Standard | Essential Question | Bloom’s Taxonomy Activities | Vocabulary | Pacing |

G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance, along a line, and distance around a circular arc. | How are points and lines related to each other? | -Create a graphic organizer involving various geometric shapes and their definitions | -Angle -Circle -Perpendicular line -Parallel line -Line segment -Arc -Point -Line | 2 days |

G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g. translation versus horizontal stretch). | How does the knowledge of transformations assist with graphic design? | -Create a poster on an approved topic of your choice demonstrating various transformations throughout your project. | -Transformations -Translations -Dilation -Reflections -Rotations -Rigid motion | 2 days |

G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. | How does angle measure relate to reflections and rotations of objects? | -Create models demonstrating how rotations and reflections of various shapes operate. | - Angle measure | 2 days |

G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. | What is the relationship between translations and other forms of geometric measurement and expression? | -Design a visual expression of rotations, reflections, and translations across circles, lines and segments. | -Perpendicular lines -Parallel lines | 2 days |

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. | How are translations interrelated? | -Use multiple forms of media to create a visual display demonstrating a series of transformations. | -Glide reflection | 2 days |

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 | What is the relationship between congruence and rigid motion. | -Predict the relationship of two given figures based on the knowledge of how transformations relate to congruence. | -Congruence -Rigid motions | 2 days |

G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. | How do corresponding sides and angles relate to congruence? | -Visually demonstrate how corresponding sides and angles relate to congruence in triangles. | -Corresponding sides -Corresponding angles -Triangles | 2 days |

G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions | How are the triangle congruence theorems interrelated? | -Support the definitions of triangle congruence algebraically or graphically. | -ASA congruence -SAS congruence -SSS congruence | 2 days |

G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent, when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints | How are alternate interior angle, vertical angles, and corresponding angles related? | Create shapes similar to tangrams to prove that corresponding angles are congruent, alternate interior angles are congruent, and vertical angles are congruent. | -Corresponding angles - Alternate interior angles -Vertical angles | 2 days |

G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. | How is knowledge of angle measure in an triangle related to determining the total number of degrees in a regular polygon? | Complete the Common Core-Aligned Task: Company Logo | -Theorems -Isosceles -Congruent -Medians | 2 days |

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. | What is the relationship between quadrilaterals, parallelograms, squares, and rectangles? | Create a poster highlighting qualities of parallelograms. Display your poster in the classroom or hallway. | -Parallelograms -Congruent -Opposite sides -Diagonals | 2 days |

G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.) Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line | How are various tools utilized in constructing geometric objects? | -Construct the following geometric constructions: 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 | -Bisector -Segment -Perpendicular bisector | 1 day |

G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle | How does knowledge of angle measure relate to the ability to draw geometric constructions? | - Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle | -Equilateral triangle -Regular hexagon -Inscribed in a circle | 1 day |

Standard | Essential Question | Bloom’s Taxonomy Activities | Vocabulary | Pacing |

G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: G.SRT.1a. 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. G.SRT.1b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. | How does scale factor relate to similarity in triangles? | Complete the interactive activities in the following site and write a reflection based on what you learned. http://bit.ly/xa7x4d | -Dilations -Scale | 2 days |

G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. | How is knowledge of scale factor important to graphic designers? | Using a paint program… create similar figures using the scale feature, and create dissimilar figures by adjusting the size manually. Using a ruler, support that your first set of figures are similar and your second set are not. | -Similarity | 2 days |

G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. | What is the least amount of information needed to prove two triangles similar? How do you know? | Using a ruler and a protractor, prove AA similarity. | -AA similarity | 2 days |

G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity | How does the Pythagorean Theorem support the case for triangle similarity? | -View the video below and create a visual proving the Pythagorean Theorem using similarity | -Proof -Triangle -Similarity | 2 days |

G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. | How does angle measure relate to triangle similarity? | -Create a table highlighting the six examples of similar triangles utilizing the resource listed below. | -Congruence -Similarity | 2 days |

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. | How are mnemonics helpful in learning trig ratios? | -Create a worksheet of six real-world examples which use trigonometric ratios. | -Sine -Cosine -Tangent -CoTangent | 2 days |

G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. | What is the relationship between cosine and sine in relation to complementary angles? | -Construct a table demonstrating the relationship between sine and cosine of complementary angles. | -Complementary angles | 2 days |

G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* | How do trigonometric ratios assist in determining height and depth when angle and distance to a given point are known? | -Complete the exercise listed below and create two problems demonstrating trigonometric ratios | -Trigonometric ratios | 2 days |

G.SRT.9 (+) Derive the formula A=1/2 absin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. | When could a contractor use the equation A=1/2 absin(C) to find the area of a triangle? | -Create a poster showing no less than three examples of finding the area of a non-right triangle using the formula A=1/2 absin(C) | -Area -Triangle -Right triangle | 1 day |

G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. | When is it appropriate to use Law of Sines instead of Law of Cosines? | -Research video examples of Law of Sines and Law of Cosines and create a short video teaching others how to use Law of Sines or Law of Cosines | -Law of Sines -Law of Cosines | 1 day |

