ACADEMIC STANDARDS AND BENCHMARKS MATHEMATICS

Transcription

1 Table f Cntents Mathematics... 4 By the end f Grade By the end f Grade By the end f Grade By the end f Grade Cmmunicatin Grade Grade Grade Grade Grade Cnnectins Grade Grade Grade Grade Grade Data Analysis Pre-Kindergarten Kindergarten Grade Grade Grade Grade Grade Gemetry and Measurement BACK TO TOP 1

2 Grade Grade Grade Grade Grade Measurement Pre-Kindergarten Kindergarten Number Operatins Grade Grade Grade Grade Grade Number Sense Pre-Kindergarten Kindergarten Grade Grade Grade Grade Grade Patterns/Algebra Pre-Kindergarten Kindergarten Grade Grade Grade BACK TO TOP 2

3 Grade Grade Prblem Slving Grade Grade Grade Grade Grade Reasning Grade Grade Grade Grade Grade Representatin Grade Grade Grade Grade Grade Shape and Space Pre-Kindergarten Kindergarten BACK TO TOP 3

4 Mathematics By the end f Grade 3 1. Students will apply a wide variety f mathematical cncepts, prcesses, and reasning skills t investigate, evaluate, justify and slve a brad range f prblems in varius cntent areas and everyday situatins. a. Frmulate a prblem, determine infrmatin required t slve the prblem, chse methds fr btaining this infrmatin, and set limits fr acceptable slutins. b. Demnstrate that there may be multiple ways t slve a prblem and explain why this is s. c. Understand that there is n ne right way t slve mathematical prblems but that different methds (e.g., wrking backward frm a slutin, using a similar prblem type, identifying a pattern) have different advantages and disadvantages. d. Transfer strategies frm a prir prblem t a new situatin. e. Use trial and errr and the prcess f eliminatin t slve prblems. f. Verify the crrectness and reasnableness f simple mathematical results. 2. Students will select and use a wide variety f tls and technlgy t supprt and validate mathematical results. a. Represent and examine mathematical situatins using cncrete materials and cmputers. b. Use a fur-functin r fractin calculatr t cnfirm cmputatins and t explre patterns. c. Use a variety f standard tls (e.g., rulers, clcks, measuring tapes, thermmeters) and nn-standard bjects (e.g., cunters, sticks, blts), t measure mathematical and physical bjects in the envirnment. 3. Students will understand and apply numbers, ways f representing numbers, relatinships amng numbers, and number systems. a. Cnnect physical, verbal, and symblic representatins f whle numbers, fractins and mixed numbers. b. Use drawings, diagrams, and mdels t shw the cncept f fractins as part f a whle and part f a set. c. Explain hw numbers are used in varius ways, including cunting, rdering, representing quantities, measuring, labeling, and indicating lcatin. d. Explain hw numbers are used in varius ways, including cunting, rdering, representing quantities, measuring, labeling, and indicating lcatin. BACK TO TOP 4

5 e. Apply place-value cncepts and numeratin t describe, cmpare, cunt, rder, and grup numbers. f. Explain the cnnectins between peratins. g. Use cncrete bjects t cunt, rder, grup, and demnstrate ne-t-ne crrespndence with whle numbers beynd 100. h. Identify patterns in number sequences (identify even and dd numbers, cunt by 2s, 3s, 5s, 10s, and 25s). i. Read, write, and rder numbers t 10, Students will estimate, cmpute, and assess reasnableness f slutins. a. Demnstrate prficiency with and memrize additin and subtractin facts thrugh 20 and multiplicatin facts thrugh 10. b. Add and subtract single- and multi-digit whle numbers with regruping. c. Apply additin and subtractin in a variety f situatins (such as cmputing perimeter, extending functins). d. Multiply multi-digit whle numbers by single-digit numbers. e. Divide tw-digit whle numbers by single-digit numbers. f. Demnstrate the cncept f multiplicatin as repeated additin and arrays; demnstrate the cncept f divisin as repeated subtractin and as sharing. g. Understand and apprpriately use the vcabulary f estimatin (such as abut, near, between). h. Use a variety f mental cmputatinal methds, strategies, and estimatin skills t find slutins and t determine the reasnableness f calculated answers, including thse invlving cncrete and abstract items and situatins, such as time and mney. i. Determine the value f a set f hst cuntry currency and U.S. currency. j. Read, write, add, and subtract with decimal ntatin in situatins invlving mney. 5. Students will estimate and measure t a required degree f accuracy and precisin by selecting and using apprpriate units, tls, and technlgies. a. Estimate befre measuring t determine the reasnableness f a slutin. b. Estimate and measure length, time, temperature, and weight t the nearest unit using custmary, metric, and nnstandard measurement. c. Cmpare and rder measurable characteristics (fr example, time, temperature, length, weight, capacity, area, perimeter) f different bjects n the same dimensins. d. Tell time t the minute with bth analg and digital clcks. e. Determine elapsed time t the hur using AM and PM. BACK TO TOP 5

