Building to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve realworld and mathematical problems.


 Joanna Sullivan
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1 Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve realwrld 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 realwrld 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 realwrld 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 realwrld 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 twdimensinal 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 twdimensinal figures using crdinates. 8.G.4. Understand that a twdimensinal 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 twdimensinal 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. NCN.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. NCN.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. NCN.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. NVM.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). NVM.2. (+) Find the cmpnents f a vectr by subtracting the crdinates f an initial pint frm the crdinates f a terminal pint. NVM.3. (+) Slve prblems invlving velcity and ther quantities that can be represented by vectrs. Perfrm peratins n vectrs. NVM.4. (+) Add and subtract vectrs. Add vectrs endtend, cmpnentwise, 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 cmpnentwise. NVM.5. (+) Multiply a vectr by a scalar.
4 Represent scalar multiplicatin graphically by scaling vectrs and pssibly reversing their directin; perfrm scalar multiplicatin cmpnentwise, 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. FBF.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. FBF.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 nninvertible functin by restricting the dmain. FBF.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. FTF.4. (+) Use the unit circle t explain symmetry (dd and even) and peridicity f trignmetric functins. Mdel peridic phenmena with trignmetric functins. FTF.5. Chse trignmetric functins t mdel peridic phenmena with specified amplitude, frequency, and midline. FTF.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. FTF.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.