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Michigan K-12 Standards
Mathematics
R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N H I P S • R I G O R • R E L E V A N C E R E L A T I O N H I P S • R I G O R • R E L E V A N C E • R E L A T I O N H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P • R I G O R • • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N S H I P S • R I G O R R E L E V A N C E • R E L A T I O N S H I P S • R I G O R • R E L E V A N C E • R E L A T I O N H I P S • R I G O R • R E L E V A N C E
Michigan State Board of Education
Kathleen N. Straus, President
Bloomfield Township
John C. Austin, Vice President Ann Arbor
Carolyn L. Curtin, Secretary
Evart
Marianne Yared McGuire, Treasurer Detroit
Nancy Danhof, NASBE Delegate
East Lansing
Elizabeth W. Bauer Birmingham
Daniel Varner
Detroit
Casandra E. Ulbrich Rochester Hills
Governor Jennifer M. Granholm
Ex Officio
Michael P. Flanagan, Chairman Superintendent of Public Instruction
Ex Officio
MDE Staff
Sally Vaughn, Ph.D.
Deputy Superintendent and Chief Academic Officer
Linda Forward, Director Office of Education Improvement and Innovation
Welcome
Welcome to the Michigan K-12 Standards for Mathematics, adopted by the State Board of Education in 2010. With the reauthorization of the 2001 Elementary and Secondary Education Act (ESEA), commonly known as No Child Left Behind (NCLB), Michigan embarked on a standards revision process, starting with the K-8 mathematics and ELA standards that resulted in the Grade Level Content Expectations (GLCE). These were intended to lay the framework for the grade level testing in these subject areas required under NCLB. These were followed by GLCE for science and social studies, and by High School Content Expectations (HSCE) for all subject areas. Seven years later the revision cycle continued with Michigan working with other states to build on and refine current state standards that would allow states to work collaboratively to develop a repository of quality resources based on a common set of standards. These standards are the result of that collaboration.
Michigan’s K–12 academic standards serve to outline learning expectations for Michigan’s students and are intended to guide local curriculum development. Because these Mathematics standards are shared with other states, local districts have access to a broad set of resources they can call upon as they develop their local curricula and assessments. State standards also serve as a platform for state-level assessments, which are used to measure how well schools are providing opportunities for all students to learn the content required to be career– and college–ready.
Linda Forward, Director, Office of Education Improvement and Innovation
Vanessa Keesler, Deputy Superintendent, Division of Education Services
Mike Flanagan, Superintendent of Public Instruction
Mathematics experiences in early childhood settings should concentrate on (1) number (which includes whole number, operations, and relations) and (2) geometry, spatial relations, and measurement, with more mathematics learning time devoted to number than to other topics. Mathematical process goals should be integrated in these content areas.
The composite standards [of Hong Kong, Korea and Singapore] have a number of features that can inform an international benchmarking process for the development of K–6 mathematics standards in the U.S. First, the composite standards concentrate the early learning of mathematics on the number, measurement, and geometry strands with less emphasis on data analysis and little exposure to algebra. The Hong Kong standards for grades 1–3 devote approximately half the targeted time to numbers and almost all the time remaining to geometry and measurement.
—Ginsburg,LeinwandandDecker,2009
Because the mathematics concepts in [U.S.] textbooks are often weak, the presentation becomes more mechanical than is ideal. We looked at both traditional and non-traditional textbooks used in the US and found this conceptual weakness in both.
—Ginsburgetal.,2005
There are many ways to organize curricula. The challenge, now rarely met, is to avoid those that distort mathematics and turn off students.
articulated over time as a sequence of topics and performances that are logical and reflect, where appropriate, the sequential or hierarchical nature of the disciplinary content from which the subject matter derives. That is, what and how students are taught should reflect not only the topics that fall within a certain academic discipline, but also the key ideas that determine how knowledge is organized and generated within that discipline. This implies
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that to be coherent, a set of content standards must evolve from particulars (e.g., the meaning and operations of whole numbers, including simple math facts and routine computational procedures associated with whole numbers and fractions) to deeper structures inherent in the discipline. These deeper structures then serve as a means for connecting the particulars (such as an understanding of the rational number system and its properties). (emphasis
Connecting the Standards for Mathematical Practice to the Standards for Mathematical ContentTheStandardsforMathematicalPracticedescribewaysinwhichdevelopingstudent
a. Whencountingobjects,saythenumbernamesinthestandardorder,pairingeachobjectwithoneandonlyonenumbernameandeachnumbernamewithoneandonlyoneobject.
b. Understandthatthelastnumbernamesaidtellsthenumberofobjectscounted.Thenumberofobjectsisthesameregardlessoftheirarrangementortheorderinwhichtheywerecounted.
c. Understandthateachsuccessivenumbernamereferstoaquantitythatisonelarger.
2. Directlycomparetwoobjectswithameasurableattributeincommon,toseewhichobjecthas“moreof”/“lessof”theattribute,anddescribethedifference.For example, directly compare the heights of twochildren and describe one child as taller/shorter.
Classify objects and count the number of objects in each category.
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
1. Describeobjectsintheenvironmentusingnamesofshapes,anddescribetherelativepositionsoftheseobjectsusingtermssuchasabove,below,beside,in front of,behind,andnext to.
Understand and apply properties of operations and the relationship between addition and subtraction.
3. Applypropertiesofoperationsasstrategiestoaddandsubtract.3Examples:If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property ofaddition.) To add 2 + 6 + 4, the second two numbers can be added to makea ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
4. Understandsubtractionasanunknown-addendproblem.For example,subtract 10 – 8 by finding the number that makes 10 when added to 8.
7. Understandthemeaningoftheequalsign,anddetermineifequationsinvolvingadditionandsubtractionaretrueorfalse.For example, whichof the following equations are true and which are false? 6 = 6, 7 = 8 – 1,5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
8. Determinetheunknownwholenumberinanadditionorsubtractionequationrelatingthreewholenumbers.For example, determine theunknown number that makes the equation true in each of the equations 8 +? = 11, 5 = � – 3, 6 + 6 = �.
2. Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofsame-sizelengthunitsthatspanitwithnogapsoroverlaps.Limit tocontexts where the object being measured is spanned by a whole number oflength units with no gaps or overlaps.
8. Solvewordproblemsinvolvingdollarbills,quarters,dimes,nickels,andpennies,using$and¢symbolsappropriately.Example: If you have 2dimes and 3 pennies, how many cents do you have?
3. Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,half of,a third of,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.
Represent and solve problems involving multiplication and division.
1. Interpretproductsofwholenumbers,e.g.,interpret5×7asthetotalnumberofobjectsin5groupsof7objectseach.For example, describea context in which a total number of objects can be expressed as 5 × 7.