6 f. Find the perimeter and area f rectangles with direct methds, including using cncrete bjects as tls. g. Recgnize the need fr a unifrm unit f measure. 6. Students will use algebraic methds t represent, analyze, and slve abstract and practical mathematical situatins invlving patterns and functinal relatinships. a. Recgnize, reprduce, extend, create, and describe repeating and increasing patterns and sequences using a variety f materials. b. Use tables, graphic rganizers, verbal rules, and pen sentences t describe patterns and ther relatinships. c. Generate and slve simple functins by identifying and applying additin and subtractin patterns. d. Generate, write, and slve pen sentences using infrmal methds (such as using manipulatives, drawing, r acting ut the slutin). e. Use cncrete bjects and symbls t mdel the cncepts f variables, expressins, equatins, and inequalities (fr example, find the missing number, symbl, r peratin sign). f. Identify and describe numeric patterns and make predictins based n them (e.g., 1 bicycle = 2 wheels, s 6 bicycles = hw many wheels). 7. Students will use spatial reasning and apply the prperties and relatinships f gemetric figures t represent, investigate, analyze, and slve prblems. a. Use cmparative directinal and psitinal wrds (such as abve, inside, left, hrizntal, middle). b. Describe, name, and label related gemetric tw- and three-dimensinal shapes (such as circle and sphere, square and cube, triangle and pyramid, rectangle and prism). c. Draw tw-dimensinal gemetric shapes and cnstruct rectangles, squares, and triangles using tls (such as gebards, grid paper, ruler, cmpass), including representatin f side, tp, and bttm views f the bject. d. Cnstruct three-dimensinal gemetric shapes, including bxes and triangular prisms. e. Identify and describe gemetric figures in the envirnment. f. Identify and create examples f line symmetry. g. Order simple gemetric figures by size. h. Estimate and determine the perimeter and area f gemetric figures using manipulatives; demnstrate cnservatin f area. i. Describe, identify, and mdel slides, flips, and turns with gemetric figures. 8. Students will use statistical methds t questin, cllect, rganize, analyze, and represent data in rder t make decisins and predictins. BACK TO TOP 6

7 a. Pse a questin and cllect data by bserving, measuring, surveying, and cunting b. Cnstruct, read, interpret, and label graphs, including pictgraphs, simple bar and line graphs, and pie charts. c. Interpret data by lking fr patterns and relatinships, determining range, cnsidering cause and effect, then drawing cnclusins and answering related questins. 9. Students will understand and apply basic cncepts f prbability. By the end f Grade 5 a. Predict and measure the utcme f events, and explain why the results f an experiment may nt match predicted utcmes. b. Use cncepts f certainty, fairness, and chance t discuss the prbability f actual events. 1. Students will apply a wide variety f mathematical cncepts, prcesses, and reasning skills t investigate, evaluate, justify and slve a brad range f prblems in varius cntent areas and everyday situatins. a. Knw hw t select and use mathematical tls and methds (such as manipulatives, mental math, calculatr, cmputer, and paper-and-pencil techniques) as a part f the prblem-slving prcess. b. Develp and apply a variety f prblem-slving strategies (fr example, make an rganized list, guess-and-check) and justify chice f strategies. c. Interpret results in the cntext f the prblem being slved (fr example, when determining the number f buses necessary t transprt students, the remainder must be runded up). d. Differentiate between relevant and irrelevant infrmatin. e. Understand hw t break a cmplex prblem int simpler parts. 2. Students will select and use a wide variety f tls and technlgy t supprt and validate mathematical results. a. Use calculatrs r sftware t verify estimatins and in prblem-slving situatins. b. Use technlgy such as spreadsheets, cameras, science prbe, r calculatrs t gather, analyze, and display mathematical data and infrmatin. 3. Students will understand and apply numbers, ways f representing numbers, relatinships amng numbers, and number systems. a. Mdel and cnnect physical, verbal, and symblic representatins f fractins,decimals, percentages, whle numbers, and mixed numbers. b. Order fractins, decimals, and whle numbers using physical, verbal, and symblic representatins. c. Recgnize the relatinship amng fractins, decimals, and percentages. BACK TO TOP 7

8 d. Use cncepts f negative numbers in cncrete situatins (such as n a number line, with temperature). e. Identify and describe different uses fr the same numerical representatin (fr f. Use, mdel, and identify place value and describe its relatinship t magnitude. example, 1/4 can represent a fractin, a divisin prblem, r a rati) and different representatins fr the same number (fr example, 2,343 is the same as 2, ; and 1 equals 16/16). g Demnstrate that mathematical peratins can represent a variety f prblem situatins (fr example, multiplicatin can represent repeated additin and a mdel fr finding area). h. Explain the relative effect f peratins with fractins and decimals (fr example, what happens t 10 when yu divide by 1/2 r multiply by.75). i. Explain, derive, cmpare, and use prperties f peratins and relatinships amng peratins. i. Explain and apply number thery cncepts (such as primes, multiples, and cmpsites). j. Read, write, and rder numbers t a millin. 4. Students will estimate, cmpute, and assess reasnableness f slutins. a. Demnstrate prficiency with and memrize multiplicatin and divisin facts thrugh 12. b. Select and use the mst efficient cmputatinal methds, chsing amng cncrete materials, paper and pencil, estimatin, mental cmputatin, and calculatrs. c. Create and slve practical prblems invlving additin, subtractin, multiplicatin and divisin. d. Additin, subtractin, multiplicatin, and divisin f whle numbers, fractins, and mixed numbers. e. Develp, analyze, and cmpare algrithms fr cmputing with fractins, decimals, percents, and integers and cmpute with them efficiently and accurately, including in multi-step prblems that require applicatin f rder f peratins. f. Knw and cnvert amng fractins, decimals, and percents fr 1/10, 1/5, 1/4, 1/2, and 3/4. g. Apply beginning number thery including identifying and using multiples, factrs, divisibility, prperties f identity (zer and ne), and prime and cmpsite numbers. h. Apply, explain, and assess the apprpriateness f a variety f estimatin strategies (such as runding t cmpatible numbers). BACK TO TOP 8

