigid motions; apply similarity in right triangles to understand right triangle trigonometry; use formulas to find the volume of three-dimensional objects; and use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities. Strands/Domains 1. Geometry a. Congruence b. Similarity, Right Triangles, and Trigonometry c. Geometric Measurement and Dimension d. 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- (extension topic) 10. Circles – Part 2 – (extension topic) 11. Probability 12. Constructions Involving Circles – (extension topic) Interdisciplinary Themes 1. 2. 3. 4. 5. Patterns Cause and Effect Scale, Proportion, and Quantity Systems and Systems Models Structure and Function 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Experiment with transformations in the plane Standard: 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. Standard Code: G-CO.1. DOK Target for this standard: 1=Recall 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Rotations / reflections and translations are based on the notions of point, line, distance along a line and distance around circular arc. Know? Define: • • • • • • • • angle circle perpendicular parallel line segment point line arc Be able to do? • name an angle, line, line segment, ray, arc, 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Experiment with transformations in the plane Standard: 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). Standard Code: G-CO.2. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Rotations / reflections and translations are based on the notions of point, line, distance along a line and distance around circular arc. Know? • • • • • • • • • • rotations reflections triangle rectangle parallelogram trapezoid regular polygon symmetry dilation translation Be able to do? • identify translations, rotations and reflections in real world situations • identify dilations, translations, rotations and reflections of triangles, rectangles, parallelograms, trapezoids or regular polygons in the coordinate plane • plot a transformation given a set of points to be translated • compare transformations that preserve size/length to those that do not 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Experiment with transformations in the plane Standard: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Standard Code: G-CO.3. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • What should students… Understand? Rotations / reflections and translations are based on the notions of point, line, distance along a line and distance around circular arc. Know? different ways to do transformations in a plane functions Be able to do? • describe rotations and reflections that map a polygon onto itself • identify lines/axes of symmetry 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Experiment with transformations in the plane Standard: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Standard Code: G-CO.4. DOK Target for this standard: 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • • • • • • What should students… Understand? Rotations / reflections and translations are based on the notions of point, line, distance along a line and distance around circular arc. Know? rigid transformations rotations reflections translation angles circles perpendicular lines parallel lines line segment Be able to do? • create definitions of rotations, reflections, and translations as rigid transformations • visualize and identify rotations, reflections, and translations that map a preimage to an image • understand properties that are preserved in rotations, reflections and translations 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Experiment with transformations in the plane Standard: 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. Standard Code: G-CO.5. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • What should students… Understand? Rotations / reflections and translations are based on the notions of point, line, distance along a line and distance around circular arc. Know? rigid transformations rotations translations reflections Be able to do? • draw rotations / reflections / translations of a geometric figure using manipulatives • recognize and draw compositions of transformations including mapping onto itself • identify rotation / reflections / translations on a coordinate plane • rotate / reflect / translate / given figures in the coordinate plane 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Understand congruence in terms of rigid motions Standard: 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. Standard Code: G-CO.6. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • What should students… Understand? Rigid motions and their properties can be used to establish the triangle congruence criteria, which can then be used to prove other theorems. Know? rigid motion congruence transformation tessellation Be able to do? • determine if two figures are congruent • determine the effect of a given rigid motion • transformation figures using geometric descriptions of rigid motion • use transformations to create patterns including tessellations • identify figures that tessellate 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Understand congruence in terms of rigid motions Standard: 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. Standard Code: G-CO.7. DOK Target for this standard: 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? 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. Know? • • • • congruence angles rigid motion corresponding angles Be able to do? • verify two triangles are congruent • show that the triangles are congruent given triangles that have been transformed 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Understand congruence in terms of rigid motions Standard: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Standard Code: G-CO.8. DOK Target for this standard: 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • • What should students… Understand? 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. Know? ASA SAS SSS congruence distance formula Be able to do? • use the definitions of congruence, based on rigid motion, to develop and explain the triangle congruence criteria • complete proofs involving ASA, SAS, SSS 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Prove geometric theorems Standard: Prove theorems about lines and Standard Code: G-CO.9. 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. DOK Target for this standard: 3=Strategic 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking Thinking LEARNING TARGETS • • • • • • • • • • What should students… Understand? 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. Know? vertical angles alternate interior angles corresponding angles transversal parallel lines perpendicular perpendicular bisector equidistance segment Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Prove geometric theorems Standard: 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. Standard Code: G-CO.10. DOK Target for this standard: 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • What should students… Understand? 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. Know? isosceles triangle midpoint median triangle Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Prove geometric theorems Standard: 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. Standard Code: G-CO.11. DOK Target for this standard: 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? 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. Know? • • • • • • congruent angles parallelogram bisector rectangle diagonals Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Make geometric constructions Standard: 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. Standard Code: G-CO.12. DOK Target for this standard: 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Geometric constructions can be created using a variety of tools. • • • • • Know? segment angle bisector perpendicular parallel Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Conditional Statements Standard: Write and conceptually understand the structure of conditional statements and their converses. Standard Code: MA.G.CO.01 DOK Target for this standard: 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Conditional statements and their converses may have different truth values. • • • Know? conditional statements If-then truth value Be able to do? • write a statement in conditional form • write the converse of a conditional statement • determine the truth value for a conditional statement and its converse 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Prove geometric theorems. Standard: Derive and apply the Segment Addition and Angle Addition Postulates. DOK Target for this standard: 1=Recall 2=Skill/Concept Standard Code: MA.G.CO.02 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Adding collinear segments and adjacent angles create new segments and angles. Know? • • • • • • segments collinear angles adjacent postulate solving equations Be able to do? • derive the Segment Addition and Angle Addition Postulates • solve for lengths of segments and measures of angles using these postulates 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Congruence Cluster: Understand congruence in terms of rigid motions Standard: 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. Standard Code: MA.G.CO.03 DOK Target for this standard: 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • What should students… Understand? Corresponding parts of congruent triangles can be used to prove a second pair of triangles are congruent. Know? ways to prove two triangles are congruent (SSS, SAS, ASA, AAS, HL) corresponding parts of congruent triangles are congruent Be able to do? • identify which triangles need to be proven congruent first to show two other triangle are congruent • prove two triangles are congruent and use CPCTC to prove two other triangles congruent 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Understand similarity in terms of similarity transformations Standard: Verify experimentally the properties of dilations given by a center and a scale factor: a. 