Mth217 Geometry Mth114 Honors Geometry

MTH217 Geometry/MTH114 Honors GeometryPage 10PARSIPPANY-TROY HILLS TOWNSHIP SCHOOLSCOURSE PROFICIENCIESCourse:MTH217, MTH114Title:GEOMETRY, HONORS GEOMETRYIn accordance with district policy as mandated by the New Jersey Administrative Code and the New Jersey Student LearningStandards Mathematics, the following are proficiencies required for the successful completion of the above named course.The student will:GEOMETRY1. 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.2. represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take pointsin 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).3. given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.4. .develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.5. given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, orgeometry software. Specify a sequence of transformations that will carry a given figure onto another.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 twofigures, use the definition of congruence in terms of rigid motions to decide if they are congruent.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 sidesand corresponding pairs of angles are congruent.8. explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.9. prove theorems about lines an angles. Theorems include: vertical angles are congruent, when a transversal crosses parallel lines, alternateinterior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly thoseequidistant from the segment’s endpoints.10. prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees: base angles o isoscelestriangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the mediansof a triangle meet at a point.

MTH217 Geometry/MTH114 Honors GeometryPage 11PROFICIENCIES (cont’d.)11. prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of aparallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.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; constructingperpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a pointnot on the line.13. construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.14. verify experimentally the properties of dilations given by a center and a scale actor: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through thecenter unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.15. given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similaritytransformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of allcorresponding pairs of sides.16. use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.17. prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, andconversely; the Pythagorean Theorem proved using triangle similarity.18. use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.19. understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometricratios for acute angles.20. explain and use the relationship between the sine and cosine of complementary angles.21. use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.22. prove that all circles are similar.23. identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, andcircumscribed angles; inscribed angles on a diameter are right angles; the radium of a circle is perpendicular to the tangent where the radiusintersects the circle.24. construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.25. 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 ofthe angle as the constant of proportionality; derive the formula for the area of a sector.26. derive the equation of a circle of a given center and radius using the Pythagorean Theorem; complete the square to find the center and radiusof a circle given by an equation.

MTH217 Geometry/MTH114 Honors GeometryPage 12PROFICIENCIES (cont’d.)27. use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given pointsin the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing thepoint (0, 2).28. prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallelor perpendicular to a given line that passes through a given point).29. find the point on a directed line segment between two given points that partitions the segment in a given ratio.30. use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.31. give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Usedissection arguments, Cavalien’s principle, and informal limit arguments.32. use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.33. identify the shapes of two-dimensional cross-sections of three-dimensional objects and identify three-dimensional objects.34. use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder.)*35. apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot.)*36. 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 rations).*

MTH217 Geometry/MTH114 Honors GeometryPROFICIENCY / OBJECTIVESTANDARDSPage 13SUGGESTED ACTIVITYStudents will:Activities incorporatedthroughout the course:PARCC preparation isincorporated on a regular basis.Materials: PARCC tutorial website andpractice examsSAT preparation is incorporatedon a regular basis.Materials: Standardized Test Preparationin text Teacher Resource CD-ROMfrom text SAT Reference SheetThe graphing calculator isincorporated on a regular basis.Geogebra could be used for udents will:Geometry (MTH 217)Informal assessmentsshould include, but arenot limited to Level A,B and C.Honors Geometry(MTH 114H) Informalassessments include,but are not limited toLevel B and C.Technology Standard8.1 is incorporatedthroughout thiscourse.

