To what extent do the Modules align with the information provided in the Progressions for Grade 8? If perfect alignment is a 10 and no alignment is a 1, what

Progressions for the Common Core State Standards in Mathematics (draft) c©The Common Core Standards Writing Team

2 July 2013

Suggested citation: Common Core Standards Writing Team. (2013, March 1). Progressions for the Common Core State Stan- dards in Mathematics (draft). Grade 8, High School, Functions. Tucson, AZ: Institute for Mathematics and Education, University of Arizona. For updates and more information about the Progressions, see http://ime.math.arizona.edu/ progressions. For discussion of the Progressions and related top- ics, see the Tools for the Common Core blog: http: //commoncoretools.me.

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Grade 8, High School, Functions* Overview Functions describe situations in which one quantity is determined by another. The area of a circle, for example, is a function of its ra- dius. When describing relationships between quantities, the defin- ing characteristic of a function is that the input value determines the output value or, equivalently, that the output value depends upon the input value.

The mathematical meaning of function is quite different from some common uses of the word, as in, “One function of the liver is to remove toxins from the body,” or “The party will be held in the function room at the community center.” The mathematical meaning of function is close, however, to some uses in everyday language. For example, a teacher might say, “Your grade in this class is a function of the effort you put into it.” A doctor might say, “Some ill- nesses are a function of stress.” Or a meteorologist might say, “After a volcano eruption, the path of the ash plume is a function of wind and weather.” In these examples, the meaning of “function” is close to its mathematical meaning.

In some situations where two quantities are related, each can be viewed as a function of the other. For example, in the context of rectangles of fixed perimeter, the length can be viewed as depending upon the width or vice versa. In some of these cases, a problem context may suggest which one quantity to choose as the input variable.

*The study of functions occupies a large part of a student’s high school career, and this document does not treat in detail all of the material studied. Rather it gives some general guidance about ways to treat the material and ways to tie it together. It notes key connections among standards, points out cognitive difficulties and pedagogical solutions, and gives more detail on particularly knotty areas of the mathematics.

The high school standards specify the mathematics that all students should study in order to be college and career ready. Additional material corresponding to (+) standards, mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics, is indicated by plus signs in the left margin.

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Undergraduate mathematics may involve functions of more than one variable. The area of a rectangle, for example, can be viewed as a function of two variables: its width and length. But in high school mathematics the study of functions focuses primarily on real-valued functions of a single real variable, which is to say that both the input and output values are real numbers. One exception is in high school geometry, where geometric transformations are considered to be functions.• For example, a translation T, which moves the plane • G-CO.2 . . . [D]escribe transformations as functions that take

points in the plane as inputs and give other points as outputs. . . .3 units to the right and 2 units up might be represented by T : px, yq ÞÑ px � 3, y� 2q. Sequences and functions Patterns are sequences, and sequences are functions with a domain consisting of whole numbers. How- ever, in many elementary patterning activities, the input values are not given explicitly. In high school, students learn to use an index

The problem with patterns3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction"and“improperfraction"initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

Students are asked to continue the pattern 2, 4, 6, 8, . . . . Here are some legitimate responses:

• Cody: I am thinking of a “plus 2 pattern,” so it continues 10, 12, 14, 16, . . . .

• Ali: I am thinking of a repeating pattern, so it continues 2, 4, 6, 8, 2, 4, 6, 8, . . . .

• Suri: I am thinking of the units digit in the multiples of 2, so it continues 0, 2, 4, 6, 8, 0, 2, . . . .

• Erica: If gpnq is any polynomial, then fpnq � 2n� pn� 1qpn� 2qpn� 3qpn� 4qgpnq describes a continuation of this sequence.

• Zach: I am thinking of that high school cheer, “Who do we appreciate?”