9 i. Use varius frms f estimatin, including runding, t determine the reasnableness f calculated answers; determine if an estimate is t high r t lw. j. Use a variety f strategies t make change and slve prblems using U.S. and hst cuntry s currency, and t cnvert between hst cuntry and U.S. currency. 5. Students will estimate and measure t a required degree f accuracy and precisin by selecting and using apprpriate units, tls, and technlgies. a. Select and use apprpriate instruments and custmary and metric units fr measuring quantities, including perimeter, vlume, area, weight, time, and temperature, with specified accuracy; match tls with the attribute they measure (fr example, rulers measure length, thermmeters measure temperature). b. Understand and apply frmulas fr finding perimeter, vlume f simple slids (excluding cylinders), and area. c. Add and subtract measurements (e.g., 12 m. 6.2 m.). d. Identify and use equivalent measurements as required by the situatin (fr example, 60 minutes = 1 hur, 7 days = 1 week). e. Identify the apprximate size f basic standard units f measurement and the relatinship between them (fr example, there are 100 centimeters in a meter). f. Slve calendar prblems invlving days, weeks, mnths, and years. g. Determine and cmpare elapsed time using AM and PM and a 24-hur clck. 6. Students will use algebraic methds t represent, analyze, and slve abstract and practical mathematical situatins invlving patterns and functinal relatinships. a. Use patterns and their extensins t make predictins and slve prblems. b. Use rules and variables t describe patterns, functins, and ther relatinships and t slve equatins. c. Generate and slve simple functins by identifying and applying multiplicatin and divisin patterns. d. Find slutins t inequalities frm a given replacement set. e. Slve simple equatins using methds such as inverse peratins, mental math, and guess-and-check. f. Use cncrete bjects and cmbinatins f symbls and numbers t create expressins that mdel mathematical situatins. g. Understand the basic characteristics f a 2-dimensinal crdinate system. 7. Students will use spatial reasning and apply the prperties and relatinships f gemetric figures t represent, investigate, analyze, and slve prblems. BACK TO TOP 9

10 a. Lcate and describe bjects in terms f their psitin with and withut cmpass directins; identify crdinates fr a given pint r lcate pints f given crdinates n a single quadrant grid. b. Cmpare, cntrast, and describe plane and slid figures and shapes using their attributes (such as number f sides, parallel r perpendicular sides, number f vertices, classificatin f angles). c. Sketch and identify line segments, midpint, intersectins, and parallel and perpendicular lines. d. Identify, draw, and measure, using a prtractr, right, btuse, and acute angles and their parts, including rays, pints, and vertices. e. Identify and mdel gemetric figures that are cngruent, similar, r symmetrical r sme cmbinatin f these prperties. f. Identify the diameter, radius, chrd, and circumference f a circle. g. Determine area and perimeter, finding bth using a variety f methds. h. Analyze and mdel transfrmatins f gemetric figures and rtatins f line segments, describing the mtins as slides, flips, r rtatins. 8. Students will use statistical methds t questin, cllect, rganize, analyze, and represent data in rder t make decisins and predictins. a. Slve prblems that invlve systematically cllecting, rganizing, and analyzing data. b. Discuss the apprpriateness f different types f data displays, and use a variety f displays (such as tables, histgrams, graphs). c. Interpret data, using the arithmetic mean, median, mde, range, and make cnvincing arguments based n data analysis and previus experiences. d. Find all pssible utcmes f a simple experiment using straightfrward methds (such as rganized lists, tree diagrams). 9. Students will understand and apply basic cncepts f prbability. By the end f Grade 8 a. Make predictins based n intuitive, experimental, and theretical prbabilities. b. Cnduct simple prbability experiments using cncrete materials (e.g. tssing ne r mre cins, spinning a spinner f even r uneven divisins, drawing bjects frm a cntainer with and withut replacement) and represent the results using fractins and prbability. 1. Students will apply a wide variety f mathematical cncepts, prcesses, and reasning skills t investigate, evaluate, justify and slve a brad range f prblems in varius cntent areas and everyday situatins. BACK TO TOP 10

11 a. Pse, explre, and slve a variety f prblems, including thse that are nnrutine r have a variety f pssible strategies r slutins r bth, in rder t build new mathematical knwledge. b. Develp, mdify, and apply an increasing variety f prblem-slving techniques t slve prblems (fr example, wrking backward, infrmatin rganizers, r slving a similar but simpler prblem). c. Try varius prblem-slving appraches befre selecting and using a strategy and reflect n different strategies used when a task is cmplete. 2. Students will select and use a wide variety f tls and technlgy t supprt and validate mathematical results. a. Use a variety f technlgies, including cmputers, scientific calculatrs, graphing calculatrs, science prbes, and digital cameras t evaluate and validate prblem slutins. b. Recgnize situatins when calculatr use is nt apprpriate (fr example, when slving a simple quadratic equatin which culd be factred) r when it yields misleading results (fr example, when a nn-linear curve appears linear). 3. Students will understand and apply numbers, ways f representing numbers, relatinships amng numbers, and number systems. a. Demnstrate a cnceptual understanding f irratinal numbers and be able t slve prblems invlving ratinal numbers (e.g., area f circles, wrking with radicals). b. Describe hw percent, rati, and prprtin apply t mathematical situatins (such as rate, similar triangles). c. Recgnize and apply multiple representatins f ratinal and irratinal numbers, expnents, abslute values, and scientific ntatin; cmpare these numbers accurately, find their apprximate lcatins n a number line, and chse apprpriate frms f these numbers. d. Demnstrate an understanding f numbers that represent large and small values, including the use f benchmarks t cmprehend their magnitude; and recgnize, understand, and apprpriately use varius representatins fr large numbers. e. Represent and explain the effect f peratins n psitive and negative numbers. f. Add, subtract, multiply, and divide fractins, decimals, percents, integers, and nnnegative whle number expnents. g. Understand and use the inverse relatinships f additin and subtractin, multiplicatin and divisin, and squaring and finding square rts t simplify cmputatins and slve prblems. h. Use factrs, multiples, and prime factrizatin t slve prblems. BACK TO TOP 11