2. Interpretwhole-numberquotientsofwholenumbers,e.g.,interpret56÷8asthenumberofobjectsineachsharewhen56objectsarepartitionedequallyinto8shares,orasanumberofshareswhen56objectsarepartitionedintoequalsharesof8objectseach.Forexample, describe a context in which a number of shares or a number ofgroups can be expressed as 56 ÷ 8.
4. Determinetheunknownwholenumberinamultiplicationordivisionequationrelatingthreewholenumbers.For example, determine theunknown number that makes the equation true in each of the equations 8× ? = 48, 5 = � ÷ 3, 6 × 6 = ?.
Understand properties of multiplication and the relationship between multiplication and division.
5. Applypropertiesofoperationsasstrategiestomultiplyanddivide.2Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known.(Commutative property of multiplication.) 3 × 5 × 2 can be found by 3× 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associativeproperty of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, onecan find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributiveproperty.)
6. Understanddivisionasanunknown-factorproblem.For example, find32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
9. Identifyarithmeticpatterns(includingpatternsintheadditiontableormultiplicationtable),andexplainthemusingpropertiesofoperations.For example, observe that 4 times a number is always even, and explainwhy 4 times a number can be decomposed into two equal addends.
a. Representafraction1/bonanumberlinediagrambydefiningtheintervalfrom0to1asthewholeandpartitioningitintobequalparts.Recognizethateachparthassize1/bandthattheendpointofthepartbasedat0locatesthenumber1/bonthenumberline.
b. Representafractiona/bonanumberlinediagrambymarkingoffalengths1/bfrom0.Recognizethattheresultingintervalhassizea/bandthatitsendpointlocatesthenumbera/bonthenumberline.
a. Understandtwofractionsasequivalent(equal)iftheyarethesamesize,orthesamepointonanumberline.
b. Recognizeandgeneratesimpleequivalentfractions,e.g.,1/2=2/4,4/6=2/3.Explainwhythefractionsareequivalent,e.g.,byusingavisualfractionmodel.
c. Expresswholenumbersasfractions,andrecognizefractionsthatareequivalenttowholenumbers.Examples: Express 3 in the form3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same pointof a number line diagram.
d. Comparetwofractionswiththesamenumeratororthesamedenominatorbyreasoningabouttheirsize.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswiththesymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.
Measurement and Data 3.MD
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
3. Drawascaledpicturegraphandascaledbargraphtorepresentadatasetwithseveralcategories.Solveone-andtwo-step“howmanymore”and“howmanyless”problemsusinginformationpresentedinscaledbargraphs.For example, draw a bar graph in which each square inthe bar graph might represent 5 pets.
a. Findtheareaofarectanglewithwhole-numbersidelengthsbytilingit,andshowthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.
b. Multiplysidelengthstofindareasofrectangleswithwhole-numbersidelengthsinthecontextofsolvingrealworldandmathematicalproblems,andrepresentwhole-numberproductsasrectangularareasinmathematicalreasoning.
c. Usetilingtoshowinaconcretecasethattheareaofarectanglewithwhole-numbersidelengthsaandb+cisthesumofa×banda×c.Useareamodelstorepresentthedistributivepropertyinmathematicalreasoning.
d. Recognizeareaasadditive.Findareasofrectilinearfiguresbydecomposingthemintonon-overlappingrectanglesandaddingtheareasofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
2. Partitionshapesintopartswithequalareas.Expresstheareaofeachpartasaunitfractionofthewhole.For example, partition a shape into 4parts with equal area, and describe the area of each part as 1/4 of the areaof the shape.
5. Generateanumberorshapepatternthatfollowsagivenrule.Identifyapparentfeaturesofthepatternthatwerenotexplicitintheruleitself.For example, given the rule “Add 3” and the starting number 1, generateterms in the resulting sequence and observe that the terms appear toalternate between odd and even numbers. Explain informally why thenumbers will continue to alternate in this way.
number and operations in Base ten2 4.nBt
Generalize place value understanding for multi-digit whole numbers.
1. Recognizethatinamulti-digitwholenumber,adigitinoneplacerepresentstentimeswhatitrepresentsintheplacetoitsright.Forexample, recognize that 700 ÷ 70 = 10 by applying concepts of place valueand division.
c. Addandsubtractmixednumberswithlikedenominators,e.g.,byreplacingeachmixednumberwithanequivalentfraction,and/orbyusingpropertiesofoperationsandtherelationshipbetweenadditionandsubtraction.
d. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewholeandhavinglikedenominators,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.
a. Understandafractiona/basamultipleof1/b.For example, usea visual fraction model to represent 5/4 as the product 5 × (1/4),recording the conclusion by the equation 5/4 = 5 × (1/4).
b. Understandamultipleofa/basamultipleof1/b,andusethisunderstandingtomultiplyafractionbyawholenumber.Forexample, use a visual fraction model to express 3 × (2/5) as 6 × (1/5),recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solvewordproblemsinvolvingmultiplicationofafractionbyawholenumber,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, if each person at a party willeat 3/8 of a pound of roast beef, and there will be 5 people at theparty, how many pounds of roast beef will be needed? Between whattwo whole numbers does your answer lie?
Understand decimal notation for fractions, and compare decimal fractions.
5. Expressafractionwithdenominator10asanequivalentfractionwithdenominator100,andusethistechniquetoaddtwofractionswithrespectivedenominators10and100.4For example, express 3/10 as30/100, and add 3/10 + 4/100 = 34/100.
6. Usedecimalnotationforfractionswithdenominators10or100.Forexample, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate0.62 on a number line diagram.
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
1. Knowrelativesizesofmeasurementunitswithinonesystemofunitsincludingkm,m,cm;kg,g;lb,oz.;l,ml;hr,min,sec.Withinasinglesystemofmeasurement,expressmeasurementsinalargerunitintermsofasmallerunit.Recordmeasurementequivalentsinatwo-columntable.For example, know that 1 ft is 12 times as long as 1 in.Express the length of a 4 ft snake as 48 in. Generate a conversion table forfeet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...
3. Applytheareaandperimeterformulasforrectanglesinrealworldandmathematicalproblems.For example, find the width of a rectangularroom given the area of the flooring and the length, by viewing the areaformula as a multiplication equation with an unknown factor.
Represent and interpret data.
4. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,1/4,1/8).Solveproblemsinvolvingadditionandsubtractionoffractionsbyusinginformationpresentedinlineplots.For example,from a line plot find and interpret the difference in length between thelongest and shortest specimens in an insect collection.
Geometric measurement: understand concepts of angle and measure angles.
a. Anangleismeasuredwithreferencetoacirclewithitscenteratthecommonendpointoftherays,byconsideringthefractionofthecirculararcbetweenthepointswherethetworaysintersectthecircle.Ananglethatturnsthrough1/360ofacircleiscalleda“one-degreeangle,”andcanbeusedtomeasureangles.
b. Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.
