1 Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define a crdinate system, with the intersectin f the lines (the rigin) arranged t cincide with the 0 n each line and a given pint in the plane lcated by using an rdered pair f numbers, called its crdinates. Understand that the first number indicates hw far t travel frm the rigin in the directin f ne axis, and the secnd number indicates hw far t travel in the directin f the secnd axis, with the cnventin that the names f the tw axes and the crdinates crrespnd (e.g., x- axis and x- crdinate, y- axis and y- crdinate). Grade 6: Gemetry Slve real-wrld and mathematical prblems invlving area, surface area, and vlume. 6.G.3. Draw plygns in the crdinate plane given crdinates fr the vertices; use crdinates t find the length f a side jining pints with the same first crdinate r the same secnd crdinate. Apply these techniques in the cntext f slving real- wrld and mathematical prblems. Grade 6: The Number System Apply and extend previus understandings f numbers t the system f ratinal numbers. 6.NS.5. Understand that psitive and negative numbers are used tgether t describe quantities having ppsite directins r values (e.g., temperature abve/belw zer, elevatin abve/belw sea level, credits/debits, psitive/negative electric charge); use psitive and negative numbers t represent quantities in real-wrld cntexts, explaining the meaning f 0 in each situatin. 6.NS.6. Understand a ratinal number as a pint n the number line. Extend number line diagrams and crdinate axes familiar frm previus grades t represent pints n the line and in the plane with negative number crdinates. Recgnize ppsite signs f numbers as indicating lcatins n ppsite sides f 0 n the number line; recgnize that the ppsite f the ppsite f a number is the number itself, e.g., ( 3) = 3, and that 0 is its wn ppsite.
2 Understand signs f numbers in rdered pairs as indicating lcatins in quadrants f the crdinate plane; recgnize that when tw rdered pairs differ nly by signs, the lcatins f the pints are related by reflectins acrss ne r bth axes. Find and psitin integers and ther ratinal numbers n a hrizntal r vertical number line diagram; find and psitin pairs f integers and ther ratinal numbers n a crdinate plane. 6.NS.8. Slve real-wrld and mathematical prblems by graphing pints in all fur quadrants f the crdinate plane. Include use f crdinates and abslute value t find distances between pints with the same first crdinate r the same secnd crdinate. Grade 8: Gemetry Understand cngruence and similarity using physical mdels, transparencies, r gemetry sftware. 8.G.1. Verify experimentally the prperties f rtatins, reflectins, and translatins: a. Lines are taken t lines, and line segments t line segments f the same length. b. Angles are taken t angles f the same measure. c. Parallel lines are taken t parallel lines. 8.G.2. Understand that a tw-dimensinal figure is cngruent t anther if the secnd can be btained frm the first by a sequence f rtatins, reflectins, and translatins; given tw cngruent figures, describe a sequence that exhibits the cngruence between them. 8.G.3. Describe the effect f dilatins, translatins, rtatins, and reflectins n tw-dimensinal figures using crdinates. 8.G.4. Understand that a tw-dimensinal figure is similar t anther if the secnd can be btained frm the first by a sequence f rtatins, reflectins, translatins, and dilatins; given tw similar tw-dimensinal figures, describe a sequence that exhibits the similarity between them.
3 High Schl: Number and Quantity Represent cmplex numbers and their peratins n the cmplex plane. N-CN.4. (+) Represent cmplex numbers n the cmplex plane in rectangular and plar frm (including real and imaginary numbers), and explain why the rectangular and plar frms f a given cmplex number represent the same number. N-CN.5. (+) Represent additin, subtractin, multiplicatin, and cnjugatin f cmplex numbers gemetrically n the cmplex plane; use prperties f this representatin fr cmputatin. Fr example, ( i) 3 = 8 because ( i) has mdulus 2 and argument 120. N-CN.6. (+) Calculate the distance between numbers in the cmplex plane as the mdulus f the difference, and the midpint f a segment as the average f the numbers at its endpints. Represent and mdel with vectr quantities. N-VM.1. (+) Recgnize vectr quantities as having bth magnitude and directin. Represent vectr quantities by directed line segments, and use apprpriate symbls fr vectrs and their magnitudes (e.g., v, v, v, v). N-VM.2. (+) Find the cmpnents f a vectr by subtracting the crdinates f an initial pint frm the crdinates f a terminal pint. N-VM.3. (+) Slve prblems invlving velcity and ther quantities that can be represented by vectrs. Perfrm peratins n vectrs. N-VM.4. (+) Add and subtract vectrs. Add vectrs end-t-end, cmpnent-wise, and by the parallelgram rule. Understand that the magnitude f a sum f tw vectrs is typically nt the sum f the magnitudes. Given tw vectrs in magnitude and directin frm, determine the magnitude and directin f their sum. Understand vectr subtractin v w as v + ( w), where w is the additive inverse f w, with the same magnitude as w and pinting in the ppsite directin. Represent vectr subtractin graphically by cnnecting the tips in the apprpriate rder, and perfrm vectr subtractin cmpnent-wise. N-VM.5. (+) Multiply a vectr by a scalar.