MTH217 Geometry/MTH114 Honors GeometryI.Page 14CONGRUENCEEssential s):a)Geometric relationships provide a means to make sense of variety of phenomena.b)Angles formed by intersecting lines, parallel lines, and perpendicular lines have special relationships.c)Properties of triangles and parallelograms can be described using the relationship between angles andsegments.d)Shape and area can be conserved during mathematical transformations.e)Geometric properties can be used to construct geometric figures.How can spatial relationship be described by careful use of geometric symbols and language?Which transformations preserve distance and angle measurement?What are the relationships between pairs of angles formed by parallel lines and a transversal?How do you describe a perpendicular bisector?What are the properties and relationships of angles, mid-segments, and medians in a triangle?What are the properties of parallelograms in terms of sides, angles, and diagonals?How does rigid motion of a figure affect the shape and area of the figure?How would you make geometric constructions with a variety of tools and methods?UNIT FOCUSProperties of equality and the laws of logic will be used to prove theorems relating to congruence and angles. Proving triangles congruentthrough two-column, paragraph and flow diagram proofs will be a primary focus.

MTH217 Geometry/MTH114 Honors GeometryPage 15CONGRUENCEPROFICIENCY/OBJECTIVEProficiencies are in parenthesis:Students will:A. Experiment withTransformations in a Plane(1, 6, 7, 12)1. Basic Definitionsa) Points, Lines andPlanesSUGGESTED ACTIVITYSTANDARDSTEACHERNOTESStudents will be able to:G.CO.12G.CO.1MP68.1.12.F.18.2.12.C.4 b) Segments and PE.6 c) Angles and TheirMeasuresG.CO.1G.CO.12d) Segments and AngleBisectorsG.CO.12G.GPE.6e) Angle PairRelationshipsG.CO.1 use basic constructions andGeogebra to reinforcevocabulary and concepts.use basic constructions andGeogebra to create, manipulateand classify angles as acute,right or obtuse.find the midpoint of a segmentand find the measure of anangle that has been bisected.Make a sketch of AB, DE , KL, and PQR1. Find the distancebetween (6,-8) and (10,15).1. Write two names for theangle:ABC1. Find the midpointbetween (-8,5) andbe able to apply the relationship(6,10).between angles such as:vertical, complementary andsupplementary, and linear pairs.define parallel andperpendicular lines.MTH217-69 daysMTH114H-62daysUse straightedgeconstruction and/orGeogebra.

MTH217 Geometry/MTH114 Honors GeometryPage 16CONGRUENCEPROFICIENCY/OBJECTIVEProficiencies are in parenthesis:Students will:f) Parallel andPerpendicular Lines2. Congruence and TrianglesSUGGESTED Students will be able to:G.CO.18.1.12.F.18.2.12.C.4 be able to apply vocabularyinvolving transformations.G.CO.7G.CO.8MP3 prove triangles are congruentand identify correspondingparts.1. Write a congruencestatement for thetriangles below .Identifyall pairs of congruentcorresponding parts.(MTH232)Prove:VWXYZ MNJKL2. Write a congruencestatement for the figurebelow: (MTH114H)Studentscould make avocabularyjournal.

MTH217 Geometry/MTH114 Honors GeometryPage 17CONGRUENCEPROFICIENCY/OBJECTIVEProficiencies are in parenthesis:Students will:3. Proving TrianglesCongruenta. SSS and SASSUGGESTED Students will be able to: identify the theorem needed toprove the given triangles arecongruent.G.CO.8MP3 use various types of proofs toprove triangles are congruent bySSS and SAS (MTH217)1. determine if there isenough information toprove:AB CD, AB CD2. prove:ABC CDAgiven:AB CD, AB CD Stations activity will givea variety of proofpractice.See Appendix H.

MTH217 Geometry/MTH114 Honors GeometryPage 18CONGRUENCEPROFICIENCY/OBJECTIVESUGGESTED ACTIVITYSTANDARDSProficiencies are in parenthesis:Students will:EVALUATION/ASSESSMENTStudents will be able to: use various types of proofs toprove triangles are congruent bySSS, SAS and HL.3. Determine if triangleWVT is congruent totriangle WTU. Writea paragraph of proofto justify yourconclusion.(MTH 114H) WVT WTUb. proving triangles arecongruent by angle-sideangle (ASA), Angle-AngleSide (AAS) and Hypotenuseleg (HL).G.CO.8 use various types of proofs toprove triangles congruent byASA, AAS and HL.1. proveBDM CFMgiven: B C D F M is the midpoint of DFTEACHERNOTES