2. Writesimpleexpressionsthatrecordcalculationswithnumbers,andinterpretnumericalexpressionswithoutevaluatingthem.For example,express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7).Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921,without having to calculate the indicated sum or product.
Analyze patterns and relationships.
3. Generatetwonumericalpatternsusingtwogivenrules.Identifyapparentrelationshipsbetweencorrespondingterms.Formorderedpairsconsistingofcorrespondingtermsfromthetwopatterns,andgraphtheorderedpairsonacoordinateplane.For example, given therule “Add 3” and the starting number 0, and given the rule “Add 6” and thestarting number 0, generate terms in the resulting sequences, and observethat the terms in one sequence are twice the corresponding terms in theother sequence. Explain informally why this is so.
Use equivalent fractions as a strategy to add and subtract fractions.
1. Addandsubtractfractionswithunlikedenominators(includingmixednumbers)byreplacinggivenfractionswithequivalentfractionsinsuchawayastoproduceanequivalentsumordifferenceoffractionswithlikedenominators.For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (Ingeneral, a/b + c/d = (ad + bc)/bd.)
2. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewhole,includingcasesofunlikedenominators,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.Usebenchmarkfractionsandnumbersenseoffractionstoestimatementallyandassessthereasonablenessofanswers.Forexample, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that3/7 < 1/2.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
3. Interpretafractionasdivisionofthenumeratorbythedenominator(a/b=a÷b).Solvewordproblemsinvolvingdivisionofwholenumbersleadingtoanswersintheformoffractionsormixednumbers,e.g.,byusingvisualfractionmodelsorequationstorepresenttheproblem.For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
a. Interprettheproduct(a/b)×qasapartsofapartitionofqintobequalparts;equivalently,astheresultofasequenceofoperationsa×q÷b.For example, use a visual fraction model toshow (2/3) × 4 = 8/3, and create a story context for this equation. Dothe same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Findtheareaofarectanglewithfractionalsidelengthsbytilingitwithunitsquaresoftheappropriateunitfractionsidelengths,andshowthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.Multiplyfractionalsidelengthstofindareasofrectangles,andrepresentfractionproductsasrectangularareas.
5. Interpretmultiplicationasscaling(resizing),by:
a. Comparingthesizeofaproducttothesizeofonefactoronthebasisofthesizeoftheotherfactor,withoutperformingtheindicatedmultiplication.
b. Explainingwhymultiplyingagivennumberbyafractiongreaterthan1resultsinaproductgreaterthanthegivennumber(recognizingmultiplicationbywholenumbersgreaterthan1asafamiliarcase);explainingwhymultiplyingagivennumberbyafractionlessthan1resultsinaproductsmallerthanthegivennumber;andrelatingtheprincipleoffractionequivalencea/b=(n×a)/(n×b)totheeffectofmultiplyinga/bby1.
andcomputesuchquotients.For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient.Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpretdivisionofawholenumberbyaunitfraction,andcomputesuchquotients.For example, create a story context for4 ÷ (1/5), and use a visual fraction model to show the quotient. Usethe relationship between multiplication and division to explain that4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solverealworldproblemsinvolvingdivisionofunitfractionsbynon-zerowholenumbersanddivisionofwholenumbersbyunitfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, how much chocolate will eachperson get if 3 people share 1/2 lb of chocolate equally? How many1/3-cup servings are in 2 cups of raisins?
measurement and data 5.md
Convert like measurement units within a given measurement system.
2. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,1/4,1/8).Useoperationsonfractionsforthisgradetosolveproblemsinvolvinginformationpresentedinlineplots.For example,given different measurements of liquid in identical beakers, find theamount of liquid each beaker would contain if the total amount in all thebeakers were redistributed equally.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
a. Findthevolumeofarightrectangularprismwithwhole-numbersidelengthsbypackingitwithunitcubes,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengths,equivalentlybymultiplyingtheheightbytheareaofthebase.Representthreefoldwhole-numberproductsasvolumes,e.g.,torepresenttheassociativepropertyofmultiplication.
b. ApplytheformulasV=l×w×handV=b×hforrectangularprismstofindvolumesofrightrectangularprismswithwhole-numberedgelengthsinthecontextofsolvingrealworldandmathematicalproblems.
c. Recognizevolumeasadditive.Findvolumesofsolidfigurescomposedoftwonon-overlappingrightrectangularprismsbyaddingthevolumesofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.
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Geometry 5.G
Graph points on the coordinate plane to solve real-world and mathematical problems.
Classify two-dimensional figures into categories based on their properties.
3. Understandthatattributesbelongingtoacategoryoftwo-dimensionalfiguresalsobelongtoallsubcategoriesofthatcategory.For example, all rectangles have four right angles and squares arerectangles, so all squares have four right angles.
Understand ratio concepts and use ratio reasoning to solve problems.
1. Understandtheconceptofaratioanduseratiolanguagetodescribearatiorelationshipbetweentwoquantities.For example, “The ratioof wings to beaks in the bird house at the zoo was 2:1, because forevery 2 wings there was 1 beak.” “For every vote candidate A received,candidate C received nearly three votes.”
2. Understandtheconceptofaunitratea/bassociatedwitharatioa:bwithb≠0,anduseratelanguageinthecontextofaratiorelationship.For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar,so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15hamburgers, which is a rate of $5 per hamburger.”1
a. Maketablesofequivalentratiosrelatingquantitieswithwhole-numbermeasurements,findmissingvaluesinthetables,andplotthepairsofvaluesonthecoordinateplane.Usetablestocompareratios.
b. Solveunitrateproblemsincludingthoseinvolvingunitpricingandconstantspeed.For example, if it took 7 hours to mow 4 lawns, thenat that rate, how many lawns could be mowed in 35 hours? At whatrate were lawns being mowed?
c. Findapercentofaquantityasarateper100(e.g.,30%ofaquantitymeans30/100timesthequantity);solveproblemsinvolvingfindingthewhole,givenapartandthepercent.
d. Useratioreasoningtoconvertmeasurementunits;manipulateandtransformunitsappropriatelywhenmultiplyingordividingquantities.
the number System 6.nS
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
1. Interpretandcomputequotientsoffractions,andsolvewordproblemsinvolvingdivisionoffractionsbyfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.Forexample, create a story context for (2/3) ÷ (3/4) and use a visual fractionmodel to show the quotient; use the relationship between multiplicationand division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3.(In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each personget if 3 people share 1/2 lb of chocolate equally? How many 3/4-cupservings are in 2/3 of a cup of yogurt? How wide is a rectangular strip ofland with length 3/4 mi and area 1/2 square mi?
