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GEOMETRY Congruence Experiment with transformations in the plane. 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-CO1] 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). [G-CO2] 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [G-CO3] 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [G-CO4] 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. [G-CO5] Core What Does It Mean? Pre-Requisite Skills What Does Mastery Future Math Standard Look Like? 1 Know precise definitions -Exposure to (recognize -Students will be Conics of angle, circle, picture of or be able to able to define, Unit Circle perpendicular line, parallel draw a picture of) an draw, and Systems of equations line, and line segment, angle, circle, recognize picture of (graphing ) based on the undefined perpendicular line, selected definitions notions of point, line, parallel line, line (an angle, circle, distance along a line, and segment, point, line, perpendicular line, distance around a circular distance along a line, parallel line, line arc. and distance around a segment, point, circular arc. line, distance along a line, and distance around a circular arc). 2 Represent transformations -Vocabulary -Students should be Vectors using a variety of methods (transformation, able to complete a Graph transformations (ELMO, Geometer’s coordinate, distance, representation of a Matrices Sketchpad); using the angles, vertices) transformation and coordinates of two figures, -Coordinate system (x- compare distance describe the change in x axis, y-axis, origin, plot, and angles of both and y values; compare point) figures distance and angles of -Introduction to both figures Geometer’s Sketchpad 3 Given a rectangle, -Vocabulary (rectangle, -Students should be Calculus rotations ( about parallelogram, trapezoid, parallelogram, able to describe the axes ) or regular polygon, trapezoid, regular rotations and Pre-Cal/algebra graph describe the rotations and polygon, rotations, and reflections on the transformations reflections that carry it reflections) coordinate plane. Matrices onto itself. -Coordinate system (x- axis, y-axis, origin, plot, point) -Basic degree changes -Know how to describe changes in the x and y values 4 Develop definitions of -Vocabulary (angles, -Students should be Calculus rotations ( about rotations, reflections, and circles, perpendicular able to identify the axes ) translations in terms of lines, parallel lines, and examples of Pre-Cal/algebra graph angles, circles, line segments) rotations, transformations perpendicular lines, -Recognize examples reflections, and Matrices parallel lines, and line of rotations, translations and segments. translations, and figures of angles, reflections circles, -Recognize figures of perpendicular lines, angles, circles, parallel lines, and perpendicular lines, line segments to parallel lines, and line develop their own segments definitions. 5 Given a geometric figure -Recognize geometric -Students should be Calculus rotations ( about and a rotation, reflection, figures. able to draw a the axes ) or translation, draw the -Recognize rotation, transformed figure Pre-Cal/algebra graph transformed figure using, reflection, and using graph paper, transformations e.g., graph paper, tracing translation. tracing paper, or Matrices paper, or geometry -Coordinate system (x- geometry software. software. Specify a axis, y-axis, origin, plot, -Students should be sequence of point) able to describe the transformations that will order of carry a given figure onto another. transformations that will carry a given figure onto another on the coordinate plane. Understand congruence in terms of rigid motions. (Build on rigid motions as a familiar starting point for development of concept of geometric proof) 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-CO6] 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-CO7] 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. [G-CO8] Core What Does It Mean? Pre-Requisite Skills What Does Mastery Future Math Standard Look Like? 6 Use translation, rotation, -translation Students will Pre-Engineering reflection -rotation demonstrate that Computer Science -reflection they can take a figure and Animation -Congruence and apply a transformation. 7 Show that a triangle that -Vocabulary (rotation, -Students should be Pre-Engineering has been rotated, reflection, translation, able to explain why a Computer Science reflected, and translated is angles, corresponding triangle that has and Animation still congruent to the sides, corresponding been rotated, original because they have angles, congruency) reflected, or corresponding pairs of translated is still sides and angles. congruent to the original. 