MTH217 Geometry/MTH114 Honors GeometryPage 19CONGRUENCEPROFICIENCY/OBJECTIVEProficiencies are in parenthesis:Students will:STANDARDSSUGGESTED ACTIVITYEVALUATION/ASSESSMENTTEACHERNOTESStudents will be able to:2. determine and explain ifit is possible to prove thetriangles are congruent,and if so, state thepostulate or theoremthey would use.(MTH 217)3. write a two-columnproof or a paragraph toprove: TQS RSQgiven: TQS RSQ R TUse congruent trianglesto prove correspondingparts of congruenttriangles are congruent(CPCTC)

MTH217 Geometry/MTH114 Honors GeometryPage 20CONGRUENCEPROFICIENCY/OBJECTIVEProficiencies are in parenthesis:Students will:B. Prove Geometric Theorems1. Prove Theoremsa. Lines and Angles(1,9,10,11,12,26,27,28,30)b. Proof andPerpendicularLinesSUGGESTED Students will be able to:G.CO.1G.CO.9G.CO.12MP2 use Geogebra to reinforce thetheorems and construct keyterms. identify the angles created whenparallel lines are cut by atransversal. identify a pair of skew lines. sketch and identifyperpendicular lines.1. sketch a set of parallellines cut by atransversal and label apair of alternate interiorangles.1. show and explain whylines are perpendicular.2. write a proof of theCongruent SupplementsTheorem and show andexplain why lines areperpendicular.Discuss the VerticalAnglesTheorem, along withother anglesrelationships.

MTH217 Geometry/MTH114 Honors GeometryPage 21CONGRUENCEPROFICIENCY/OBJECTIVEProficiencies are in parenthesis:Students will:c. Parallel Lines andTransversalsEVALUATION/ASSESSMENTSUGGESTED ACTIVITYSTANDARDSTEACHERNOTESStudents will be able to:G.CO.9MP7 use Geogebra to investigateparallel lines being cut by atransversal.1. Find the value of x.Use constructions with arule and protractor toverify relationshipsbetween angles.x 203x - 4See Appendix H.2.oxo (x 20)70o3. (MTH114H)x2 145x 20

MTH217 Geometry/MTH114 Honors GeometryPage 22CONGRUENCEPROFICIENCY/OBJECTIVEProficiencies are in parenthesis:Students will:d. Proving Lines AreParallelSUGGESTED Students will be able to:G.CO.9G.CO.12MP2 prove lines are parallel byusing the converse ofcorresponding angle postulate,converse of alternate interiorangle, alternate exterior angle,and consecutive interior angletheorems.1. find the value of x thatmakes a b.(2x 1) o(3x – 5)o2. prove: 1 2given: 1 and 2 aresupplementary.l11l22Teacher may want togroup by level to achievemastery.

MTH217 Geometry/MTH114 Honors GeometryPage 23CONGRUENCEPROFICIENCY/OBJECTIVEProficiencies are in parenthesis:Students will:STANDARDSSUGGESTED ACTIVITYEVALUATION/ASSESSMENTStudents will be able to:3. to shoot the snow as far aspossible, each snowmaker below is set at a 45oangle. The axles of thesnowmakers are allparallel, but the proof isdifficult in 3 dimensions.To simplify the problem,think of the illustration asa flat image on a piece ofpaper. The axles andbarrels are represented inthe diagram to the right.Lines j and 2 intersect atC.Prove: j kGiven: 1 2 , m A m B 45oTEACHERNOTES

MTH217 Geometry/MTH114 Honors GeometryPage 24CONGR

MTH217 GEOMETRY MTH114 HONORS GEOMETRY . A Mathematics Course Outline . Approved by the Board of Education _September 12, 2013_ . contra-positive and bi-conditional, and determine the truth table. (MTH 114H, MTH 217) 2. gain experience in the art of conjecturing, exploring and discovering geometric relationships.