Compute fluently with multi-digit numbers and find common factors and multiples.
a. Recognizeoppositesignsofnumbersasindicatinglocationsonoppositesidesof0onthenumberline;recognizethattheoppositeoftheoppositeofanumberisthenumberitself,e.g.,–(–3)=3,andthat0isitsownopposite.
b. Understandsignsofnumbersinorderedpairsasindicatinglocationsinquadrantsofthecoordinateplane;recognizethatwhentwoorderedpairsdifferonlybysigns,thelocationsofthepointsarerelatedbyreflectionsacrossoneorbothaxes.
c. Findandpositionintegersandotherrationalnumbersonahorizontalorverticalnumberlinediagram;findandpositionpairsofintegersandotherrationalnumbersonacoordinateplane.
a. Interpretstatementsofinequalityasstatementsabouttherelativepositionoftwonumbersonanumberlinediagram.For example,interpret –3 > –7 as a statement that –3 is located to the right of –7 ona number line oriented from left to right.
b. Write,interpret,andexplainstatementsoforderforrationalnumbersinreal-worldcontexts.For example, write –3 oC > –7 oC toexpress the fact that –3 oC is warmer than –7 oC.
c. Understandtheabsolutevalueofarationalnumberasitsdistancefrom0onthenumberline;interpretabsolutevalueasmagnitudeforapositiveornegativequantityinareal-worldsituation.Forexample, for an account balance of –30 dollars, write |–30| = 30 todescribe the size of the debt in dollars.
d. Distinguishcomparisonsofabsolutevaluefromstatementsaboutorder.For example, recognize that an account balance less than –30dollars represents a debt greater than 30 dollars.
a. Writeexpressionsthatrecordoperationswithnumbersandwithlettersstandingfornumbers.For example, express the calculation“Subtract y from 5” as 5 – y.
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b. Identifypartsofanexpressionusingmathematicalterms(sum,term,product,factor,quotient,coefficient);viewoneormorepartsofanexpressionasasingleentity.For example, describe theexpression 2 (8 + 7) as a product of two factors; view (8 + 7) as botha single entity and a sum of two terms.
c. Evaluateexpressionsatspecificvaluesoftheirvariables.Includeexpressionsthatarisefromformulasusedinreal-worldproblems.Performarithmeticoperations,includingthoseinvolvingwhole-numberexponents,intheconventionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).For example, use the formulas V = s3 and A = 6 s2 to find the volumeand surface area of a cube with sides of length s = 1/2.
3. Applythepropertiesofoperationstogenerateequivalentexpressions.For example, apply the distributive property to the expression 3 (2 + x) toproduce the equivalent expression 6 + 3x; apply the distributive propertyto the expression 24x + 18y to produce the equivalent expression6 (4x + 3y); apply properties of operations to y + y + y to produce theequivalent expression 3y.
4. Identifywhentwoexpressionsareequivalent(i.e.,whenthetwoexpressionsnamethesamenumberregardlessofwhichvalueissubstitutedintothem).For example, the expressions y + y + y and 3yare equivalent because they name the same number regardless of whichnumber y stands for.
Reason about and solve one-variable equations and inequalities.
Represent and analyze quantitative relationships between dependent and independent variables.
9. Usevariablestorepresenttwoquantitiesinareal-worldproblemthatchangeinrelationshiptooneanother;writeanequationtoexpressonequantity,thoughtofasthedependentvariable,intermsoftheotherquantity,thoughtofastheindependentvariable.Analyzetherelationshipbetweenthedependentandindependentvariablesusinggraphsandtables,andrelatethesetotheequation.For example, in aproblem involving motion at constant speed, list and graph ordered pairsof distances and times, and write the equation d = 65t to represent therelationship between distance and time.
Geometry 6.G
Solve real-world and mathematical problems involving area, surface area, and volume.
2. Findthevolumeofarightrectangularprismwithfractionaledgelengthsbypackingitwithunitcubesoftheappropriateunitfractionedgelengths,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengthsoftheprism.ApplytheformulasV = l w h and V = b htofindvolumesofrightrectangularprismswithfractionaledgelengthsinthecontextofsolvingreal-worldandmathematicalproblems.
1. Recognizeastatisticalquestionasonethatanticipatesvariabilityinthedatarelatedtothequestionandaccountsforitintheanswers.Forexample, “How old am I?” is not a statistical question, but “How old are thestudents in my school?” is a statistical question because one anticipatesvariability in students’ ages.
b. Describingthenatureoftheattributeunderinvestigation,includinghowitwasmeasuredanditsunitsofmeasurement.
c. Givingquantitativemeasuresofcenter(medianand/ormean)andvariability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.
d. Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.
Analyze proportional relationships and use them to solve real-world and mathematical problems.
1. Computeunitratesassociatedwithratiosoffractions,includingratiosoflengths,areasandotherquantitiesmeasuredinlikeordifferentunits.For example, if a person walks 1/2 mile in each 1/4 hour, computethe unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2miles per hour.
a. Decidewhethertwoquantitiesareinaproportionalrelationship,e.g.,bytestingforequivalentratiosinatableorgraphingonacoordinateplaneandobservingwhetherthegraphisastraightlinethroughtheorigin.
b. Identifytheconstantofproportionality(unitrate)intables,graphs,equations,diagrams,andverbaldescriptionsofproportionalrelationships.
c. Representproportionalrelationshipsbyequations.For example, iftotal cost t is proportional to the number n of items purchased ata constant price p, the relationship between the total cost and thenumber of items can be expressed as t = pn.
d. Explainwhatapoint (x, y) onthegraphofaproportionalrelationshipmeansintermsofthesituation,withspecialattentiontothepoints(0,0)and(1, r) where r istheunitrate.
3. Useproportionalrelationshipstosolvemultistepratioandpercentproblems.Examples: simple interest, tax, markups and markdowns,gratuities and commissions, fees, percent increase and decrease, percenterror.
the number System 7.nS
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
a. Describesituationsinwhichoppositequantitiescombinetomake0.For example, a hydrogen atom has 0 charge because its twoconstituents are oppositely charged.
b. Understandp+qasthenumberlocatedadistance|q|fromp,inthepositiveornegativedirectiondependingonwhetherqispositiveornegative.Showthatanumberanditsoppositehaveasumof0(areadditiveinverses).Interpretsumsofrationalnumbersbydescribingreal-worldcontexts.
c. Understandsubtractionofrationalnumbersasaddingtheadditiveinverse,p–q=p+(–q).Showthatthedistancebetweentworationalnumbersonthenumberlineistheabsolutevalueoftheirdifference,andapplythisprincipleinreal-worldcontexts.
d. Applypropertiesofoperationsasstrategiestoaddandsubtractrationalnumbers.
a. Understandthatmultiplicationisextendedfromfractionstorationalnumbersbyrequiringthatoperationscontinuetosatisfythepropertiesofoperations,particularlythedistributiveproperty,leadingtoproductssuchas(–1)(–1)=1andtherulesformultiplyingsignednumbers.Interpretproductsofrationalnumbersbydescribingreal-worldcontexts.