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. Standard Code: Standard: G-SRT.1. DOK Target for this standard: 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • • • • What should students… Understand? A dilation is an enlargement or reduction of a pre-image through a center point. 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 Be able to do? • • • • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Understand similarity in terms of similarity transformations Standard: 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. Standard Code: G-SRT.2. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • What should students… Understand? Similar figures have congruent corresponding sides and proportional sides. Triangles can be similar by various theorems. Know? definition of similar definition of proportions definition of corresponding parts of changes Be able to do? • 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 ~ 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Understand similarity in terms of similarity transformations Standard: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Standard Code: G-SRT.3. DOK Target for this standard: 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • What should students… Understand? Two pairs of congruent angles are sufficient to prove two triangles are similar. (AA) 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 Be able to do? • show that the triangles are similar given two pairs of congruent angles in two triangles • derive the Third Angles Theorem 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Prove theorems involving similarity Standard: 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. Standard Code: G-SRT.4. DOK Target for this standard: 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • What should students… Understand? Similarity is used to prove theorems about triangles. Know? properties of proportions recognize the 3 similar triangles when an altitude is drawn from the right angle of a right triangle Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Prove theorems involving similarity Standard: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Standard Code: G-SRT.5. DOK Target for this standard: 1=Recall 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Non-triangular geometric figures can be shown to be congruent or similar in the same way triangles are. Know? • congruent figures are similar figures with a scale factor of 1 Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles Standard: 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. Standard Code: G-SRT.6. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • What should students… Understand? Similar right triangles are used to generate ratios between sides, leading to trigonmetric functions. Know? similar triangles right triangles ratio proportion Be able to do? • 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 trigonmetric ratios (sine, cosine and tangent) • discover the relationships in special right triangles 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles Standard: Explain and use the relationship between the sine and cosine of complementary angles. Standard Code: G-SRT.7. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? The sine and cosine of complementary angles are equivalent. • • • Know? Sine Cosine complementary Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles Standard: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Standard Code: G-SRT.8. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • • • • • • What should students… Understand? Six parts of right triangles are interdependent. All missing parts of a right triangle can be found using trigonometric ratios and/or Pythagorean Theorem. Know? right triangles SOH CAH TOA Pythagorean theorem square roots inverse trigonometry opposite and adjacent legs hypotenuse angle of elevation and angle of depression Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Apply trigonometry to general triangles Standard: (+) 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. Standard Code: G-SRT.9. DOK Target for this standard: 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? The area of oblique (non-right) triangles can be found by A=1/2 ab sin C. • • • Know? definition of oblique triangles Sine formula for the area of a triangle Be able to do? • 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 C A h a c B 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Apply trigonometry to general triangles Standard: (+) Prove the Laws of Sines and Cosines and use them to solve problems. Standard Code: G-SRT.10. DOK Target for this standard: 1=Recall 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? The Law of Sines and Law of Cosines are used to find missing pieces of oblique (non-right) triangles. Know? • • • Sine Cosine when to use Law of Sines vs. Law of Cosines vs. SOH CAH TOA Be able to do? • prove the Law of Sines and Law of Cosines using: b A h C c B a • use Law of Sines and Law of Cosines to solve oblique triangles 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Similarity, Right Triangles, and Trigonometry Cluster: Apply trigonometry to general triangles Standard: (+) 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). DOK Target for this standard: 1=Recall 2=Skill/Concept Standard Code: G-SRT.11. 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? The Law of Sines and Cosines can be used in applied problems to find missing sides and angles of any type of triangle. Know? • • • Law of Sines Law of Cosines when to use the Law of Sines vs. the Law of Cosines Be able to do? • use Law of Sines and Cosines to find unknown measures of right and oblique triangles in real-world problems 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Circles Cluster: Understand and apply theorems about circles Standard: Prove that all circles are similar. Standard Code: G-C.1. DOK Target for this standard: 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Circles are similar and therefore, useful ratios are created. • • • • • • • • Know? formula(s) for circumference radius diameter circle circumference similarity ratio proportions Be able to do? • find ratio of similarity using circumference/diameter and identify that the ratio is 𝜋 • use similarity ratios to find missing information 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Circles Cluster: Understand and apply theorems about circles Standard: 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. Standard Code: G-C.2. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Segments drawn in circles create relationships between arcs and angles. • • • • • • • • Know? chords tangent arc measure inscribed angle central angle diameter secant arc length Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Circles Cluster: Find arc lengths and areas of sectors of circles Standard: 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. Standard Code: G-C.5. DOK Target for this standard: 1=Recall 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Circles are similar, and therefore, useful ratios are created. • • • • Know? circumference of a circle area of a circle definition of an arc/arc length definition of a sector Be able to do? • 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) 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Expressing Geometric Properties with Equations Cluster: Translate between the geometric description and Standard: 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. DOK Target for this standard: 1=Recall 2=Skill/Concept 3=Strategic Thinking Standard Code: G-GPE.1. the equation for a conic section 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • • What should students… Understand? 