Because the task provides no structure, all of these answers must be considered correct. Without any structure, continuing the pattern is simply speculation—a guessing game. Because there are infinitely many ways to continue a sequence, patterning problems should provide enough structure so that the sequence is well defined.

to indicate which term is being discussed. In the example in the margin, Erica handles this issue by deciding that the term 2 would correspond to an index value of 1. Then the terms 4, 6, and 8 would correspond to input values of 2, 3, and 4, respectively. Erica could have decided that the term 2 would correspond to a different index value, such as 0. The resulting formula would have been different, but the (unindexed) sequence would have been the same. Functions and Modeling In modeling situations, knowledge of the context and statistics are sometimes used together to find a func- tion defined by an algebraic expression that best fits an observed relationship between quantities. (Here “best” is assessed informally, see the Modeling Progression and high school Statistics and Prob- ability Progression for further discussion and examples.) Then the algebraic expressions can be used to interpolate (i.e., approximate or predict function values between and among the collected data values) and to extrapolate (i.e., to approximate or predict function values beyond the collected data values). One must always ask whether such approximations are reasonable in the context.

In school mathematics, functional relationships are often given by algebraic expressions. For example, f pnq � n2 for n ¥ 1 gives the nth square number. But in many modeling situations, such as the temperature at Boston’s Logan Airport as a function of time, algebraic expressions may not be suitable. Functions and Algebra See the Algebra Progression for a discus- sion of the connection and distinctions between functions, on the one hand, and algebra and equation solving, on the other. Perhaps the most productive connection is that solving equations can be seen as finding the intersections of graphs of functions.A-REI.11

A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y � fpxq and y � gpxq intersect are the solutions of the equation fpxq � gpxq; find the solutions approx- imately, e.g., using technology to graph the functions, make ta- bles of values, or find successive approximations. Include cases where fpxq and/or gpxq are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.K–7 foundations for functions Before they learn the term “func-

tion,” students begin to gain experience with functions in elementary

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grades. In Kindergarten, they use patterns with numbers such as the 5� n pattern to learn particular additions and subtractions.

A trickle of pattern standards in Grades 4 and 5 continues the preparation for functions.4.OA.5, 5.OA.3 Note that in both these stan-

4.OA.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not ex- plicit in the rule itself.

5.OA.3Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

dards a rule is explicitly given. Traditional pattern activities, where students are asked to continue a pattern through observation, are not a mathematical topic, and do not appear in the Standards in their own right.1

The Grade 4–5 pattern standards expand to the domain of Ratios Experiences with functions before Grade 83Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction"and“improperfraction"initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

Kindergarten Operations and Algebraic Thinking

fpnq � 5� n Grade 3 Operations and Algebraic Thinking

1� 9 � 9

2� 9 � 2� p10� 1q � p2� 10q � p2� 1q � 20� 2 � 18

3� 9 � 3� p10� 1q � p3� 10q � p3� 1q � 30� 3 � 27, fpnq � 9� n � 10� n� n

Grade 4 Geometric Measurement feet 0 1 2 3

inches 0 12 24 fptq � 12t

Grade 6 Ratios and Proportional Relationships

d meters 3 6 9 12 15 3 2 1 2 4

t seconds 2 4 6 8 10 1 2 3

4 3

8 3fptq � 3

2 t

and Proportional Relationships in Grades 6–7. In Grade 6, as they work with collections of equivalent ratios, students gain experience with tables and graphs, and correspondences between them. They attend to numerical regularities in table entries and corresponding geometrical regularities in their graphical representations as plotted points.MP.8 In Grade 7, students recognize and represent an impor-

MP.8 “Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.”

tant type of regularity in these numerical tables—the multiplicative relationship between each pair of values—by equations of the form y � cx , identifying c as the constant of proportionality in equations and other representations7.RP.2 (see the Ratios and Proportional Re- lationships Progression).