8 Use rotation, reflection, -Vocabulary (rotation, -Students should be Pre-Engineering and translation to prove reflection, translation, able to prove a Computer Science triangle congruence (ASA, congruency, angle, triangle that has and Animation SAS, and SSS). side) been rotated, Trigonometry ( -Coordinate system (x- reflected, or understanding law axis, y-axis, origin, translated is of sine, cosine) plot, point) congruent to the Unit Circle original by ASA, SAS, or SSS. Prove geometric theorems. (Focus on validity of underlying reasoning while using variety of ways of writing proofs) 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-CO9] 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180º; 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-CO10] 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. [G-CO11] Core What Does It Mean? Pre-Requisite Skills What Does Mastery Future math Standard Look Like? 9 Recognize angle pairs and -Vocabulary Students will be Architecture what is true when the lines Vertical angles able to develop and are parallel. Use theorems Transversal solve algebraic and definition of bisectors. Parallel lines equations based on Congruent the relationships Alternate interior that exist between angles the angles / Corresponding segments angles Perpendicular Students will be Bisector able to prove the Segment alternate exterior Equidistant angle theorem by End point using the Same-Side interior corresponding angles angles postulate Alternate exterior and vertical angles. angles Supplementary Students should be -Developing able to use the equations based bisector definition on known and bisector relationships theorems in related proofs. 10 Properties of triangles will -Vocabulary Students will be Unit Circle be discovered through Interior angles able to use more Trigonometry logical reasoning. This Base angles than one method to Word problems includes the fact that the Isosceles triangles prove the triangle Engineering sum of the interior angles Midsegment sum theorem; Architecture of a triangle equal 180 Medians Students will be Art degrees, the base angles able to prove the of an isosceles triangle are base angles congruent, and a theorem; prove the midsegment is parallel to midsegment the third side and half its theorem length. 11 Prove theorems about -Vocabulary Students will be Word problems parallelograms Parallelogram able to use a variety Engineering Congruent of methods to Architecture Opposite angles prove relationships Art Diagonals about Bisect parallelograms Rectangles Opposite Sides -Triangle congruence theorems AAS, SAS etc - CPCTC 12 Use a compass and -Vocabulary Students will be Engineering straight edge, paper Circle Center able to accurately CAD folding, reflectors, Arc do the following Architecture software (Geometer’s Angle constructions: Construction/design Sketchpad) to create Perpendicular - Copy a geometric constructions Bisector segment Parallel - Copy an Segment angle Radius - Bisect a -Skills segment Use a compass - Bisect an and straight edge angle - Perpendicul ar lines - Parallel lines ( through a pt not on the line) - Perpendicul ar bisector of a segment 13 Construct an equilateral -Vocabulary Students will be Unit circle triangle, square, and Equilateral able to accurately Calculus regular hexagon inscribed triangle construct an Engineering in a circle Square equilateral triangle, Regular square, and regular Inscribed hexagon inscribed Circle in a circle. Hexagon 14 Verify experimentally the -Vocabulary Students will be Engineering properties of dilations Scale factor able to draw and CAD given by a center and a Dilation describe dilations ( Architecture scale factor Center of dilation properties) using Graphing Parallel Line various tools ( Transformations Ratio software/compass & straight edge) 15 Determine if two figures -Vocabulary Students will be Daily life are similar, explain why Similarity able to determine (proportion two triangles are similar. Similarity and explain why concept) transformation two triangles are All other Math Similar similar using classes Proportional transformations Corresponding and corresponding angles parts. Corresponding sides Using cross products to solve proportions 16 Use the properties of -Vocabulary Students will be CAD/ Engineering similarity transformations Similar able to discover, Design to establish the AA Transformation recognize, and use College Algebra criterion for two triangles -sum of angles of a the AA criterion for to be similar triangle triangle similarity. 17 Prove theorems about SAS, AAS, SAS, SSS, Students will be Proof concepts used triangles HL able to prove in all future math Vocabulary theorems about courses/life -Hypotenuse triangles including Unit Circle -Angle but not limited to Calculus -Side the Triangle angle Trigonometry -Congruent bisector theorem Physics -proportion and the -similarity Pythagorean -Pythagorean theorem using Theorem triangle similarity. 18 Using the fact that two -proving triangles Students will be Calculus triangles are congruent or are congruent able to produce a Proof concepts used similar to prove other facts -Using CPCTC proof in which in all future math about various geometric -Recognizing several steps are courses figures. properties of other required including geometric figures using congruent and / or similar triangles. 19 Trigonometric ratio -right triangle Students will be Algebra II definitions can be derived -sine,cosine, able to Precalculus from the similarity of the tangent demonstrate the Calculus side ratios in right -opposite, connection Probability/Stats triangles as properties of adjacent, between the Alg. Connections the angles. hypotenuse trigonometric ratios Trigonometry -ratio and the side ratios -similarity in right triangles as a property of the acute angles. 20 Explain and use the -sine Students will Unit Circle relationship between the -cosine demonstrate that Trigonometry sine and cosine of -complementary the sine of one of Calculus complementary angles -opposite, the two acute Algebra II adjacent, angles is the cosine Alg. Connections hypotenuse of the other and vice versa. 