GTPS Curriculum Geometry

Transcription

1 3 weeks Topic: 1-Points, Lines, Planes, and Angles /Enduring 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. 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 will students use the definitions of the most basic elements and terms of Geometry to solve Geometric problems using critical thinking and deductive reasoning? To use correct vocabulary Determine the distance between two points Determine the type of angle by its measurement Sketch a diagram that represents a give Geometric problem Find the measure of an angle Algebraically Find the measure of an angle with a protractor Name and sketch adjacent angles Determine the locus of points that are a solution set Use Geometric Symbolic Notation correctly Apply Geometric Theorems to solve problems Determine the difference between equals and congruence 1.1 A Game and Some Geometry 1.2 Points, Lines, and Planes 1.3 Segments, Rays, and Distance 1.4 Angles 1.5 Postulates and Theorems Relating Points, Lines, and Planes Chapter tests Quizzes

2 3 Weeks Topic: 2-Deductive Reasoning 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. 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. 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. /Enduring How are proofs used to develop conjectures in Mathematics? Use correct vocabulary Recognize the Hypothesis and Conclusion of an if-then statement State the converse of an if-then statement Use a counterexample to disprove an ifthen statement Apply if and only if statements to biconditional statements Apply the properties of algebra and congruence in proofs Use the Midpoint Theorem and Angle Bisector Theorem Complete a two column proof Apply the definitions of complementary and supplementary angles in proofs and in algebraic solutions Apply the definitions and theorems of vertical angles and perpendicular lines in proofs and in algebraic solutions 2.1 If-then Statements; Converses 2.2 Properties from Algebra 2.3 Proving Theorems 2.4 Special Pairs of Angles 2.5 Perpendicular Lines 2.6 Planning a Proof

3 3 Weeks Topic: 3-Parallel Lines and Planes 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. G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. /Enduring How does slope influence lines intersection, parallelism, and skewness? Use correct vocabulary when discussing parallel lines Distinguish the difference between intersecting, parallel, and skew lines Identify the angles formed by parallel lines. Identify parallel and skew planes Complete proofs involving parallel lines. Use Algebra and Geometric theorems to solve angles of parallel lines Prove lines are parallel by Theorems, Postulates, and Algebra. Name a Triangle by sides and angles. Identify the exterior and remote interior angles of a triangle Solve for the angles of a triangle Use inductive reasoning to make conclusions. 3.1 Definitions 3.2 Properties of Parallel Lines 3.3 Proving Lines Parallel 3.4 Angles of a Triangle 3.5 Angles of a Polygon 3.6 Inductive Reasoning

4 3 Weeks Topic: 4-Congruent Triangles 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. 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. 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. /Enduring How can you determine if two triangles are congruent, or use congruent triangles to prove segments and angles are congruent? Use correct vocabulary when discussing congruent triangles Determine if two objects are congruent Label corresponding parts of congruent objects Name a pair of congruent objects in corresponding order Prove two triangles are congruent by SSS, SAS, ASA, AAS methods Use congruent triangles in a proof Determine if a triangle is isosceles Use HL, LL, HA, LA in proofs involving right triangles Complete proofs involving overlapping triangles Identify an altitude, median, and a perpendicular bisector of a triangle Solve problems involving altitudes, medians, and bisectors 4.1 Congruent Figures 4.2 Some Ways to Prove Triangles Congruent 4.3 Using Congruent Triangles 4.4 The Isosceles Triangle Theorems 4.5 Other Methods of Proving Triangles Congruent 4.6 Using More than One Pair of Congruent Triangles 4.7 Medians, Altitudes, and Perpendicular Bisectors

5 3 Weeks Topic: 5-Quadrilaterals 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. /Enduring How can the properties of parallelograms be used to determine the name of a given four sided polygon? Use correct vocabulary when discussing quadrilaterals Apply the properties of parallelograms to proofs Determine the lengths of segments of parallelograms Determine the measure of angles in parallelograms Determine the length of a segment of two sides of a triangle Use the property of the median of a trapezoid to determine the length of the bases or the median Complete proofs involving parallelograms and trapezoids Materials: McDougal Little Geometry 5.1 Properties of Parallelograms 5.2 Ways to Prove that Quadrilaterals are Parallelograms 5.3 Theorems Involving Parallel Lines 5.4 Special Parallelograms 5.5 Trapezoids