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. Know? Be able to do? distance formula • derive the equation of a circle given center and radius using Pythagorean Theorem Pythagorean Theorem (distance formula) properties of radicals • complete the square to find the center completing the square and radius of a circle factoring • manipulate the equations of circles from vertex to standard form 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Expressing Geometric Properties with Equations Cluster: Translate between the geometric description and the equation for a conic section Standard: Derive the equation of a parabola given a focus and directrix. Standard Code: G-GPE.2. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Given a focus and directrix they can derive the equation of a parabola. • • • • Know? distance formula standard form vertex form FOIL Be able to do? • derive the equation of a parabola given a focus and directrix • find the equation of a parabola given a focus and directrix 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically Standard: 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). Standard Code: G-GPE.4. DOK Target for this standard: 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Algebra can be applied to geometric proofs. 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 Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically Standard: 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). Standard Code: G-GPE.5. DOK Target for this standard: 1=Recall 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? The slope criteria for parallel and perpendicular lines can be used to solve geometric problems. • • • • • • • • Know? slope parallel lines perpendicular lines slope-intercept form point-slope form perpendicular bisector altitude midpoint Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically Standard: Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Standard Code: G-GPE.6. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Line segments can be partitioned proportionally. Know? • • • • • • • ratio distance formula midpoint formula proportion endpoint line segment triangle proportionality theorem Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically Standard: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. DOK Target for this standard: 1=Recall 2=Skill/Concept Standard Code: G-GPE.7. 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • • What should students… Understand? The area or perimeter of a figure can be found by applying geometric concepts to points on a coordinate plane. Know? distance formula area formula perimeter simplify radicals identify polygons Be able to do? • • • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Geometric Measurement and Dimension Cluster: Explain volume formulas and use them to solve problems Standard: 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. Standard Code: G-GMD.1. DOK Target for this standard: 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Perimeter, area and volume of two dimensional and three dimensional shapes can be derived. • • Know? volume formulas area formulas Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Geometric Measurement and Dimension Cluster: Explain volume formulas and use them to solve problems Standard: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. DOK Target for this standard: 1=Recall 2=Skill/Concept Standard Code: G-GMD.3. 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • What should students… Understand? Volume formulas are useful for solving real-world problems. Know? volume formulas for: - cylinders - pyramids - spheres - cones area formulas for: rectangles circles triangles Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Geometric Measurement and Dimension Cluster: Visual relationships between twodimensional and three-dimensional objects Standard: Identify the shapes of twodimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Standard Code: G-GMD.4. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? There is a relationship between two and three dimensional shapes. • • • • Know? names of the 2-D shapes names of the 3-D shapes definition of cross-section definition of rotation Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Modeling with Geometry Cluster: Apply geometric concepts in modeling situations Standard: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Standard Code: G-MG.1. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • What should students… Understand? Real life objects can be modeled using two dimensional and three dimensional geometric shapes. Know? Be able to do? two dimensional shapes properties • recognize two dimensional and three dimensional shapes in real life situations three dimensional shapes properties • create three dimensional objects and discuss their properties 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Modeling with Geometry Cluster: Apply geometric concepts in modeling situations Standard: Standard Code: DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS What should students… Understand? • Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTU’s per cubic foot). Know? Be able to do? • formulas for calculating area • determine which formula(s) should be used in a given situation • formulas for calculating volume • draw 2-dimensional and 3-dimensional • appropriate units of measurement for figures that model a given situation specific quantities • solve application problems that require • 2-D and 3-D shapes finding area and volume • proportions • density • unit analysis 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Modeling with Geometry Cluster: Apply geometric concepts in modeling situations Standard Code: G-MG.3. Standard: 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). DOK Target for this standard: 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Real life objects can be modeled using 2-D and 3-D geometric shapes. Know? • • 2-D and 3-D shapes and their properties formulas for area, surface area and volume of 2-D and 3-D shapes Be able to do? • • • • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence and conditional probability and use them to interpret data Standard: 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”). Standard Code: S-CP.1. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Independence and conditional probability can be used to interpret data. 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 Be able to do? • • identify sample space and events within a sample space identify subsets from within the sample space 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence and conditional probability and use them to interpret data Standard: 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. Standard Code: S-CP.2. DOK Target for this standard: 1=Recall 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Independence and conditional probability can be used to interpret data. Know? Definitions • independent events • conditional probability Be able to do? • determine if two events are independent 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence and conditional probability and use them to interpret data Standard: 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. Standard Code: S-CP.3. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Independence and conditional probability can be used to interpret data. Know? • multiplication principle Be able to do? • use the multiplication principle to calculate conditional probabilities 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence and conditional probability and use them to interpret data Standard: 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. Standard Code: S-CP.4. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS What should students… Understand? • Independence and conditional probability can be used to interpret data. • • • • Know? sample space two-way table conditional probability independent events Be able to do? • 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 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence and conditional probability and use them to interpret data Standard: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Standard Code: S-CP.5. DOK Target for this standard: 1=Recall 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Independence and conditional probability can be used to interpret data. • • Know? independent events conditional probability Be able to do? • recognize and explain the concepts of conditional probability and independence in a real-life setting 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model Standard: 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. Standard Code: S-CP.6. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Independence and conditional probability can be used to interpret data. Know? • conditional probability formula Be able to do? • calculate and interpret conditional probability of A given B 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model Standard: 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. Standard Code: S-CP.7. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Different probability formulas can use be used to calculate and interpret real world phenomena. Know? Definitions • disjoint events • mutually exclusive events addition rule of probability Be able to do? • calculate and interpret probabilities using the addition rule 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model Standard: (+) 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. Standard Code: S-CP.8. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Different probability formulas can use be used to calculate and interpret real world phenomena. • • • Know? multiplication rule of probability conditional probability independent events Be able to do? • calculate and interpret a probability using the multiplication rule 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Conditional Probability and the Rules of Probability Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model Standard: (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Standard Code: S-CP.9. DOK Target for this standard: 1=Recall 2=Skill/Concept 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • What should students… Understand? Different probability formulas can use be used to calculate and interpret real world phenomena. Know? Definitions • factorials • combination • permutation Formulas to calculate probabilities of a • combination • permutation Be able to do? • determine the difference between a permutation and a combination • calculate probabilities using the appropriate permutation or combination formula 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Using Probability to Make Decisions Cluster: Use probability to evaluate outcomes of decisions Standard: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). DOK Target for this standard: 1=Recall 2=Skill/Concept Standard Code: S-MD.6. 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • What should students… Understand? Probabilities exhibit relationships that can be extended, described, and generalized to make decisions. Know? definition of random how to use a random number generator Be able to do? • understand factors that make decisions fair and random o toss a die o flip a coin o use a spinner 2015 Algebraic Geometry UNPACKING THE STANDARDS Course: Geometry Standards Used: MLS Strand/Domain: Using Probability to Make Decisions Cluster: Use probability to evaluate outcomes of decisions Standard: (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Standard Code: S-MD.7. DOK Target for this standard: Recall 2=Skill/Concept 3=Strategic Thinking 1=Recall 2=Skill/Concept 3=Strategic Thinking 4=Extended Thinking LEARNING TARGETS • • • • • What should students… Understand? Probabilities exhibit relationships that can be extended, described, and generalized to make decisions. Know? multiplication rule addition rule permutations combinations Be able to do? • 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 2015 Algebraic Geometry COURSE: Geometry UNIT TITLE: Introduction to Geometry and Basic Constructions SUGGESTED UNIT TIMELINE: 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 STANDARDS: WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? 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. UNIT DESCRIPTION: 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. UNIT VOCABULARY SUPPORTING STANDARD (S) 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 UNIT SCORING GUIDE MAJOR STANDARD (M) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. 2015 Algebraic Geometry FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # G-CO.1 TEACHER INSTRUCTION Use constructions to develop formal definitions of midpoint, angle bisector, segment bisector, perpendicular bisector and congruence STUDENT LEARNING 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 Practice Worksheets 9.1-9.6, Glencoe Book Rotation Demonstration (Geogebra) Reflection Demonstration Activity Reflection Demonstration (y=x) (Geogebra) Reflection Demonstration (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. Rotational symmetry activity for students HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS 2015 Algebraic Geometry Discover how a reflection or series of reflections can map a figure onto itself. 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 G-CO.5 G-CO.12 New York Curriculum Lesson 18 (p. 131) Discovering parallel lines using reflection Compositions of Transformations Use technology (Geogebra, SMART Notebook, Core Math Tools, etc.) to contstruct Construction Tutorials Reflection discovery activity for students Video introducing transformations 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 2015 Algebraic Geometry Transformations Review Packet Website with additional resources (under construction) DISCOVERY EDUCATION RESOURCES STANDARD: G.CO.1 Video Math Explanation Model Lesson STANDARD: G.CO.2 Video Math Overview 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. Circles: Formulas and Definitions Defining Circles Defining Circumference, Radius and Diameter Circular Structures: Design and Architecture Geometry: Introduction to Angles: Definitions Geometry: Angle Relationships: Definitions Geometry: Points, Lines, and Planes: Definitions/Examples Geometry: Points, Lines, and Planes: A Line Perpendicular to a Plane Geometry: Congruence, Symmetry, and Transformations 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). Example 2: Rotation and Translation -- The London Eye and Boating Example 3: Reflection -- Beach Flag Example 3: Geometric Functions -- Art Computer Coordinates: Playing with Pixels Geometry: Using Matrices to Perform Transformations Geometry: Translations and Glide Reflections Geometry: Reflections 2015 Algebraic Geometry Math Explanation Geometry: Translations and Glide Reflections: Coordinate Notation Model Lesson Geometry: Congruence, Symmetry, and Transformations Congruence and Proof 2015 Algebraic Geometry STANDARD: G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Video Symmetry and Transformations Reflections Beach Bag Math Explanation Geometry: Reflections: Angles of Rotation Geometry: Reflections: Lines and Points of Symmetry Math Overview Geometry: Reflections Model Lesson Geometry: Congruence, Symmetry, and Transformations Activity Symmetry and Transformations STANDARD: G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Video Geometric Transformations Math Explanation Geometry: Translations and Glide Reflections: Invariant Points Geometry: Congruent Triangles and Congruence Transformations: Transformations Math Overview Congruent Triangles and Congruence Transformations Geometry: Reflections Model Lesson Geometry: Congruence, Symmetry, and Transformations 2015 Algebraic Geometry 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 Video Example 1: Translations Example 2: Rotations Math Explanation Geometry: Translations and Glide Reflections: Graphing Translations of Segments Geometry: Translations and Glide Reflections: Sketching Glide Reflections Geometry: Reflections: Coordinate Plane Double Reflections Geometry: Reflections: Double Reflections, Part Two Model Lesson Geometry: Congruence, Symmetry, and Transformations Activity Algebra Rules and Combinations STANDARD: G.CO.12 Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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. Video Example 2: Constructing an Angle Bisector Example 3: Constructing a Perpendicular Bisector Example 2: Drawing a Perpendicular Line -- Planning a Wall Example 3: Bisecting an Angle -- American Indian Museum Math Overview Geometry: Constructions: Parallel and Perpendicular Lines Math Explanation Geometry: Segment Length and Precision: Constructing a Bisector 2015 Algebraic Geometry COURSE: 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 STANDARDS: WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? 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. UNIT DESCRIPTION: 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. MAJOR SUPPORTING STANDARD STANDARD (M) (S) UNIT VOCABULARY 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 # G-CO.9 TEACHER INSTRUCTION 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 2015 Algebraic Geometry 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 Perpendicular bisector practice Glencoe book pg 327 Parallel lines and angle relationship Parallel lines task 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/ 2015 Algebraic Geometry DISCOVERY EDUCATION RESOURCES 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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. Video Section B: Angles and Their Theorems Section C: Parallel Lines and Angles Math Overview Geometry: Introduction to Proofs Geometry: Proving Two Lines Are Parallel Math Explanation Geometry: Proving Two Lines Are Parallel: Proving Lines Are Parallel Geometry: Inscribed Angles: Proofs Model Lesson Activity Congruence and Proof Congruence Theorems 2015 Algebraic Geometry COURSE: Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Triangle Congruence ESSENTIAL QUESTION(S): 3 weeks What processes are valid to prove two triangles are congruent? What can you conclude about two triangles that are congruent? WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? REFERENCE/ STANDARDS: STANDARD 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 G-CO.6 G-CO.7 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. G-CO.8 UNIT DESCRIPTION: Students will be able to determine if and prove that two triangles are congruent. UNIT VOCABULARY MAJOR SUPPORTING STANDARD STANDARD (M) (S) Rigid motion, corresponding parts, ASA, SAS,SSS,ASA,SAS, SSS, HL, CPCTC 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.6 G-CO.7 TEACHER INSTRUCTION Page 232-307 Glencoe Geometry Textbook Section 4.3 – Identify congruent figures and name by corresponding parts STUDENT LEARNING Worksheet – Triangle Angle Sum Triangle Congruence Skills Checks – Page 2 Video Showing Triangle Angle Worksheet – Sum Theorem – cutting angles Corresponding parts to form a line Multiple Choice Questions Show Triangles are Activity to Discover Congruent – Also links to SSS, Triangle Congruences – HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS Best Strategies by Benson – p. 27 #46 2015 Algebraic Geometry SAS, AAS, ASA, HL 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 BensonProofs on p. 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 Best Strategies by Benson – p. 9 #12 2015 Algebraic Geometry DISCOVERY EDUCATION RESOURCES 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. Math Overview Geometry: Congruent Triangles and Congruence Transformations Geometry: Space Figures and Drawings Math Explanation Geometry: Translations and Glide Reflections: Isometries Geometry: Congruent Triangles and Congruence Transformations: Identifying Congruent Triangles Model Lesson STANDARD: G.CO.7 Math Explanation Model Lesson STANDARD: G.CO.8 Congruence and 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. Geometry: Congruent Triangles and Congruence Transformations: Naming Congruent Sides and Angles Geometry: Congruent Triangles and Congruence Transformations: Naming Congruent Angles and Sides Geometry: Congruent Triangles and Congruence Transformations: Finding Congruent Triangles in Patterns Congruence and Proof Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions. Math Explanation Geometry: Congruent Triangles and Congruence Transformations: Transformations Math Overview Geometry Proving Angles are Congruent Geometry Proving Angles are Congruent 2015 Algebraic Geometry Model Lesson Congruence and Proof 2015 Algebraic Geometry COURSE: Geometry UNIT TITLE: Triangles and Coordinate Proofs ESSENTIAL QUESTION(S): SUGGESTED UNIT TIMELINE: 2 weeks 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: STANDARD A listing of all standards included in the unit # G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°, base angles of isosceles triangles G-GPE.4 G-GPE.5 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: The student will define midsegment, median, centroid, perpendicular and angle bisectors, and altitude of triangles. The student will apply and prove properties of these parts of triangles. Students will use the coordinate plane to complete proofs. MAJOR SUPPORTING STANDARD STANDARD (M) (S) UNIT VOCABULARY Altitude, Angle bisector, Centroid, Equilateral, Isosceles, Median, Midsegment, Parallel, Perpendicular, Perpendicular bisector, Scalene 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 HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? 2015 Algebraic Geometry 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 INTERVENTIONS EXTENSIONS -Calibrating Consoles (Problem-Based 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 283291 section 4-6 G-CO.10 Define and apply midsegment, median, centroid, perpendicular and angle bisectors, and altitudes of triangles. -Finding Centroid and Orthocenter -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) -Median of a Triangle Notes -Orthocenters and Altitudes (Best Strategies by Benson, #156) -Concurrent Medians Construction In Glencoe text on page 322291 sections 5-1 and 5-2 G-GPE.5 -Finding Medians (Best Strategies by Benson, #149) Review slope-intercept form, point-slope form, perpendicular bisector, altitude, and midpoint. In Glencoe text on page 196204 sections 3-4 -Finding Bisectors, Medians and Altitudes (Geometry Stations, Pg 50-54) -9 Point Circle Project -Review distance with triangles (Best Strategies by Benson, #69) -Review distance in coordinate plane (Best Strategies by Benson, #70) 2015 Algebraic Geometry Introduce slopes of parallel and perpendicular lines. In Glencoe text on page 186195 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. -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 Applications using midsegment, median, centroid, perpendicular and angle bisectors, and altitude. -Sailing Centroid (Problem-Based Tasks: Math II, Pg 189) -Median Application Problems -Median Application Problems G-CO.10 Coordinate proofs involving midsegment, median, centroid, perpendicular and -Coordinate Proofs -Constellation Coordinate Proof 2015 Algebraic Geometry angle bisectors, and altitude. -Coordinate Proofs G-GPE.4 Coordinate proofs to determine if a triangle is isosceles, equilateral, scalene or right using distance. -Coordinate Proofs -Right Triangle Proofs (#2 and #6) -Classify Triangle by Sides -Coordinate Proofs -Prove a triangle is isosceles ADDITIONAL UNIT RESOURCES DISCOVERY EDUCATION RESOURCES STANDARD: G.CO.10 Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180o, 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, and the medians of a triangle meet at a point. Video Section B: Proving Triangles Congruent The Angles and Their Degrees Example 2: Geometric Proof -- Hawaii Section A: Proving the Similarity of Triangles Math Overview Geometry: Proving Triangles Are Congruent: SSS and SAS Postulates Geometry: Proving Triangles Are Congruent: ASA Postulate and the AAS Theorem Math Explanation Geometry: Introduction to Proofs: Developing Proofs Geometry: Introduction to Proofs: Developing Proofs Geometry: Introduction to Proofs: Flow Proofs Model Lesson Congruence and Proof 2015 Algebraic Geometry Activity STANDARD: G.GPE.4 Video Proving Theorems about Triangles Use coordinates to prove simple geometric theorems algebraically. 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). Geometric Interpretation of the Theorem of Pythagoras Introduction: Washington D.C. and the Coordinate Plane Example 1: The Cartesian Coordinate System -- DC Map Example 2: Slopes and Relationships of Lines -- DC Monuments Example 3: Distances and Midpoints -- DC Tourists The Distance Formula & the Midpoint Formula Math Overview Geometry: Midpoints and Distance Math Explanation Algebra I: The Distance Formula: Understanding the Relationship Between the Distance Formula and the Pythagorean Theorem Geometry: Midpoints and Distance: Finding Length of a Segment Part 2 Model Lesson Coordinate Geometry and How It's Used Activity You've Got a Point 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). 2015 Algebraic Geometry Video Section C: Properties of Linear Graphs Properties of Parallel Lines Math Overview Geometry: The Slope of Parallel and Perpendicular Lines Math Explanation Geometry: Writing Equations of Lines: Parallel & Perpendicular Lines Geometry: The Slope of Parallel and Perpendicular Lines: Writing Equations for a Perpendicular Line Geometry: The Slope of Parallel and Perpendicular Lines: Checking For Perpendicular Lines - 1 Model Lesson Coordinate Geometry and How It's Used Activity Parallel and Perpendicular Lines 2015 Algebraic Geometry COURSE: Geometry UNIT TITLE: Similarity ESSENTIAL QUESTION(S): SUGGESTED UNIT TIMELINE: 3 weeks 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 when solving problems about similar figure? WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? REFERENCE/ STANDARDS: STANDARD A listing of all standards included in the unit # Verify experimentally the properties of dilations given by a center and a scale factor: G-SRT.1 a. 