Geometry Enduring Understandings Students will understand 1. that all circles are similar.


 Godfrey Townsend
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1 High School  Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps, grids, charts, spreadsheets) 3. How can geometry be used to solve problems about realworld situations, spatial relationships, and logical reasoning? Essential Vocabulary: inscribed angles, radii, chords, central angles, circumscribed angles, diameter, perpendicular, tangent, inscribed, circumscribed, quadrilateral, tangent line, arc length, radian measure, constant proportionality, area of a sector HS.GC.1: Prove that all circles are similar. 1. circle 2. similarity 1. that all circles are similar. 1. prove that all circles are similar. HS.GC.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. 1. inscribed angles 2. radii 3. chords 1. that central, inscribed, and circumscribed angles are related. 2. that angles inscribed on a diameter are right angles. 3. that the the radius of a circle is perpendicular to the tangent where the radius intersects the circle. 1. identify inscribed angles, radii, and chords. 2. describe the relationship among angles, radii, and chords. HS.GC.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 1. inscribed triangle, circumscribed triangle, inscribed quadrilateral 1. that there is a relationship between inscribed and circumscribed. 2. that angles have properties in quadrilaterals inscribed in a circle. 1. construct an inscribed and circumscribed circles with triangles and quadrilaterals. 2. prove the properties of angles for a quadrilateral inscribed in a circle.
2 HS.GC.4: (+) Construct a tangent line from a point outside a given circle to the circle. 1. tangent line 1. that a tangent line can be constructed from 2. point a point outside a given circle to the circle. circle 1. construct a tangent line. HS.GC.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. 1. arc length 2. radian measure 3. constant proportionality 4. area of a sector 1. that the length of the arc intercepted by an angle is proportional to the radius. 1. derive the formula for area of a sector. 2. derive the proportionality between arc length and radius..
3 High School  Congruence Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps, grids, charts, spreadsheets) 3. How can geometry be used to solve problems about realworld situations, spatial relationships, and logical reasoning? Essential Vocabulary: angle, circle, perpendicular line, parallel line, and line segment, rotation, reflection, translation, rigid transformation, nonrigid transformation, dilation, rotation, reflection, line of reflection, symmetry, congruence, rigid motion, corresponding parts, rigid motion, triangle, corresponding, biconditional statement, triangle congruence, ASA, SAS, SSS, theorem, vertical angles, transversal, alternate interior angles, corresponding angles, perpendicular bisector, equidistant, triangle, interior angles, base angles, isosceles triangles, midpoint, median of a triangle, parallelogram, diagonal, construction, compass, straightedge, equilateral triangle, square, regular hexagon, circle HS.GCO.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. 1. angle 2. circle 3. perpendicular line 4. parallel line 5. line segment. 1. that the the undefined notions of point, line, distance along a line, and distance around a circular arc relate to the definitions of angle, circle, perpendicular line, parallel line, and line segment. 1. define angle, circle, perpendicular line, parallel line, and line segment.
4 HS.GCO.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). CPM Geometry Only 1. transformations, 2. geometry software 3. functions 4. points 5. planes 6. distance 7. angle 8. translation 1. that transformations can be represented using a variety of media. 2. that coordinate points can be transformed as functions. 3. that there is a difference between a rigid and nonrigid transformations. 1. represent transformations in a plane. 2. describe transformations as functions. 3. compare transformations. HS.GCO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. CPM Geometry Only 1. rectangle 2. parallelogram 3. trapezoid 4. regular polygon 5. rotations 6. reflections 1. that symmetry can be describes in rotations and reflections. 1. describe rotations and reflections.
5 HS.GCO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 1. rotations 2. reflections 3. translations 4. angles 5. circles 6. perpendicular lines 7. parallel lines 8. line segments 1. that rigid transformations can be defined using basic geometry terms. 1. define rotations, reflections, and transformations in multiple terms. HS.GCO.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. Only 1. geometric figure 2. rotation 3. reflection 4. translation 5. transformed figure 6. graph paper 7. tracing paper 8. geometry software 9. sequence 10. transformations 11. figure 1. that a sequence of transformations will carry a given figure onto another. 1. draw multistep transformations. 2. explain the types of transformations. 3. use a variety of methods.
6 HS.GCO.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. Only 1. congruence 2. rigid motion 3. figures 1. that rigid motion does not change the size or shape of an object. 1. compare figures to determine congruence. 2. use transformations on a single figure to find corresponding parts and congruence. HS.GCO.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. 1. congruence 2. rigid motions 3. triangle 4. corresponding pairs of sides 5. corresponding pairs of angles 1. that rigid motion does not change the size or shape of an object. 1. show congruence in terms of rigid motion. HS.GCO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. 1. triangle congruence 2. ASA, SAS, SSS 1. that triangles can be proved congruent using ASA, SAS, and SSS. 1. explain the criteria for triangle concgruence.