21 *Use the trigonometric -sine Students will be Unit Circle ratios and the -cosine able to use the Trigonometry Pythagorean theorem to -tangent trigonometric ratios Calculus solve right triangles in -acute angles to solve real world Algebra II applied problems. -right triangle problems such as Alg. Connections measuring ‘tall’ objects, finding a distance, etc. 22 +Derive the formula -vocabulary Students will be Algebra II A=1/2abSin(C) by drawing -sine able to draw an Precalculus an auxiliary line from a -auxiliary line auxiliary line in a Calculus vertex perpendicular to -vertex triangle in such a Trigonmetry the opposite side -perpendicular way that the Physics -opposite side formula -derive A=1/2abSin(C) is derived. 23 +Prove the laws of Sines -sine Students will be Algebra II and Cosines and use them -cosine able to prove the Precalculus to solve problems -acute angle Law of Sines and Calculus Cosines. Trigonometry Additionally, they Physics will be able to use them to solve problems. 24 +Understand and apply -Law of Sines Students will be Algebra II the Law of Sines and -Law of Cosines able to use these Precalculus Cosines to find unknown -Acute angles Laws to solve Calculus measurements in any problems such as Trigonometry triangle. surveying problems, Physics resultant forces, etc. The student should be able to use any kind of triangle. 25 Prove that all circles are -circle Students will be Conics similar -radius able to Trigonometry -diameter demonstrate or PreCalculus -circumference explain why all -pi circles are similar. 26 Identify and describe -radius Students will be Engineering relationships among -chord able to describe and Computer inscribed angles, radii, and -circle use the relationship Animations chords -inscribed between inscribed, Vectors -circumscribed central, and Unit Circle -central angle circumscribed Calculus -right angle angles, show that Trigonometry -perpendicular angles inscribed in a Physics -tangent semicircle are right -intersection angles (inscribed on a diameter), radius is tangent to a tangent at the intersection point. 27 Construct the inscribed -inscribed Students will be Engineering and circumscribed circles -circumscribed able to use either CAD of a triangle and prove -quadrilateral compass and Architecture properties of angles for a -opposite, straight edge or quadrilateral inscribed in a consecutive angles software to prove circle. properties of inscribed and circumscribed triangles. 28 +Construct a tangent line -tangent Students will use a Calculus from a point outside a -circle compass and PreCalculus given circle to the circle straight edge or software to construct the tangent to a circle from a point outside the circle. 29 Derive using similarity the -similarity Students will be College Algebra fact that the length of the -sector ( Area ) able to show that Unit Circle arc intercepted by an -arc ( length ) the length of the arc angle is proportional to -circle is proportional to the radius and define the -proportion the radius and the radian measure of the -radius radian measure is angle as the constant of -intercepted arc the constant of proportionality; derive the -radius proportionality. formula for the area of a -radian measure Students should be sector. -constant able to derive the formula for the area of a sector. 30 Derive the equation of a -circle equation Students will be Algebra II circle given a center and -completing the able to derive the PreCalculus radius using the square equation of a circle Conics Pythagorean theorem. -Pythagorean with given radius Complete the square to theorem and center using find the center and radius -radius the Pythagorean of a circle given by an -circle center theorem. Students equation. -diameter will be able to complete the square to find the center and radius of a circle. 31 Use coordinates to prove -solving equations Students should be Algebra II simple geometric -coordinate proofs able to use algebra Precalculus theorems algebraically. -slope and algebraic -parallel & formulas to prove perpendicular lines or disprove -Pythagorean properties of circles Theorem and quadrilaterals. -distance formula ( between points) -quadrilateral definitions -quadrilateral properties -graphing circles 32 Prove the slope criteria for -slope Students will be All future math parallel and perpendicular -parallel able to determine courses lines and use them to -perpendicular and use the solve geometric problems. -graphing lines relationship -equations of lines between the slopes of parallel and perpendicular lines. 33 Find the point on a -line segment Students will be Algebra II directed line segment -point able to locate a between two given points -ratio point on a segment that partitions the -proportion using a ratios and segment in a given ratio. proportions 34 Use coordinates to -distance formula Students will be All future math compute perimeters of -perimeter able to calculate courses polygons and areas of -polygon perimeters of triangles and rectangles. -area polygons and areas -triangle of rectangles and -rectangle triangles by using the distance formula for example. 35 Determine areas and -perimeter Students will be All future math perimeters of regular -area able to calculate the courses polygons including -inscribed perimeter and areas inscribed and -circumscribed of regular polygons. circumscribed polygons, -polygon given the coordinates of -vertex vertices or other characteristics 36 Give an informal argument -circle Students will be Algebra II for the formulas for the -circumference able to use Physics circumference of a circle, -area dissection Calculus area of a circle, volume of -volume arguments, PreCalculus a cylinder, pyramid, and -cylinder cavalieri’s principle, cone. -pyramid and informal limit -cone arguments to derive or defend the formulas for circumference, area, and volume. 