6 3 Weeks Topic: 6- Inequalities in Geometry A-REI.12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. /Enduring How do inequalities in triangles affect the angle and the sides opposite those angles? Use correct vocabulary when discussing inequalities Apply the exterior angle inequality theorem to triangles Determine if one segment or angle is larger than another Use inequalities and their properties in proofs Write the inverse and contrapositive of an if-then statement Complete an indirect proof Apply the triangle inequality theorem to determine the relationship between the sides of a triangle Apply the SAS inequality and SSS inequality methods to determine the relationship between two corresponding parts of different triangles 6.1 Inequalities 6.2 Inverses and Contrapositives 6.3 Indirect Proof 6.4 Inequalities for One Triangle 6.5 Inequalities for Two Triangle

7 3 Weeks Topic: 7- Ratio, Proportion, and Similarity 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. 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. G-SRT.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 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. G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. /Enduring How can the properties of similarity be used to determine if two polygons are similar or to find the measures of corresponding parts of similar polygons? Use correct vocabulary when discussing similar polygons Set up, simplify, and solve a proportion Apply the properties of proportions to solve equations Use proportions to find parts of similar triangles Determine if two polygons are similar Determine the scale factor between two similar polygons Determine if 2 triangles are similar using AA similarity, SAS similarity, or SSS similarity Use proportions and similarity to complete proofs Determine the lengths of segments that have been divided proportionally 7.1 Ratio and Proportion 7.2 Properties of Proportions 7.3 Similar Polygons 7.4 A Postulate for Similar Triangles 7.5 Theorems for Similar Triangles 7.6 Proportional Lengths

8 3 Weeks Topic: 8-Right Triangles 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. G-SRT.7. Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 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. G-SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. 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). /Enduring How can the properties of right triangles and trigonometry be used to solve mathematical and real world problems? Use correct vocabulary when discussing right triangles Express radicals in simplest form Apply Geometric Means to right triangles Use the Pythagorean theorem to solve right triangles Use the converse of the Pythagorean Theorem to determine the type of triangle (acute, right, obtuse) Use the basic trigonometric functions to solve for missing parts of right triangles Apply the Law of Sines and Law of Cosines to solve for parts of oblique triangles 8.1 Similarity in Right Angles 8.2 The Pythagorean Theorem 8.3 The Converse of the Pythagorean Theorem 8.4 Special Right Triangles 8.5 The Tangent Ratio 8.6 The Sine and Cosine Ratios 8.7 Applications of Right Triangle Trigonometry

9 3 Weeks Topic: 9- Circles G-C.1. Prove that all circles are similar. 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. G-C.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G-C.4. (+) Construct a tangent line from a point outside a given circle to the circle. 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. /Enduring How do arcs, central angles, segments, and circumference relate to one another within a circle? Use correct vocabulary when working with circles Use properties of tangent lines to find unknown lengths or angles Use the properties of arcs to find unknown measures of angles Use the properties of chords to discover lengths of segments or angle measurements Apply properties to complete a proof Use properties of inscribed angles to find the measures of arcs Understand the relationships among the arcs of a circle and the angles formed by chords, secants, and tangents Use and understand theorems that state relationships involving products of parts of chords, parts of secants, and parts of tangents and secants 9.1 Basic Terms 9.2 Tangents 9.3 Arcs and Central Angles 9.4 Arc and Chords 9.5 Inscribed Angles 9.6 Other Angles 9.7 Circles and Length of Segments

10 3 Weeks Topic: 10- Constructions and Loci 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. G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. /Enduring How can we construct geometric figures by knowing the properties of them? How can loci help our understanding of points? Use correct vocabulary when constructing figures Create congruent segments and angles Construct an angle bisector and a perpendicular bisector Construct parallel lines Construct a circle with a tangent line Construct a circle circumscribed about a given polygon Construct a circle inscribed in a given polygon Understand the properties of loci 10.1 What Construction Means 10.2 Perpendiculars and Parallels 10.3 Concurrent Lines 10.4 Circles 10.5 Special Segments 10.6 The Meaning of Locus 10.7 Locus Problems 10.8 Locus and Construction

