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Geometry South Central BOCES Grade HS UNIT 1 Tools for the Trade (8 weeks) UNIT 2 Identical Twins and Mini-Me (5 weeks) What happens to the coordinates of the vertices of shapes when different transformations are applied in the plane? Geometric Definitions and Terms S.4-GLE.1-EO.a.i S.4-GLE.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of EO.a.vi point, line, distance along a line, and distance around a circular arc. (MA10-GR.HS-) Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (MA10-GR.HS-) How would the idea of congruency be used outside of mathematics? What does it mean for two things to the same? Are there different degrees of sameness? Rotations of Polygons S.4-GLE.1EO.a.v Given a rectangle, parallelogram, trapezoid, or regular polygon, -S.4-GLE.1describe the rotations and reflections that carry it onto itself. EO.b.i, ii 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. Transformations Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using. Specify a sequence of transformations that will carry a given figure onto another Represent transformations in the plane using; 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 S.4-GLE.1EO.a.vii S.4-GLE.1EO.a.viii S.4-GLE.1EO.a.ii, iii S.4-GLE.1EO.a.iv Congruence 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. S.4-GLE.1EO.b.iii S.4-GLE.1EO.b.iv Theorems and Proofs Prove theorems about lines, angles, triangles, and parallelograms. Use coordinates to prove simple geometric theorems algebraically. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. S.4-GLE.1-EO.c S.4-GLE.3EO.a.ii.1 S.4-GLE.3EO.a.ii.2 Similarity Verify experimentally the properties of dilations given by a center and a 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. S.4-GLE2-EO.a.i S.4-GLE2EO.a.ii, iii) S.4-GLE2EO.a.iv S.4-GLE2EO.b.iii S.4-GLE2-EO.b.ii Distance Formula, Areas, and Perimeter Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use the distance formula on coordinates to compute perimeters of polygons and areas of triangles and rectangles S.4-GLE.3EO.a.ii.3 S.4-GLE.3EO.a.ii.4 Constructions Make formal geometric constructions with a variety of tools and methods. S.4-GLE.1-EO.d.i 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 that all circles are similar. UNIT 3 3 Rights Don’t Make A … (4 weeks) UNIT 4 What Goes Around (5 weeks) How can you determine the measure of something that you cannot measure physically? Do perfect circles naturally occur in the physical world? If so, how do we model them? Why are circles at the foundation of constructions? Circle, Radii, Chords S.4-GLE.2-EO.e.i Identify and describe relationships among inscribed angles, radii, S.4-GLE.2-EO.e.ii, iii and chords. S.4-GLE.2-EO.f Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 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. Similar Triangles and Similarity Prove theorems about similar triangles. Understand through similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. S.4-GLE.2EO.b.i S.4-GLE.2EO.c.i Sine, Cosine, and Pythagorean Theorem 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. S.4-GLE.2EO.c.ii S.4-GLE.2EO.c.iii Equation of Circle and Equilateral Triangle 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. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. S.4-GLE.3EO.a.i.1, 2 S.4-GLE.1EO.d.ii) UNIT 5 On the Cat Walk (4 weeks) How might surface area and volume be used to explain biological differences in animals? Geometric Shapes S.4-GLE.5EO.a.i Use geometric shapes, their measures, and their properties to S.4-GLE.5describe objects. EO.a.ii Apply concepts of density based on area and volume in modeling S.4-GLE.5situations. EO.a.iii Apply geometric methods to solve design problems. Formulas for Circumference, Area, Volume 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 volume formulas for cylinders, pyramids, cones, and spheres to solve problems. S.4-GLE.4EO.a.i S.4-GLE.4EO.a.ii Identifying 2 and 3 Dimensional Objects Identify the shapes of two-dimensional cross-sections of threedimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects S.4-GLE.4EO.b.i