7 HS.GCO.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. 1. theorems 2. lines 3. angles 4. vertical angles 5. transversal 6. parallel lines 7. alternate interior angles 8. corresponding angles 9. perpendicular bisector 10. equidistant 11. segment 12. endpoints 1. that congruence can be used to prove theorems about lines and angles. 1. prove theorems about lines and angles. HS.GCO.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. 1. theorems 2. triangle 3. interior angles 4. base angles 5. isosceles triangles 6. midpoints 7. median of a triangle 1. that congruence can be used to prove theorems about triangles. 1. prove theorems about triangles.
8 HS.GCO.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. 1. theorems 2. parallelograms 3. opposite sides 4. opposite angles 5. diagonals 1. that congruence can be used to prove theorems about parallelograms. 1. prove theorems about parallelograms. 2. bisect parallelograms. HS.GCO.12: Make formal geometric constructions, including those representing Montana American Indians, 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. 1. constructions 2. compass 3. straightedge 4. reflective devices 5. geometric software 1. that geometric constructions can be completed in multiple ways. 1. make geometric constructions. 2. copy segments and angles. 3. bisect segments and angles 4. construct parallel lines.
9 HS.GCO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 1. equilateral triangle 2. square 3. regular hexagon 4. circle 1. that basic regular polygons can be constructed inscribed in a circle. 1. construct regular polygons.
10 High School Geometric Measurement and Dimension Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps, grids, charts, spreadsheets) 3. How can geometry be used to solve problems about realworld situations, spatial relationships, and logical reasoning? Essential Vocabulary: cylinder, pyramid, cone, Cavalieri s principle, crosssection HS.GGMD.1: Give an informational 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. 1. formula 2. circumference 3. circle 4. area 5. volume 6. cylinder 7. pyramid 8. cone 1. that arguments can be given to explain circumference, area and volume formulas for circle, cylinder, pyramid, and cone. 1. give informational arguments (in more than one way) for formulas. HS.GGMD.2: (+) Give an informal argument using Cavalieri s principle for the formulas for the volume of a sphere and other solid figures. Not in current curriculum 1. Cavalieri s principle 2. formula 3. volume 4. sphere 5. solid figures 1. that Cavalieri s principle can be used for the formulas for volume of a sphere and other solid figures. 2. give an informal arguments for formulas.
11 HS.GGMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* 1. volume 2. cylinder 3. pyramid 4. cone sphere 1. that volume formulas can be used for cylinders, pyramids, cones, and spheres. 1. use formulas to solve problems. HS.GGMD.4: Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects. 1. shapes 2. twodimensional, 3. threedimensional 4. crosssections 5. rotations 1. that 2D and 3D objects can be used to identify one another. 1. identify the shapes of twodimensional crosssections of threedimensional objects. 2. identify threedimensional objects generated by rotations of twodimensional objects.
12 High School Expressing Geometric Properties with Equations Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps, grids, charts, spreadsheets) 3. How can geometry be used to solve problems about realworld situations, spatial relationships, and logical reasoning? Essential Vocabulary: complete the square, Pythagorean Theorem, parabola, focus, directix, ellipse, hyperbola, foci, slope, parallel, perpendicular, point, line segment, partition, ratio, distance formula, perimeter, area HS.GGPE.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. 1. circle 2. center 3. radius 4. Pythagorean Theorem 5. complete the square 1. that the Pythagorean Theorem can be used to derive the equation of a circle. 1. derive the equation of a circle of given center and radius using the Pythagorean Theorem. 2. find the center and radius of a circle by completing the square. HS.GGPE.2: Derive the equation of a parabola given a focus and directrix. Algebra III 1. equation 1. that the equation of a parabola can be 2. parabola derived. 3. focus 4. directix 1. derive the equation of a parabola given a focus and directix.
13 HS.GGPE.3: (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Algebra III 1. ellipse 2. hyperbola 3. foci 1. that the equation of ellipses and hyperbolas can be derived given the foci. 1. derive the equations of ellipses and hyperbolas given the foci. HS.GGPE.4: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). 1. coordinates 2. geometric theorems 1. that coordinates can be used to prove simple geometric theorems algebraically. 1. use coordinates to prove theorems algebraically. HS.GGPE.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). 1. slope, 2. parallel 3. perpendicular 1. that slope criteria can be used to solve geometric problems. 1. prove the slope of parallel and perpendicular lines to use them to solve geometric problems.