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: Students will identify and apply similarity properties. UNIT VOCABULARY 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 • • • • MAJOR SUPPORTING STANDARD STANDARD (M) (S) 2015 Algebraic Geometry 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 G-SRT.2 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 Book - Geometry Station Activities Book pg. 112 G-SRT.3 G-SRT.5 Worksheet pages 27-29 G-SRT.4 Worksheet pages 34-36 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 Task Problems - CCSS Problem Based Tasks for Mathematics II – Down, Down, Down pg. 226 Suddenly Sinking pg. 230 Geometry Station HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS 2015 Algebraic Geometry 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 DISCOVERY EDUCATION RESOURCES STANDARD: G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor. a. 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. Video Example 3: Dilations Similar Figures: Scaled Down Geometry: Dilations Math Overview Geometry: Dilations Geometry: Understanding Similar Polygons Math Explanation Geometry: Dilations: Origin Centered Dilations Geometry: Dilations: Scale Factors 2015 Algebraic Geometry STANDARD: G.SRT.2 Video Math Explanation STANDARD: G.SRT.3 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. Using Proportions to Create Similar Figures Congruent and Similarity Transformations Similar Figures: Scaled Down Geometry: Properties of Similar Trangles Geometry: Determining and Using Similiar Triangles: Finding Similiar Triangles in a Figure Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. Video Part Two: Special Triangles Part Two: Special Triangles (continued from Program Four) Math Overview Geometry: Properties of Similar Triangles Geometry: Determining and Using Similar Triangles Math Explanation Geometry: Determining and Using Similar Triangles: Finding Similar Triangles in a Figure Model Lesson STANDARD: G.SRT.5 Math Explanation Model Lesson Building Bridges with Similar Triangles Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures Geometry: Dilations: Dilation Proofs Geometry: Building Bridges with Trigonometry Building Bridges with Similar Triangles 2015 Algebraic Geometry 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; and the Pythagorean Theorem proved using triangle similarity. Math Overview Math Explanation Geometry: Congruent Right Triangles Geometry: Determining and Using Similar Triangles: Proving with Similar Triangles Geometry: Determining and Using Similar Triangles: Proofs 2015 Algebraic Geometry COURSE: Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: 2 ½ 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 STANDARDS: WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? 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. UNIT DESCRIPTION: Students will find missing sides and angles of a triangle using trigonometry. • • • • • 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. UNIT VOCABULARY 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 HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT UNIT SCORING GUIDE (link) MAJOR SUPPORTING STANDARD STANDARD (M) (S) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. 2015 Algebraic Geometry FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # G-SRT.6 TEACHER INSTRUCTION STUDENT LEARNING Instructional strategies, 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 45-45-90 Triangle 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 G-SRT.8 G-SRT.10 Worksheet – Special Triangles (Answer Key) Worksheet – Special Right Triangles (Answer Key) Exit Slip – Special Right Triangles (Answer Key) Special Right Triangles Problems Geometry Station Activity for Common Core pgs. 139-150 Sine and Cosine of Complementary Angles Inverse Function Notes Complimentary Angles Activity Inverse Function WS Angle of Elevation and Depression Prove Law of Sines and Law of Cosines Angle of Elevation and Depression WS Glencoe Secondary Math Aligned to the CC Pgs. 12-16 #1-6, 8-20, 22-27, 3142, 47-50 Glencoe Secondary Math Aligned to the CC Pg. 15 #45 and #46 Law of Sines Problems Law of Cosines Problems HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS 2015 Algebraic Geometry G-SRT.11 Glencoe 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 Glencoe Secondary Math Aligned to the CC Pgs. 12-16 #7, 21, 28, 29, 30, 4344, 51-53 Apply formula to find area of oblique triangles DISCOVERY EDUCATION RESOURCES STANDARD: G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* Video Example 2: Pythagorean Theorem Proof -- History Example 3: Using the Pythagorean Theorem -- Carpentry Example 1: Pythagorean Theorem -- Video Game Design Example 2: Sine, Cosine, Tangent -- Firefighting Example 3: A Small Angle -- Flying Introducing the Pythagorean Theorem The Pythagorean Theorem: What It Is & How to Use It Using the Pythagorean Theorem Math Overview Geometry: The Pythagorean Theorem and Its Converse Skill Builder The Pythagorean Theorem Model Lesson Building Bridges with Similar Triangles Activity Numbers for Finding your Triangle 2015 Algebraic Geometry 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. Video Introduction: Structures and Triangles Example 1: Special Right Triangles -- Bridges Introduction: Jumping off of Right Triangles Section B: Six Trig Functions Math Overview Geometry: Trigonometry Model Lesson Building Bridges with Similar Triangles Activity Finding Ratios in Right Triangles STANDARD: G.SRT.7 Math Explanation Explain and use the relationship between the sine and cosine of complementary angles. Trigonometry: Right Triangle Trigonometry: Using the Complementary Angle Theorem and Given Trigonometric Values to Evaluate the Trigonometric Expression Trigonometry: Right Triangle Trigonometry: Using the Fundamental Identities and Complementary Angle Theorem to Evaluate Trigonometric Expressions Trigonometry: Right Triangle Trigonometry: Using the Complementary Angle Theorem to Evaluate Trigonometric Expressions Trigonometry: Right Triangle Trigonometry: Finding Angles that Satisfy Trigonometric Equations Precalculus: Trigonometric Ratios in Right Triangles Model Lesson Building Bridges with Similar Triangles Activity Obstacle Course 2015 Algebraic Geometry STANDARD: G.SRT.9 Derive the formula A=1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Math Overview Geometry: Working with the Law of Cosines Algebra II: Law of Sines Algebra II: Law of Cosines Math Explanation Geometry: Working with the Law of Cosines: Reasons in Proofs STANDARD: G.SRT.10 Math Overview Math Explanation STANDARD: G.SRT.11 Prove the Law of Sines and the Law of Cosines to solve problems. Geometry: Working with the Law of Cosines Algebra II: Law of Sines Algebra II: Law of Cosines Geometry: Working with the Law of Cosines: Reasons in Proofs 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). Math Overview Geometry: Using the Law of Sines in Non-Right Triangles Math Explanation Geometry: Using the Law of Sines in Non-Right Triangles: The Law of Sines Geometry: Using the Law of Sines in Non-Right Triangles: Solving Triangles-Law of Sines Geometry: Working with the Law of Cosines: Using the Law of Cosines to Find Missing Side Geometry: Working with the Law of Cosines: Solving Triangles Given Two Sides and Included Angle Model Lesson Geometry: Building Bridges with Trigonometry 2015 Algebraic Geometry Activity Bridge Surveying Solving Triangles in Bridge Designs 2015 Algebraic Geometry COURSE: Geometry UNIT TITLE: Quadrilaterals and Coordinate Geometry ESSENTIAL QUESTION(S): SUGGESTED UNIT TIMELINE: 2 weeks 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: STANDARD A listing of all standards included in the unit # Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a G-CO.11 G-GPE.4 G-GPE.5 G-GPE.6 G-GPE.7 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: 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. MAJOR SUPPORTING STANDARD STANDARD (M) (S) UNIT VOCABULARY 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 # TEACHER INSTRUCTION STUDENT LEARNING HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS 2015 Algebraic Geometry G-CO.11 Instructional strategies, 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 Tasks, activities, links to practice, etc. NCSM Great Tasks p.145148- 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 Have the students discover the Example problems-Best properties of quadrilaterals Strategies by Benson #’s using a discovery activity like 29, 30, 31, 32, 33 the NCSM Great tasks or the discovery examples below Lesson- Discover properties about special quadrilaterals using variable coordinates on the coordinate plane. 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 2015 Algebraic Geometry DISCOVERY EDUCATION RESOURCES 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. Video Euclid's Proposition 41 Math Overview Geometry: Proving That a Quadrilateral Is a Parallelogram Math Explanation Geometry: Proving that a Quadrilateral is a Parallelogram: Proofs for Parallelograms Geometry: Proving that a Quadrilateral is a Parallelogram: Parallelograms and Flow Proofs Geometry: Properties of Parallelograms: Parallelogram Proofs Geometry: Trapezoids and Kites: Proofs with Trapezoids and Kites Model Lesson Congruence and Proof Activity Parallelogram Proof Cards Proving Theorems about Parallelograms STANDARD: G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. 