12 i. Recgnize and use the assciative and cmmutative prperties f additin and multiplicatin, and the distributive prperty f multiplicatin ver additin t simplify cmputatins with ratinal numbers. j. Mdel and cnnect physical, verbal, and symblic representatins f real numbers by hand and/r with a graphing calculatr. 4. Students will estimate, cmpute, and assess reasnableness f slutins. a. Slve prblems using rates and understand rate as a unit f measure. b. Use algrithms fr cmputing with fractins, decimals, percents, and integers and cmpute them efficiently and accurately with and withut a calculatr. c. Use multi-step cmputatinal prcedures with ratinal and irratinal numbers. d. Estimate the value f irratinal numbers. e. Develp, analyze, and explain methds t slve prblems invlving prprtins and percents (such as scaling, finding equivalent ratis). f. Cmpute circumference, area, surface area, and vlume f gemetric figures; find missing dimensins f right triangles using the Pythagrean therem. g. Estimate the value f tips, discunts, and taxes using hst cuntry and U.S. currency. h. Explain and apply the rules f divisibility, square numbers, prime factrizatin, and the prperties f zer with the rder f peratins. i. Determine what a reasnable degree f accuracy wuld be in particular situatins (e.g., great precisin is required in scientific experiments, but much less is required in estimating a grcery bill). 5. Students will estimate and measure t a required degree f accuracy and precisin by selecting and using apprpriate units, tls, and technlgies. a. Estimate and measure angles and use frmulas t find perimeter, area, and circumference f plane figures and the vlume and surface area f prisms, pyramids, and cylinders t a specified degree f accuracy. b. Select and use apprpriate units and tls t measure length, area, vlume, angle, and weight t apprpriate levels f precisin. c. Cnvert measurements within and between mnetary systems and within and between metric and custmary systems and demnstrate an understanding f the relatinship between units in metric and custmary systems. d. Select and apply indirect methds f measurement including frmulae, gemetric ideas and relatinships (such as cngruence, similarity and the Pythagrean therem) and algrithms fr mathematical relatinships (such as scale, prprtins, rate). e. Select and use a variety f methds and tls t cnstruct and cmpare plane figures f given measures. f. Apply infrmatin abut time znes and elapsed time t slve prblems. BACK TO TOP 12

13 6. Students will use algebraic methds t represent, analyze, and slve abstract and practical mathematical situatins invlving patterns and functinal relatinships. a. Analyze, create, and generalize numeric and visual patterns. b. Describe relatinships between symblic expressins and graphs n the crdinate plane, with particular attentin t the hrizntal and vertical intercepts, pints f intersectin, and slpe (fr linear relatins). c. Analyze functinal relatinships t explain hw a change in ne quantity results in a change in anther (fr example, the relatinship amng length, area, and vlume). d. Find the value f a variable by evaluating frmulas and algebraic expressins fr given values (fr example, if an bject has a length f 4 and an area f 28, what is the bject s width?). e. Rewrite frmulas in terms f the missing variable (fr example, "if an bject has a length f 4 and an area f 28, what is the bject s width?" can be expressed as 4W=28 r as W=28/4). f. Create expressins, equatins, and inequalities t represent prblem situatins and t slve prblems invlving linear relatinships. g. Demnstrate fluency in generating equivalent expressins fr simple algebraic expressins and in slving linear equatins and inequalities. h. Demnstrate a basic understanding f rate f change, including cnnectins between slpe f a line and cnstant rate f change and their meaning in cntext. 7. Students will use spatial reasning and apply the prperties and relatinships f gemetric figures t represent, investigate, analyze, and slve prblems. a. Describe the relatinship between an equatin and its graph. b. Use crdinate gemetry t represent and interpret relatinships defined by equatins and frmulas (fr example, distance, mid-pint), translating amng rdered pairs, graphs, and equatins. c. Mdel, classify, cmpare, and sketch a variety f tw- and three-dimensinal regular and irregular figures. d. Apply prperties f equality and prprtinality t slve prblems invlving cngruent r similar shapes. e. Describe and apply gemetric prperties and relatinships (such as cngruence, perpendicularity). f. Describe and apply a variety f strategies fr determining circumference, perimeter, area, surface area, angle measure, and vlume. g. Explain and apply the Pythagrean therem. h. Draw and describe the results f transfrmatins, including translatins, rtatins, reflectins, and dilatins (shrinking r enlarging), using prper ntatin. BACK TO TOP 13

14 8. Students will use statistical methds t questin, cllect, rganize, analyze, and represent data in rder t make decisins and predictins. a. Frmulate and slve prblems by cllecting, rganizing, analyzing (including cmparing and cntrasting), and displaying data in a variety f ways (including stem and leaf plts, histgrams, whisker plts, surveys, circle graphs) by hand and with technlgy. b. Find, describe, and interpret the arithmetic mean, median, and mde as measures f the center f a data set; select the apprpriate measure in particular situatins t mst accurately and fairly represent the data. c. Describe ways t define a sample grup; analyze a sample t make inferences abut a ppulatin. d. Design and cnduct a simulatin t study a prblem and cmmunicate the results. 9. Students will understand and apply basic cncepts f prbability. By the end f Grade 12 a. Determine theretical prbability using a variety f methds, including creating a sample space; cmpare theretical expectatins t experimental results. b. Design, cnduct, and analyze the results f prbability experiments. 1. Students will apply a wide variety f mathematical cncepts, prcesses, and reasning skills t investigate, evaluate, justify and slve a brad range f prblems in varius cntent areas and everyday situatins. a. Explre the validity and efficiency f varius prblem-psing and prblemslving strategies; develp alternative strategies and generalizatins as needed. b. Mnitr prgress tward slutins. c. Generalize strategies and reflect n their prficiency and merit. 2. Students will select and use a wide variety f tls and technlgy t supprt and validate mathematical results. a. Use graphing calculatrs and cmputer sftware effectively and efficiently t define and slve varius types f prblems. 3. Students will understand and apply numbers, ways f representing numbers, relatinships amng numbers, and number systems. a. Explain the effect f peratins n measurements (fr example, the imprecise nature f measurement is amplified with multiplicatin). b. Understands the cncept f infinity. 4. Students will estimate, cmpute, and assess reasnableness f slutins. a. Manipulate algebraic prcedures with real and cmplex numbers. BACK TO TOP 14