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b. Understandthatintegerscanbedivided,providedthatthedivisorisnotzero,andeveryquotientofintegers(withnon-zerodivisor)isarationalnumber.Ifpandqareintegers,then–(p/q)=(–p)/q=p/(–q).Interpretquotientsofrationalnumbersbydescribingreal-worldcontexts.
c. Applypropertiesofoperationsasstrategiestomultiplyanddividerationalnumbers.
d. Convertarationalnumbertoadecimalusinglongdivision;knowthatthedecimalformofarationalnumberterminatesin0soreventuallyrepeats.
2. Understandthatrewritinganexpressionindifferentformsinaproblemcontextcanshedlightontheproblemandhowthequantitiesinitarerelated.For example, a + 0.05a = 1.05a means that “increase by5%” is the same as “multiply by 1.05.”
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
3. Solvemulti-stepreal-lifeandmathematicalproblemsposedwithpositiveandnegativerationalnumbersinanyform(wholenumbers,fractions,anddecimals),usingtoolsstrategically.Applypropertiesofoperationstocalculatewithnumbersinanyform;convertbetweenformsasappropriate;andassessthereasonablenessofanswersusingmentalcomputationandestimationstrategies.For example: If a womanmaking $25 an hour gets a 10% raise, she will make an additional 1/10 ofher salary an hour, or $2.50, for a new salary of $27.50. If you want to placea towel bar 9 3/4 inches long in the center of a door that is 27 1/2 incheswide, you will need to place the bar about 9 inches from each edge; thisestimate can be used as a check on the exact computation.
a. Solvewordproblemsleadingtoequationsoftheformpx+q=randp(x+q)=r,wherep,q,andrarespecificrationalnumbers.Solveequationsoftheseformsfluently.Compareanalgebraicsolutiontoanarithmeticsolution,identifyingthesequenceoftheoperationsusedineachapproach.For example, the perimeter of arectangle is 54 cm. Its length is 6 cm. What is its width?
b. Solvewordproblemsleadingtoinequalitiesoftheformpx+q>rorpx+q<r,wherep,q,andrarespecificrationalnumbers.Graphthesolutionsetoftheinequalityandinterpretitinthecontextoftheproblem.For example: As a salesperson, you are paid $50 perweek plus $3 per sale. This week you want your pay to be at least$100. Write an inequality for the number of sales you need to make,and describe the solutions.
Geometry 7.G
Draw, construct, and describe geometrical figures and describe the relationships between them.
2. Usedatafromarandomsampletodrawinferencesaboutapopulationwithanunknowncharacteristicofinterest.Generatemultiplesamples(orsimulatedsamples)ofthesamesizetogaugethevariationinestimatesorpredictions.For example, estimate the mean word length ina book by randomly sampling words from the book; predict the winner ofa school election based on randomly sampled survey data. Gauge how faroff the estimate or prediction might be.
Draw informal comparative inferences about two populations.
3. Informallyassessthedegreeofvisualoverlapoftwonumericaldatadistributionswithsimilarvariabilities,measuringthedifferencebetweenthecentersbyexpressingitasamultipleofameasureofvariability.For example, the mean height of players on the basketballteam is 10 cm greater than the mean height of players on the soccer team,about twice the variability (mean absolute deviation) on either team; ona dot plot, the separation between the two distributions of heights isnoticeable.
4. Usemeasuresofcenterandmeasuresofvariabilityfornumericaldatafromrandomsamplestodrawinformalcomparativeinferencesabouttwopopulations.For example, decide whether the words in a chapterof a seventh-grade science book are generally longer than the words in achapter of a fourth-grade science book.
Investigate chance processes and develop, use, and evaluate probability models.
6. Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthatproducesitandobservingitslong-runrelativefrequency,andpredicttheapproximaterelativefrequencygiventheprobability.For example, when rolling a number cube 600 times, predictthat a 3 or 6 would be rolled roughly 200 times, but probably not exactly200 times.
a. Developauniformprobabilitymodelbyassigningequalprobabilitytoalloutcomes,andusethemodeltodetermineprobabilitiesofevents.For example, if a student is selected atrandom from a class, find the probability that Jane will be selectedand the probability that a girl will be selected.
b. Developaprobabilitymodel(whichmaynotbeuniform)byobservingfrequenciesindatageneratedfromachanceprocess.For example, find the approximate probability that a spinning pennywill land heads up or that a tossed paper cup will land open-enddown. Do the outcomes for the spinning penny appear to be equallylikely based on the observed frequencies?
a. Understandthat,justaswithsimpleevents,theprobabilityofacompoundeventisthefractionofoutcomesinthesamplespaceforwhichthecompoundeventoccurs.
b. Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tablesandtreediagrams.Foraneventdescribedineverydaylanguage(e.g.,“rollingdoublesixes”),identifytheoutcomesinthesamplespacewhichcomposetheevent.
c. Designanduseasimulationtogeneratefrequenciesforcompoundevents.For example, use random digits as a simulationtool to approximate the answer to the question: If 40% of donorshave type A blood, what is the probability that it will take at least 4donors to find one with type A blood?
2. Userationalapproximationsofirrationalnumberstocomparethesizeofirrationalnumbers,locatethemapproximatelyonanumberlinediagram,andestimatethevalueofexpressions(e.g.,π2).For example,by truncating the decimal expansion of √2, show that √2 is between 1 and2, then between 1.4 and 1.5, and explain how to continue on to get betterapproximations.
3. Usenumbersexpressedintheformofasingledigittimesanintegerpowerof10toestimateverylargeorverysmallquantities,andtoexpresshowmanytimesasmuchoneisthantheother.For example,estimate the population of the United States as 3 × 108 and the populationof the world as 7 × 109, and determine that the world population is morethan 20 times larger.
Understand the connections between proportional relationships, lines, and linear equations.
5. Graphproportionalrelationships,interpretingtheunitrateastheslopeofthegraph.Comparetwodifferentproportionalrelationshipsrepresentedindifferentways.For example, compare a distance-timegraph to a distance-time equation to determine which of two movingobjects has greater speed.
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solvelinearequationsinonevariable.
a. Giveexamplesoflinearequationsinonevariablewithonesolution,infinitelymanysolutions,ornosolutions.Showwhichofthesepossibilitiesisthecasebysuccessivelytransformingthegivenequationintosimplerforms,untilanequivalentequationoftheformx=a,a=a,ora=bresults(whereaandbaredifferentnumbers).
b. Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressionsusingthedistributivepropertyandcollectingliketerms.
a. Understandthatsolutionstoasystemoftwolinearequationsintwovariablescorrespondtopointsofintersectionoftheirgraphs,becausepointsofintersectionsatisfybothequationssimultaneously.
b. Solvesystemsoftwolinearequationsintwovariablesalgebraically,andestimatesolutionsbygraphingtheequations.Solvesimplecasesbyinspection.For example, 3x + 2y = 5 and 3x +2y = 6 have no solution because 3x + 2y cannot simultaneously be 5and 6.
c. Solvereal-worldandmathematicalproblemsleadingtotwolinearequationsintwovariables.For example, given coordinates for twopairs of points, determine whether the line through the first pair ofpoints intersects the line through the second pair.
2. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).For example, given a linear function represented by a tableof values and a linear function represented by an algebraic expression,determine which function has the greater rate of change.
3. Interprettheequationy=mx+basdefiningalinearfunction,whosegraphisastraightline;giveexamplesoffunctionsthatarenotlinear.For example, the function A = s2 giving the area of a square as a functionof its side length is not linear because its graph contains the points (1,1),(2,4) and (3,9), which are not on a straight line.
Use functions to model relationships between quantities.
5. Useinformalargumentstoestablishfactsabouttheanglesumandexteriorangleoftriangles,abouttheanglescreatedwhenparallellinesarecutbyatransversal,andtheangle-anglecriterionforsimilarityoftriangles.For example, arrange three copies of the same triangle so thatthe sum of the three angles appears to form a line, and give an argumentin terms of transversals why this is so.
3. Usetheequationofalinearmodeltosolveproblemsinthecontextofbivariatemeasurementdata,interpretingtheslopeandintercept.For example, in a linear model for a biology experiment, interpret a slopeof 1.5 cm/hr as meaning that an additional hour of sunlight each day isassociated with an additional 1.5 cm in mature plant height.
4. Understandthatpatternsofassociationcanalsobeseeninbivariatecategoricaldatabydisplayingfrequenciesandrelativefrequenciesinatwo-waytable.Constructandinterpretatwo-waytablesummarizingdataontwocategoricalvariablescollectedfromthesamesubjects.Userelativefrequenciescalculatedforrowsorcolumnstodescribepossibleassociationbetweenthetwovariables.For example, collectdata from students in your class on whether or not they have a curfew onschool nights and whether or not they have assigned chores at home. Isthere evidence that those who have a curfew also tend to have chores?
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mathematics Standards for High SchoolThehighschoolstandardsspecifythemathematicsthatallstudentsshould
Extend the properties of exponents to rational exponents.
1. Explainhowthedefinitionofthemeaningofrationalexponentsfollowsfromextendingthepropertiesofintegerexponentstothosevalues,allowingforanotationforradicalsintermsofrationalexponents.For example, we define 51/3 to be the cube root of 5because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
a. Addvectorsend-to-end,component-wise,andbytheparallelogramrule.Understandthatthemagnitudeofasumoftwovectorsistypicallynotthesumofthemagnitudes.
b. Giventwovectorsinmagnitudeanddirectionform,determinethemagnitudeanddirectionoftheirsum.
c. Understandvectorsubtractionv–wasv+(–w),where–wistheadditiveinverseofw,withthesamemagnitudeaswandpointingintheoppositedirection.Representvectorsubtractiongraphicallybyconnectingthetipsintheappropriateorder,andperformvectorsubtractioncomponent-wise.
5. (+)Multiplyavectorbyascalar.
a. Representscalarmultiplicationgraphicallybyscalingvectorsandpossiblyreversingtheirdirection;performscalarmultiplicationcomponent-wise,e.g.,asc(v
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b. Computethemagnitudeofascalarmultiplecvusing||cv||=|c|v.Computethedirectionofcvknowingthatwhen|c|v≠0,thedirectionofcviseitheralongv(forc>0)oragainstv(forc<0).
Perform operations on matrices and use matrices in applications.
mathematics | High School—algebraExpressions.Anexpressionisarecordofacomputationwithnumbers,symbolsthatrepresentnumbers,arithmeticoperations,exponentiation,and,atmoreadvancedlevels,theoperationofevaluatingafunction.Conventionsabouttheuseofparenthesesandtheorderofoperationsassurethateachexpressionisunambiguous.Creatinganexpressionthatdescribesacomputationinvolvingageneralquantityrequirestheabilitytoexpressthecomputationingeneralterms,abstractingfromspecificinstances.
Equations and inequalities.Anequationisastatementofequalitybetweentwoexpressions,oftenviewedasaquestionaskingforwhichvaluesofthevariablestheexpressionsoneithersideareinfactequal.Thesevaluesarethesolutionstotheequation.Anidentity,incontrast,istrueforallvaluesofthevariables;identitiesareoftendevelopedbyrewritinganexpressioninanequivalentform.
Connections to Functions and Modeling. Expressionscandefinefunctions,andequivalentexpressionsdefinethesamefunction.Askingwhentwofunctionshavethesamevalueforthesameinputleadstoanequation;graphingthetwofunctionsallowsforfindingapproximatesolutionsoftheequation.Convertingaverbaldescriptiontoanequation,inequality,orsystemoftheseisanessentialskillinmodeling.
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Seeing Structure in Expressions
• Interpret the structure of expressions
• Write expressions in equivalent forms to solveproblems
Arithmetic with Polynomials and Rational Expressions
• Perform arithmetic operations on polynomials
• Understand the relationship between zeros andfactors of polynomials
• Use polynomial identities to solve problems
• rewrite rational expressions
Creating Equations
• Create equations that describe numbers orrelationships
Reasoning with Equations and Inequalities
• Understand solving equations as a process ofreasoning and explain the reasoning
• Solve equations and inequalities in one variable
• Solve systems of equations
• represent and solve equations and inequalitiesgraphically
a. Interpretpartsofanexpression,suchasterms,factors,andcoefficients.
b. Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.For example, interpretP(1+r)nas the productof P and a factor not depending on P.
2. Usethestructureofanexpressiontoidentifywaystorewriteit.Forexample, see x4–y4as(x2)2–(y2)2,thus recognizing it as a difference ofsquares that can be factored as(x2–y2)(x2+y2).
Write expressions in equivalent forms to solve problems
a. Factoraquadraticexpressiontorevealthezerosofthefunctionitdefines.
b. Completethesquareinaquadraticexpressiontorevealthemaximumorminimumvalueofthefunctionitdefines.
c. Usethepropertiesofexponentstotransformexpressionsforexponentialfunctions. For example the expression1.15tcan berewritten as(1.151/12)12t≈1.01212tto reveal the approximate equivalentmonthly interest rate if the annual rate is 15%.
4. Provepolynomialidentitiesandusethemtodescribenumericalrelationships.For example, the polynomial identity(x2+y2)2=(x2–y2)2+(2xy)2can be used to generate Pythagorean triples.
Create equations that describe numbers or relationships
1. Createequationsandinequalitiesinonevariableandusethemtosolveproblems.Include equations arising from linear and quadraticfunctions, and simple rational and exponential functions.
3. Representconstraintsbyequationsorinequalities,andbysystemsofequationsand/orinequalities,andinterpretsolutionsasviableornon-viableoptionsinamodelingcontext.For example, represent inequalitiesdescribing nutritional and cost constraints on combinations of differentfoods.