37 Use volume formulas for -volume Students will be Calculus cylinders, pyramids, cones, -cylinder able to use the PreCalculus and spheres to solve -cone volume formulas for problems -sphere cylinders, cones, and spheres to solve problems. 38 Determine the relationship -similarity Students will be Calculus between surface areas of -area able to use the fact Construction similar figures and -volume that two figures are Algebra II volumes of similar figures -ratio similar to calculate Word Problems -proportion volume or area. Architecture 39 Identify the shapes of two- -circle Students will be Computer dimensional cross-sections -ellipse able to determine Animation of three-dimensional -parabola the names of cross Computer Science objects, and identify 3-D -hyperbola sections of 3-D CAD objects generated by -square objects. Airforce/military rotations of two- -rectangle tests dimensional objects. -parallelogram Conics -triangle -various and sundry 3-D objects 40 Use geometric shapes, -solid geometry Students will be CAD their measures, and their shapes able to use Computer Science properties to describe -volume composite figures Art objects -area to model and Architecture -measuring estimate area and volume of various 3-D objects. 41 Apply concepts of density -density Students will be Physics based on area and volume -ratios able to model Chemistry in modeling situations ( -proportions density using Precalculus persons per square miles, -area geometric concepts. Calculus BTUs per cubic feet) -volume 42 Apply geometric methods ( possible ) Students will be Physics to solve design problems -area able to use Architecture (e.g., designing an object -perimeter geometric concepts Engineering or structure to satisfy -minimum to design solutions CAD physical constraints or -maximum to real life Design/planning minimize cost; working -distance problems. This Real life budget with typographic grid -measurements would be a good management systems based on ratios).* opportunity to use graphing calculators, spreadsheets, or GSP 43 Understand the -Probability Students will be Algebra II conditional probability of -independent able to calculate the PreCalculus A given B as P(A and variable probability of Probability/Stats B)/P(B), and interpret -dependent independent of independence of A and B variable events. as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 44 Construct and interpret -frequency table Students will be Statistics two-way frequency tables -tally marks able to collect, tally, Economics/Govern of data when two -probability and interpret ment/Social Studies categories are associated -independent statistical data. with each object being variable classified. Use the two- -dependent way table as a sample variable space to decide if events -random are independent and to approximate conditional probabilities. 45 Recognize and explain the -probability Students should be Statistics concepts of conditional -independent able to compare Business probability and variable and analyze Probability independence in everyday -dependent statistical data to Economics language and everyday variable determine if events situations. are dependent or independent. 46 Find the conditional -conditional Students should be Statistics probability of A given B as probability able to find the Business the fraction of B's -model/modeling probability of Probability outcomes that also belong -fractions dependent events. Economics to A, and interpret the -outcome answer in terms of the model. 47 Apply the Addition Rule, -probability Students will be Statistics P(A or B) = P(A) + P(B) - P(A -independent able to model and Business and B), and interpret the events also show Probability answer in terms of the -dependent events mathematically the Economics model. -fractions probability of events A or B. 48 (+) Apply the general -probability Students will be Statistics Multiplication Rule in a -independent able to model and Business uniform probability model, events also show Probability P(A and B) = P(A)P(B|A) = -dependent events mathematically the Economics P(B)P(A|B), and interpret -fractions probability of the answer in terms of the events A and B. model. 49 (+) Use permutations and -combination Students should be Computer Science ( combinations to compute -permutation able to calculate a Gaming industry) probabilities of compound -probability combination and a Statistics events and solve permutation and Probability problems. solve related Business problems 50 (+) Use probabilities to -probability Students should be Government make fair decisions (e.g., -random able to use Statistics drawing by lots, using a probability to make Economics random number a fair decision in a Probability generator). given random situation. 51 (+) Analyze decisions and -probability Students will be Government strategies using probability able to use Statistics concepts (e.g., product probability to Economics/ testing, medical testing, determine Marketing pulling a hockey goalie at appropriate Probability the end of a game). strategies. Sports and/or Medicine

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posted: | 3/2/2012 |

language: | English |

pages: | 11 |

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