15 b. Apply factrials, expnents, and matrices t slve practical prblems. c. Cmpute permutatins and cmbinatins. d. Assess the errr resulting frm estimatin and runding, using bth custmary and metric units. e. Estimate algebraic slutins. f. Determine when t use exact value slutins and distinguish between exact value and apprximate values. 5. Students will estimate and measure t a required degree f accuracy and precisin by selecting and using apprpriate units, tls, and technlgies. a. Incrprate units int all aspects f measurement prblems and determine the apprpriateness f a slutin based upn dimensinal analysis. b. Explain the relatinship amng errr, precisin, and accuracy in measurement, including the cmpunding f errrs in calculatins. c. Evaluate the accuracy and precisin f measurements resulting frm the measuring tls and methds chsen. d. Apply indirect methds, such as ratis and trignmetry, t find missing dimensins. e. Interpret varius internatinal measurement systems (such as the Richter Scale, decibels) t describe phenmena and slve prblems. 6. Students will use algebraic methds t represent, analyze, and slve abstract and practical mathematical situatins invlving patterns and functinal relatinships. a. Define functins and their prperties and find the inverse f a functin; understand the relatinship between a functin and its inverse. b. Create and slve linear and quadratic equatins and inequalities. c. Add, subtract, multiply, divide, and simplify ratinal and irratinal expressins; add, subtract, multiply and divide plynmials. d. Identify, graph, and describe the graphs f basic families f functins including linear, abslute value, quadratic, expnential, and trignmetric functins and explain why a variety f phenmena can be mdeled by the same type f functin. e. Slve systems f equatins and inequalities. f. Frmulate expressins, equatins, inequalities, systems f equatins, systems f inequalities, and matrices t mdel prblem situatins. g. Use matrices t rganize and manipulate data, including matrix additin, subtractin, multiplicatin, and scalar multiplicatin. 7. Students will use spatial reasning and apply the prperties and relatinships f gemetric figures t represent, investigate, analyze, and slve prblems. BACK TO TOP 15

16 a. Be familiar with the graphs f the fllwing equatins and be able t apply them t prblem slving: y = sinx; y = csx; y = tanx. b. Use crdinate gemetry t graph linear and quadratic equatins, determine slpes f lines, identify parallel and perpendicular lines, and find pssible slutins t sets f equatins. c. Create tw-dimensinal representatins f three-dimensinal bjects (e.g., draw a basic cube). d. Cnstruct gemetric mdels, transfrmatins, and scale drawings using a variety f methds and tls (such as paper flding r prtractr). e. Identify cngruent and similar figures; apply this infrmatin t slve prblems f. Use basic trignmetric ratis and trignmetric laws (triangle trignmetry) t slve prblems invlving indirect measurement. g. Use vectr methds, matrices, and transfrmatins 8. Students will use statistical methds t questin, cllect, rganize, analyze, and represent data in rder t make decisins and predictins. a. Describe the attributes f several cmmn distributins (e.g., nrmal, unifrm,pissn, expnential), indicating data sets that wuld be expected t fllw each type f distributin b. Determine regressin equatins t mdel and draw inferences frm data; summarize and interpret single-variable data by chsing measures f central tendency and dispersin. c. Analyze the validity f statistical cnclusins and the use, misuse, and abuse f data caused by chices f scale, inapprpriate chices f central tendency, incrrect curve fitting, r inapprpriate use f cntrl grups. d. Pse questins; cllect, rganize, and represent data t answer thse questins. 9. Students will understand and apply basic cncepts f prbability. a. Describe the nrmal curve in general terms and use its prperties t answer questins abut sets f data. b. Find the prbability f simple events, cmpund events, and independent events using a variety f methds including the fundamental cunting principle. Cmmunicatin Grade 1 1. Express mathematical ideas cherently and clearly t peers, teachers, and thers (e.g., with verbal ideas, mdels r manipulatives, pictures, r symbls). BACK TO TOP 16

17 2. Extend mathematical knwledge by cnsidering the thinking and strategies f thers (e.g., agree r disagree, rephrase anther student s explanatin, analyze anther student s explanatin). 3. Relate manipulatives, pictures, diagrams, and symbls t mathematical ideas. 4. Represent, discuss, write, and read mathematical ideas and cncepts. Start by relating everyday language t mathematical language and symbls and prgress tward the use f apprpriate terminlgy (e.g., add mre becmes plus, balance the equatin becmes slve the equatin ). Grade 2 Grade 3 1. Express mathematical ideas cherently and clearly t peers, teachers, and thers (e.g., with verbal ideas, mdels r manipulatives, pictures, r symbls). 2. Extend mathematical knwledge by cnsidering the thinking and strategies f thers (e.g., agree r disagree, rephrase anther student s explanatin, analyze anther student s explanatin). 3. Relate manipulatives, pictures, diagrams, and symbls t mathematical ideas. 4. Represent, discuss, write, and read mathematical ideas and cncepts. Start by relating everyday language t mathematical language and symbls and prgress tward the use f apprpriate terminlgy (e.g., add mre becmes plus, repeated additin becmes multiplicatin, fair share becmes divide, balance the equatin becmes slve the equatin ). 1. Express mathematical ideas cherently and clearly t peers, teachers, and thers (e.g., with verbal ideas, mdels r manipulatives, pictures, r symbls). 2. Extend mathematical knwledge by cnsidering the thinking and strategies f thers (e.g., agree r disagree, rephrase anther student s explanatin, analyze anther student s explanatin). 3. Relate manipulatives, pictures, diagrams, and symbls t mathematical ideas. 4. Represent, discuss, write, and read mathematical ideas and cncepts. Start by relating everyday language t mathematical language and symbls and prgress tward the use f apprpriate terminlgy (e.g., add mre becmes plus, repeated additin becmes multiplicatin, fair share becmes divide, balance the equatin becmes slve the equatin ). BACK TO TOP 17