G.SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g. surveying problems, resultant forces.) | What are three real-world applications of Law of Sines and Law of Cosines? | -Using the internet, find a real-world example of Law of Sines or Law of Cosines and evaluate which formula would be appropriate for use; justify your answers. | -Law of Sines -Law of Cosines | 1 day |

Standard | Essential Question | Bloom’s Taxonomy Activities | Vocabulary | Pacing |

G.C.1 Prove that all circles are similar | Why is it beneficial to know that all circles are similar? | -Complete the following activity and write one paragraph discussing why you believe all circles are similar. http://illuminations.nctm.org/ActivityDetail.aspx?ID=116 | -Circles -Similar | 1 day |

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. | How is knowledge of the properties of circles and cylinders beneficial to the manufacturing industry? | -Solve the following problem and create similar problems for your peers to complete. An audio CD spins at a rate of 200 rotations per minute. If the radius of a CD is 6 cm, how far does a point on the outer edge travel during the playing of a 57-minute CD | -Inscribed angles -Radii -Chords -Central angles -Circumscribed angles -Tangents | 1 day |

G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. | Where are inscribed circles and circumscribed used in everyday life? | -Research practical uses of inscribed and circumscribed circles pertaining to triangles and create a visual demonstrating the results of your research. | -Inscribed angles -Circumscribed circles | 1 day |

G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle. | Why would landscape artists need to know how to create a tangent line? | -Design a garden using circles and lines, include at least two tangents in your diagram. | -Compass -Ruler | 1 day |

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. | How is area of a sector related to the length of the arc? | -Watch the following video and create a similar problem for your peers to complete. http://www.5min.com/Video/How-to-Find-the-Area-of-a-Sector-of-a-Circle-275614332 | -Similarity -Arc -Radian | 2 days |

Standard | Essential Question | Bloom’s Taxonomy Activities | Vocabulary | Pacing |

G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. | How are radii related to the sides of an inscribed right triangle? | -Watch the following video as a guide to derive the equation of a circle using the Pythagorean theorem: http://www.youtube.com/watch?v=pZVufpgozCw -Create a worksheet for your peers with three sample problems | -Center -Circle -Pythagorean theorem -Distance formula -Radius -Complete the square | 3 days |

G.GPE.2 Derive the equation of a parabola given a focus *and directrix | What is the relationship between focus and directrix to the parabola formula? | -In teams, construct a poster of various graphs of parabolas with the focus and directrix color-coded to match the corresponding components in each given equation. | -Parabola -Focus -Directrix | 3 days |

Ggpe3 | ||||

G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plan 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). | How can the use of geometry and related algebraic equations assist in proving geometric theorems? | -Support geometric theorems algebraically and graphically. | -Proof -Coordinates | 3 days |

G.GPE.5 Prove the slope criteria for parallel and perpendicular lines; 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). | How is slope related to parallel and perpendicular lines? | -Evaluate if lines are parallel, intersecting, or perpendicular based on slope criteria. | -Slope | 3 days |

G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. | How does distance formula relate to determining ratios in a line segment? | -Create a problem on a coordinate plane in which a line segment is divided after a certain ratio is completed; solve your problem. For example, 1/3 of the distance between (2,3) and (-3,5) | -Line segment -Ratio -Distance formula | 3 days |

G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ | Why is knowledge of geometry and algebra in relation to mathematics in the real world? | -Design visual representations of the link between algebra and geometry in relation to distance and area. | -Distance formula | 3 days |

Standard | Essential Question | Bloom’s Taxonomy Activities | Vocabulary | Pacing |

G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. | How is height related to the area of a circle and the volume of a similar cylinder? | -Create a visual demonstrating the relationship between circumference, area, height, and volume of a circle and its related cylinder | -Informal argument -Circumference -Volume | 1 day |

Ggmd2 | ||||

G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* | In what contexts are volume formulas necessary for math after high school? | -Create a problem for your classmates to solve in which they must use at least three volume formulas | -Volume -Cone -Cylinder -Pyramids -Sphere | 1 day |

G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. | Where is multidimensional math most studied? What could it express about our universe? | -Create a 2 sets of PowerPoint slides showing the progression of dimension from 1 ^{st} to 4^{th} | -Dimension -Conic -Triangle -Cone -Circle -Sphere | 3 days |

Standard | Essential Question | Bloom’s Taxonomy Activities | Vocabulary | Pacing |

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).* | How do artists consider geometric shapes when creating masterpieces? | -Find a picture of an object of interest to you. Use a program such as Paint or PowerPoint to create the object using only geometric shapes | -Geometric shapes -Cylinder -Sphere -Rectangular prism -Parabola | 3 days |

G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g. persons per square mile, BTUs per cubic foot).* | How does population density in NYC assist in the saving of the world’s endangered languages? | -Research the U.S. at night over time. Create a visual with no less than four graphics showing the changes in population density based on images from the evening sky | -Density -Area -Volume | 3 days |

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).* | How is it possible for an object o have a finite area but unlimited perimeter? | -Use a computer program to investigate fractals and discuss the apparent limitations of the objects | -Fractals | 3 days |

### Geometry

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