4. Rearrangeformulastohighlightaquantityofinterest,usingthesamereasoningasinsolvingequations.For example, rearrange Ohm’s law V =IR to highlight resistance R.
reasoning with equations and Inequalities a-reI
Understand solving equations as a process of reasoning and explain the reasoning
a. Usethemethodofcompletingthesquaretotransformanyquadraticequationinxintoanequationoftheform(x–p)2=qthathasthesamesolutions.Derivethequadraticformulafromthisform.
b. Solvequadraticequationsbyinspection(e.g.,forx2=49),takingsquareroots,completingthesquare,thequadraticformulaandfactoring,asappropriatetotheinitialformoftheequation.Recognizewhenthequadraticformulagivescomplexsolutionsandwritethemasa±biforrealnumbersaandb.
7. Solveasimplesystemconsistingofalinearequationandaquadraticequationintwovariablesalgebraicallyandgraphically.For example,find the points of intersection between the line y=–3xandthecirclex2+y2=3.
mathematics | High School—functionsFunctionsdescribesituationswhereonequantitydeterminesanother.Forexample,thereturnon$10,000investedatanannualizedpercentagerateof4.25%isafunctionofthelengthoftimethemoneyisinvested.Becausewecontinuallymaketheoriesaboutdependenciesbetweenquantitiesinnatureandsociety,functionsareimportanttoolsintheconstructionofmathematicalmodels.
3. Recognizethatsequencesarefunctions,sometimesdefinedrecursively,whosedomainisasubsetoftheintegers.For example, theFibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) +f(n-1) for n ≥ 1.
Interpret functions that arise in applications in terms of the context
4. Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship.Key features include: intercepts; intervals where thefunction is increasing, decreasing, positive, or negative; relative maximumsand minimums; symmetries; end behavior; and periodicity.★
5. Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitativerelationshipitdescribes.For example, if the functionh(n) gives the number of person-hours it takes to assemble n engines in afactory, then the positive integers would be an appropriate domain for thefunction.★
a. Usetheprocessoffactoringandcompletingthesquareinaquadraticfunctiontoshowzeros,extremevalues,andsymmetryofthegraph,andinterprettheseintermsofacontext.
b. Usethepropertiesofexponentstointerpretexpressionsforexponentialfunctions.For example, identify percent rate of changein functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, andclassify them as representing exponential growth or decay.
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9. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).For example, given a graph of one quadratic function andan algebraic expression for another, say which has the larger maximum.
Building functions f-Bf
Build a function that models a relationship between two quantities
a. Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.
b. Combinestandardfunctiontypesusingarithmeticoperations.Forexample, build a function that models the temperature of a coolingbody by adding a constant function to a decaying exponential, andrelate these functions to the model.
c. (+)Composefunctions. For example, if T(y) is the temperature inthe atmosphere as a function of height, and h(t) is the height of aweather balloon as a function of time, then T(h(t)) is the temperatureat the location of the weather balloon as a function of time.
3. Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecificvaluesofk(bothpositiveandnegative);findthevalueofkgiventhegraphs.Experimentwithcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Include recognizing even and odd functions from their graphs andalgebraic expressions for them.
4. Findinversefunctions.
a. Solveanequationoftheformf(x)=cforasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.Forexample, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.
b. (+)Verifybycompositionthatonefunctionistheinverseofanother.
c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhasaninverse.
d. (+)Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain.
mathematics | High School—modelingModelinglinksclassroommathematicsandstatisticstoeverydaylife,work,anddecision-making.Modelingistheprocessofchoosingandusingappropriatemathematicsandstatisticstoanalyzeempiricalsituations,tounderstandthembetter,andtoimprovedecisions.Quantitiesandtheirrelationshipsinphysical,economic,publicpolicy,social,andeverydaysituationscanbemodeledusingmathematicalandstatisticalmethods.Whenmakingmathematicalmodels,technologyisvaluableforvaryingassumptions,exploringconsequences,andcomparingpredictionswithdata.
modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★).
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mathematics | High School—GeometryAnunderstandingoftheattributesandrelationshipsofgeometricobjectscanbeappliedindiversecontexts—interpretingaschematicdrawing,estimatingtheamountofwoodneededtoframeaslopingroof,renderingcomputergraphics,ordesigningasewingpatternforthemostefficientuseofmaterial.
Connections to Equations. Thecorrespondencebetweennumericalcoordinatesandgeometricpointsallowsmethodsfromalgebratobeappliedtogeometryandviceversa.Thesolutionsetofanequationbecomesageometriccurve,makingvisualizationatoolfordoingandunderstandingalgebra.Geometricshapescanbedescribedbyequations,makingalgebraicmanipulationintoatoolforgeometricunderstanding,modeling,andproof.
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Congruence
• experiment with transformations in the plane
• Understand congruence in terms of rigidmotions
• Prove geometric theorems
• make geometric constructions
Similarity, Right Triangles, and Trigonometry
• Understand similarity in terms of similaritytransformations
• Prove theorems involving similarity
• define trigonometric ratios and solve problemsinvolving right triangles
• apply trigonometry to general triangles
Circles
• Understand and apply theorems about circles
• find arc lengths and areas of sectors of circles
Expressing Geometric Properties with Equations
• translate between the geometric descriptionand the equation for a conic section
• Use coordinates to prove simple geometrictheorems algebraically
Geometric Measurement and Dimension
• explain volume formulas and use them to solveproblems
• Visualize relationships between two-dimensional and three-dimensional objects
9. Provetheoremsaboutlinesandangles. Theorems include: verticalangles are congruent; when a transversal crosses parallel lines, alternateinterior angles are congruent and corresponding angles are congruent;points on a perpendicular bisector of a line segment are exactly thoseequidistant from the segment’s endpoints.
10. Provetheoremsabouttriangles.Theorems include: measures of interiorangles of a triangle sum to 180°; base angles of isosceles triangles arecongruent; the segment joining midpoints of two sides of a triangle isparallel to the third side and half the length; the medians of a trianglemeet at a point.
11. Provetheoremsaboutparallelograms.Theorems include: oppositesides are congruent, opposite angles are congruent, the diagonalsof a parallelogram bisect each other, and conversely, rectangles areparallelograms with congruent diagonals.
Make geometric constructions
12. Makeformalgeometricconstructionswithavarietyoftoolsandmethods(compassandstraightedge,string,reflectivedevices,paperfolding,dynamicgeometricsoftware,etc.).Copying a segment;copying an angle; bisecting a segment; bisecting an angle; constructingperpendicular lines, including the perpendicular bisector of a line segment;and constructing a line parallel to a given line through a point not on theline.
4. Provetheoremsabouttriangles.Theorems include: a line parallel to oneside of a triangle divides the other two proportionally, and conversely; thePythagorean Theorem proved using triangle similarity.