4 Represent scalar multiplicatin graphically by scaling vectrs and pssibly reversing their directin; perfrm scalar multiplicatin cmpnent-wise, e.g., as c(v x, v y ) = (cv x, cv y ). Cmpute the magnitude f a scalar multiple cv using cv = c v. Cmpute the directin f cv knwing that when c v 0, the directin f cv is either alng v (fr c > 0) r against v (fr c < 0). High Schl: Functins Build new functins frm existing functins. F-BF.3. Identify the effect n the graph f replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) fr specific values f k (bth psitive and negative); find the value f k given the graphs. Experiment with cases and illustrate an explanatin f the effects n the graph using technlgy. Include recgnizing even and dd functins frm their graphs and algebraic expressins fr them. F-BF.4. Find inverse functins. Slve an equatin f the frm f(x) = c fr a simple functin f that has an inverse and write an expressin fr the inverse. Fr example, f(x) =2 x 3 r f(x) = (x+1)/(x 1) fr x 1. (+) Verify by cmpsitin that ne functin is the inverse f anther. (+) Read values f an inverse functin frm a graph r a table, given that the functin has an inverse. (+) Prduce an invertible functin frm a nn-invertible functin by restricting the dmain. F-BF.5. (+) Understand the inverse relatinship between expnents and lgarithms and use this relatinship t slve prblems invlving lgarithms and expnents. Extend the dmain f trignmetric functins using the unit circle. F-TF.4. (+) Use the unit circle t explain symmetry (dd and even) and peridicity f trignmetric functins. Mdel peridic phenmena with trignmetric functins. F-TF.5. Chse trignmetric functins t mdel peridic phenmena with specified amplitude, frequency, and midline. F-TF.6. (+) Understand that restricting a trignmetric functin t a dmain n which it is always increasing r always decreasing allws its inverse t be cnstructed. F-TF.7. (+) Use inverse functins t slve trignmetric equatins that arise in mdeling cntexts; evaluate the slutins using technlgy, and interpret them in terms f the cntext.
5 High Schl: Gemetry Experiment with transfrmatins in the plane G.CO.1. Knw precise definitins f angle, circle, perpendicular line, parallel line, and line segment, based n the undefined ntins f pint, line, distance alng a line, and distance arund a circular arc. G- CO.2. Represent transfrmatins in the plane using, e.g., transparencies and gemetry sftware; describe transfrmatins as functins that take pints in the plane as inputs and give ther pints as utputs. Cmpare transfrmatins that preserve distance and angle t thse that d nt (e.g., translatin versus hrizntal stretch). G- CO.3. Given a rectangle, parallelgram, trapezid, r regular plygn, describe the rtatins and reflectins that carry it nt itself. G- CO.4. Develp definitins f rtatins, reflectins, and translatins in terms f angles, circles, perpendicular lines, parallel lines, and line segments. G- CO.5. Given a gemetric figure and a rtatin, reflectin, r translatin, draw the transfrmed figure using, e.g., graph paper, tracing paper, r gemetry sftware. Specify a sequence f transfrmatins that will carry a given figure nt anther. Understand cngruence in terms f rigid mtins G- CO.6. Use gemetric descriptins f rigid mtins t transfrm figures and t predict the effect f a given rigid mtin n a given figure; given tw figures, use the definitin f cngruence in terms f rigid mtins t decide if they are cngruent. G- CO.7. Use the definitin f cngruence in terms f rigid mtins t shw that tw triangles are cngruent if and nly if crrespnding pairs f sides and crrespnding pairs f angles are cngruent. G- CO.8. Explain hw the criteria fr triangle cngruence (ASA, SAS, and SSS) fllw frm the definitin f cngruence in terms f rigid mtins. Prve gemetric therems G- CO.9. Prve therems abut lines and angles. Therems include: vertical angles are cngruent; when a transversal crsses parallel lines, alternate interir angles are cngruent and crrespnding angles are cngruent; pints n a perpendicular bisectr f a line segment are exactly thse equidistant frm the segment s endpints. G- CO.10. Prve therems abut triangles. Therems include: measures f interir angles f a triangle sum t 180 ; base angles f issceles triangles are cngruent; the segment jining midpints f tw sides f a triangle is parallel t the third side and half the length; the medians f a triangle meet at a pint. G- CO.11. Prve therems abut parallelgrams. Therems include: ppsite sides are cngruent, ppsite angles are cngruent, the diagnals f a parallelgram bisect each ther, and cnversely, rectangles are parallelgrams with cngruent diagnals.