18 Grade 4 1. Relate manipulatives, pictures, diagrams, and symbls t mathematical ideas. 2. Represent, discuss, write, and read mathematical ideas and cncepts. Start by relating everyday language t mathematical language and symbls and prgress tward the use f apprpriate terminlgy (e.g., add mre becmes plus, repeated additin becmes multiplicatin, fair share becmes divide, balance the equatin becmes slve the equatin ). Grade 5 1. Express mathematical ideas cherently and clearly t peers, teachers, and thers (e.g., with verbal ideas, mdels r manipulatives, pictures, r symbls). 2. Extend mathematical knwledge by cnsidering the thinking and strategies f thers (e.g., agree r disagree, rephrase anther student s explanatin, analyze anther student s explanatin). 3. Relate manipulatives, pictures, diagrams, and symbls t mathematical ideas. 4. Represent, discuss, write, and read mathematical ideas and cncepts. Start by relating everyday language t mathematical language and symbls and prgress tward the use f apprpriate terminlgy (e.g., add mre becmes plus, repeated additin becmes multiplicatin, fair share becmes divide, balance the equatin becmes slve the equatin ). Cnnectins Grade 1 1. Relate varius cncrete and pictrial mdels f cncepts and prcedures t ne anther (e.g., use tw clrs f cubes t represent additin facts fr the number 5, cunt by 2s, 5s, 10s) 2. Link cncepts t prcedures and eventually t symblic ntatin (e.g., represent actins like snap, clap, clap with symbls A B B, divide a candy bar int 3 equal pieces that represent ne piece as 1/3). 3. Recgnize relatinships amng different tpics within mathematics (e.g., the length f an bject can be represented by a number) 4. Use mathematical strategies t slve prblems that relate t ther curriculum areas and the real wrld (e.g., use a timeline t sequence events, BACK TO TOP 18

19 use symmetry in art wrk, explre fractins in quilt designs and t describe pizza slices). Grade 2 Grade 3 Grade 4 1. Relate varius cncrete and pictrial mdels f cncepts and prcedures t ne anther (e.g., use tw clrs f cubes t represent additin facts fr the number 5, relate patterns n a hundreds chart t multiples, use base-10 blcks t represent decimals). 2. Use mathematical strategies t slve prblems that relate t ther curriculum areas and the real wrld (e.g., use a timeline t sequence events, use symmetry in art wrk, explre fractins in quilt designs and t describe pizza slices). 1. Relate varius cncrete and pictrial mdels f cncepts and prcedures t ne anther (e.g., use tw clrs f cubes t represent additin facts fr the number 5, relate patterns n a hundreds chart t multiples, use base-10 blcks t represent decimals). 2. Recgnize relatinships amng different tpics within mathematics (e.g., the length f an bject can be represented by a number, multiplicatin facts can be mdeled with gemetric arrays, ½ can be written as.5 and 50%). 3. Use mathematical strategies t slve prblems that relate t ther curriculum areas and the real wrld (e.g., use a timeline t sequence events, use symmetry in art wrk, explre fractins in quilt designs and t describe pizza slices). 1. Relate varius cncrete and pictrial mdels f cncepts and prcedures t ne anther (e.g., use tw clrs f cubes t represent additin facts fr the number 5, relate patterns n a hundreds chart t multiples, use base-10 blcks t represent decimals). 2. Link cncepts (e.g., fractins, multiplicatin, divisin) t prcedures and eventually t symblic ntatin (e.g., demnstrate 3 4 with a gemetric array, divide a candy bar int 3 equal pieces that represent ne piece as 1/3). 3. Recgnize relatinships amng different tpics within mathematics (e.g., the length f an bject can be represented by a number, multiplicatin facts can be mdeled with gemetric arrays, ½ can be written as.5 and 50%). BACK TO TOP 19

20 4. Use mathematical strategies t slve prblems that relate t ther curriculum areas and the real wrld (e.g., use a timeline t sequence events, use symmetry in art wrk, explre fractins in quilt designs and t describe pizza slices). Grade 5 1. Relate varius cncrete and pictrial mdels f cncepts and prcedures t ne anther (e.g., use tw clrs f cubes t represent additin facts fr the number 5, relate patterns n a hundreds chart t multiples, use base-10 blcks t represent decimals). 2. Link cncepts t prcedures and eventually t symblic ntatin (e.g., represent actins like snap, clap, clap with symbls A B B, demnstrate 3 4 with a gemetric array, divide a candy bar int 3 equal pieces that represent ne piece as ). 3. Recgnize relatinships amng different tpics within mathematics (e.g., the length f an bject can be represented by a number, multiplicatin facts can be mdeled with gemetric arrays, can be written as.5 and 50%). 4. Use mathematical strategies t slve prblems that relate t ther curriculum areas and the real wrld (e.g., use a timeline t sequence events, use symmetry in art wrk, explre fractins in quilt designs and t describe pizza slices). Data Analysis Pre-Kindergarten 1. Develp grwing abilities t cllect, describe, and recrd infrmatin thrugh a variety f means, including discussin, drawings, maps, charts and graphs. 2. Describe similarities and differences between bjects Kindergarten 1. Cllect and analyze infrmatin abut bjects and events in the envirnment. 2. Create and verbally explain a data display r graph (e.g., real bject graph, pictrial graphs) BACK TO TOP 20

21 Grade 1 Grade 2 Grade 3 Grade 4 1. Organize, describe, and display data using cncrete bjects, pictures, grids, and numbers. 2. Frmulate and slve prblems that invlve cllecting and analyzing data cmmn t children s lives (e.g., clr f shes, numbers f pets, favrite fds). 1. Cllect, srt, rganize, and display data in charts, bar graphs, and tables (e.g., cllect data n teeth lst and display results in a chart). 2. Summarize and interpret data in charts, bar graphs, and tables. 3. Make predictins and estimates t describe data (e.g., predict what data n teeth lst might lk like fr yunger children and/r lder children). 1. Pse questins, cllect, recrd, and interpret data t help answer questins (e.g., Which was the mst ppular bth at ur carnival?). 2. Read graphs and charts; identify the main idea, draw cnclusins, make predictins based n the data (e.g., predict hw many children will bring their lunch based n a menu). 3. Cnstruct a bar graph r pictgraph with labels and a title frm a set f data. 1. Examine data displays such as tallies, tables, charts and graphs and use the bservatins t pse and answer questins (e.g., chse a table in scial studies f ppulatin data and write prblems). 2. Cllect, rganize and recrd data in tables and graphs (e.g., bar, pictgraph, line plts). 3. Investigate and recrd prbabilities by experimenting with devices that generate randm utcmes (e.g., cins, number cubes, spinners). BACK TO TOP 21