14 HS.GGPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 1. point 2. line segment 3. partition 4. ratio 1. that points partition line segments in a given ratio. 1. find a point on a segment that creates a bisector or any other given ratio. HS.GGPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. * 1. coordinates 2. perimeters 3. polygons 4. area of triangle 5. area of rectangle 6. distance formula 1. that coordinates can be used to compute perimeters of polygons and areas of triangles and rectangles. 1. use coordinates to compute perimeters and areas of triangles and rectangles. 2. use the distance formula.
15 High School Modeling with Geometry Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps, grids, charts, spreadsheets) 3. How can geometry be used to solve problems about realworld situations, spatial relationships, and logical reasoning? Essential Vocabulary : geometric shapes, measure, properties, density, area, volume, geometric methods HS.GMG.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; modeling a Montana American Indian tipi as a cone).* 1. geometric shapes measures 2. properties 1. that objects can be described using a variety of characteristics. 1. use geometric shapes, their measures, and their properties to describe objects. HS.GMG.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* 1. density area 2. volume 1. that concepts of density are based on area and volume. 1. apply concepts of density based on area and volume in modeling situations HS.GMG.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).* 1. geometric methods 1. that geometric methods can be used to solve design problems. 1. apply geometric methods to solve design problems.
16 High School Similarity, Right Triangles, and Trigonometry Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps, grids, charts, spreadsheets) 3. How can geometry be used to solve problems about realworld situations, spatial relationships, and logical reasoning? Essential Vocabulary : scale factor, similarity, similarity transformations, proportionality, AA, criterion, Pythagorean Theorem, similarity, side ratios, trigonometric ratios, sine, cosine, complementary angles, trigonometric ratios, auxiliary line, vertex, Law of Sines, Law of Cosines, resultant forces HS.GSRT.1: 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. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 1. properties of dilations 2. center 3. scale factor 1. that a dilation changes a figure s size based on scale factor. 1. verify the properties of dilations. HS.GSRT.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. 1. similarity 2. similarity transformations 3. proportionality 4. corresponding angles 5. corresponding sides 1. that transformations can be used to decide if two figures are similar. 1. determine if two figures are similar using similarity transformations. 2. explain why two figures are similar using similarity transformations
17 HS.GSRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 1. similarity transformations 2. AA 3. criterion 1. that triangles can be shown to be similar using the AA criterion. 1. use properties of transformations to establish the AA criterion. HS.GSRT.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. 1. theorems, 2. triangle 3. parallel, 4. Pythagorean Theorem 1. that similarity can be used to complete proofs about triangles. 1. prove theorems about triangles. HS.GSRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 1. congruence 2. similarity 3. triangles 4. geometric figures 1. that congruence and similarity can be used for triangles to complete proofs about geometric figures. 1. solve problems using congruence and similarity. 2. prove relationships in geometric figures. HS.GSRT.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. 1. side ratios 2. right triangles 3. trigonometric ratios 4. acute angles 1. that side ratios in right triangles are properties of the angles in the triangle. 1. define trigonometric ratios. 2. explain the relationship between side ratios and trigonometric ratios.
18 HS.GSRT.7: Explain and use the relationship between the sine and cosine of complementary angles. 1. sine 2. cosine 3. complementary angles 1. that there is a relationship between sine and cosine of complementary angles. 1. explain how sine and cosine are related. HS.GSRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Geometry Enduring Understanding 1. trigonometric ratios 2. Pythagorean Theorem 1. that the Pythagorean Theorem and trigonometric ratios can be used to solve right triangles. 1. apply trigonometric ratios and the Pythagorean Theorem to right triangle. 2. model trigonometric ratios and the Pythagorean Theorem. HS.GSRT.9: (+) Derive the formula A = ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Algebra II/Trig 1. area formula 2. auxiliary line 3. vertex 4. perpendicular 5. opposite side 1. that sine can be used to derive the formula of a triangle using sine. 1. derive the formula of a triangle using sine. HS.GSRT.10: (+) Prove the Laws of Sines and Cosines and use them to solve problems. 1. law of sines 2. law of cosines 1. that the laws of sines and cosines can be used to solve problems. 1. prove the laws of sines and cosines. 2. use the laws to solve problems
19 HS.GSRT.11: (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces). Only 1. Law of Sines 2. Law of Cosines 3. measurements 4. resultant forces 1. that the Laws of Sines and Cosines can be used on right and nonright triangles in application based problems. 1. apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles.
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