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). Video Geometric Interpretation of the Theorem of Pythagoras Introduction: Washington D.C. and the Coordinate Plane Example 1: The Cartesian Coordinate System -- DC Map Example 2: Slopes and Relationships of Lines -- DC Monuments Example 3: Distances and Midpoints -- DC Tourists The Distance Formula & the Midpoint Formula 2015 Algebraic Geometry Math Overview Geometry: Midpoints and Distance Math Explanation Algebra I: The Distance Formula: Understanding the Relationship Between the Distance Formula and the Pythagorean Theorem Geometry: Midpoints and Distance: Finding Length of a Segment Part 2 Model Lesson Coordinate Geometry and How It's Used 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). Video Section C: Properties of Linear Graphs Properties of Parallel Lines Example 2: Slopes and Relationships of Lines __ DC Monuments Math Overview Geometry: The Slope of Parallel and Perpendicular Lines Algebra I: Parallel and Perpendicular Lines Math Explanation Geometry: The Slope of Parallel and Perpendicular Lines: Checking For Perpendicular Lines - 1 Trigonometry: Parallel and Perpendicular Lines: Proving Perpendicularity Trigonometry: Parallel and Perpendicular Lines: Finding Slopes of Parallel and Perpendicular Lines Model Lesson Coordinate Geometry and How It's Used Activity Parallel and Perpendicular Lines STANDARD: G.GPE.6 Math Overview Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Geometry: Understanding Parallel Lines and Their Proportional Parts 2015 Algebraic Geometry Math Explanation Geometry: Understanding Parallel Lines and Their Proportions: Real World Applications Geometry: Determining and Using Similar Triangles: Determining Unknown Values! Model Lesson Coordinate Geometry and How It's Used Activity You've Got a Point STANDARD: G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Math Explanation Geometry: Midpoints and Distance: Perimeter and Distance Formula Geometry: Midpoints and Distance: Finding Perimeter in the Coordinate Plane Geometry: Coordinates in Space: Distance Between Points - 2 Model Lesson Coordinate Geometry and How It's Used Activity The Midpoint and Other Divisions The Distance Formula and the Midpoint Formula 2015 Algebraic Geometry COURSE: Geometry UNIT TITLE: Two-Dimension vs. Three-Dimension ESSENTIAL QUESTION(S): SUGGESTED UNIT TIMELINE: 3 ½ weeks 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? WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? REFERENCE/ STANDARDS: STANDARD 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 G-GMD.1 G-GMD.3 G-GMD.4 G-MG.1 G-MG.2 G-MG.3 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). UNIT DESCRIPTION: This unit investigates area and volume paying particular attention to modeling situations. MAJOR SUPPORTING STANDARD STANDARD (M) (S) UNIT VOCABULARY 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 STANDARD # TEACHER INSTRUCTION Instructional strategies, links to websites, resources, etc. STUDENT LEARNING Tasks, activities, links to practice, etc. HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS 2015 Algebraic Geometry G-GMD.1 Anything that will help teacher provide instruction related to the standard(s) Relate diameter and circumference Glencoe 1.6 Informally prove the area of a circle Glencoe 11.3 G-GMD.3 Area of Circles Calculate volume of prisms and cylinders using the Cavalieri principle Glencoe 12.4 Cavalieri's Principle Worksheet 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 Area of Prisms, Pyramids, Cylinders, and Cones Solve real-world problems involving pyramids Glencoe 12.3 and 12.5 Solve real-world problems involving cylinders Glencoe 12.2 and 12.4 G-GMD.4 Circle Poster Circumference Solve real-world problems involving spheres Glencoe 12.6 Visualize cross-sections of prisms Surface Area and Volume - All Surface Area and Volume – Prisms and Cylinders Surface Area and Volume – Spheres Online Activity - Volume of Cones, Cylinders, and Spheres 2015 Algebraic Geometry 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 G-MG.2 2D vs. 3D - Volume.ks-ig Prism and Cylinders LA SA G-MG.3 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 2015 Algebraic Geometry DISCOVERY EDUCATION RESOURCES STANDARD: G.GMD.1 Video Math Overview Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. Section C: Circleville (Circumference of Circles) Section F: Volume Area and Volume The Volume of Rectangular Solids The Volume of Cylindrical Solids Geometry: Area of Regular Polygons and Circles Geometry: Volumes of Prisms and Cylinders Geometry: Volumes of Pyramids and Cones Math Explanation Geometry: Volumes of Prisms and Cylinders: Cavalieri's Principle Geometry: Volumes of Prisms and Cylinders: Experimenting with Prisms and Cylinders Model Lesson Geometry: Three-Dimensional Shapes STANDARD: G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Introduction: Geometric Quantities and Fantastic Animation Example 1: Surface Area -- Boxes and Cans Video Example 2: Volume -- Pools and Cans Example 3: Surface Area and Volume -- Cheese Geometry: Volumes of Prisms and Cylinders: Hollow Solids Math Geometry: Volumes of Prisms and Cylinders: Prism Volume Explanation Geometry: Volumes of Prisms and Cylinders: Triangular Prism Volume 2015 Algebraic Geometry Geometry: Volumes of Prisms and Cylinders: Cylinder Volume Geometry: Volumes of Pyramids and Cones: Volume of a Cone and Cavalier's Principle Model Lesson Geometry: Three-Dimensional Shapes Activity Turn up the Volume 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.) Video Bridges Triangles: Bridges of Support Geometric Constructions Model Lesson Geometry: Building Bridges With Trigonometry Activity Bridge Surveying Looking at Bridges STANDARD: G.MG.2 Video STANDARD: G.GMD.4 Video Apply concepts of density based on area and volume in modeling situations (e.g. persons per square mile, BTU’s per cubic foot.) Definition of Density in Physics Measurement: Fluid Volume Use Ratio to Calculate Population per Square Mile (BGL) Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Example 1: Scale Drawings--Maps Example 2: Blueprints--Museum Example 3: Planar Cross-Sections--Earth and MRI 2015 Algebraic Geometry Math Overview Geometry: Space Figures and Drawings Math Explanation Geometry: Space Figures and Drawings: Cube Cross Sections Geometry: Space Figures and Drawings: Foundation and Orthographic Drawings Geometry: Surface Areas of Prisms and Cylinders: Solids of Revolution Geometry: Volumes of Prisms and Cylinders: Solids of Revolution Geometry: Surface Areas of Pyramids and Cones: Solids of Revolution Geometry: Volumes of Pyramids and Cones: Rotating Shapes and Volume 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). Video Terri Norstrand: Skatepark Designer Topographic Maps Model Lesson Geometry: Building Bridges with Trigonometry World Goes Round in Circles The Real Number System: Heavenly Observations Activity Building a Scale Model 2015 Algebraic Geometry COURSE: Geometry UNIT TITLE: Circles – Part 1 SUGGESTED UNIT TIMELINE: 2 ½ weeks 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? 4. How can the equations of circles be derived using the Pythagorean Theorem? 5. How can coordinate geometry be used to solve real-life problems? REFERENCE/ STANDARD # G-C.1 G-C.5 G-GPE.1 G-GPE.4 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. 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. 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). 