22 Grade 5 1. Analyze data t create and interpret tables and graphs. 2. Justify the selectin f the type f table r graph (e.g., a line graph may be mre apprpriate than a bar graph when displaying the height f a persn ver time). 3. Cmpare and translate between displays f data (e.g., multiple sets f data n the same graph, Venn diagrams, a cmbinatin f diagrams, charts, tables, graphs). 4. Frmulate questins, design investigatins, cnsider samples, and cllect, rganize, and analyze data using bservatin, measurement, surveys, r experiments (e.g., hw far can 5th graders thrw a sftball based n where it first hits the grund?). 5. Determine the range (spread) and the mean (average) f a set f data. 6. Determine the prbability f events ccurring in familiar cntexts r experiments and express prbabilities as fractins (e.g., find the fractinal prbability f an event given a biased spinner). 7. List permutatins and cmbinatins f up t five items. Gemetry and Measurement Grade 1 1. Srt and identify cngruent shapes. 2. Cmpare bjects by size and quantity (e.g., mre than, less than, equal t). 3. Identify tw-dimensinal gemetric shapes and bjects in everyday situatins (e.g., the face f a rund clck is a circle, a desktp is a rectangle). 4. Use language t describe relatinships f bjects in space (e.g., abve, belw, behind, between). 5. Tell time n digital and analg clcks t the hur and half-hur. 6. Identify and name the value f lcal currency 7. Use the calendar t develp the cncepts f days, weeks, and mnths. 8. Identify slid figures and srt by faces and vertices. Grade 2 BACK TO TOP 22

23 1. Srt and classify symmetric and cngruent figures. 2. Identify tw-dimensinal gemetric shapes in everyday situatins (e.g., a stp sign is an ctagn). 3. Measure bjects with nnstandard and standard units (e.g., use a human ft [nnstandard] then a ruler [standard] t measure length t the nearest inch). 4. Select and use apprpriate units f measurement in prblem slving and everyday situatins. 5. Tell time n digital and analg clcks t the quarter-hur. 6. Identify and cunt mney; cnnect cins and bills with place value. Grade 3 Grade 4 1. Describe and cmpare tw- and three-dimensinal shapes (e.g., cunt the edges and faces f a cube, cmbine r divide basic shapes t frm new shapes, identify and draw cngruent shapes). 2. Identify lcatins n a grid with rdered pairs (e.g., give the lcatin f a ship n a grid by selecting D, 1). 3. Slve prblems with custmary units invlving length using half-inch and quarter-inch measurements and weight using pund and unce. 4. Slve prblems with metric units invlving length using meter and centimeter and mass using gram and kilgram. 5. Use manipulatives t develp the cncept f perimeter and area (e.g., cver a shape with pattern blcks t find area). 6. Develp and use strategies t estimate measurements (e.g., use parts f the bdy as benchmarks fr measuring length). 7. Tell time n digital and analg clcks t 5 minutes and use infrmatin t slve prblems invlving time and temperature (e.g., read a thermmeter). 1. Identify, draw, and cnstruct mdels f intersecting, parallel, and perpendicular lines (e.g., use spaghetti, straws, tthpicks). 2. Identify and cmpare angles equal t, less than, r greater than 90 degrees (e.g., use right angles t determine the apprximate size f ther angles; make a variety f angles using flexible straws and cmpare). 3. Identify basic characteristics f the rectangular crdinate system and find the distance between hrizntal and vertical lines f a rectangular crdinate system (e.g., the x-axis is the hrizntal axis). BACK TO TOP 23

24 4. Describe the effects n tw- and three-dimensinal bjects when they slide (translate), flip (reflect), and turn (rtate) (e.g., tessellatins). 5. Establish benchmarks fr custmary and metric units and estimate the measures f a variety f bjects and cmpare units (e.g., mass: the mass f a raisin is abut 1 gram, length: the width f a finger is abut 1 centimeter). 6. Select apprpriate custmary and metric units f measure t slve applicatin prblems invlving length, weight, mass, and vlume. 7. Slve applicatin prblems invlving mney, time and temperature (e.g., elapsed time). Grade 5 1. Identify and describe the basic prperties f figures (e.g., tw r three-dimensinality, symmetry, number f faces, types f angles). 2. Find the perimeter f simple plygns and area f a rectangle (e.g., use 1-inch tiles t build rectangles f different perimeters and areas). 3. Use nnstandard units (beans, rice, candies) and standard units (centimeter cubes, 1-inch cubes) t find the vlume f rectangular slids and estimate the vlume f ther slids. 4. Use the apprpriate units and tls t estimate and measure temperature, distance, length, weight, and angles. 5. Cnvert basic measurements f vlume, weight and distance within the same system fr metric and custmary units (e.g., inches t feet, hurs t minutes, centimeters t meters). Measurement Pre-Kindergarten 1. Measure bjects using nn-standard units f measurement (e.g., pencil, paper clip, blck) 2. Cmpare bjects accrding t bservable attributes (e.g., lng, lnger, lngest) 3. Develp an awareness f simple time cncepts within his/her daily life (e.g., yesterday, tmrrw) 4. Sequence events in the cntext f daily activities Kindergarten BACK TO TOP 24