11 3 Weeks Topic: 11- Areas of Plane Figures 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. 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. /Enduring How can the number of sides of a polygon be used to determine the figure s interior and exterior angle sums? Which various methods can be used to find the area of various polygons? Use correct vocabulary when discussing polygons and area Find the area of any rectangle, parallelogram, square, rhombus, or trapezoid Determine the area of any triangle, right and non-right Apply the Area Addition Postulate to any polygon Use trigonometric functions to find the missing parts needed to determine the area of a given polygon using the Area Addition Postulate Determine the circumference, arc length, area, or area of a sector of a circle Compare the areas of polygons using scale factors and ratios Use area to find the theoretical probability of an event occurring Materials: McDougal Little Geometry 11.1 Areas of Rectangles 11.2 Areas of Parallelograms, Triangles and Rhombuses 11.3 Areas of Trapezoids 11.4 Areas of Regular Polygons 11.5 Circumferences and Areas of Circles 11.6 Arc Lengths and Areas of Sectors 11.7 Ratios of Areas 11.8 Geometric Probability

12 Grade 8 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. G-GMD.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G-GMD.4. Identify the shapes of twodimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. 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). 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). 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). Topic: 12- Areas and Volumes of Solids /Enduring How can knowledge of right prisms, right pyramids, right cylinders, right cones, and spheres help determine the surface area and volume of any given object of no particular shape? Use correct vocabulary when discussing solids Identify the parts of a prism, pyramid, cylinder, sphere Find the lateral area of a solid Find the surface area of a solid Determine the scale factor between two similar solids Find the area and volume of two similar solids Materials: McDougal Little Geometry 12.1 Prisms 12.2 Pyramids 12.3 Cylinders and Cones 12.4 Spheres 12.5 Areas and Volumes of Similar Solids

13 3 Weeks Topic: 13- Coordinate Geometry 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. G-GPE.2. Derive the equation of a parabola given a focus and directrix. G-GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). 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). G-GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G-GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. /Enduring How is coordinate algebra used when writing geometric proofs? Use correct vocabulary when discussing coordinate geometry Evaluate the distance between any two points Write the equation of a circle given its center and radius Identify the center and radius of a circle given its equation Determine the slope of a line Determine if points are collinear Determine if lines are perpendicular, parallel, or neither from their algebraic equations Find the equation of lines, perpendicular lines, parallel lines Graph lines and vectors Find the ordered pair and magnitude that represents a vector Find the sum of two vectors Find the midpoint of any two points Supply the coordinates needed to complete a geometric proof Complete a coordinate proof Materials: McDougal Little Geometry 13.1 The Distance Formula 13.2 Slope of a Line 13.3 Parallel and Perpendicular Lines 13.4 Vectors 13.5 The Midpoint Formula 13.6 Graphing Linear Equations 13.7 Writing Linear Equations 13.8 Organizing Coordinate Proofs 13.9 Coordinate Geometry Proofs

14 3 Weeks Topic: 14- Transformations G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 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. Understand congruence in terms of rigid motions 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. /Enduring Can movements of objects be explained and modeled with mathematics using reflections, rotations, translations, and dilations? Use correct vocabulary and notation when discussing transformations Determine the image or pre-image of a mapping Determine if a mapping is isometric Reflect a point, segment, angle, object over a line Determine the transformation between two points Complete a rotation of a point or object about another point Find the image of an object under a dilation as either an expansion or contraction Describe the translation of an object to its image or from an image to its preimage Find the inverse of a mapping Locate the images of composite mappings Materials: McDougal Little Geometry 14.1 Mapping and Functions 14.2 Reflections 14.3 Translations and Glide Reflections 14.4 Rotations 14.5 Dilations 14.6 Composites of Mappings 14.7 Inverses and the Identity 14.8 Symmetry in Plane and in Space