Geometric Relationships



Geometry Outline: Updated 2/2012Blue: Identifying StandardsGreen: New AdditionsRed: Not in the new standardsGeometric Relationships ≈ 4 days G.CO.1Lines and Planes Undefined terms: point, line, planeNotation for point, line, plane, parallel, perpendicularParallel PlanesIdentify parts of postulate: hypothesis, conclusionCollinear, coplanarVertical angles, complementary, supplementaryParallel lines cut by transversals & the angle relationships that ariseApplication of line perpendicular to planeProperties and proofs of parallel lines cut by transversals G.CO.9 (2/12)Review & Assessment Informal and Formal Proofs ≈ 78 + daysLogic – 12 daysLogic Conceptstranslate written expressions to symbolic representation (p, q, V, →, etc)negationconjunction disjunctionconditional, hypothesis, conclusion, hidden conditionalbiconditionaltruth values – tables & wordsconverse (change order)inverse (insert negation)contrapositive (change order and negate) logically equivalentLogic Proofscontrapositivedetachmentsyllogism/chain rule/modus tollensDeMorgan’sdisjunctive inferenceReview & Assessment Triangles - Congruent ≈ 25 days G.CO.7, G.CO.8, G.SRT.5Review & expand (see resources)Corresponding parts (sides, angles, appropriate naming)Identify included side, included angleCongruent () – define & recognizeEstablish the format of a two-column proofSSSSASASAAASHL – right triangleCorresponding Parts of Congruent Triangles are Congruent (CPCTC)Review & AssessmentTriangles – Similar ≈ 15 days G.SRT.5Similar triangles – dilation (conceptual comparison) G.SRT.1, G.SRT.2AA Similar Proofs G.SRT.3SSS Similar Proofs G.SRT.2SAS Similar Proofs G.SRT.2Pythagorean theorem & converse G.SRT.4Mean proportional in Right Triangles (Geometric Mean) G.SRT.4Corresponding Sides of Similar Triangle are in Proportion (CPSTP) G.SRT.2, G.SRT.4Triangle Midsegments G.CO.10Understand 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.6G.SRT.7- sine and cosine relationship of complementary angles (2/12)G.SRT.8- Pythagorean theorem and Trig ratios in applied problems (2/12)Review & AssessmentTriangle Properties ≈ 7 days G.CO.10Angle sum in a triangle (measuring angles with protractors & software)Review classification using acute, obtuse, right, scalene, isosceles & equilateralIsosceles Triangle and PropertiesInvestigate & applyProof & indirect proofExterior Angle TheoremInvestigate & applyProof & indirect proofTriangle Inequality TheoremInvestigate & apply proof & indirect proofReview & AssessmentPolygons 3 daysSum of the measures of the interior angles of a polygonSum of the measures of the exterior angles of a polygonMeasure of an interior angle of a regular polygonMeasure of an exterior angle of a regular polygonClassify polygons by the number of sides and/or anglesReview & AssessmentQuadrilaterals 16 daysProperties of special quadrilaterals G.SRT.5ParallelogramRectangleRhombusSquareTrapezoidIsosceles TrapezoidSolving linear equations using props of quadsSolving quadratic equations using props of quadsGiven a specific quad (parallelogram, rectangle, square, rhombus, trapezoid), prove a specified congruence (angleangle, segmentsegment) G.CO.11Given specific properties, prove a quad is a parallelogram, rectangle, rhombus, square, trapezoid, isosceles trapezoid)Review and Assessment Coordinate Geometry ≈ 15 DaysLinear: Parallel and Perpendicular ≈ 3 days G.GPE.5Parallel lines have equal slopesPerpendicular lines have negative reciprocal slopesVertical and Horizontal lines are perpendicularPoint-Slope form of a lineSlope-Intercept form of a lineLinear: Distance and Midpoint ≈ 3 days *G.GPE.7Length (distance) of a line segment Midpoint of a line segmentFind endpoint given one endpoint and the midpointPerpendicular bisector of a line segment G.CO.9Find the point on a directed line segment between two given points that partitions the segment in a given ratio G.GPE.6Linear: Applications/Informal Proofs ≈ 3 days G.GPE.4Justify algebraically the properties of quadrilaterals and trianglesUsing the distance formula to find lengths of sides and diagonals to classify triangles and quadrilateralsUsing the distance formula to compute perimeter of polygons and areas of triangles and rectangles (G.GPE.7) 2/12Using the midpoint formula:(e.g. show that the diagonals bisect each other)Using the slope formula to find parallel/perpendicular sides/diagonalsCircles ≈ 2 days The center of a circle is an ordered pairThe relationship between the distance formula and the equation of a circleEquation of a circleGraphing a circleDerive the equation of a circle of a given center and radius using the Pythagorean Theorem. G.GPE.1Complete the square to find the center and radius of a circle given by an equation. G.GPE.1Quadratic-Linear Systems ≈ 1 dayLinear EquationQuadratic Equation (including the Circle)Systems of Equations solved GraphicallyChecking SolutionsDerive the equation of a parabola given a focus and a directrix. G.GPE.2Review and Assessment ≈ 3 daysTransformational Geometry ≈ 12 days Reflections ≈ 2 days G.CO.3, G.CO.5, G.CO.6, G.CO.2, G.CO.4Line reflections (x-axis, y-axis, x=a, y=b, , )Point reflections Symmetry G.SRT.2Orientation, preservation