25 Number Operatins Grade 1 1. Measure bjects using nnstandard units f measurement (e.g., pencil, paper clip, blck) 2. Cmpare bjects accrding t bservable attributes (e.g., lng, lnger, lngest, shrt, shrter, shrtest, big, bigger, biggest, small, medium and large) 3. Cmpare and rder bjects in graduated rder (e.g., shrtest t tallest, thinnest t thickest) 4. Identify the apprpriate instrument used t measure length (ruler), weight (scale), time (clck: digital/analg, calendar: day, mnth, year, seasn), and temperature (thermmeter) 5. Identify days f the week and mnths f the year 1. Use mdels t cnstruct additin and subtractin facts thrugh 18 (e.g., cunters, cubes). 2. Perfrm additin by jining sets f bjects and subtractin by separating and by cmparing sets f bjects. 3. Demnstrate fluency with basic additin and subtractin facts (i.e., memrize and apply additin and subtractin facts) thrugh 18. Cmpute efficiently and accurately with single-digit numbers, duble digit with n carrying. 4. Recgnize and apply the cmmutative and identity prperties f additin using mdels and manipulatives t develp cmputatinal skills (e.g., = 4 + 2, = 3). 5. Write additin and subtractin number sentences fr prblem-slving situatins. 6. Acquire strategies fr making cmputatins (e.g., use estimatin, number sense t judge reasnableness, cunting n). Grade 2 1. Demnstrate fluency with basic additin and subtractin facts (i.e., memrize and apply additin and subtractin facts) and fact families t Slve tw-digit additin and subtractin prblems with and withut regruping using a variety f techniques (e.g., cncrete, paper and pencil, mental math). BACK TO TOP 25

26 3. Develp peratin sense by applying the assciative prperty f additin (e.g., 3 + (2 +1) = (3 + 2) + 1 ). 4. Describe the relatinship between additin and subtractin (e.g., additin and subtractin are inverse peratins). 5. Use mental strategies (r decmpsitin strategies) fr additin and subtractin (e.g., make a grup f 10 bjects and 2 bjects frm a grup f 7 bjects and 5 bjects). 6. Cmplete additin number sentences with a missing addend and use t slve everyday prblems. Grade 3 1. Estimate, find the sum and difference, with and withut regruping, f 3- and 4-digit numbers t slve applicatin prblems. 2. Demnstrate fluency with basic multiplicatin facts and fact families (i.e., memrize and apply multiplicatin facts). 3. Develp multiplicatin algrithms (e.g., use physical materials t shw 4 grups f 3 bjects, shw multiplicatin as repeated additin). 4. Estimate the prduct f 2-digit numbers by runding t the nearest multiple f 10 t slve applicatin prblems. 5. Recgnize and apply the cmmutative and identity prperties f multiplicatin using mdels and manipulatives t develp cmputatinal skills (e.g., 3 5 = 5 3, 7 1 = 7). 6. Slve prblems invlving mney that require additin and subtractin. Grade 4 1. Estimate and find the prduct f 2- and 3-digit numbers t slve applicatin prblems. 2. Demnstrate fluency with basic divisin facts and fact families (i.e., memrize and apply divisin facts). 3. Develp divisin algrithms (e.g., use physical materials t shw 12 bjects arranged in 3 grups, shw divisin as repeated subtractin and as the inverse f multiplicatin). 4. Estimate and find the qutient (with and withut remainders) with a 1-digit divisr and a 2- r 3-digit dividend t slve applicatin prblems. BACK TO TOP 26

27 5. Apply a variety f estimatin and mental math techniques t simplify cmputatins (e.g., use runding t estimate is abut r 20, use 30 5 t find the prduct f 300 5). 6. Develp peratin sense by applying the assciative prperty f multiplicatin (e.g., 6 (2 3) = (6 2) 3). Grade 5 1. Use estimatin skills t determine slutins t prblems invlving decimals. 2. Apply estimatin skills t slve prblems invlving cmmn percents and equivalent 3. Add and subtract decimal numbers with the same and different place values (e.g., ) t slve prblems. 4. Multiply and divide whle numbers and decimal numbers with 1- r 2-digit multipliers r divisrs t slve prblems. 5. Add and subtract fractins and mixed numbers t slve prblems using a variety f methds (e.g., use fractin strips, find the least cmmn denminatr [LCD]). Number Sense Pre-Kindergarten 1. Demnstrate ne-ne crrespndence in cunting bjects and matching grups f bjects 2. Cunt in sequence t ten in the cntext f daily activities and play 3. Cunt bjects in a set ne-by-ne frm ne thrugh five 4. Identify and create sets f bjects ne thrugh five 5. Identify and name numerals 0-10 Kindergarten 1. Cmpare a grup r set t anther grup, set r numerical quantity and verbally explain which has mre, less r equivalent quantities 2. Pair and cunt bjects using ne-ne crrespndence (e.g., ne napkin fr each child at snack time) 3. Cunt frward t twenty and backward frm ten BACK TO TOP 27

28 4. Cunt bjects in a set ne-by-ne frm ne thrugh ten 5. Identify and create sets f bjects zer thrugh ten 6. Identify and write numerals zer thrugh ten, in and ut f sequence. Children may still be reversing sme numerals 7. Identify the rdinal psitin (first thrugh fifth) f bjects 8. Cmbine and remve bjects frm sets and verbally describe the result (e.g., adding bjects t a set makes the set larger, subtracting bjects frm a set makes the set smaller) Grade 1 Grade 2 1. Use cncrete mdels f tens and nes t develp the cncept f place value. 2. Cmpare and rder bjects 3. Use rdinal numbers first thrugh tenth t rder bjects. 4. Read and write numerals t Cunt as many as 100 bjects by nes, tws, fives, and tens. 6. Investigate cncepts f fractinal parts (halves, thirds, furths). 1. Use cncrete mdels f hundreds, tens, and nes t develp the cncepts f place value. 2. Demnstrate (using cncrete bjects, pictures, and numerical symbls) fractinal parts including halves, thirds, and furths. 3. Link place value cncepts t the reading and writing f numbers (e.g., base-10 blcks). 4. Represent a number in a variety f ways (e.g., write the calendar day in different ways, write 15 as 8 + 7, write 25 as 2 tens + 5 nes r as 1 ten + 15 nes). 5. Write a number sentence t cmpare numbers less than 100 (e.g., 5 is mre than 2, 3 is less than 7, page 51 cmes after 50, and 51 is between 50 and 60). 6. Develp and use strategies f estimatin (e.g., cmpse, decmpse and regrup numbers, use knwledge f 10 t estimate quantities and sums [tw numbers less than 10 can nt add up t mre than 20], use bdy parts t estimate measurements). 7. Determine whether a number is dd r even. BACK TO TOP 28