2. Identifyanddescriberelationshipsamonginscribedangles,radii,andchords.Include the relationship between central, inscribed, andcircumscribed angles; inscribed angles on a diameter are right angles;the radius of a circle is perpendicular to the tangent where the radiusintersects the circle.
Use coordinates to prove simple geometric theorems algebraically
4. Usecoordinatestoprovesimplegeometrictheoremsalgebraically.Forexample, prove or disprove that a figure defined by four given points in thecoordinate plane is a rectangle; prove or disprove that the point (1, √3) lieson the circle centered at the origin and containing the point (0, 2).
Connections to Functions and Modeling.Functionsmaybeusedtodescribedata;ifthedatasuggestalinearrelationship,therelationshipcanbemodeledwitharegressionline,anditsstrengthanddirectioncanbeexpressedthroughacorrelationcoefficient.
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Interpreting Categorical and Quantitative Data
• Summarize, represent, and interpret data on asingle count or measurement variable
• Summarize, represent, and interpret data ontwo categorical and quantitative variables
• Interpret linear models
Making Inferences and Justifying Conclusions
• Understand and evaluate random processesunderlying statistical experiments
• make inferences and justify conclusions fromsample surveys, experiments and observationalstudies
Conditional Probability and the Rules of Prob-ability
• Understand independence and conditionalprobability and use them to interpret data
• Use the rules of probability to computeprobabilities of compound events in a uniformprobability model
Using Probability to Make Decisions
• Calculate expected values and use them tosolve problems
• Use probability to evaluate outcomes ofdecisions
a. Fitafunctiontothedata;usefunctionsfittedtodatatosolveproblemsinthecontextofthedata.Use given functions or choosea function suggested by the context. Emphasize linear, quadratic, andexponential models.
b. Informallyassessthefitofafunctionbyplottingandanalyzingresiduals.
c. Fitalinearfunctionforascatterplotthatsuggestsalinearassociation.
2. Decideifaspecifiedmodelisconsistentwithresultsfromagivendata-generatingprocess,e.g.,usingsimulation.For example, a modelsays a spinning coin falls heads up with probability 0.5. Would a result of 5tails in a row cause you to question the model?
Make inferences and justify conclusions from sample surveys, experiments, and observational studies
4. Constructandinterprettwo-wayfrequencytablesofdatawhentwocategoriesareassociatedwitheachobjectbeingclassified.Usethetwo-waytableasasamplespacetodecideifeventsareindependentandtoapproximateconditionalprobabilities.For example, collectdata from a random sample of students in your school on their favoritesubject among math, science, and English. Estimate the probability that arandomly selected student from your school will favor science given thatthe student is in tenth grade. Do the same for other subjects and comparethe results.
5. Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageandeverydaysituations.Forexample, compare the chance of having lung cancer if you are a smokerwith the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability model
3. (+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichtheoreticalprobabilitiescanbecalculated;findtheexpectedvalue.For example, find the theoretical probabilitydistribution for the number of correct answers obtained by guessing onall five questions of a multiple-choice test where each question has fourchoices, and find the expected grade under various grading schemes.
4. (+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichprobabilitiesareassignedempirically;findtheexpectedvalue.For example, find a current data distribution on thenumber of TV sets per household in the United States, and calculate theexpected number of sets per household. How many TV sets would youexpect to find in 100 randomly selected households?
a. Findtheexpectedpayoffforagameofchance.For example, findthe expected winnings from a state lottery ticket or a game at a fast-food restaurant.
b. Evaluateandcomparestrategiesonthebasisofexpectedvalues.For example, compare a high-deductible versus a low-deductibleautomobile insurance policy using various, but reasonable, chances ofhaving a minor or a major accident.
Addition and subtraction within 5, 10, 20, 100, or 1000.Additionorsubtractionoftwowholenumberswithwholenumberanswers,andwithsumorminuendintherange0-5,0-10,0-20,or0-100,respectively.Example:8+2=10isanadditionwithin10,14–5=9isasubtractionwithin20,and55–18=37isasubtractionwithin100.
First quartile. ForadatasetwithmedianM,thefirstquartileisthemedianofthedatavalueslessthanM.Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},thefirstquartileis6.2See also:median,thirdquartile,interquartilerange.
Independently combined probability models.Twoprobabilitymodelsaresaidtobecombinedindependentlyiftheprobabilityofeachorderedpairinthecombinedmodelequalstheproductoftheoriginalprobabilitiesofthetwoindividualoutcomesintheorderedpair.
1AdaptedfromWisconsinDepartmentofPublicInstruction,http://dpi.wi.gov/standards/mathglos.html,accessedMarch2,2010.2Manydifferentmethodsforcomputingquartilesareinuse.ThemethoddefinedhereissometimescalledtheMooreandMcCabemethod.SeeLangford,E.,“QuartilesinElementaryStatistics,”Journal of Statistics EducationVolume14,Number3(2006).
Mean absolute deviation.Ameasureofvariationinasetofnumericaldata,computedbyaddingthedistancesbetweeneachdatavalueandthemean,thendividingbythenumberofdatavalues.Example:Forthedataset{2,3,6,7,10,12,14,15,22,120},themeanabsolutedeviationis20.
Multiplication and division within 100.Multiplicationordivisionoftwowholenumberswithwholenumberanswers,andwithproductordividendintherange0-100.Example:72÷8=9.
Number line diagram. Adiagramofthenumberlineusedtorepresentnumbersandsupportreasoningaboutthem.Inanumberlinediagramformeasurementquantities,theintervalfrom0to1onthediagramrepresentstheunitofmeasureforthequantity.
Percent rate of change.Arateofchangeexpressedasapercent.Example:ifapopulationgrowsfrom50to55inayear,itgrowsby5/50=10%peryear.
Probability distribution.Thesetofpossiblevaluesofarandomvariablewithaprobabilityassignedtoeach.
Probability model. Aprobabilitymodelisusedtoassignprobabilitiestooutcomesofachanceprocessbyexaminingthenatureoftheprocess.Thesetofalloutcomesiscalledthesamplespace,andtheirprobabilitiessumto1.See also: uniformprobabilitymodel.
Random variable. Anassignmentofanumericalvaluetoeachoutcomeinasamplespace.
Third quartile.ForadatasetwithmedianM,thethirdquartileisthemedianofthedatavaluesgreaterthanM.Example:Forthedataset{2,3,6,7,10,12,14,15,22,120},thethirdquartileis15.See also:median,firstquartile,interquartilerange.
Transitivity principle for indirect measurement. IfthelengthofobjectAisgreaterthanthelengthofobjectB,andthelengthofobjectBisgreaterthanthelengthofobjectC,thenthelengthofobjectAisgreaterthanthelengthofobjectC.Thisprincipleappliestomeasurementofotherquantitiesaswell.
Uniform probability model.Aprobabilitymodelwhichassignsequalprobabilitytoalloutcomes.See also:probabilitymodel.
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