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. UNIT VOCABULARY Tangents, secants, arc, chords, ratio, diameter, radius, Pythagorean Theorem, Conic, Completing the Square, Distance Formula, Properties of Radicals, Factoring. In addition, students use their prior algebraic understanding of the quadratic equation and Pythagorean Theorem to find the equation of a circle. Students will understand how to utilize the coordinate plane to determine whether a given point is on a circle. HOW DO WE KNOW STUDENTS HAVE LEARNED? UNIT ASSESSMENT BLUEPRINT MAJOR SUPPORTING STANDARD STANDARD (M) (S) 2015 Algebraic Geometry 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-C.1 G-C.5 G-C.5 G-C.1 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 G-C.5 Glencoe Geometry Baseball book section 11-3 G-C.5 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 Lesson- Equations of Circles. Teaches the basics of circles. (see teacher exchange files) G-C.5 G-C.1 G-C.1 G-C.5 G.GPE.1 Lesson- Completing the 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 Merry-GoRound Practice for sectors and segments Activity investigating arc length and area of a sector Practice – Equations of Circles Day 1 Worksheet (see teacher exchange files) Practice – Completing the HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS 2015 Algebraic Geometry G.GPE.4 square with circles (see teacher exchange files) Square with Circles Worksheet (see teacher exchange files) Lesson plan - Deriving the equation of a circle using the Pythagorean Theorem Practice – see lesson plan Lesson – Determining whether a point is on a circle (see teacher exchange files) Practice – Determining whether a point is on a circle worksheet (see teacher exchange files) Lesson plan – Proving whether a point is on a circle. ADDITIONAL UNIT RESOURCES www.learnzillion.com Practice – see lesson plan Online practice that covers the entire Unit. Multiple Choice practice over Circles (PDF available) DISCOVERY EDUCATION RESOURCES STANDARD: G.C.1 Prove that all circles are similar. Video Recap: Similarity Circular Structures: Design and Architecture Example 3: Circle- Pools Activity Deriving the Equation for a Circle Sector Area Model Lesson Circles: Understanding Structure 2015 Algebraic Geometry 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. Video Defining & Problem Solving with Sectors Math Overview Geometry: Special Segments in Circles Geometry: Geometric Probability Math Geometry: Geometric Probability: Probability in Sector of a Circle Explanation Model Lesson STANDARD: G.GPE.1 Math Overview Math Explanation Model Lesson Circles: Understanding Structure 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. Geometry: Equations of Circles Geometry: Exploring Circles Geometry: Equations of Circles: Center and Radius Geometry: Equations of Circles: Equation of Circle Given Center and Radius Geometry: Equations of Circles: Equation of Circle Given Points Geometry: Exploring Circles: Determining the Radius, Diameter, and Circumference Trigonometry: Circles: Understanding Graphs of Circles and Writing the Equation from the Center and Radius Coordinate Geometry and How It's Used Circles: Understanding Structure World Goes Round in Circles 2015 Algebraic Geometry COURSE: Geometry UNIT TITLE: SUGGESTED UNIT TIMELINE: Circles – Part 2 ESSENTIAL QUESTION(S): 2 weeks What are the relationships between parts of a circle? Can those relationships be used to find unknown parts of a circle? WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? REFERENCE/ STANDARDS: STANDARD 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 G-C.2 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. 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 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 STUDENT LEARNING Geometry Station Activities for Common Core State Standards Pages 151-165 Problem Based Tasks for Math II (orange book) HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS 2015 Algebraic Geometry G-C.2 G-C.2 G-C.2 G-C.2 G-C.2 G-C.2 G-C.2 G-C.2 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 10-7 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 Website for special segments in circles (Sketchpad) Website for special segments are two intersecting lines and a circle 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 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) 2015 Algebraic Geometry DISCOVERY EDUCATION RESOURCES 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. Video Sines and Chord Lengths Ancient Astronomy: The Connection Between Sines and Chords and Circles Math Overview Geometry: Inscribed Angles Geometry: Arcs and Chords Math Explanation Geometry: Inscribed Angles: Angle Measures and Variables Geometry: Inscribed Angles: Proofs Geometry: Arcs and Chords: Center of Circles Geometry: Arcs and Chords: Proofs Geometry: Arcs and Chords: Theorems Model Lesson Circles: Understanding Structure Activity Arc Length and Radius 2015 Algebraic Geometry COURSE: Geometry UNIT TITLE: Probability ESSENTIAL QUESTION(S): SUGGESTED UNIT TIMELINE: 3 weeks 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 STANDARDS: WHAT DO WE WANT STUDENTS TO KNOW, UNDERSTAND, AND BE ABLE TO DO? 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). UNIT DESCRIPTION: In this unit students will use conditional probability, represent sample space, use permutations and combinations, and find probabilities of compound events. UNIT VOCABULARY 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 MAJOR SUPPORTING STANDARD STANDARD (M) (S) A compilation of the proficiency levels and exemplars for the unit that defines mastery of the standards. 2015 Algebraic Geometry FACILITATING ACTIVITIES Strategies and methods for teaching and learning STANDARD # S-CP.1 TEACHER INSTRUCTION Introduce sample space, outcome, classical/empirical with deck of cards, coins, dice, skittles, spinners Lesson over sample space STUDENT LEARNING Empirical probability: Activities with concrete manipulatives Spinner Activity Probability and Data Analysis Activities 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 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 HOW WILL WE RESPOND WHEN STUDENTS HAVE NOT LEARNED? INTERVENTIONS HOW WILL WE RESPOND WHEN STUDENTS HAVE ALREADY LEARNED? EXTENSIONS 2015 Algebraic Geometry S-CP. 6 Using probability to make fair decisions Resource for teachers Problem Based Tasks for Math II (orange book) Making Decisions pg. 358 Conditional probability demonstrated Worksheets for fair decisions Interactive Activities for students S-CP.7 Lesson Decision Trees Help with Addition Rule S-CP.7 S-CP.8 Additional lesson on decision trees Lesson on Addition Rule Video explaining Multiplication 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 Problem Based Tasks for Math II (orange book) The Multiplication Rule pg. 345 Explanation of Multiplication Rule S-CP.9 Lesson over Permutations and Problem Based Tasks for Combinations 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 2015 Algebraic Geometry DISCOVERY EDUCATION RESOURCES 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"). Video The Statistics of Sampling Sampling Techniques Math Explanation Algebra II: Compound Events: Compliments of Events Algebra II: Compound Events: Applications of Event Complements Model Lesson What's the Chance? 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. Video Example 1: Independent Events-- Ultimate Frisbee Understanding the Odds Joint Probability: Understanding the Odds Math Overview Algebra I: Independent Events Model Lesson What's the Chance? Activity Frequency Suduko 2015 Algebraic Geometry 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. Video Example 3: Compound Events -- Life Expectancy and Insurance Model Lesson What's the Chance? 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. Video Frequency Distribution Example 2: Frequency Distribution and Line Graphs-- High Temperature Math Explanation Algebra I: Measures of Central Tendency: Finding Mean, Median,... Activity Frequency Suduko STANDARD: S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Video Example 3: Conditional Probabilities -- Baseball Batting Order Example 1: Independent Events-- Ultimate Frisbee Model Lesson What's the Chance? Activity Mad Chemist 2015 Algebraic Geometry 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. Video Section B: The Uniform Distribution Model Lesson What's the Chance? 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. Math Overview Algebra II: Compound Events Math Explanation Algebra II: Compound Events: Finding Missing Probabilities Algebra II: Compound Events: Probability of A and B Algebra II: Dependent and Independent Probabilities: Probability of Independent Events Model Lesson What's the Chance? STANDARD: S.CP.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. Math Explanation Algebra II: Dependent and Independent Probabilities: Dependent Probabilities Model Lesson What's the Chance? 2015 Algebraic Geometry STANDARD: S.CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems. Video Permutations and Combinations: Part 1 Permutations and Combinations: Part 2 Probabilities of Compound Events Math Overview Algebra II: Counting and Combinations Math Explanation Algebra II: Basic Probability: Combinations vs. Permutations Algebra II: Counting and Combinations: Combinations vs. Permutations Model Lesson What's the Chance? STANDARD: S.MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator) Video What Are the Odds?: Heart Attack Discovering Math: Advanced: Probability Math Explanation Algebra II: Basic Probability: Probability With Random Numbers Algebra II: Dependent and Independent Probabilities: Independent.. 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) Video What Are the Odds?: Car Accidents Math Starters: Shopping Mall Starters Analyzing Everyday Risks, Benefits, and Alternatives in Decisions What Are the Odds?: House Fires Math Overview Algebra II: Basic Probability Algebra II: Dependent and Independent Probabilities 2015 Algebraic Geometry