of distance, preservation of angle measure, parallelism, perpendicularity, collinearity, midpointOpposite (indirect) isometryDescribe how to reflect a quadrilateral/ regular polygon onto itself (G.CO.3) 2/12Rotations ≈ 2 days G.CO.3, G.CO.5, G.CO.6, G.CO.2, G.CO.4About the origin of and , clockwise and counterclockwiseOrientation, preservation of distance, preservation of angle measure, parallelism, perpendicularity, collinearity, midpointPoint reflection, half-turnDirect isometryDescribe how to rotate a quadrilateral/ regular polygon onto itself (G.CO.3)Translations ≈ 2 days G.CO.5, G.CO.6, G.CO.2, G.CO.4Orientation, preservation of distance, preservation of angle measure, parallelism, perpendicularity, collinearity, midpointDirect isometryDilations ≈ 2 days G.SRT.1a, G.CO.2Similarity (Corresponding sides are proportional, corresponding angles are congruent) G.SRT.2, G.SRT.3Factor of dilation(positive/negative, enlargement/reduction) G.SRT.1, G.SRT.1a, G.SRT.1bComposition of Transformations (proper notation required) ≈ 2 days G.CO.6, G.CO.2Glide ReflectionOrientation, preservation of distance, preservation of angle measure, parallelism, perpendicularity, co-linearity, midpoint2 or more of a reflection, rotation, translation, or dilationDeconstruction of Compositions (Writing a rule) G.CO.5Review and Assessment ≈ 2 daysCircles ≈ 20 DaysProve that all circles are similar G.C.1Relationships of angles and arcs of a circle G.C.2CentralInscribedInterior angle formed by 2 intersecting chordsExterior angle formed by secants and/or tangentsAngle formed by tangent and chord on the circleAngle formed by tangent and diameter/radiusRelationship of segments that intersect circlesIntersection of two chords in the circleFind the measures of secants and tangents (including problems involving quadratic equations)Two tangents to the same circle from the same exterior pointRelationship of Arcs and ChordsArc measure (degrees & radians G.C.5) G.CO.1Derive the formula for the area of a sector G.C.5Congruent chords and arcsChords equidistant from the center of the circlePerpendicular bisector of a chordCommon tangents of two non-intersecting or tangent circlesInscribed and Circumscribed polygons G.C.3 (2/12)Review and Assessment ≈ 3 daysGeometric Relationships ≈ 7 daysSolids *G.MG.1Use nets to describe prisms, pyramids and conesAnalysis of 2 and 3 dimensional units of measureProperties, area, volume, (surface area, lateral area review from 7th Gr) of: G.GMD.1, *G.GMD3ConeCylinderPrismsPyramidsSphere Analyze and solve verbal problemsIdentify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects G.GMD.4 (Modeling?)Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* G.MG.2Apply geometric methods to design problems (G.MG.3)Review & Assessment Constructions and Locus ≈ ? daysConstructions ≈ 3 days G.CO.12Angle bisectordefinition of angle bisectorcongruenceRadiiAngleTriangle (CPCTC)measurementTools of construction and their useProtractor (validating conjectures)Perpendicular bisectordefinition of perpendicular linesbisect a segmentright anglesproperties of isosceles triangles (median, altitude, angle bisector to base)measurementtools of construction and their useprotractor (validating conjectures)Parallel or perpendicular lines thru a given point G.CO.92 points determine a unique lineMethod of proving lines parallelCopy an angle (construct congruent angles)If alternate interior angles are congruent, then the lines are parallel.Construct lines perpendicular using:Point on the linePoint NOT on the line Equilateral triangles Interior angle sum of a triangleTypes of trianglesRadii of the same circle are congruentEquilateral vs. equiangularInscribed in a circle G.CO.13Equilateral TriangleSquareRegular HexagonLocus ≈ 7 days Centers related to a triangle Complete appropriate constructions to locate the:Incenter (angle bisectors)Centroid (medians) G.CO.10Orthocenter (altitudes)Circumcenter (perpendicular bisectors)Identify Euler’s line (optional)Compound loci 5 Fundamental LociGiven distance from a pointEquidistant from 2 pointsGiven distance from a lineEquidistant from 2 parallel linesEquidistant from 2 intersecting linesSolve problems involving 2 or more fundamental loci, including centers of triangles (reference G.G.21)Review and Assessment ≈ 2 daysApplication of Probablility~10 days1.??????? Probability (7 days) a. Define and solve problems using the following1.???? sample space 2.???? simple probability of a single event3.???? probability with “and” (single event)4.???? probability with “or” (single event)5.???? complement6.???? empirical probability (based on specific sample data)* 7.???? impossible events8.???? certain events9.???? Fundamental Counting Principleb. Analyze a set of events to determine when:1. some or all are equally likely to occur2. one is more likely to occur3. an event is certain to happen or not to happen c. Conditional Probability 1. with replacement2. without replacement d. Probability of a series of independent and dependent events 1.???? and2.???? or 3.???? mutually exclusive4.???? not mutually exclusive4.???? Review and Assessment (3 days) ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download