The notion of a function is introduced in Grade 8. Linear functions are a major focus, but note that students are also expected to give examples of functions that are not linear.8.F.3 In high school, students deepen their understanding of the notion of function, expanding their repertoire to include quadratic and exponential functions, and in- creasing their understanding of correspondences between geomet- ric transformations of graphs of functions and algebraic transforma- tions of the associated equations.F-BF.3 The trigonometric functions

F-BF.3 Identify the effect on the graph of replacing fpxq by fpxq� k , kfpxq, fpkxq, and fpx�kq for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

are another important class of functions. In high school, students study trigonometric ratios in right triangles.G-SRT.6 Understanding

G-SRT.6 Understand that by similarity, side ratios in right trian- gles are properties of the angles in the triangle, leading to defini- tions of trigonometric ratios for acute angles.

radian measure of an angle as arc length on the unit circle enables students to build on their understanding of trigonometric ratios as- sociated with acute angles, and to explain how these ratios extend to trigonometric functions whose domains are included in the real numbers.

The (+) standards for the conceptual categories of Geometry and Functions detail further trigonometry addressed to students who intend to take advanced mathematics courses such as calculus. This includes the Law of Sines and Law of Cosines, as well as further study of the values and properties of trigonometric functions.

1This does not exclude activities where patterns are used to support other stan- dards, as long as the case can be made that they do so.

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Grade 8 Define, evaluate, and compare functions Since the elementary grades, students have been describing patterns and expressing re-

8.F.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding out- put.Function notation is not required in Grade 8.

8.F.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

MP.1 “Mathematically proficient students can explain correspon- dences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.”

lationships between quantities. These ideas become semi-formal in Grade 8 with the introduction of the concept of function: a rule that assigns to each input exactly one output.8.F.1 Formal language, such as domain and range, and function notation may be postponed until high school.

Building on their earlier experiences with graphs and tables in Grades 6 and 7, students a routine of exploring functional relation- ships algebraically, graphically, numerically in tables, and through verbal descriptions.8.F.2 They explain correspondences between equa- tions, verbal descriptions, tables, and graphs (MP.1). Repeated rea- soning about entries in tables or points on graphs results in equa- tions for functional relationships (MP.8). To develop flexibility in interpreting and translating among these various representations, students compare two functions represented in different ways, as illustrated by “Battery Charging” in the margin.

Battery Charging 3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction"and“improperfraction"initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

Sam wants to take his MP3 player and his video game player on a car trip. An hour before they plan to leave, he realized that he forgot to charge the batteries last night. At that point, he plugged in both devices so they can charge as long as possible before they leave.

Sam knows that his MP3 player has 40% of its battery life left and that the battery charges by an additional 12 percentage points every 15 minutes.

His video game player is new, so Sam doesn’t know how fast it is charging but he recorded the battery charge for the first 30 minutes after he plugged it in.

time charging in minutes 0 10 20 30 percent player battery charged 20 32 44 56

1. If Sam’s family leaves as planned, what percent of the battery will be charged for each of the two devices when they leave?

2. How much time would Sam need to charge the battery 100% on both devices?

Task from Illustrative Mathematics. For solutions and discussion, see illustrativemathematics.org/illustrations/641.

The main focus in Grade 8 is linear functions, those of the form y � mx � b, where m and b are constants.8.F.3 Students learn to

8.F.3Interpret the equation y � mx�b as defining a linear func- tion, whose graph is a straight line; give examples of functions that are not linear.

recognize linearity in a table: when constant differences between input values produce constant differences between output values. And they can use the constant rate of change appropriately in a verbal description of a context.

The proof that y � mx � b is also the equation of a line, and hence that the graph of a linear function is a line, is an important piece of reasoning connecting algebra with geometry in Grade 8. See the Expressions and Equations Progression. Connection to Algebra and Geometry In high school, after stu- dents have become fluent with geometric transformations and have worked with similarity, another connection between algebra and geometry can be made in the context of linear functions.

The figure in the margin shows a “slope triangle” with one red side formed by the vertical intercept and the point on the line with x-coordinate equal to 1. The larger triangle is formed from the inter- Dilation of a “slope triangle”3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction"and“improperfraction"initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodec